Are Small-Scale SVARs Useful for Business Cycle Analysis? Revisiting Nonfundamentalness

Are Small-Scale SVARs Useful for Business Cycle Analysis? Revisiting Nonfundamentalness Abstract Nonfundamentalness arises when current and past values of the observables do not contain enough information to recover structural vector autoregressive (SVAR) disturbances. Using Granger causality tests, the literature suggested that several small-scale SVAR models are nonfundamental and thus not necessarily useful for business cycle analysis. We show that causality tests are problematic when SVAR variables cross-sectionally aggregate the variables of the underlying economy or proxy for nonobservables. We provide an alternative testing procedure, illustrate its properties with Monte Carlo simulations, and re-examine a prototypical small-scale SVAR model. (JEL: C5, C32, E5) The editor in charge of this paper was Claudio Michelacci. Acknowledgments: We thank the editor (Claudio Michelacci), five anonymous referees, Carlos Velasco, Jesus Gonzalo, Hernan Seoane, Gianni Amisano, Jordi Gali, Davide de Bortoli, Luca Sala, Benjamin Holcblat, Domenico Giannone, Lutz Kilian, Luca Gambetti, and Mario Forni and the participants of seminars at BI Norwegian Business School, UPF, Bocconi, Humboldt, Luiss, University of Glasgow, University of Helsinki, and Federal Reserve Bank of New York, Federal Reserve Board, and of the 2016 IAAE and 2016 ESEM conferences for comments and discussions. Canova acknowledges the financial support from the Spanish Ministerio de Economia y Competitividad through the grants ECO2012-33247 and ECO2015-68136-P and FEDER, UE. Canova is a Research Fellow at CEPR. 1. Introduction Structural vector autoregressive (SVAR) models have been extensively used over the last 30 years to study sources of cyclical fluctuations. The methodology hinges on the assumption that structural shocks can be obtained from linear combinations of current and past values of the observables. Nonfundamentalness arises when this is not the case. In a nonfundamental system, structural shocks obtained via standard identification procedures may have little to do with the true disturbances, even when identification is correctly performed, making SVAR evidence unreliable. Because likelihood or spectral estimation procedures cannot distinguish fundamental versus nonfundamental Gaussian systems (see e.g. Canova 2007, p. 114), it is conventional in applied work to rule out all the nonfundamental representations that possess the same second-order structure of the data. However, this choice is arbitrary. There are rational expectation models (Hansen and Sargent 1991), optimal prediction models (Hansen and Hodrick 1980), permanent income models (Fernández-Villaverde et al. 2007), news shocks models (Forni, Gambetti, and Sala 2014), and fiscal foresight models (Leeper, Walker, and Yang 2013), where optimal decisions may generate nonfundamental solutions. In addition, nonobservability of certain states or particular choices of observables may make fundamental systems nonfundamental. Despite the far-reaching implications it has for applied work, little is known on how to empirically detect nonfundamentalness. Following the lead of Lutkepohl (1991), Giannone and Reichlin (2006), and Forni and Gambetti (2014) (henceforth, FG) suggest that, under fundamentalness, external information should not Granger cause VAR variables. Using such a methodology, FG and Forni et al. (2014) argued that several small-scale SVARs are nonfundamental, thus implicitly questioning the economic conclusions that are obtained. Considering the popularity of small-scale SVARs in macroeconomics, this result is disturbing. This paper shows that Granger causality diagnostics may lead to spurious results in common and relevant situations. Why are there problems? Because of small samples, instabilities, identification, or interpretation difficulties, one typically uses a small-scale SVAR to examine the transmission of relevant disturbances, even if the process generating the data (DGP) features many more variables and shocks. But the shocks recovered by such SVAR systems are linear combinations of a potentially larger set of primitive structural shocks driving the economy. Thus, any variable excluded from the SVAR, but containing information about these primitive disturbances, predicts SVAR shocks (and thus Granger cause the endogenous variables), regardless of whether the model is fundamental or not. To illustrate the point, suppose we want to measure the effects of technology shocks on economic activity. Small-scale SVARs designed for this purpose typically include an aggregate measure of labor productivity, hours, and a few other aggregate variables. Suppose that what drives the economy are sector-specific, serially correlated productivity disturbances. The technology shock recovered from an SVAR will be a linear transformation of current and past sectoral productivity shocks. Because, for example, sectoral capital or sectoral labor productivity has information about sectoral disturbances, they will predict SVAR technology shocks, both when the model is fundamental and when it is not. A similar problem occurs when the SVAR features a proxy variable. For example, total factor productivity (TFP) is latent and typical estimates are obtained from output, capital, and hours-worked data. If capital and hours worked are excluded from the SVAR, any variable that predicts them will Granger cause estimated TFP, regardless of whether the model is fundamental or not. In general, whenever a small-scale SVAR is used, aggregation rather than nonfundamentalness may be the reason for why Granger causality tests find predictability. Thus, if nonfundamentalness is of interest, it is crucial to have a testing approach that is robust to aggregation and nonobservability problems. We propose an alternative procedure, based on ideas of Sims (1972), which has this property and exploits the fact that, under nonfundamentalness, future SVAR shocks predict a vector of variables excluded from the SVAR. We perform Monte Carlo simulations using a version of the model of Leeper et al. (2013) as DGP with capital tax, income tax, and productivity disturbances. We assume that the SVAR includes capital and an aggregate tax variable (or an aggregate tax rate computed from revenues and output data) and show that our approach has good small sample properties. In contrast, spurious nonfundamentalness arises with standard diagnostics. Absent aggregation problems, our approach, and Granger causality tests have similar small sample properties. We re-examine the small-scale SVAR employed by Beaudry and Portier (2006) designed to measure the macroeconomic effects of news. We find that the model is fundamental according to our test but nonfundamental according to a Granger causality diagnostic. We show that the rejection of the null with the latter is due to aggregation: once coarsely disaggregated TFP data are used in the SVAR, Granger causality no longer rejects the null of fundamentalness. The dynamics responses to news shocks in the systems with aggregated and disaggregated TFP measures are however similar (see also Beaudry et al. 2015). Thus, the SVAR disturbances the two systems recover are likely to be similar combinations of the primitive structural shocks and, thus, not necessarily economically interpretable. Two caveats need to be mentioned. First, our analysis is concerned with Gaussian macroeconomic variables. For non-Gaussian situations, see Hamidi Saneh (2014) or Gourieroux and Monfort (2015). Second, although we focus on SVARs, our procedure also works for structural vector autoregressive moving average (SVARMA) models, as long as the largest moving average (MA) root is sufficiently away from unity. The rest of the paper is organized as follows. Section 2 provides examples of nonfundamental systems and highlights the reasons for why problem occurs. Section 3 shows why standard tests may fail and propose an alternative approach. Section 4 examines the performance of various procedures using Monte Carlo simulations. Section 5 investigates the properties of a small-scale SVAR system. Section 6 concludes. 2. A Few Example of Nonfundamental Systems As Kilian and Lutkepohl (2016) highlighted, the literature has primarily focused on nonfundamentalness driven by a mismatch between agents and econometricians information sets, because of omitted variables (see e.g. Giannone and Reichlin 2006; Kilian and Murphy 2014) or the timing of news revelation (see e.g. Forni et al. 2014; Leeper et al. 2013). However, there may be other reasons for why it emerges. First, nonfundamentalness may be intrinsic to the optimization process and to the modeling choices an investigator makes (see e.g. Hansen and Sargent 1980, 1991). Optimizing models producing nonfundamental solutions are numerous; the next example shows one. Example 1. Suppose the dividend process is dt = et − aet−1, where a < 1, and suppose stock prices are expected discounted future dividends: pt = Et∑jβjdt+j, 0 < β < 1. The equilibrium value of pt in terms of the dividends innovations is   \begin{equation} p_{t}=(1-\beta a)e_{t}-ae_{t-1}. \end{equation} (1)Thus, even though the dividends process is fundamental (a < 1), the process for stock prices could be nonfundamental if |(1 − βa)/a| < 1, which occurs when 1/1 + β < a. If a ≥ 0.5, any economically reasonable value of β will make stock prices nonfundamental. On the other hand, if we allow stock prices to have a bubble component $$e_{t}^{b}$$ whose expected value is zero, the vector $$(e_{t},e_{t}^{b})$$ is fundamental for (dt, pt), regardless of the value of β. Thus, allowing for bubbles in theory makes a difference as far as recovering dividend shocks from the data. Second, nonfundamentalness may be due to nonobservability of some of the endogenous variables of a fundamental model. The next example illustrates how this is possible. Example 2. Suppose the production function (in logs) is   \begin{equation} Y_t = K_t + e_t, \end{equation} (2)and the law of motion of capital is   \begin{equation} K_t = (1-\delta ) K_{t-1}+ a e_t. \end{equation} (3)If both (Kt, Yt) are observable, this is just a bivariate restricted VAR(1) and et is fundamental for both (kt, yt). However, if the capital stock is unobservable, (2) becomes   \begin{equation} Y_t -(1-\delta ) Y_{t-1}= (1+a) e_t+ (1-\delta )e_{t-1}. \end{equation} (4)Clearly, if a < 0 and |a| < |δ|, et cannot be expressed as a convergent sum of current and past values of Yt and (4) is nonfundamental. In addition, if δ and a are both small, (4) has an MA root close to unity and a finite-order VAR for Yt poorly approximates the underlying bivariate process; see also Ravenna (2007) and Giacomini (2013). Third, a particular variable selection may induce nonfundamentalness, even if the system is, in theory, fundamental. Hansen and Hodrick (1980) showed that this happens when forecast errors are used in a VAR. The next example shows a less known situation. Example 3. Consider a standard consumption-saving problem. Let income Yt = et be a white noise. Let β = 1/R < 1 be the discount factor and assume quadratic preferences. Then   \begin{equation} C_{t}=C_{t-1}+(1-R^{-1})e_{t}. \end{equation} (5)Thus, the growth rate of consumption has a fundamental representation. However, if we set up the empirical model in terms of savings, St ≡ Yt − Ct, the solution is   \begin{equation} S_{t}-S_{t-1}=R^{-1}e_{t}-e_{t-1}, \end{equation} (6)and the growth rate of saving is nonfundamental. In sum, there may be many reasons for why an empirical model may be nonfundamental. Assuming away nonfundamentalness is problematic. Focusing on omitted variable or anticipation problems is, on the other hand, reductive. One ought to have procedures able to detect whether an SVAR is fundamental and, if it is not, whether violations are intrinsic to theory or due to applied investigators choices. 3. The Setup Because in this section we need to distinguish the structural disturbances driving the fluctuations in the DGP from the shocks an SVAR may recover, we use the convention that “primitive” structural shocks are the disturbances of the DGP and “SVAR” structural shocks those obtained with the empirical model. We assume that the DGP for the observables can be represented by an n-dimensional vector of stationary variables χt driven by s ≥ n serial and mutually uncorrelated primitive structural shocks ςt. Assumption 1. The vector χt satisfies   \begin{equation*} \chi _{t}=\Gamma (L)C\varsigma _{t}, \end{equation*} where C is an n × s matrix, $$\Gamma (L)=\sum _{i=0}^{\infty }\Gamma _{i}L^{i}$$, Γ0 = I, Γi’s are (n × n) matrices each i, L is the lag operator, and $$\sum _{i=0}^{\infty }\Gamma _{i}^{2}<\infty$$. The DGP in (1) is quite general and covers, for example, stationary dynamics general equilibrium (DSGE) models solved around a deterministic steady state or nonstationary DSGEs solved around a deterministic or a stochastic balanced growth path. Stationarity is assumed for convenience; the arguments we present are independent of whether χt stochastically drifts or not. Assumption 1 places mild restrictions on the roots of Γ(L). In theory, ςt could be fundamental for χt or not. Given a typical sample, n the dimension of χt is generally large and Γ(L) is of infinite dimension. Thus, for estimation and inferential purposes, an applied investigator typically confines attention to an m-dimensional vector xt, where $$\mathcal {H}_{t}^{x} \subset \mathcal {H}_{t}^{\chi }$$, and $$\mathcal {H}_{t}^{j}$$ is the closed linear span of {js: s ≤ t}, jt = (xt, χt).1 Assumption 2. The vector xt is driven by an  m × 1 vector of mutually and serially uncorrelated SVAR structural shocks ςx, t:   \begin{eqnarray} x_{t} &=& \Gamma _{x}(L)C_{x}\varsigma _{t} \end{eqnarray} (7)  \begin{eqnarray} \qquad\qquad\quad\quad &\equiv &\Pi (L)u_{t}=\Pi (L)D\varsigma _{x,t}, \end{eqnarray} (8)where m < n, Γx(L) is an m × m matrix for every L, Cx is also an m × s matrix, $$\Pi (L)=\sum _{i=0}^{\infty }\Pi _{i}L^{i}$$, Πi are m × m matrices for each i, Π0 = I, $$\sum _{i=0}^{\infty }\Pi _{i}^{2}<\infty$$, D is an m × m matrix. Equation (7) covers many cases of interest in macroeconomics. For example, xt may contain a subset of the variables belonging to χt, linear combinations, regression residuals, or forecast errors computed from the elements of χt. Thus, the framework includes the case of a variable belonging to the DGP but unobserved and thus omitted from the empirical model (as in Example 2); the situation where the DGP has disaggregated variables but the empirical model is set up in terms of aggregated variables; the case where the DGP has an unobservable variable (e.g., total factor productivity) proxied by a linear combination of observables (i.e., output, capital and labor); and the case where all DGP variables are observables (e.g., we have consumption data) but the empirical model contains linear combinations of the observables (i.e., savings as in Example 3). Because the dimension of ςt is larger than the dimension of xt, cross-sectional aggregation occurs. That is, the econometrician estimating an SVAR may be able to recover the m × 1 vector ςx, t from the reduced form residuals ut, but never the s × 1 vector ςt. For example, the DGP may describe a small open economy subject to external shocks coming from many countries, whereas the empirical model is specified so that only rest of the world variables are used. If Γ(L) has a block exogenous structure, it may be possible to aggregate the vector external shocks into one shock without contamination from other disturbances, see, for example, Faust and Leeper (1988). However, even in this case, it is clearly impossible to recover the full vector of country-specific external disturbances. Next, we provide the definition of fundamentalness for the empirical model (8) (see also Alessi, Barigozzi, and Capasso 2011; Rozanov 1967). Definition 1. An uncorrelated process {ut} is xt-fundamental if $$\mathcal {H}_{t}^{u}=\mathcal {H}_{t}^{x}$$ for all t. It is nonfundamental if $$\mathcal {H}_{t}^{u} \subset \mathcal {H}_{t}^{x}$$ and $$\mathcal {H}_{t}^{u}\ne \mathcal {H}_{t}^{x}$$, for at least one t. The empirical model (8) is fundamental if and only if all the roots of the determinant of the Π(L) polynomial lie outside the unit circle in the complex plane—in this case $$\mathcal {H}_{t}^{u}=\mathcal {H}_{t}^{x}$$, for all t. Alternatively, the model is fundamental if it is possible to express ut as a convergent sum of current and past xt’s. Fundamentalness is closely related to the concept of invertibility: The latter requires that no root of the determinant of Π(L) is on or inside the unit circle. Because we consider stationary variables, the two concepts are equivalent in our framework. In standard situations, there is a one-to-one mapping between ut and ςt and thus examining the fundamentalness of ut provides information about the fundamentalness of ςt. When the mapping is not one to one but the relationship between ut and ςt has a particular structure, it may be possible to find conditions insuring that when ut is fundamental for xt, ςt is fundamental for χt (see e.g. Forni et al. 2009). In all other situations, many of which are of interest, knowing the properties of ut for xt may tell us little about the properties of the primitive shocks ςt for χt. Note that, although ςx, t are linear combination of ςt, they may still be economically interesting. An aggregate TFP shock may be meaningful, even if the sectoral TFP shocks drive the economy, as long as several sectoral TFP disturbances produce similar dynamics for the variables of the SVAR. On the other hand, it is not generally true that a fundamental shock is necessarily structurally interpretable (this occurs e.g. when the wrong D matrix is used to recover ςx, t from a fundamental ut). 3.1. Standard Approaches To Detect Nonfundamentalness Checking whether a Gaussian VAR is fundamental or not is complicated because the likelihood function or the spectral density cannot distinguish between a fundamental and a nonfundamental representations. Earlier work by Lippi and Reichlin (1993, 1994) informally compared the dynamics produced by fundamental and selected nonfundamental representations. Giannone and Reichlin (2006) proposed to use Granger causality tests. The procedure works as follows. Suppose we augment xt with a vector of variables yt:   \begin{equation} {\left[\begin{array}{c}x_{t} \\ y_{t}\end{array}\right]} = {\left[\begin{array}{cc}\Pi (L) & 0 \\ B(L) & C(L)\end{array}\right]} {\left[\begin{array}{c}u_{t} \\ v_{t} \end{array}\right]} , \end{equation} (9)where vt are specific to yt and orthogonal to ut. Assume that all the roots of the determinant of B(L) are outside the unit circle. If (8) is fundamental, ut = Π(L)−1xt, and   \begin{equation} y_{t}=B(L)\Pi (L)^{-1}x_{t}+C(L)v_{t}, \end{equation} (10)where B(L)Π(L)−1 is one  sided in the nonnegative powers of L. Thus, under fundamentalness, yt is a function of current and past values of xt, but xt does not depend on yt. Hence, to detect nonfundamentalness, one can check whether xt is predicted by lags of yt. Although such an approach is useful to examine whether there are variables omitted from the empirical model, it is not clear whether it can reliably detect nonfundamentalness when shock aggregation is present. The reason is that cross-sectional aggregation is not innocuous. For example, Chang and Hong (2006) show that aggregate and sectoral technology shocks behave quite differently, and Sbrana and Silvestrini (2010) show that volatility predictions are quite different depending on the degree of cross-sectional aggregation of the portfolio one considers. The next example shows that aggregation may lead to spurious conclusions when using Granger causality to test for fundamentalness in small-scale SVARs. Example 4. Suppose the DGP is given by the following trivariate process:   \begin{eqnarray} \,\,\chi _{1t} &=\varsigma _{1t}+b_{1}\varsigma _{1t-1}+a\varsigma _{2t}+a\varsigma _{3t} \end{eqnarray} (11)  \begin{eqnarray} \qquad\quad\chi _{2t} &=a\varsigma _{1t}+\varsigma _{2t}+b_{2}\varsigma _{2t-1}+a\varsigma _{3t}+ \varsigma _{4t} \end{eqnarray} (12)  \begin{eqnarray} \qquad\quad\chi _{3t} &=a\varsigma _{1t}+a\varsigma _{2t}+\varsigma _{3t}+b_{3}\varsigma _{3t-1}- \varsigma _{4t}, \end{eqnarray} (13)where ςt = [ς1t, ς2t, ς3t, ς4t]΄ ∼ iid(0, diag(Σς)) and a ≤ 1. Suppose an econometrician sets up a bivariate empirical model with x1t = χ1t and x2t = 0.5(χ2t + χ3t). Thus, the second variable is an aggregated version of the last two variables of the DGP. The process generating xt is   \begin{eqnarray} x_{t} &=\left( \begin{array}{c}x_{1t} \\ x_{2t} \end{array} \right) =\left[ \begin{array}{c@{\quad}c@{\quad}c}1+b_{1}L & a & a \\ a & 0.5((a+1)+b_2 L) & 0.5((a+1)+b_3 L) \end{array} \right] \left( \begin{array}{c}\varsigma _{1t} \\ \varsigma _{2t} \\ \varsigma _{3t} \end{array} \right). \nonumber\\ \end{eqnarray} (14) Because with two endogenous variables one can recover at most two shocks, the econometrician implicitly estimates   \begin{eqnarray} x_{t} &=\left( \begin{array}{c}x_{1t} \\ x_{2t} \end{array} \right) =\left[ \begin{array}{c@{\quad}c}1+b_{1}L & a \\ a & 1+c L \end{array} \right] \left( \begin{array}{c}u_{1t} \\ u_{2t} \\ \end{array} \right), \end{eqnarray} (15)where $$\sigma ^2_{u1}=\sigma ^2_{\varsigma 1}$$. Letting ρ0 + ρ1L ≡ [0.5(a + 1) 0.5(a + 1)] + [0.5b2 0.5b3]L, and $$\hat{\Sigma }_{\varsigma } = \text{diag} \left\lbrace \sigma ^2_{\varsigma 2}, \sigma ^2_{\varsigma 3} \right\rbrace$$, c and $$\sigma ^2_{u2}$$ are obtained from   \begin{eqnarray} \mathrm{E}(x_{2t}x_{2t}^{\prime })&\equiv & \gamma (0) = \rho _0 \hat{\Sigma }_{\varsigma } \rho _0^{\prime }+ \rho _1 \hat{\Sigma }_{\varsigma } \rho _1^{\prime }= (1+c^2) \sigma ^2_{u2}, \end{eqnarray} (16)  \begin{eqnarray} \mathrm{E}(x_{2t}x_{2t-1}^{\prime }) &\equiv & \gamma (1) = \rho _1 \hat{\Sigma }_{\varsigma } \rho _0^{\prime }= c \sigma ^2_{u2}. \phantom{2_{u2}= (1+1\,c^2) \sigma ^2_{u2},} \end{eqnarray} (17)These two conditions can be combined to obtain the quadratic equation   \begin{equation} c^2\gamma (1)-c \gamma (0)+\gamma (1)=0. \end{equation} (18)Given γ(0), γ(1), (18) can be used to compute the solution for c and then $$\sigma ^2_{u2}=c^{-1}\gamma (1)$$. Because ut in (15) is a white noise, it is unpredictable using ut−s (or xt−s), s > 0. However, it can be predicted using ςt−s, even when ut is fundamental. In fact, letting c* be the fundamental solution of (18) and using (14) and (15) have   \begin{eqnarray} u_{2t} &=&(1+c^{\ast }L)^{-1}[\rho _{0}\hat{\varsigma }_{t}+\rho _{1}\hat{\varsigma }_{t-1}] \nonumber \\ &=&\rho _{0}\hat{\varsigma }_{t}+c^{\ast }\rho _{0}\hat{\varsigma }_{t-1}+(c^{\ast })^{2}\rho _{0}\hat{\varsigma }_{t-2}+(c^{\ast })^{3}\rho _{0}\hat{\varsigma }_{t-3}+\cdots \nonumber \\ &&{}\quad +\rho _{1}\hat{\varsigma }_{t-1}+c^{\ast }\rho _{1}\hat{\varsigma }_{t-2}+(c^{\ast })^{2}\rho _{1}\hat{\varsigma }_{t-3}+(c^{\ast })^{3}\rho _{1}\hat{\varsigma }_{t-4}+\cdots , \end{eqnarray} (19)where $$\hat{\varsigma }=[\varsigma _{2t},\varsigma _{3t}]^{\prime }$$. Because χ2t−s and χ3t−s carry information about ςt−s, lags of yt = [χ2t, χ3t] predict ut, and thus xt. Notice that in terms of equation (9), ς4t plays the role of vt. To gain intuition for why predictability tests give spurious results, notice that (19) implies $$(1+c^{\ast }L)u_{2t}=\rho _{0}\hat{\varsigma }_{t}+\rho _{1}\hat{\varsigma }_{t-1}$$. Thus, under aggregation, estimated SVAR shocks are linear functions of current and past primitive structural shocks, making them predictable using any variable that has information about the lags of the primitive structural shocks. This occurs even if the VAR is correctly specified (i.e., there are sufficient lags to recover ut as in (15)). In standard SVARs with no aggregation, the condition corresponding to (19) is ut = ρςt. Thus, absent misspecification, lags of yt will not predict ut. Granger causality tests have been used by many as a tool to detect misspecification in small-scale VARs. For example, if a serially correlated variable is omitted from the VAR, ut, the econometrician recovers are serially correlated and thus predictable using any variable correlated with the omitted one, see, for example, Canova, Michelacci, and Lopez Salido (2010). When they are applied to systems like those in Example 4, causality tests detect misspecification but for the wrong reason. The VAR system is fundamental, ut derived from (15) are white noise, but Granger causality tests reject the predictability null because aggregation has created a particular correlation structure in SVAR shocks. Example 4 also clearly highlights that the concepts of predictable, fundamental, and structural shocks are distinct. ut’s in (15) are predictable, regardless of whether they are fundamental or not. In addition, ut = ςx, t are structural, in the sense that the responses of x1t to ut and to ςit, i = 1, 2, 3, are similar, even ut are predictable. Finally, ut may be nonfundamental (if c, the nonfundamental solution of (18) is used in (19)), even if they are structural. A similar outcome obtains if the empirical model contains, for example, an estimated proxy for an observable variable or residuals computed from the elements of χt. Suppose (x1t = χ1t, x2t = χ1t − γ1χ2t − γ2χ3t)΄, and γ1, γ2 are (estimated) parameters. For example, x2t are Solow residuals and γ1, γ2 are the labor and the capital shares. The process generating xt is   \begin{eqnarray*} &&x_{t} = \nonumber \\ && \left[ \begin{array}{c@{\,\,}c@{\,\,}c@{\,\,}c}1+b_{1}L & a & a & 0 \\ (1-\gamma a-(1-\gamma )a)-b_{1}L & (a-\gamma -a(1-\gamma ))-b_{2}L & (a-\gamma a-(1-\gamma ))-b_{3}L) & -\gamma _{1}+\gamma _{2}\end{array}\right]\\ && \quad \times \,\left( \begin{array}{c}\varsigma _{1t} \\ \varsigma _{2t} \\ \varsigma _{3t} \\ \varsigma _{4t}\end{array} \right). \end{eqnarray*} As before, the econometrician estimates (15). Also in this situation, ut is unpredictable using ut−s or xt−s. However, lags of any yt constructed as noisy linear transformation of [χ2t, χ3t] predict ut, even when it is fundamental for xt. In sum, the existence of variables that Granger cause xt may have nothing to do with fundamentalness. What is crucial to create spurious results is that SVAR shocks linearly aggregate the information contained in current and past primitive structural shocks. Although to some readers Example 4 may look special, it is not. We next formally show that predictability obtains, in general, under linear cross-sectional aggregation. This together with the fact that small-scale SVARs are generally used in business cycle analysis, even when the DGP may feature a large number of primitive structural shocks, should convince skeptical readers of the relevance of Example 4. Proposition 1 shows that the class of moving average models is closed with respect to linear transformations and Proposition 2 that aggregated moving average models are predictable. Proposition 1. Let χ1t be a zero-mean MA(q1) process   \begin{equation} \chi _{1t} = \varsigma _{1t} + \Phi _{1}\varsigma _{1t-1} + \Phi _{2}\varsigma _{1t-2} + \cdots + \Phi _{q_{1}}\varsigma _{1t-q_{1}} \equiv \Phi (L)\varsigma _{1t}, \end{equation} (20)with $$\mathrm{E}(\varsigma _{1t}\varsigma _{1t-j})=\sigma _{1}^{2}$$ if j = 0 and 0 otherwise, and let χ2t be a zero-mean MA(q2) process:   \begin{equation} \chi _{2t}=\varsigma _{2t}+\Psi _{1}\varsigma _{2t-1}+\Psi _{2}\varsigma _{2t-2}+\cdots +\Psi _{q_{2}}\varsigma _{2t-{q_{2}}}\equiv \Psi (L)\varsigma _{2t}, \end{equation} (21)with $$\mathrm{E}(\varsigma _{2t}\varsigma _{2t-j})= \sigma _{2}^{2}$$ if j = 0 and 0 otherwise. Assume that χ1t and χ2t are independent at all leads and lags. Then   \begin{equation} x_{t}=\chi _{1t}+\gamma \chi _{2t}=u_{t}+\Pi _{1}u_{t-1}+\Pi _{2}u_{t-2}+\cdots +\Pi _{q}u_{t-q}\equiv \Pi (L)u_{t}, \end{equation} (22)where q = max {q1, q2}, γ is a vector of constants, and ut is a white noise process. Proof. The proof follows from Hamilton (1994, p. 106). Proposition 2. Let xt be an m-dimensional process obtained as in Proposition 1. Then, ς1t−s and ς2t−s, s ≥ 1 Granger cause xt. Proof. It is enough to show that   \begin{equation*} \mathbb {P}\big [x_{t}|x_{t-1},x_{t-2},\ldots ,\varsigma _{1t-1},\varsigma _{1t-2},\ldots ,\varsigma _{2t-1},\varsigma _{2t-2},\ldots \ \big ]\ne \mathbb {P}\big [x_{t}|x_{t-1},x_{t-2},\ldots \big ], \end{equation*} when the model is fundamental, where $$\mathbb {P}$$ is the linear projection operator. Here, $$\mathcal {H}_{t}^{x}=\mathcal {H}_{t}^{u}$$. Hence, it suffices to show that ut is Granger caused by lagged values of ς1t and ς2t. That is   \begin{equation*} \mathbb {P}\big [u_{t}|u_{t-1},u_{t-2},\ldots ,\varsigma _{1t-1},\varsigma _{1t-2},\ldots ,\varsigma _{2t-1},\varsigma _{2t-2},\ldots \ \big ]\ne \mathbb {P}\big [u_{t}|u_{t-1},u_{t-2},\ldots \big ]. \end{equation*} From Proposition 1, we have that Π(L)ut = Φ(L)ς1t + Ψ(L)ς2t, and therefore ut = Π(L)−1Φ(L)ς1t + Π(L)−1Ψ(L)ς2t, where Π(L)−1 exists because the model is fundamental. Hence, Π(L)−1Φ(L) and Π(L)−1Ψ(L) are one-sided polynomial in the nonnegative powers of L and   \begin{multline*} \mathbb {P}[u_{t}|u_{t-1},u_{t-2},\ldots ,\varsigma _{1t-1},\varsigma _{1t-2},\ldots ,\varsigma _{2t-1},\varsigma _{2t-2},\ldots \ ]\\ =\mathbb {P}[u _{t}|\varsigma _{1t-1},\varsigma _{1t-2},\ldots ,\varsigma _{2t-1},\varsigma _{2t-2},\ldots \ ]\ne 0, \end{multline*} where the equality follows from ut being a white noise process. Thus, although ut in (22) is unpredictable given own lagged values, it can be predicted using lagged values of ς1t and ς2t because the information contained in the histories of ς1t and ς2t is not optimally aggregated into ut. Although the analysis is so far concerned with the fundamentalness of the vector ut, it is common in the VAR literature to focus attention on just one shock, see, for example, Christiano, Eichenbaum, and Evans (1999) or Galí (1999). The next example shows when one can recover a shock from current and past values of the observables, even when the system is nonfundamental. Example 5. Consider the following systems:   \begin{eqnarray} x_{1,t}& =&u_{1t} \\ x_{2,t}& =&u_{1t}+ u_{2t}-3 u_{2t-1} \nonumber \end{eqnarray} (23)  \begin{eqnarray} x_{1,t}& =&u_{1t}-2u_{2t-1} \\ x_{2,t}& =&u_{1t-1}+u_{2t-1}. \nonumber \end{eqnarray} (24)Both systems are nonfundamental—the determinants of the MA matrix are 1 − 3L, and L(1 − 2L), respectively, and they both vanish for L < 1. Thus, it is impossible to recover ut = (u1t, u2t) from current and lagged xt = (x1, t, x2, t)΄. However, although in the first system u1t can be obtained from x1, t, in the second system no individual shock can be obtained from linear combinations of current and past xt’s. A necessary condition for an SVAR shock to be an innovation is that it is orthogonal to the past values of the observables. FG suggest that a shock derived as in the first system of Example 5 is fundamental if it is unpredictable using (orthogonal to the) lags of the principal components obtained from variables belonging to the econometrician’s information set. Three important points need to be made about such an approach. First, fundamentalness is a property of a system not of a single shock. Thus, orthogonality tests are, in general, insufficient to assess fundamentalness. Second, as it is clear from Example 5, when one shock can be recovered, it is not the shock that creates nonfundamentalness in the first place. Finally, an orthogonality test has the same shortcomings as a Granger causality test. It will reject the null of unpredictability of an SVAR shock using disaggregated variables or factors providing noisy information about them, when the SVAR shock is a linear combinations of primitive disturbances, for exactly the same reasons that Granger causality tests fail. 3.2. An Alternative Approach In this section, we propose an alternative testing approach that we expect to have better properties in the situations of interest in this paper. To see what the procedure involves suppose we still augment (8) with a vector of additional variables yt = B(L)ut + C(L)vt. If (8) is fundamental, ut can be obtained as from current and past values of xt:   \begin{equation} u_{t}=x_{t}-\sum _{j=1}^{r}\omega _{j}x_{t-j}, \end{equation} (25)where ω(L) = Π(L)−1 and r is generally finite. Thus, under fundamentalness, yt only depends on current and past values of ut. If instead (8) is nonfundamental, ut cannot be recovered from the current and past values of xt. A VAR econometrician can only recover $$u_{t}^{\ast }=x_{t}-\sum _{j=1}^{r}\omega ^*_{j}x_{t-j}$$, where ω(L)* = Π(L)−1θ(L)−1, which is related to ut via   \begin{equation} u_{t}^{\ast }=\theta (L)u_{t}, \end{equation} (26)where θ(L) is a Blaschke matrix.2 Thus, the relationship between yt and the shocks recovered by the econometrician is $$y_{t}=B(L)\theta (L)^{-1} \theta (L)u_{t}+C(L)v_{t}\equiv B(L)^{\ast } u_t^* + C(L) v_t$$. Because B(L)* is generally a two-sided polynomial, yt depends on current, past, and future values of $$u_{t}^*$$. This proves the following proposition. Proposition 3. The system (8) is fundamental if $$u_{t+j}^{\ast },\ j \ge 1$$ fails to predict yt. Example 6. To illustrate proposition 3, let xt = (1 − 2.0L)ut, and then   \begin{eqnarray} x_t &=&(1-2.0L)\frac{(1-0.5L)}{(1-2.0L)}\frac{(1-2.0L)}{(1-0.5L)}u_{t}\equiv (1-0.5L)u_{t}^{\ast }, \end{eqnarray} (27)where   \begin{equation*} u_{t}^{\ast }=\frac{(1-2.0L)}{(1-0.5L)}u_{t}. \end{equation*} Let yt = (1 − 0.5L)ut + (1 − 0.6L)vt. Then   \begin{eqnarray} y_{t} &=&(1-0.5L)\frac{(1-0.5L)}{(1-2.0L)}u_{t}^{\ast }+(1-0.6L)v_{t} \nonumber \\ &=&\sum _{j=0}^{\infty }(1/2)^{j}((1-0.5L)^{2}u_{t+j}^{\ast })+(1-0.6L)v_{t-j}. \end{eqnarray} (28) Two points about our testing procedure need to be stressed. First, Sims (1972) has shown that xt is exogenous with respect to yt if future values of xt do not help to explain yt. Similarly here, a VAR system is fundamental if future values of xt (ut) do not help to predict the variables yt, excluded from the empirical model. Thus, although the null tested here and with Granger causality is the same, aggregation/nonobservability problems may make the testing results different. Second, our approach is likely to have better size properties, when SVAR shocks are linear functions of lags of primitive shocks, because yt generally contains more information than xt—under fundamentalness, future values of ut will not predict yt. Note also that our test is sufficiently general to detect nonfundamentalness due to structural causes, omitted variables, or the use of proxy indicators. 4. Some Monte Carlo Evidence To evaluate the small sample properties of traditional predictability tests and of our new procedure, we carry out a simulation study using a version of the model of Leeper et al. (2013), with two sources of tax disturbances. The representative household maximizes   \begin{equation} \mathrm{E}_{0}\sum _{t=0}^{\infty }\beta ^{t}\log (C_{t}) \end{equation} (29)subject to   \begin{equation} C_{t}+(1-\tau _{t,k})K_{t}+T_{t}\le (1-\tau _{t,y})A_{t}K_{t-1}^{\alpha }=(1-\tau _{t,y})Y_{t}, \end{equation} (30)where Ct, Kt, Yt, Tt, τt, k and τt, y denote time-t consumption, capital, output, lump-sum transfers, investment tax and income tax rates, respectively; At is a technology disturbance and Et is the conditional expectation operator. To keep the setup tractable, we assume full capital depreciation. The government sets tax rates randomly and adjusts transfers to satisfy Tt = τt, yYt + τt, kKt. The Euler equation and the resource constraints are   \begin{eqnarray} \frac{1}{C_{t}} &=&\alpha \beta \mathrm{E}_{t}\Big [\frac{(1-\tau _{t+1,y})}{(1-\tau _{t,k})}\frac{1}{C_{t+1}}\frac{A_{t+1}K_{t}^{\alpha }}{K_{t}}\Big ] \end{eqnarray} (31)  \begin{eqnarray} C_{t}+K_{t} &=A_{t}K_{t-1}^{\alpha }. \end{eqnarray} (32)Log linearizing, combining (31) and (32), we have   \begin{equation} \hat{K}_{t}=\alpha \hat{K}_{t-1}+\sum _{i=0}^{\infty }\theta ^{i}\mathrm{E} _{t}\hat{A}_{t+i+1}-\kappa _{k}\sum _{i=0}^{\infty }\theta ^{i}\mathrm{E}_{t} \hat{\tau }_{t+i,k}-\kappa _{y}\sum _{i=0}^{\infty }\theta ^{i}\mathrm{E}_{t} \hat{\tau }_{t+i+1,y}, \end{equation} (33)where   \begin{equation*} \kappa _{k}=\frac{\tau _{k}(1-\theta )}{(1-\tau _{k})},\quad \kappa _{y}=\frac{\tau _{y}(1-\theta )}{(1-\tau _{y})}, \quad \theta =\alpha \beta \frac{1-\tau _{y}}{1-\tau _{k}}, \quad \hat{K}_{t}\equiv \log (K_{t})-\log (K), \end{equation*}   \begin{equation*} \hat{A}_{t}\equiv \log (A_{t})-\log (A),\quad \hat{\tau }_{t,k}\equiv \log (\tau _{t,k})-\log (\tau _{k}), \quad \hat{\tau }_{t,y}\equiv \log (\tau _{t,y})-\log (\tau _{y}), \end{equation*} and lower case letters denote percentage deviations from steady states. We posit that technology and investment tax shocks are iid: $$\hat{A}_{t}=\varsigma _{t,A},\hat{\tau }_{t,k}=\varsigma _{t,k}$$; and that the income tax shock is an MA(1) process: $$\hat{\tau }_{t,y}=\varsigma _{t,y}+b\varsigma _{t-1,y}$$. Then, (33) is   \begin{equation} \hat{K}_{t}=\alpha \hat{K}_{t-1}+\varsigma _{t,a}-\kappa _{k}\varsigma _{t,k}-\kappa _{y}b\varsigma _{t,y}. \end{equation} (34)We assume that an econometrician observes $$\hat{K}_{t}$$ and an aggregate tax variable   \begin{equation} \hat{\tau }_{t}=\omega \hat{\tau }_{t,y}+\hat{\tau }_{t,k}=\varsigma _{t,k}+\omega (\varsigma _{t,y}+b\varsigma _{t-1,y}), \end{equation} (35)where ω controls the relative weight of income taxes in the aggregate. Alternatively, one can assume that investment and income tax revenues are both observables, but an econometrician works with a weighted sum of them. If $$(\hat{K}_{t},\hat{\tau }_{t})$$ are the variables the econometrician uses in the VAR, our design covers both the cases of aggregation and of a relevant latent variable. In fact, the DGP for the observables is   \begin{equation} {\left[\begin{array}{c}(1-\alpha L)\hat{K}_{t} \\ \hat{\tau }_{t} \end{array}\right]} = {\left[\begin{array}{c@{\quad}c@{\quad}c}1 & -\kappa _{k} & -\kappa _{y}b \\ 0 & 1 & \omega (1+bL) \end{array}\right]} {\left[\begin{array}{c}\varsigma _{t,a} \\ \varsigma _{t,k} \\ \varsigma _{t,y} \end{array}\right]} \equiv \Gamma _{x}(L)C_{x}\varsigma _{t}, \end{equation} (36) whereas the process recoverable by the econometrician is   \begin{equation} {\left[\begin{array}{c}(1-\alpha L)\hat{K}_{t} \\ \hat{\tau }_{t} \end{array}\right]} = {\left[\begin{array}{c@{\quad}c}1 & \rho \\ 0 & 1+cL \end{array}\right]} {\left[\begin{array}{c}u_{t,1} \\ u_{t,2} \end{array}\right]} \equiv \Pi (L)u_{t}, \end{equation} (37)where $$\sigma _{1}^{2}=\sigma _{a}^{2}$$, whereas $$c,\sigma _{2}^{2},\rho$$ are obtained from   \begin{eqnarray} c^{2}-c\left(\left(1+b^{2}\right)/b+\sigma _{k}^{2}/\left(\omega ^{2}b\sigma _{y}^{2}\right)\right)+1 &=0 \end{eqnarray} (38)  \begin{eqnarray} \sigma _{2}^{2} &=b\omega ^{2}\sigma _{y}^{2}/c \end{eqnarray} (39)  \begin{eqnarray} \rho &=-\sqrt{\left(\omega ^{2}\kappa _{y}^{2}b^{2}\sigma _{y}^{2}+\kappa _{k}^{2}\sigma _{k}^{2}\right)/\sigma _{2}^{2}}. \end{eqnarray} (40)By comparing (37) and (36), one can see that the aggregate tax shock ut, 2 will produce the same qualitative dynamic response in $$\hat{K}_{t}$$ as the investment and the income tax shocks but the scale of the effect will be altered. Depending on the size of ω, the aggregate shock will look more like the income or the investment tax shock. For the exercises we present, we let ςt, a, ςt, k, ςt, y ∼ iid N(0, 1); set α = 0.36, β = 0.99, τy = 0.25, τk = 0.1, ω = 1 and vary b so that c ∈ (0.1, 0.8) (fundamentalness region) or c ∈ (2, 9) (nonfundamentalness region). To perform the tests, we need additional data not used in the empirical model (37). We assume that an econometrician observes a panel of 30 time series generated by   \begin{equation} (1-0.9 L)y_{i,t}=\varsigma _{t,a}+\gamma _{i} \varsigma _{t,y}+(1-\gamma _i) \varsigma _{t,k}+\xi _{i,t},\quad i=1,\ldots ,30, \end{equation} (41)where $$\xi _{i,t}\sim \,\,\text{iid}\,\,N\left(0,\sigma ^2_{\xi }\right)$$, and γi is Bernoulli, taking value 1 with probability 0.5. The properties of our procedure, denoted by CH, are examined with the regression   \begin{equation} f_{t}=\sum _{j=1}^{p_1}\varphi _{j}f_{t-j} +\sum _{j=0}^{p_2}\psi _{-j}u_{t-j} +\sum _{j=1}^{q}\psi _{j}u_{t+j}+e_{t}, \end{equation} (42)where ft is an s × 1 vector of principal components of (41) and ut is estimated using   \begin{equation} x_t=\sum _{j=1}^r \rho _{j}x_{t-j}+ u_t, \end{equation} (43)where $$x_{t}=(\hat{\tau } _{t},\hat{K}_{t})^{\prime }$$. The null is $$\mathbb {H}_{0}^{CH}:R\Psi =0$$, where Ψ = Vec[ψ1, ψ2, …, ψq], R is a matrix of zeros and ones. We report the results for p1 = 4, p2 = 0, q = 2, r = 4. To examine the properties of Granger causality tests, denoted by GC, we employ   \begin{equation} x_{t}=\sum _{j=0}^{p_1} \varphi _{j}x_{t-j}+\sum _{j=1}^{p_2}\varphi _{j}f_{t-j}+e_{t}, \end{equation} (44)where again $$x_{t}=(\hat{\tau } _{t},\hat{K}_{t})^{\prime }$$. The null is $$\mathbb {H}_{0}^{GC}:R\varPhi=0$$ where $$\varPhi=\mathrm{Vec}{[\varphi _{1},\varphi _{2},\ldots ,\varphi _{p_2}]}$$ and R is a matrix of zeros and ones. We report results for p1 = 4, p2 = 2. To perform an orthogonality test, denoted by OR, we first estimate (43) with r = 4. The tax shock, ut, τ, is identified as the only one affecting $$\hat{\tau }_{t}$$. Then, in the regression   \begin{equation} u_{t,\tau }=\sum _{j=1}^{p_{2}}\lambda _{j}f_{t-j}+e_{t}, \end{equation} (45)the ortogonality null is $$\mathbb {H}_{0}^{OR}:R\Lambda =0$$ where Λ = Vec[λ1, λ2, …, λq] and R is a matrix of zeros and ones. We report results for p2 = 2. To maintain comparability, all null hypotheses are tested using an F-test, setting s = 3 and $$\sigma _{\xi }^{2}=1$$ and no correction for generated regressors in (42) and (45). The appendix present results for the CH test when other values of p2, $$\sigma _{\xi }^{2}$$, s, and q are used. We set T = 200, which is the length of the time series used in Section 5, and T = 2, 000. To better understand the properties of the tests, we also run an experiment with no aggregation problems. Here, τk, t = 0, ∀t so that the DGP for capital and taxes is   \begin{equation} {\left[\begin{array}{c}(1-\alpha L)\hat{K}_{t} \\ \hat{\tau }_{t}\end{array}\right]} = {\left[\begin{array}{c@{\quad}c}1 & -\kappa _{y}b \\ 0 & (1+bL) \end{array}\right]} {\left[\begin{array}{c}\varsigma _{t,a} \\ \varsigma _{t,y} \\ \end{array}\right]}, \end{equation} (46)and the process for the additional data is   \begin{equation} (1-0.9 L)y_{i,t}=\varsigma _{t,a}+\gamma _{i} \varsigma _{t,y}+\xi _{i,t},\quad i=1,\ldots ,n. \end{equation} (47) The percentage of rejections of the null in 1,000 replications when the model is fundamental are in Tables 1 and 2. Our procedure is undersized (it rejects less than expected from the nominal size) but its performance is independent of the nominal confidence level and the sample size. Granger causality and orthogonality tests are prone to spurious nonfundamentalness. This is clear when T = 2,000; in the smaller sample, predictability due to aggregation is somewhat harder to detect. Table 1. Size of the tests: aggregation, T = 200.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  1.5  1.3  0.5  0.6  1.4  1.7  1.2  1.7  5%  0.8  0.5  0.1  0.3  0.3  0.9  0.4  1.1  1%  0.1  0.1  0.1  0.2  0.1  0.2  0.1  0.1  GC  10%  13.1  15.1  16.5  15.8  19.5  27.4  38.9  55.1  5%  7.5  8.2  9.5  9.2  11.2  15.5  26.7  42.2  1%  2.0  2.7  2.2  3.1  4.3  5.8  10.9  19.6  OR  10%  5.2  5.7  5.3  6.2  6.5  6.2  8.5  13.2  5%  2.9  2.3  2.9  2.5  3.5  2.3  4.2  6.5  1%  0.1  0.5  0.6  0.2  0.2  0.7  0.7  1.6    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  1.5  1.3  0.5  0.6  1.4  1.7  1.2  1.7  5%  0.8  0.5  0.1  0.3  0.3  0.9  0.4  1.1  1%  0.1  0.1  0.1  0.2  0.1  0.2  0.1  0.1  GC  10%  13.1  15.1  16.5  15.8  19.5  27.4  38.9  55.1  5%  7.5  8.2  9.5  9.2  11.2  15.5  26.7  42.2  1%  2.0  2.7  2.2  3.1  4.3  5.8  10.9  19.6  OR  10%  5.2  5.7  5.3  6.2  6.5  6.2  8.5  13.2  5%  2.9  2.3  2.9  2.5  3.5  2.3  4.2  6.5  1%  0.1  0.5  0.6  0.2  0.2  0.7  0.7  1.6  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 1. Size of the tests: aggregation, T = 200.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  1.5  1.3  0.5  0.6  1.4  1.7  1.2  1.7  5%  0.8  0.5  0.1  0.3  0.3  0.9  0.4  1.1  1%  0.1  0.1  0.1  0.2  0.1  0.2  0.1  0.1  GC  10%  13.1  15.1  16.5  15.8  19.5  27.4  38.9  55.1  5%  7.5  8.2  9.5  9.2  11.2  15.5  26.7  42.2  1%  2.0  2.7  2.2  3.1  4.3  5.8  10.9  19.6  OR  10%  5.2  5.7  5.3  6.2  6.5  6.2  8.5  13.2  5%  2.9  2.3  2.9  2.5  3.5  2.3  4.2  6.5  1%  0.1  0.5  0.6  0.2  0.2  0.7  0.7  1.6    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  1.5  1.3  0.5  0.6  1.4  1.7  1.2  1.7  5%  0.8  0.5  0.1  0.3  0.3  0.9  0.4  1.1  1%  0.1  0.1  0.1  0.2  0.1  0.2  0.1  0.1  GC  10%  13.1  15.1  16.5  15.8  19.5  27.4  38.9  55.1  5%  7.5  8.2  9.5  9.2  11.2  15.5  26.7  42.2  1%  2.0  2.7  2.2  3.1  4.3  5.8  10.9  19.6  OR  10%  5.2  5.7  5.3  6.2  6.5  6.2  8.5  13.2  5%  2.9  2.3  2.9  2.5  3.5  2.3  4.2  6.5  1%  0.1  0.5  0.6  0.2  0.2  0.7  0.7  1.6  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 2. Size of the tests: aggregation, T = 2,000.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.1  0.4  0.2  0.1  0.2  0.1  1.9  9.5  5%  0.1  0.2  0.2  0.1  0.1  0.1  1.0  4.2  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.8  GC  10%  83.3  86.6  88.8  92.4  98.1  99.9  100  100  5%  75.3  75.1  76.3  83.9  96.1  99.8  100  100  1%  49.7  44.7  46.0  58.7  83.9  98.6  100  100  OR  10%  34.0  30.1  29.4  30.2  41.7  52.1  81.0  99.1  5%  21.4  18.5  18.5  19.0  27.9  36.0  66.4  96.5  1%  7.4  6.8  7.3  6.4  9.0  13.0  34.9  81.8    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.1  0.4  0.2  0.1  0.2  0.1  1.9  9.5  5%  0.1  0.2  0.2  0.1  0.1  0.1  1.0  4.2  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.8  GC  10%  83.3  86.6  88.8  92.4  98.1  99.9  100  100  5%  75.3  75.1  76.3  83.9  96.1  99.8  100  100  1%  49.7  44.7  46.0  58.7  83.9  98.6  100  100  OR  10%  34.0  30.1  29.4  30.2  41.7  52.1  81.0  99.1  5%  21.4  18.5  18.5  19.0  27.9  36.0  66.4  96.5  1%  7.4  6.8  7.3  6.4  9.0  13.0  34.9  81.8  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 2. Size of the tests: aggregation, T = 2,000.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.1  0.4  0.2  0.1  0.2  0.1  1.9  9.5  5%  0.1  0.2  0.2  0.1  0.1  0.1  1.0  4.2  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.8  GC  10%  83.3  86.6  88.8  92.4  98.1  99.9  100  100  5%  75.3  75.1  76.3  83.9  96.1  99.8  100  100  1%  49.7  44.7  46.0  58.7  83.9  98.6  100  100  OR  10%  34.0  30.1  29.4  30.2  41.7  52.1  81.0  99.1  5%  21.4  18.5  18.5  19.0  27.9  36.0  66.4  96.5  1%  7.4  6.8  7.3  6.4  9.0  13.0  34.9  81.8    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.1  0.4  0.2  0.1  0.2  0.1  1.9  9.5  5%  0.1  0.2  0.2  0.1  0.1  0.1  1.0  4.2  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.8  GC  10%  83.3  86.6  88.8  92.4  98.1  99.9  100  100  5%  75.3  75.1  76.3  83.9  96.1  99.8  100  100  1%  49.7  44.7  46.0  58.7  83.9  98.6  100  100  OR  10%  34.0  30.1  29.4  30.2  41.7  52.1  81.0  99.1  5%  21.4  18.5  18.5  19.0  27.9  36.0  66.4  96.5  1%  7.4  6.8  7.3  6.4  9.0  13.0  34.9  81.8  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Why are traditional predictability tests rejecting the null much more than one would expect from the nominal size? The answer is obtained recalling equation (19). ut are linear combinations of current and past values of $$\hat{A}_{t}, \hat{\tau }_{t,k}, \hat{\tau }_{t,y}$$, whereas ft are linear combinations of $$\hat{A}_{t}, \hat{\tau }_{t,k}, \hat{\tau }_{t,y}$$ and ξi, t, ı = 1, …, 30. Because $$\hat{\tau }_{t,k}$$ is serially correlated, lags of ft may help to predict xt even when lags of xt are included, in particular, when the draws for γi are small. It is known that Granger causality tests have poor size properties when xt is persistent, see, for example, Ohanian (1988). Tables 3 and 4 disentangle aggregation from persistence problems: Because they have been constructed absent aggregation, they report size distortions due to persistent data. It is clear that, when b > 0.6, the size of Granger causality tests is distorted. To properly run such tests, the lag length p1 of the testing equation must be made function of the (unknown) persistence of the DGP. However, when b > 0.8, distortions are present even if p1 = 10. The orthogonality test performs better because it preliminary filters xt with a VAR. Thus, high serial correlation in xt is less of a problem. Table 3. Size of the tests: no aggregation, T = 200.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  2.5  1.7  2.5  1.6  1.5  2.3  2.8  2.6  5%  1.1  0.6  0.6  1.2  0.5  0.8  1.2  1.0  1%  0.1  0.1  0.2  0.1  0.1  0.1  0.2  0.3  GC  10%  11.4  10.5  13.3  13.5  10.9  14.8  15.5  28.4  5%  5.6  5.0  6.2  8.2  5.3  7.4  9.2  19.8  1%  1.3  1.0  1.6  1.6  1.1  2.0  2.4  6.1  OR  10%  4.4  5.1  5.3  4.7  4.0  6.4  6.2  8.9  5%  1.7  1.3  2.8  2.2  1.3  2.3  2.6  4.9  1%  0.2  0.1  0.5  0.3  0.1  0.6  0.6  1.8    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  2.5  1.7  2.5  1.6  1.5  2.3  2.8  2.6  5%  1.1  0.6  0.6  1.2  0.5  0.8  1.2  1.0  1%  0.1  0.1  0.2  0.1  0.1  0.1  0.2  0.3  GC  10%  11.4  10.5  13.3  13.5  10.9  14.8  15.5  28.4  5%  5.6  5.0  6.2  8.2  5.3  7.4  9.2  19.8  1%  1.3  1.0  1.6  1.6  1.1  2.0  2.4  6.1  OR  10%  4.4  5.1  5.3  4.7  4.0  6.4  6.2  8.9  5%  1.7  1.3  2.8  2.2  1.3  2.3  2.6  4.9  1%  0.2  0.1  0.5  0.3  0.1  0.6  0.6  1.8  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 3. Size of the tests: no aggregation, T = 200.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  2.5  1.7  2.5  1.6  1.5  2.3  2.8  2.6  5%  1.1  0.6  0.6  1.2  0.5  0.8  1.2  1.0  1%  0.1  0.1  0.2  0.1  0.1  0.1  0.2  0.3  GC  10%  11.4  10.5  13.3  13.5  10.9  14.8  15.5  28.4  5%  5.6  5.0  6.2  8.2  5.3  7.4  9.2  19.8  1%  1.3  1.0  1.6  1.6  1.1  2.0  2.4  6.1  OR  10%  4.4  5.1  5.3  4.7  4.0  6.4  6.2  8.9  5%  1.7  1.3  2.8  2.2  1.3  2.3  2.6  4.9  1%  0.2  0.1  0.5  0.3  0.1  0.6  0.6  1.8    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  2.5  1.7  2.5  1.6  1.5  2.3  2.8  2.6  5%  1.1  0.6  0.6  1.2  0.5  0.8  1.2  1.0  1%  0.1  0.1  0.2  0.1  0.1  0.1  0.2  0.3  GC  10%  11.4  10.5  13.3  13.5  10.9  14.8  15.5  28.4  5%  5.6  5.0  6.2  8.2  5.3  7.4  9.2  19.8  1%  1.3  1.0  1.6  1.6  1.1  2.0  2.4  6.1  OR  10%  4.4  5.1  5.3  4.7  4.0  6.4  6.2  8.9  5%  1.7  1.3  2.8  2.2  1.3  2.3  2.6  4.9  1%  0.2  0.1  0.5  0.3  0.1  0.6  0.6  1.8  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 4. Size of the tests: no aggregation, T = 2,000.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.9  1.3  1.0  0.9  0.5  1.0  1.3  6.0  5%  0.3  0.5  0.5  0.3  0.2  0.3  0.5  3.8  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  1.7  GC  10%  13.2  13.3  15.6  14.8  18.2  26.7  53.8  99.8  5%  7.4  8.0  8.7  9.0  11.1  16.3  41.2  99.3  1%  1.6  1.9  2.5  3.1  3.0  5.2  19.4  95.3  OR  10%  3.9  5.2  5.6  4.8  4.2  6.8  8.7  20.9  5%  1.3  3.2  1.7  1.8  1.7  3.2  4.8  17.8  1%  0.3  0.7  0.4  0.4  0.2  0.4  1.2  6.0    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.9  1.3  1.0  0.9  0.5  1.0  1.3  6.0  5%  0.3  0.5  0.5  0.3  0.2  0.3  0.5  3.8  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  1.7  GC  10%  13.2  13.3  15.6  14.8  18.2  26.7  53.8  99.8  5%  7.4  8.0  8.7  9.0  11.1  16.3  41.2  99.3  1%  1.6  1.9  2.5  3.1  3.0  5.2  19.4  95.3  OR  10%  3.9  5.2  5.6  4.8  4.2  6.8  8.7  20.9  5%  1.3  3.2  1.7  1.8  1.7  3.2  4.8  17.8  1%  0.3  0.7  0.4  0.4  0.2  0.4  1.2  6.0  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 4. Size of the tests: no aggregation, T = 2,000.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.9  1.3  1.0  0.9  0.5  1.0  1.3  6.0  5%  0.3  0.5  0.5  0.3  0.2  0.3  0.5  3.8  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  1.7  GC  10%  13.2  13.3  15.6  14.8  18.2  26.7  53.8  99.8  5%  7.4  8.0  8.7  9.0  11.1  16.3  41.2  99.3  1%  1.6  1.9  2.5  3.1  3.0  5.2  19.4  95.3  OR  10%  3.9  5.2  5.6  4.8  4.2  6.8  8.7  20.9  5%  1.3  3.2  1.7  1.8  1.7  3.2  4.8  17.8  1%  0.3  0.7  0.4  0.4  0.2  0.4  1.2  6.0    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.9  1.3  1.0  0.9  0.5  1.0  1.3  6.0  5%  0.3  0.5  0.5  0.3  0.2  0.3  0.5  3.8  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  1.7  GC  10%  13.2  13.3  15.6  14.8  18.2  26.7  53.8  99.8  5%  7.4  8.0  8.7  9.0  11.1  16.3  41.2  99.3  1%  1.6  1.9  2.5  3.1  3.0  5.2  19.4  95.3  OR  10%  3.9  5.2  5.6  4.8  4.2  6.8  8.7  20.9  5%  1.3  3.2  1.7  1.8  1.7  3.2  4.8  17.8  1%  0.3  0.7  0.4  0.4  0.2  0.4  1.2  6.0  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Comparing the size tables constructed with and without aggregation, one can see that the properties of the CH test do not depend on the presence of aggregation or the persistence of the DGP. On the other hand, aggregation makes the properties of Granger causality and orthogonality tests significantly worse. Tables 5 and 6 report the empirical power of the tests when T = 200 with and without aggregation. All tests are similarly powerful to detect nonfundamentalness when it is present, regardless of the confidence level and the nature of the DGP. Although not reported for reasons of space, the power of the three tests is unchanged when T = 2,000. Table 5. Power of the tests: aggregation, T = 200.   c  2  3  4  5  6  7  8  9  CH  10%  99.9  100  100  100  100  100  100  100  5%  99.5  100  100  100  100  100  100  100  1%  98.7  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100    c  2  3  4  5  6  7  8  9  CH  10%  99.9  100  100  100  100  100  100  100  5%  99.5  100  100  100  100  100  100  100  1%  98.7  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 5. Power of the tests: aggregation, T = 200.   c  2  3  4  5  6  7  8  9  CH  10%  99.9  100  100  100  100  100  100  100  5%  99.5  100  100  100  100  100  100  100  1%  98.7  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100    c  2  3  4  5  6  7  8  9  CH  10%  99.9  100  100  100  100  100  100  100  5%  99.5  100  100  100  100  100  100  100  1%  98.7  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 6. Power of the tests: no aggregation, T = 200.   c  2  3  4  5  6  7  8  9  CH  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100    c  2  3  4  5  6  7  8  9  CH  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 6. Power of the tests: no aggregation, T = 200.   c  2  3  4  5  6  7  8  9  CH  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100    c  2  3  4  5  6  7  8  9  CH  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large The additional tables in the appendix indicate that the size properties of the CH test are insensitive to the selection of three nuisance parameters: the variance of the shocks to the additional data $$\sigma _{\xi }^{2}$$, the number of principal components used in the testing equation s, and the number of leads of the first-stage residuals used in the testing equation q. On the other hand, the choice of p2, the number of lags of the first-stage residuals used in the testing equations, matters. This is true, in particular, when the persistence of the DGP increases and is due to the fact that with high persistence, r = 4 is insufficient to whiten the first-stage residual, and the presence of serial correlation in ut makes its future values spuriously significant. To avoid this problem in practice, we recommend users to specify the testing equation with only leads of ut. Alternatively, if lags of ut are included, r should be large to insure that serial correlation in the first-stage residuals is negligible. 5. Reconsidering A Small-Scale SVAR Standard business cycle theories assume that economic fluctuations are driven by surprises in current fundamentals, such as aggregate productivity or the monetary policy rule. Motivated by the idea that changes in expectations about future fundamentals may drive business fluctuations, Beaudry and Portier (2006) study the effect of news shocks on the real economy using an SVAR that contains stock prices and TFP. Because models featuring news shocks have solutions displaying moving average components, empirical models with a finite number of lags may be unable to capture the underlying dynamics, making the SVARs considered in the literature prone to nonfundamentalness. In addition, Forni et al. (2014) provide a stylized Lucas tree model where perfectly predictable news to the dividend process may induce nonfundamentalness in a VAR system comprising the growth rate of stock prices and the growth rate of dividends. The solution of their model when news come two periods in advance is   \begin{equation} {\left[\begin{array}{c}\Delta d_{t} \\ \Delta p_{t} \end{array}\right]} = {\left[\begin{array}{c@{\quad}c}L^{2} & 1 \\ \frac{\beta ^{2}}{1-\beta }+\beta L & \frac{\beta }{1-\beta } \end{array}\right]} {\left[\begin{array}{c}\varsigma _{1t} \\ \varsigma _{2t} \end{array}\right]} \equiv C(L)\varsigma _{t}, \end{equation} (48)where dt are dividends, pt are stock prices, and 0 < β < 1 is the discount factor. Because C(L) vanishes for L = 1 and L = −β, ut is nonfundamental for (Δdt, Δpt). Intuitively, this occurs because agents’ information set, which includes current and past values of structural shocks, is not aligned with the econometrician’s information set, which includes current and past values of the growth rate of dividends and stock prices. The fundamental and nonfundamental dynamics this model generates in response to news shocks are similar because the root generating nonfundamentalness (L = −β) is near unity, see also Beaudry et al. (2015). In general, the properties of the SVAR the econometrician considers depend on the process describing the information flows, on the variables observed by the econometrician and those included in the SVAR. To re-examine the evidence, we estimate a VAR with the growth rates of capacity-adjusted TFP and of stock prices for the period 1960Q1–2010Q4, both of which are taken from Beaudry and Portier (2014), and we use the same principal components as in Forni et al. (2014). Table 7 reports the p-values of the tests, varying the number of principal components employed in the auxiliary regression, which enter in first difference in all the tests. In the CH test, the testing model has four lags of the PC and we are examining the predictive power of two leads of the VAR residuals. In the GC test, the lag length of the VAR is chosen by BIC and two lags of the principal components are used in the tests. Table 7. Testing fundamentalness: VAR with TFP growth and stock prices growth.   PC = 3  PC = 4  PC = 5  PC = 6  PC = 7  PC = 8  PC = 9  PC = 10  Sample 1960–2010  CH  0.05  0.08  0.03  0.15  0.13  0.08  0.14  0.13  GC  0.02  0.00  0.04  0.01  0.00  0.00  0.00  0.00  Fernald data, sample 1960–2005  GC(agg)  0.02  0.16  0.22  0.00  0.00  0.02  0.01  0.01  GC(dis)  0.17  0.52  0.54  0.11  0.09  0.17  0.25  0.34  Wang data, sample 1960–2009  GC(agg)  0.05  0.02  0.11  0.03  0.00  0.00  0.00  0.00  GC(dis)  0.37  0.38  0.51  0.40  0.27  0.28  0.27  0.23    PC = 3  PC = 4  PC = 5  PC = 6  PC = 7  PC = 8  PC = 9  PC = 10  Sample 1960–2010  CH  0.05  0.08  0.03  0.15  0.13  0.08  0.14  0.13  GC  0.02  0.00  0.04  0.01  0.00  0.00  0.00  0.00  Fernald data, sample 1960–2005  GC(agg)  0.02  0.16  0.22  0.00  0.00  0.02  0.01  0.01  GC(dis)  0.17  0.52  0.54  0.11  0.09  0.17  0.25  0.34  Wang data, sample 1960–2009  GC(agg)  0.05  0.02  0.11  0.03  0.00  0.00  0.00  0.00  GC(dis)  0.37  0.38  0.51  0.40  0.27  0.28  0.27  0.23  Notes: The table reports the p-value of the tests; CH is the test proposed in this paper; GC is the Granger causality test; the row GC(agg) reports the results of the test using aggregate data, the row GC(dis) the results of the test using disaggregated data; PC is the number of principal component in the auxiliary regression. In CH test, the number of leads tested is two and the preliminary VAR has four lags. In GC test, the lag length of the VAR is chosen with BIC and two lags of the principal components are used in the tests. View Large Table 7. Testing fundamentalness: VAR with TFP growth and stock prices growth.   PC = 3  PC = 4  PC = 5  PC = 6  PC = 7  PC = 8  PC = 9  PC = 10  Sample 1960–2010  CH  0.05  0.08  0.03  0.15  0.13  0.08  0.14  0.13  GC  0.02  0.00  0.04  0.01  0.00  0.00  0.00  0.00  Fernald data, sample 1960–2005  GC(agg)  0.02  0.16  0.22  0.00  0.00  0.02  0.01  0.01  GC(dis)  0.17  0.52  0.54  0.11  0.09  0.17  0.25  0.34  Wang data, sample 1960–2009  GC(agg)  0.05  0.02  0.11  0.03  0.00  0.00  0.00  0.00  GC(dis)  0.37  0.38  0.51  0.40  0.27  0.28  0.27  0.23    PC = 3  PC = 4  PC = 5  PC = 6  PC = 7  PC = 8  PC = 9  PC = 10  Sample 1960–2010  CH  0.05  0.08  0.03  0.15  0.13  0.08  0.14  0.13  GC  0.02  0.00  0.04  0.01  0.00  0.00  0.00  0.00  Fernald data, sample 1960–2005  GC(agg)  0.02  0.16  0.22  0.00  0.00  0.02  0.01  0.01  GC(dis)  0.17  0.52  0.54  0.11  0.09  0.17  0.25  0.34  Wang data, sample 1960–2009  GC(agg)  0.05  0.02  0.11  0.03  0.00  0.00  0.00  0.00  GC(dis)  0.37  0.38  0.51  0.40  0.27  0.28  0.27  0.23  Notes: The table reports the p-value of the tests; CH is the test proposed in this paper; GC is the Granger causality test; the row GC(agg) reports the results of the test using aggregate data, the row GC(dis) the results of the test using disaggregated data; PC is the number of principal component in the auxiliary regression. In CH test, the number of leads tested is two and the preliminary VAR has four lags. In GC test, the lag length of the VAR is chosen with BIC and two lags of the principal components are used in the tests. View Large The CH test finds the system fundamental and, in general, the number of PC included in the testing equations does not matter. In contrast, a Granger causality test rejects the null of fundamentalness. Because the VAR includes TFP, which is a latent variable, and estimates are obtained from an aggregated production function, differences in the results could be due to aggregation and/or nonobservability problems. To verify this possibility, we consider a VAR where in place of utilization-adjusted aggregated TFP, we consider two different utilization-adjusted sectoral TFP measures. The first was constructed by John Fernald at the Federal Reserve Bank of San Francisco, and is obtained using the methodology of Basu et al. (2013), which produces time series for private consumption TFP, private investment TFP, government consumption and investment TFP and “net trade” TFP. The second panel of Table 7 presents results obtained in a VAR that includes consumption TFP (obtained aggregating private and public consumption), investment TFP (obtained aggregating private and public investments), and net trade TFP, all in log growth rates, and the growth rate of stock prices. Because the data end in 2005, the first row of the panel reports the p-values of a Granger causality test for the original bivariate system restricted to the 1960–2005 sample. As an alternative, we use the utilization-adjusted industry TFP data constructed by Christina Wang at the Federal Reserve Bank of Boston. We reaggregate industry TFPs into manufacturing, services, and “others” sectors, convert the data from annual to quarterly using a polynomial regression, and use the growth rate of these three variables together with the growth rate of stock prices in the VAR. The third panel of Table 7 presents results obtained with this VAR. Because the data end in 2009, the first row of the panel reports the p-values of a Granger causality test for the original bivariate system restricted to the 1960–2009 sample. Granger causality tests applied to the original bivariate system estimated over the two new samples still find the VAR nonfundamental. When the test is used in the VARs with sectoral/industry TFP measures, the null of nonfundamentalness is instead not rejected for all choices of vectors of principal components. Because this result holds when we enter the sectoral/industry TFP variables in level rather than growth rates, when we allow for a break in the TFP series, and when we use only two sectoral/industry TFP variables in the VAR, the conclusion is that a Granger causality test rejects the null in the original VAR because of aggregation problems. The diagnostic of this paper, being robust to aggregation problems, correctly identifies the original bivariate VAR as fundamental. Clearly, if the DGP is a truly sectoral model, the shocks and the dynamics produced by both the bivariate and the four variable VAR systems are likely to be averages of the shocks and dynamics of the primitive economy, which surely includes more than two or four disturbances. The interesting question is whether the news shocks extracted in the two and four variable systems produce different TFP responses. For illustration, Figure 1 reports the responses of stock prices and of TFP to standardized technology news shocks in the original VAR and in the four variable VAR with Fernald disaggregated TFP measures. For the four variable VAR, we only present the responses of investment TFP because the responses of the other two TFP variables are insignificantly different from zero. It is clear that the conditional dynamics in the two systems are qualitatively similar and statistically indistinguishable. Nevertheless, median responses are smaller, uncertainty is more pervasive, and the hump in the TFP response muted in the larger system. Hence, cross-sectional aggregation does not change much the dynamics but makes TFP responses artificially large and more precisely estimated. Researchers often construct models to quantitatively match the dynamics induced by shocks in small-scale VARs. Figure 1 suggests that the size and the persistence of the structural shocks needed to produce the aggregate evidence are probably smaller than previously agreed upon. Figure 1. View largeDownload slide Responses to technology news shocks. The dotted regions report pointwise 68% credible intervals; the solid line is the pointwise median response. The x-axis reports quarters and the y-axis the response of the level of the variable in deviation from the predictable path. Figure 1. View largeDownload slide Responses to technology news shocks. The dotted regions report pointwise 68% credible intervals; the solid line is the pointwise median response. The x-axis reports quarters and the y-axis the response of the level of the variable in deviation from the predictable path. 6. Conclusions Small-scale SVAR models are often used in empirical business cycle analyses even though the economic model one thinks has generated the data has a larger number of variables and shocks. In this situation, SVAR shocks are linear transformations of current and past primitive structural shocks perturbing the economy. SVAR shocks might be fundamental or nonfundamental, depending on the details of the economy, the information set available to the econometrician, and the variables chosen in the empirical analysis. However, variables providing noisy information about the primitive structural shocks will Granger cause SVAR shocks, even when the SVAR is fundamental. A similar problem arises when SVAR variables proxy for latent variables. We conduct a simulation study illustrating that spurious nonfundamentalness may indeed occur when the SVAR used for the empirical analysis is of smaller scale than the DGP of the data. We propose an alternative testing procedure that has the same power properties as existing diagnostics when nonfundamentalness is present but does not face aggregation or nonobservability problems when the system is fundamental. We also show that the procedure is robust to specification issues and to nuisance features. We demonstrate that a Granger causality diagnostic finds that a bivariate SVAR measuring the impact of news is nonfundamental, whereas our test finds it fundamental. The presence of an aggregated TFP measure in the SVAR explains the discrepancy. When sectoral TFP measures are used, a Granger causality diagnostic also finds the SVAR fundamental. A few lessons can be learned from our paper. First, Granger causality tests may give misleading conclusions when testing for fundamentalness whenever aggregation or nonobservability problems are present. Second, to derive reliable conclusions, one should have fundamentalness tests that are insensitive to specification and nuisance features. The test proposed in this paper satisfies both criteria; those present in the literature do not. Finally, if one is willing to assume that the DGP is a particular structural model, the procedure described (Sims and Zha 2006) can be used to check if a particular VAR shock can be recovered from current and past values of the observables, therefore bypassing the need to check for fundamentalness. However, when the DGP is unknown, the structural model one employs misspecified, or the exact mapping from the DGP and the estimated SVAR hard to construct, procedures like ours can help researchers to understand whether small-scale SVARs are good starting points to undertake informative business cycle analyses. Appendix This appendix reports the size of the CH test when nuisance parameters are varied. We change the number of lags of first-stage residuals in the auxiliary regression p2; the variance of the error in the DGP for the additional variables $$\sigma ^2_{\xi }$$; the number of principal components used in the auxiliary regressions s; and the number of leads of the first-stage residuals in the auxiliary regression q. Power tables are omitted, because they are identical to those in the text. Table A.1. Size of the CH test, aggregation, varying p2.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  p2=4  10%  11.2  13.5  14.5  13.3  14.8  20.1  29.0  44.1  5%  2.5  2.3  2.5  2.2  2.9  4.6  6.1  11.9  1%  1.6  1.9  1.2  1.6  2.2  4.1  6.2  12.3  p2=2  10%  10.5  13.2  12.1  12.5  14.1  19.3  27.0  40.8  5%  5.8  7.1  5.4  6.0  7.6  12.2  15.9  29.7  1%  1.8  2.0  0.9  1.1  2.1  3.2  5.7  12.5    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  p2=4  10%  11.2  13.5  14.5  13.3  14.8  20.1  29.0  44.1  5%  2.5  2.3  2.5  2.2  2.9  4.6  6.1  11.9  1%  1.6  1.9  1.2  1.6  2.2  4.1  6.2  12.3  p2=2  10%  10.5  13.2  12.1  12.5  14.1  19.3  27.0  40.8  5%  5.8  7.1  5.4  6.0  7.6  12.2  15.9  29.7  1%  1.8  2.0  0.9  1.1  2.1  3.2  5.7  12.5  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; p2 represents the number of lags in the testing equation (42). View Large Table A.1. Size of the CH test, aggregation, varying p2.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  p2=4  10%  11.2  13.5  14.5  13.3  14.8  20.1  29.0  44.1  5%  2.5  2.3  2.5  2.2  2.9  4.6  6.1  11.9  1%  1.6  1.9  1.2  1.6  2.2  4.1  6.2  12.3  p2=2  10%  10.5  13.2  12.1  12.5  14.1  19.3  27.0  40.8  5%  5.8  7.1  5.4  6.0  7.6  12.2  15.9  29.7  1%  1.8  2.0  0.9  1.1  2.1  3.2  5.7  12.5    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  p2=4  10%  11.2  13.5  14.5  13.3  14.8  20.1  29.0  44.1  5%  2.5  2.3  2.5  2.2  2.9  4.6  6.1  11.9  1%  1.6  1.9  1.2  1.6  2.2  4.1  6.2  12.3  p2=2  10%  10.5  13.2  12.1  12.5  14.1  19.3  27.0  40.8  5%  5.8  7.1  5.4  6.0  7.6  12.2  15.9  29.7  1%  1.8  2.0  0.9  1.1  2.1  3.2  5.7  12.5  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; p2 represents the number of lags in the testing equation (42). View Large Table A.2. Size of the CH-test, aggregation, varying $$ \sigma ^2_ \xi$$.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  2.20  1.80  1.70  2.10  1.60  2.10  1.80  3.00  $$\sigma ^2_{\xi }=4$$  5%  1.10  0.70  0.40  0.60  0.50  1.00  0.60  0.90    1%  0.30  0.10  0.10  0.00  0.00  0.20  0.20  0.10    10%  1.00  0.70  0.20  0.80  0.50  1.50  0.60  1.10  $$\sigma ^2_{\xi }=0.25$$  5%  0.50  0.40  0.10  0.20  0.40  0.50  0.30  0.30    1%  0.00  0.20  0.00  0.10  0.00  0.00  0.10  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  2.20  1.80  1.70  2.10  1.60  2.10  1.80  3.00  $$\sigma ^2_{\xi }=4$$  5%  1.10  0.70  0.40  0.60  0.50  1.00  0.60  0.90    1%  0.30  0.10  0.10  0.00  0.00  0.20  0.20  0.10    10%  1.00  0.70  0.20  0.80  0.50  1.50  0.60  1.10  $$\sigma ^2_{\xi }=0.25$$  5%  0.50  0.40  0.10  0.20  0.40  0.50  0.30  0.30    1%  0.00  0.20  0.00  0.10  0.00  0.00  0.10  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; $$\sigma ^2_{\xi }$$ is the variance of the idiosyncratic error in the DGP for additional data. View Large Table A.2. Size of the CH-test, aggregation, varying $$ \sigma ^2_ \xi$$.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  2.20  1.80  1.70  2.10  1.60  2.10  1.80  3.00  $$\sigma ^2_{\xi }=4$$  5%  1.10  0.70  0.40  0.60  0.50  1.00  0.60  0.90    1%  0.30  0.10  0.10  0.00  0.00  0.20  0.20  0.10    10%  1.00  0.70  0.20  0.80  0.50  1.50  0.60  1.10  $$\sigma ^2_{\xi }=0.25$$  5%  0.50  0.40  0.10  0.20  0.40  0.50  0.30  0.30    1%  0.00  0.20  0.00  0.10  0.00  0.00  0.10  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  2.20  1.80  1.70  2.10  1.60  2.10  1.80  3.00  $$\sigma ^2_{\xi }=4$$  5%  1.10  0.70  0.40  0.60  0.50  1.00  0.60  0.90    1%  0.30  0.10  0.10  0.00  0.00  0.20  0.20  0.10    10%  1.00  0.70  0.20  0.80  0.50  1.50  0.60  1.10  $$\sigma ^2_{\xi }=0.25$$  5%  0.50  0.40  0.10  0.20  0.40  0.50  0.30  0.30    1%  0.00  0.20  0.00  0.10  0.00  0.00  0.10  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; $$\sigma ^2_{\xi }$$ is the variance of the idiosyncratic error in the DGP for additional data. View Large Table A.3. Size of the CH test, aggregation, varying s.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.10  1.10  0.30  0.60  0.80  1.00  1.00  1.70  s = 2  5%  0.50  0.50  0.00  0.30  0.40  0.50  0.10  0.60    1%  0.10  0.10  0.00  0.10  0.00  0.10  0.00  0.10    10%  1.70  1.80  0.70  1.80  1.40  1.90  1.40  2.50  s = 4  5%  0.80  0.70  0.10  0.60  0.50  0.60  0.50  1.10    1%  0.20  0.10  0.00  0.10  0.00  0.20  0.10  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.10  1.10  0.30  0.60  0.80  1.00  1.00  1.70  s = 2  5%  0.50  0.50  0.00  0.30  0.40  0.50  0.10  0.60    1%  0.10  0.10  0.00  0.10  0.00  0.10  0.00  0.10    10%  1.70  1.80  0.70  1.80  1.40  1.90  1.40  2.50  s = 4  5%  0.80  0.70  0.10  0.60  0.50  0.60  0.50  1.10    1%  0.20  0.10  0.00  0.10  0.00  0.20  0.10  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; s is the length of the vector of factors in the testing equation (42). View Large Table A.3. Size of the CH test, aggregation, varying s.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.10  1.10  0.30  0.60  0.80  1.00  1.00  1.70  s = 2  5%  0.50  0.50  0.00  0.30  0.40  0.50  0.10  0.60    1%  0.10  0.10  0.00  0.10  0.00  0.10  0.00  0.10    10%  1.70  1.80  0.70  1.80  1.40  1.90  1.40  2.50  s = 4  5%  0.80  0.70  0.10  0.60  0.50  0.60  0.50  1.10    1%  0.20  0.10  0.00  0.10  0.00  0.20  0.10  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.10  1.10  0.30  0.60  0.80  1.00  1.00  1.70  s = 2  5%  0.50  0.50  0.00  0.30  0.40  0.50  0.10  0.60    1%  0.10  0.10  0.00  0.10  0.00  0.10  0.00  0.10    10%  1.70  1.80  0.70  1.80  1.40  1.90  1.40  2.50  s = 4  5%  0.80  0.70  0.10  0.60  0.50  0.60  0.50  1.10    1%  0.20  0.10  0.00  0.10  0.00  0.20  0.10  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; s is the length of the vector of factors in the testing equation (42). View Large Table A.4. Size of the CH test, aggregation, varying q.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.80  3.10  2.40  1.90  2.00  2.60  1.60  3.70  q = 1  5%  0.70  1.40  0.80  0.30  0.70  1.50  0.70  2.10    1%  0.00  0.10  0.00  0.00  0.40  0.10  0.30  0.50    10%  1.20  0.80  0.50  0.70  0.90  1.20  0.60  1.80  q = 2  5%  0.40  0.20  0.20  0.30  0.30  0.50  0.30  0.80    1%  0.00  0.00  0.00  0.00  0.00  0.10  0.00  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.80  3.10  2.40  1.90  2.00  2.60  1.60  3.70  q = 1  5%  0.70  1.40  0.80  0.30  0.70  1.50  0.70  2.10    1%  0.00  0.10  0.00  0.00  0.40  0.10  0.30  0.50    10%  1.20  0.80  0.50  0.70  0.90  1.20  0.60  1.80  q = 2  5%  0.40  0.20  0.20  0.30  0.30  0.50  0.30  0.80    1%  0.00  0.00  0.00  0.00  0.00  0.10  0.00  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; q represents the number of leads in the testing equation (42). View Large Table A.4. Size of the CH test, aggregation, varying q.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.80  3.10  2.40  1.90  2.00  2.60  1.60  3.70  q = 1  5%  0.70  1.40  0.80  0.30  0.70  1.50  0.70  2.10    1%  0.00  0.10  0.00  0.00  0.40  0.10  0.30  0.50    10%  1.20  0.80  0.50  0.70  0.90  1.20  0.60  1.80  q = 2  5%  0.40  0.20  0.20  0.30  0.30  0.50  0.30  0.80    1%  0.00  0.00  0.00  0.00  0.00  0.10  0.00  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.80  3.10  2.40  1.90  2.00  2.60  1.60  3.70  q = 1  5%  0.70  1.40  0.80  0.30  0.70  1.50  0.70  2.10    1%  0.00  0.10  0.00  0.00  0.40  0.10  0.30  0.50    10%  1.20  0.80  0.50  0.70  0.90  1.20  0.60  1.80  q = 2  5%  0.40  0.20  0.20  0.30  0.30  0.50  0.30  0.80    1%  0.00  0.00  0.00  0.00  0.00  0.10  0.00  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; q represents the number of leads in the testing equation (42). View Large 1 The linear span is the smallest closed subspace that contains the subspaces. 2 Blaschke matrices are complex-valued filters. The main property of Blaschke matrices is that they take orthonormal white noises into orthonormal white noises. See Lippi and Reichlin (1994) for more details. References Alessi Lucia, Barigozzi Matteo, Capasso Marco ( 2011). “Non-Fundamentalness in Structural Econometric Models: A Review.” International Statistical Review , 79, 16– 47. Google Scholar CrossRef Search ADS   Basu Susanto, Fernald John, Fisher Jonas, Kimball Miles ( 2013). “Sector-Specific Technical Change.” Tech. rep. , National Bureau of Economic Research. Beaudry Paul, Feve Patrick, Guay Alain, Portier Franck ( 2015). “When is Non-Fundamentalness in VARs a Real Problem? An Application to News Shocks.” Tech. rep. , University of Toulouse. Beaudry Paul, Portier Franck ( 2006). “Stock Prices, News, and Economic Fluctuations.” American Economic Review , 96 (4), 1293– 1307. 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Are Small-Scale SVARs Useful for Business Cycle Analysis? Revisiting Nonfundamentalness

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Abstract

Abstract Nonfundamentalness arises when current and past values of the observables do not contain enough information to recover structural vector autoregressive (SVAR) disturbances. Using Granger causality tests, the literature suggested that several small-scale SVAR models are nonfundamental and thus not necessarily useful for business cycle analysis. We show that causality tests are problematic when SVAR variables cross-sectionally aggregate the variables of the underlying economy or proxy for nonobservables. We provide an alternative testing procedure, illustrate its properties with Monte Carlo simulations, and re-examine a prototypical small-scale SVAR model. (JEL: C5, C32, E5) The editor in charge of this paper was Claudio Michelacci. Acknowledgments: We thank the editor (Claudio Michelacci), five anonymous referees, Carlos Velasco, Jesus Gonzalo, Hernan Seoane, Gianni Amisano, Jordi Gali, Davide de Bortoli, Luca Sala, Benjamin Holcblat, Domenico Giannone, Lutz Kilian, Luca Gambetti, and Mario Forni and the participants of seminars at BI Norwegian Business School, UPF, Bocconi, Humboldt, Luiss, University of Glasgow, University of Helsinki, and Federal Reserve Bank of New York, Federal Reserve Board, and of the 2016 IAAE and 2016 ESEM conferences for comments and discussions. Canova acknowledges the financial support from the Spanish Ministerio de Economia y Competitividad through the grants ECO2012-33247 and ECO2015-68136-P and FEDER, UE. Canova is a Research Fellow at CEPR. 1. Introduction Structural vector autoregressive (SVAR) models have been extensively used over the last 30 years to study sources of cyclical fluctuations. The methodology hinges on the assumption that structural shocks can be obtained from linear combinations of current and past values of the observables. Nonfundamentalness arises when this is not the case. In a nonfundamental system, structural shocks obtained via standard identification procedures may have little to do with the true disturbances, even when identification is correctly performed, making SVAR evidence unreliable. Because likelihood or spectral estimation procedures cannot distinguish fundamental versus nonfundamental Gaussian systems (see e.g. Canova 2007, p. 114), it is conventional in applied work to rule out all the nonfundamental representations that possess the same second-order structure of the data. However, this choice is arbitrary. There are rational expectation models (Hansen and Sargent 1991), optimal prediction models (Hansen and Hodrick 1980), permanent income models (Fernández-Villaverde et al. 2007), news shocks models (Forni, Gambetti, and Sala 2014), and fiscal foresight models (Leeper, Walker, and Yang 2013), where optimal decisions may generate nonfundamental solutions. In addition, nonobservability of certain states or particular choices of observables may make fundamental systems nonfundamental. Despite the far-reaching implications it has for applied work, little is known on how to empirically detect nonfundamentalness. Following the lead of Lutkepohl (1991), Giannone and Reichlin (2006), and Forni and Gambetti (2014) (henceforth, FG) suggest that, under fundamentalness, external information should not Granger cause VAR variables. Using such a methodology, FG and Forni et al. (2014) argued that several small-scale SVARs are nonfundamental, thus implicitly questioning the economic conclusions that are obtained. Considering the popularity of small-scale SVARs in macroeconomics, this result is disturbing. This paper shows that Granger causality diagnostics may lead to spurious results in common and relevant situations. Why are there problems? Because of small samples, instabilities, identification, or interpretation difficulties, one typically uses a small-scale SVAR to examine the transmission of relevant disturbances, even if the process generating the data (DGP) features many more variables and shocks. But the shocks recovered by such SVAR systems are linear combinations of a potentially larger set of primitive structural shocks driving the economy. Thus, any variable excluded from the SVAR, but containing information about these primitive disturbances, predicts SVAR shocks (and thus Granger cause the endogenous variables), regardless of whether the model is fundamental or not. To illustrate the point, suppose we want to measure the effects of technology shocks on economic activity. Small-scale SVARs designed for this purpose typically include an aggregate measure of labor productivity, hours, and a few other aggregate variables. Suppose that what drives the economy are sector-specific, serially correlated productivity disturbances. The technology shock recovered from an SVAR will be a linear transformation of current and past sectoral productivity shocks. Because, for example, sectoral capital or sectoral labor productivity has information about sectoral disturbances, they will predict SVAR technology shocks, both when the model is fundamental and when it is not. A similar problem occurs when the SVAR features a proxy variable. For example, total factor productivity (TFP) is latent and typical estimates are obtained from output, capital, and hours-worked data. If capital and hours worked are excluded from the SVAR, any variable that predicts them will Granger cause estimated TFP, regardless of whether the model is fundamental or not. In general, whenever a small-scale SVAR is used, aggregation rather than nonfundamentalness may be the reason for why Granger causality tests find predictability. Thus, if nonfundamentalness is of interest, it is crucial to have a testing approach that is robust to aggregation and nonobservability problems. We propose an alternative procedure, based on ideas of Sims (1972), which has this property and exploits the fact that, under nonfundamentalness, future SVAR shocks predict a vector of variables excluded from the SVAR. We perform Monte Carlo simulations using a version of the model of Leeper et al. (2013) as DGP with capital tax, income tax, and productivity disturbances. We assume that the SVAR includes capital and an aggregate tax variable (or an aggregate tax rate computed from revenues and output data) and show that our approach has good small sample properties. In contrast, spurious nonfundamentalness arises with standard diagnostics. Absent aggregation problems, our approach, and Granger causality tests have similar small sample properties. We re-examine the small-scale SVAR employed by Beaudry and Portier (2006) designed to measure the macroeconomic effects of news. We find that the model is fundamental according to our test but nonfundamental according to a Granger causality diagnostic. We show that the rejection of the null with the latter is due to aggregation: once coarsely disaggregated TFP data are used in the SVAR, Granger causality no longer rejects the null of fundamentalness. The dynamics responses to news shocks in the systems with aggregated and disaggregated TFP measures are however similar (see also Beaudry et al. 2015). Thus, the SVAR disturbances the two systems recover are likely to be similar combinations of the primitive structural shocks and, thus, not necessarily economically interpretable. Two caveats need to be mentioned. First, our analysis is concerned with Gaussian macroeconomic variables. For non-Gaussian situations, see Hamidi Saneh (2014) or Gourieroux and Monfort (2015). Second, although we focus on SVARs, our procedure also works for structural vector autoregressive moving average (SVARMA) models, as long as the largest moving average (MA) root is sufficiently away from unity. The rest of the paper is organized as follows. Section 2 provides examples of nonfundamental systems and highlights the reasons for why problem occurs. Section 3 shows why standard tests may fail and propose an alternative approach. Section 4 examines the performance of various procedures using Monte Carlo simulations. Section 5 investigates the properties of a small-scale SVAR system. Section 6 concludes. 2. A Few Example of Nonfundamental Systems As Kilian and Lutkepohl (2016) highlighted, the literature has primarily focused on nonfundamentalness driven by a mismatch between agents and econometricians information sets, because of omitted variables (see e.g. Giannone and Reichlin 2006; Kilian and Murphy 2014) or the timing of news revelation (see e.g. Forni et al. 2014; Leeper et al. 2013). However, there may be other reasons for why it emerges. First, nonfundamentalness may be intrinsic to the optimization process and to the modeling choices an investigator makes (see e.g. Hansen and Sargent 1980, 1991). Optimizing models producing nonfundamental solutions are numerous; the next example shows one. Example 1. Suppose the dividend process is dt = et − aet−1, where a < 1, and suppose stock prices are expected discounted future dividends: pt = Et∑jβjdt+j, 0 < β < 1. The equilibrium value of pt in terms of the dividends innovations is   \begin{equation} p_{t}=(1-\beta a)e_{t}-ae_{t-1}. \end{equation} (1)Thus, even though the dividends process is fundamental (a < 1), the process for stock prices could be nonfundamental if |(1 − βa)/a| < 1, which occurs when 1/1 + β < a. If a ≥ 0.5, any economically reasonable value of β will make stock prices nonfundamental. On the other hand, if we allow stock prices to have a bubble component $$e_{t}^{b}$$ whose expected value is zero, the vector $$(e_{t},e_{t}^{b})$$ is fundamental for (dt, pt), regardless of the value of β. Thus, allowing for bubbles in theory makes a difference as far as recovering dividend shocks from the data. Second, nonfundamentalness may be due to nonobservability of some of the endogenous variables of a fundamental model. The next example illustrates how this is possible. Example 2. Suppose the production function (in logs) is   \begin{equation} Y_t = K_t + e_t, \end{equation} (2)and the law of motion of capital is   \begin{equation} K_t = (1-\delta ) K_{t-1}+ a e_t. \end{equation} (3)If both (Kt, Yt) are observable, this is just a bivariate restricted VAR(1) and et is fundamental for both (kt, yt). However, if the capital stock is unobservable, (2) becomes   \begin{equation} Y_t -(1-\delta ) Y_{t-1}= (1+a) e_t+ (1-\delta )e_{t-1}. \end{equation} (4)Clearly, if a < 0 and |a| < |δ|, et cannot be expressed as a convergent sum of current and past values of Yt and (4) is nonfundamental. In addition, if δ and a are both small, (4) has an MA root close to unity and a finite-order VAR for Yt poorly approximates the underlying bivariate process; see also Ravenna (2007) and Giacomini (2013). Third, a particular variable selection may induce nonfundamentalness, even if the system is, in theory, fundamental. Hansen and Hodrick (1980) showed that this happens when forecast errors are used in a VAR. The next example shows a less known situation. Example 3. Consider a standard consumption-saving problem. Let income Yt = et be a white noise. Let β = 1/R < 1 be the discount factor and assume quadratic preferences. Then   \begin{equation} C_{t}=C_{t-1}+(1-R^{-1})e_{t}. \end{equation} (5)Thus, the growth rate of consumption has a fundamental representation. However, if we set up the empirical model in terms of savings, St ≡ Yt − Ct, the solution is   \begin{equation} S_{t}-S_{t-1}=R^{-1}e_{t}-e_{t-1}, \end{equation} (6)and the growth rate of saving is nonfundamental. In sum, there may be many reasons for why an empirical model may be nonfundamental. Assuming away nonfundamentalness is problematic. Focusing on omitted variable or anticipation problems is, on the other hand, reductive. One ought to have procedures able to detect whether an SVAR is fundamental and, if it is not, whether violations are intrinsic to theory or due to applied investigators choices. 3. The Setup Because in this section we need to distinguish the structural disturbances driving the fluctuations in the DGP from the shocks an SVAR may recover, we use the convention that “primitive” structural shocks are the disturbances of the DGP and “SVAR” structural shocks those obtained with the empirical model. We assume that the DGP for the observables can be represented by an n-dimensional vector of stationary variables χt driven by s ≥ n serial and mutually uncorrelated primitive structural shocks ςt. Assumption 1. The vector χt satisfies   \begin{equation*} \chi _{t}=\Gamma (L)C\varsigma _{t}, \end{equation*} where C is an n × s matrix, $$\Gamma (L)=\sum _{i=0}^{\infty }\Gamma _{i}L^{i}$$, Γ0 = I, Γi’s are (n × n) matrices each i, L is the lag operator, and $$\sum _{i=0}^{\infty }\Gamma _{i}^{2}<\infty$$. The DGP in (1) is quite general and covers, for example, stationary dynamics general equilibrium (DSGE) models solved around a deterministic steady state or nonstationary DSGEs solved around a deterministic or a stochastic balanced growth path. Stationarity is assumed for convenience; the arguments we present are independent of whether χt stochastically drifts or not. Assumption 1 places mild restrictions on the roots of Γ(L). In theory, ςt could be fundamental for χt or not. Given a typical sample, n the dimension of χt is generally large and Γ(L) is of infinite dimension. Thus, for estimation and inferential purposes, an applied investigator typically confines attention to an m-dimensional vector xt, where $$\mathcal {H}_{t}^{x} \subset \mathcal {H}_{t}^{\chi }$$, and $$\mathcal {H}_{t}^{j}$$ is the closed linear span of {js: s ≤ t}, jt = (xt, χt).1 Assumption 2. The vector xt is driven by an  m × 1 vector of mutually and serially uncorrelated SVAR structural shocks ςx, t:   \begin{eqnarray} x_{t} &=& \Gamma _{x}(L)C_{x}\varsigma _{t} \end{eqnarray} (7)  \begin{eqnarray} \qquad\qquad\quad\quad &\equiv &\Pi (L)u_{t}=\Pi (L)D\varsigma _{x,t}, \end{eqnarray} (8)where m < n, Γx(L) is an m × m matrix for every L, Cx is also an m × s matrix, $$\Pi (L)=\sum _{i=0}^{\infty }\Pi _{i}L^{i}$$, Πi are m × m matrices for each i, Π0 = I, $$\sum _{i=0}^{\infty }\Pi _{i}^{2}<\infty$$, D is an m × m matrix. Equation (7) covers many cases of interest in macroeconomics. For example, xt may contain a subset of the variables belonging to χt, linear combinations, regression residuals, or forecast errors computed from the elements of χt. Thus, the framework includes the case of a variable belonging to the DGP but unobserved and thus omitted from the empirical model (as in Example 2); the situation where the DGP has disaggregated variables but the empirical model is set up in terms of aggregated variables; the case where the DGP has an unobservable variable (e.g., total factor productivity) proxied by a linear combination of observables (i.e., output, capital and labor); and the case where all DGP variables are observables (e.g., we have consumption data) but the empirical model contains linear combinations of the observables (i.e., savings as in Example 3). Because the dimension of ςt is larger than the dimension of xt, cross-sectional aggregation occurs. That is, the econometrician estimating an SVAR may be able to recover the m × 1 vector ςx, t from the reduced form residuals ut, but never the s × 1 vector ςt. For example, the DGP may describe a small open economy subject to external shocks coming from many countries, whereas the empirical model is specified so that only rest of the world variables are used. If Γ(L) has a block exogenous structure, it may be possible to aggregate the vector external shocks into one shock without contamination from other disturbances, see, for example, Faust and Leeper (1988). However, even in this case, it is clearly impossible to recover the full vector of country-specific external disturbances. Next, we provide the definition of fundamentalness for the empirical model (8) (see also Alessi, Barigozzi, and Capasso 2011; Rozanov 1967). Definition 1. An uncorrelated process {ut} is xt-fundamental if $$\mathcal {H}_{t}^{u}=\mathcal {H}_{t}^{x}$$ for all t. It is nonfundamental if $$\mathcal {H}_{t}^{u} \subset \mathcal {H}_{t}^{x}$$ and $$\mathcal {H}_{t}^{u}\ne \mathcal {H}_{t}^{x}$$, for at least one t. The empirical model (8) is fundamental if and only if all the roots of the determinant of the Π(L) polynomial lie outside the unit circle in the complex plane—in this case $$\mathcal {H}_{t}^{u}=\mathcal {H}_{t}^{x}$$, for all t. Alternatively, the model is fundamental if it is possible to express ut as a convergent sum of current and past xt’s. Fundamentalness is closely related to the concept of invertibility: The latter requires that no root of the determinant of Π(L) is on or inside the unit circle. Because we consider stationary variables, the two concepts are equivalent in our framework. In standard situations, there is a one-to-one mapping between ut and ςt and thus examining the fundamentalness of ut provides information about the fundamentalness of ςt. When the mapping is not one to one but the relationship between ut and ςt has a particular structure, it may be possible to find conditions insuring that when ut is fundamental for xt, ςt is fundamental for χt (see e.g. Forni et al. 2009). In all other situations, many of which are of interest, knowing the properties of ut for xt may tell us little about the properties of the primitive shocks ςt for χt. Note that, although ςx, t are linear combination of ςt, they may still be economically interesting. An aggregate TFP shock may be meaningful, even if the sectoral TFP shocks drive the economy, as long as several sectoral TFP disturbances produce similar dynamics for the variables of the SVAR. On the other hand, it is not generally true that a fundamental shock is necessarily structurally interpretable (this occurs e.g. when the wrong D matrix is used to recover ςx, t from a fundamental ut). 3.1. Standard Approaches To Detect Nonfundamentalness Checking whether a Gaussian VAR is fundamental or not is complicated because the likelihood function or the spectral density cannot distinguish between a fundamental and a nonfundamental representations. Earlier work by Lippi and Reichlin (1993, 1994) informally compared the dynamics produced by fundamental and selected nonfundamental representations. Giannone and Reichlin (2006) proposed to use Granger causality tests. The procedure works as follows. Suppose we augment xt with a vector of variables yt:   \begin{equation} {\left[\begin{array}{c}x_{t} \\ y_{t}\end{array}\right]} = {\left[\begin{array}{cc}\Pi (L) & 0 \\ B(L) & C(L)\end{array}\right]} {\left[\begin{array}{c}u_{t} \\ v_{t} \end{array}\right]} , \end{equation} (9)where vt are specific to yt and orthogonal to ut. Assume that all the roots of the determinant of B(L) are outside the unit circle. If (8) is fundamental, ut = Π(L)−1xt, and   \begin{equation} y_{t}=B(L)\Pi (L)^{-1}x_{t}+C(L)v_{t}, \end{equation} (10)where B(L)Π(L)−1 is one  sided in the nonnegative powers of L. Thus, under fundamentalness, yt is a function of current and past values of xt, but xt does not depend on yt. Hence, to detect nonfundamentalness, one can check whether xt is predicted by lags of yt. Although such an approach is useful to examine whether there are variables omitted from the empirical model, it is not clear whether it can reliably detect nonfundamentalness when shock aggregation is present. The reason is that cross-sectional aggregation is not innocuous. For example, Chang and Hong (2006) show that aggregate and sectoral technology shocks behave quite differently, and Sbrana and Silvestrini (2010) show that volatility predictions are quite different depending on the degree of cross-sectional aggregation of the portfolio one considers. The next example shows that aggregation may lead to spurious conclusions when using Granger causality to test for fundamentalness in small-scale SVARs. Example 4. Suppose the DGP is given by the following trivariate process:   \begin{eqnarray} \,\,\chi _{1t} &=\varsigma _{1t}+b_{1}\varsigma _{1t-1}+a\varsigma _{2t}+a\varsigma _{3t} \end{eqnarray} (11)  \begin{eqnarray} \qquad\quad\chi _{2t} &=a\varsigma _{1t}+\varsigma _{2t}+b_{2}\varsigma _{2t-1}+a\varsigma _{3t}+ \varsigma _{4t} \end{eqnarray} (12)  \begin{eqnarray} \qquad\quad\chi _{3t} &=a\varsigma _{1t}+a\varsigma _{2t}+\varsigma _{3t}+b_{3}\varsigma _{3t-1}- \varsigma _{4t}, \end{eqnarray} (13)where ςt = [ς1t, ς2t, ς3t, ς4t]΄ ∼ iid(0, diag(Σς)) and a ≤ 1. Suppose an econometrician sets up a bivariate empirical model with x1t = χ1t and x2t = 0.5(χ2t + χ3t). Thus, the second variable is an aggregated version of the last two variables of the DGP. The process generating xt is   \begin{eqnarray} x_{t} &=\left( \begin{array}{c}x_{1t} \\ x_{2t} \end{array} \right) =\left[ \begin{array}{c@{\quad}c@{\quad}c}1+b_{1}L & a & a \\ a & 0.5((a+1)+b_2 L) & 0.5((a+1)+b_3 L) \end{array} \right] \left( \begin{array}{c}\varsigma _{1t} \\ \varsigma _{2t} \\ \varsigma _{3t} \end{array} \right). \nonumber\\ \end{eqnarray} (14) Because with two endogenous variables one can recover at most two shocks, the econometrician implicitly estimates   \begin{eqnarray} x_{t} &=\left( \begin{array}{c}x_{1t} \\ x_{2t} \end{array} \right) =\left[ \begin{array}{c@{\quad}c}1+b_{1}L & a \\ a & 1+c L \end{array} \right] \left( \begin{array}{c}u_{1t} \\ u_{2t} \\ \end{array} \right), \end{eqnarray} (15)where $$\sigma ^2_{u1}=\sigma ^2_{\varsigma 1}$$. Letting ρ0 + ρ1L ≡ [0.5(a + 1) 0.5(a + 1)] + [0.5b2 0.5b3]L, and $$\hat{\Sigma }_{\varsigma } = \text{diag} \left\lbrace \sigma ^2_{\varsigma 2}, \sigma ^2_{\varsigma 3} \right\rbrace$$, c and $$\sigma ^2_{u2}$$ are obtained from   \begin{eqnarray} \mathrm{E}(x_{2t}x_{2t}^{\prime })&\equiv & \gamma (0) = \rho _0 \hat{\Sigma }_{\varsigma } \rho _0^{\prime }+ \rho _1 \hat{\Sigma }_{\varsigma } \rho _1^{\prime }= (1+c^2) \sigma ^2_{u2}, \end{eqnarray} (16)  \begin{eqnarray} \mathrm{E}(x_{2t}x_{2t-1}^{\prime }) &\equiv & \gamma (1) = \rho _1 \hat{\Sigma }_{\varsigma } \rho _0^{\prime }= c \sigma ^2_{u2}. \phantom{2_{u2}= (1+1\,c^2) \sigma ^2_{u2},} \end{eqnarray} (17)These two conditions can be combined to obtain the quadratic equation   \begin{equation} c^2\gamma (1)-c \gamma (0)+\gamma (1)=0. \end{equation} (18)Given γ(0), γ(1), (18) can be used to compute the solution for c and then $$\sigma ^2_{u2}=c^{-1}\gamma (1)$$. Because ut in (15) is a white noise, it is unpredictable using ut−s (or xt−s), s > 0. However, it can be predicted using ςt−s, even when ut is fundamental. In fact, letting c* be the fundamental solution of (18) and using (14) and (15) have   \begin{eqnarray} u_{2t} &=&(1+c^{\ast }L)^{-1}[\rho _{0}\hat{\varsigma }_{t}+\rho _{1}\hat{\varsigma }_{t-1}] \nonumber \\ &=&\rho _{0}\hat{\varsigma }_{t}+c^{\ast }\rho _{0}\hat{\varsigma }_{t-1}+(c^{\ast })^{2}\rho _{0}\hat{\varsigma }_{t-2}+(c^{\ast })^{3}\rho _{0}\hat{\varsigma }_{t-3}+\cdots \nonumber \\ &&{}\quad +\rho _{1}\hat{\varsigma }_{t-1}+c^{\ast }\rho _{1}\hat{\varsigma }_{t-2}+(c^{\ast })^{2}\rho _{1}\hat{\varsigma }_{t-3}+(c^{\ast })^{3}\rho _{1}\hat{\varsigma }_{t-4}+\cdots , \end{eqnarray} (19)where $$\hat{\varsigma }=[\varsigma _{2t},\varsigma _{3t}]^{\prime }$$. Because χ2t−s and χ3t−s carry information about ςt−s, lags of yt = [χ2t, χ3t] predict ut, and thus xt. Notice that in terms of equation (9), ς4t plays the role of vt. To gain intuition for why predictability tests give spurious results, notice that (19) implies $$(1+c^{\ast }L)u_{2t}=\rho _{0}\hat{\varsigma }_{t}+\rho _{1}\hat{\varsigma }_{t-1}$$. Thus, under aggregation, estimated SVAR shocks are linear functions of current and past primitive structural shocks, making them predictable using any variable that has information about the lags of the primitive structural shocks. This occurs even if the VAR is correctly specified (i.e., there are sufficient lags to recover ut as in (15)). In standard SVARs with no aggregation, the condition corresponding to (19) is ut = ρςt. Thus, absent misspecification, lags of yt will not predict ut. Granger causality tests have been used by many as a tool to detect misspecification in small-scale VARs. For example, if a serially correlated variable is omitted from the VAR, ut, the econometrician recovers are serially correlated and thus predictable using any variable correlated with the omitted one, see, for example, Canova, Michelacci, and Lopez Salido (2010). When they are applied to systems like those in Example 4, causality tests detect misspecification but for the wrong reason. The VAR system is fundamental, ut derived from (15) are white noise, but Granger causality tests reject the predictability null because aggregation has created a particular correlation structure in SVAR shocks. Example 4 also clearly highlights that the concepts of predictable, fundamental, and structural shocks are distinct. ut’s in (15) are predictable, regardless of whether they are fundamental or not. In addition, ut = ςx, t are structural, in the sense that the responses of x1t to ut and to ςit, i = 1, 2, 3, are similar, even ut are predictable. Finally, ut may be nonfundamental (if c, the nonfundamental solution of (18) is used in (19)), even if they are structural. A similar outcome obtains if the empirical model contains, for example, an estimated proxy for an observable variable or residuals computed from the elements of χt. Suppose (x1t = χ1t, x2t = χ1t − γ1χ2t − γ2χ3t)΄, and γ1, γ2 are (estimated) parameters. For example, x2t are Solow residuals and γ1, γ2 are the labor and the capital shares. The process generating xt is   \begin{eqnarray*} &&x_{t} = \nonumber \\ && \left[ \begin{array}{c@{\,\,}c@{\,\,}c@{\,\,}c}1+b_{1}L & a & a & 0 \\ (1-\gamma a-(1-\gamma )a)-b_{1}L & (a-\gamma -a(1-\gamma ))-b_{2}L & (a-\gamma a-(1-\gamma ))-b_{3}L) & -\gamma _{1}+\gamma _{2}\end{array}\right]\\ && \quad \times \,\left( \begin{array}{c}\varsigma _{1t} \\ \varsigma _{2t} \\ \varsigma _{3t} \\ \varsigma _{4t}\end{array} \right). \end{eqnarray*} As before, the econometrician estimates (15). Also in this situation, ut is unpredictable using ut−s or xt−s. However, lags of any yt constructed as noisy linear transformation of [χ2t, χ3t] predict ut, even when it is fundamental for xt. In sum, the existence of variables that Granger cause xt may have nothing to do with fundamentalness. What is crucial to create spurious results is that SVAR shocks linearly aggregate the information contained in current and past primitive structural shocks. Although to some readers Example 4 may look special, it is not. We next formally show that predictability obtains, in general, under linear cross-sectional aggregation. This together with the fact that small-scale SVARs are generally used in business cycle analysis, even when the DGP may feature a large number of primitive structural shocks, should convince skeptical readers of the relevance of Example 4. Proposition 1 shows that the class of moving average models is closed with respect to linear transformations and Proposition 2 that aggregated moving average models are predictable. Proposition 1. Let χ1t be a zero-mean MA(q1) process   \begin{equation} \chi _{1t} = \varsigma _{1t} + \Phi _{1}\varsigma _{1t-1} + \Phi _{2}\varsigma _{1t-2} + \cdots + \Phi _{q_{1}}\varsigma _{1t-q_{1}} \equiv \Phi (L)\varsigma _{1t}, \end{equation} (20)with $$\mathrm{E}(\varsigma _{1t}\varsigma _{1t-j})=\sigma _{1}^{2}$$ if j = 0 and 0 otherwise, and let χ2t be a zero-mean MA(q2) process:   \begin{equation} \chi _{2t}=\varsigma _{2t}+\Psi _{1}\varsigma _{2t-1}+\Psi _{2}\varsigma _{2t-2}+\cdots +\Psi _{q_{2}}\varsigma _{2t-{q_{2}}}\equiv \Psi (L)\varsigma _{2t}, \end{equation} (21)with $$\mathrm{E}(\varsigma _{2t}\varsigma _{2t-j})= \sigma _{2}^{2}$$ if j = 0 and 0 otherwise. Assume that χ1t and χ2t are independent at all leads and lags. Then   \begin{equation} x_{t}=\chi _{1t}+\gamma \chi _{2t}=u_{t}+\Pi _{1}u_{t-1}+\Pi _{2}u_{t-2}+\cdots +\Pi _{q}u_{t-q}\equiv \Pi (L)u_{t}, \end{equation} (22)where q = max {q1, q2}, γ is a vector of constants, and ut is a white noise process. Proof. The proof follows from Hamilton (1994, p. 106). Proposition 2. Let xt be an m-dimensional process obtained as in Proposition 1. Then, ς1t−s and ς2t−s, s ≥ 1 Granger cause xt. Proof. It is enough to show that   \begin{equation*} \mathbb {P}\big [x_{t}|x_{t-1},x_{t-2},\ldots ,\varsigma _{1t-1},\varsigma _{1t-2},\ldots ,\varsigma _{2t-1},\varsigma _{2t-2},\ldots \ \big ]\ne \mathbb {P}\big [x_{t}|x_{t-1},x_{t-2},\ldots \big ], \end{equation*} when the model is fundamental, where $$\mathbb {P}$$ is the linear projection operator. Here, $$\mathcal {H}_{t}^{x}=\mathcal {H}_{t}^{u}$$. Hence, it suffices to show that ut is Granger caused by lagged values of ς1t and ς2t. That is   \begin{equation*} \mathbb {P}\big [u_{t}|u_{t-1},u_{t-2},\ldots ,\varsigma _{1t-1},\varsigma _{1t-2},\ldots ,\varsigma _{2t-1},\varsigma _{2t-2},\ldots \ \big ]\ne \mathbb {P}\big [u_{t}|u_{t-1},u_{t-2},\ldots \big ]. \end{equation*} From Proposition 1, we have that Π(L)ut = Φ(L)ς1t + Ψ(L)ς2t, and therefore ut = Π(L)−1Φ(L)ς1t + Π(L)−1Ψ(L)ς2t, where Π(L)−1 exists because the model is fundamental. Hence, Π(L)−1Φ(L) and Π(L)−1Ψ(L) are one-sided polynomial in the nonnegative powers of L and   \begin{multline*} \mathbb {P}[u_{t}|u_{t-1},u_{t-2},\ldots ,\varsigma _{1t-1},\varsigma _{1t-2},\ldots ,\varsigma _{2t-1},\varsigma _{2t-2},\ldots \ ]\\ =\mathbb {P}[u _{t}|\varsigma _{1t-1},\varsigma _{1t-2},\ldots ,\varsigma _{2t-1},\varsigma _{2t-2},\ldots \ ]\ne 0, \end{multline*} where the equality follows from ut being a white noise process. Thus, although ut in (22) is unpredictable given own lagged values, it can be predicted using lagged values of ς1t and ς2t because the information contained in the histories of ς1t and ς2t is not optimally aggregated into ut. Although the analysis is so far concerned with the fundamentalness of the vector ut, it is common in the VAR literature to focus attention on just one shock, see, for example, Christiano, Eichenbaum, and Evans (1999) or Galí (1999). The next example shows when one can recover a shock from current and past values of the observables, even when the system is nonfundamental. Example 5. Consider the following systems:   \begin{eqnarray} x_{1,t}& =&u_{1t} \\ x_{2,t}& =&u_{1t}+ u_{2t}-3 u_{2t-1} \nonumber \end{eqnarray} (23)  \begin{eqnarray} x_{1,t}& =&u_{1t}-2u_{2t-1} \\ x_{2,t}& =&u_{1t-1}+u_{2t-1}. \nonumber \end{eqnarray} (24)Both systems are nonfundamental—the determinants of the MA matrix are 1 − 3L, and L(1 − 2L), respectively, and they both vanish for L < 1. Thus, it is impossible to recover ut = (u1t, u2t) from current and lagged xt = (x1, t, x2, t)΄. However, although in the first system u1t can be obtained from x1, t, in the second system no individual shock can be obtained from linear combinations of current and past xt’s. A necessary condition for an SVAR shock to be an innovation is that it is orthogonal to the past values of the observables. FG suggest that a shock derived as in the first system of Example 5 is fundamental if it is unpredictable using (orthogonal to the) lags of the principal components obtained from variables belonging to the econometrician’s information set. Three important points need to be made about such an approach. First, fundamentalness is a property of a system not of a single shock. Thus, orthogonality tests are, in general, insufficient to assess fundamentalness. Second, as it is clear from Example 5, when one shock can be recovered, it is not the shock that creates nonfundamentalness in the first place. Finally, an orthogonality test has the same shortcomings as a Granger causality test. It will reject the null of unpredictability of an SVAR shock using disaggregated variables or factors providing noisy information about them, when the SVAR shock is a linear combinations of primitive disturbances, for exactly the same reasons that Granger causality tests fail. 3.2. An Alternative Approach In this section, we propose an alternative testing approach that we expect to have better properties in the situations of interest in this paper. To see what the procedure involves suppose we still augment (8) with a vector of additional variables yt = B(L)ut + C(L)vt. If (8) is fundamental, ut can be obtained as from current and past values of xt:   \begin{equation} u_{t}=x_{t}-\sum _{j=1}^{r}\omega _{j}x_{t-j}, \end{equation} (25)where ω(L) = Π(L)−1 and r is generally finite. Thus, under fundamentalness, yt only depends on current and past values of ut. If instead (8) is nonfundamental, ut cannot be recovered from the current and past values of xt. A VAR econometrician can only recover $$u_{t}^{\ast }=x_{t}-\sum _{j=1}^{r}\omega ^*_{j}x_{t-j}$$, where ω(L)* = Π(L)−1θ(L)−1, which is related to ut via   \begin{equation} u_{t}^{\ast }=\theta (L)u_{t}, \end{equation} (26)where θ(L) is a Blaschke matrix.2 Thus, the relationship between yt and the shocks recovered by the econometrician is $$y_{t}=B(L)\theta (L)^{-1} \theta (L)u_{t}+C(L)v_{t}\equiv B(L)^{\ast } u_t^* + C(L) v_t$$. Because B(L)* is generally a two-sided polynomial, yt depends on current, past, and future values of $$u_{t}^*$$. This proves the following proposition. Proposition 3. The system (8) is fundamental if $$u_{t+j}^{\ast },\ j \ge 1$$ fails to predict yt. Example 6. To illustrate proposition 3, let xt = (1 − 2.0L)ut, and then   \begin{eqnarray} x_t &=&(1-2.0L)\frac{(1-0.5L)}{(1-2.0L)}\frac{(1-2.0L)}{(1-0.5L)}u_{t}\equiv (1-0.5L)u_{t}^{\ast }, \end{eqnarray} (27)where   \begin{equation*} u_{t}^{\ast }=\frac{(1-2.0L)}{(1-0.5L)}u_{t}. \end{equation*} Let yt = (1 − 0.5L)ut + (1 − 0.6L)vt. Then   \begin{eqnarray} y_{t} &=&(1-0.5L)\frac{(1-0.5L)}{(1-2.0L)}u_{t}^{\ast }+(1-0.6L)v_{t} \nonumber \\ &=&\sum _{j=0}^{\infty }(1/2)^{j}((1-0.5L)^{2}u_{t+j}^{\ast })+(1-0.6L)v_{t-j}. \end{eqnarray} (28) Two points about our testing procedure need to be stressed. First, Sims (1972) has shown that xt is exogenous with respect to yt if future values of xt do not help to explain yt. Similarly here, a VAR system is fundamental if future values of xt (ut) do not help to predict the variables yt, excluded from the empirical model. Thus, although the null tested here and with Granger causality is the same, aggregation/nonobservability problems may make the testing results different. Second, our approach is likely to have better size properties, when SVAR shocks are linear functions of lags of primitive shocks, because yt generally contains more information than xt—under fundamentalness, future values of ut will not predict yt. Note also that our test is sufficiently general to detect nonfundamentalness due to structural causes, omitted variables, or the use of proxy indicators. 4. Some Monte Carlo Evidence To evaluate the small sample properties of traditional predictability tests and of our new procedure, we carry out a simulation study using a version of the model of Leeper et al. (2013), with two sources of tax disturbances. The representative household maximizes   \begin{equation} \mathrm{E}_{0}\sum _{t=0}^{\infty }\beta ^{t}\log (C_{t}) \end{equation} (29)subject to   \begin{equation} C_{t}+(1-\tau _{t,k})K_{t}+T_{t}\le (1-\tau _{t,y})A_{t}K_{t-1}^{\alpha }=(1-\tau _{t,y})Y_{t}, \end{equation} (30)where Ct, Kt, Yt, Tt, τt, k and τt, y denote time-t consumption, capital, output, lump-sum transfers, investment tax and income tax rates, respectively; At is a technology disturbance and Et is the conditional expectation operator. To keep the setup tractable, we assume full capital depreciation. The government sets tax rates randomly and adjusts transfers to satisfy Tt = τt, yYt + τt, kKt. The Euler equation and the resource constraints are   \begin{eqnarray} \frac{1}{C_{t}} &=&\alpha \beta \mathrm{E}_{t}\Big [\frac{(1-\tau _{t+1,y})}{(1-\tau _{t,k})}\frac{1}{C_{t+1}}\frac{A_{t+1}K_{t}^{\alpha }}{K_{t}}\Big ] \end{eqnarray} (31)  \begin{eqnarray} C_{t}+K_{t} &=A_{t}K_{t-1}^{\alpha }. \end{eqnarray} (32)Log linearizing, combining (31) and (32), we have   \begin{equation} \hat{K}_{t}=\alpha \hat{K}_{t-1}+\sum _{i=0}^{\infty }\theta ^{i}\mathrm{E} _{t}\hat{A}_{t+i+1}-\kappa _{k}\sum _{i=0}^{\infty }\theta ^{i}\mathrm{E}_{t} \hat{\tau }_{t+i,k}-\kappa _{y}\sum _{i=0}^{\infty }\theta ^{i}\mathrm{E}_{t} \hat{\tau }_{t+i+1,y}, \end{equation} (33)where   \begin{equation*} \kappa _{k}=\frac{\tau _{k}(1-\theta )}{(1-\tau _{k})},\quad \kappa _{y}=\frac{\tau _{y}(1-\theta )}{(1-\tau _{y})}, \quad \theta =\alpha \beta \frac{1-\tau _{y}}{1-\tau _{k}}, \quad \hat{K}_{t}\equiv \log (K_{t})-\log (K), \end{equation*}   \begin{equation*} \hat{A}_{t}\equiv \log (A_{t})-\log (A),\quad \hat{\tau }_{t,k}\equiv \log (\tau _{t,k})-\log (\tau _{k}), \quad \hat{\tau }_{t,y}\equiv \log (\tau _{t,y})-\log (\tau _{y}), \end{equation*} and lower case letters denote percentage deviations from steady states. We posit that technology and investment tax shocks are iid: $$\hat{A}_{t}=\varsigma _{t,A},\hat{\tau }_{t,k}=\varsigma _{t,k}$$; and that the income tax shock is an MA(1) process: $$\hat{\tau }_{t,y}=\varsigma _{t,y}+b\varsigma _{t-1,y}$$. Then, (33) is   \begin{equation} \hat{K}_{t}=\alpha \hat{K}_{t-1}+\varsigma _{t,a}-\kappa _{k}\varsigma _{t,k}-\kappa _{y}b\varsigma _{t,y}. \end{equation} (34)We assume that an econometrician observes $$\hat{K}_{t}$$ and an aggregate tax variable   \begin{equation} \hat{\tau }_{t}=\omega \hat{\tau }_{t,y}+\hat{\tau }_{t,k}=\varsigma _{t,k}+\omega (\varsigma _{t,y}+b\varsigma _{t-1,y}), \end{equation} (35)where ω controls the relative weight of income taxes in the aggregate. Alternatively, one can assume that investment and income tax revenues are both observables, but an econometrician works with a weighted sum of them. If $$(\hat{K}_{t},\hat{\tau }_{t})$$ are the variables the econometrician uses in the VAR, our design covers both the cases of aggregation and of a relevant latent variable. In fact, the DGP for the observables is   \begin{equation} {\left[\begin{array}{c}(1-\alpha L)\hat{K}_{t} \\ \hat{\tau }_{t} \end{array}\right]} = {\left[\begin{array}{c@{\quad}c@{\quad}c}1 & -\kappa _{k} & -\kappa _{y}b \\ 0 & 1 & \omega (1+bL) \end{array}\right]} {\left[\begin{array}{c}\varsigma _{t,a} \\ \varsigma _{t,k} \\ \varsigma _{t,y} \end{array}\right]} \equiv \Gamma _{x}(L)C_{x}\varsigma _{t}, \end{equation} (36) whereas the process recoverable by the econometrician is   \begin{equation} {\left[\begin{array}{c}(1-\alpha L)\hat{K}_{t} \\ \hat{\tau }_{t} \end{array}\right]} = {\left[\begin{array}{c@{\quad}c}1 & \rho \\ 0 & 1+cL \end{array}\right]} {\left[\begin{array}{c}u_{t,1} \\ u_{t,2} \end{array}\right]} \equiv \Pi (L)u_{t}, \end{equation} (37)where $$\sigma _{1}^{2}=\sigma _{a}^{2}$$, whereas $$c,\sigma _{2}^{2},\rho$$ are obtained from   \begin{eqnarray} c^{2}-c\left(\left(1+b^{2}\right)/b+\sigma _{k}^{2}/\left(\omega ^{2}b\sigma _{y}^{2}\right)\right)+1 &=0 \end{eqnarray} (38)  \begin{eqnarray} \sigma _{2}^{2} &=b\omega ^{2}\sigma _{y}^{2}/c \end{eqnarray} (39)  \begin{eqnarray} \rho &=-\sqrt{\left(\omega ^{2}\kappa _{y}^{2}b^{2}\sigma _{y}^{2}+\kappa _{k}^{2}\sigma _{k}^{2}\right)/\sigma _{2}^{2}}. \end{eqnarray} (40)By comparing (37) and (36), one can see that the aggregate tax shock ut, 2 will produce the same qualitative dynamic response in $$\hat{K}_{t}$$ as the investment and the income tax shocks but the scale of the effect will be altered. Depending on the size of ω, the aggregate shock will look more like the income or the investment tax shock. For the exercises we present, we let ςt, a, ςt, k, ςt, y ∼ iid N(0, 1); set α = 0.36, β = 0.99, τy = 0.25, τk = 0.1, ω = 1 and vary b so that c ∈ (0.1, 0.8) (fundamentalness region) or c ∈ (2, 9) (nonfundamentalness region). To perform the tests, we need additional data not used in the empirical model (37). We assume that an econometrician observes a panel of 30 time series generated by   \begin{equation} (1-0.9 L)y_{i,t}=\varsigma _{t,a}+\gamma _{i} \varsigma _{t,y}+(1-\gamma _i) \varsigma _{t,k}+\xi _{i,t},\quad i=1,\ldots ,30, \end{equation} (41)where $$\xi _{i,t}\sim \,\,\text{iid}\,\,N\left(0,\sigma ^2_{\xi }\right)$$, and γi is Bernoulli, taking value 1 with probability 0.5. The properties of our procedure, denoted by CH, are examined with the regression   \begin{equation} f_{t}=\sum _{j=1}^{p_1}\varphi _{j}f_{t-j} +\sum _{j=0}^{p_2}\psi _{-j}u_{t-j} +\sum _{j=1}^{q}\psi _{j}u_{t+j}+e_{t}, \end{equation} (42)where ft is an s × 1 vector of principal components of (41) and ut is estimated using   \begin{equation} x_t=\sum _{j=1}^r \rho _{j}x_{t-j}+ u_t, \end{equation} (43)where $$x_{t}=(\hat{\tau } _{t},\hat{K}_{t})^{\prime }$$. The null is $$\mathbb {H}_{0}^{CH}:R\Psi =0$$, where Ψ = Vec[ψ1, ψ2, …, ψq], R is a matrix of zeros and ones. We report the results for p1 = 4, p2 = 0, q = 2, r = 4. To examine the properties of Granger causality tests, denoted by GC, we employ   \begin{equation} x_{t}=\sum _{j=0}^{p_1} \varphi _{j}x_{t-j}+\sum _{j=1}^{p_2}\varphi _{j}f_{t-j}+e_{t}, \end{equation} (44)where again $$x_{t}=(\hat{\tau } _{t},\hat{K}_{t})^{\prime }$$. The null is $$\mathbb {H}_{0}^{GC}:R\varPhi=0$$ where $$\varPhi=\mathrm{Vec}{[\varphi _{1},\varphi _{2},\ldots ,\varphi _{p_2}]}$$ and R is a matrix of zeros and ones. We report results for p1 = 4, p2 = 2. To perform an orthogonality test, denoted by OR, we first estimate (43) with r = 4. The tax shock, ut, τ, is identified as the only one affecting $$\hat{\tau }_{t}$$. Then, in the regression   \begin{equation} u_{t,\tau }=\sum _{j=1}^{p_{2}}\lambda _{j}f_{t-j}+e_{t}, \end{equation} (45)the ortogonality null is $$\mathbb {H}_{0}^{OR}:R\Lambda =0$$ where Λ = Vec[λ1, λ2, …, λq] and R is a matrix of zeros and ones. We report results for p2 = 2. To maintain comparability, all null hypotheses are tested using an F-test, setting s = 3 and $$\sigma _{\xi }^{2}=1$$ and no correction for generated regressors in (42) and (45). The appendix present results for the CH test when other values of p2, $$\sigma _{\xi }^{2}$$, s, and q are used. We set T = 200, which is the length of the time series used in Section 5, and T = 2, 000. To better understand the properties of the tests, we also run an experiment with no aggregation problems. Here, τk, t = 0, ∀t so that the DGP for capital and taxes is   \begin{equation} {\left[\begin{array}{c}(1-\alpha L)\hat{K}_{t} \\ \hat{\tau }_{t}\end{array}\right]} = {\left[\begin{array}{c@{\quad}c}1 & -\kappa _{y}b \\ 0 & (1+bL) \end{array}\right]} {\left[\begin{array}{c}\varsigma _{t,a} \\ \varsigma _{t,y} \\ \end{array}\right]}, \end{equation} (46)and the process for the additional data is   \begin{equation} (1-0.9 L)y_{i,t}=\varsigma _{t,a}+\gamma _{i} \varsigma _{t,y}+\xi _{i,t},\quad i=1,\ldots ,n. \end{equation} (47) The percentage of rejections of the null in 1,000 replications when the model is fundamental are in Tables 1 and 2. Our procedure is undersized (it rejects less than expected from the nominal size) but its performance is independent of the nominal confidence level and the sample size. Granger causality and orthogonality tests are prone to spurious nonfundamentalness. This is clear when T = 2,000; in the smaller sample, predictability due to aggregation is somewhat harder to detect. Table 1. Size of the tests: aggregation, T = 200.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  1.5  1.3  0.5  0.6  1.4  1.7  1.2  1.7  5%  0.8  0.5  0.1  0.3  0.3  0.9  0.4  1.1  1%  0.1  0.1  0.1  0.2  0.1  0.2  0.1  0.1  GC  10%  13.1  15.1  16.5  15.8  19.5  27.4  38.9  55.1  5%  7.5  8.2  9.5  9.2  11.2  15.5  26.7  42.2  1%  2.0  2.7  2.2  3.1  4.3  5.8  10.9  19.6  OR  10%  5.2  5.7  5.3  6.2  6.5  6.2  8.5  13.2  5%  2.9  2.3  2.9  2.5  3.5  2.3  4.2  6.5  1%  0.1  0.5  0.6  0.2  0.2  0.7  0.7  1.6    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  1.5  1.3  0.5  0.6  1.4  1.7  1.2  1.7  5%  0.8  0.5  0.1  0.3  0.3  0.9  0.4  1.1  1%  0.1  0.1  0.1  0.2  0.1  0.2  0.1  0.1  GC  10%  13.1  15.1  16.5  15.8  19.5  27.4  38.9  55.1  5%  7.5  8.2  9.5  9.2  11.2  15.5  26.7  42.2  1%  2.0  2.7  2.2  3.1  4.3  5.8  10.9  19.6  OR  10%  5.2  5.7  5.3  6.2  6.5  6.2  8.5  13.2  5%  2.9  2.3  2.9  2.5  3.5  2.3  4.2  6.5  1%  0.1  0.5  0.6  0.2  0.2  0.7  0.7  1.6  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 1. Size of the tests: aggregation, T = 200.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  1.5  1.3  0.5  0.6  1.4  1.7  1.2  1.7  5%  0.8  0.5  0.1  0.3  0.3  0.9  0.4  1.1  1%  0.1  0.1  0.1  0.2  0.1  0.2  0.1  0.1  GC  10%  13.1  15.1  16.5  15.8  19.5  27.4  38.9  55.1  5%  7.5  8.2  9.5  9.2  11.2  15.5  26.7  42.2  1%  2.0  2.7  2.2  3.1  4.3  5.8  10.9  19.6  OR  10%  5.2  5.7  5.3  6.2  6.5  6.2  8.5  13.2  5%  2.9  2.3  2.9  2.5  3.5  2.3  4.2  6.5  1%  0.1  0.5  0.6  0.2  0.2  0.7  0.7  1.6    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  1.5  1.3  0.5  0.6  1.4  1.7  1.2  1.7  5%  0.8  0.5  0.1  0.3  0.3  0.9  0.4  1.1  1%  0.1  0.1  0.1  0.2  0.1  0.2  0.1  0.1  GC  10%  13.1  15.1  16.5  15.8  19.5  27.4  38.9  55.1  5%  7.5  8.2  9.5  9.2  11.2  15.5  26.7  42.2  1%  2.0  2.7  2.2  3.1  4.3  5.8  10.9  19.6  OR  10%  5.2  5.7  5.3  6.2  6.5  6.2  8.5  13.2  5%  2.9  2.3  2.9  2.5  3.5  2.3  4.2  6.5  1%  0.1  0.5  0.6  0.2  0.2  0.7  0.7  1.6  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 2. Size of the tests: aggregation, T = 2,000.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.1  0.4  0.2  0.1  0.2  0.1  1.9  9.5  5%  0.1  0.2  0.2  0.1  0.1  0.1  1.0  4.2  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.8  GC  10%  83.3  86.6  88.8  92.4  98.1  99.9  100  100  5%  75.3  75.1  76.3  83.9  96.1  99.8  100  100  1%  49.7  44.7  46.0  58.7  83.9  98.6  100  100  OR  10%  34.0  30.1  29.4  30.2  41.7  52.1  81.0  99.1  5%  21.4  18.5  18.5  19.0  27.9  36.0  66.4  96.5  1%  7.4  6.8  7.3  6.4  9.0  13.0  34.9  81.8    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.1  0.4  0.2  0.1  0.2  0.1  1.9  9.5  5%  0.1  0.2  0.2  0.1  0.1  0.1  1.0  4.2  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.8  GC  10%  83.3  86.6  88.8  92.4  98.1  99.9  100  100  5%  75.3  75.1  76.3  83.9  96.1  99.8  100  100  1%  49.7  44.7  46.0  58.7  83.9  98.6  100  100  OR  10%  34.0  30.1  29.4  30.2  41.7  52.1  81.0  99.1  5%  21.4  18.5  18.5  19.0  27.9  36.0  66.4  96.5  1%  7.4  6.8  7.3  6.4  9.0  13.0  34.9  81.8  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 2. Size of the tests: aggregation, T = 2,000.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.1  0.4  0.2  0.1  0.2  0.1  1.9  9.5  5%  0.1  0.2  0.2  0.1  0.1  0.1  1.0  4.2  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.8  GC  10%  83.3  86.6  88.8  92.4  98.1  99.9  100  100  5%  75.3  75.1  76.3  83.9  96.1  99.8  100  100  1%  49.7  44.7  46.0  58.7  83.9  98.6  100  100  OR  10%  34.0  30.1  29.4  30.2  41.7  52.1  81.0  99.1  5%  21.4  18.5  18.5  19.0  27.9  36.0  66.4  96.5  1%  7.4  6.8  7.3  6.4  9.0  13.0  34.9  81.8    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.1  0.4  0.2  0.1  0.2  0.1  1.9  9.5  5%  0.1  0.2  0.2  0.1  0.1  0.1  1.0  4.2  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.8  GC  10%  83.3  86.6  88.8  92.4  98.1  99.9  100  100  5%  75.3  75.1  76.3  83.9  96.1  99.8  100  100  1%  49.7  44.7  46.0  58.7  83.9  98.6  100  100  OR  10%  34.0  30.1  29.4  30.2  41.7  52.1  81.0  99.1  5%  21.4  18.5  18.5  19.0  27.9  36.0  66.4  96.5  1%  7.4  6.8  7.3  6.4  9.0  13.0  34.9  81.8  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Why are traditional predictability tests rejecting the null much more than one would expect from the nominal size? The answer is obtained recalling equation (19). ut are linear combinations of current and past values of $$\hat{A}_{t}, \hat{\tau }_{t,k}, \hat{\tau }_{t,y}$$, whereas ft are linear combinations of $$\hat{A}_{t}, \hat{\tau }_{t,k}, \hat{\tau }_{t,y}$$ and ξi, t, ı = 1, …, 30. Because $$\hat{\tau }_{t,k}$$ is serially correlated, lags of ft may help to predict xt even when lags of xt are included, in particular, when the draws for γi are small. It is known that Granger causality tests have poor size properties when xt is persistent, see, for example, Ohanian (1988). Tables 3 and 4 disentangle aggregation from persistence problems: Because they have been constructed absent aggregation, they report size distortions due to persistent data. It is clear that, when b > 0.6, the size of Granger causality tests is distorted. To properly run such tests, the lag length p1 of the testing equation must be made function of the (unknown) persistence of the DGP. However, when b > 0.8, distortions are present even if p1 = 10. The orthogonality test performs better because it preliminary filters xt with a VAR. Thus, high serial correlation in xt is less of a problem. Table 3. Size of the tests: no aggregation, T = 200.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  2.5  1.7  2.5  1.6  1.5  2.3  2.8  2.6  5%  1.1  0.6  0.6  1.2  0.5  0.8  1.2  1.0  1%  0.1  0.1  0.2  0.1  0.1  0.1  0.2  0.3  GC  10%  11.4  10.5  13.3  13.5  10.9  14.8  15.5  28.4  5%  5.6  5.0  6.2  8.2  5.3  7.4  9.2  19.8  1%  1.3  1.0  1.6  1.6  1.1  2.0  2.4  6.1  OR  10%  4.4  5.1  5.3  4.7  4.0  6.4  6.2  8.9  5%  1.7  1.3  2.8  2.2  1.3  2.3  2.6  4.9  1%  0.2  0.1  0.5  0.3  0.1  0.6  0.6  1.8    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  2.5  1.7  2.5  1.6  1.5  2.3  2.8  2.6  5%  1.1  0.6  0.6  1.2  0.5  0.8  1.2  1.0  1%  0.1  0.1  0.2  0.1  0.1  0.1  0.2  0.3  GC  10%  11.4  10.5  13.3  13.5  10.9  14.8  15.5  28.4  5%  5.6  5.0  6.2  8.2  5.3  7.4  9.2  19.8  1%  1.3  1.0  1.6  1.6  1.1  2.0  2.4  6.1  OR  10%  4.4  5.1  5.3  4.7  4.0  6.4  6.2  8.9  5%  1.7  1.3  2.8  2.2  1.3  2.3  2.6  4.9  1%  0.2  0.1  0.5  0.3  0.1  0.6  0.6  1.8  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 3. Size of the tests: no aggregation, T = 200.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  2.5  1.7  2.5  1.6  1.5  2.3  2.8  2.6  5%  1.1  0.6  0.6  1.2  0.5  0.8  1.2  1.0  1%  0.1  0.1  0.2  0.1  0.1  0.1  0.2  0.3  GC  10%  11.4  10.5  13.3  13.5  10.9  14.8  15.5  28.4  5%  5.6  5.0  6.2  8.2  5.3  7.4  9.2  19.8  1%  1.3  1.0  1.6  1.6  1.1  2.0  2.4  6.1  OR  10%  4.4  5.1  5.3  4.7  4.0  6.4  6.2  8.9  5%  1.7  1.3  2.8  2.2  1.3  2.3  2.6  4.9  1%  0.2  0.1  0.5  0.3  0.1  0.6  0.6  1.8    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  2.5  1.7  2.5  1.6  1.5  2.3  2.8  2.6  5%  1.1  0.6  0.6  1.2  0.5  0.8  1.2  1.0  1%  0.1  0.1  0.2  0.1  0.1  0.1  0.2  0.3  GC  10%  11.4  10.5  13.3  13.5  10.9  14.8  15.5  28.4  5%  5.6  5.0  6.2  8.2  5.3  7.4  9.2  19.8  1%  1.3  1.0  1.6  1.6  1.1  2.0  2.4  6.1  OR  10%  4.4  5.1  5.3  4.7  4.0  6.4  6.2  8.9  5%  1.7  1.3  2.8  2.2  1.3  2.3  2.6  4.9  1%  0.2  0.1  0.5  0.3  0.1  0.6  0.6  1.8  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 4. Size of the tests: no aggregation, T = 2,000.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.9  1.3  1.0  0.9  0.5  1.0  1.3  6.0  5%  0.3  0.5  0.5  0.3  0.2  0.3  0.5  3.8  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  1.7  GC  10%  13.2  13.3  15.6  14.8  18.2  26.7  53.8  99.8  5%  7.4  8.0  8.7  9.0  11.1  16.3  41.2  99.3  1%  1.6  1.9  2.5  3.1  3.0  5.2  19.4  95.3  OR  10%  3.9  5.2  5.6  4.8  4.2  6.8  8.7  20.9  5%  1.3  3.2  1.7  1.8  1.7  3.2  4.8  17.8  1%  0.3  0.7  0.4  0.4  0.2  0.4  1.2  6.0    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.9  1.3  1.0  0.9  0.5  1.0  1.3  6.0  5%  0.3  0.5  0.5  0.3  0.2  0.3  0.5  3.8  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  1.7  GC  10%  13.2  13.3  15.6  14.8  18.2  26.7  53.8  99.8  5%  7.4  8.0  8.7  9.0  11.1  16.3  41.2  99.3  1%  1.6  1.9  2.5  3.1  3.0  5.2  19.4  95.3  OR  10%  3.9  5.2  5.6  4.8  4.2  6.8  8.7  20.9  5%  1.3  3.2  1.7  1.8  1.7  3.2  4.8  17.8  1%  0.3  0.7  0.4  0.4  0.2  0.4  1.2  6.0  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 4. Size of the tests: no aggregation, T = 2,000.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.9  1.3  1.0  0.9  0.5  1.0  1.3  6.0  5%  0.3  0.5  0.5  0.3  0.2  0.3  0.5  3.8  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  1.7  GC  10%  13.2  13.3  15.6  14.8  18.2  26.7  53.8  99.8  5%  7.4  8.0  8.7  9.0  11.1  16.3  41.2  99.3  1%  1.6  1.9  2.5  3.1  3.0  5.2  19.4  95.3  OR  10%  3.9  5.2  5.6  4.8  4.2  6.8  8.7  20.9  5%  1.3  3.2  1.7  1.8  1.7  3.2  4.8  17.8  1%  0.3  0.7  0.4  0.4  0.2  0.4  1.2  6.0    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  CH  10%  0.9  1.3  1.0  0.9  0.5  1.0  1.3  6.0  5%  0.3  0.5  0.5  0.3  0.2  0.3  0.5  3.8  1%  0.1  0.1  0.1  0.1  0.1  0.1  0.1  1.7  GC  10%  13.2  13.3  15.6  14.8  18.2  26.7  53.8  99.8  5%  7.4  8.0  8.7  9.0  11.1  16.3  41.2  99.3  1%  1.6  1.9  2.5  3.1  3.0  5.2  19.4  95.3  OR  10%  3.9  5.2  5.6  4.8  4.2  6.8  8.7  20.9  5%  1.3  3.2  1.7  1.8  1.7  3.2  4.8  17.8  1%  0.3  0.7  0.4  0.4  0.2  0.4  1.2  6.0  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Comparing the size tables constructed with and without aggregation, one can see that the properties of the CH test do not depend on the presence of aggregation or the persistence of the DGP. On the other hand, aggregation makes the properties of Granger causality and orthogonality tests significantly worse. Tables 5 and 6 report the empirical power of the tests when T = 200 with and without aggregation. All tests are similarly powerful to detect nonfundamentalness when it is present, regardless of the confidence level and the nature of the DGP. Although not reported for reasons of space, the power of the three tests is unchanged when T = 2,000. Table 5. Power of the tests: aggregation, T = 200.   c  2  3  4  5  6  7  8  9  CH  10%  99.9  100  100  100  100  100  100  100  5%  99.5  100  100  100  100  100  100  100  1%  98.7  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100    c  2  3  4  5  6  7  8  9  CH  10%  99.9  100  100  100  100  100  100  100  5%  99.5  100  100  100  100  100  100  100  1%  98.7  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 5. Power of the tests: aggregation, T = 200.   c  2  3  4  5  6  7  8  9  CH  10%  99.9  100  100  100  100  100  100  100  5%  99.5  100  100  100  100  100  100  100  1%  98.7  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100    c  2  3  4  5  6  7  8  9  CH  10%  99.9  100  100  100  100  100  100  100  5%  99.5  100  100  100  100  100  100  100  1%  98.7  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 6. Power of the tests: no aggregation, T = 200.   c  2  3  4  5  6  7  8  9  CH  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100    c  2  3  4  5  6  7  8  9  CH  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large Table 6. Power of the tests: no aggregation, T = 200.   c  2  3  4  5  6  7  8  9  CH  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100    c  2  3  4  5  6  7  8  9  CH  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  GC  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  OR  10%  100  100  100  100  100  100  100  100  5%  100  100  100  100  100  100  100  100  1%  100  100  100  100  100  100  100  100  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is no aggregation; CH is the test proposed in this paper; OR is the orthogonality test; GC is the Granger causality test. The length of the vector of principal components used in the testing equation is s = 3. View Large The additional tables in the appendix indicate that the size properties of the CH test are insensitive to the selection of three nuisance parameters: the variance of the shocks to the additional data $$\sigma _{\xi }^{2}$$, the number of principal components used in the testing equation s, and the number of leads of the first-stage residuals used in the testing equation q. On the other hand, the choice of p2, the number of lags of the first-stage residuals used in the testing equations, matters. This is true, in particular, when the persistence of the DGP increases and is due to the fact that with high persistence, r = 4 is insufficient to whiten the first-stage residual, and the presence of serial correlation in ut makes its future values spuriously significant. To avoid this problem in practice, we recommend users to specify the testing equation with only leads of ut. Alternatively, if lags of ut are included, r should be large to insure that serial correlation in the first-stage residuals is negligible. 5. Reconsidering A Small-Scale SVAR Standard business cycle theories assume that economic fluctuations are driven by surprises in current fundamentals, such as aggregate productivity or the monetary policy rule. Motivated by the idea that changes in expectations about future fundamentals may drive business fluctuations, Beaudry and Portier (2006) study the effect of news shocks on the real economy using an SVAR that contains stock prices and TFP. Because models featuring news shocks have solutions displaying moving average components, empirical models with a finite number of lags may be unable to capture the underlying dynamics, making the SVARs considered in the literature prone to nonfundamentalness. In addition, Forni et al. (2014) provide a stylized Lucas tree model where perfectly predictable news to the dividend process may induce nonfundamentalness in a VAR system comprising the growth rate of stock prices and the growth rate of dividends. The solution of their model when news come two periods in advance is   \begin{equation} {\left[\begin{array}{c}\Delta d_{t} \\ \Delta p_{t} \end{array}\right]} = {\left[\begin{array}{c@{\quad}c}L^{2} & 1 \\ \frac{\beta ^{2}}{1-\beta }+\beta L & \frac{\beta }{1-\beta } \end{array}\right]} {\left[\begin{array}{c}\varsigma _{1t} \\ \varsigma _{2t} \end{array}\right]} \equiv C(L)\varsigma _{t}, \end{equation} (48)where dt are dividends, pt are stock prices, and 0 < β < 1 is the discount factor. Because C(L) vanishes for L = 1 and L = −β, ut is nonfundamental for (Δdt, Δpt). Intuitively, this occurs because agents’ information set, which includes current and past values of structural shocks, is not aligned with the econometrician’s information set, which includes current and past values of the growth rate of dividends and stock prices. The fundamental and nonfundamental dynamics this model generates in response to news shocks are similar because the root generating nonfundamentalness (L = −β) is near unity, see also Beaudry et al. (2015). In general, the properties of the SVAR the econometrician considers depend on the process describing the information flows, on the variables observed by the econometrician and those included in the SVAR. To re-examine the evidence, we estimate a VAR with the growth rates of capacity-adjusted TFP and of stock prices for the period 1960Q1–2010Q4, both of which are taken from Beaudry and Portier (2014), and we use the same principal components as in Forni et al. (2014). Table 7 reports the p-values of the tests, varying the number of principal components employed in the auxiliary regression, which enter in first difference in all the tests. In the CH test, the testing model has four lags of the PC and we are examining the predictive power of two leads of the VAR residuals. In the GC test, the lag length of the VAR is chosen by BIC and two lags of the principal components are used in the tests. Table 7. Testing fundamentalness: VAR with TFP growth and stock prices growth.   PC = 3  PC = 4  PC = 5  PC = 6  PC = 7  PC = 8  PC = 9  PC = 10  Sample 1960–2010  CH  0.05  0.08  0.03  0.15  0.13  0.08  0.14  0.13  GC  0.02  0.00  0.04  0.01  0.00  0.00  0.00  0.00  Fernald data, sample 1960–2005  GC(agg)  0.02  0.16  0.22  0.00  0.00  0.02  0.01  0.01  GC(dis)  0.17  0.52  0.54  0.11  0.09  0.17  0.25  0.34  Wang data, sample 1960–2009  GC(agg)  0.05  0.02  0.11  0.03  0.00  0.00  0.00  0.00  GC(dis)  0.37  0.38  0.51  0.40  0.27  0.28  0.27  0.23    PC = 3  PC = 4  PC = 5  PC = 6  PC = 7  PC = 8  PC = 9  PC = 10  Sample 1960–2010  CH  0.05  0.08  0.03  0.15  0.13  0.08  0.14  0.13  GC  0.02  0.00  0.04  0.01  0.00  0.00  0.00  0.00  Fernald data, sample 1960–2005  GC(agg)  0.02  0.16  0.22  0.00  0.00  0.02  0.01  0.01  GC(dis)  0.17  0.52  0.54  0.11  0.09  0.17  0.25  0.34  Wang data, sample 1960–2009  GC(agg)  0.05  0.02  0.11  0.03  0.00  0.00  0.00  0.00  GC(dis)  0.37  0.38  0.51  0.40  0.27  0.28  0.27  0.23  Notes: The table reports the p-value of the tests; CH is the test proposed in this paper; GC is the Granger causality test; the row GC(agg) reports the results of the test using aggregate data, the row GC(dis) the results of the test using disaggregated data; PC is the number of principal component in the auxiliary regression. In CH test, the number of leads tested is two and the preliminary VAR has four lags. In GC test, the lag length of the VAR is chosen with BIC and two lags of the principal components are used in the tests. View Large Table 7. Testing fundamentalness: VAR with TFP growth and stock prices growth.   PC = 3  PC = 4  PC = 5  PC = 6  PC = 7  PC = 8  PC = 9  PC = 10  Sample 1960–2010  CH  0.05  0.08  0.03  0.15  0.13  0.08  0.14  0.13  GC  0.02  0.00  0.04  0.01  0.00  0.00  0.00  0.00  Fernald data, sample 1960–2005  GC(agg)  0.02  0.16  0.22  0.00  0.00  0.02  0.01  0.01  GC(dis)  0.17  0.52  0.54  0.11  0.09  0.17  0.25  0.34  Wang data, sample 1960–2009  GC(agg)  0.05  0.02  0.11  0.03  0.00  0.00  0.00  0.00  GC(dis)  0.37  0.38  0.51  0.40  0.27  0.28  0.27  0.23    PC = 3  PC = 4  PC = 5  PC = 6  PC = 7  PC = 8  PC = 9  PC = 10  Sample 1960–2010  CH  0.05  0.08  0.03  0.15  0.13  0.08  0.14  0.13  GC  0.02  0.00  0.04  0.01  0.00  0.00  0.00  0.00  Fernald data, sample 1960–2005  GC(agg)  0.02  0.16  0.22  0.00  0.00  0.02  0.01  0.01  GC(dis)  0.17  0.52  0.54  0.11  0.09  0.17  0.25  0.34  Wang data, sample 1960–2009  GC(agg)  0.05  0.02  0.11  0.03  0.00  0.00  0.00  0.00  GC(dis)  0.37  0.38  0.51  0.40  0.27  0.28  0.27  0.23  Notes: The table reports the p-value of the tests; CH is the test proposed in this paper; GC is the Granger causality test; the row GC(agg) reports the results of the test using aggregate data, the row GC(dis) the results of the test using disaggregated data; PC is the number of principal component in the auxiliary regression. In CH test, the number of leads tested is two and the preliminary VAR has four lags. In GC test, the lag length of the VAR is chosen with BIC and two lags of the principal components are used in the tests. View Large The CH test finds the system fundamental and, in general, the number of PC included in the testing equations does not matter. In contrast, a Granger causality test rejects the null of fundamentalness. Because the VAR includes TFP, which is a latent variable, and estimates are obtained from an aggregated production function, differences in the results could be due to aggregation and/or nonobservability problems. To verify this possibility, we consider a VAR where in place of utilization-adjusted aggregated TFP, we consider two different utilization-adjusted sectoral TFP measures. The first was constructed by John Fernald at the Federal Reserve Bank of San Francisco, and is obtained using the methodology of Basu et al. (2013), which produces time series for private consumption TFP, private investment TFP, government consumption and investment TFP and “net trade” TFP. The second panel of Table 7 presents results obtained in a VAR that includes consumption TFP (obtained aggregating private and public consumption), investment TFP (obtained aggregating private and public investments), and net trade TFP, all in log growth rates, and the growth rate of stock prices. Because the data end in 2005, the first row of the panel reports the p-values of a Granger causality test for the original bivariate system restricted to the 1960–2005 sample. As an alternative, we use the utilization-adjusted industry TFP data constructed by Christina Wang at the Federal Reserve Bank of Boston. We reaggregate industry TFPs into manufacturing, services, and “others” sectors, convert the data from annual to quarterly using a polynomial regression, and use the growth rate of these three variables together with the growth rate of stock prices in the VAR. The third panel of Table 7 presents results obtained with this VAR. Because the data end in 2009, the first row of the panel reports the p-values of a Granger causality test for the original bivariate system restricted to the 1960–2009 sample. Granger causality tests applied to the original bivariate system estimated over the two new samples still find the VAR nonfundamental. When the test is used in the VARs with sectoral/industry TFP measures, the null of nonfundamentalness is instead not rejected for all choices of vectors of principal components. Because this result holds when we enter the sectoral/industry TFP variables in level rather than growth rates, when we allow for a break in the TFP series, and when we use only two sectoral/industry TFP variables in the VAR, the conclusion is that a Granger causality test rejects the null in the original VAR because of aggregation problems. The diagnostic of this paper, being robust to aggregation problems, correctly identifies the original bivariate VAR as fundamental. Clearly, if the DGP is a truly sectoral model, the shocks and the dynamics produced by both the bivariate and the four variable VAR systems are likely to be averages of the shocks and dynamics of the primitive economy, which surely includes more than two or four disturbances. The interesting question is whether the news shocks extracted in the two and four variable systems produce different TFP responses. For illustration, Figure 1 reports the responses of stock prices and of TFP to standardized technology news shocks in the original VAR and in the four variable VAR with Fernald disaggregated TFP measures. For the four variable VAR, we only present the responses of investment TFP because the responses of the other two TFP variables are insignificantly different from zero. It is clear that the conditional dynamics in the two systems are qualitatively similar and statistically indistinguishable. Nevertheless, median responses are smaller, uncertainty is more pervasive, and the hump in the TFP response muted in the larger system. Hence, cross-sectional aggregation does not change much the dynamics but makes TFP responses artificially large and more precisely estimated. Researchers often construct models to quantitatively match the dynamics induced by shocks in small-scale VARs. Figure 1 suggests that the size and the persistence of the structural shocks needed to produce the aggregate evidence are probably smaller than previously agreed upon. Figure 1. View largeDownload slide Responses to technology news shocks. The dotted regions report pointwise 68% credible intervals; the solid line is the pointwise median response. The x-axis reports quarters and the y-axis the response of the level of the variable in deviation from the predictable path. Figure 1. View largeDownload slide Responses to technology news shocks. The dotted regions report pointwise 68% credible intervals; the solid line is the pointwise median response. The x-axis reports quarters and the y-axis the response of the level of the variable in deviation from the predictable path. 6. Conclusions Small-scale SVAR models are often used in empirical business cycle analyses even though the economic model one thinks has generated the data has a larger number of variables and shocks. In this situation, SVAR shocks are linear transformations of current and past primitive structural shocks perturbing the economy. SVAR shocks might be fundamental or nonfundamental, depending on the details of the economy, the information set available to the econometrician, and the variables chosen in the empirical analysis. However, variables providing noisy information about the primitive structural shocks will Granger cause SVAR shocks, even when the SVAR is fundamental. A similar problem arises when SVAR variables proxy for latent variables. We conduct a simulation study illustrating that spurious nonfundamentalness may indeed occur when the SVAR used for the empirical analysis is of smaller scale than the DGP of the data. We propose an alternative testing procedure that has the same power properties as existing diagnostics when nonfundamentalness is present but does not face aggregation or nonobservability problems when the system is fundamental. We also show that the procedure is robust to specification issues and to nuisance features. We demonstrate that a Granger causality diagnostic finds that a bivariate SVAR measuring the impact of news is nonfundamental, whereas our test finds it fundamental. The presence of an aggregated TFP measure in the SVAR explains the discrepancy. When sectoral TFP measures are used, a Granger causality diagnostic also finds the SVAR fundamental. A few lessons can be learned from our paper. First, Granger causality tests may give misleading conclusions when testing for fundamentalness whenever aggregation or nonobservability problems are present. Second, to derive reliable conclusions, one should have fundamentalness tests that are insensitive to specification and nuisance features. The test proposed in this paper satisfies both criteria; those present in the literature do not. Finally, if one is willing to assume that the DGP is a particular structural model, the procedure described (Sims and Zha 2006) can be used to check if a particular VAR shock can be recovered from current and past values of the observables, therefore bypassing the need to check for fundamentalness. However, when the DGP is unknown, the structural model one employs misspecified, or the exact mapping from the DGP and the estimated SVAR hard to construct, procedures like ours can help researchers to understand whether small-scale SVARs are good starting points to undertake informative business cycle analyses. Appendix This appendix reports the size of the CH test when nuisance parameters are varied. We change the number of lags of first-stage residuals in the auxiliary regression p2; the variance of the error in the DGP for the additional variables $$\sigma ^2_{\xi }$$; the number of principal components used in the auxiliary regressions s; and the number of leads of the first-stage residuals in the auxiliary regression q. Power tables are omitted, because they are identical to those in the text. Table A.1. Size of the CH test, aggregation, varying p2.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  p2=4  10%  11.2  13.5  14.5  13.3  14.8  20.1  29.0  44.1  5%  2.5  2.3  2.5  2.2  2.9  4.6  6.1  11.9  1%  1.6  1.9  1.2  1.6  2.2  4.1  6.2  12.3  p2=2  10%  10.5  13.2  12.1  12.5  14.1  19.3  27.0  40.8  5%  5.8  7.1  5.4  6.0  7.6  12.2  15.9  29.7  1%  1.8  2.0  0.9  1.1  2.1  3.2  5.7  12.5    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  p2=4  10%  11.2  13.5  14.5  13.3  14.8  20.1  29.0  44.1  5%  2.5  2.3  2.5  2.2  2.9  4.6  6.1  11.9  1%  1.6  1.9  1.2  1.6  2.2  4.1  6.2  12.3  p2=2  10%  10.5  13.2  12.1  12.5  14.1  19.3  27.0  40.8  5%  5.8  7.1  5.4  6.0  7.6  12.2  15.9  29.7  1%  1.8  2.0  0.9  1.1  2.1  3.2  5.7  12.5  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; p2 represents the number of lags in the testing equation (42). View Large Table A.1. Size of the CH test, aggregation, varying p2.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  p2=4  10%  11.2  13.5  14.5  13.3  14.8  20.1  29.0  44.1  5%  2.5  2.3  2.5  2.2  2.9  4.6  6.1  11.9  1%  1.6  1.9  1.2  1.6  2.2  4.1  6.2  12.3  p2=2  10%  10.5  13.2  12.1  12.5  14.1  19.3  27.0  40.8  5%  5.8  7.1  5.4  6.0  7.6  12.2  15.9  29.7  1%  1.8  2.0  0.9  1.1  2.1  3.2  5.7  12.5    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  p2=4  10%  11.2  13.5  14.5  13.3  14.8  20.1  29.0  44.1  5%  2.5  2.3  2.5  2.2  2.9  4.6  6.1  11.9  1%  1.6  1.9  1.2  1.6  2.2  4.1  6.2  12.3  p2=2  10%  10.5  13.2  12.1  12.5  14.1  19.3  27.0  40.8  5%  5.8  7.1  5.4  6.0  7.6  12.2  15.9  29.7  1%  1.8  2.0  0.9  1.1  2.1  3.2  5.7  12.5  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; p2 represents the number of lags in the testing equation (42). View Large Table A.2. Size of the CH-test, aggregation, varying $$ \sigma ^2_ \xi$$.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  2.20  1.80  1.70  2.10  1.60  2.10  1.80  3.00  $$\sigma ^2_{\xi }=4$$  5%  1.10  0.70  0.40  0.60  0.50  1.00  0.60  0.90    1%  0.30  0.10  0.10  0.00  0.00  0.20  0.20  0.10    10%  1.00  0.70  0.20  0.80  0.50  1.50  0.60  1.10  $$\sigma ^2_{\xi }=0.25$$  5%  0.50  0.40  0.10  0.20  0.40  0.50  0.30  0.30    1%  0.00  0.20  0.00  0.10  0.00  0.00  0.10  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  2.20  1.80  1.70  2.10  1.60  2.10  1.80  3.00  $$\sigma ^2_{\xi }=4$$  5%  1.10  0.70  0.40  0.60  0.50  1.00  0.60  0.90    1%  0.30  0.10  0.10  0.00  0.00  0.20  0.20  0.10    10%  1.00  0.70  0.20  0.80  0.50  1.50  0.60  1.10  $$\sigma ^2_{\xi }=0.25$$  5%  0.50  0.40  0.10  0.20  0.40  0.50  0.30  0.30    1%  0.00  0.20  0.00  0.10  0.00  0.00  0.10  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; $$\sigma ^2_{\xi }$$ is the variance of the idiosyncratic error in the DGP for additional data. View Large Table A.2. Size of the CH-test, aggregation, varying $$ \sigma ^2_ \xi$$.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  2.20  1.80  1.70  2.10  1.60  2.10  1.80  3.00  $$\sigma ^2_{\xi }=4$$  5%  1.10  0.70  0.40  0.60  0.50  1.00  0.60  0.90    1%  0.30  0.10  0.10  0.00  0.00  0.20  0.20  0.10    10%  1.00  0.70  0.20  0.80  0.50  1.50  0.60  1.10  $$\sigma ^2_{\xi }=0.25$$  5%  0.50  0.40  0.10  0.20  0.40  0.50  0.30  0.30    1%  0.00  0.20  0.00  0.10  0.00  0.00  0.10  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  2.20  1.80  1.70  2.10  1.60  2.10  1.80  3.00  $$\sigma ^2_{\xi }=4$$  5%  1.10  0.70  0.40  0.60  0.50  1.00  0.60  0.90    1%  0.30  0.10  0.10  0.00  0.00  0.20  0.20  0.10    10%  1.00  0.70  0.20  0.80  0.50  1.50  0.60  1.10  $$\sigma ^2_{\xi }=0.25$$  5%  0.50  0.40  0.10  0.20  0.40  0.50  0.30  0.30    1%  0.00  0.20  0.00  0.10  0.00  0.00  0.10  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; $$\sigma ^2_{\xi }$$ is the variance of the idiosyncratic error in the DGP for additional data. View Large Table A.3. Size of the CH test, aggregation, varying s.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.10  1.10  0.30  0.60  0.80  1.00  1.00  1.70  s = 2  5%  0.50  0.50  0.00  0.30  0.40  0.50  0.10  0.60    1%  0.10  0.10  0.00  0.10  0.00  0.10  0.00  0.10    10%  1.70  1.80  0.70  1.80  1.40  1.90  1.40  2.50  s = 4  5%  0.80  0.70  0.10  0.60  0.50  0.60  0.50  1.10    1%  0.20  0.10  0.00  0.10  0.00  0.20  0.10  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.10  1.10  0.30  0.60  0.80  1.00  1.00  1.70  s = 2  5%  0.50  0.50  0.00  0.30  0.40  0.50  0.10  0.60    1%  0.10  0.10  0.00  0.10  0.00  0.10  0.00  0.10    10%  1.70  1.80  0.70  1.80  1.40  1.90  1.40  2.50  s = 4  5%  0.80  0.70  0.10  0.60  0.50  0.60  0.50  1.10    1%  0.20  0.10  0.00  0.10  0.00  0.20  0.10  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; s is the length of the vector of factors in the testing equation (42). View Large Table A.3. Size of the CH test, aggregation, varying s.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.10  1.10  0.30  0.60  0.80  1.00  1.00  1.70  s = 2  5%  0.50  0.50  0.00  0.30  0.40  0.50  0.10  0.60    1%  0.10  0.10  0.00  0.10  0.00  0.10  0.00  0.10    10%  1.70  1.80  0.70  1.80  1.40  1.90  1.40  2.50  s = 4  5%  0.80  0.70  0.10  0.60  0.50  0.60  0.50  1.10    1%  0.20  0.10  0.00  0.10  0.00  0.20  0.10  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.10  1.10  0.30  0.60  0.80  1.00  1.00  1.70  s = 2  5%  0.50  0.50  0.00  0.30  0.40  0.50  0.10  0.60    1%  0.10  0.10  0.00  0.10  0.00  0.10  0.00  0.10    10%  1.70  1.80  0.70  1.80  1.40  1.90  1.40  2.50  s = 4  5%  0.80  0.70  0.10  0.60  0.50  0.60  0.50  1.10    1%  0.20  0.10  0.00  0.10  0.00  0.20  0.10  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; s is the length of the vector of factors in the testing equation (42). View Large Table A.4. Size of the CH test, aggregation, varying q.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.80  3.10  2.40  1.90  2.00  2.60  1.60  3.70  q = 1  5%  0.70  1.40  0.80  0.30  0.70  1.50  0.70  2.10    1%  0.00  0.10  0.00  0.00  0.40  0.10  0.30  0.50    10%  1.20  0.80  0.50  0.70  0.90  1.20  0.60  1.80  q = 2  5%  0.40  0.20  0.20  0.30  0.30  0.50  0.30  0.80    1%  0.00  0.00  0.00  0.00  0.00  0.10  0.00  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.80  3.10  2.40  1.90  2.00  2.60  1.60  3.70  q = 1  5%  0.70  1.40  0.80  0.30  0.70  1.50  0.70  2.10    1%  0.00  0.10  0.00  0.00  0.40  0.10  0.30  0.50    10%  1.20  0.80  0.50  0.70  0.90  1.20  0.60  1.80  q = 2  5%  0.40  0.20  0.20  0.30  0.30  0.50  0.30  0.80    1%  0.00  0.00  0.00  0.00  0.00  0.10  0.00  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; q represents the number of leads in the testing equation (42). View Large Table A.4. Size of the CH test, aggregation, varying q.   c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.80  3.10  2.40  1.90  2.00  2.60  1.60  3.70  q = 1  5%  0.70  1.40  0.80  0.30  0.70  1.50  0.70  2.10    1%  0.00  0.10  0.00  0.00  0.40  0.10  0.30  0.50    10%  1.20  0.80  0.50  0.70  0.90  1.20  0.60  1.80  q = 2  5%  0.40  0.20  0.20  0.30  0.30  0.50  0.30  0.80    1%  0.00  0.00  0.00  0.00  0.00  0.10  0.00  0.10    c  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8    10%  1.80  3.10  2.40  1.90  2.00  2.60  1.60  3.70  q = 1  5%  0.70  1.40  0.80  0.30  0.70  1.50  0.70  2.10    1%  0.00  0.10  0.00  0.00  0.40  0.10  0.30  0.50    10%  1.20  0.80  0.50  0.70  0.90  1.20  0.60  1.80  q = 2  5%  0.40  0.20  0.20  0.30  0.30  0.50  0.30  0.80    1%  0.00  0.00  0.00  0.00  0.00  0.10  0.00  0.10  Notes: The table reports the percentage of rejections of the null hypothesis in 1,000 replications when there is aggregation, T = 200, and three principal components of the large dataset are considered; q represents the number of leads in the testing equation (42). 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Published: Sep 25, 2017

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