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Abstract We incorporate a latent stochastic volatility factor and macroeconomic expectations in an affine model for the term structure of nominal and real rates. We estimate the model over 1999–2016 on U.S. data for nominal and TIPS yields, the realized and implied volatility of T-bonds, and survey forecasts of GDP growth and inflation. We find relatively stable inflation risk premia averaging at 40 basis points at the long-end, and which are strongly related to the volatility factor and conditional mean of output growth. We also document real risk premia that turn negative in the post-crisis period, and a non-negligible variance risk premium. Analyzing the dynamics of real interest rates, inflation expectations, and inflation risk premia is relevant for a wide array of financial decisions. Central banks, for example, may use the information from inflation-indexed bond prices to infer inflation expectations and determine the conduct of monetary policy. Index-linked bonds are also particularly suitable financial instruments for the investment strategies of institutional investors, such as pension funds and insurance companies. Observing the dynamics of bond prices denominated in both nominal and real terms enables the nominal yield of any maturity to be separated into its individual components, that is, the real interest rate, the expected inflation rate, and the inflation risk premium. The latter arises from the fact that investors holding nominal bonds are exposed to unanticipated changes in future inflation, and therefore command a premium to bear such risk. In principle, the size and sign of the premium depends on the covariance between shocks to consumption and inflation. If this covariance is negative, meaning that consumption tends to be low when inflation is high, then nominal assets bear more risk and investors require a positive premium to hold them. If the covariance is positive, then nominal bonds become a hedging instrument for negative shocks in consumption and the inflation risk premium can be negative. In the United States, the issuance of inflation-linked bonds (denominated TIPS) has started only in 1997, but the market has since then grown very rapidly.1 The empirical evidence on the properties of the U.S. inflation risk premia from the TIPS market is, however, rather mixed. A strand of the literature documents medium- to long-term inflation risk premia that are mainly positive, with the 10-year premium averaging between 40 and 60 basis points (bps) (see e.g., Durham, 2006; Adrian and Wu, 2009; Chen, Liu, and Cheng, 2010; D’Amico, Kim, and Wei, 2010; Haubrich, Pennacchi, and Ritchken, 2012). On the other hand, other studies claim that long-term inflation risk premia are, on average, close to zero (see e.g., Christensen, Lopez, and Rudebusch, 2010; Hordahl and Tristani, 2010; Grishchenko and Huang, 2013). In general, there is an agreement about the fact that short-term inflation risk premia should be either very small or negative. Other empirical works analyze the dynamics of inflation risk premia in the United States over longer time periods, but in doing so do not make use of data on inflation-linked bond prices (see, e.g., Buraschi and Jiltsov, 2005; Ang, Bekaert, and Wei, 2008; Chernov and Mueller, 2012; Ajello, Benzoni, and Chyruk, 2014).2 In this paper, we contribute to both the modeling and estimation of inflation risk premia. On the former, we develop a novel model where the nominal and real term structures, and their volatilities, are explicitly linked to latent stochastic volatility and economic factors. The existing studies on inflation risk premia either focus on the role of volatility (such as Adrian and Wu, 2009; Haubrich, Pennacchi, and Ritchken, 2012) or on measures of real economic activity (such as Hordahl and Tristani, 2010). To the best of our knowledge, no study simultaneously takes into account the joint influence of these two factors on yields and inflation risk premia. We fill this void by casting aggregate stochastic volatility and macroeconomic conditions as captured by the conditional mean of output growth in a parsimonious no-arbitrage affine term structure model. The model delivers endogenous closed-form solutions for the term structures of nominal and real rates, their volatilities, output growth, and inflation expectations under the physical probability measure, and inflation and real risk premia. We relate nominal and real rates to four economic drivers. The instantaneous real rate and inflation expectations naturally derive from the Fisher theorem as key determinants of the real and nominal term structure. Benninga and Protopapadakis (1983) show that under uncertainty the term structure of real and nominal interest rates is related to the short-term real rate, inflation rate, and risk premia due to the variability of money prices and the purchasing power riskiness of nominal bonds. Our third factor is represented by expected output (real GDP) growth. A number of studies document significant role for macroeconomic variables in explaining the term structure of interest rates (see Ang and Piazzesi, 2003; Diebold, Rudebusch, and Aruoba, 2006). This role extends over and beyond the standard level, slope, and curvature components of the cross-section of (nominal) yields. Among others, Ludvigson and Ng (2009), Bikbov and Chernov (2010), and Joslin, Priebsch, and Singleton (2014) show that macroeconomic variables help explaining bond risk premia, the level of yields, or both.3 Finally, we incorporate a fourth stochastic volatility factor that is responsible for time-varying conditional second moment in the other state variables. There is a voluminous literature investigating the relation between yields and volatility [see among others Heston (1993); Collin-Dufresne, Goldstein, and Jones (2009); Jacobs and Karoui (2009), and more recently Cieslak and Povala (2016) and Feldhütter, Heyerdahl-Larsen, and Illeditsch (2016)]. In the context of affine models, our approach is similar in spirit to Almeida, Graveline, and Joslin (2011), who use interest rate caps data and show that the inclusion of stochastic volatility improves the fit of bond risk premia. Also closely related is Creal and Wu (2017), who simultaneously model the first and second moments of macro variables and yields and show that stochastic volatility may impact their conditional means. On a similar vein, Bansal and Shaliastovich (2012) develop a long-run risk model where the volatilities of inflation and real growth are treated separately and affect bond risk premia. None of these studies, however, looks at aggregate stochastic volatility with the purpose of understanding inflation risk premia. In addition, we account for the possibility that TIPS yields (which we use in our empirical analysis) are occasionally inaccurate proxies of real rates due to lack in the liquidity of this market [see Campbell, Shiller, and Viceira (2009), D’Amico, Kim, and Wei (2010), and Grishchenko and Huang (2013)]. We accomplish this by introducing a liquidity factor that enters the pricing of TIPS, but not that of nominal yields. Our empirical contribution is to fit the model using observable proxies of yields volatility and market expectations about economic activity. To be precise, we estimate the model over the 1999–2016 period by maximum likelihood using a Kalman filter algorithm. Our observation equations consist of monthly observations of nominal Treasury yields, TIPS yields, and surveys of professional forecasts (SPF) for GDP growth and inflation. Moreover, we also require the model to fit the term structure of realized volatilities and the implied volatility of the 10-year Treasury-Note future options. Thus, we attempt to match simultaneously the cross-section and time-series properties of both the level of yields (which include risk compensation) and their second moment, and link them to observable macro expectations. The T-note series plays a key role in enhancing our identification of both stochastic volatility and the risk premium parameters. In these respects, our approach is similar to Cieslak and Povala (2016) who document that informative second-moment data (realized and implied) improve the precision of the filtering.4 In terms of goodness of fit, we find that the model passes nearly all standard moment specification tests and delivers reasonable levels for the maximal attainable Sharpe Ratio (see Duffee, 2010). In particular, we cannot reject the null hypothesis of zero differences between the model-implied and realized first and second moment of nominal yields, TIPS, yield volatilities, and GDP forecasts. As a further validation exercise, we document a similar performance when applying the 1999–2016 model estimates to the 1985–1998 period, for which data on TIPS are not available. Turning to the model-implied estimates, during the whole sample period the term structure of inflation risk premia is generally positively sloped, with average premia raising from 12 bps at the two-year to about 44 bps at the 10-year maturity. The premia are highest in the pre-crisis period, get closer to zero at the peak of the crisis, and finally revert back to their pre-crisis levels. In terms of level, our 10-year series is in between those reported by Grishchenko and Huang (2013) and Haubrich, Pennacchi, and Ritchken (2012). However, unlike Haubrich, Pennacchi, and Ritchken (2012) our short-term inflation risk premia do not plummet into negative territory at the peak of the crisis. In contrast with other studies that feature macro factors (Hordahl and Tristani, 2010) or rely on yield-only approaches (Christensen, Lopez, and Rudebusch, 2010), we estimate inflation risk premia that are less volatile and noisy. This result resonates with the evidence in Almeida, Graveline, and Joslin (2011) that incorporating stochastic volatility in an affine model helps improving the precision with which risk premia are estimated. In terms of economic significance of the factors, we find that stochastic volatility and (especially) expected output growth are key drivers of inflation risk premia. However, while the effect of the volatility factor increases with maturities, the impact of expected output growth is positive, hump-shaped, and economically large both at the short and long end. Together, these two factors explain nearly 35% of the variance of 10-year inflation risk premia. The model also reveals quite rich dynamics for real risk premia. The term structure is positively sloped in the pre-crisis periods, but becomes U-shaped starting with the third quarter of 2008, when we observe a marked downward trend across all maturities. From the mid-2010, real risk premia turn negative and remain such until the end of the sample. This pattern is consistent with the negative nominal term premium reported by Durham (2013) and Adrian, Crump, and Moench (2013).5 The inclusion of the macro expectations is a key to capture the marked decline in real risk premia in the last part of the sample, as this appears to be missed by models featuring only stochastic volatility, such as Haubrich, Pennacchi, and Ritchken (2012). Notably, inflation and real risk premia are only weakly correlated, and load differently on standard predictors of bond risk premia. We also find a significant link between inflation expectations (under the physical measure) and stochastic volatility. Thus, our paper contributes to the literature that studies the determinants of expected inflation, and its relation with risk premia (see e.g., Ang, Bekaert, and Wei, 2008; Chernov and Mueller, 2012; Ajello, Benzoni, and Chyruk, 2014; Cieslak and Povala, 2015; Fleckenstein, Longstaff, and Lustig, 2017). In the final part of the paper, we present a series of model extensions in several directions. In particular, we squarely look at the effect of the financial crisis by re-estimating the model on the 2007–2010 period. During this period, the relation between risk premia and volatility turns negative at short horizons, as expected during flight-to-safety episodes that alter the risk-return tradeoff to stocks and bonds (see e.g., Campbell, Shiller, and Viceira, 2009). We further use inflation swaps data in place of TIPS as an alternative proxy for real rates in the post-2004 period. For inflation risk premia, the estimated term structure using either data is almost overlapping, whereas for real risk premia we find a similar shape of the term structure, but the level differs by some 20 bps at the long end. The estimated dynamics of the risk premia and the conclusions we draw regarding the role of the volatility and macroeconomic factors also remain valid. 1 The Model In this section, we outline our term structure model. Section 1.1 describes the data-generating process for the driving latent variables and risk premia. Next, Section 1.2 presents the implied no-arbitrage term structure for nominal and real rates, and risk premia. 1.1 State Variables and Macro Factors We assume that the economy is driven by the following four economic forces: the real interest rate, r; the expected inflation rate, π; the conditional mean of output growth, μ, which reflects investors’ expectations regarding the evolution of the real economy; and a variance factor, v, which drives the conditional volatility of all other variables. In addition to being jointly related in the diffusion component through v, the factors potentially affect each other in the conditional mean (drift). As argued above, various studies link either of these factors to the dynamics of real and nominal yields. We consider all of them simultaneously to explain the cross-section and time-series of bond prices (yields) and their volatilities. Furthermore, we introduce a fifth state variable ℓ that accounts for specific features of the market for TIPSs, which we use later in our estimation strategy. As documented by D’Amico, Kim, and Wei (2010), during the first few years after its creation in 1997 the TIPS market suffered from a lack of liquidity with respect to the market of nominal Treasury bonds. A similar view is advanced by Campbell, Shiller, and Viceira (2009) and Christensen and Gillan (2011), who examine the abnormal behavior of the TIPS market at the peak of the financial crisis. This evidence suggests that occasional disruptions and liquidity dry-ups may break the link between TIPS and real rates. Ignoring these effects in our modeling framework may mar our inference on the other factors when the model is asked to fit TIPS data. Therefore, we let ℓ enter the pricing of TIPS, but not that of nominal and real yields. In order to separately identify this component, we allow its conditional volatility and (potentially) mean to be related to the volatility factor only. We collect the five latent factors in the vector X=(v μ π r ℓ)′ . In the notation of Dai and Singleton (2000), the dynamic of X(t) under the physical probability measure evolves according to the following A1(5) specification: dX(t)=K(Θ−X(t))dt+ΞS(t)dW(t). (1) In the (5 × 5) matrix K of mean-reversion coefficients, we impose that all off-diagonal elements involving the factor ℓ except k5,1 are zero6—that is, as mentioned above, we allow only the volatility factor to affect the drift of ℓ . The vector Θ of long-run means is (5 ×1); Ξ is diagonal (5 × 5); and S(t) is diagonal (5 × 5) with the element in position (i, i) given by [S(t)]ii=βi′X(t) , with βi denoting the i-th column of the (5 × 5) matrix β which has ones in the first row and zeros elsewhere. The Brownian shocks to the four economic factors are allowed to be correlated, but they are orthogonal to shocks to the liquidity factor. We denote Ω the resulting (5 × 5) covariance matrix of dW(t). If we define Σ= Chol(ΞΩΞ) , where Chol is the Cholesky decomposition, we can rewrite the model as: dX(t)=K(Θ−X(t))dt+ΣS(t)dz(t), (2) where now dz(t) denotes a vector of independent Brownian motions. Our specification thus implies that v follows a non-negative square-root process which drives the conditional volatility of the other, conditionally Gaussian state variables. Therefore, stochastic volatility enters the expectation of future interest rates either through the drift term or by introducing conditional heteroskedasticity. The possibility that stochastic volatility feeds back to the conditional mean of the macro series and yields is consistent with the model and the empirical evidence presented in Creal and Wu (2017). We explicitly model the dynamics of the price level and output growth. In common with most term structure models including inflation (such as, e.g., Pennacchi, 1991; Ang, Bekaert, and Wei, 2008), we assume that the exogenously given process for the price level is supported by the underlying equilibrium in the money market. For the real economy, we assume that there exists a single technology producing a single physical good and that production output follows a stochastic process with a time-varying conditional mean. The expected inflation rate π and the conditional mean of output growth μ are then defined as the stochastic drift components of the price level and production output processes, which follow correlated Ito processes whose variance is affine in the variance factor v: dpp=π(t)dt+σ0,pdz0,p+σ1,pv(t)dz1,p, (3) dqq=μ(t)dt+ξ(σ0,qdz0,p+σ1,qv(t)dz1,p)+1−ξ2(σ0,qdz0,q+σ1,qvdz1,q), (4) with z0,p,z1,p,z0,q , and z1,q as uncorrelated Brownian motions. This choice implies that the stochastic volatility factor also captures time-variation in macroeconomic risk. We collect the corresponding parameters in the vector Φ=(σ0,p,σ1,p,σ0,q,σ1,q,ξ) . To close the model, we need to specify the functional form of the instantaneous market price for risk. We adopt the general “essentially affine” specification of Duffee (2002) (see also Duarte, 2004): Ψ(t)=S−(t) (Λ0+Λ1 X(t)), (5) where S−(t) denotes the inverse of S(t), Λ0 is a (5×1) vector of constant risk premia, and Λ1 is (5×5) with the same off-diagonal zero constraints as in K. This formulation is particularly appealing as it allows risk premia to vary over time, and potentially to change sign. Extant studies find risk premia to nominal bond that exhibit significant time variation and may even turn negative (see e.g., Cochrane and Piazzesi, 2005; Ludvigson and Ng, 2009). The affine specification in Equations (2)–(5) leads to a stochastic process of X(t) under the risk-adjusted probability measure ℚ whose drift and diffusion terms are also affine. In fact, Girsanov’s theorem implies that the dynamics of dX(t) under ℚ follows: dX(t)=(K˜Θ˜−K˜X(t))dt+ΣS(t)dz˜(t), (6) where K˜=K+ΣΛ1, K˜Θ˜=KΘ−ΣΛ0 , and dz˜(t) are the risk-neutral Brownian motions.7 1.2 Term Structure and Risk Premia Following Benninga and Protopapadakis (1983), we obtain the model’s implication for the pricing of nominal and real bonds by imposing that the instantaneous nominal interest rate y equals the sum of the instantaneous expected inflation rate π, real interest rate r, and inflation risk premium irp. The last term is related to the correlation between output and price level and to the volatility of the price level. Combining (2) with Equations (3) and (4) delivers an instantaneous inflation risk premium that is linear in the variance factor v: irp=−1dt(Covt{dpp,dqq}+Vart{dpp})=−σ0,p(ξσ0,q+σ0,p)−σ1,p(ξσ1,q+σ1,p)v≡ɛ0+ɛv. (7) This result allows us to write the instantaneous nominal interest rate y as an affine function of the state vector: y(t)=δ0+δ′X(t), (8) where δ0=ɛ0 and δ′=(ɛ 0 1 1 0) . The affine risk-neutral dynamics in Equation (6) together with Equation (8) imply that the model falls in the class of affine term structure models (Piazzesi, 2010). The equilibrium arbitrage-free price of a nominal unit discount bond with time to maturity τ at time t has an exponentially affine closed-form solution: F(τ;t)=exp[A¯Y(τ)−B¯Y′(τ)X(t)]. (9) The coefficients A¯Y(τ) and B¯Y(τ) depend on the underlying model parameters and solve the following system of ordinary differential equations (ODEs): dA¯Y(τ)dτ=−(K˜ Θ˜)′B¯Y(τ)−δ0. dB¯Y(τ)dτ=−K˜′B¯Y(τ)+12∑i=15[Σ′B¯Y(τ)]i2βi+δ. The nominal term structure is therefore affine in the state vector: Y(τ;t)=AY(τ)+BY′(τ)X(t), (10) where AY(τ)≡−A¯Y(τ)/τ and BY(τ)≡B¯Y(τ)/τ . From Expression (8), we obtain the equilibrium real rates R(τ;t) by solving the system of ODEs (1.2) subject to the constraints δ0=R0 and δ′=R(0 0 0 1 0) : R(τ;t)=AR(τ)+BR′(τ)X(t). (11) The breakeven rate is defined as H(τ,t)=Y(τ,t)−R(τ;t) . Note that by the definition of δ′ , the liquidity factor does not enter the pricing of either nominal, real, or breakeven rates. We create a wedge between TIPS and real rates by allowing the factor ℓ to affect the pricing of TIPS. That is, equilibrium TIPS rates T(τ;t) obtain by solving the system of ODEs (1.2) subject to the constraints δ0=T0 and δ′=T(0 0 0 1 1) : T(τ;t)=AT(τ)+BT′(τ)X(t). (12) This expression clarifies that for a given τ, higher values of ℓ are associated with periods when TIPS are more imperfect proxies for real rates. Turning to second moments, the diffusion term in the risk-adjusted dynamics of X(t) is affine in v, which implies that nominal yield volatilities are time-varying and are driven by a single factor. More formally, the term structure of the variance of nominal yield changes under the risk-adjusted measure is given by: V(τ;t)=BY′(τ)(Σ S(t)Σ′)BY(τ). (13) Equally, the model delivers a closed-form solution for the term structure of the volatility of real interest rates that is also affine in v. From the closed-form expressions above, we obtain the term structure of inflation and real risk premia. Similarly to Haubrich, Pennacchi, and Ritchken (2012), we define the inflation risk premium IRP(τ;t) as the difference between the breakeven rate under the risk-adjusted and physical probability measure, or IRP(τ;t)=H(τ;t)−Hℙ(τ;t). (14) where Hℙ(τ;t)=−1τln(Eℙ[e−∫tt+τ(ys−rs)ds|It]) . This difference depends crucially on the covariance between inflation and output growth rates, which captures the bond’s ability to act as a hedge against a decrease in consumption. Similarly, let Rℙ(τ;t) be the implied yield of a real zero coupon bond when the expectation is taken under the physical measure, or Rℙ(τ;t)=−1τln(Eℙ[e−∫tt+τrsds|It]) . We then define the real risk premium RRP(τ;t) for a given maturity τ as the difference between the real rate R(τ;t) and Rℙ(τ;t) , or RRP(τ;t)=R(τ;t)−Rℙ(τ;t). (15) Since the process for the state vector is affine under both measures, risk premia are linear functions of Xt. Moreover, it is important to note that unlike the instantaneous inflation risk premia in Equation (7), state variables other than v can affect IRP(τ;t) through their ability to predict future realizations of v under the ℙ or ℚ measure (i.e., to the extent they enter K, K˜ , or both). 2 Data and Preliminary Statistics Our empirical analysis combines data on yields, yield volatilities, and macroeconomic forecasts. First, we obtain data on U.S. Treasury and TIPS from Gurkaynak, Sack, and Wright (2007, 2008). We use end-of-month observations on annualized zero coupon yields with maturities ranging from 2 to 10 years over the sample period from January 1999 to December 2016. For TIPS yields, the shortest observed maturity was five years before January 2004 and two years afterward. Figure 1 displays the time series of the 2-, 5-, and 10-year nominal Treasury yields (Panel A) and TIPS yields (Panel B). The term structure of the nominal yields is moderately upward sloping (sometimes downward sloping) over the 1999–2000 and 2005–2007 periods. However, it becomes very steep during the 2001–2004 and 2008–2014 periods, with the spread between the 10-year and the one-year rate rising above 250 bps. Figure 1. View largeDownload slide Time series of input data. This figure displays the time series of nominal yields (Panel A), TIPS yields (Panel B), and realized volatility of nominal yield (Panel C) for the 2-year, 5-year (dashed line), and 10-year (thick line) maturity. Panel D displays the implied volatility of the option on the 10-year T-note future. Panels E and F display, respectively, the median professional forecasts for one-year ahead inflation (GDP deflator) and real GDP growth. Figure 1. View largeDownload slide Time series of input data. This figure displays the time series of nominal yields (Panel A), TIPS yields (Panel B), and realized volatility of nominal yield (Panel C) for the 2-year, 5-year (dashed line), and 10-year (thick line) maturity. Panel D displays the implied volatility of the option on the 10-year T-note future. Panels E and F display, respectively, the median professional forecasts for one-year ahead inflation (GDP deflator) and real GDP growth. In general, a declining trend in long-term nominal rates can be observed, which seems to be mainly originating from the behavior of real rates. In fact, apart from the months following Lehman’s default in September 2008, when there is a sudden and temporary increase, long-term TIPS yields decline from about 4% in 1999 to a range between −1% and +1% in 2016. As a consequence of the 2008 financial crisis and the subsequent expansionary monetary policy, short-term nominal rates remain close to zero during the post-2009 sample period. Second, we construct estimates of realized yield volatility. A distinct feature of the model is the assumption of a latent factor driving the term structure of nominal, real, and inflation volatilities. To identify the factor, we rely on a standard realized volatility estimator computed as the standard deviation of daily changes in nominal Treasury yields within a given month. Panel C of Figure 1 displays these realized volatilities for the 2-, 5-, and 10-year bonds. Yield volatilities vary in the 50–150 bps range for most of the sample, but experience a peak at about 300 bps during 2008. Also noteworthy is the fact that the term structure of volatilities is downward sloping until 2008, and steeply upward sloping in the post-crisis period. We augment realized volatilities with end-of-month quotations of the implied volatility of the 10-year Treasury-Note future options. This series avails ourselves of risk-adjusted market expectations of future nominal yield volatility, and therefore considerably enhances our identification of both v and the risk premium parameters. The series is displayed in Panel D of the figure. Third and finally, we capture the dynamics of expected inflation and real growth through the Philadelphia SPF data. To be precise, we use the median one-year-ahead forecasts of annual GDP deflator and annual real GDP growth rates. These data are available on a quarterly basis. Panels E and F of the figure display the corresponding time series. Expected inflation rates increase by almost 1% in the 1999–2000 period. They then drop to 1.5% around the time of the 2001 recession, before starting to rise up to 2.5% in 2007–2008. A sharp decline follows the Lehman default, with inflation expectations near 1%, before returning to a value close to 2% in the final part of the sample. The behavior of expected real growth is similar, but with a much more pronounced drop in 2001 (from 3% in the early 2000 to about 1%) and especially at the end of 2008 (from about 3.5% in mid-2006 to a minimum of about −1%). In Panel A of Table 1 we collect summary statistics of the data. Notably, on average, the term structures of nominal and TIPS yields are upward sloping, whereas the term structure of the volatility of nominal yields flattens at the long end. However, as stated above, there is great variability in the level and slope of the curves across the sample period. Inflation and GDP growth expectations average out at 1.85% and 2.62%, respectively, with GDP growth rates that are nearly three times more volatile. Table 1. Summary statistics and factor analysis Panel A: Summary statistics Series Average Standard deviation Min. Max. Nominal yields 2-year 230 194 19 665 5-year 303 161 63 663 10-year 390 136 150 670 TIPS yields 2-year (from January 2004) 7 144 −212 502 5-year 120 158 −169 428 10-year 172 132 −79 429 Realized volatility nominal yields 2-year 75 43 10 311 5-year 92 36 32 259 10-year 93 32 36 215 Macro forecasts SPF inflation 185 31 101 249 SPF real GDP growth 262 85 −109 410 Panel A: Summary statistics Series Average Standard deviation Min. Max. Nominal yields 2-year 230 194 19 665 5-year 303 161 63 663 10-year 390 136 150 670 TIPS yields 2-year (from January 2004) 7 144 −212 502 5-year 120 158 −169 428 10-year 172 132 −79 429 Realized volatility nominal yields 2-year 75 43 10 311 5-year 92 36 32 259 10-year 93 32 36 215 Macro forecasts SPF inflation 185 31 101 249 SPF real GDP growth 262 85 −109 410 Panel B: Principal component analysis Series PC1 PC2 PC3 PC4 Nominal yields (9) 97.19 2.72 0.08 0.00 TIPS yields (6) 99.65 0.34 0.01 0.00 TIPS yields (9) since January 2004 97.06 2.81 0.13 0.01 Nominal and TIPS (15) 95.22 3.45 1.26 0.05 Nominal and TIPS (18) since January 2004 90.19 6.50 2.97 0.28 Realized volatility nominal yields (9) 90.23 9.24 0.49 0.04 Nominal and TIPS and realized volatility and macro forecasts (26) 56.86 33.18 4.37 2.91 Panel B: Principal component analysis Series PC1 PC2 PC3 PC4 Nominal yields (9) 97.19 2.72 0.08 0.00 TIPS yields (6) 99.65 0.34 0.01 0.00 TIPS yields (9) since January 2004 97.06 2.81 0.13 0.01 Nominal and TIPS (15) 95.22 3.45 1.26 0.05 Nominal and TIPS (18) since January 2004 90.19 6.50 2.97 0.28 Realized volatility nominal yields (9) 90.23 9.24 0.49 0.04 Nominal and TIPS and realized volatility and macro forecasts (26) 56.86 33.18 4.37 2.91 Notes: Panel A reports summary statistics (in basis points) for the time series of nominal yields, TIPS yields, realized volatility of nominal yield, and macroeconomic forecasts that are used in the empirical analysis. The sample period is from January 1999 to December 2016 for all series but TIPS, whose maturities shorter than five-year start in January 2004. Panel B reports the percentage of the total variation in the correlation matrix of the set of variables described in the first column which is explained by the first four principal components (PC1–PC4). For TIPS we include only maturities from the five-year onward in the second and fourth row, and all maturities starting January 2004 in the third and fifth row. In parentheses, we report the total number of series. View Large Table 1. Summary statistics and factor analysis Panel A: Summary statistics Series Average Standard deviation Min. Max. Nominal yields 2-year 230 194 19 665 5-year 303 161 63 663 10-year 390 136 150 670 TIPS yields 2-year (from January 2004) 7 144 −212 502 5-year 120 158 −169 428 10-year 172 132 −79 429 Realized volatility nominal yields 2-year 75 43 10 311 5-year 92 36 32 259 10-year 93 32 36 215 Macro forecasts SPF inflation 185 31 101 249 SPF real GDP growth 262 85 −109 410 Panel A: Summary statistics Series Average Standard deviation Min. Max. Nominal yields 2-year 230 194 19 665 5-year 303 161 63 663 10-year 390 136 150 670 TIPS yields 2-year (from January 2004) 7 144 −212 502 5-year 120 158 −169 428 10-year 172 132 −79 429 Realized volatility nominal yields 2-year 75 43 10 311 5-year 92 36 32 259 10-year 93 32 36 215 Macro forecasts SPF inflation 185 31 101 249 SPF real GDP growth 262 85 −109 410 Panel B: Principal component analysis Series PC1 PC2 PC3 PC4 Nominal yields (9) 97.19 2.72 0.08 0.00 TIPS yields (6) 99.65 0.34 0.01 0.00 TIPS yields (9) since January 2004 97.06 2.81 0.13 0.01 Nominal and TIPS (15) 95.22 3.45 1.26 0.05 Nominal and TIPS (18) since January 2004 90.19 6.50 2.97 0.28 Realized volatility nominal yields (9) 90.23 9.24 0.49 0.04 Nominal and TIPS and realized volatility and macro forecasts (26) 56.86 33.18 4.37 2.91 Panel B: Principal component analysis Series PC1 PC2 PC3 PC4 Nominal yields (9) 97.19 2.72 0.08 0.00 TIPS yields (6) 99.65 0.34 0.01 0.00 TIPS yields (9) since January 2004 97.06 2.81 0.13 0.01 Nominal and TIPS (15) 95.22 3.45 1.26 0.05 Nominal and TIPS (18) since January 2004 90.19 6.50 2.97 0.28 Realized volatility nominal yields (9) 90.23 9.24 0.49 0.04 Nominal and TIPS and realized volatility and macro forecasts (26) 56.86 33.18 4.37 2.91 Notes: Panel A reports summary statistics (in basis points) for the time series of nominal yields, TIPS yields, realized volatility of nominal yield, and macroeconomic forecasts that are used in the empirical analysis. The sample period is from January 1999 to December 2016 for all series but TIPS, whose maturities shorter than five-year start in January 2004. Panel B reports the percentage of the total variation in the correlation matrix of the set of variables described in the first column which is explained by the first four principal components (PC1–PC4). For TIPS we include only maturities from the five-year onward in the second and fourth row, and all maturities starting January 2004 in the third and fifth row. In parentheses, we report the total number of series. View Large A relevant feature of the model is the presence of strong common components that drive time-series fluctuations in the level of nominal and real yields, their conditional volatility, and inflation and output growth expectations. To formally explore this assumption, in Panel B of the table we look at the correlation structure of the data. In particular, we report the percentage of the total variation in the correlation matrix explained by the first four principal components. We first look at the level of nominal and TIPS yields. For the whole term structure of nominal yields, three factors explain about 99% of the total variation, with the first factor being responsible for more than 97%. The long end of the term structure of TIPS yields (maturities from five years onward) is almost entirely spanned by a single factor, whose role however decreases to 97% when including also the short end (maturities from two years onward since January 2004). When combining nominal and TIPS yields, three factors (post January 2004) capture nearly 100% of the total variation across the 18 series. There is also a strong factor structure in the term structure of realized volatilities of nominal yields, as more than 90% of their variation is explained by the first principal component. The second factor also accounts for a significant 10%. Finally, the last row of the panel combines the nominal and TIPS (maturities from five years onward) yields, nominal yield volatilities, and the macroeconomic forecasts—a total of 26 series. The first two principal components account for nearly 90% of total variation, and the third factor explains an additional 4%. This result suggests that a low-dimensional state vector is responsible for the large bulk of fluctuations across such a relatively wide array of series. 3 Empirical Results In this section, we present the main empirical results of the paper. We first discuss the econometric approach in Section 3.1. We present the resulting estimates of the model parameters and state variables in Section 3.2, followed by statistics on the goodness of fit in Section 3.3. In Sections 3.4–3.6, we set out the properties of the estimated term structure of inflation expectations and risk premia. 3.1 Estimation Method The model is estimated via quasi maximum likelihood using the Kalman filter. This methodology has become a standard approach for the estimation of term structure models that feature unobservable state variables [see Duffee and Stanton (2012) for a review]. We briefly describe here the overall setup of the estimation and the data that enter the filter. A formal description of the set of state and observation equations together with details on the implementation of the filter is presented in the Appendix. Our system of observation equations includes the following series: (i) nominal yields; (ii) TIPS yields; (iii) the realized variance of nominal yield changes; (iv) the implied (risk-neutral) variance of the 10-year Treasury-Note future option; and (v) macro expectations, as captured by the SPF forecasts.8 In the end, the system consists of 27 equations prior to January 2004, and 30 equations afterward as the two-, three-, and four-year TIPS yields become available. The observation equations are obtained by adding to the variables’ model-implied expression an observation error, which is assumed to be normally distributed and homoskedastic. Also, the filter accommodates the fact that the SPF forecasts are available only at the quarterly frequency. The five state equations are composed of the discrete time (monthly) equivalent of the continuous-time model in Equation (2). In the estimation, we impose the cross-equation restrictions that originate from the (affine) expression for the conditional covariance matrix of the shocks to the state vector (see Equation (A.5) in the Appendix). This is a constraint that helps us to better identify the volatility coefficients of the model. The inclusion of the implied variance of nominal yields among the observable variables fitted by the model also improves the identification of the variance coefficients of the stochastic processes of the real interest rate and the expected inflation rate, as well as the unobservable variance factor. This is crucial for capturing the time-varying behavior of risk premia. 3.2 Estimated Parameters and State Variables Panel A of Table 2 reports the estimated parameter values, with underneath bootstrapped p-values in parentheses. Similarly to Haubrich, Pennacchi, and Ritchken (2012), we find that the expected inflation rate and the real interest rate exhibit significant mean reversion, although in our case the half-life for a shock in the variables to return to its steady state is well above one year. We also find a relevant mean reversion in the volatility factor, and in the conditional mean of the output growth. The TIPS liquidity factor is the least persistent among the variables. The variance factor negatively affects the drift of all the other state variables, while expected output growth has a positive effect on the conditional mean of v and a negative one on that of π and r. Finally, the expected inflation and real rates positively affect each other in the drift and exhibit a similar sensitivity of their volatilities to the variance factor. The correlation between their instantaneous shocks is instead negative around −0.5. The correlation between shocks to v and shocks to all other factors is positive, albeit not largely so. The estimated Λ1 are all significant and negative in the main diagonal, which implies that risk-adjustment decreases the speed of mean reversion. We return to a specification test for risk premia in the next section. Table 2. Maximum-likelihood estimates Panel A: Maximum-likelihood estimates Mean reversion K Θ 0.3422 0.0097 −0.0410 −0.0075 – 0.0042 (0.0030) (0.2994) (0.1042) (0.7105) (0.0332) −0.0610 0.3861 −0.0471 −0.2418 – 0.0496 (0.0057) (0.0064) (0.0140) (0.0026) (0.0018) −3.2412 −0.0149 0.4507 0.0367 – 0.0362 (0.0080) (0.2430) (0.0056) (0.0146) (0.0036) −5.3006 −0.0305 0.0623 0.3507 – 0.0726 (0.0036) (0.0511) (0.0093) (0.0137) (0.0002) −0.5014 – – – 0.1277 0.0014 (0.0111) (0.0021) (0.7937) Panel A: Maximum-likelihood estimates Mean reversion K Θ 0.3422 0.0097 −0.0410 −0.0075 – 0.0042 (0.0030) (0.2994) (0.1042) (0.7105) (0.0332) −0.0610 0.3861 −0.0471 −0.2418 – 0.0496 (0.0057) (0.0064) (0.0140) (0.0026) (0.0018) −3.2412 −0.0149 0.4507 0.0367 – 0.0362 (0.0080) (0.2430) (0.0056) (0.0146) (0.0036) −5.3006 −0.0305 0.0623 0.3507 – 0.0726 (0.0036) (0.0511) (0.0093) (0.0137) (0.0002) −0.5014 – – – 0.1277 0.0014 (0.0111) (0.0021) (0.7937) Volatilities and correlations diag(Ξ) Ω Φ 0.0543 1 0.0037 0.0968 0.0033 – σ0,p 0.0060 (0.1559) (0.0485) (0.0022) (0.2932) (0.0189) 0.3652 0.0037 1 −0.0334 −0.1105 – σ1,p 0.4181 (0.0082) (0.0485) (0.0085) (0.0013) (0.0087) 0.5257 0.0968 −0.0334 1 −0.5119 – σ0,q 0.0005 (0.0038) (0.0022) (0.0085) (0.0019) (0.1109) 0.5182 0.0033 −0.1105 −0.5119 1 – σ1,q 1.3991 (0.0061) (0.2932) (0.0013) (0.0019) (0.0013) 0.1512 – – – – 1 ξ 0.6170 (0.0012) (0.0151) Volatilities and correlations diag(Ξ) Ω Φ 0.0543 1 0.0037 0.0968 0.0033 – σ0,p 0.0060 (0.1559) (0.0485) (0.0022) (0.2932) (0.0189) 0.3652 0.0037 1 −0.0334 −0.1105 – σ1,p 0.4181 (0.0082) (0.0485) (0.0085) (0.0013) (0.0087) 0.5257 0.0968 −0.0334 1 −0.5119 – σ0,q 0.0005 (0.0038) (0.0022) (0.0085) (0.0019) (0.1109) 0.5182 0.0033 −0.1105 −0.5119 1 – σ1,q 1.3991 (0.0061) (0.2932) (0.0013) (0.0019) (0.0013) 0.1512 – – – – 1 ξ 0.6170 (0.0012) (0.0151) Risk premia Λ0 Λ1 0.0002 −0.5132 −0.1079 −0.0246 0.0370 – (0.7978) (0.5770) (0.0084) (0.0136) (0.0097) −0.0008 0.3535 −0.7939 0.7111 −0.6731 – (0.7881) (0.0116) (0.0067) (0.0051) (0.0187) −0.0005 −0.1184 −0.0294 −0.0529 −0.0214 – (0.7948) (0.0089) (0.0240) (0.0087) (0.0828) −0.0003 −0.2112 −0.0972 0.4084 −0.4682 – (0.7975) (0.0043) (0.0061) (0.0138) (0.0054) 0.0055 −0.0755 – – – −0.7129 (0.6027) (0.0106) (0.0092) Risk premia Λ0 Λ1 0.0002 −0.5132 −0.1079 −0.0246 0.0370 – (0.7978) (0.5770) (0.0084) (0.0136) (0.0097) −0.0008 0.3535 −0.7939 0.7111 −0.6731 – (0.7881) (0.0116) (0.0067) (0.0051) (0.0187) −0.0005 −0.1184 −0.0294 −0.0529 −0.0214 – (0.7948) (0.0089) (0.0240) (0.0087) (0.0828) −0.0003 −0.2112 −0.0972 0.4084 −0.4682 – (0.7975) (0.0043) (0.0061) (0.0138) (0.0054) 0.0055 −0.0755 – – – −0.7129 (0.6027) (0.0106) (0.0092) View Large Table 2. Maximum-likelihood estimates Panel A: Maximum-likelihood estimates Mean reversion K Θ 0.3422 0.0097 −0.0410 −0.0075 – 0.0042 (0.0030) (0.2994) (0.1042) (0.7105) (0.0332) −0.0610 0.3861 −0.0471 −0.2418 – 0.0496 (0.0057) (0.0064) (0.0140) (0.0026) (0.0018) −3.2412 −0.0149 0.4507 0.0367 – 0.0362 (0.0080) (0.2430) (0.0056) (0.0146) (0.0036) −5.3006 −0.0305 0.0623 0.3507 – 0.0726 (0.0036) (0.0511) (0.0093) (0.0137) (0.0002) −0.5014 – – – 0.1277 0.0014 (0.0111) (0.0021) (0.7937) Panel A: Maximum-likelihood estimates Mean reversion K Θ 0.3422 0.0097 −0.0410 −0.0075 – 0.0042 (0.0030) (0.2994) (0.1042) (0.7105) (0.0332) −0.0610 0.3861 −0.0471 −0.2418 – 0.0496 (0.0057) (0.0064) (0.0140) (0.0026) (0.0018) −3.2412 −0.0149 0.4507 0.0367 – 0.0362 (0.0080) (0.2430) (0.0056) (0.0146) (0.0036) −5.3006 −0.0305 0.0623 0.3507 – 0.0726 (0.0036) (0.0511) (0.0093) (0.0137) (0.0002) −0.5014 – – – 0.1277 0.0014 (0.0111) (0.0021) (0.7937) Volatilities and correlations diag(Ξ) Ω Φ 0.0543 1 0.0037 0.0968 0.0033 – σ0,p 0.0060 (0.1559) (0.0485) (0.0022) (0.2932) (0.0189) 0.3652 0.0037 1 −0.0334 −0.1105 – σ1,p 0.4181 (0.0082) (0.0485) (0.0085) (0.0013) (0.0087) 0.5257 0.0968 −0.0334 1 −0.5119 – σ0,q 0.0005 (0.0038) (0.0022) (0.0085) (0.0019) (0.1109) 0.5182 0.0033 −0.1105 −0.5119 1 – σ1,q 1.3991 (0.0061) (0.2932) (0.0013) (0.0019) (0.0013) 0.1512 – – – – 1 ξ 0.6170 (0.0012) (0.0151) Volatilities and correlations diag(Ξ) Ω Φ 0.0543 1 0.0037 0.0968 0.0033 – σ0,p 0.0060 (0.1559) (0.0485) (0.0022) (0.2932) (0.0189) 0.3652 0.0037 1 −0.0334 −0.1105 – σ1,p 0.4181 (0.0082) (0.0485) (0.0085) (0.0013) (0.0087) 0.5257 0.0968 −0.0334 1 −0.5119 – σ0,q 0.0005 (0.0038) (0.0022) (0.0085) (0.0019) (0.1109) 0.5182 0.0033 −0.1105 −0.5119 1 – σ1,q 1.3991 (0.0061) (0.2932) (0.0013) (0.0019) (0.0013) 0.1512 – – – – 1 ξ 0.6170 (0.0012) (0.0151) Risk premia Λ0 Λ1 0.0002 −0.5132 −0.1079 −0.0246 0.0370 – (0.7978) (0.5770) (0.0084) (0.0136) (0.0097) −0.0008 0.3535 −0.7939 0.7111 −0.6731 – (0.7881) (0.0116) (0.0067) (0.0051) (0.0187) −0.0005 −0.1184 −0.0294 −0.0529 −0.0214 – (0.7948) (0.0089) (0.0240) (0.0087) (0.0828) −0.0003 −0.2112 −0.0972 0.4084 −0.4682 – (0.7975) (0.0043) (0.0061) (0.0138) (0.0054) 0.0055 −0.0755 – – – −0.7129 (0.6027) (0.0106) (0.0092) Risk premia Λ0 Λ1 0.0002 −0.5132 −0.1079 −0.0246 0.0370 – (0.7978) (0.5770) (0.0084) (0.0136) (0.0097) −0.0008 0.3535 −0.7939 0.7111 −0.6731 – (0.7881) (0.0116) (0.0067) (0.0051) (0.0187) −0.0005 −0.1184 −0.0294 −0.0529 −0.0214 – (0.7948) (0.0089) (0.0240) (0.0087) (0.0828) −0.0003 −0.2112 −0.0972 0.4084 −0.4682 – (0.7975) (0.0043) (0.0061) (0.0138) (0.0054) 0.0055 −0.0755 – – – −0.7129 (0.6027) (0.0106) (0.0092) View Large Table 2. Continued Panel B: Summary statistics of state variables Average Standard deviation Min. Max. v 0.0004 0.0003 0.0000 0.0020 μ 0.0303 0.0095 −0.0133 0.0486 π 0.0182 0.0057 −0.0015 0.0296 r 0.0040 0.0201 −0.0217 0.0498 ℓ 0.0030 0.0059 −0.0074 0.0302 Panel B: Summary statistics of state variables Average Standard deviation Min. Max. v 0.0004 0.0003 0.0000 0.0020 μ 0.0303 0.0095 −0.0133 0.0486 π 0.0182 0.0057 −0.0015 0.0296 r 0.0040 0.0201 −0.0217 0.0498 ℓ 0.0030 0.0059 −0.0074 0.0302 Notes: Panel A reports the maximum-likelihood estimates of the term structure model outlined in Section 1. The coefficients are ordered as [v;μ;π;r;ℓ] . Underneath the estimates, bootstrapped p-values are reported in parentheses. Panel B reports summary statistics for the filtered state variables. View Large Table 2. Continued Panel B: Summary statistics of state variables Average Standard deviation Min. Max. v 0.0004 0.0003 0.0000 0.0020 μ 0.0303 0.0095 −0.0133 0.0486 π 0.0182 0.0057 −0.0015 0.0296 r 0.0040 0.0201 −0.0217 0.0498 ℓ 0.0030 0.0059 −0.0074 0.0302 Panel B: Summary statistics of state variables Average Standard deviation Min. Max. v 0.0004 0.0003 0.0000 0.0020 μ 0.0303 0.0095 −0.0133 0.0486 π 0.0182 0.0057 −0.0015 0.0296 r 0.0040 0.0201 −0.0217 0.0498 ℓ 0.0030 0.0059 −0.0074 0.0302 Notes: Panel A reports the maximum-likelihood estimates of the term structure model outlined in Section 1. The coefficients are ordered as [v;μ;π;r;ℓ] . Underneath the estimates, bootstrapped p-values are reported in parentheses. Panel B reports summary statistics for the filtered state variables. View Large Turning our attention to the estimated latent state variables, Figure 2 plots their time series, while Panel B of Table 2 reports summary statistics. We note that the variance factor (top-left plot) tracks quite closely the dynamics of yield volatilities, with distinct spikes during the early 2000s and the 2008–2009 crisis periods. The average monthly volatility is about 1.9%, but varies significantly throughout the period with a standard deviation of 0.60% and a maximum of about 4.5%. Expected output growth (top-right plot) and the expected inflation rate (mid-left plot) also follow patterns similar to their corresponding SPF forecasts, averaging about 3% and 2%, respectively. The series are positively correlated at 0.55, although this number drops to 0.41 if we exclude the last quarter of 2008 and the first quarter of 2009 when both of them (and in particular, expected growth) turn negative. In contrast, the correlation between v and μ is negative at −0.44, which implies that volatility generally tends to increase during business cycle downturns. Figure 2. View largeDownload slide Estimates of latent state variables. This figure plots the Kalman Filter time-series estimates of the five latent state variables in the model: the variance factor ( v ), expected output growth (μ), expected inflation rate (π), the real rate ( r ), and the (il)liquidity factor of TIPS ( ℓ ). Figure 2. View largeDownload slide Estimates of latent state variables. This figure plots the Kalman Filter time-series estimates of the five latent state variables in the model: the variance factor ( v ), expected output growth (μ), expected inflation rate (π), the real rate ( r ), and the (il)liquidity factor of TIPS ( ℓ ). The real interest rate (mid-right plot) averages at a meager 0.40% during the period. This number, however, is the combination of decreasing real rates in the 0–5% range until 2004, when the Federal Reserve aggressive monetary policy brought nominal short-term interest rates down from 6% to 1%, compressing real rates between 0% and 3% during the 2004–2008 period. The instantaneous real rate remains negative from 2009 onward in correspondence with the near zero-rate FED monetary policy. We also note a positive trend in the later part of the sample when the FED started to release its quantitative easing policies. The TIPS instantaneous (il)liquidity factor ℓ is displayed in the bottom-left plot of Figure 2. The variable averages about 1.5% in the early part of the sample, turns to zero starting in 2005, spikes at 3% around Lehman’s default, and then finally reverts back toward a 0.60% average. This pattern is entirely consistent with the arguments in D’Amico, Kim, and Wei (2010); Campbell, Shiller, and Viceira (2009); and Christensen and Gillan (2011) about liquidity issues at market inception and surrounding Lehman’s collapse. Such correspondence is noteworthy given that ℓ is treated as latent variable in the filter and is not explicitly linked to an empirical proxy. In Appendix Table A.1, we collect the model-implied loadings on the state vector [i.e., the B(τ) coefficients in the affine functions] for yields and risk premia. To ease their economic interpretation, in Table 3 we report the fraction of the variance of fitted yields and risk premia (at the τ= 2-, 5-, and 10-year maturity) that is accounted for by each factor.9 In Panel A, we see that the real interest rate (75%) and expected inflation (30%) are mostly responsible for variations in the two-year nominal yield. However, the role of stochastic volatility increases with maturity and, for τ equal to 10 years, it becomes comparable to that of the real rate at about 30%, while most of the variability of long-term nominal yields is explained by inflation expectations. The importance of v at long maturities is confirmed for real yields in Panel B, where it accounts for about one-third of the overall variance and is the second driving factor beyond r. For TIPS (Panel C), we see that the liquidity factor ℓ explains about one-third of their overall variance. The term structure of the B(τ) coefficients on ℓ is downward sloping, which implies that the liquidity component of TIPS yields is indeed only 12 bps on average across maturities. Table 3. Model-implied variance decomposition Panel A: Nominal yields, Y(τ) Panel B: Real yields, R(τ) τ v μ π r v μ π r 2 −4.30 0.29 29.12 74.89 −1.13 0.00 0.04 101.09 5 16.97 0.08 27.41 55.54 14.05 −0.03 0.16 85.82 10 27.14 −0.54 39.69 33.71 33.24 −0.47 9.01 58.22 Panel A: Nominal yields, Y(τ) Panel B: Real yields, R(τ) τ v μ π r v μ π r 2 −4.30 0.29 29.12 74.89 −1.13 0.00 0.04 101.09 5 16.97 0.08 27.41 55.54 14.05 −0.03 0.16 85.82 10 27.14 −0.54 39.69 33.71 33.24 −0.47 9.01 58.22 Panel C: TIPS yields, T(τ) Panel D: Expected inflation, Π(τ) τ v μ π r ℓ v μ π r 2 −3.04 −0.07 0.57 69.60 32.93 −2.88 −0.13 103.12 −0.11 5 6.39 0.17 −1.53 58.48 36.50 5.07 −1.31 96.12 0.12 10 18.47 0.60 0.99 40.57 39.36 12.28 −2.42 89.78 0.36 Panel C: TIPS yields, T(τ) Panel D: Expected inflation, Π(τ) τ v μ π r ℓ v μ π r 2 −3.04 −0.07 0.57 69.60 32.93 −2.88 −0.13 103.12 −0.11 5 6.39 0.17 −1.53 58.48 36.50 5.07 −1.31 96.12 0.12 10 18.47 0.60 0.99 40.57 39.36 12.28 −2.42 89.78 0.36 Panel E: Inflation risk premium, IRP (τ) Panel F: Real risk premium, RRP (τ) τ v μ π r v μ π r 2 3.14 20.83 75.27 0.76 0.47 −0.55 30.52 69.56 5 9.22 27.54 63.23 0.01 3.47 −0.40 19.40 77.53 10 13.47 21.76 64.62 0.15 16.17 1.83 1.27 80.73 Panel E: Inflation risk premium, IRP (τ) Panel F: Real risk premium, RRP (τ) τ v μ π r v μ π r 2 3.14 20.83 75.27 0.76 0.47 −0.55 30.52 69.56 5 9.22 27.54 63.23 0.01 3.47 −0.40 19.40 77.53 10 13.47 21.76 64.62 0.15 16.17 1.83 1.27 80.73 Notes: This table reports the model-implied contribution (in percentage) of each state variable to the overall variance of nominal yields, real yields, inflation expectations, inflation risk premia, and real risk premia at the 2-, 5-, and 10-year maturity. For all series but TIPS, the contribution of the liquidity factor ℓ is zero. View Large Table 3. Model-implied variance decomposition Panel A: Nominal yields, Y(τ) Panel B: Real yields, R(τ) τ v μ π r v μ π r 2 −4.30 0.29 29.12 74.89 −1.13 0.00 0.04 101.09 5 16.97 0.08 27.41 55.54 14.05 −0.03 0.16 85.82 10 27.14 −0.54 39.69 33.71 33.24 −0.47 9.01 58.22 Panel A: Nominal yields, Y(τ) Panel B: Real yields, R(τ) τ v μ π r v μ π r 2 −4.30 0.29 29.12 74.89 −1.13 0.00 0.04 101.09 5 16.97 0.08 27.41 55.54 14.05 −0.03 0.16 85.82 10 27.14 −0.54 39.69 33.71 33.24 −0.47 9.01 58.22 Panel C: TIPS yields, T(τ) Panel D: Expected inflation, Π(τ) τ v μ π r ℓ v μ π r 2 −3.04 −0.07 0.57 69.60 32.93 −2.88 −0.13 103.12 −0.11 5 6.39 0.17 −1.53 58.48 36.50 5.07 −1.31 96.12 0.12 10 18.47 0.60 0.99 40.57 39.36 12.28 −2.42 89.78 0.36 Panel C: TIPS yields, T(τ) Panel D: Expected inflation, Π(τ) τ v μ π r ℓ v μ π r 2 −3.04 −0.07 0.57 69.60 32.93 −2.88 −0.13 103.12 −0.11 5 6.39 0.17 −1.53 58.48 36.50 5.07 −1.31 96.12 0.12 10 18.47 0.60 0.99 40.57 39.36 12.28 −2.42 89.78 0.36 Panel E: Inflation risk premium, IRP (τ) Panel F: Real risk premium, RRP (τ) τ v μ π r v μ π r 2 3.14 20.83 75.27 0.76 0.47 −0.55 30.52 69.56 5 9.22 27.54 63.23 0.01 3.47 −0.40 19.40 77.53 10 13.47 21.76 64.62 0.15 16.17 1.83 1.27 80.73 Panel E: Inflation risk premium, IRP (τ) Panel F: Real risk premium, RRP (τ) τ v μ π r v μ π r 2 3.14 20.83 75.27 0.76 0.47 −0.55 30.52 69.56 5 9.22 27.54 63.23 0.01 3.47 −0.40 19.40 77.53 10 13.47 21.76 64.62 0.15 16.17 1.83 1.27 80.73 Notes: This table reports the model-implied contribution (in percentage) of each state variable to the overall variance of nominal yields, real yields, inflation expectations, inflation risk premia, and real risk premia at the 2-, 5-, and 10-year maturity. For all series but TIPS, the contribution of the liquidity factor ℓ is zero. View Large We also compute the correlation of monthly changes in the economic factors with the first three principal components obtained from monthly changes in nominal yields. We observe that the first principal component (the “level” factor) is mostly related to expected inflation and real rates, with correlations of 0.53 and 0.66, respectively. The correlation with the second principal component (the “slope” factor) is positive for the real rate and expected output growth, a result in line with the stream of the literature linking output expectations to the slope of the term structure (see, e.g., Harvey, 1988). Finally, the variance factor has a negative correlation at −0.14 with the third principal component (the “curvature” factor). Interestingly, μ is positively correlated (around 0.2) with the fifth principal component, a result that is consistent with Adrian, Crump, and Moench (2013) and Joslin, Priebsch, and Singleton (2014). 3.3 Specification Test and Goodness of Fit To assess the model performance, we first look at the standard deviation of pricing errors, which are defined as the difference between actual and model-implied series. The first column of Table 4 reports this statistic for several combinations of the data. Across all maturities, the average standard deviation is 20 bps for nominal yields, 12 bps for TIPS yields, and 13.5 bps for the volatility (realized and implied) of nominal yields.10 The standard deviation of pricing errors is quite larger at 46 bps for SPF inflation, while SPF real GDP growth is quite precisely estimated with a standard deviation of only 4 bps. When combining all series together in the last row of the table, the overall standard deviation amounts to less than 16 bps. Table 4. Goodness of fit and specification tests Series Average standard deviation (ε^) Spec. test, p-value First moment Second moment Nominal yields 20.14 0.99 0.99 TIPS yields 11.94 0.99 0.78 Nominal and TIPS 16.04 0.99 0.99 Volatility nominal yields 13.54 0.65 0.13 Nominal and TIPS and volatility 15.15 0.99 0.76 SPF inflation 46.20 0.57 0.00 SPF real GDP growth 4.00 0.67 0.93 Nominal and TIPS and volatility and SPF 15.81 0.99 0.04 Series Average standard deviation (ε^) Spec. test, p-value First moment Second moment Nominal yields 20.14 0.99 0.99 TIPS yields 11.94 0.99 0.78 Nominal and TIPS 16.04 0.99 0.99 Volatility nominal yields 13.54 0.65 0.13 Nominal and TIPS and volatility 15.15 0.99 0.76 SPF inflation 46.20 0.57 0.00 SPF real GDP growth 4.00 0.67 0.93 Nominal and TIPS and volatility and SPF 15.81 0.99 0.04 Notes: This table reports the average standard deviation of pricing errors (the differences between actual and model-implied series, ɛ^) and the p-values of specification tests based on the model estimates for different series. The p-values are from the point statistic M=(m−m¯)′Q(m−m¯)∼χ2(n) , where: m and m¯ indicate, respectively, the sample and model-implied unconditional moments; Q is the covariance matrix of the sample estimates of the unconditional moments, estimated through GMM with the Newey and West (1987) correction; and n is the number of over-identifying restrictions. The test is on the first and second moment. View Large Table 4. Goodness of fit and specification tests Series Average standard deviation (ε^) Spec. test, p-value First moment Second moment Nominal yields 20.14 0.99 0.99 TIPS yields 11.94 0.99 0.78 Nominal and TIPS 16.04 0.99 0.99 Volatility nominal yields 13.54 0.65 0.13 Nominal and TIPS and volatility 15.15 0.99 0.76 SPF inflation 46.20 0.57 0.00 SPF real GDP growth 4.00 0.67 0.93 Nominal and TIPS and volatility and SPF 15.81 0.99 0.04 Series Average standard deviation (ε^) Spec. test, p-value First moment Second moment Nominal yields 20.14 0.99 0.99 TIPS yields 11.94 0.99 0.78 Nominal and TIPS 16.04 0.99 0.99 Volatility nominal yields 13.54 0.65 0.13 Nominal and TIPS and volatility 15.15 0.99 0.76 SPF inflation 46.20 0.57 0.00 SPF real GDP growth 4.00 0.67 0.93 Nominal and TIPS and volatility and SPF 15.81 0.99 0.04 Notes: This table reports the average standard deviation of pricing errors (the differences between actual and model-implied series, ɛ^) and the p-values of specification tests based on the model estimates for different series. The p-values are from the point statistic M=(m−m¯)′Q(m−m¯)∼χ2(n) , where: m and m¯ indicate, respectively, the sample and model-implied unconditional moments; Q is the covariance matrix of the sample estimates of the unconditional moments, estimated through GMM with the Newey and West (1987) correction; and n is the number of over-identifying restrictions. The test is on the first and second moment. View Large To provide a visual inspection of the fit, Appendix Figure A.1 plots the actual and model-implied series for the five-year maturity nominal yield, TIPS yield and realized yield volatility, the 10-year implied yield variance, the one-year SPF inflation, and real GDP expectations. Absolute errors on the nominal and TIPS yields are generally below 20 bps. The fit of SPF series is quite different. For inflation, the model-implied series matches the drop in the actual forecasts at the peak of the crisis, but misses part of the variability afterward. The model-implied output growth tracks instead very closely the corresponding SPF series. Since the model imposes several moment restrictions, we test its adequacy through a standard specification test (see, e.g., Duffee, 2002; Ang, Bekaert, and Wei, 2008). In particular, we apply a GMM-type test to assess the closeness of the estimated unconditional moments to the sample moments. The test is based on the point statistic M=(m−m¯)′Ξ−1(m−m¯) , where m¯ are the sample estimates of the unconditional moments, m are the model-implied unconditional moments, and Ξ is the covariance matrix of the sample estimates of the unconditional moments, which is estimated using GMM with the Newey and West (1987) correction for heteroskedasticity and autocorrelation. Under the null hypothesis, the statistic is distributed as χ2(n) , where n is the number of over-identifying restrictions. The last two columns in Table 4 report the p-value of the test for the first and second moment, respectively, for the same combinations of series. Overall, the performance of the model is remarkably good for both the level and the volatilities of yields. The model also captures quite well the sample moments of output growth and the average SPF inflation. Only in the case of the second moment of SPF inflation is the model rejected. When considering all series together, the p-value is 0.99 for the first moment and 0.04 for the second moment.11 Since the key parameters for the estimation of inflation and real risk premia are the market price of risk parameters Λ0 and Λ1 , following Adrian, Crump, and Moench (2013), we apply a Wald test for the null hypothesis that the different rows of the vector and the matrix are equal to zero. In particular, we alternatively test the null hypothesis that (i) the i-th row of Λ0 and the i-th row of Λ1 are jointly equal to zero, which would imply that the corresponding factor risk is not priced in the model; and (ii) the i-th row of Λ1 is equal to zero, that is, a test on the time variation of the market price of risk associated with the corresponding factor. We find that both null hypotheses are strongly rejected for all state variables, with p-values below 1%. As a further metric of interest, we look at the model-implied maximal Sharpe ratio, defined as the Sharpe ratio that can be attained by a portfolio of bonds that span the payoff of the stochastic discount factor. Duffee (2010) documents that flexible affine term structure models featuring four or five Gaussian factors generate implausibly high Sharpe ratios, in the order of 1030. In Figure 3, we plot the time series of the maximal Sharpe ratio of simple monthly bond returns corresponding to the estimates of Table 2 and the filtered state variables of Figure 2.12 We note that the series does not take on extreme values, with a maximum of 1.10 and a full-sample average of 0.37. These numbers mimic quite closely those reported by Adrian, Crump, and Moench (2013) for their five-factor model. Figure 3. View largeDownload slide Maximal Sharpe ratio. This figure displays the time series of the estimated maximal conditional model-implied Sharpe ratio for monthly simple bond returns, defined as in Section 3.3. Figure 3. View largeDownload slide Maximal Sharpe ratio. This figure displays the time series of the estimated maximal conditional model-implied Sharpe ratio for monthly simple bond returns, defined as in Section 3.3. We conclude that despite a low-dimensional state vector, the model is capable of generating reasonable pricing errors. It is able to match well sample moment of nominal yields, real yields, nominal volatilities, and macro forecasts, without generating implausible risk compensations. We return to a robustness test of the model performance in Section 4.3. 3.4 Inflation Expectations The left plot of Panel A of Figure 4 displays the time series of the model-implied inflation expectations under the physical probability measure. The term structure of inflation expectations is generally downward sloping, with the average spread between the 10-year and the one-year maturity expected inflation rate being around −12 bps. Short-term expectations are relatively volatile, whereas long-term expectations are quite stable and fluctuate in a range between 115 and 215 bps. A sudden change occurs at the peak of the financial crisis, when the shape of the term structure of inflation expectations experiences a tilt. Following Lehman’s default, short-term inflation expectations collapse and get close to zero, reflecting the market’s fear of the possibility of a prolonged recession. The average term structure assumes a steep shape, with the spread between the 10-year and the one-year maturity expected inflation rate rising to 100 bps. Figure 4. View largeDownload slide Inflation expectations, inflation risk premium, and real risk premium. This figure plots the model-implied ℙ-measure expected inflation (Panel A), inflation risk premium (Panel B), and real risk premium (Panel C). Within each panel, the left plot displays the estimated 2-, 5-, and 10-year series, whereas the right plot displays the impulse-response function (IRF) at the 10-year maturity following a one-standard-deviation positive monthly shock to each state variable. Figure 4. View largeDownload slide Inflation expectations, inflation risk premium, and real risk premium. This figure plots the model-implied ℙ-measure expected inflation (Panel A), inflation risk premium (Panel B), and real risk premium (Panel C). Within each panel, the left plot displays the estimated 2-, 5-, and 10-year series, whereas the right plot displays the impulse-response function (IRF) at the 10-year maturity following a one-standard-deviation positive monthly shock to each state variable. To gauge the economic significance of the latent factors, we look at the impulse-response function (IRF) calculated as the reaction of the 10-year series to a positive one-standard-deviation shock to the state vector. The right plot in the panel reports the IRF for the 10-year inflation expectation. As expected, we observe a significant positive response to shocks in the expected inflation rate and in the volatility factor, whereas shocks in the real interest rate and in the conditional mean of output growth have a much smaller impact. Analogous conclusions emerge by looking at the variance decomposition for inflation expectations in Panel D of Table 3. The expected inflation rate is by far the predominant factor in explaining the variance of two-year inflation expectations. However, at the 10-year maturity the variance factor accounts for an economically significant 12%. The result that stochastic volatility has an important role in driving the long end of the curve lends further support to our modeling framework. Our estimates reveal a change in the relation between expected inflation and real rates fol