TY - JOUR AU - Wu,, Ximing AB - Summary We develop a specification test of predictive densities, based on the fact that the generalized residuals of correctly specified predictive density models are independent and identically distributed uniform. The proposed sequential test examines the hypotheses of serial independence and uniformity in two stages, wherein the first‐stage test of serial independence is robust to violation of uniformity. The approach of the data‐driven smooth test is employed to construct the test statistics. The asymptotic independence between the two stages facilitates proper control of the overall type I error of the sequential test. We derive the asymptotic null distribution of the test, which is free of nuisance parameters, and we establish its consistency. Monte Carlo simulations demonstrate excellent finite sample performance of the test. We apply this test to evaluate some commonly used models of stock returns. 1. Introduction Density forecast is of fundamental importance for decision‐making under uncertainty, wherein good point estimates might not be adequate. Accurate density forecasts of key macroeconomic and financial variables, such as inflation, unemployment rate, stock returns and exchange rate, facilitate informed decision‐making of policy‐makers and financial managers, particularly when a forecaster's loss function is asymmetric and the underlying process is non‐Gaussian. Given the importance of density forecast, great caution should be exercised in judging the quality of density forecast models. In a seminal paper, Diebold et al. (1998) introduced the method of dynamic probability integral transformation (PIT) to evaluate out‐of‐sample density forecasts. The transformed data are often called the generalized residuals of a forecast model. Given a time series {Yt} ⁠, denote by {Zt} the generalized residuals associated with some density forecast model, which is defined in the next section. Diebold et al. (1998) showed that if a forecast model is correctly specified, {Zt} is independent and identically distributed (i.i.d.) uniformly on [0, 1]. The serial independence signifies the correct dynamic structure while uniformity characterizes the correct specification of the unconditional distribution. Subsequently, many formal tests have been developed based on this approach, extending the original method to accommodate issues such as the influence of nuisance parameters, dynamic misspecification errors, multiple‐step‐ahead forecasts, etc. For general overviews of this body of literature, see Corradi and Swanson (2006c, 2012) and references therein. Suppose that {Zt} is a strictly stationary process with an invariant marginal distribution G0. Let P0 be the joint distribution of Zt−j and Zt ⁠, where j is a positive integer. According to the theorem of Sklar (1959), there exists a copula function C0:[0,1]2→[0,1] such that P0(Zt−j,Zt)=C0(G0(Zt−j),G0(Zt)),(1.1) where C0 completely characterizes the dependence structure between Zt−j and Zt ⁠. Most of the existing work evaluates density forecasts by simultaneously testing the serial independence and uniformity hypotheses based on the left‐hand side of (1.1), comparing the density of P0 with the product of two standard uniform densities. However, if the joint null hypothesis is rejected, the simultaneous test ‘generally provides no guidance as to why’ (Diebold et al., 1998). Is the rejection attributable to a violation of the uniformity of the unconditional distribution of {Zt} ⁠, a violation of the serial independence of {Zt} ⁠, or both? To address this issue, we propose a sequential test for the i.i.d. uniformity of the generalized residuals. This sequential test, based on the copula representation of a joint distribution as in the right‐hand side of (1.1), examines first whether C0 is the independent copula and then whether G0 is the uniform distribution. Rejection of the independence hypothesis effectively terminates the test; otherwise, a subsequent uniformity test for the unconditional distribution is conducted. We establish the large‐sample properties of the proposed tests, which are distribution‐free asymptotically. An appealing feature of our sequential test is that the first‐stage test of independent copula is constructed to be robust to misspecification of marginal distributions. Therefore, it remains valid even if {Zt} is not uniformly distributed (because of misspecified marginal distributions). However, because the first‐stage independence test is consistent with asymptotic power equal to 1, the uniformity test (if necessary) is not affected by violation of independence asymptotically. Generally, the overall type I error of sequential tests can be difficult to control. Nonetheless, we establish that the two stages of the proposed sequential test are asymptotically independent and we suggest a simple method to properly control the overall type I error of our test. Compared with simultaneous tests, the proposed sequential test enjoys certain advantages. The first‐stage robust test for serial independence can be easily constructed as parameter estimation uncertainty does not affect its limiting distribution. Upon the rejection of serial independence, the testing procedure terminates. The testing task is simplified in such cases. More importantly, it facilitates the diagnosis of the source of misspecification. Tay and Wallis (2000) show that serial dependence in {Zt} may signal poorly captured dynamics, whereas non‐uniformity may indicate improper distributional assumptions, or poorly captured dynamics, or both. If the serial independence test is not rejected and the uniformity test is rejected, we may improve deficient density forecast by calibration; see Diebold et al. (1999). However, if the serial independence of the generalized residuals is rejected, we can further apply uniformity tests that are robust to violations of independence to examine the specification of the marginal distributions.1 In addition, our serial independence test is based on copulas, which are invariant to strictly monotonic transformations of random variables. Therefore, the rich information captured by the copula of {Zt} can be used to diagnose the dynamic structure of the forecast model, in case the serial independence is rejected; see, e.g. Chen et al. (2004). Below, we first briefly review the relevant literature on the specification test of predictive densities and Neyman's smooth tests. We present, in Section 3, the sequential test of correct density forecasts and its theoretical properties. In Section 4, we use a series of Monte Carlo simulations to demonstrate the excellent finite sample performance of the proposed tests under a variety of circumstances. We then apply these tests to evaluate a host of commonly used models of stock market returns in Section 5. We conclude in Section 6. Technical assumptions and proofs of theorems are relegated to the Appendix. 2. Background 2.1. Specification test of predictive densities For simplicity, consider the one‐step‐ahead forecast of the conditional density f0t(·|Ωt−1) of Yt ⁠, where Ωt−1 represents the information set available at time t−1 ⁠.2 We split a sample of N observations {Yt}t=1N into an in‐sample subset of size R for model estimation and an out‐of‐sample subset of size n=N−R for forecast performance evaluation. Denote by Ft(·|Ωt−1,θ) and ft(·|Ωt−1,θ) some conditional distribution and density functions of Yt given Ωt−1 ⁠, where θ∈Θ⊂Rq ⁠. The dynamic PIT of the data {Yt}t=R+1N ⁠, with respect to the density forecast ft(·|Ωt−1,θ) ⁠, is defined as Zt(θ)=Ft(Yt|Ωt−1,θ)=∫−∞Ytft(v|Ωt−1,θ)dv,t=R+1,…,N.(2.1) The transformed data, Zt(θ) ⁠, are often called the generalized residuals of a forecast model. Suppose that the density forecast model is correctly specified in the sense that there exists some θ0 such that f0t(y|Ωt−1)=ft(y|Ωt−1,θ0) almost surely (a.s.) and for all t. Under this condition, Diebold et al. (1998) showed that the generalized residuals {Zt(θ0)} should be i.i.d. uniform on [0, 1]. Therefore, the test of a generic conditional density function ft(·|Ωt−1,θ) is equivalent to a test of the joint hypothesis H0:{Zt(θ0)}isasequenceofi.i.d.uniformrandomvariablesforsomeθ0∈Θ⊂Rq.(2.2) The alternative hypothesis is the negation of the null (2.2). Hereafter, we shall write Zt(θ) as Zt for simplicity whenever there is no ambiguity. Because the generalized residuals are constructed using out‐of‐sample predictions, tests based on generalized residuals are out‐of‐sample tests. Diebold et al. (1998) used some intuitive graphical methods to separately examine the serial independence and uniformity of the generalized residuals. Subsequently, many authors have adopted the approach of PIT to develop formal specification tests of predictive densities. Diebold et al. (1999) extend the method to bivariate data. Berkowitz (2001) further transformed the generalized residuals to Φ−1(Zt) ⁠, where Φ−1(·) is the inverse of the standard normal distribution function, and proposed tests of serial independence under the assumption of linear autoregressive dependence. Chen et al. (2004) suggested copula‐based tests of serial independence of the generalized residuals against alternative parametric copulas. These tests do not consider the effect of parameter estimation in their test statistics. By explicitly accounting for the impact of parameter estimation uncertainty, Bai (2003) proposed Kolmogorov‐type tests. Hong et al. (2007) constructed nonparametric tests by comparing kernel estimate of the joint density of (Zt−j,Zt) with the product of two uniform densities. Park and Zhang (2010) proposed data‐driven smooth tests, which simultaneously test the uniformity and independence. Chen (2011) considered a family of moment‐based tests. Recently, Corradi and Swanson (2006a,c,b) proposed Kolmogorov‐type tests that allow for dynamic misspecification. Rossi and Sekhposyan (2015) proposed a new test wherein parameter estimation error is preserved under the null hypothesis. 2.2. Neyman's smooth test Omnibus tests are desirable in goodness‐of‐fit testing, wherein the alternative hypotheses are often vague. Some classic omnibus tests, such as the Kolmogorov–Smirnov test or Cramér–von Mises test, are known to be consistent but only have good powers to detect a few deviations from the null hypothesis under moderate sample sizes; see, e.g. Fan (1996). In this study, we adopt Neyman's smooth test, which enjoys attractive theoretical and finite sample properties and can be tailored to adapt to unknown underlying distributions; see Rayner and Best (1990) for a general review. Here, we briefly review the smooth test. For simplicity, suppose for now that {Zt}t=1n is an i.i.d. sample from a distribution G0 defined on the unit interval. To test the uniformity hypothesis, Neyman (1937) considered an alternative family of smooth distributions given by g(z)=exp∑i=1kbiψi(z)+b0,z∈[0,1].(2.3) Here, b0 is a normalization constant, such that g integrates to unity, and ψi are shifted Legendre polynomials, given by ψi(z)=2i+1i!didzi(z2−z)i,i=1,…,k,(2.4) which are orthonormal with respect to the standard uniform distribution. Consequently, E[ψi(z)]=0 for all i and E[ψi(z)ψj(z)]=0 ⁠, i≠j if z follows the standard uniform distribution. Under the assumption that G0 is a member of (2.3), testing uniformity amounts to testing the hypothesis B≡(b1,…,bk)′=0 ⁠, to which the likelihood ratio test can be readily applied. Alternatively, one can construct a score test, which is asymptotically locally optimal and also computationally easy. Define ψ̂i=n−1∑t=1nψi(Zt) and ψ̂(k)=(ψ̂1,…,ψ̂k)′ ⁠. Neyman's smooth test for uniformity is constructed as Nk=nψ̂(k)′ψ̂(k).(2.5) Under uniformity, Nk converges in distribution to the χ2 distribution with k degrees of freedom as n→∞ ⁠. The performance of the smooth test depends on the choice of k. Ledwina (1994) proposed a data‐driven approach to select a proper k. Various aspects of this adaptive smooth test are studied by Kallenberg and Ledwina (1995, 1997), Inglot et al. (1997) and Claeskens and Hjort (2004). For applications of smooth tests in econometrics, see, e.g. Bera and Ghosh (2002), Bera et al. (2013) and Lin and Wu (2015), and references therein. 3. Sequential Test of Correct Density Forecasts In this study, we propose a sequential procedure for evaluating density forecasts, taking advantage of the copula representation of a joint distribution. A copula is a multivariate probability distribution with standard uniform margins. Copulas provide a natural way to separately examine the marginal behaviour of {Zt} and its serial dependence structure. This separation permits us to test the serial independence and uniformity of {Zt} sequentially. Rejection of serial independence effectively terminates the procedure; otherwise, a subsequent test on uniformity is conducted. Below, we start with the copula‐based test of serial independence, followed by the uniformity test of the univariate marginal distributions. We then explain the rationale of this sequential test, why we place the independence test in the first stage and, lastly, how to obtain desired overall type I error of the test. 3.1. Robust test of serial independence Copula completely characterizes the dependence structure among random variables. Thus, testing for serial independence between Zt−j and Zt can be based on their copula function. In particular, testing their independence is equivalent to testing the hypothesis that their copula density is constant at unity.3 Given a density forecast model ft(·|Ωt−1,θ) ⁠, define the generalized residuals Ẑt≡Zt(θ̂t)=∫−∞Ytft(v|Ωt−1,θ̂t)dv,t=R+1,…,N,(3.1) where θ̂t is the maximum likelihood estimate (MLE) of θ given by θ̂t≡θ̂[t0:t1]=argmaxθ∑t=t0t1lnft(yt|Ωt−1,θ),1≤t0ζlogn(3.7) where Ψ̂c,k(j) is the kth element of Ψ̂c,(k)(j) and ζ=2.4 ⁠. Note that this penalty is ‘adaptive’ in the sense that either the AIC or BIC is adopted in a data‐driven manner, depending on the empirical evidence pertinent to the magnitude of deviation from independence.7 Next we present the asymptotic properties of the proposed test Q̂C(j) based on a set of basis functions ΨC selected according to the procedure described above. The first part of the following theorem provides the asymptotic distribution of the test statistic under the null hypothesis and the second part establishes its consistency. Theorem 3.2. Let KC(j) be selected according to (3.6). Suppose that the assumptions for Theorem 3.1 hold. (a) Suppose that C0(·,·) is the independent copula. Then, limn→∞ Pr(KC(j)=1)=1 and Q̂C(j)→dχ12 as n→∞ ⁠. (b) Let P be an alternative and let G0 be the marginal distribution of Zt under P ⁠. Suppose that EP[ψi1i2(G0(Zt−j),G0(Zt))]≠0 for some i1, i2 in 1,…,M ⁠. Then Q̂C(j)→∞ as n→∞ ⁠. The test Q̂C(j) is designed to detect serial dependence between the residuals j periods apart. In practice, it is desirable to test the independence hypothesis jointly at a number of lags. Therefore, we consider the following portmanteau test ŴQ(p)=∑j=1pQ̂C(j),(3.8) where p is the longest prediction horizon of interest. One limitation of this test is that the selection of p can be arbitrary. In order to address this limitation, Escanciano and Lobato (2009) proposed an adaptive portmanteau Box–Pierce test for serial correlation, which selects the unknown order of autocorrelation in a data‐driven manner. This strategy has been employed to test the correct specification of a vector autoregression model by Escanciano et al. (2013). In a similar spirit, we further consider an adaptive portmanteau test and we select the optimal number of lags p according to the following criterion p∼=min{k:ŴQ(k)−Γ2(k,n,ζ)≥ŴQ(s)−Γ2(s,n,ζ),1≤k,s≤p},(3.9) where the complexity penalty Γ2(k,n,ζ) is the same as (3.7) with max1≤j≤pmax1≤i≤K(j)|nΨ̂C,i(j)| taking the place of max1≤k≤|Ψc||nΨ̂c,k(j)|. The asymptotic properties of ŴQ(p∼) follow readily from Theorem 3.2. Theorem 3.3. Let p∼ be selected according to (3.9). (a) Suppose that the assumptions for Theorem 3.2(a) hold for j=1,…,p ⁠. Then limn→∞Pr(p∼=1)=1 and ŴQ(p∼)→dχ12 as n→∞ ⁠. (b) Suppose instead that the assumptions for Theorem 3.2(b) hold for at least one j in j=1,…,p ⁠. Then ŴQ(p∼)→∞ as n→∞ ⁠. We conclude this section by noting that when the serial independence of the generalized residuals is rejected, it is often of interest to explore whether the serial dependence comes primarily through the conditional mean or higher conditional moments (see Diebold et al., 1998). In order to address this issue, Hong et al. (2007) proposed separate out‐of‐sample inference procedures that can detect serial dependence of {Yt} in terms of the level, volatility, skewness, kurtosis and leverage effect, etc. Similarly, we also propose a simple separate test for this purpose. Let μk=E[(U−1/2)k] and σk2= var [(U−1/2)k] ⁠, where U is a standard uniform random variable. Denote Φ̂k,l(j)=1n−j∑t=R+j+1N(Ût−j−1/2)k−μkσk(Ût−1/2)l−μlσl. We further consider the following test R̂k,l(j)=(n−j)Φ̂k,l(j)2.(3.10) Although similar to the independence test of Diebold et al. (1998), the current test uses Ût=Ĝn(Ẑt) ⁠, rather than Ẑt ⁠, to construct the test statistic (3.10). This assures that the asymptotic distribution of R̂k,l(j) is not affected by the estimation of θ0, which is needed to obtain Ẑt ⁠; see also Chen (2011). Following Hong et al. (2007), we consider (k,l)=(1,1),(2,2),(3,3),(4,4),(1,2),(2,1) in (3.10) in order to detect possible autocorrelations in level, volatility, skewness, kurtosis, ARCH‐in‐mean and leverage effects of {Yt} ⁠, respectively. For example, we can use R̂2,2(j) to test autocorrelation in the volatility of {Yt} ⁠. Similarly to ŴQ(p∼) ⁠, we can further construct a data‐driven portmanteau test ŴR(k,l)(p∼) based on R̂k,l(j) ⁠. Using Theorems 3.2 and 3.3, it is straightforward to show that under the null hypothesis, R̂k,l(j)→dχ12 and ŴR(k,l)(p∼)→dχ12 as n→∞ ⁠. The critical value of R̂k,l(j) can be tabulated and that of ŴR(k,l)(p∼) can be calculated following the simulation approach proposed for ŴQ(p∼) ⁠, which is given below. The consistency of these tests can also be established in a similar manner. 3.2. Test of uniformity Testing correct unconditional distribution of the forecast model is equivalent to testing the hypothesis that the generalized residuals {Ẑt} are uniformly distributed. Because Zt−j and Zt share the same marginal distribution G0(·) under (1.1), testing the uniformity of {Ẑt} is equivalent to testing the uniformity of {Ẑt−j} ⁠. However, conducting the test on {Ẑt−j} is advantageous as it allows us to construct a test in the second stage that is independent to the first‐stage test in the presence of estimated parameter uncertainty. Therefore, in this section, we shall conduct the uniformity test on {Ẑt−j} ⁠. Let ΨU={ΨU,1,…,ΨU,KU} be a non‐empty subset of a candidate set ψU={ψ1,… ⁠, ψM} ⁠, where KU≡|ΨU| ⁠. We consider a smooth alternative distribution given by g(z)=exp∑i=1KUbU,iΨU,i(z)+bU,0,z∈[0,1],(3.11) where bU,0 is a normalization constant. Let BU=(bU,1,…,bU,KU)′ ⁠. Clearly, BU=0 yields g(z)=1 ⁠, coinciding with the uniform density. Under the assumption that {Ẑt−j} are distributed according to (3.11), testing for uniformity is equivalent to testing the following hypothesis: H0U:BU=0. Correspondingly, one can construct a smooth test based on the sample moments Ψ̂U(j)=(Ψ̂U,1(j),…,Ψ̂U,KU(j))′ ⁠, where Ψ̂U,i(j)=(n−j)−1∑t=R+j+1NΨU,i(Ẑt−j) for i=1,… ⁠, KU ⁠. Compared with test (2.5) derived under a simple hypothesis, the present test is complicated by the presence of nuisance parameters θ̂t and it depends on the estimation scheme used in the density forecast. Proper adjustments are required to account for their influences. Let st=(∂/∂θ)lnft(Yt|Ωt−1,θ) be the gradient of the predictive density. Define s0,t−j=st−j|θ=θ0,Z0,t−j=Zt−j(θ0),A=E[s0,t−js0,t−j′],D=E[ΨU(Z0,t−j)s0,t−j′].(3.12) We assume that R,n→∞ as N→∞ and limN→∞n/R=τ ⁠, for some fixed number 0≤τ<∞ ⁠. Also define η=τ,fixed,0,recursive,−(τ2/3),rolling(τ≤1),−1+(2/3τ),rolling(τ>1).(3.13) Below we show that the asymptotic variance of Ψ̂U(j) is given by VU=IKU+ηDA−1D′, where IKU is a KU‐dimensional identity matrix. Next define η̂=η|τ=n/R,ŝt=∂∂θlnft(Yt|Ωt−1,θ)|θ=θ̂t,Â=1n−j∑t=R+j+1Nŝt−jŝt−j′,D̂=1n−j∑t=R+j+1NΨU(Ẑt−j)ŝt−j′.(3.14) We can estimate VU consistently using its sample counterpart: V̂U=IKU+η̂D̂Â−1D̂′.(3.15) We then construct a smooth test of uniformity as follows: N̂U(j)=(n−j)Ψ̂U(j)′V̂U−1Ψ̂U(j).(3.16) Note here the dependence on j is made explicit in the notations to emphasize that N̂U(j) is constructed based on {Ẑt−j} ⁠. Applying the results of West and McCracken (1998) to this test and following the arguments of Chen (2011), we establish the following results. Theorem 3.4. Suppose that Assumptions A.1–A.6 given in Appendix A hold. Under H0U as n→∞ ⁠: (a) Ψ̂U(j)→p0 and n−jΨ̂U(j)→dN(0,VU) ⁠; (b) the test statistic N̂U(j)→dχKU2 ⁠. Remark 3.1. When the parameter θ in the forecast density model ft(·|Ωt−1,θ) is known, VU is reduced to IKU ⁠, the variance obtained under the simple hypothesis. The adjustment DA−1D′ for nuisance parameters is multiplied by a factor η that reflects the estimation scheme of the density forecast. Remark 3.2. West and McCracken (1998) required the moment functions ψi to be continuously differentiable. McCracken (2000) extended the results of West and McCracken (1998) to allow for non‐differentiable moment functions, but their expectations are still required to be continuously differentiable with respect to θ. We note that the basis functions considered in this study, such as the Legendre polynomials and cosine series, satisfy the regularity conditions given in West and McCracken (1998). Like the copula test of independence presented above, the test of uniformity (3.16) depends crucially on the configuration of ΨU to capture potential deviations from uniformity. In the spirit of Kallenberg and Ledwina (1999), we start with a candidate set Ψu such that Ψu,1=ψ1 and the rest of the set correspond to the elements of {ψ2,…,ψM} arranged in descending order according to their corresponding entries in the vector V̂u−1/2|n−jΨ̂u| ⁠, where V̂u is the estimated covariance matrix of n−jΨ̂u ⁠.8 Given the ordered candidate set Ψu ⁠, we proceed to use an information criterion to select ΨU ⁠. Denote the subset of Ψu with its first k elements by Ψu,(k)={Ψu,1,…,Ψu,k} ⁠, k=1,…,M and the corresponding Vu,(k) and Nu,(k)(j) ⁠, as given in (3.15) and (3.16), are similarly defined; their sample analogues are denoted by Ψ̂u,(k)(j),V̂u,(k) and N̂u,(k)(j) ⁠, respectively. For each k, let Ψ̂u,(k)*(j)=V̂u,(k)−1/2Ψ̂u,(k)(j) ⁠. Following Inglot and Ledwina (2006), we use the following criterion to select a suitable ΨU ⁠, whose cardinality is denoted by KU(j) ⁠: KU(j)=min{k:N̂u,(k)(j)−Γ(k,n−j,ζ)≥N̂u,(s)(j)−Γ(s,n−j,ζ),1≤k,s≤M},(3.17) where the penalty Γ(k,n,ζ) is the same as (3.7) with max1≤k≤M|nΨ̂u,k∗(j)| taking the place of max1≤k≤|Ψc||nΨ̂c,k(j)|. The following theorem characterizes the asymptotic behaviour of N̂U(j) under the null hypothesis and its consistency. Theorem 3.5. Let KU(j) be selected according to (3.17). Suppose that Assumptions A.1–A.6 given in Appendix A hold. (a) Suppose that Z0,t−j given by (3.12) follows a uniform distribution. Then limn→∞Pr(KU(j)=1)=1 and N̂U(j)→dχ12 as n→∞ ⁠. (b) Suppose instead that Zt−j is distributed according to an alternative distribution P such that EP[ψS(Zt−j)]≠0 for some S∈{1,2,…,M} ⁠. Then N̂U(j)→∞ as n→∞ ⁠. 3.3. Construction of sequential test and inference We have presented two separate smooth tests for the serial independence and uniformity of the generalized residuals {Zt} ⁠. Here we proceed to construct a sequential test for the hypothesis of correct density forecast, which is equivalent to the i.i.d. uniformity of {Zt} ⁠. As is indicated in the introduction, our sequential test facilitates the diagnostics of misspecification in density forecast. Under the null hypothesis of i.i.d. uniformity, a sequential test is valid regardless of whether the independence test or the uniformity test comes in first. However, this invariance may be compromised if either serial independence or uniformity does not hold. The test for uniformity is constructed under the assumption of serial independence. In the presence of dynamic misspecification, serial independence of {Zt} is violated and the uniformity test suffers size distortion. In contrast, the robust test of independent copula is asymptotically invariant to possible deviations from uniformity due to misspecified marginal distributions. Therefore, we choose to place the independence test in the first stage of the sequential test. The test is terminated if the independence hypothesis is rejected; a subsequent test on uniformity is conducted only when the independence hypothesis is not rejected. This arrangement assures that the uniformity test (if necessary) is not compromised by possible violation of serial independence. The sequential nature of the proposed test complicates its inference: ignoring the two‐stage nature of the design can sometimes inflate the type I error. Suppose that the significance levels for the first and second stages of a sequential test are set at α1 and α2, respectively. Denote by p2|1 the probability of rejecting the second‐stage hypothesis, conditional on not rejecting the first‐stage hypothesis. The overall type I error of the two‐stage test, denoted by α, is given by α=α1+p2|1(1−α1).(3.18) If the tests from the first and second stages are independent, (3.18) is simplified to α=α1+α2(1−α1).(3.19) Next, we show that the proposed tests on serial independence and uniformity are asymptotically independent under the null hypothesis. Theorem 3.6. Under the null hypothesis of correct specification of density forecast, the test statistics Q̂C(j) and N̂U(j) are asymptotically independent for j=1,…,p ⁠; similarly, ŴQ(p) (ŴQ(p∼)) and N̂U(p) are asymptotically independent. The asymptotic independence suggested by this theorem facilitates the control of type I error α via proper choice of α1 and α2 based on (3.19). Qiu and Sheng (2008) suggest that, in the absence of a priori guidance on the significance levels of the two stages, a natural choice is to set α1=α2 ⁠. It follows that α1=α2=1−1−α. For instance, setting α=5% yields α1=α2≈2.53%. After α1 and α2 have been determined, the overall p‐value is given by p-value=p1,ifp1≤α1,α1+p2(1−α1),otherwise,(3.20) where p1 and p2 are p‐values of the first and second stages. A two‐stage test using the p‐value (3.20) rejects the overall null hypothesis when either of the following occurs: (a) the first‐stage null hypothesis is rejected (i.e. p1≤α1 ⁠); (b) the first‐stage null hypothesis is not rejected and the second stage null hypothesis is rejected (i.e. p1>α1 and p2≤α2 ⁠). We conclude this section with a procedure to calculate the critical values for our sequential tests. Because the proposed independence test and uniformity test are constructed in a data‐driven fashion, the number of functions selected is random even under the null hypothesis. Consequently, the χ12 distribution does not provide an adequate approximation to the distribution of the test statistics under moderate sample sizes. To deal with this problem, we propose a simple simulation procedure to obtain the critical values. Because both tests derived in the previous sections are asymptotically distribution‐free, their limiting distributions can be approximated via simple simulations by drawing repeatedly from the standard uniform distribution. The procedures are described below. Serial independence test Step 1. Generate an i.i.d. random sample {Zl,t}t=1n from the standard uniform distribution. Step 2. Calculate the empirical distribution of {Zl,t}t=1n ⁠, denoted by {Ûl,t}t=1n ⁠. Step 3. For j=1,…,p ⁠, select a set of basis functions ΨC according to (3.6); calculate Q̂C(l)(j) according to (3.5). Step 4. Select the optimal number of lags p∼ according to (3.9); calculate the adaptive portmanteau test ŴQ(l)(p∼)=∑j=1p∼Q̂C(l)(j) ⁠. Step 5. Repeat Steps 1–4 L times to obtain {Q̂C(l)(j)}l=1L ⁠, j=1,…,p and {ŴQ(l)(p∼)}l=1L ⁠. Use the (1−α1)th percentile of {Q̂C(l)(j)}l=1L as the (1−α1)th percent critical value of Q̂C(j) for j=1,…,p ⁠; use the (1−α1)th percentile of {ŴQ(l)(p∼)}l=1L as the (1−α1)th percent critical value of ŴQ(p∼) ⁠. Uniformity test (if necessary) Step 1. Generate an i.i.d. random sample {Zl,t}t=1n from the standard uniform distribution. Step 2. Select a set of basis functions ΨU according to the selection rule given in (3.17); compute the corresponding test statistic N̂U(l)(j) ⁠. Step 3. Repeat Steps 1 and 2 L times to obtain {N̂U(l)(j)}l=1L ⁠. Use the (1−α2)th percentile of {N̂U(l)(j)}l=1L to approximate the (1−α2)th percent critical value of N̂U(j) ⁠. Note that the approximated critical values are obtained via simple simulations based on the uniform distribution. These procedures do not require sampling from the data; nor do they entail any estimation based on the data. Numerical experiments reported in Section 4 suggest that the simulated critical values provide good size performance under small sample sizes. 4. Monte Carlo Simulations In this section, we use numerical simulations to examine the finite sample performance of the proposed sequential test. To facilitate comparison with the existing body of literature, we closely follow the experiment design of Hong et al. (2007) in our simulations. In particular, we generate random samples of length N=R+n and we split the samples into R in‐sample observations for estimation and n out‐of‐sample observations for density forecast evaluation. We consider three out‐of‐sample sizes : n=250 ⁠, 500 and 1,000; for each n, we consider four estimation–evaluation ratios: R/n=1 ⁠, 2 and 3.9 We repeat each experiment 3,000 times. We set the confidence level at α=5% and, for each n, we calculate the critical values using the simulation procedures described above, with 10,000 repetitions under the null hypothesis of i.i.d. uniformity of {Zt} ⁠. Following Hong et al. (2007), we use the following two commonly used models to assess the size of the proposed tests: the random‐walk‐normal model (RW‐N) Yt=2.77εt,εt∼i.i.d.N(0,1);(4.1) and the GARCH(1,1)‐normal model(GARCH‐N) Yt=htεt,εt∼i.i.d.N(0,1),ht=0.76+0.14Yt−12+0.77ht−1.(4.2) We also focus on testing the correctness of RW‐N model against the following alternative data‐generating processes (DGPs): Dgp1. Random‐walk‐T model (RW‐T)10 Yt=2.78εt,εt∼i.i.d.ν−2νt(ν),ν=3.39.(4.3) Dgp2. GARCH‐N model defined in (4.2). Dgp3. Regime‐switching‐T model (RS‐T) Yt=σ(st)εt,εt∼m.d.s.ν(st)−2ν(st)t(ν(st)), where st=1 or 2, and m.d.s ⁠. stands for martingale difference sequence. The transition probability between the two regimes is defined as P(st=l|st−1=l)=11+exp(−cl),l=1,2, where (σ(1),σ(2),ν(1),ν(2))=(1.81,3.67,6.92,3.88) and (c1,c2)=(3.12,2.76) ⁠. The construction of a data‐driven smooth test starts with a candidate set of basis functions. As shown by Ledwina (1994) and Kallenberg and Ledwina (1995), the type of orthogonal basis functions makes little difference; e.g. the shifted Legendre polynomials defined on [0, 1] and the cosine series, given by 2cos(iπz),i=1,2,… ⁠, provide largely identical results. Moreover, the test statistics are not sensitive to the size of the candidate set ψU or ψC as defined in Section 3. We use the shifted Legendre polynomials in our simulations. Following Kallenberg and Ledwina (1999), we set M=2 and KC(j)≤2 in the first‐stage copula test of serial independence. In the second‐stage test on uniformity (if necessary), we set M=10 ⁠, following Ledwina (1994). In either test, we then apply its corresponding information criterion prescribed in the previous sections to select a suitable set of basis functions, based on which the test statistic is calculated. In the copula test of independence, we consider the single‐lag test Q̂C(j) with j=1 ⁠, 5 and 10, the portmanteau tests ŴQ(p) with p=5 ⁠, 10 and 20 and the automatic portmanteau test ŴQ(p∼) ⁠. In the uniformity test, we use N̂U(20) ⁠, which is constructed based on Ẑt−20 ⁠. All tests provide satisfactory results. Because the single‐lag tests with j=5 or 10 are generally dominated by those with j=1 and the portmanteau tests, we choose not to report them to save space. Table 1 reports the empirical sizes of the sequential test, hereafter SQT. For comparison, we also report the results of the nonparametric omnibus test by Hong et al. (2007), hereafter HLZ, and of the simultaneous data‐driven smooth test by Park and Zhang (2010), hereafter PZ. Hong et al. (2007) constructed nonparametric tests that jointly test the uniformity and serial independence of {Zt} by comparing kernel estimator of the joint density of (Zt−j,Zt) with the product of two uniform densities. Because the nuisance parameters converge at a root‐n rate while the test statistics converge at nonparametric rates, the effects of nuisance parameter estimation are asymptotically negligible. The convenience of not having to directly account for the parameter estimation error is gained at the prices of bandwidth selection for kernel densities and slower convergence rates. Park and Zhang (2010) adopted the data‐driven smooth test to evaluate the accuracy of the conditional density function. Unlike the proposed test, their test is a simultaneous test of the uniformity and independence that relies on bootstrap critical values. This test is found to have low power against misspecification of the marginal distributions, which is discussed in more detail below. Table 1 Simulation results: empirical sizes (fixed estimation scheme) DGP . . Test . n=250 . n=500 . n= 1,000 . . . . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . RW‐N R/n=1 SQT 4.9 4.9 4.7 5.2 4.9 5.7 6.1 5.1 4.9 5.5 5.5 5.0 5.4 5.0 5.1 HLZ 7.3 7.5 8.0 7.9 7.5 7.9 8.2 7.8 7.5 7.9 8.2 7.8 PZ 5.2 5.2 5.7 5.1 5.1 4.8 5.5 5.6 5.7 R/n=2 SQT 4.9 5.7 5.1 5.0 5.2 4.7 5.2 5.1 5.3 4.8 4.6 4.9 4.9 5.1 4.7 HLZ 5.6 6.3 6.3 6.2 5.9 6.6 6.2 6.4 5.9 6.6 6.2 6.4 PZ 4.9 5.2 4.7 4.2 4.8 4.8 4.9 4.5 5.0 R/n=3 SQT 4.8 4.3 4.3 4.5 4.4 4.9 5.1 4.7 4.8 5.0 6.0 5.3 5.9 5.6 6.2 HLZ 4.8 5.5 5.8 5.9 5.3 5.8 5.8 5.9 5.7 5.8 5.6 5.9 PZ 5.4 4.7 5.2 6.1 5.9 5.8 4.6 4.2 4.5 GARCH‐N R/n=1 SQT 7.6 7.8 7.2 7.0 7.5 7.3 7.2 7.1 6.9 7.6 6.6 6.3 6.1 5.8 7.2 HLZ 10.4 12.4 12.1 12 10.7 10.9 10.7 10.7 8.2 9.8 10.1 9.9 PZ 6.2 6.0 6.0 5.3 5.8 6.2 4.6 5.7 6.3 R/n=2 SQT 6.5 6.7 6.7 6.4 6.4 6.8 5.9 6.6 6.5 5.5 5.7 5.5 6.1 5.3 5.4 HLZ 6.6 7.8 7.9 8 7.0 7.2 7.8 7.7 6.2 6.8 6.9 6.8 PZ 6.3 6.4 5.4 6.8 6.4 5.9 4.6 4.3 4.6 R/n=3 SQT 5.3 5.0 5.0 4.5 5.0 5.4 4.8 5.4 5.1 5.1 5.9 5.4 5.7 5.5 5.5 HLZ 5.8 6.1 6.5 6.7 6.4 7.0 7.2 6.8 6.0 6.3 6.2 5.7 PZ 5.6 4.9 5.5 4.8 5.2 5.1 4.6 4.3 4.4 DGP . . Test . n=250 . n=500 . n= 1,000 . . . . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . RW‐N R/n=1 SQT 4.9 4.9 4.7 5.2 4.9 5.7 6.1 5.1 4.9 5.5 5.5 5.0 5.4 5.0 5.1 HLZ 7.3 7.5 8.0 7.9 7.5 7.9 8.2 7.8 7.5 7.9 8.2 7.8 PZ 5.2 5.2 5.7 5.1 5.1 4.8 5.5 5.6 5.7 R/n=2 SQT 4.9 5.7 5.1 5.0 5.2 4.7 5.2 5.1 5.3 4.8 4.6 4.9 4.9 5.1 4.7 HLZ 5.6 6.3 6.3 6.2 5.9 6.6 6.2 6.4 5.9 6.6 6.2 6.4 PZ 4.9 5.2 4.7 4.2 4.8 4.8 4.9 4.5 5.0 R/n=3 SQT 4.8 4.3 4.3 4.5 4.4 4.9 5.1 4.7 4.8 5.0 6.0 5.3 5.9 5.6 6.2 HLZ 4.8 5.5 5.8 5.9 5.3 5.8 5.8 5.9 5.7 5.8 5.6 5.9 PZ 5.4 4.7 5.2 6.1 5.9 5.8 4.6 4.2 4.5 GARCH‐N R/n=1 SQT 7.6 7.8 7.2 7.0 7.5 7.3 7.2 7.1 6.9 7.6 6.6 6.3 6.1 5.8 7.2 HLZ 10.4 12.4 12.1 12 10.7 10.9 10.7 10.7 8.2 9.8 10.1 9.9 PZ 6.2 6.0 6.0 5.3 5.8 6.2 4.6 5.7 6.3 R/n=2 SQT 6.5 6.7 6.7 6.4 6.4 6.8 5.9 6.6 6.5 5.5 5.7 5.5 6.1 5.3 5.4 HLZ 6.6 7.8 7.9 8 7.0 7.2 7.8 7.7 6.2 6.8 6.9 6.8 PZ 6.3 6.4 5.4 6.8 6.4 5.9 4.6 4.3 4.6 R/n=3 SQT 5.3 5.0 5.0 4.5 5.0 5.4 4.8 5.4 5.1 5.1 5.9 5.4 5.7 5.5 5.5 HLZ 5.8 6.1 6.5 6.7 6.4 7.0 7.2 6.8 6.0 6.3 6.2 5.7 PZ 5.6 4.9 5.5 4.8 5.2 5.1 4.6 4.3 4.4 Note: This table reports the empirical sizes of the SQT under the fixed estimation scheme with Q̂C(1) ⁠, ŴQ(p) ⁠, p=5,10,20 or ŴQ(p∼) in the first stage. The sizes of the HLZ and PZ tests, rounded to the first decimal place, are also reported. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab Table 1 Simulation results: empirical sizes (fixed estimation scheme) DGP . . Test . n=250 . n=500 . n= 1,000 . . . . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . RW‐N R/n=1 SQT 4.9 4.9 4.7 5.2 4.9 5.7 6.1 5.1 4.9 5.5 5.5 5.0 5.4 5.0 5.1 HLZ 7.3 7.5 8.0 7.9 7.5 7.9 8.2 7.8 7.5 7.9 8.2 7.8 PZ 5.2 5.2 5.7 5.1 5.1 4.8 5.5 5.6 5.7 R/n=2 SQT 4.9 5.7 5.1 5.0 5.2 4.7 5.2 5.1 5.3 4.8 4.6 4.9 4.9 5.1 4.7 HLZ 5.6 6.3 6.3 6.2 5.9 6.6 6.2 6.4 5.9 6.6 6.2 6.4 PZ 4.9 5.2 4.7 4.2 4.8 4.8 4.9 4.5 5.0 R/n=3 SQT 4.8 4.3 4.3 4.5 4.4 4.9 5.1 4.7 4.8 5.0 6.0 5.3 5.9 5.6 6.2 HLZ 4.8 5.5 5.8 5.9 5.3 5.8 5.8 5.9 5.7 5.8 5.6 5.9 PZ 5.4 4.7 5.2 6.1 5.9 5.8 4.6 4.2 4.5 GARCH‐N R/n=1 SQT 7.6 7.8 7.2 7.0 7.5 7.3 7.2 7.1 6.9 7.6 6.6 6.3 6.1 5.8 7.2 HLZ 10.4 12.4 12.1 12 10.7 10.9 10.7 10.7 8.2 9.8 10.1 9.9 PZ 6.2 6.0 6.0 5.3 5.8 6.2 4.6 5.7 6.3 R/n=2 SQT 6.5 6.7 6.7 6.4 6.4 6.8 5.9 6.6 6.5 5.5 5.7 5.5 6.1 5.3 5.4 HLZ 6.6 7.8 7.9 8 7.0 7.2 7.8 7.7 6.2 6.8 6.9 6.8 PZ 6.3 6.4 5.4 6.8 6.4 5.9 4.6 4.3 4.6 R/n=3 SQT 5.3 5.0 5.0 4.5 5.0 5.4 4.8 5.4 5.1 5.1 5.9 5.4 5.7 5.5 5.5 HLZ 5.8 6.1 6.5 6.7 6.4 7.0 7.2 6.8 6.0 6.3 6.2 5.7 PZ 5.6 4.9 5.5 4.8 5.2 5.1 4.6 4.3 4.4 DGP . . Test . n=250 . n=500 . n= 1,000 . . . . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . RW‐N R/n=1 SQT 4.9 4.9 4.7 5.2 4.9 5.7 6.1 5.1 4.9 5.5 5.5 5.0 5.4 5.0 5.1 HLZ 7.3 7.5 8.0 7.9 7.5 7.9 8.2 7.8 7.5 7.9 8.2 7.8 PZ 5.2 5.2 5.7 5.1 5.1 4.8 5.5 5.6 5.7 R/n=2 SQT 4.9 5.7 5.1 5.0 5.2 4.7 5.2 5.1 5.3 4.8 4.6 4.9 4.9 5.1 4.7 HLZ 5.6 6.3 6.3 6.2 5.9 6.6 6.2 6.4 5.9 6.6 6.2 6.4 PZ 4.9 5.2 4.7 4.2 4.8 4.8 4.9 4.5 5.0 R/n=3 SQT 4.8 4.3 4.3 4.5 4.4 4.9 5.1 4.7 4.8 5.0 6.0 5.3 5.9 5.6 6.2 HLZ 4.8 5.5 5.8 5.9 5.3 5.8 5.8 5.9 5.7 5.8 5.6 5.9 PZ 5.4 4.7 5.2 6.1 5.9 5.8 4.6 4.2 4.5 GARCH‐N R/n=1 SQT 7.6 7.8 7.2 7.0 7.5 7.3 7.2 7.1 6.9 7.6 6.6 6.3 6.1 5.8 7.2 HLZ 10.4 12.4 12.1 12 10.7 10.9 10.7 10.7 8.2 9.8 10.1 9.9 PZ 6.2 6.0 6.0 5.3 5.8 6.2 4.6 5.7 6.3 R/n=2 SQT 6.5 6.7 6.7 6.4 6.4 6.8 5.9 6.6 6.5 5.5 5.7 5.5 6.1 5.3 5.4 HLZ 6.6 7.8 7.9 8 7.0 7.2 7.8 7.7 6.2 6.8 6.9 6.8 PZ 6.3 6.4 5.4 6.8 6.4 5.9 4.6 4.3 4.6 R/n=3 SQT 5.3 5.0 5.0 4.5 5.0 5.4 4.8 5.4 5.1 5.1 5.9 5.4 5.7 5.5 5.5 HLZ 5.8 6.1 6.5 6.7 6.4 7.0 7.2 6.8 6.0 6.3 6.2 5.7 PZ 5.6 4.9 5.5 4.8 5.2 5.1 4.6 4.3 4.4 Note: This table reports the empirical sizes of the SQT under the fixed estimation scheme with Q̂C(1) ⁠, ŴQ(p) ⁠, p=5,10,20 or ŴQ(p∼) in the first stage. The sizes of the HLZ and PZ tests, rounded to the first decimal place, are also reported. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab Table 2 Simulation results: empirical powers (fixed estimation scheme) DGP . Test . . n=250 . n=500 . n= 1,000 . . . . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . RW‐T R/n=1 SQT 82.7 82.6 82.8 82.8 82.7 98.6 98.6 98.5 98.5 98.5 100 100 100 100 100 HLZ 36.8 39.8 38.8 37.9 49.3 50.4 47.1 43.2 65 68.2 61.9 PZ 24.2 24.3 25.1 40.3 48.1 56 60 67.2 74.2 R/n=2 SQT 86.6 86.6 86.6 86.5 86.8 99.5 99.5 99.5 99.5 99.6 100 100 100 100 100 HLZ 33.4 34.5 33.3 32.2 45.1 47.3 43.5 38.9 65.2 66.4 60 PZ 28.3 32.7 36 40 46 54.8 61.6 70.9 77 R/n=3 SQT 87.4 87.6 87.4 87.5 87.4 99.5 99.5 99.5 99.5 99.5 100 100 100 100 100 HLZ 30.6 33 31.2 30.1 41.6 44.6 41.6 36.8 64.1 65.6 58.2 PZ 26.3 34 37.8 40.1 47.6 54.1 61.6 71.7 80.1 GARCH‐N R/n=1 SQT 41.5 48.1 45.8 42.5 44.2 69 76.8 74.9 69.8 73.1 91.9 96.3 96.1 94.1 95.8 HLZ 36.8 30.4 39.8 38.8 37.9 36.9 35 50.4 47.1 52.5 43.4 PZ 46.3 45 39.9 84.4 81.8 74.5 98.3 97.9 96.4 R/n=2 SQT 42.3 47.6 46.3 43.1 44.9 70.4 77.3 75.7 70 74.4 92.7 96.8 96.5 94.7 96.5 HLZ 33.4 34.5 33.3 32.2 45.1 47.3 43.5 38.9 65.2 66.4 60 PZ 55.6 54.6 48.5 85 81.2 75 98.2 97.7 96.1 R/n=3 SQT 43.6 49.4 48 43.9 46 69.5 77 75.9 70.2 74 92.7 96.5 96.6 94.9 96.4 HLZ 30.6 33 31.2 30.1 41.6 44.6 41.6 36.8 64.1 65.6 58.2 PZ 54.7 50.9 46.4 81.1 78.2 71.3 98.2 97.8 95.8 RS‐T R/n=1 SQT 80.5 80.7 80.6 80.2 80.4 98.4 98.5 98.6 98.5 98.4 100 100 100 100 100 HLZ 72.67 78.77 79.4 79.83 96.07 97.83 98.17 98.1 99.97 100 100 PZ 60.3 55.3 49.4 87.7 89.4 89 98.8 99.3 98.9 R/n=2 SQT 85.6 85.9 85.9 85.7 85.7 99.3 99.3 99.3 99.3 99.3 100 100 100 100 100 HLZ 80.63 86.17 86.83 87.1 98.23 99.13 99.27 99.33 100 100 100 PZ 66.8 70.2 68.3 87.8 89.2 89.5 99.1 99.5 99.1 R/n=3 SQT 87.5 87.7 87.6 87.5 87.6 99.3 99.2 99.2 99.2 99.2 100 100 100 100 100 HLZ 82.13 87.87 88.43 88.93 98.9 99.33 99.4 99.43 100 100 100 PZ 65.9 68.2 66.6 88.4 89.6 89.3 98.9 99.3 99.4 DGP . Test . . n=250 . n=500 . n= 1,000 . . . . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . RW‐T R/n=1 SQT 82.7 82.6 82.8 82.8 82.7 98.6 98.6 98.5 98.5 98.5 100 100 100 100 100 HLZ 36.8 39.8 38.8 37.9 49.3 50.4 47.1 43.2 65 68.2 61.9 PZ 24.2 24.3 25.1 40.3 48.1 56 60 67.2 74.2 R/n=2 SQT 86.6 86.6 86.6 86.5 86.8 99.5 99.5 99.5 99.5 99.6 100 100 100 100 100 HLZ 33.4 34.5 33.3 32.2 45.1 47.3 43.5 38.9 65.2 66.4 60 PZ 28.3 32.7 36 40 46 54.8 61.6 70.9 77 R/n=3 SQT 87.4 87.6 87.4 87.5 87.4 99.5 99.5 99.5 99.5 99.5 100 100 100 100 100 HLZ 30.6 33 31.2 30.1 41.6 44.6 41.6 36.8 64.1 65.6 58.2 PZ 26.3 34 37.8 40.1 47.6 54.1 61.6 71.7 80.1 GARCH‐N R/n=1 SQT 41.5 48.1 45.8 42.5 44.2 69 76.8 74.9 69.8 73.1 91.9 96.3 96.1 94.1 95.8 HLZ 36.8 30.4 39.8 38.8 37.9 36.9 35 50.4 47.1 52.5 43.4 PZ 46.3 45 39.9 84.4 81.8 74.5 98.3 97.9 96.4 R/n=2 SQT 42.3 47.6 46.3 43.1 44.9 70.4 77.3 75.7 70 74.4 92.7 96.8 96.5 94.7 96.5 HLZ 33.4 34.5 33.3 32.2 45.1 47.3 43.5 38.9 65.2 66.4 60 PZ 55.6 54.6 48.5 85 81.2 75 98.2 97.7 96.1 R/n=3 SQT 43.6 49.4 48 43.9 46 69.5 77 75.9 70.2 74 92.7 96.5 96.6 94.9 96.4 HLZ 30.6 33 31.2 30.1 41.6 44.6 41.6 36.8 64.1 65.6 58.2 PZ 54.7 50.9 46.4 81.1 78.2 71.3 98.2 97.8 95.8 RS‐T R/n=1 SQT 80.5 80.7 80.6 80.2 80.4 98.4 98.5 98.6 98.5 98.4 100 100 100 100 100 HLZ 72.67 78.77 79.4 79.83 96.07 97.83 98.17 98.1 99.97 100 100 PZ 60.3 55.3 49.4 87.7 89.4 89 98.8 99.3 98.9 R/n=2 SQT 85.6 85.9 85.9 85.7 85.7 99.3 99.3 99.3 99.3 99.3 100 100 100 100 100 HLZ 80.63 86.17 86.83 87.1 98.23 99.13 99.27 99.33 100 100 100 PZ 66.8 70.2 68.3 87.8 89.2 89.5 99.1 99.5 99.1 R/n=3 SQT 87.5 87.7 87.6 87.5 87.6 99.3 99.2 99.2 99.2 99.2 100 100 100 100 100 HLZ 82.13 87.87 88.43 88.93 98.9 99.33 99.4 99.43 100 100 100 PZ 65.9 68.2 66.6 88.4 89.6 89.3 98.9 99.3 99.4 Note: This table reports the empirical powers of the SQT with Q̂C(1) ⁠, ŴQ(p) ⁠, p=5,10,20 or ŴQ(p∼) in the first stage. The powers of the HLZ and PZ tests, rounded to the first decimal place, are also reported. The results of W(20) for HLZ are left blank because they are not reported in HLZ. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab Table 2 Simulation results: empirical powers (fixed estimation scheme) DGP . Test . . n=250 . n=500 . n= 1,000 . . . . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . RW‐T R/n=1 SQT 82.7 82.6 82.8 82.8 82.7 98.6 98.6 98.5 98.5 98.5 100 100 100 100 100 HLZ 36.8 39.8 38.8 37.9 49.3 50.4 47.1 43.2 65 68.2 61.9 PZ 24.2 24.3 25.1 40.3 48.1 56 60 67.2 74.2 R/n=2 SQT 86.6 86.6 86.6 86.5 86.8 99.5 99.5 99.5 99.5 99.6 100 100 100 100 100 HLZ 33.4 34.5 33.3 32.2 45.1 47.3 43.5 38.9 65.2 66.4 60 PZ 28.3 32.7 36 40 46 54.8 61.6 70.9 77 R/n=3 SQT 87.4 87.6 87.4 87.5 87.4 99.5 99.5 99.5 99.5 99.5 100 100 100 100 100 HLZ 30.6 33 31.2 30.1 41.6 44.6 41.6 36.8 64.1 65.6 58.2 PZ 26.3 34 37.8 40.1 47.6 54.1 61.6 71.7 80.1 GARCH‐N R/n=1 SQT 41.5 48.1 45.8 42.5 44.2 69 76.8 74.9 69.8 73.1 91.9 96.3 96.1 94.1 95.8 HLZ 36.8 30.4 39.8 38.8 37.9 36.9 35 50.4 47.1 52.5 43.4 PZ 46.3 45 39.9 84.4 81.8 74.5 98.3 97.9 96.4 R/n=2 SQT 42.3 47.6 46.3 43.1 44.9 70.4 77.3 75.7 70 74.4 92.7 96.8 96.5 94.7 96.5 HLZ 33.4 34.5 33.3 32.2 45.1 47.3 43.5 38.9 65.2 66.4 60 PZ 55.6 54.6 48.5 85 81.2 75 98.2 97.7 96.1 R/n=3 SQT 43.6 49.4 48 43.9 46 69.5 77 75.9 70.2 74 92.7 96.5 96.6 94.9 96.4 HLZ 30.6 33 31.2 30.1 41.6 44.6 41.6 36.8 64.1 65.6 58.2 PZ 54.7 50.9 46.4 81.1 78.2 71.3 98.2 97.8 95.8 RS‐T R/n=1 SQT 80.5 80.7 80.6 80.2 80.4 98.4 98.5 98.6 98.5 98.4 100 100 100 100 100 HLZ 72.67 78.77 79.4 79.83 96.07 97.83 98.17 98.1 99.97 100 100 PZ 60.3 55.3 49.4 87.7 89.4 89 98.8 99.3 98.9 R/n=2 SQT 85.6 85.9 85.9 85.7 85.7 99.3 99.3 99.3 99.3 99.3 100 100 100 100 100 HLZ 80.63 86.17 86.83 87.1 98.23 99.13 99.27 99.33 100 100 100 PZ 66.8 70.2 68.3 87.8 89.2 89.5 99.1 99.5 99.1 R/n=3 SQT 87.5 87.7 87.6 87.5 87.6 99.3 99.2 99.2 99.2 99.2 100 100 100 100 100 HLZ 82.13 87.87 88.43 88.93 98.9 99.33 99.4 99.43 100 100 100 PZ 65.9 68.2 66.6 88.4 89.6 89.3 98.9 99.3 99.4 DGP . Test . . n=250 . n=500 . n= 1,000 . . . . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . Q(1) . W(5) . W(10) . W(20) . W(p∼) . RW‐T R/n=1 SQT 82.7 82.6 82.8 82.8 82.7 98.6 98.6 98.5 98.5 98.5 100 100 100 100 100 HLZ 36.8 39.8 38.8 37.9 49.3 50.4 47.1 43.2 65 68.2 61.9 PZ 24.2 24.3 25.1 40.3 48.1 56 60 67.2 74.2 R/n=2 SQT 86.6 86.6 86.6 86.5 86.8 99.5 99.5 99.5 99.5 99.6 100 100 100 100 100 HLZ 33.4 34.5 33.3 32.2 45.1 47.3 43.5 38.9 65.2 66.4 60 PZ 28.3 32.7 36 40 46 54.8 61.6 70.9 77 R/n=3 SQT 87.4 87.6 87.4 87.5 87.4 99.5 99.5 99.5 99.5 99.5 100 100 100 100 100 HLZ 30.6 33 31.2 30.1 41.6 44.6 41.6 36.8 64.1 65.6 58.2 PZ 26.3 34 37.8 40.1 47.6 54.1 61.6 71.7 80.1 GARCH‐N R/n=1 SQT 41.5 48.1 45.8 42.5 44.2 69 76.8 74.9 69.8 73.1 91.9 96.3 96.1 94.1 95.8 HLZ 36.8 30.4 39.8 38.8 37.9 36.9 35 50.4 47.1 52.5 43.4 PZ 46.3 45 39.9 84.4 81.8 74.5 98.3 97.9 96.4 R/n=2 SQT 42.3 47.6 46.3 43.1 44.9 70.4 77.3 75.7 70 74.4 92.7 96.8 96.5 94.7 96.5 HLZ 33.4 34.5 33.3 32.2 45.1 47.3 43.5 38.9 65.2 66.4 60 PZ 55.6 54.6 48.5 85 81.2 75 98.2 97.7 96.1 R/n=3 SQT 43.6 49.4 48 43.9 46 69.5 77 75.9 70.2 74 92.7 96.5 96.6 94.9 96.4 HLZ 30.6 33 31.2 30.1 41.6 44.6 41.6 36.8 64.1 65.6 58.2 PZ 54.7 50.9 46.4 81.1 78.2 71.3 98.2 97.8 95.8 RS‐T R/n=1 SQT 80.5 80.7 80.6 80.2 80.4 98.4 98.5 98.6 98.5 98.4 100 100 100 100 100 HLZ 72.67 78.77 79.4 79.83 96.07 97.83 98.17 98.1 99.97 100 100 PZ 60.3 55.3 49.4 87.7 89.4 89 98.8 99.3 98.9 R/n=2 SQT 85.6 85.9 85.9 85.7 85.7 99.3 99.3 99.3 99.3 99.3 100 100 100 100 100 HLZ 80.63 86.17 86.83 87.1 98.23 99.13 99.27 99.33 100 100 100 PZ 66.8 70.2 68.3 87.8 89.2 89.5 99.1 99.5 99.1 R/n=3 SQT 87.5 87.7 87.6 87.5 87.6 99.3 99.2 99.2 99.2 99.2 100 100 100 100 100 HLZ 82.13 87.87 88.43 88.93 98.9 99.33 99.4 99.43 100 100 100 PZ 65.9 68.2 66.6 88.4 89.6 89.3 98.9 99.3 99.4 Note: This table reports the empirical powers of the SQT with Q̂C(1) ⁠, ŴQ(p) ⁠, p=5,10,20 or ŴQ(p∼) in the first stage. The powers of the HLZ and PZ tests, rounded to the first decimal place, are also reported. The results of W(20) for HLZ are left blank because they are not reported in HLZ. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab All three tests use the fixed estimation scheme in their estimation. Both the SQT and PZ tests use the approach of the data‐driven smooth test; their sizes are generally close to the 5% theoretical value and do not seem to vary across the R/n ratios. The sizes of the HLZ test vary noticeably with the R/n ratio. With R/n=1 ⁠, their sizes average around 8% and 10% for the RW‐N and GARCH‐N models, respectively. This oversize problem improves with the R/n ratio but seems to persist as sample size increases. The empirical powers of the SQT test, together with those of the HLZ and PZ tests, are reported in Table 2. The SQT test generally outperforms the HLZ and PZ tests. In particular, under the RW‐T DGP, the SQT test dominates the other two tests by substantial margins; under the GARCH‐N DGP, the SQT and PZ tests are comparable and dominate the HLZ test; under the RS‐T DGP, the SQT and HLZ tests are comparable and dominate the PZ test. Some remarks are in order. (a) The PZ test essentially focuses on testing the copula density, implicitly assuming that the marginal distributions are uniform. Consequently, it has good power against misspecification in the serial dependence (GARCH‐N versus RW‐N in our simulations). However, it has weak power against misspecification in the marginal distributions. In the presence of misspecified marginal distributions, the PZ test is dominated by the other two tests, especially so when the misspecification only occurs in the marginal distributions (RW‐T versus RW‐N). (b) The HLZ test, based on the joint density of (⁠ Ẑt−j,Ẑt ⁠), is seen to provide good powers against the RS‐T alternative, which deviates from the null in both the marginal distributions and serial dependence. However, it has relatively low power when the violation only occurs in one aspect, as in the RW‐T or GARCH‐N case. (c) The proposed SQT tests Q̂C(1),ŴQ(5) ⁠, ŴQ(10) ⁠, ŴQ(20) and ŴQ(p∼) provide largely similar performance, with slight variations in a few cases. Because the adaptive portmanteau test ŴQ(p∼) performs well across all alternatives, it is recommended, unless there are strong reasons to focus on a specific lag or time horizon in the testing. Therefore, we focus on this test in the following discussion of independence test and the empirical investigations. As discussed above, the first‐stage robust test can serve as a stand‐alone test for serial independence of the generalized residuals. Table 3 reports simulation results of the robust test ŴQ(p∼) of the independent copula, focusing on the hypothesis of the RW‐N model. The first three columns reflect the empirical sizes, which are centred about the nominal 5% significance level. The middle three columns show the substantial powers of the proposed tests against the alternative of the GARCH‐N model. The last three columns report the results against the RW‐T model, which are correctly centred about the 5% level despite the misspecification of the unconditional distribution under the null hypothesis of the RW‐N model. This experiment confirms our theoretical analysis that the rank‐based copula test of serial independence is robust to misspecification in the unconditional distribution. Table 3 Robust tests for serial independence (fixed estimation scheme) DGP . RW‐N . GARCH‐N . RW‐T . . R/n=1 . R/n=2 . R/n=3 . R/n=1 . R/n=2 . R/n=3 . R/n=1 . R/n=2 . R/n=3 . n=250 4.7 4.9 5.5 39.8 39.7 40.1 4.9 5 5 n=500 5.1 5.4 5.3 73.1 73.5 72.6 4.4 5.3 5.3 n= 1,000 4.8 5.1 5.9 95.8 96 95.6 5.2 5.8 5.0 DGP . RW‐N . GARCH‐N . RW‐T . . R/n=1 . R/n=2 . R/n=3 . R/n=1 . R/n=2 . R/n=3 . R/n=1 . R/n=2 . R/n=3 . n=250 4.7 4.9 5.5 39.8 39.7 40.1 4.9 5 5 n=500 5.1 5.4 5.3 73.1 73.5 72.6 4.4 5.3 5.3 n= 1,000 4.8 5.1 5.9 95.8 96 95.6 5.2 5.8 5.0 Note: This table reports the empirical sizes and powers of the data‐driven portmanteau test statistic ŴQ(p∼) for serial independence. The null hypothesis is that the data are generated from the RW‐N model. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab Table 3 Robust tests for serial independence (fixed estimation scheme) DGP . RW‐N . GARCH‐N . RW‐T . . R/n=1 . R/n=2 . R/n=3 . R/n=1 . R/n=2 . R/n=3 . R/n=1 . R/n=2 . R/n=3 . n=250 4.7 4.9 5.5 39.8 39.7 40.1 4.9 5 5 n=500 5.1 5.4 5.3 73.1 73.5 72.6 4.4 5.3 5.3 n= 1,000 4.8 5.1 5.9 95.8 96 95.6 5.2 5.8 5.0 DGP . RW‐N . GARCH‐N . RW‐T . . R/n=1 . R/n=2 . R/n=3 . R/n=1 . R/n=2 . R/n=3 . R/n=1 . R/n=2 . R/n=3 . n=250 4.7 4.9 5.5 39.8 39.7 40.1 4.9 5 5 n=500 5.1 5.4 5.3 73.1 73.5 72.6 4.4 5.3 5.3 n= 1,000 4.8 5.1 5.9 95.8 96 95.6 5.2 5.8 5.0 Note: This table reports the empirical sizes and powers of the data‐driven portmanteau test statistic ŴQ(p∼) for serial independence. The null hypothesis is that the data are generated from the RW‐N model. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab 5. Empirical Application In this section, we apply the proposed smooth tests to evaluate various forecast models of stock returns. In particular, we study the daily value‐weighted S&P500 returns, with dividends, from 3 July 1962 to 29 December 1995. These data have been analysed by Diebold et al. (1998) and Chen (2011), among others. Diebold et al. (1998) proposed some intuitive graphical methods to assess separately the serial independence and uniformity of the generalized residuals. Although simple, their graphical approach does not account for the influence of nuisance parameters. Chen (2011) proposed a generalized PIT‐based moment test to unify the existing uniformity tests and serial independence tests. In particular, he considered tests based on some pre‐determined moment functions and accounted for the parameter estimation uncertainty using the West–McCracken method for out‐of‐sample tests. Following Diebold et al. (1998), we divide the sample roughly into two halves. Observations from 3 July 1962 to 29 December 1978 (with a total of 4,133 observations) are used for estimation, while those from 2 January 1979 to 29 December 1995 (with a total of 4,298 observations) are used for evaluation. Diebold et al. (1998) considered three models: i.i.d. normal, MA(1)‐GARCH(1,1)‐N and MA(1)‐GARCH(1,1)‐T. They found that the GARCH models significantly outperform the i.i.d. normal model. Their results lend support to the MA(1)‐GARCH(1,1)‐T model. In addition to the three models studied in Diebold et al. (1998), Chen (2011) also considered the MA(1)‐EGARCH(1,1)‐N and MA(1)‐EGARCH(1,1)‐T models. His results suggest that the GARCH‐T model outperforms the GARCH‐N model in the uniformity test; however, both the GARCH and EGARCH models fail to correctly predict the dynamics of the return series in the forecast period. We revisit this empirical study and consider the following eight models: RW‐N, RW‐T, GARCH(1,1)‐N, GARCH(1,1)‐T, EGARCH(1,1)‐N, EGARCH(1,1)‐T, RiskMetrics‐N (RM‐N) and RiskMetrics‐T (RM‐T). This particular set of competitors is considered in this empirical investigation because they facilitate comparison with Diebold et al. (1999) and Chen (2011) and they represent the most commonly used models for stock returns. Note that the last two RiskMetrics models were also considered in Hong et al. (2007). In accordance with Diebold et al. (1998) and Chen (2011), we use the fixed estimation scheme and adopt an MA(1) specification in the conditional mean for all GARCH‐type and RiskMetrics models. After calculating the generalized residuals according to density forecast models, we test their i.i.d. uniformity using the proposed sequential tests. To save space, we focus on the sequential test with the automatic portmanteau test ŴQ(p∼) in the first stage. Recall that the significance levels of both stages are set at 2.53% to obtain a 5% overall significance level of the sequential test. The simulated critical value of ŴQ(p∼) at the 2.53% significance level is 9.12. The test results are reported in Table 4. In order to provide useful information about the specification of marginal distributions, the results for the stand‐alone uniformity test N̂U(20) are also reported.11 The simulated critical value of N̂U(20) at the 5% significance level is 5.64. It transpires that the hypothesis of correct density forecast is rejected for all models. Table 4 Test results for estimated density forecast models . RW‐N . RW‐T . GARCH‐N . GARCH‐T . EGARCH‐N . EGARCH‐T . RM‐N . RM‐T . ŴQ(p∼) 838.5 838.5 186.5 189.5 179.9 194.5 182.5 211 N̂U(20) 370.8 241.6 79.9 11.6 70.8 32.5 116.7 47 . RW‐N . RW‐T . GARCH‐N . GARCH‐T . EGARCH‐N . EGARCH‐T . RM‐N . RM‐T . ŴQ(p∼) 838.5 838.5 186.5 189.5 179.9 194.5 182.5 211 N̂U(20) 370.8 241.6 79.9 11.6 70.8 32.5 116.7 47 Note: The parameters of the models are estimated from the estimation sample (from 3 July 1962 to 29 December 1978 with a total of 4,133 observations). The generalized residuals are obtained using the evaluation sample (from 2 January 1979 to 29 December 1995 with a total of 4,298 observations.) The hypothesis of the independent copula is rejected for all models, effectively terminating the tests. Open in new tab Table 4 Test results for estimated density forecast models . RW‐N . RW‐T . GARCH‐N . GARCH‐T . EGARCH‐N . EGARCH‐T . RM‐N . RM‐T . ŴQ(p∼) 838.5 838.5 186.5 189.5 179.9 194.5 182.5 211 N̂U(20) 370.8 241.6 79.9 11.6 70.8 32.5 116.7 47 . RW‐N . RW‐T . GARCH‐N . GARCH‐T . EGARCH‐N . EGARCH‐T . RM‐N . RM‐T . ŴQ(p∼) 838.5 838.5 186.5 189.5 179.9 194.5 182.5 211 N̂U(20) 370.8 241.6 79.9 11.6 70.8 32.5 116.7 47 Note: The parameters of the models are estimated from the estimation sample (from 3 July 1962 to 29 December 1978 with a total of 4,133 observations). The generalized residuals are obtained using the evaluation sample (from 2 January 1979 to 29 December 1995 with a total of 4,298 observations.) The hypothesis of the independent copula is rejected for all models, effectively terminating the tests. Open in new tab Figure 1. Open in new tabDownload slide Histogram of Ẑt ⁠: MA(1)‐GARCH(1,1)‐T model. [Color figure can be viewed at wileyonlinelibrary.com] Figure 1. Open in new tabDownload slide Histogram of Ẑt ⁠: MA(1)‐GARCH(1,1)‐T model. [Color figure can be viewed at wileyonlinelibrary.com] Examination of the test results provides the following insights. (a) The first stage of the robust test decisively rejects the hypothesis of serial independence, indicating that none of the models in consideration can adequately describe the dynamics of the daily S&P500 returns. Similar findings are reported in Chen (2011). (b) Comparison among the models suggests that the GARCH‐type and RiskMetrics models generally outperform random‐walk models, emphasizing the importance of accounting for volatility clustering. Allowing asymmetric behaviour in volatility through the EGARCH model, or applying the exponential smoothing technique through the RiskMetrics model, does not seem to improve the specification of the dynamic structure. (c) The rejection of serial independence in the first stage of the sequential test effectively terminates the test. Nonetheless, we report the results of the stand‐alone uniformity test as they may provide useful information regarding the goodness‐of‐fit of the unconditional distributions. All models with a normal innovation are rejected decisively, while those with a t innovation are marginally rejected. This finding is consistent with the general consensus that the distributions of stock returns are fat‐tailed. (d) Both the robust test and stand‐alone uniformity test seem to favour the GARCH‐T model among all models under consideration, which agrees with the findings of Diebold et al. (1998). We plot the histograms of Ẑt and the correlograms of (Ẑt−Z¯)i with Z¯=n−1∑t=R+1NẐt and i=1,2,3,4 for the MA(1)‐GARCH(1,1)‐T model in Figures 1 and 2. Consistent with the test on uniformity, the histogram of Ẑt is nearly uniform. However, the sample autocorrelations of (Ẑt−Z¯) and (Ẑt−Z¯)3 are significantly different from zero at lag one, indicating that the GARCH(1,1) model fails to adequately characterize the dynamic structure of stock returns. Figure 2. Open in new tabDownload slide Correlograms of the powers of Ẑt ⁠: MA(1)‐GARCH(1,1)‐T model. [Color figure can be viewed at wileyonlinelibrary.com] Figure 2. Open in new tabDownload slide Correlograms of the powers of Ẑt ⁠: MA(1)‐GARCH(1,1)‐T model. [Color figure can be viewed at wileyonlinelibrary.com] Lastly, we conduct the out‐of‐sample separate inference tests, as described at the end of Section 3.1, for {Ẑt} derived from MA(1)‐GARCH(1,1)‐T model. At the 5% significance level, the critical values of R̂k,l(1) at lag one and ŴR(k,l)(p∼) are 3.84 and 3.97, respectively. The test statistics R̂k,k(1) with k=1,2,3,4 are 153.9, 3.2, 59.2 and 5.3, respectively and the corresponding data‐driven portmanteau test statistics ŴR(k,k)(p∼) are 170.9, 18.6, 70.6 and 21.1, respectively. These results are consistent with the correlograms reported in Figure 2. Those figures show that at lag one, there are strong autocorrelations in level and skewness, minor autocorrelation in kurtosis and no autocorrelation in variance of {Ẑt} ⁠. Jointly considering the first 20 lags, there exist strong autocorrelations in the level and skewness of {Ẑt} and small autocorrelations in variance and kurtosis of {Ẑt} ⁠. 6. Concluding Remarks We have proposed a sequential test for the specification of predictive density models. The proposed test is shown to have a nuisance‐parameter‐free asymptotic distribution under the null hypothesis of correct specification of predictive density. One attractive feature of the test is that it facilitates the diagnosis of the potential sources of misspecification by separating the independence test and uniformity test of the generalized residuals. Monte Carlo simulations demonstrate excellent performances of the test. We have focused on testing correct density forecast models in the present study. All models of stock returns considered in the previous section are rejected by our tests. Although an MA‐GARCH‐T model is preferred, it is not clear if this model is significantly better than some of its competitors. For this purpose, formal model selection procedure is needed. In fact, another equally important subject of the predictive density literature is how to select a best model from a set of competing models that might all be misspecified. We conjecture that the methods proposed in the present study can be extended to formal model comparison and model selection. We leave these topics for future study. Acknowledgements The authors are grateful to Dennis Kristensen (co‐editor) and an anonymous referee for many suggestions that greatly improved this paper. We are also grateful to Xiaohong Chen, Yi‐Ting Chen, Zaichao Du, Wilbert Kallenberg, Teresa Ledwina and Anthony Tay for their valuable input. J. Lin acknowledges support from the National Natural Science Foundation of China (grant 71501164). Footnotes " Several uniformity tests for dependent data have been proposed in the literature. For example, Munk et al. (2011) extended the data‐driven smooth test by Ledwina (1994) to time series, adjusting for the estimation of cumulative autocovariance. Corradi and Swanson (2006a) and Rossi and Sekhposyan (2015) proposed tests for uniformity robust to violations of independence that allow for dynamic misspecification under the null. " Following Diebold et al. (1998), our approach may be extended to handle h‐step‐ahead density forecasts by partitioning the generalized residuals into groups that are h‐periods apart and using Bonferroni bounds. " Although the independence test developed here is similar to that of Kallenberg and Ledwina (1999), the purposes of these two tests are rather different: our test is designed to detect the temporal dependence of a univariate time series, while their test is for the contemporaneous dependence between two univariate variables. " Kallenberg and Ledwina (1999) considered two configurations: the ‘diagonal’ test includes only terms of the form ψii,i=1,2,… ⁠, while the ‘mixed’ test allows both diagonal and off‐diagonal entries. In this study we focus on the latter, which is more general. " Note that Escanciano and Lobato (2009) and Escanciano et al. (2013) also considered a given upper bound in the selection criterion. " When Ψc contains only ψ11, the corresponding test statistic is proportional to the square of Spearman's rank correlation coefficient. " Given its asymptotic equivalence to the AIC, the method of cross‐validation can also be used for moment selection. We opt for AIC/BIC in this study mainly because of their ease of implementation. " Note that standardization of the moments is necessary for our tests as the elements of Ψ̂u are generally correlated and differ in variance. In contrast, the test in Kallenberg and Ledwina (1999) utilizes rank‐based sample moments that are free of nuisance parameters and asymptotically orthonormal under the null hypothesis. Therefore, their procedure does not require standardization of the moments. " As suggested by a referee, we also examine the case of R/n=1/2 ⁠. No comparable results are available from Hong et al. (2007) or Park and Zhang (2010). To save space, the results are reported in Appendix B. " We also experimented with the RW‐T DGP with the degrees of freedom being 4, 5 or 6. The power of our tests is good in general. To save space, these results are not reported here but they are available from the authors upon request. " Note that here we conduct the uniformity test under the null hypotheses of uniformity and independence of {Zt} ⁠. Therefore, the test results cannot be interpreted solely as the deviation from the uniformity of {Zt} ⁠. Nonetheless, they may provide useful insights. References Bai , J. ( 2003 ). Testing parametric conditional distributions of dynamic models . Review of Economics and Statistics 85 , 531 – 49 . Google Scholar Crossref Search ADS WorldCat Bera , A. K. and A. Ghosh ( 2002 ). Neyman's smooth test and its applications in econometrics. In A. Ullah, A. Wan and A. Chaturvedi (Eds.), Handbook of Applied Econometrics and Statistical Inference , 177 – 230 . Boca Raton, FL : CRC Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Bera , A. K. , A. Ghosh and Z. Xiao ( 2013 ). A smooth test for the equality of distributions . Econometric Theory 29 , 419 – 46 . Google Scholar Crossref Search ADS WorldCat Berkowitz , J. ( 2001 ). Testing density forecasts, with applications to risk management . Journal of Business and Economic Statistics 19 , 465 – 74 . Google Scholar Crossref Search ADS WorldCat Chen , X. , Y. Fan and A. Patton ( 2004 ). Simple tests for models of dependence between multiple financial time series, with applications to us equity returns and exchange rates . Working Paper 483, Financial Markets Group, London School of Economics. Chen , Y. , ‐T., ( 2011 ). Moment tests for density forecast evaluation in the presence of parameter estimation uncertainty . Journal of Forecasting 30 , 409 – 50 . Google Scholar Crossref Search ADS WorldCat Claeskens , G. and N. L. Hjort ( 2004 ). Goodness of fit via non‐parametric likelihood ratios . Scandinavian Journal of Statistics 31 , 487 – 513 . Google Scholar Crossref Search ADS WorldCat Corradi , V. and N. R. Swanson ( 2006a ). Bootstrap conditional distribution tests in the presence of dynamic misspecification . Journal of Econometrics 133 , 779 – 806 . Google Scholar Crossref Search ADS WorldCat Corradi , V. and N. R. Swanson ( 2006b ). Predictive density and conditional confidence interval accuracy tests . Journal of Econometrics 135 , 187 – 228 . Google Scholar Crossref Search ADS WorldCat Corradi , V. and N. R. Swanson ( 2006c ). Predictive density evaluation. In C. W. Granger, G. Elliot and A. Timmerman (Eds.), Handbook of Economic Forecasting , 197 – 284 . Amsterdam : Elsevier . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Corradi , V. and N. R. Swanson ( 2012 ). A survey of recent advances in forecast accuracy comparison testing, with an extension to stochastic dominance. In X. C. Chen and N. R. Swanson (Eds.), Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis: Essays in Honor of Halbert L. White Jr , 121 – 44 . Berlin : Springer . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC de Wet , T. and R. Randles ( 1987 ). On the effects of substituting parameter estimators in limiting χ2u and v statistics . Annals of Statistics 15 , 398 – 412 . Google Scholar Crossref Search ADS WorldCat Diebold , F. X. , T. A. Gunther and A. S. Tay ( 1998 ). Evaluating density forecasts with applications to financial risk management . International Economic Review 39 , 863 – 83 . Google Scholar Crossref Search ADS WorldCat Diebold , F. X. , J. Hahn and A. S. Tay ( 1999 ). Multivariate density forecast evaluation and calibration in financial risk management: high‐frequency returns on foreign exchange . Review of Economics and Statistics 81 , 661 – 73 . Google Scholar Crossref Search ADS WorldCat Escanciano , J. C. and I. N. Lobato ( 2009 ). An automatic portmanteau test for serial correlation . Journal of Econometrics 151 , 140 – 49 . Google Scholar Crossref Search ADS WorldCat Escanciano , J. C. , I. N. Lobato and L. Zhu ( 2013 ). Automatic specification testing for vector autoregressions and multivariate nonlinear time series models . Journal of Business and Economic Statistics 31 , 426 – 37 . Google Scholar Crossref Search ADS WorldCat Escanciano , J. C. and J. Olmo ( 2010 ). Backtesting parametric value‐at‐risk with estimation risk . Journal of Business and Economic Statistics 28 , 36 – 51 . Google Scholar Crossref Search ADS WorldCat Fan , J. ( 1996 ). Test of significance based on wavelet thresholding and Neyman's truncation . Journal of the American Statistical Association 91 , 674 – 88 . Google Scholar Crossref Search ADS WorldCat Genest , C. , K. Ghoudi and L. Rivest ( 1995 ). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions . Biometrika 82 , 543 – 52 . Google Scholar Crossref Search ADS WorldCat Hong , Y. , H. Li and F. Zhao ( 2007 ). Can the random walk model be beaten in out‐of‐sample density forecasts? Evidence from intraday foreign exchange rates . Journal of Econometrics 141 , 736 – 76 . Google Scholar Crossref Search ADS WorldCat Inglot , T. , W. C. Kallenberg and T. Ledwina ( 1997 ). Data‐driven smooth tests for composite hypotheses . Annals of Statistics 25 , 1222 – 50 . Google Scholar Crossref Search ADS WorldCat Inglot , T. and T. Ledwina ( 2006 ). Towards data‐driven selection of a penalty function for data‐driven Neyman tests . Linear Algebra and its Applications 417 , 124 – 33 . Google Scholar Crossref Search ADS WorldCat Jennrich , R. I. ( 1969 ). Asymptotic properties of non‐linear least squares estimators . Annals of Mathematical Statistics 40 , 633 – 43 . Google Scholar Crossref Search ADS WorldCat Kallenberg , W. C. and T. Ledwina ( 1995 ). Consistency and Monte Carlo simulation of a data‐driven version of smooth goodness‐of‐fit tests . Annals of Statistics 23 , 1594 – 608 . Google Scholar Crossref Search ADS WorldCat Kallenberg , W. C. and T. Ledwina ( 1997 ). Data‐driven smooth tests when the hypothesis is composite . Journal of the American Statistical Association 92 , 1094 – 104 . Google Scholar Crossref Search ADS WorldCat Kallenberg , W. C. and T. Ledwina ( 1999 ). Data‐driven rank tests for independence . Journal of the American Statistical Association 94 , 285 – 301 . Google Scholar Crossref Search ADS WorldCat Ledwina , T. ( 1994 ). Data‐driven version of the Neyman smooth test of fit . Journal of American Statistical Association 89 , 1000 – 5 . Google Scholar Crossref Search ADS WorldCat Lin , J. and X. Wu ( 2015 ). Smooth tests of copula specifications . Journal of Business and Economic Statistics 33 , 128 – 43 . Google Scholar Crossref Search ADS WorldCat McCracken , M. W. ( 2000 ). Robust out‐of‐sample inference . Journal of Econometrics 99 , 195 – 223 . Google Scholar Crossref Search ADS WorldCat Munk , A. , J.‐P. Stockis, J. Valeinis and G. Giese ( 2011 ). Neyman smooth goodness‐of‐fit tests for the marginal distribution of dependent data . Annals of the Institute of Statistical Mathematics 63 , 939 – 59 . Google Scholar Crossref Search ADS WorldCat Neyman , J. ( 1937 ). Smooth test for goodness of fit . Scandinavian Aktuarietidskr 20 , 149 – 99 . OpenURL Placeholder Text WorldCat Park , S. Y. and Y. Zhang ( 2010 ). Density forecast evaluation using data‐driven smooth test . Working paper, Chinese University of Hong Kong . Qiu , P. and J. Sheng ( 2008 ). A two‐stage procedure for comparing hazard rate functions . Journal of the Royal Statistical Society , Series B 70 , 191 – 208 . OpenURL Placeholder Text WorldCat Randles , R. ( 1984 ). On tests applied to residuals . Journal of American Statistical Association 79 , 349 – 54 . Google Scholar Crossref Search ADS WorldCat Rayner , J. and D. Best ( 1990 ). Smooth tests of goodness of fit: an overview . International Statistical Review 58 , 9 – 17 . Google Scholar Crossref Search ADS WorldCat Rossi , B. and T. Sekhposyan ( 2015 ). Alternative tests for correct specification of conditional predictive densities . Working paper, ICREA – Universität Pompeu Fabra. Sklar , A. ( 1959 ). Fonctions de répartition à n dimensions et leurs marges . Publications de l'Institut de Statistique de l'Université de Paris 8 , 229 – 31 . OpenURL Placeholder Text WorldCat Tay , A. S. and K. F. Wallis ( 2000 ). Density forecasting: a survey . Journal of Forecasting 19 , 235 – 54 . Google Scholar Crossref Search ADS WorldCat West , K. D. and M. W. McCracken ( 1998 ). Regression‐based tests of predictive ability . International Economic Review 39 , 817 – 40 . Google Scholar Crossref Search ADS WorldCat Appendix A: Proof of Results We first introduce some notations. For a vector a=(a1,…,ak)′ and r>0 ⁠, define |a|r=(∑i=1k|ai|r)1/r ⁠, |a|=|a|2 and |a|∞=max1≤i≤k|ai| ⁠. For a matrix A=(aij)1≤i≤m,1≤j≤n ⁠, let ∥A∥ denote the Euclidean norm and ∥A∥∞=max1≤i≤m,1≤j≤n|aij| ⁠. For convenience, define nj=n−j ⁠. Some assumptions are required to establish the asymptotic properties of the proposed selection rule and test statistics. These assumptions should hold in an arbitrary neighbourhood of θ0∈Θ ⁠. This region will then be called Θ0. We use C to denote a generic constant, which might vary from one place to another. Assumption A.1. {Yt}t=1N is generated from an unknown conditional probability density function f0t(v|Ωt−1) ⁠, where Ωt−1 is the information set available at time t−1 ⁠. Assumption A.2. The generalized residuals {Zt}t=R+1N of the density forecast model f0t(v|Ωt−1) are a sample of strictly stationary and ergodic process. Assumption A.3. Let Θ be a finite‐dimensional parameter space: (a) for each θ∈Θ ⁠, ft(v|Ωt−1,θ) is a conditional density model for {Yt}t=1N ⁠, and is a measurable function of (v,Ωt−1) ⁠; (b) ft(v|Ωt−1,θ) is twice‐continuously differentiable with respect to θ in Θ0 with probability one. Assumption A.4. The expected moment function E[ψi(G0(Zt(θ)))] is differentiable with respect to θ. Assumption A.5. θ̂t is a t‐consistent estimator for θ0. Moreover, θ̂t satisfies θ̂t−θ0=Ξ(t)−1S(t) ⁠, where: (a) Ξ(t)→Ξ ⁠, a matrix of rank q; (b) depending on the forecasting scheme, S(t) is a q×1 vector such that S(t)=(t−1)−1∑k=1t−1s0k (recursive), S(t)=R−1∑k=t−Rt−1s0k (rolling) or S(t)=R−1∑k=1Rs0k (fixed), where s0t is defined in (3.12); (c) E[s0k]=0 (a.s.). Assumption A.6. VU=IKU+ηDA−1D′ is finite and positive definite, where D, A and η are defined in (3.12) and (3.13). Assumption A.7. Denote Gt(θ,z)=Pr(Zt(θ)≤z|Ωt−1) ⁠. Gt(θ,z) is continuously differentiable in θ and z (a.s.). Moreover, Esupθ∈Θ0,z∈R|∂Gt(θ,z)∂z|0 is an arbitrary but fixed constant. By (A.1) of Escanciano and Olmo (2010), we can show supz∈R|Kn(ĉ,z)−Kn(0,z)|=op(1) for any ĉ=Op(1) ⁠. Set ĉ=maxR+1≤t≤Nt(θ̂t−θ0) ⁠. It follows that nsupz|Ĝn(z)−Gn(z)|=supz|1n∑t=R+1N{I(Ẑt≤z)−I(Z0t≤z)}|=supz|1n∑t=R+1N{Gt(θ̂t,z)−Gt(θ0,z)}|+op(1).(A.1) Next, by the mean value theorem and interchanging expectation and differentiation, we have Dn:=supz|1n∑t=R+1N{Gt(θ̂t,z)−E[Gt(θ̂t,z)]−Gt(θ0,z)+E[Gt(θ0,z)]}|=supz|1n∑t=R+1N∂Gt(θ∼t,z)∂θ−E∂Gt(θ∼t,z)∂θ(θ̂t−θ0)|, where θ∼t is between θ̂t and θ0. Note that under Assumptions A.5 and A.7, by the ULLN of Jennrich (1969, Theorem 2), we can show that Dn=op(1) ⁠. Hence, by (A.1), we have nsupz|Ĝn(z)−Gn(z)|=supz|E∂Gt(θ0,z)∂θ1n∑t=R+1N(θ̂t−θ0)|+op(1). This completes the proof of Lemma A.1. □ Proof of Theorem 3.1. Recall that Z0t=Zt(θ0) ⁠, Ẑt=Zt(θ̂t) ⁠, Ut=G0(Z0t) ⁠, Ĝn(z)=(n+1)−1∑t=R+1NI(Zt(θ̂t)≤z) and Gt(θ,z)=E[I(Zt(θ)≤z)|Ωt−1] ⁠. Let Gn(z)=(n+1)−1∑t=R+1NI(Zt(θ0)≤z) ⁠. By the mean‐value theorem, we expand Ψ̂C(j) to obtain njΨ̂C(j)=1nj∑t=R+j+1N{ΨC(Ut−j,Ut)+ΨC(1)(U∼t−j,U∼t)[Ĝn(Ẑt−j)−G0(Z0,t−j)],+ΨC(2)(U∼t−j,U∼t)[Ĝn(Ẑt)−G0(Z0,t)]}=A1n+A2n+A3n,(A.2) where U∼t=G∼n(Zt(θ∼)) is some random value between Ĝn(Ẑt) and G0(Z0t) ⁠, ΨC(l)(u1,u2)=∂ΨC(u1,u2)/∂ul ⁠, l=1,2 ⁠, and A1n=1nj∑t=R+j+1NΨC(Ut−j,Ut),(A.3) A2n=1nj∑t=R+j+1N{ΨC(1)(Ut−j,Ut)[Ĝn(Ẑt−j)−G0(Z0,t−j)]}+1nj∑t=R+j+1N{ΨC(2)(Ut−j,Ut)[Ĝn(Ẑt)−G0(Z0t)]}=B1n+B2n(A.4) A3n=1nj∑t=R+j+1N{[ΨC(1)(U∼t−j,U∼t)−ΨC(1)(Ut−j,Ut)][Ĝn(Ẑt−j)−G0(Z0,t−j)]+[ΨC(2)(U∼t−j,U∼t)−ΨC(2)(Ut−j,Ut)][Ĝn(Ẑt)−G0(Z0t)]},(A.5) where B1n and B2n are implicitly defined. In order to show that njΨ̂C(j)=nj−1/2∑t=R+j+1NΨC(Ut−j,Ut)+op(1) ⁠, it suffices to show that A2n=op(1) and A3n=op(1) ⁠. We first consider A2n ⁠. By applying Lemma A.1 to B1n defined in (A.4), we have B1n=1nj∑t=R+j+1NΨC(1)(Ut−j,Ut)[Gn(Z0,t−j)−G0(Z0,t−j)]+1nj∑t=R+j+1NΨC(1)(Ut−j,Ut)h(θ0,Z0,t−j)1nj∑s=R+j+1N(θ̂s−j−θ0)+op(1)=C1n+C2n+op(1),(A.6) where C1n and C2n are implicitly defined. Under H0C ⁠, following the proof of Proposition 2.1 of Genest et al. (1995), we can show that C1n=∫01∫01ΨC(1)(u1,u2)[I(Ut−j≤u1)−u1]du1du2+op(1).(A.7) Recall that ΨC is the tensor product of basis functions, which are orthonormal with respect to uniform distribution. We can rewrite ΨC(1)(u1,u2)=Ψ1′(u1)∘Ψ2(u2),(A.8) where Ψ1 and Ψ2 are vectors of basis functions, Ψ1′(u)=∂Ψ1(u)/∂u and ○ denotes the Hadamard product. We then have C1n=∫01Ψ1′(u1)[I(Ut−j≤u1)−u1]du1∘∫01Ψ2(u2)du2+op(1)=0+op(1),(A.9) where the last equality holds due to the orthogonality of Ψ2 with respect to the standard uniform distribution. Under Assumptions A.5 and A.7, by the ULLN, we have C2n=EZ[ΨC(1)(Ut−j,Ut)h(θ0,Z0,t−j)]1nj∑s=R+j+1N(θ̂s−j−θ0)+op(1).(A.10) By the same proof as that for (A.9), we have EZ[ΨC(1)(Ut−j,Ut)h(θ0,Z0,t−j)]=0 as E[Ψ2(U)]=0 ⁠, where U is uniformly distributed on [0, 1]. Thus, C2n=op(1) ⁠. We note here that the estimation effect of θ can be ignored because Ut are defined based on the ranks of the generalized residuals and therefore are exactly uniformly distributed. For general treatments of rank‐based tests with nuisance parameters, see Randles (1984) and de Wet and Randles (1987). By (A.6), it follows that B1n=op(1) ⁠. Similarly, we can show B2n=op(1) ⁠. Then, by (A.4), we have A2n=op(1) ⁠. Now, turn to A3n ⁠. By (A.1), it is straightforward to show that A3n=op(1) ⁠. Therefore, we have njΨ̂C(j)=nj−1/2∑t=R+j+1NΨC(Ut−j,Ut)+op(1) ⁠. Under H0C ⁠, it is easy to show that nj−1/2∑t=R+j+1NΨC(Ut−j,Ut)→dN(0,IKC) ⁠. It follows that njΨ̂C(j)→dN(0,IKC) ⁠. This completes the proof of Theorem 3.1. □ 3.2 and 3.3 Proof of Theorems The proof of Theorems 3.2 and 3.3 are similar to that of Theorem 3.5 and therefore omitted. □ Proof of Theorem 3.4. We introduce some additional notation: Υ=∂∂θ′E[ΨU(Zt−j)]|θ=θ0,Ξ=E∂∂θ′st−j|θ=θ0,V0=∑k=−∞∞E[ΨU(Z0,t−j)ΨU(Z0,t−j−k)′],D0=∑k=−∞∞E[ΨU(Z0,t−j)s0,t−j−k′],A0=∑k=−∞∞E[s0,t−js0,t−j−k′]. Note that the gradient function s0,t−j is defined in (3.12) and the matrix Ξ is defined in Assumption A.5. Recall that Ψ̂U is a KU‐dimensional vector of sample moments. By applying Lemmata 4.1 and 4.2 of West and McCracken (1998), under Assumptions A.2 and A.5, we can obtain the asymptotic normality of Ψ̂U(j) ⁠, njΨ̂U(j)→dN(0,Ω), where Ω=V0−η1(D0Ξ−1Υ′+ΥΞ−1D0′)+η2ΥΞ−1A0Ξ−1Υ′, in which η1 and η2 depend on the estimation scheme as follows: η1=0,fixed,1−(1/τ)ln(1+τ),recursive,(τ/2),rolling(τ≤1),1−(1/2τ),rolling(τ>1),η2=τ,fixed,2(1−(1/τ)ln(1+τ)),recursive,τ−(τ2/3),rolling(τ≤1),1−(1/3τ),rolling(τ>1).(A.11) Under the null hypothesis of i.i.d. and uniformity of Zt ⁠, following the proofs of Chen (2011, p. 423), by the law of iterated expectations and the martingale‐difference conditions, E[ΨU(Z0,t−j)|Ωt−j−1]=0 and E[s0,t−j|Ωt−j−1]=0 for all (t−j) ⁠, we can show that ∀k≠0 ⁠, E[ΨU(Z0,t−j)ΨU(Z0,t−j−k)′]=0 ⁠, E[ΨU(Z0,t−j)s0,t−j−k′]=0 and E[s0,t−js0,t−j−k′]=0 ⁠. Accordingly, we can obtain the simplified results V0=IKU ⁠, D0=D and A0=A ⁠, and hence Ω=IKU−η1(DΞ−1Υ′+ΥΞ−1D′)+η2ΥΞ−1AΞ−1Υ′ ⁠, where D and A are defined in (3.12). Furthermore, we can use the generalized information matrix equality to write Ξ+A=0 and Υ+D=0 under H0. Therefore, we can further simplify Ω to VU:=IKU+ηDA−1D ⁠, where η are defined in (3.13). Furthermore, because of the consistency of n/R for τ and the MLE θ̂t for θ0 under H0, we can show that Ψ̂U→pΨU and V̂U→pVU by the uniform law of large number theorem (ULLN) of Jennrich (1969, Theorem 2). Under Assumption A.6, it follows that N̂U(j)→dχKU2 ⁠. □ Proof of Theorem 3.5. Define the simplified BIC as follows SKU=min{k:N̂u,(k)(j)−klognj≥N̂u,(s)(j)−slognj,1≤k,s≤M}, where M is a given number that is sufficiently large. In order to prove Theorem 3.5(a), we need to establish that, under the null hypothesis, limn→∞Pr(KU(j)=SKU)=1,(A.12) and limn→∞Pr(SKU=1)=1.(A.13) We start by proving (A.12). Recall that Ψ̂u,(k)*(j)=V̂u,(k)−1/2Ψ̂u,(k)(j) ⁠, which is a k×1 vector. Define the event An(ζ)={nj|Ψ̂u,(k)*(j)|∞>ζlognj}. By the definition of Γ(k,n,ζ) defined in (3.17), in order to prove (A.12), it suffices to prove P(An(ζ))=o(1) ⁠. Define Ψ∼u,(k)*(j)=Vu,(k)−1/2[nj−1∑t=R+j+1NΨu,(k)(Z0,t−j)] ⁠. We can rewrite Ψ̂u,(k)*(j) as njΨ̂u,(k)*(j)=njV̂u,(k)−1/2Vu,(k)1/2Ψ∼u,(k)*(j)+Rn(k), where Rn(k)=njV̂u,(k)−1/2{Ψ̂u,(k)(j)−1nj∑t=R+j+1NΨu,(k)(Z0,t−j)}.(A.14) Because |θ̂t−θ0|=op(t−1/2) by Assumption A.5, we have Ψ̂u,(k)(j)=nj−1∑t=R+1NΨu,(k) (Z0,t−j)+op(n−1/2) ⁠. By (A.14), we obtain |Rn(k)|∞=op(1) ⁠. Therefore, for any ε>0 ⁠, we have P(|Rn(k)|∞>2−1ζlognj)≤ε/2 for a sufficiently large n. Next define the event AJ={∥V̂u,(k)−1/2Vu,(k)1/2∥≤J}, where J is a positive constant to be defined later. Under Assumption A.2, by the consistency of θ̂t and the ULLN, we have ∥V̂u,(k)−Vu,(k)∥=op(1) ⁠. Under Assumption A.6, it is straightforward to show that there exists a positive constant J such that P(AJc)<ε/2 for all ε>0 ⁠, where AJc is the complementary set of AJ ⁠. Notice that P(An(ζ))≤P(An(ζ)∩AJ)+P(AJc)≤P(nj|V̂u,(k)−1/2Vu,(k)1/2Ψ∼u,(k)*(j)|∞>2−1ζlognj,AJ)+P(|Rn(k)|∞>2−1ζlognj)+ε/2≤P(nj∥V̂u,(k)−1/2Vu,(k)1/2∥|Ψ∼u,(k)*(j)|>2−1ζlognj,AJ)+ε≤P(nj|Ψ∼u,(k)*(j)|>(2J)−1ζlognj)+ε. Because nj|Ψ∼u,(k)*(j)|=op(1) ⁠, it follows that P(nj|Ψ∼u,(k)*(j)|>(2J)−1ζlognj)→0 as n→∞ ⁠. We then have P(An(ζ))≤ε for an arbitrary ε, from which (A.12) follows. We next prove (A.13). Note that Pr(SKU=1)=1−∑k=2MPr(SKU=k) ⁠. Because SKU=k implies that dimension k ‘beats’ dimension 1 and N̂u,(1)(j)≥0 ⁠, we obtain by the definition of SKU the following Pr(SKU=k)≤Pr(N̂u,(k)(j)−klognj≥N̂u,(1)(j)−lognj)=Pr(N̂u,(k)(j)≥(k−1)lognj). It follows that Pr(N̂u,(k)(j)≥(k−1)lognj)≤Q1n+Q2n, where Q1n=Pr(Nu,(k)(j)≥(k−1)lognj/2) and Q2n=Pr(nj|Ψ̂u,(k)∗(j)−Ψ∼u,(k)∗(j)|2≥(k−1)lognj/2) ⁠. Because, under the null, Nu,(k)(j) converges to a non‐degenerate (⁠ χk2‐distributed) random variable for any k, it is straightforward to show that Q1n→0 as n→∞ ⁠. Because nj|Ψ̂u,(k)∗(j)−Ψ∼u,(k)∗(j)|=op(1) ⁠, Q2n→0 as n→∞ ⁠. It follows immediately that limn→∞Pr(SKU=1)=1 ⁠. Further note that Pr(N̂U(j)≤x)=Pr(N̂u,(1)(j)≤x)−Pr(N̂u,(1)(j)≤x,SKU≥2)+Pr(N̂U(j)≤x,SKU≥2). Because, under the null, N̂u,(1)(j)→dχ12 and Pr(SKU≥2)→p0 as n→∞ ⁠, it follows immediately that N̂U(j)→dχ12 ⁠. We now proceed to Theorem 3.5(b). Define the simplified AIC as AKU=min{k:N̂u,(k)(j)−2k≥N̂u,(s)(j)−2s,1≤k,s≤M}. In order to prove Theorem 3.5(b), we need to establish that under the alternative distribution P ⁠, limn→∞Pr(KU(j)=AKU)=1(A.15) and Pr(AKU≥S)→1.(A.16) We denote by An(ζ)c the complementary set of An(ζ) ⁠. Under the alternative distribution P ⁠, there exists an S≤M such that EP[ψS(Zt−j)]≠0 ⁠. Denote by S′ the corresponding index of ψS in the ordered set Ψu ⁠, where S′≤M ⁠. It follows that EP[Ψ∼u,(S′)∗(j)]≠0 ⁠. We then have P(An(ζ)c)≤Pnj|Ψ̂u,(S′)∗(j)|<ζlognj→0, as Ψ̂u,(S′)∗(j)=Ψ∼u,(S′)∗(j)+op(1) by the ULLN. Hence, (A.15) holds. Now, according to the ULLN, for k=1,…,S′−1 ⁠, |Ψ̂u,(k)∗(j)|2→0 and |Ψ̂u,(S′)∗(j)|2→|Ψ∼u,(S′)∗(j)|2>0 ⁠. We obtain P(AKU=k)≤P(N̂u,(k)(j)−2k≥N̂u,(S′)(j)−2S′)≤P(nj|Ψ̂u,(k)∗(j)|2≥2(k−S′)+nj|Ψ̂u,(S′)∗(j)|2)→0. Therefore, (A.16) also holds. For a generic constant C>0 ⁠, we have Pr(N̂U(j)≤C)=Pr(N̂U(j)≤C,AKU≥S′)+o(1)≤P(nj|Ψ̂u,(S′)∗|2≤C)+o(1)=o(1), where the last equality follows the ULLN. Then, N̂U(j)→∞ as n→∞ ⁠. □ Proof of Theorem 3.6. We first establish the asymptotic independence between Q̂C(j) and N̂U(j) for a given j. Recall that Ψ̂C(j)=(Ψ̂C,1(j),…,Ψ̂C,KC(j))′ and Ψ̂U(j)=(Ψ̂U,1(j),…,Ψ̂U,KU(j))′ ⁠. We show that (njΨ̂C(j)′,njΨ̂U(j)′)′ has a multivariate normal distribution and they are uncorrelated. Let X(N)=∑t=R+j+1NΨC(Ut−j,Ut)Y(N)=∑t=R+j+1NΨU(Z0,t−j)+E∂ΨU(Z0,t−j)∂θΞ−1S(t−j). Following reasoning analogous to that for Theorems 3.1 and 3.4, under Assumption A.5, we can show that njΨ̂C(j)=nj−1/2X(N)+op(1) ⁠, njΨ̂U(j)=nj−1/2Y(N)+op(1) and nj−1/2X(N)nj−1/2Y(N)→dN0KC×10KU×1,IKCΣ12Σ21VU, where 0k×l is a k‐by‐l matrix of zeros, Ik is the identity matrix of size k, IKC ⁠, VU are covariance matrices of nj−1/2X(N) and nj−1/2Y(N) ⁠, whereas Σ12=Σ21′ is the cross‐covariance matrix of nj−1/2X(N) and nj−1/2Y(N) ⁠. Now we compute Σ12. By applying the results from Lemmata 4.1 and 4.2 of West and McCracken (1998), following the proof of Theorem 3.4, we can show Σ12=limn→∞Cov(nj−1/2X(N),nj−1/2Y(N))=E[ΨC(Ut−j,Ut)ΨU(Z0,t−j)′]−η1Π0Ξ−1Υ′, where η1 is defined in (A.11) and Π0=∑k=−∞∞E[ΨC(Ut−j,Ut)s0,t−j−k′]. By using the same notation as (A.8), we rewrite ΨC(Ut−j,Ut)=Ψ1(Ut−j)∘Ψ2(Ut) ⁠. Therefore, under the null hypothesis that {Zt(θ0)} is i.i.d. uniform, by applying the law of iterative expectation and using the fact E[Ψ2(Ut)]=0 ⁠, we can show that E[ΨC(Ut−j,Ut)ΨU(Z0,t−j)′]=0.(A.17) Define Π=E[ΨC(Ut−j,Ut)s0,t−j′] ⁠. Following the same argument as the proof in Theorem 3.4, we can show that Π0=Π ⁠. Note that s0,t−j is a function of Ωt−j ⁠. Following the same arguments used to derive (A.17), we can show that Π=0 ⁠. Combining with (A.17), we have shown that Σ21=0KC×KU ⁠. Similarly, we can show that Σ21=0KU×KC ⁠. So, (nj−1/2X(N)′,nj−1/2Y(N)′)′ has a multivariate normal distribution and they are uncorrelated. It follows that (njΨ̂C(j)′,njΨ̂U(j)′)′ converges in distribution to a multivariate normal distribution and they are uncorrelated. Therefore, njΨ̂C(j) and njΨ̂U(j) are asymptotically independent of each other, and so are Q̂C(j) and N̂U(j) ⁠. Because this result holds for an arbitrary j, the asymptotic independence between the (automatic) portmanteau test ŴQ(p) (ŴQ(p∼)) and N̂U(p) follows immediately. The proof is now finished. □ Appendix B: Additional Simulation Results In this appendix, we report results of some additional simulations suggested by a referee. First, we consider the case of R/n=1/2 ⁠. The results under the fixed estimation scheme are reported in Table Table B.1.. We set the out‐of‐sample evaluation period n=100 ⁠, 250, 500 and 1,000. The overall size and power performances are satisfactory and improve with sample size. Second, we examine the empirical sizes of SQT under the rolling scheme. As shown in Table Table B.2., the tests have better overall size under the rolling scheme compared with those under the fixed scheme. Table B.1. Empirical sizes and powers for R/n=1/2 (fixed estimation scheme) DGP . RW‐N . GARCH‐N . RW‐T . GARCH‐N . RS‐T . NULL . RW‐N . GARCH‐N . RW‐N . RW‐N . RW‐N . n=100 Q(1) 4.1 8.5 34.2 16.1 31.1 W(5) 3.5 7.5 34.1 19.8 32.2 W(10) 4.3 6.5 34.0 18.4 31.8 W(20) 4.2 6.1 34.2 17.2 31.3 WQ(p∼) 4.1 6.0 34.0 16.2 30.5 n=250 Q(1) 4.0 8.5 75.9 40.2 72.9 W(5) 4.5 8.4 76.2 46.9 73.5 W(10) 4.6 8.0 75.9 45.4 73.4 W(20) 4.5 7.6 76.1 41.3 73.1 WQ(p∼) 4.7 7.6 75.9 43.6 73.0 n=500 Q(1) 4.8 8.1 96.2 66.2 95.8 W(5) 4.7 8.1 96.2 75.9 95.9 W(10) 4.8 9.0 96.1 74.0 95.8 W(20) 4.8 7.6 96.1 67.4 95.6 WQ(p∼) 4.4 8.7 96.2 73.8 95.8 n= 1,000 Q(1) 5.0 8.9 99.97 92.0 99.97 W(5) 4.5 9.3 99.97 96.5 99.97 W(10) 5.0 9.0 99.97 96.1 99.97 W(20) 4.3 7.9 99.97 93.6 99.97 WQ(p∼) 4.3 8.7 99.97 95.5 99.97 DGP . RW‐N . GARCH‐N . RW‐T . GARCH‐N . RS‐T . NULL . RW‐N . GARCH‐N . RW‐N . RW‐N . RW‐N . n=100 Q(1) 4.1 8.5 34.2 16.1 31.1 W(5) 3.5 7.5 34.1 19.8 32.2 W(10) 4.3 6.5 34.0 18.4 31.8 W(20) 4.2 6.1 34.2 17.2 31.3 WQ(p∼) 4.1 6.0 34.0 16.2 30.5 n=250 Q(1) 4.0 8.5 75.9 40.2 72.9 W(5) 4.5 8.4 76.2 46.9 73.5 W(10) 4.6 8.0 75.9 45.4 73.4 W(20) 4.5 7.6 76.1 41.3 73.1 WQ(p∼) 4.7 7.6 75.9 43.6 73.0 n=500 Q(1) 4.8 8.1 96.2 66.2 95.8 W(5) 4.7 8.1 96.2 75.9 95.9 W(10) 4.8 9.0 96.1 74.0 95.8 W(20) 4.8 7.6 96.1 67.4 95.6 WQ(p∼) 4.4 8.7 96.2 73.8 95.8 n= 1,000 Q(1) 5.0 8.9 99.97 92.0 99.97 W(5) 4.5 9.3 99.97 96.5 99.97 W(10) 5.0 9.0 99.97 96.1 99.97 W(20) 4.3 7.9 99.97 93.6 99.97 WQ(p∼) 4.3 8.7 99.97 95.5 99.97 Note: This table reports the empirical sizes and powers of the SQT with Q̂C(1) ⁠, ŴQ(p) ⁠, p=5,10,20 or ŴQ(p∼) in the first stage. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab Table B.1. Empirical sizes and powers for R/n=1/2 (fixed estimation scheme) DGP . RW‐N . GARCH‐N . RW‐T . GARCH‐N . RS‐T . NULL . RW‐N . GARCH‐N . RW‐N . RW‐N . RW‐N . n=100 Q(1) 4.1 8.5 34.2 16.1 31.1 W(5) 3.5 7.5 34.1 19.8 32.2 W(10) 4.3 6.5 34.0 18.4 31.8 W(20) 4.2 6.1 34.2 17.2 31.3 WQ(p∼) 4.1 6.0 34.0 16.2 30.5 n=250 Q(1) 4.0 8.5 75.9 40.2 72.9 W(5) 4.5 8.4 76.2 46.9 73.5 W(10) 4.6 8.0 75.9 45.4 73.4 W(20) 4.5 7.6 76.1 41.3 73.1 WQ(p∼) 4.7 7.6 75.9 43.6 73.0 n=500 Q(1) 4.8 8.1 96.2 66.2 95.8 W(5) 4.7 8.1 96.2 75.9 95.9 W(10) 4.8 9.0 96.1 74.0 95.8 W(20) 4.8 7.6 96.1 67.4 95.6 WQ(p∼) 4.4 8.7 96.2 73.8 95.8 n= 1,000 Q(1) 5.0 8.9 99.97 92.0 99.97 W(5) 4.5 9.3 99.97 96.5 99.97 W(10) 5.0 9.0 99.97 96.1 99.97 W(20) 4.3 7.9 99.97 93.6 99.97 WQ(p∼) 4.3 8.7 99.97 95.5 99.97 DGP . RW‐N . GARCH‐N . RW‐T . GARCH‐N . RS‐T . NULL . RW‐N . GARCH‐N . RW‐N . RW‐N . RW‐N . n=100 Q(1) 4.1 8.5 34.2 16.1 31.1 W(5) 3.5 7.5 34.1 19.8 32.2 W(10) 4.3 6.5 34.0 18.4 31.8 W(20) 4.2 6.1 34.2 17.2 31.3 WQ(p∼) 4.1 6.0 34.0 16.2 30.5 n=250 Q(1) 4.0 8.5 75.9 40.2 72.9 W(5) 4.5 8.4 76.2 46.9 73.5 W(10) 4.6 8.0 75.9 45.4 73.4 W(20) 4.5 7.6 76.1 41.3 73.1 WQ(p∼) 4.7 7.6 75.9 43.6 73.0 n=500 Q(1) 4.8 8.1 96.2 66.2 95.8 W(5) 4.7 8.1 96.2 75.9 95.9 W(10) 4.8 9.0 96.1 74.0 95.8 W(20) 4.8 7.6 96.1 67.4 95.6 WQ(p∼) 4.4 8.7 96.2 73.8 95.8 n= 1,000 Q(1) 5.0 8.9 99.97 92.0 99.97 W(5) 4.5 9.3 99.97 96.5 99.97 W(10) 5.0 9.0 99.97 96.1 99.97 W(20) 4.3 7.9 99.97 93.6 99.97 WQ(p∼) 4.3 8.7 99.97 95.5 99.97 Note: This table reports the empirical sizes and powers of the SQT with Q̂C(1) ⁠, ŴQ(p) ⁠, p=5,10,20 or ŴQ(p∼) in the first stage. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab Table B.2. Empirical sizes of the sequential tests under the rolling estimation scheme DGP . Q(1) . W(5) . W(10) . W(20) . W(p∼) . n=250 RW‐N R/n=1 5.5 5.8 5.4 5.2 5.6 R/n=2 5.3 5.4 5.2 5.4 5.3 R/n=3 5.5 5.5 5.6 6.0 5.9 GARCH‐N R/n=1 7.0 7.7 7.9 8.1 7.6 R/n=2 5.1 4.9 4.9 5.2 5.1 R/n=3 5.7 5.2 5.1 6.1 5.3 n=500 RW‐N R/n=1 5.5 5.3 5.5 5.9 5.8 R/n=2 4.9 5.1 4.8 5.3 5.2 R/n=3 5.2 5.2 5.8 5.9 5.6 GARCH‐N R/n=1 5.4 5.3 5.6 5.6 5.7 R/n=2 5.2 4.8 5.6 5.7 5.4 R/n=3 5.5 5.5 5.6 5.8 5.8 n= 1,000 RW‐N R/n=1 5.7 5.9 5.8 5.7 5.4 R/n=2 5.2 4.8 5 5.1 4.7 R/n=3 4.6 4.9 4.8 4.8 5.5 GARCH‐N R/n=1 4.9 5.2 5.5 5.7 5.3 R/n=2 4.6 5.0 4.9 5.4 4.8 R/n=3 5.2 5.2 5.6 5.8 4.7 DGP . Q(1) . W(5) . W(10) . W(20) . W(p∼) . n=250 RW‐N R/n=1 5.5 5.8 5.4 5.2 5.6 R/n=2 5.3 5.4 5.2 5.4 5.3 R/n=3 5.5 5.5 5.6 6.0 5.9 GARCH‐N R/n=1 7.0 7.7 7.9 8.1 7.6 R/n=2 5.1 4.9 4.9 5.2 5.1 R/n=3 5.7 5.2 5.1 6.1 5.3 n=500 RW‐N R/n=1 5.5 5.3 5.5 5.9 5.8 R/n=2 4.9 5.1 4.8 5.3 5.2 R/n=3 5.2 5.2 5.8 5.9 5.6 GARCH‐N R/n=1 5.4 5.3 5.6 5.6 5.7 R/n=2 5.2 4.8 5.6 5.7 5.4 R/n=3 5.5 5.5 5.6 5.8 5.8 n= 1,000 RW‐N R/n=1 5.7 5.9 5.8 5.7 5.4 R/n=2 5.2 4.8 5 5.1 4.7 R/n=3 4.6 4.9 4.8 4.8 5.5 GARCH‐N R/n=1 4.9 5.2 5.5 5.7 5.3 R/n=2 4.6 5.0 4.9 5.4 4.8 R/n=3 5.2 5.2 5.6 5.8 4.7 Note: This table reports the empirical sizes of the SQT under the rolling scheme with Q̂C(1) ⁠, ŴQ(p) ⁠, p=5,10,20 or ŴQ(p∼) in the first stage. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab Table B.2. Empirical sizes of the sequential tests under the rolling estimation scheme DGP . Q(1) . W(5) . W(10) . W(20) . W(p∼) . n=250 RW‐N R/n=1 5.5 5.8 5.4 5.2 5.6 R/n=2 5.3 5.4 5.2 5.4 5.3 R/n=3 5.5 5.5 5.6 6.0 5.9 GARCH‐N R/n=1 7.0 7.7 7.9 8.1 7.6 R/n=2 5.1 4.9 4.9 5.2 5.1 R/n=3 5.7 5.2 5.1 6.1 5.3 n=500 RW‐N R/n=1 5.5 5.3 5.5 5.9 5.8 R/n=2 4.9 5.1 4.8 5.3 5.2 R/n=3 5.2 5.2 5.8 5.9 5.6 GARCH‐N R/n=1 5.4 5.3 5.6 5.6 5.7 R/n=2 5.2 4.8 5.6 5.7 5.4 R/n=3 5.5 5.5 5.6 5.8 5.8 n= 1,000 RW‐N R/n=1 5.7 5.9 5.8 5.7 5.4 R/n=2 5.2 4.8 5 5.1 4.7 R/n=3 4.6 4.9 4.8 4.8 5.5 GARCH‐N R/n=1 4.9 5.2 5.5 5.7 5.3 R/n=2 4.6 5.0 4.9 5.4 4.8 R/n=3 5.2 5.2 5.6 5.8 4.7 DGP . Q(1) . W(5) . W(10) . W(20) . W(p∼) . n=250 RW‐N R/n=1 5.5 5.8 5.4 5.2 5.6 R/n=2 5.3 5.4 5.2 5.4 5.3 R/n=3 5.5 5.5 5.6 6.0 5.9 GARCH‐N R/n=1 7.0 7.7 7.9 8.1 7.6 R/n=2 5.1 4.9 4.9 5.2 5.1 R/n=3 5.7 5.2 5.1 6.1 5.3 n=500 RW‐N R/n=1 5.5 5.3 5.5 5.9 5.8 R/n=2 4.9 5.1 4.8 5.3 5.2 R/n=3 5.2 5.2 5.8 5.9 5.6 GARCH‐N R/n=1 5.4 5.3 5.6 5.6 5.7 R/n=2 5.2 4.8 5.6 5.7 5.4 R/n=3 5.5 5.5 5.6 5.8 5.8 n= 1,000 RW‐N R/n=1 5.7 5.9 5.8 5.7 5.4 R/n=2 5.2 4.8 5 5.1 4.7 R/n=3 4.6 4.9 4.8 4.8 5.5 GARCH‐N R/n=1 4.9 5.2 5.5 5.7 5.3 R/n=2 4.6 5.0 4.9 5.4 4.8 R/n=3 5.2 5.2 5.6 5.8 4.7 Note: This table reports the empirical sizes of the SQT under the rolling scheme with Q̂C(1) ⁠, ŴQ(p) ⁠, p=5,10,20 or ŴQ(p∼) in the first stage. The nominal size is 0.05. R/n denotes the estimation–evaluation ratio. Results are based on 3,000 replications. Open in new tab © 2017 Royal Economic Society. TI - A sequential test for the specification of predictive densities JF - Econometrics Journal DO - 10.1111/ectj.12085 DA - 2017-06-01 UR - https://www.deepdyve.com/lp/oxford-university-press/a-sequential-test-for-the-specification-of-predictive-densities-JOAkbglLrs SP - 190 VL - 20 IS - 2 DP - DeepDyve ER -