Generalized spectral testing for multivariate continuous-time modelsBin Chen a,∗, Yongmiao Hong b,c
a Department of Economics, University of Rochester, Rochester, NY, 14627, United Statesb Department of Economics & Department of Statistical Science, Cornell University, Ithaca, NY 14850, United Statesc Wang Yanan Institute for Studies in Economics (WISE) & MOE Key Laboratory in Econometrics, Xiamen University, Xiamen 361005, China
a r t i c l e i n f o
Article history:Received 17 December 2008Received in revised form21 May 2011Accepted 1 June 2011Available online 15 June 2011
JEL classification:C4E4G0
Keywords:Affine jump–diffusion modelConditional characteristic functionDiscrete-time distribution modelGeneralized cross-spectrumLévy processesModel specification testMultivariate continuous-time model
a b s t r a c t
We develop an omnibus specification test for multivariate continuous-timemodels using the conditionalcharacteristic function, which often has a convenient closed-form or can be accurately approximated formany multivariate continuous-time models in finance and economics. The proposed test fully exploitsthe information in the joint conditional distribution of underlying economic processes and hence isexpected to have good power in amultivariate context. A class of easy-to-interpret diagnostic proceduresis supplemented to gauge possible sources of model misspecification. Our tests are also applicable todiscrete-time distribution models. Simulation studies show that the tests provide reliable inference infinite samples.
Multivariate continuous-time models have proved to be versa-tile and productive tools in finance and economics (e.g., Andersenet al. (2002), Chernov et al. (2003), Dai and Singleton (2003), Panand Singleton (2008), Jarrow et al. (2010), and Piazzesi (2010)).Compared with that of discrete-time models, the econometricanalysis of continuous-time models is often challenging. In thepast decade or so, substantial progress has been made in devel-oping estimationmethods for continuous-timemodels.1 However,relatively little effort has been devoted to specification and eval-uation of continuous-time models. A continuous-time model es-sentially specifies the transition density of underlying processes.Model misspecification generally renders inconsistent parameterestimators and their variance–covariancematrix estimators, yield-ing misleading conclusions in inference and hypothesis testing.
(Y. Hong).1 Sundaresan (2001) points out that ‘‘perhaps the most significant development
in the continuous-time field during the last decade has been the innovations ineconometric theory and in the estimation techniques for models in continuoustime.’’ For other reviews of this literature, see (e.g.) Tauchen (1997) and Ait-Sahalia(2007).
Correct model specification is also crucial for valid economic inter-pretations of model parameters. More importantly, a misspecifiedmodel can lead to large errors in pricing, hedging and managingrisk. However, economic theories usually do not suggest any con-crete functional form for continuous-time models; the choice of amodel is somewhat arbitrary. A prime example of this practice isthe pricing and hedging literature, where continuous-time mod-els are generally assumed to have a functional form that resultsin a closed-form pricing formula. The models, though convenient,are often incorrect or suboptimal. To avoid this pitfall, the devel-opment of reliable specification tests for continuous-time modelsis necessary.
In a pioneer paper, Ait-Sahalia (1996a) develops a nonparamet-ric test for univariate diffusion models. By observing that the driftand diffusion functions completely characterize the stationary (i.e.,marginal) density of a diffusion model, Ait-Sahalia (1996a) com-pares the model-implied stationary density with a smoothed ker-nel density estimator based on discretely sampled data.2 Gao andKing (2004) develop a simulation procedure to improve the finitesample performance of Ait-Sahalia’s (1996a) test. These tests are
2 Ait-Sahalia (1996a) also proposes a transition density-based test that exploitsthe ‘‘transition discrepancy’’ characterized by the forward and backward Kol-mogorov equations, although the marginal density-based test is most emphasized.
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 269
convenient to implement. Nevertheless, they may pass over a mis-specified model that has a correct stationary density.
Hong and Li (2005) develop an omnibus nonparametricspecification test for continuous-time models. The test uses thetransition density,which depicts the full dynamics of a continuous-time process. When a univariate continuous-time model iscorrectly specified, the probability integral transform (PIT) of datavia the model-implied transition density is i.i.d. U[0, 1]. Hong andLi (2005) check the joint hypothesis of i.i.d. U[0, 1] by using asmoothed kernel estimator of the joint density of the PIT series.
However, Hong and Li’s (2005) approach cannot be extendedto a multivariate model. The PIT series with respect to a model-implied multivariate transition density are no longer i.i.d U[0, 1],even if the model is correctly specified. In an application, Hongand Li (2005) evaluate multivariate affine term structure mod-els (ATSMs) for interest rates by using the marginal PIT for eachstate variable. This is analogous to Singleton’s (2001) conditionalcharacteristic function-based limited-information maximum like-lihood (LML-CCF) estimator in a related context, which is basedon the conditional density function of an individual state variable.As pointed out by Singleton (2001), ‘‘the LML-CCF estimator fullyexploits the information in the conditional likelihood function ofthe individual yj,t+1, but not the information in the joint condi-tional distribution of yt+1.’’ This generally leads to an asymptoticefficiency loss in estimation. Similarly, Hong and Li’s (2005) test,when applied to each state variable of a multivariate process, mayfail to detect misspecification in the joint dynamics of state vari-ables. In particular, the test may easily overlook misspecificationin the conditional correlations between state variables. Moreover,the use of the transition density may not be computationally con-venient because the transition densities of most continuous-timemodels have no closed-form.
Gallant and Tauchen (1996) propose a class of Efficient Methodof Moments (EMM) tests that can be used to test multivariatecontinuous-time models. They propose a χ2 test for modelmisspecification, and a class of appealing diagnostic t-tests thatcan be used to gauge possible sources formodel failure. Since thesetests are by-products of the EMM algorithm, they cannot be usedwhen the model is estimated by other methods. This may limit thescope of these tests’ otherwise useful applications. Bhardwaj et al.(2008) consider a simulation-based test, which is an extensionof Andrews’ (1997) conditional Kolmogorov test, for multivariatediffusion processes. The limiting distribution of the test statisticis not nuisance parameter free and hence asymptotically criticalvalues must be obtained via the block bootstrap, which may betime-consuming.
There have been other tests for univariate diffusion models inthe recent literature. Ait-Sahalia et al. (2009) propose some tests bycomparing the model-implied transition density and distributionfunction with their nonparametric counterparts. Chen et al. (2008)also propose a transition density-based test using a nonparametricempirical likelihood approach. Li (2007) focuses on the parametricspecification of the diffusion function by measuring the distancebetween the model-implied diffusion function and its kernelestimator. These approaches could be extended to multivariatecontinuous-time models. However, all these tests maintain theMarkov assumption for the data generating process (DGP), andconsider the finite order lag only. If the DGP is non-Markov, thesetests may miss some dynamic misspecifications.
This paper proposes a newapproach to testing the adequacy of amultivariate continuous-timemodel that uses the full informationof the joint dynamics of state variables. In a multivariate context,modeling the joint dynamics of state variables is important inmany applications (e.g., Alexander (1998)). For example, as theconditional correlations between asset returns change over time,the specificweight allocated to each assetwithin a portfolio should
be adjusted accordingly. Similarly, hedging requires knowledge ofconditional correlations between the returns on different assetswithin the hedge. Conditional correlations are also important inpricing structured products such as rainbow options, which arebased onmultiple underlying assetswhose prices are correlated. Inthe term structure literature,models of interest rate term structureimpose dynamic cross-sectional restrictions, as implied by theno-arbitrage condition on bond yields of different maturities.This joint dynamics can be used to investigate the transmissionmechanism that transfers the impact of government policy fromspot rates to longer-term yields. There is a conflict between theflexibilities of modeling instantaneous conditional variances andinstantaneous conditional correlations of bond yields. Dai andSingleton (2000) find that not only do swap rate data consistentlycall for negative conditional correlations between bond yields, butalso factor correlations help explain the shape of the term structureof bond yields’ volatilities. Indeed, as Engle (2002) points out,‘‘the quest for reliable estimates of correlations between financialvariables has been the motivation for countless academic articles,practitioner conferences and Wall Street research’’.
There has been a long history of using the CF in estimation andhypotheses testing in statistics and econometrics. To name a few,Koutrouvelis (1980) constructs a chi-squared goodness-of-fit testfor simple null hypotheses with empirical characteristic function(ECF). Fan (1997) takes the CF approach to testing multivariatedistributions. But both tests maintain the i.i.d. assumption andhence are not suitable for the time series data. Recently, Su andWhite (2007) test conditional independence by comparing theunrestricted and restricted CCFs via a kernel regression. All aboveworks deal with discrete-time models, in recent years, the CFapproach has attracted an increasing attention in the continuous-time literature. For most continuous-time models, the transitiondensity has no closed-form, whichmakes estimation of and testingfor continuous-time models rather challenging. However, for ageneral class of affine jump–diffusion (AJD) models (e.g., Duffieet al. (2000)) and time-changed Lévy processes (e.g., Carr and Wu(2003, 2004)), the CCF has a closed-form as an exponential-affinefunction of state variables up to a system of ordinary differentialequations. This fact has been exploited to develop new estimationmethods for multifactor continuous-time models in the literature.Specifically, Chacko and Viceira (2003) suggest a spectral GMMestimator based on the average of the differences between theECF and the model-implied CF. Jiang and Knight (2002) derive theunconditional joint CF of an AJD model and use it to develop someGMM and ECF estimation procedures. Singleton (2001) proposesboth time-domain estimators based on the Fourier transformof theCCF, and frequency-domain estimators directly based on the CCF.By extending Carrasco and Florens (2000), Carrasco et al. (2007)propose GMM estimators with a continuum of moment conditions(C-GMM) via the CF. All these estimation methods differ in theiruses of the conditional information set. As of yet, no attempt hasbeen made in the literature to use the CF to test continuous-time models, although Carrasco et al. (2007) do mention that ‘‘thefuture work will have to refine our results on estimation of non-Markovian processes and latent states as well as develop tests inthe framework of CF-based continuum of moment conditions’’.
In light of the convenient closed-form of the CCF of AJDmodels,we provide a new test of the adequacy of multivariate continuous-time models. The multivariate continuous-time models caninclude jumps and the underlying DGPs of observable statevariables need not be Markov. Naturally, as a special case, ourtest can be used to check univariate continuous-time models.Compared with the existing tests for continuous-time models inthe literature, our approach has several main advantages.
First, because the CCF is the Fourier transform of the transitiondensity, our omnibus test fully exploits the information in the
270 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
joint transition density of state variables rather than only theinformation in the transition densities of individual state variables.Thus, it can detect misspecifications in the joint transition densityeven if the transition density of each state variable is correctlyspecified. When the underlying multivariate continuous-timeprocess is Markov, our omnibus test is consistent against anymodel misspecification. This is unattainable by some existing testsfor multivariate continuous-timemodels. Moreover, we have useda novel generalized cross-spectral approach, which embeds theCCF in a spectral framework, thus enjoying the appealing featuresof spectral analysis. For example, it checks many lags. This isparticularly useful when the DGP is non-Markov.
Second, besides the omnibus test, we propose a class ofdiagnostic tests by differentiating the generalized cross-spectrumof the state vector. These tests can evaluate howwell a continuous-time model captures various specific aspects of the joint dynamicsand they are easy to interpret. In particular, these tests can providevaluable information about neglected dynamics in conditionalmeans, conditional variances, and conditional correlations of statevariables, respectively. Therefore, they complement Gallant andTauchen’s (1996) popular EMM-based individual t-tests. All ouromnibus test and diagnostic tests are derived from a unifiedframework.
Third, our tests are applicable to a wide variety of continuous-time models and discrete-time multivariate distribution models,since we impose regularity conditions on the CCF of discretelyobserved samples with some fixed sample frequency, rather thanon stochastic differential equations (SDEs). By using the CCFto characterize the adequacy of a model, our tests are mostconvenient whenever the model has a closed-form CCF and manypopular continuous-time models in finance (e.g., the class ofmultivariate AJD models and the class of time-changed Lévyprocesses) have a closed-form CCF, although they have no closed-form transition density. Of course, our tests can also evaluatethe multivariate continuous-time models with no closed-formCCF. In this case, we need to recover the model-implied CCF byusing inverse Fourier transforms or simulation methods. Unliketests based on CFs in the statistical literature, which often havenonstandard asymptotic distributions, our tests have a convenientnull asymptotic N(0, 1) distribution.
Fourth, we do not require a particular estimation method.Any
√T -consistent parametric estimators can be used. Parameter
estimation uncertainty does not affect the asymptotic distributionof our test statistics. One can proceed as if true model parameterswere known and equal to parameter estimates. This makes ourtests easy to implement, particularly in view of the notoriousdifficulty of estimating multivariate continuous-time models. Theonly inputs needed to calculate the test statistics are the discretelyobserved data and the model-implied CCF.
In Section 2, we introduce the framework, state the hypotheses,and characterize the correct specification of a multivariatecontinuous-time model. In Section 3, we propose a generalizedcross-spectral omnibus test, and in Section 4 we derive theasymptotic null distribution of our omnibus test and discuss itsasymptotic power property. In Section 5, we develop a class ofgeneralized cross-spectral derivative tests that focuses on variousspecific aspects of the joint dynamics of a time series model. InSection 6, we assess the reliability of the asymptotic theory infinite samples by simulation. Section 7 concludes our work. Allmathematical proofs are collected in Appendix. A GAUSS code toimplement our tests is available from the authors upon request.Throughout the paper, we will use C to denote a generic boundedconstant, ‖·‖ for the Euclidean norm, and A∗ for the complexconjugate of A.
2. Hypotheses of interest
For concreteness, we focus on a multivariate continuous-timesetup.3 For a given complete probability space (Ω,F , P) and aninformation filtration (Ft), we assume that aN×1 state vectorXt isa continuous-time DGP in some state space D ⊂ RN . We permitbut do not require Xt to be Markov, which is often assumed in thecontinuous-time modeling in finance and macroeconomics.4
In financial modeling, the following classM of continuous-timemodels is often used to capture the dynamics of Xt
where Wt is an N × 1 standard Brownian motion in RN , 2 is afinite-dimensional parameter space, µ : D × 2 → RN is a driftfunction (i.e., instantaneous conditionalmean),σ : D×2 → RN×N
is a diffusion function (i.e., instantaneous conditional standarddeviation), and Jt is a pure jump process whose jump size followsa probability distribution ν: D × 2 → R+ and whose jumptimes arrive with intensity λ : D × 2 → R+.6 We allow somestate variables to be unobservable. One example is the stochasticvolatility (SV) model, where the volatility, which is a proxy for theinformation inflow, is a latent process.
The setup (2.1) is a general multivariate specification thatnests most existing continuous-time models in finance andeconomics. For example, suppose we restrict the drift µ (·, ·),the instantaneous covariance matrix σ (·, ·) σ (·, ·)′ and the jumpintensity λ (·, ·) to be affine functions of the state vector Xt ;namely,
µ(Xt , θ) = K0 + K1Xt ,[σ(Xt , θ)σ(Xt , θ)
′]jl = [H0]jl + [H1]jlXt ,
j, l = 1, . . . ,N,λ (Xt , θ) = L0 + L′
1Xt ,
(2.2)
where K0 ∈ RN , K1 ∈ RN×N , H0 ∈ RN×N , H1 ∈ RN×N×N , L0 ∈ R,and L1 ∈ RN are unknown parameters. Then we obtain the class ofAJD models of Duffie et al. (2000).
It is well known that for a continuous-timemodel characterizedby a SDE, the specification of the drift µ(Xt , θ), the diffusionσ(Xt , θ) and the jump process Jt(θ) completely determines thejoint transition density of Xt . We use p(x, t|Fs, θ) to denote themodel-implied transition density of Xt = x given Fs, where s < t .Suppose Xt has a true transition density, say p0(x, t|Fs). Then thecontinuous-time model is correctly specified for the full dynamicsof Xt if there exists some parameter value θ0 ∈ 2 such that
H0 : p(·,t|Fs, θ0) = p0(·, t|Fs)
almost surely (a.s.) and for all t, s, s < t. (2.3)
3 Our test is applicable to both continuous-time and discrete-time models butwe focus on a continuous-time setup due to the following reasons. Our approachis most convenient when the CCF has a closed-form andmany popular continuous-timemodels in finance have no closed-form transition density, but do have a closed-form CCF. On the other hand, to our knowledge, no CF-based test is available tocheck the specification of continuous-time models, although the CF approach hasbeen used in the estimation of continuous-time models. Hence, our test nicely fillsthe gap in the literature.4 However, Easley and O’Hara (1992) develop an economic structural model and
show that financial time series, such as prices and volumes, are likely non-Markov.5 Despite the fact that most continuous-time models characterized by SDEs in
the literature are Markov, it is still important to allow Xt to be non-Markov. Eventhe null model is Markov, to allow for non-Markov DGPs under the alternative willensure the power of the test against a wider range of misspecification, particularly,dynamic misspecification. On the other hand, the model may be Markov butinvolves some latent variables, as is the case of SV models. As a result, observablestate variables themselves will be non-Markov, even under the null.6 We assume that the functionsµ, σ , ν andλ are regular enough to have a unique
strong solution to (2.1). See (e.g.) Ait-Sahalia (1996a), Duffie et al. (2000) andGenon-Catalot et al. (2000) for more discussions.
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 271
Alternatively, if for all θ ∈ 2, we have
HA : p(·, t|Fs, θ) = p0(·, t|Fs)
for some s < t with positive probability measure, (2.4)
then the continuous-time model is misspecified for the fulldynamics of Xt .
The transition density characterization can be used to testcorrect specification of a continuous-time model. When Xt isunivariate, Hong and Li (2005) propose a kernel-based test for acontinuous-time model by checking whether the PIT
Zt(θ0) ≡
∫ Xt
−∞
p(x, t|It−1, θ0)dx ∼ i.i.d.U[0, 1], (2.5)
which holds under H0, where It−1 = Xt−1,Xt−21, . . . is theinformation set available at time t−1, where1 is a fixed samplinginterval for an observed sample. The i.i.d. U[0, 1] property forthe PIT has also been used in other contexts (e.g., Diebold et al.(1998)). However, there are some limitations to this approach. Forexample, for most continuous-time diffusion models (except suchsimple diffusion models as Vasicek’s (1977) model), the transitiondensities, have no closed-form. Most importantly, the PIT cannotbe applied to the multivariate joint transition density p(x, t|Fs, θ),because when N > 1,
Zt(θ0) =
∫ X1t
−∞
· · ·
∫ XNt
−∞
p(x, t|It−1, θ0)dx (2.6)
is no longer i.i.d. U[0, 1] even if H0 holds. Hong and Li (2005)suggest using the PIT for each state variable. This is valid, butit does not make full use of the information contained in thejoint distribution of Xt . In particular, it may miss important modelmisspecification in the joint dynamics of Xt . For example, considerthe DGP
dX1,tX2,t
=
κ11 0κ21 κ22
θ1 − X1,tθ2 − X2,t
dt
+
σ11 00 σ22
dW1,tW2,t
,
whereW1,t ,W2,t
are independent standard Brownian motions
and κ21 = 0. Suppose we fit the data using the model
dX1,tX2,t
=
κ11 00 κ22
θ1 − X1,tθ2 − X2,t
dt
+
σ11 00 σ22
dW1,tW2,t
.
Then this model is misspecified because it ignores correlations indrift. Now, following Hong and Li (2005), we calculate the gener-alized residuals
Z1,t , Z2,t , Z1,t−1, Z2,t−1, . . .
, where Z1,t and Z2,t
are the PITs of X1,t and X2,t with respect to the conditional densitymodels p(X1,t , t|Xt−1, X2,t , θ) and p(X2,t , t|Xt−1, θ) respectively,and θ = (κ11, κ22, θ1, θ2, σ11, σ22)
′. Then Hong and Li’s (2005) testwill have no power because each of these PITs is an i.i.d. U [0, 1]sequence respectively.
As the Fourier transform of the transition density, the CCF cancapture the full dynamics of Xt . Let ϕ(u, t, |Fs, θ) be the model-implied CCF of Xt , conditional on Fs at time s < t; that is,
ϕ(u, t|Fs, θ) ≡ Eθ
exp
iu′Xt
|Fs
=
∫RN
expiu′x
p(x, t|Fs, θ)dx, u ∈ RN , i =
√−1, (2.7)
where Eθ (·|Fs) denotes the conditional expectation under themodel-implied transition density p(x, t|Fs, θ). Note that for aMarkov model, the filtration Fs can be replaced by Xs.
Given the equivalence between the transition density and theCCF, we can write the hypotheses of interest H0 in (2.3) versus HAin (2.4) as follows:
H0 : Eexp
iu′Xt
|Fs
= ϕ(u, t|Fs, θ0) a.s. for all u ∈ RN
and for some θ0 ∈ 2 (2.8)
versus
HA : Eexp
iu′Xt
|Fs
= ϕ(u, t|Fs, θ)
with positive probability for all θ ∈ 2. (2.9)
Suppose we have a discrete random sample XtT1t=1 of size T .
For notational simplicity, we set 1 = 1 below, where time ismeasured in units of the sampling interval of data.7 Also,wedenoteIt−1 = Xt−1,Xt−2, . . ., the information set available at time t−1.Define the process
Zt(u, θ) ≡ expiu′Xt
− ϕ(u, t|It−1, θ), u ∈ RN . (2.10)
Then H0 is equivalent to the following martingale differencesequence (MDS) characterization:
E [Zt(u, θ0)|It−1] = 0 for all u ∈ RN
and some θ0 ∈ 2, a.s. (2.11)
We may call Zt(u, θ) a ‘‘CCF-based generalized residual’’ of thecontinuous-time model M. This can be seen obviously from theauxiliary regression
expiu′Xt
= ϕ(u, t|It−1, θ0)+ Zt(u, θ0), (2.12)
where ϕ(u, t|It−1, θ0) is the regression model for the dependentvariable exp
iu′Xt
conditional on information set It−1, and
Zt(u, θ0) is a MDS regression disturbance (under H0).In principle,we can always obtain the CCF by the inverse Fourier
transform, provided the transition density ofXt is given. Even if theCCF or the transition density has no closed-form,we can accuratelyapproximate the model transition density by using (e.g.) theHermite expansion method of Ait-Sahalia (2002), the simulationmethods of Brandt and Santa-Clara (2002) and Pedersen (1995),or the closed-form approximation method of Duffie et al. (2003),and then calculating the Fourier transform. Nevertheless, our testis most convenient when the CCF has a closed-form.
AJDmodels are a class of continuous-timemodelswith a closed-form CCF, developed and popularized by Dai and Singleton (2000),Duffie and Kan (1996), and Duffie et al. (2000). These models haveproven fruitful in capturing the dynamics of economic variables,such as interest rates, exchange rates and stock prices. It hasbeen shown (e.g., Duffie et al. (2000)) that for AJDs, the CCF of Xtconditional on It−1 is a closed-form exponential-affine function ofXt−1:
ϕ(u, t|It−1, θ) = expαt−1(u)+ βt−1(u)
′Xt−1, (2.13)
where αt−1 : RN→ R and βt−1 : RN
→ RN satisfy the complex-valued Riccati equations:
βt = K′
1βt +12β
′
tH1βt + L1gβt− 1
,
αt = K′
0βt +12β
′
tH0βt + L0gβt− 1
,
(2.14)
with boundary conditions βT (u) = iu and aT (u) = 0.8
7 Our approach can be adapted to the case of1 = 1 easily. See Singleton (2001,p. 117) for related discussion.8 Assuming that the spot rate is an affine function of the state vector Xt and that
Xt follows an affine diffusion, Duffie and Kan (1996) show that the yield of thezero coupon bond has a closed-form CCF. Assuming that the spot rate is a quadraticfunction of the normally distributed state vector, Ahn et al. (2002) also derive theclosed-form CCF for the yield of the zero coupon bond.
272 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
Affine SV models are another popular class of multivariatecontinuous-time models with a closed-form CCF (e.g., Heston(1993), Bates (1996, 2000), Bakshi et al. (1997), Das and Sundaram(1999)). SV models can capture salient properties of volatilitysuch as randomness and persistence. Basic affine SV models,affine SV models with Poisson jumps and affine SV models withLévy processes have been widely used in modeling asset returndynamics as they allow for closed-form solutions for Europeanoption prices (e.g., Heston (1993), Das and Sundaram (1999),Chacko and Viceira (2003), Carr and Wu (2003, 2004), and Huangand Wu (2003)).
To test SVmodels, where Vt is a latent process, we need tomod-ify the characterization (2.11) to make it operatable. Generally, weput Xt = (X′
1,t ,X′
2,t)′, where X1,t ⊂ RN1 denotes the observable
state variables, X2,t ⊂ RN2 denotes the unobservable state vari-ables, and N1 + N2 = N . Also, partition u conformably as u =u′
1,u′
2
′. Then we can define
Z1,t (u1, θ) = expiu′
1X1,t− φ(u1, t|I1,t−1, θ),
where
φ(u1, t|I1,t−1, θ) ≡ Eθ[expiu′
1X1,t|I1,t−1]
= Eθϕ[(u′
1, 0′)′, t|It−1, θ]|I1,t−1,
I1,t−1 =X1,t−1,X1,t−2, . . .
is the information set on the observ-
ables that is available at time t −1 and the second equality followsfrom the law of iterated expectations. Then we have
EZ1,t (u1, θ0) |I1,t−1
= 0, for all u1 ∈ RN1
and some θ0 ∈ 2, a.s. (2.15)
This provides a basis for constructing operational tests for mul-tivariate continuous-time models with partially observable statevariables.9
Although themodel-implied CCF ϕ (u, t|It−1, θ), where It−1 =
(I1,t−1, I2,t−1), may have a closed-form, the conditional expec-tation φ(u1, t|I1,t−1, θ) generally has no closed-form. However,one can approximate it accurately by using some simulation tech-niques. For almost all continuous-time models characterized bySDEs in the literature, the CCF ϕ
where p(x2,t−1|I1,t−1, θ) is themodel-implied transition density ofthe unobservable X2,t−1 given the observable information I1,t−1.Noting that models with latent variables are the leading examplesthat may yield non-Markov observables in the continuous-timeliterature, we discuss several popular methods to estimate themodel-implied CCF based on observables. We first considerparticle filters, which have been developed by Gordon et al. (1993),Pitt and Shephard (1999) and Johannes et al. (2009). The term‘‘particle’’ was first used by Kitagawa (1996) in this literatureto denote the simulated discrete data with random support.Particle filters are the class of simulation filters that recursivelyapproximate the filtering random variable x2,t−1|I1,t−1, θ by‘‘particles’’ X1
2,t−1, X22,t−1, . . . , X
J2,t−1 with discrete probability
mass of π1t−1, π
2t−1, . . . , π
Jt−1. Hence a continuous variable is
approximately a discrete one with random support. These discretepoints are viewed as samples from p
x2,t−1|I1,t−1, θ
and as
J → ∞, the particles can approximate the conditional densityincreasingly well.
9 This characterization has beenused in Singleton (2001, Equation 50) to estimateaffine asset pricing models with unobservable components.
The key of this method is to propagate particles Xj2,t−2
Jj=1 one
step forward to get the new particles Xj2,t−1
Jj=1. By the Bayes rule,
we havepx2,t−1|I1,t−1, θ
=
px1,t−1|x2,t−1, I1,t−2, θ
px2,t−1|I1,t−2, θ
px1,t−1|I1,t−2, θ
,
wherepx2,t−1|I1,t−2, θ
=
∫px2,t−1|x2,t−2, I1,t−2, θ
px2,t−2|I1,t−2, θ
dx2,t−2.
We can approximate px2,t−1|I1,t−1, θ
up to some proportional-
ity; namely,
px2,t−1|I1,t−1, θ
∝ p
x1,t−1|X2,t−1, I1,t−2, θ
×
J−j=1
πjt−1p
x2,t−1|X2,t−2, I1,t−2, θ
,
where p(x1,t−1|X2,t−1, I1,t−2, θ) and∑J
j=1 πjt−1p(x2,t−1|X2,t−2,
I1,t−2, θ) can be viewed as the likelihood and prior respectively.As pointed out by Gordon et al. (1993), the particle filters requirethat the likelihood function can be evaluated and that X2t−1 can besampled from p(x2,t−1|X2,t−2, I1,t−2, θ). These can be achieved byusing time-discretized solutions to the SDEs.10
To implement particle filters, we can use Johannes et al. (2009)and Pitt and Shephard (1999) algorithm. First we generate a sim-ulated sample (XM
2,t−2)jJj=1, where XM
2,t−2 = X2,t−2, X2,t−2+ 1M,
. . . , X2,t−2+ M−1M
and M is an integer. Then we simulate them onestep forward, evaluate the likelihood function, and set
πjt−1 =
p[x1,t−1|(XM2,t−1)
j, I1,t−2, θ]
J∑j=1
p[x1,t−1|(XM2,t−1)
j, I1,t−2, θ]
, j = 1, . . . , J.
Finally, we resample J particles with weights πjt−1
Jj=1 to obtain a
new random sample of size J.11 It has been shown (e.g., Bally andTalay (1996), Del Moral et al. (2001)) that with both large J andM , this algorithm sequentially generates valid simulated samplesfrom p(x2,t−1|I1,t−1, θ). Hence φ(u1, t|I1,t−1, θ) can be estimatedby Monte Carlo averages.12
The second method to approximate p(x2,t−1, t − 1|I1,t−1, θ) isGallant and Tauchen’s (1998) SNP-based reprojection technique,which can characterize the dynamic response of a partiallyobserved nonlinear system to its past observable history. First,we can generate simulated samples X1,t−1
Jt=2 and X2,t−1
Jt=2
from the continuous-time model, where J is a large integer.Then, we project the simulated data X2,t−1
Jt=2 onto a Hermite
series representation of the transition density p(x2,t−1, t −
1|X1,t−1,X1,t−2, . . . X1,t−L), where L denotes a truncation lag order.
10 There are many discretization methods in practice, such as the Euler scheme,the Milstein scheme, and the explicit strong scheme. See (e.g.) Kloeden et al. (1994)for more discussion.11 This is called sampling/importance resampling (SIR) in the literature. Alterna-tive methods include rejection sampling and Markov chain Monte Carlo (MCMC)algorithm. See Doucet et al. (2001) and Pitt and Shephard (1999) for more discus-sion.12 In a related estimation context, Chacko and Viceira (2003), Jiang and Knight(2002) and Singleton (2001) derive analytical expressions for Eθϕ[(u′
1, 0′)′,
t|It−1, θ]|I1,t−1 for some suitable subset I1,t−1 of I1,t−1 . For example, Chacko andViceira (2003) obtain a closed-formexpression for E[ϕlog S(u, t|It−1, θ)| log St−1], byintegrating out Vt−1 . This is computationallymore convenient, but it is less efficient.
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 273
With a suitable choice of L via some information criteria such asAIC or BIC, we can approximate p(x2,t−1, t −1|I1,t−1, θ) arbitrarilywell. The final step is to evaluate the estimated density function atthe observed data in the conditional information set. See Gallantand Tauchen (1998) for more discussion.
For models whose CCF is exponentially affine in Xt−1,13 we canalso adopt Bates’ (2007) approach to compute φ(u1, t|I1,t−1, θ).First, at time t = 1, we initialize the CCF of the latent vector X2,t−1conditional on I1,t−1 at its unconditional CF. Then, by exploitingthe Markov property and the affine structure of the CCF, we canevaluate the model-implied CCF conditional on data observedthrough period t , namely, Eθ[ϕ(u, t|Xt−1, θ)|I1,t−1], and thus anestimator for φ(u1, t|I1,t−1, θ) is obtained.
3. Omnibus testing
Wenowpropose an omnibus specification test for the adequacyof a multivariate continuous-time model by using the CCF char-acterization in (2.11) or (2.15). For notational convenience, belowwe focus on the case where Xt is fully observable. The proposedprocedures are readily applicable to the case where only X1,t isobservable, with Z1,t (u, θ) and I1,t−1 replacing Zt (u, θ) and It−1respectively. It is not a trivial task to check (2.11) because the MDSproperty must hold for each u ∈ RN , and because Zt (u, θ) maydisplay serial dependence in higher order conditional moments.Any test for (2.11) should be robust to time-varying conditionalheteroskedasticity and higher order moments of unknown formin Zt (u, θ). To check the MDS property of Zt (u, θ), we substan-tively extend Hong’s (1999) univariate generalized spectrum to amultivariate generalized cross-spectrum.14 The generalized spec-trum is an analytic tool for nonlinear time series that embeds the CFin a spectral framework. It can capture nonlinear dynamics whilemaintaining the nice features of spectral analysis.
Because the dimension of It−1 can be infinity, we encounter theso-called ‘‘curse of dimensionality’’ problem in checking the MDSproperty in (2.11). Fortunately, the generalized spectral approachprovides a solution to tackle this difficulty. It checks many lags ina pairwise manner, thus avoiding the ‘‘curse of dimensionality’’.
Define the generalized covariance function
Γj(u, v) = cov[Zt(u, θ), expiv′Xt−|j|
], u, v ∈ RN . (3.1)
With the generalized cross-covariance Γj (u, v), we can define thegeneralized cross-spectrum
F(ω,u, v) =12π
∞−j=−∞
Γj(u, v) exp (−ijω) ,
ω ∈ [−π, π], u, v ∈ RN , (3.2)
where ω is the frequency. This is the Fourier transform of thegeneralized covariance function
Γj(u, v)
and thus contains the
same information asΓj(u, v)
. An advantage of generalized cross-
spectral analysis is that it can capture cyclical patterns causedby both linear and nonlinear cross dependence. Examples includevolatility spillover, the comovements of tail distribution clusteringbetween state variables, and asymmetric spillover of businesscycles cross different sectors or countries. Another attractivefeature of F(ω,u, v) is that it does not require the existence of anymoment condition on Xt .
Under H0, we have Γj(u, v) = 0 for all u, v ∈ RN and all j = 0.Consequently, the generalized cross-spectrum F(ω,u, v) becomes
13 Examples include AJD models (Duffie et al., 2000) and time-changed Lévyprocesses (Carr and Wu, 2003, 2004).14 This is not a trivial extension since Hong’s (1999) test is under an i.i.d.assumption.
a ‘‘flat’’ spectrum:
F(ω,u, v) = F0(ω,u, v) ≡12πΓ0(u, v),
ω ∈ [−π, π], u, v ∈ RN . (3.3)
We can test H0 by checking whether a consistent estimator forF(ω,u, v) is flat with respect to frequency ω. Any significantdeviation from a flat generalized cross-spectrum is evidence ofmodel misspecification.
Suppose we have a discretely observed sample XtTt=1 of size
T . Then we estimate the generalized covariance Γj (u, v) by itssample analogue
Γj(u, v) =1
T − |j|
T−t=|j|+1
Zt(u, θ)exp
iv′Xt−|j|
− ϕj(v)
,
u, v ∈ RN , (3.4)
where
Zt(u, θ) ≡ expiu′Xt
− ϕ(u, t|IĎ
t−1, θ),
IĎt is the observed information set available at time t that may
involve certain initial values, θ is a√T -consistent estimator for
θ0 and ϕj(v) = (T − |j|)−1∑Tt=|j|+1 exp
iv′Xt−|j|
is the empirical
unconditional CF of Xt . We note that the information set It−1 =
Xt−1,Xt−2, . . . dates back to past infinity and is not feasible.Thus, when Xt is non-Markov, we may need to assume someinitial values in computing ϕ(u, t|It−1, θ). Therefore, we have toreplace It−1 with a truncated information set I
Ďt−1, which contains
some initial values. We provide a condition (see Assumption A.5 inSection 4) to ensure that the use of initial values has no impact onthe asymptotic distribution of the proposed test statistics.
A consistent estimator for F0(ω,u, v) is
F0(ω,u, v) =12πΓ0(u, v), ω ∈ [−π, π], u, v ∈ RN . (3.5)
Consistent estimation for F (ω,u, v) is more challenging. We use asmoothed kernel estimator
F(ω,u, v) =12π
T−1−t=1−T
(1 − |j| /T )1/2k(j/p)Γj(u, v)e−ijω,
ω ∈ [−π, π],u, v ∈ RN , (3.6)
where p ≡ p(T ) → ∞ is a bandwidth or an effective lag order, andk : R → [−1, 1] is a kernel function, assigning weights to variouslags. Examples of k(·) include the Bartlett kernel, the Parzenkernel and the Quadratic-Spectral (QS) kernel. In (3.6), the factor(1 − |j| /T )1/2 is a finite sample correction. It could be replacedby unity. Under regularity conditions, F(ω,u, v) and F0(ω,u, v)are consistent for F(ω,u, v) and F0(ω,u, v) respectively. Theseestimators converge to the same limit under H0 but they generallyconverge to different limits under HA, giving the power of the test.
We canmeasure thedistance between F(ω,u, v) and F0(ω,u, v)by the quadratic form
πT2
∫∫∫ π
−π
F(ω,u, v)− F0(ω,u, v)2 dωdW (u) dW (v)
=
T−1−j=1
k2(j/p)(T − j)∫∫ Γj(u, v)
2 dW (u) dW (v) , (3.7)
where the equality follows by Parseval’s identity,W : RN→ R+ is
a positive nondecreasing right-continuous function, and theunspecified integrals are all taken over the support of W (·). Anexample ofW (·) is the N (0, IN) CDF, where IN is a N × N identity
274 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
matrix. The function W (·) can also be a step function, analogousto the CDF of a discrete random vector.
Our omnibus test statistic for H0 against HA is a standardizedversion of (3.7):
Q (0, 0) =
T−1−j=1
k2(j/p)(T − j)
×
∫∫ Γj(u, v)2 dW (u) dW (v)− C(0, 0)
]D(0, 0), (3.8)
where the centering and scaling factors
C(0, 0) =
T−1−j=1
k2(j/p)(T − j)−1T−
t=j+1
∫∫ Zt(u, θ)2×
ψt−j(v)2 dW (u) dW (v) ,
D(0, 0) = 2T−2−j=1
T−2−l=1
k2(j/p)k2(l/p)
×
∫∫∫∫dW (u1) dW (u2) dW (v1) dW (v2)
×
1T − max(j, l)
T−t=max(j,l)+1
Zt(u1, θ)Zt(u2, θ)ψt−j(v1)ψt−l(v2)
2
,
where ψt(v) = eiv′Xt − ϕ(v), and ϕ(v) = T−1∑T
t=1 eiv′Xt . The
factors C (0, 0) and D (0, 0) are the approximately mean andvariance of the quadratic form in (3.7). They have taken intoaccount the impact of higher order dependence in the generalizedresidual Zt (u, θ0). As a result, the Q (0, 0) test is robust toconditional heteroskedasticity and time-varying higher orderconditional moments of unknown form in Zt(u, θ0).
In practice, whenW (·) is continuous, Q (0, 0) can be calculatedby numerical integration or simulation. This may be computation-ally costly when the dimension N of Xt is large. Alternatively, onecan use a finite number of grid points for u and v,which is equiva-lent to using a discrete CDF. For example, we can generate finitelymany numbers of u and v from the N (0, IN) distribution. This willreduce the computational cost, but at a cost of some power loss.
4. Asymptotic theory
To derive the null asymptotic distribution of the test statisticQ (0, 0) and investigate its asymptotic power property, we imposefollowing regularity conditions.
Assumption A.1. A discrete-time sample XtT1t=1, where1 ≡ 1 is
the sampling interval, is observed at equally spaced discrete times.
Assumption A.2. Let ϕ (u, t|It−1, θ) be the CCF ofXt given It−1 ≡
Xt−1,Xt−2, . . . for a time series model for Xt . (i) For each θ ∈
2, each u ∈ RN , and each t , ϕ (u, t|It−1, θ) is measurable withrespect to It−1; (ii) for each θ ∈ 2, each u ∈ RN , and each t ,ϕ (u, t|It−1, θ) is twice continuously differentiable with respectto θ ∈ 2 with probability one; (iii) supu∈RN limT→∞ T−1∑T
t=1
E supθ∈2 ‖∂∂θϕ(u, t|It−1, θ)‖
2≤ C and supu∈RN limT→∞ T−1∑T
t=1
E supθ∈2 ‖∂2
∂θ∂θ′ ϕ (u, t|It−1, θ) ‖ ≤ C .
Assumption A.3. θ is a parameter estimator such that√T (θ −
a strictly stationary β-mixing process with the mixing coefficient|β (l)| ≤ Cl−ν for some constant ν > 2.
Assumption A.5. Let IĎt be the observed feasible information set
available at time t that may involve certain initial values. Thensupu∈RN limT→∞
∑Tt=1 E supθ∈2 |ϕ(u, t|It−1, θ) − ϕ(u, t|IĎ
t−1,
θ)|2 ≤ C .
Assumption A.6. k : R → [−1, 1] is a symmetric function that iscontinuous at zero and all points in R except for a finite numberof points, with k (0) = 1 and k (z) ≤ c |z|−b for some b > 1
2 asz → ∞.
Assumption A.7. W : RN→ R+ is a nondecreasing right-conti-
nuous function with
RN dW (u) < ∞ and
RN ‖u‖4 dW (u) < ∞.
Furthermore, W =wdv for some nonnegative weighting func-
tion w and some measure v, where W is absolutely continuouswith respect to v andw is symmetric about the origin.
Assumption A.1 imposes some regularity conditions on the dis-cretely observed random sample. Both univariate andmultivariatecontinuous-time or discrete-time processes are covered, and weallow but do not require Xt to be Markov. This distinguishes ourtest from all other tests in the literature. We emphasize that it isimportant to allow the DGP to be non-Markov, even if one is inter-ested in testing the adequacy of a Markov model. This is becausethe misspecification of a Markov model may come from not onlythe improper specification in functional forms, but also the viola-tion of the Markov assumption. The non-Markov assumption forthe DGPs of observable state variables is also necessarywhen somestate variables are latent, as is the case of SV models.
There are two kinds of asymptotics in the literature oncontinuous-time models. The first is to let the sampling interval1 → 0. This implies that the number of observations per unit oftime tends to infinity. The second is to let the time horizon T →
∞. As argued by Ait-Sahalia (1996b), the first approach hardlymatches the way in which new data are added to the sample.Even if such ultra-high-frequency data are available, marketmicro-structural problems are likely to complicate the analysisconsiderably. Hence, like Ait-Sahalia (1996b) and Singleton (2001),we fix the sampling interval 1 and derive the asymptoticproperties of our test for an expanding sampling period (i.e., T →
∞). Unlike Ait-Sahalia (1996a,b), however, we do not imposeadditional conditions on the SDEs for Xt . Instead, we imposeconditions directly on the model-implied CCF, which is equivalentto imposing conditions on the model-implied transition density.For these reasons, our approach is more general and the proposedtests are applicable to both continuous-time and discrete-timemodels.
Assumption A.2 imposes regularity conditions on the CCF of themultivariate time series model. As the CCF is the Fourier trans-form of the transition density, we can ensure Assumption A.2 byimposing the following conditions on the model-implied transi-tion density p (x, t|It−1, θ): (i) for each t , each x ∈ D, and eachθ ∈ 2, p (x, t|It−1, θ) is measurable with respect to It−1; (ii) foreach t , each x ∈ D, and each given It−1, p (x, t|It−1, θ) is twicecontinuously differentiable with respect to θ ∈ 2 with probabil-ity one; and (iii) supx∈D limT→∞ T−1∑T
t=1 E supθ∈2 ‖∂∂θ
ln p(x, t|It−1, θ)‖
2≤ C , and supx∈D limT→∞ T−1∑T
t=1 E supθ∈2 ‖∂2
∂θ∂θ′ ln p(x, t|It−1, θ) ‖ ≤ C .
Assumption A.3 requires a√T -consistent estimator θ under
H0. Examples include asymptotically optimal and suboptimalestimators, such as Gallant and Tauchen’s (1996) EMM, Singleton’s(2001) ML-CCF and GMM-CCF, Ait-Sahalia’s (2002) and Ait-Sahaliaand Kimmel’s (2010) approximated MLE, Carrasco et al.’s (2007)C-GMM, and Chib et al.’s (2010) MCMC method. We do notrequire any asymptotically most efficient estimator or a specifiedestimator. This is attractive for practitioners given the notorious
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 275
difficulty of efficient estimation of many multivariate nonlineartime series models.
Assumption A.4 is a regularity condition on the temporal de-pendence of the process Xt , ϕ (u, t|It−1, θ0) ,
∂∂θϕ(u, t|It−1, θ0).
The β-mixing assumption is a standard condition for discretetime series analysis. In the continuous-time context, almost allcontinuous-time models in the literature are Markov. For Markovmodels, ϕ (u, t|It−1, θ0) and ∂
∂θϕ (u, t|It−1, θ0) are functions of
Xt−1. Thus, Assumption A.4 holds if the discretely observed ran-dom sample Xt
Tt=1 is a strictly stationary β-mixing process with
|β (l)| ≤ Cl−ν for some constant ν > 2.15Assumption A.5 is a start-up value condition, which ensures
that the impact of initial values (if any) assumed in IĎt−1 is
asymptotically negligible. This condition holds automatically forMarkov models and easily for many non-Markov models. Forsimplicity, we illustrate such an initial value condition by adiscrete-time GARCHmodel: Xt = εt
√ht , where ht = ω+αht−1+
βX2t−1 and εt is i.i.d.N(0, 1). Here we have ϕ (u, t|It−1, θ) =
exp(− 12u
2ht), where It−1 = Xt−1, Xt−2, . . .. Furthermore, wehave I
Ďt−1 = Xt−1, Xt−2, . . . , X1, h0, where h0 is an initial value
assumed for h0. By recursive substitution, we obtain
var (Xt |It−1, θ)− varXt |I
Ďt−1, θ
= ω + β
t−2−j=0
αjX2t−1−j
+βαt−1h0 − ω − β
t−2−j=0
αjX2t−1−j − βαt−1h0.
It follows that
supu∈R
T−t=1
E supθ∈Θ
ϕ (u, t|It−1, θ)− ϕu, t|IĎ
t−1, θ2
≤ supu∈R
T−t=1
E supθ∈2
×
1 − exp
12u
2var (Xt |It−1, θ)− var
Xt |I
Ďt−1, θ
exp
12u
2ht
2
≤ supu∈RN
T−t=1
E supθ∈2
u2βαt−1h0 − h0
exp
u2ω
≤ C
provided ω > 0, 0 < α, β < 1, α + β < 1 and Eh0 < ∞.Assumption A.6 is the regularity condition on the kernel
function. The continuity of k (·) at 0 and k (0) = 1 ensure thatthe bias of the generalized cross-spectral estimator F (ω,u, v)vanishes to zero asymptotically as T → ∞. The condition onthe tail behavior of k (·) ensures that higher order lags haveasymptotically negligible impact on the statistical properties ofF (ω,u, v). Assumption A.6 covers most commonly used kernels.For kernels with bounded support, such as the Bartlett and Parzenkernels, b = ∞. For kernels with unbounded support, b is somefinite positive real number. For example, b = 2 for the Quadratic-Spectral kernel.
Assumption A.7 imposes mild conditions on the functionW (·).The CDF of any symmetric distribution with finite fourth momentsatisfies Assumption A.7. Note that W (·) can be the CDF ofcontinuous or discrete distribution andw(·) is its Radon–Nikodym
15 Suggested by Hansen and Scheinkman (1995) and Ait-Sahalia (1996a), oneset of sufficient conditions for the β-mixing when N = 1 is: (i) limx→l or x→uσ (x, θ) π (x, θ) = 0; and (ii) limx→l or x→u |σ (x, θ) /2µ (x, θ)−σ (x, θ) [∂σ (x, θ) /∂x] < ∞, where l and u are left and right boundaries of Xt with possibly l = −∞
and/or u = +∞, and π (x, θ) is the model-implied marginal density.
derivative with respect to v. IfW (·) is the CDF of some continuousdistribution, v is Lebesgue measure; if W (·) is the CDF of somediscrete distribution, v is some counting measure. This providesa convenient way to implement our tests, because we can avoidhigh-dimensional numerical integrations by using a finite numberof grid points for u and v. This is equivalent to using the CDF of adiscrete random vector.
We now state the asymptotic distribution of the omnibus testQ (0, 0) under H0.
Theorem 1. Suppose Assumptions A.1–A.7 hold, and p = cTλ for1+δνδ
< λ < (3 +1
4b−2 )−1, where 0 < c, δ < ∞ and ν is defined
in Assumption A.4. Then Q (0, 0) d→N(0, 1) under H0 as T → ∞.
An important feature of Q (0, 0) is that the use of the estimatedgeneralized residuals Zt(u, θ) in place of the true unobservablegeneralized residuals Zt (u, θ0) has no impact on the limiting dis-tribution of Q (0, 0). One can proceed as if the true parameter valueθ0 were known and equal to θ. Intuitively, the parametric esti-mator θ converges to θ0 faster than the nonparametric estimatorF (ω,u, v) converges to F (ω,u, v) as T → ∞. Consequently, thelimiting distribution of Q (0, 0) is solely determined by F (ω,u, v),and replacing θ0 by θ has no impact asymptotically. This deliversa convenient procedure, because any
√T -consistent estimator can
be used.We allow forweakly dependent data and data dependencehas some impact on the feasible range of the bandwidth p. The con-dition on the tail behavior of the kernel function k(·) also has someimpact. For kernels with bounded support (e.g., the Bartlett andParzen), λ < 1
3 because b = ∞. For the QS kernel (b = 2), λ < 619 .
These conditions are mild.Next,we investigate the asymptotic power of Q (0, 0)underHA.
Theorem 2. Suppose Assumptions A.1–A.7 hold, and p = cTλ for0 < λ < 1
2 and 0 < c < ∞. Then as T → ∞,
p12
TQ (0, 0)→p 1
√D (0, 0)
∞−j=1
∫∫ Γj (u, v)2 dW (u) dW (v)
=π
2√D (0, 0)
∫∫∫ π
−π
|F (ω,u, v)− F0 (ω,u, v)|2
× dωdW (u) dW (v) ,
where
D (0, 0) = 2∫
∞
0k4 (z) dz
∫∫|Σ0 (u1,u2)|
2 dW (u1) dW (u2)
×
∫∫ ∞−j=−∞
Ωj (v1, v2)2 dW (v1) dW (v2) ,
and Σ0 (u, v) = covZtu, θ∗
, Zt
v, θ∗
, and Ωj (u, v) =
cov(eiu′Xt , eiv
′Xt−|j|).
The function G (ω,u, v) is the generalized spectral density of thestate vector Xt. It captures temporal dependence in Xt. Thedependence of the constant D (0, 0) on G (ω,u, v) is due to thefact that the conditioning variable exp
iv′Xt−|j|
is a time series
process.Following Bierens (1982) and Stinchcombe and White (1998),
we have that for j > 0, Γj(u, v) = 0 for all u, v ∈ RN ifand only if E
Zt(u, θ∗)|Xt−j
= 0 a.s. for all u ∈ RN . Suppose
EZt(u, θ∗)|Xt−j
= 0 at some lag j > 0 under HA. Then we have Γj (u, v)
2 dW (u) dW (v) ≥ C > 0 for any function W (·)that is positive, monotonically increasing and continuous, withunbounded support onRN . As a result, P[Q (0, 0) > C (T )] → 1 for
276 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
any sequence of constants C(T ) = o(T/p1/2). Thus Q (0, 0) hasasymptotic unit power at any given significance level α ∈ (0, 1),whenever E
Zt(u, θ∗)|Xt−j
is nonzero at some lag j > 0 underHA.
Note that for a multivariate Markov process Xt , we always haveEZtu, θ∗
|Xt−j
= 0 at least for some j > 0 under HA. Hence,
Q (0, 0) is consistent against HA when Xt is Markov. Gallant andTauchen’s (1996) EMM test does not have this property.
For a non-Markovian process Xt , the hypothesis that EZt(u, θ0)|Xt−j
= 0 a.s. for all u ∈ RN and some θ0 ∈ 2 and all
j > 0 is not equivalent to the hypothesis that E [Zt(u, θ0)|It−1] =
0 a.s. for allu ∈ RN and some θ0 ∈ 2. The latter implies the formerbut not vice versa. This is the price we have to pay for dealingwith the difficulty of the ‘‘curse of dimensionality’’. Nevertheless,our test is expected to have power against a wide range of non-Markovian processes, since we check many lag orders. The use of alarge number of lags might cause the loss of power, due to the lossof a large number of degrees of freedom. Fortunately, such powerloss is substantially alleviated for Q (0, 0), thanks to the downwardweighting by k2 (·) for higher order lags. Generally speaking, thestate vector Xt is more affected by the recent events than theremote past events. In such scenarios, equal weighting to eachlag is not expected to be powerful. Instead, downward weightingis expected to enhance better power because it discounts pastinformation. Thus, we expect that the power of our test is not sosensitive to the choice of the lag order. This is confirmed by oursimulation studies below.
5. Diagnostic testing
When a multivariate time series model M is rejected by theomnibus test Q (0, 0), it would be interesting to explore possiblesources of the rejection. For example, one may like to knowwhether the misspecification comes from conditional mean/driftdynamics, conditional variance/diffusion dynamics, or conditionalcorrelations between state variables. Such information, if any, willbe valuable in reconstructing the model.
The CCF is a convenient and useful tool to gauge possiblesources of model misspecification. As is well known, the CCF canbe differentiated to obtain conditional moments. We now developa class of diagnostic tests by differentiating the generalized cross-spectrum F(ω,u, v). This class of diagnostic tests can provideuseful information about how well a multivariate time seriesmodel can capture the dynamics of various conditional momentsand conditional cross-moments of state variables.
Define an N × 1 index vector
m = (m1,m2, . . . ,mN)′,
where mc ≥ 0 for all 1 ≤ c ≤ N , and put |m| =∑N
c=1 mc . Thenwe define the generalized cross-spectral derivative
F (0,m,0)(ω, 0, v) ≡∂m1
∂um11
· · ·∂mN
∂umNN
F(ω,u, v)u=0
=12π
∞−j=−∞
Γ(m,0)j (0, v) exp (−ijω) , (5.1)
where the derivative of the generalized cross-covariance function
Γ(m,0)j (0, v) = cov
N∏
c=1
(iXct)mc
− Eθ
N∏
c=1
(iXct)mc
It−1
, exp
iv′Xt−|j|
.
Here, as before, Eθ (·|It−1) is the conditional expectation under themodel-implied transition density p (x, t|It−1, θ).
To gain insight into the generalized cross-spectral derivativeF (0,m,0)(ω, 0, v), we consider a bivariate process Xt = (X1t , X2t)
′
and examine the cases of |m| = 1 and |m| = 2 respectively:
• Case 1: |m| = 1. We have m = (1, 0) or m = (0, 1). Ifm = (1, 0),
Γ(m,0)j (0, v) = icov
X1t − Eθ(X1t |It−1), exp
iv′Xt−|j|
.
Ifm = (0, 1), then
Γ(m,0)j (0, v) = icov
X2t − Eθ(X2t |It−1), exp
iv′Xt−|j|
.
Thus, the choice of |m| = 1 can be used to check misspecifica-tions in the conditional mean dynamics of X1t and X2t respec-tively.
• Case 2: |m| = 2. We have m = (2, 0), (0, 2) or (1, 1). Ifm = (2, 0),
Γ(m,0)j (0, v) = i2cov
X21t − Eθ(X2
1t |It−1), expiv′Xt−|j|
.
Ifm = (0, 2),
Γ(m,0)j (0, v) = i2cov
X22t − Eθ(X2
2t |It−1), expiv′Xt−|j|
.
Finally, ifm = (1, 1),
Γ(m,0)j (0, v) = i2cov
×X1tX2t − Eθ (X1tX2t |It−1) , exp
iv′Xt−|j|
. (5.2)
Thus, the choice of |m| = 2 can be used to check model mis-specifications in the conditional volatility of state variables aswell as their conditional correlations.
We now define the class of diagnostic test statistics as follows:
Q (m, 0) =
T−1−j=1
k2(j/p)(T − j)
×
∫ Γ (m,0)j (0, v)
2 dW (v)− C(m, 0)
D(m, 0), (5.3)
where the centering and scaling factors
C(m, 0) =
T−1−j=1
k2(j/p)1
T − j
×
T−t=|j|+1
∫ Z (m)t (0, θ)2 ψt−j(v)
2 dW (v) ,
D(m, 0) = 2T−2−j=1
T−2−l=1
k2(j/p)k2(l/p)
×
∫∫ 1T − max(j, l)
T−t=max(j,l)+1
Z (m)t (0, θ)2
× ψt−j(v1)ψt−j(v2)
2
dW (v1) dW (v2) ,
and
Z (m)t (0, θ) =
N∏c=1
(iXct)mc − E
θ
N∏
c=1
(iXct)mc |I
Ďt−1
.
To derive the limit distribution of Q (m, 0) underH0, we imposesome moment conditions.
Assumption A.8. (i) limT→∞ T−1∑Tt=1 E supθ∈2 ∂
∂θ
[∂m1
∂um11
· · ·∂mN
∂umNNϕ (u, t|It−1, θ) |u=0
]2 ≤ C;
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 277
(ii) limT→∞ T−1∑Tt=1 E supθ∈2 ∂2
∂θ∂θ′
[∂m1
∂um11
· · ·∂mN
∂umNNϕ (u, t|It−1, θ) |u=0
]2 ≤ C;
(iii) limT→∞ T−1∑Tt=1 E supθ∈2 ∂m1
∂um11
· · ·∂mN
∂umNNϕ (u, t|It−1, θ) |u=0
4 ≤ C;
(iv) EΠN
c=1X8(1+δ)mcct
≤ C .
Assumption A.9. Xt ,∂m1
∂um11
· · ·∂mN
∂umNNϕ (u, t|It−1, θ0) ,
∂∂θ
[∂m1
∂um11
· · ·∂mN
∂umNNϕ (u, t|It−1, θ0) |u=0] is a strictly stationary β-
mixing process with the mixing coefficient |β (l)| ≤ Cl−ν for someconstant ν > 2.
Assumption A.10. Let IĎt be the observed feasible information set
available at time t that may involve certain initial values. ThenlimT→∞
∑Tt=1 E supθ∈2 |
∂m1
∂um11
· · ·∂mN
∂umNNϕ (u, t|It−1, θ) |u=0
−∂m1
∂um11
· · ·∂mN
∂umNNϕ(u, t|IĎ
t−1, θ)|u=0|2
≤ C .
Theorem 3. Suppose Assumptions A.1, A.3 and A.6–A.10 hold forsome pre-specified m and p = cTλ for 1+δ
νδ< λ < (3 +
14b−2 )
−1,where 0 < c, δ < ∞ and ν is defined in Assumption A.9. ThenQ (m, 0) d
→N(0, 1) under H0 as T → ∞.
Like the omnibus test Q (0, 0), Q (m, 0) has a convenientasymptotic N (0, 1) distribution and parameter estimation uncer-tainty in θhas no impact on the asymptotic distribution of Q (m, 0).Any
√T -consistent estimator can be used. Moreover, different
choices of m can examine various specific dynamic aspects of thestate vector Xt and thus provide information on how well a mul-tivariate time series model fits various aspects of the conditionaldistribution of Xt .16
It should be pointed out that the generalized cross-spectralderivative tests may fail to detect certain specific model misspeci-fications. In particular, they will fail to detect some time-invariantmodelmisspecifications. For example, suppose a time seriesmodelassumes a zero conditional correlation between state variables,while there exists a nonzero but time-invariant conditional corre-lation. Then the derivative of the generalized covariance in (5.2) isidentically zero. In this case, the correlation test will fail to detectsuch time-invariant conditional correlation. Of course, the testsQ (m, 0)will generally have power against time-varying misspeci-fications.
These diagnostic tests are designed to test specification of var-ious conditional moments, i.e., whether the conditional momentsof Xt are correctly specified given the discrete sample information.We note that the first two conditional moments differ from theinstantaneous conditional mean (drift) and instantaneous condi-tional variance (squared diffusion). In general, the conditional mo-ments tested here are functions of drift, diffusion and jump (see,e.g., Eqs. (6.2) and (6.3) in Section 6). Therefore, careful interpreta-tion is needed. However, our simulation studies below show thatthe tests Q (m, 0) with |m| = 1 and 2 are very indicative of driftand diffusion misspecifications respectively.
16 In the literature, there have been diagnostic tests based on PITs to gaugepossible sources of model rejection (e.g., Diebold et al. (1998), Hong and Li(2005)). As pointed out by Bontemps and Meddahi (2010), however, the diagnostictests based on the PIT have some drawback: when one rejects a specification oftransformed variables, it is difficult to know how to modify the model for originalstate variables to obtain a correct specification. However, our tests are directlybased on state variables and thus avoid this drawback.
6. Finite sample performance
It is unclear how well the asymptotic theory can providereliable reference and guidance when applied to financial timeseries data, which is well known to display highly persistent serialdependence. We now investigate the finite sample performanceof the proposed tests for the adequacy of a multivariate affinediffusion model.
6.1. Simulation design
To examine the size of Q (m, 0) under H0, we consider thefollowing DGP:
• DGP0 (Uncorrelated Gaussian Diffusion):
d
X1tX2tX3t
=
κ11 0 00 κ22 00 0 κ33
θ1 − X1tθ2 − X2tθ3 − X3t
dt
+
σ11 0 00 σ22 00 0 σ33
d
W1tW2tW3t
. (6.1)
We set (κ11, κ22, κ33, θ1, θ2, θ3, σ11, σ22, σ33) = (0.5, 1, 2, 0,0, 0, 1, 1, 1). We use the Euler scheme to simulate 1000 data setsof the random sample Xt
T1t=1 at the monthly frequency for T =
250, 500 and 1000 respectively. These sample sizes correspondto about twenty to one hundred years of monthly data. Eachsimulated sample path is generated using 120 intervals permonth.We then discard 119 out of every 120 observations, obtainingdiscrete observations at the monthly frequency. For each data set,we use MLE to estimate model (6.1), with no restrictions on theintercept coefficients. All data are generated using the Matlab 6.1random number generator on a PC.
With a diagonal matrix κ = diag(κ11, κ22,κ33), DGP0 is an un-correlated 3-factor Gaussian diffusion process. As shown in Duffee(2002), the Gaussian diffusion model has analytic expressions forthe conditional mean and the conditional variance respectively:
E(Xt |Xs) =I − e−κ(t−s) θ + e−κ(t−s)Xs, (6.2)
var(Xt |Xs) =
∫ t
se−κ(t−m)66′e−κ′(t−m)dm
= diagσ 211
2κ11[1 − e2κ11(s−t)
],σ 222
2κ22
× [1 − e2κ22(s−t)],σ 233
2κ33[1 − e2κ33(s−t)
]
, (6.3)
where θ = (θ1, θ2, θ3)′ and 6 = diag
σ11,σ22, σ33
.
To investigate the power of Q (m, 0) in distinguishing model(6.1) from alternative processes, we also generate data from fiveaffine diffusion processes respectively:
• DGP1 [Correlated Gaussian Diffusion, with Constant Correlation inDrift]:
d
X1tX2tX3t
=
0.5 0 0−0.5 1 00.5 0.5 2
−X1t−X2t−X3t
dt
+
1 0 00 1 00 0 1
d
W1tW2tW3t
; (6.4)
• DGP2 [Uncorrelated CIR (Cox et al., 1985) Diffusion]:
d
X1tX2tX3t
=
0.5 0 00 1 00 0 2
2 − X1t1 − X2t1 − X3t
dt
278 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
+
X1t 0 00
X2t 0
0 0X3t
d
W1tW2tW3t
; (6.5)
• DGP3 [Correlated Gaussian Diffusion, with Constant Correlation inDiffusion]:
d
X1tX2tX3t
=
0.5 0 00 1 00 0 2
−X1t−X2t−X3t
dt
+
1 0 00.5 1 00.5 0.5 1
d
W1tW2tW3t
; (6.6)
• DGP4 [Correlated CIR Diffusion]:
d
X1tX2tX3t
=
0.5 0 00 1 00 0 2
2 − X1t1 − X2t1 − X3t
dt
+
X1t 0 0
0.5X1t
X2t 0
0.5X1t 0.5
X2t
X3t
d
W1tW2tW3t
. (6.7)
• DGP5 [Mixture of Gaussian and CIR Processes]:
d
X1tX2tX3t
=
0.5 0 00 1 00 0 2
2 − X1t1 − X2t−X3t
dt
+
X1t 0 0
0.5X1t
X2t 0
0 0 1
d
W1tW2tW3t
. (6.8)
With a nondiagonal matrix κ, DGP1 is a correlated 3-factorGaussian diffusion process, a special case of the canonical A0 (3)defined in Dai and Singleton (2000). Under DGP1, model (6.1) ismisspecified for the drift function but is correctly specified for thediffusion function. Eqs. (6.2) and (6.3) show that the conditionalmeans and the conditional variances of X2t and X3t (but not X1t ),and the conditional covariances between X1t , X2t and X3t aremisspecified, if model (6.1) is used to fit the data generated fromDGP1. However, the misspecifications in the conditional variancesand conditional correlations of Xt are time-invariant.
DGP2 is an uncorrelated 3-factor CIR diffusion process, whichhas a non-central χ2 transition distribution. Under DGP2, model(6.1) is correctly specified for the drift function but is misspecifiedfor the diffusion function because it fails to capture the ‘‘leveleffect’’. If model (6.1) is used to fit the data generated from DGP2,the conditional means and conditional covariances of X1t , X2t andX3t are correctly specified, but their conditional variances aremisspecified.
DGP3 is another correlated Gaussian diffusion process, wherethe correlations between state variables come from diffusionsrather than drifts. Under DGP3, model (6.1) is correctly specifiedfor the drift function but ismisspecified for the diffusion function. Ifmodel (6.1) is used to fit data generated fromDGP3, the conditionalmeans of X1t , X2t and X3t are correctly specified, but the conditionalvariances of X2t and X3t , and the conditional covariances betweenX1t , X2t and X3t are misspecified. Specifically, model (6.1) assumeszero conditional correlations between state variables, whereasthere exist nonzero but time-invariant conditional correlationsbetween state variables under DGP3.
DGP4 is a correlated 3-factor CIR diffusion process, where theconditional variances and covariances of state variables dependon state variables. If we use model (6.1) to fit data generatedfrom DGP4, the conditional means of X1t , X2t and X3t are correctly
specified, but there are dynamic misspecifications in conditionalvariances and conditional covariances of state variables.
DGP5 is a mixture of Gaussian and CIR diffusion processes,where X1t , X2t is a correlated 2-factor CIR diffusion process andX3t is a univariate Gaussian process. Under DGP5, model (6.1) ismisspecified for the conditional variances of X1t and X2t and theconditional covariance between X1t and X2t . The conditionalmeansof X1t , X2t and X3t and the conditional covariances between X1t andX3t , X2t and X3t are correctly specified.
For each of DGPs 1–5, we generate 500 data sets of the randomsample Xt
T1t=1 by the Euler scheme, for T = 250, 500 and
1000 respectively at the monthly sample frequency. For eachdata set, we estimate model (6.1) via MLE. Because model (6.1)is misspecified under all five DGPs, our omnibus test Q (0, 0) isexpected to have nontrivial power under DGPs 1–5, provided thesample size T is sufficiently large. We will also examine howdiagnostic tests Q (m, 0) for |m| > 0 can reveal information aboutvarious model misspecifications.
6.2. Monte Carlo evidence
To reduce computational costs, we generate u and v from aN (0, I3) distribution, with each u and v having 30 grid pointsin R3 respectively. We use the Bartlett kernel, which has abounded support and is computationally efficient. Our simulationexperience suggests that the choices of W (·) and k (·) have littleimpact on both size and power of the Q (m, 0) tests.17 LikeHong (1999), we use a data-driven p via a plug-in method thatminimizes the asymptotic integrated mean squared error of thegeneralized cross-spectral F (ω,u, v), with the Bartlett kernel k (·)used in some preliminary generalized cross-spectral estimators. Toexamine the sensitivity of the choice of the preliminary bandwidthp on the size and power of the tests, we consider p in the range of10–40.18
Table 1 reports the rejection rates (in terms of percentage)of Q (m, 0) under DGP0 at the 10% and 5% significance levels,using the asymptotic theory. The omnibus Q (0, 0) test tends tounderreject when T = 250, but it improves as T increases. Thesize Q (0, 0) is not very sensitive to the preliminary lag order p. Forexample, when T = 250, the rejection rate at the 5% level attainsits maximum 3.8% at p = 10, and attains its minimum 2.5% at p =
31, 33, 34. For T = 1000, the rejection rate at the 5% level attainsits maximum 6.2% at p = 17, 18, 19, 35, and attains its minimum5.6% at p = 11. We also consider the diagnostic tests Q (m, 0) for|m| = 1, 2, which check model misspecifications in conditionalmeans, conditional variances and conditional correlations of statevariables. The Q (m, 0) tests have similar size patterns to Q (0, 0),except that some of them tend to overreject a bit when T = 1000.Overall, both omnibus and diagnostic tests have reasonable sizesat both the 10% and 5% levels and the sizes are robust to the choiceof the preliminary lag order p.
Next, we turn to examine the power of Q (m, 0). Tables 2–6report the rejection rates of Q (m, 0) under DGPs 1–5 at the10% and 5% levels respectively. Under DGP1, model (6.1) ignoresnonzero constant correlations in drifts. The omnibus test Q (0, 0)is able to detect such model misspecification. The rejection rateof Q (0, 0) is about 65% at the 5% level when T = 1000. Becauseonly the misspecifications of the conditional means in X2t andX3t are time-varying when model (6.1) is used to fit the datafrom DGP1, we expect that the mean tests, Q ((0, 1, 0)′, 0) and
17 We have tried the Parzen kernel for k (·) and the Uniform(−2√3, 2
√3) CDF for
W (·), obtaining similar results.18 We have tried p in the range of 1–10 and results are similar.
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 279
Notes: (i) DGP0 is an uncorrelated Gaussian diffusion process, given in Eq. (6.1);(ii) Q (0, 0) is the omnibus test; Q1, Q2 and Q3 are conditional mean tests, conditional variance tests and conditional correlation tests respectively;(iii) The p values are based on the results of 1000 iterations.
Notes: (i) DGP1 is a correlated Gaussian diffusion process with correlation in drift, given in Eq. (6.4);(ii) Q (0, 0) is the omnibus test; Q1, Q2 and Q3 are conditional mean tests, conditional variance tests and conditional correlation tests respectively;(iii) The p values are based on the results of 500 iterations.
280 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
Notes: (i) DGP2 is an uncorrelated CIR diffusion process, given in Eq. (6.5);(ii) Q (0, 0) is the omnibus test; Q1, Q2 and Q3are conditional mean tests, conditional variance tests and conditional correlation tests respectively;(iii) The p values are based on the results of 500 iterations.
Notes: (i) DGP3 is a correlated Gaussian diffusion process with correlation in diffusion, given in Eq. (6.6);(ii) Q (0, 0) is the omnibus test; Q1, Q2 and Q3 are conditional mean tests, conditional variance tests and conditional correlation tests respectively;(iii) The p values are based on the results of 500 iterations.
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 281
Notes: (i) DGP4 is a correlated CIR diffusion process, given in Eq. (6.7);(ii) Q (0, 0) is the omnibus test; Q1, Q2 and Q3 are conditional mean tests, conditional variance tests and conditional correlation tests respectively;(iii) The p values are based on the results of 500 iterations.
Notes: (i) DGP5 is a mixture of Gaussian and CIR Processes, given in Eq. (6.8);(ii) Q (0, 0) is the omnibus test; Q1, Q2 and Q3 are conditional mean tests, conditional variance tests and conditional correlation tests respectively;(iii) The p values are based on the results of 500 iterations.
282 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
Q ((0, 0, 1)′, 0), will be powerful but the variance and covariancetests Q (m, 0), where |m| = 2, will have no power. This is indeedconfirmed in our simulation. Table 2 shows that the Q ((0, 1, 0)′, 0)and Q ((0, 0, 1)′, 0) tests are able to capture time-varying meanmisspecifications in X2t and X3t , as their rejection rates are about75% and 99% respectively at the 5% level when T = 1000.19However, the rejection rates of all variance and covariance testsQ (m, 0) for |m| = 2 are close to significance levels.
Under DGP2, model (6.1) ignores the so-called ‘‘level effect’’ indiffusions. The omnibus test Q (0, 0) has good power under DGP2,with a rejection rate of 75% at the 5% level when T = 1000. Notethat the omnibus test Q (0, 0) is less powerful than the conditionalvariance tests, because Q (0, 0) has to check all possible directionswhereas only the conditional variances are misspecified underDGP2. The variance tests Q ((2, 0, 0)′, 0), Q ((0, 2, 0)′, 0) andQ ((0, 0, 2)′, 0) have excellent power, which increases with T andapproaches unity when T = 1000. Interestingly, the powers of thediagnostic tests for conditional means and conditional covariancesare close to significance levels, indicating that these diagnostictests do not overreject correctly specified conditional means andconditional covariances of state variables.
Under DGP3, model (6.1) is correctly specified for bothconditional means and conditional variances of state variablesbut is misspecified for conditional correlations between statevariables, because it ignores the nonzero but time-invariantcorrelations in diffusions. As expected, the omnibus test Q (0, 0)has excellent power when model (6.1) is used to fit the datagenerated fromDGP3. The power of Q (0, 0) increases significantlywith the sample size T and approaches unity when T = 1000.However, our correlation diagnostic tests fail to capture themisspecifications in correlations, because they are time-invariant.
Under DGP4, model (6.1) is correctly specified for conditionalmeans of state variables but is misspecified for conditionalvariances and conditional correlations of state variables, becauseit ignores both the ‘‘level effect’’ and time-varying conditionalcorrelations in diffusions. The omnibus test Q (0, 0) has excellentpower when (6.1) is used to fit data generated from DGP4. This isconsistent with the fact that DGP4 deviates most frommodel (6.1).The Q (m, 0) tests with |m| = 2 have good power in detectingmisspecifications in conditional variances and correlations of statevariables. The mean tests Q (m, 0) with |m| = 1 have no power,because the conditional means of state variables are correctlyspecified.
Under DGP5, model (6.1) is correctly specified for conditionalmeans of state variables but is misspecified for conditionalvariances of X1t and X2t and the conditional correlation betweenX1t and X2t . Again, the omnibus test Q (0, 0) has good powerunder DGP5, with a rejection rate of 95% at the 5% level whenT = 1000. The variance tests Q ((2, 0, 0)′, 0), Q ((0, 2, 0)′, 0) andthe covariance test Q ((1, 1, 0)′, 0) have excellent power, whichincreases with T and attains unity when T = 1000. The powersof other diagnostic tests are close to significance levels, becausethose conditional moments are correctly specified.
To sum up, we observe:
• The Q (m, 0) tests have reasonable sizes in finite samples.Although the omnibus test Q (0, 0) tends to underreject whenT = 250, it improves dramatically as the sample size Tincreases. The sizes of all tests Q (m, 0) are robust to the choice
19 The power of themean test Q ((1, 0, 0) , 0) for the first state variable X1t is closeto the significance level. This is expected because the drift function ofX1t is correctlyspecified.
of a preliminary lag order used to estimate the generalizedcross-spectrum.
• The omnibus test Q (0, 0) has good omnibus power in detectingvarious model misspecifications. It has reasonable power evenwhen the sample size T is as small as 250. This demonstrates thenice feature of the proposed cross-spectral approach, which cancapture various model misspecifications.
• The diagnostic tests Q (m, 0) can check various specific aspectsof model misspecifications. Generally speaking, the mean tests,Q (m, 0), with |m| = 1, can detect misspecification indrifts; the variance and covariance tests Q (m, 0), with |m| =
2, can check misspecifications in variances and covariancesrespectively. However, the correlation tests fail to detectneglected nonzero but time-invariant conditional correlationsin diffusions.
7. Conclusion
The CCF-based estimation of multivariate continuous-timemodels has attracted increasing attention in econometrics. Wehave complemented this literature by proposing a CCF-basedomnibus specification test for the adequacy of a multivariatecontinuous-time model, which has not been attempted in theprevious literature. The proposed test can be applied to a varietyof univariate and multivariate continuous-time models, includingthose with jumps. It is also applicable to testing the adequacyof discrete-time dynamic multivariate distribution models. Themost appealing feature of our omnibus test is that it fully exploitsthe information in the joint dynamics of state variables and thuscan capture misspecification in modeling the joint dynamics,which may be easily missed by existing procedures. Indeed, whenthe underlying economic process is Markov, our omnibus testis consistent against any type of model misspecification. Weassume that the DGP of state variables may not be Markov.This not only ensures the power of the proposed tests againsta wider range of misspecification but also makes our approachapplicable to testing models with latent variables, such asSV models. Our omnibus test is supplemented by a class ofdiagnostic procedures, which is obtained by differentiating theCCF and focuses on various specific aspects of the joint dynamicssuch as whether there exists neglected dynamics in conditionalmeans, conditional variances, and conditional correlations of statevariables respectively. Such information is useful for practitionersin reconstructing a misspecified model. Our procedures are mostuseful when the CCF of a multivariate time series model hasa closed-form, as are the class of AJD models and the class oftime-changed Lévy processes that have been commonly used inthe literature. All test statistics follow a convenient asymptoticN(0, 1) distribution, and they are applicable to various estimationmethods, including suboptimal consistent estimators. Moreover,parameter estimation uncertainty has no impact on the asymptoticdistribution of the test statistics. Simulation studies show that theproposed tests perform reasonably well in finite samples.
Acknowledgments
We thank the co-editor, A. Ronald Gallant, and two refereesfor careful and constructive comments. We also thank Yacine Ait-Sahalia, Zongwu Cai, Jianqing Fan, Jerry Hausman, Roger Klein,Chung-ming Kuan, Arthur Lewbel, Norm Swanson, Yiu Kuen Tse,Chunchi Wu and Zhijie Xiao, and seminar participants at BostonCollege, Rutgers University and SingaporeManagement Universityfor their comments and discussions. Any remaining errors aresolely ours.
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 283
Appendix. Mathematical appendix
Throughout the appendix, we let Q (0, 0) be defined in the sameway as Q (0, 0) in (3.8) with the unobservable generalized residualsample Zt(u, θ0)
Tt=1 replacing the estimated generalized residual
sample Zt(u, θ)Tt=1. Also, C ∈ (1,∞) denotes a generic boundedconstant.
Proof of Theorem 1. The proof of Theorem1 consists of the proofsof Theorems A.1 and A.2 below.
Theorem A.1. Under the conditions of Theorem 1, Q (0, 0) −
Q (0, 0)p
→ 0.
Theorem A.2. Under the conditions of Theorem 1 and q = p1+1
4b−2
(ln2 T )1
2b−1 , Q (0, 0) d→N(0, 1).
Proof of Theorem A.1. Put Tj ≡ T −|j|, and let Γj(u, v) be definedin the same way as Γj(u, v) in (3.4), with Zt(u, θ) replaced byZt(u, θ0). To show Q (0, 0)− Q (0, 0)
p→ 0, it suffices to show
D−12 (0, 0)
∫∫ T−1−j=1
k2(j/p)Tj|Γj(u, v)|2 − |Γj(u, v)|2
× dW (u)dW (v)
p→ 0, (A.1)
p−1[C(0, 0) − C(0, 0)] = OP(T−
12 ), and p−1
[D(0, 0) − D(0, 0)] =
oP (1), where C(0, 0) and D(0, 0) are defined in the same way asC(0, 0) and D(0, 0) in (3.8), with Zt(u, θ) replaced by Zt(u, θ0). Forspace, we focus on the proof of (A.1); the proofs for p−1
[C(0, 0)−
C(0, 0)] = OP(T−12 ) and p−1
[D(0, 0) − D(0, 0)] = oP (1)are straightforward. We note that it is necessary to obtain theconvergence rate OP(pT−
where Re (A2) is the real part of A2 and Γj(u, v)∗ is the complexconjugate of Γj(u, v). Then, (A.1) follows from Propositions A.1 andA.2 below, and p → ∞ as T → ∞.
Proposition A.1. Under the conditions of Theorem 1, A1 = OP(1).
Proposition A.2. Under the conditions of Theorem 1, p−12 A2
p→ 0.
Proof of Proposition A.1. Put ψt(v) ≡ eiv′Xt − ϕ (v) and ϕ(v) ≡
E(eiv′Xt ). Then straightforward algebra yields that for j > 0,
Γj(u, v)− Γj(u, v) = T−1j
T−t=j+1
Zt(u, θ)− Zt(u, θ0)
ψt−j (v)
+ϕ (v)− ϕj (v)
T−1j
T−t=j+1
Zt(u, θ)− Zt (u, θ0)
= B1j(u, v)+ B2j(u, v), say. (A.3)
It follows that A1 ≤ 2∑2
a=1∑T−1
j=1 k2(j/p)Tj
|Baj(u, v)|2dW (u)dW (v). Proposition A.1 follows from Lemmas A.1 and A.2 below,and p/T → 0.
Lemma A.1.∑T−1
j=1 k2(j/p)Tj
|B1j(u, v)|2dW (u)dW (v) = OP(1).
Lemma A.2.∑T−1
j=1 k2(j/p)Tj
|B2j(u, v)|2dW (u)dW (v) = OP
(p/T ).
We now show these lemmas. Throughout, we put aT (j) ≡
k2(j/p)T−1j .
Proof of Lemma A.1. A second order Taylor series expansionyields
for some θ between θ and θ0.For the second term in (A.4), we have
T−1−j=1
k2 (j/p) Tj
∫∫ B12j (u, v)2 dW (u) dW (v)
≤ C√T (θ − θ0)
4 ∫∫ T−1
T−t=1
supu∈RN
supθ∈2
∂2
∂θ∂θ′
× ϕ (u, t|It−1, θ)
2
T−1−j=1
αT (j)
×dW (u) dW (v) = OP (p/T ) ,
where we made use of the fact thatT−1−j=1
aT (j) =
T−1−j=1
k2(j/p)T−1j = O(p/T ), (A.5)
given p = cTλ for λ ∈ (0, 1), as shown in Hong (1999, (A.15),p. 1213).
For the third term in (A.4), we haveT−1−j=1
k2 (j/p) Tj
∫∫ B13j (u, v)2 dW (u) dW (v)
≤ 4T−1−j=1
aT (j)∫∫
T−t=j+1
ϕ u, t|It−1, θ
−ϕu, t|IĎ
t−1, θ 2
dW (u) dW (v) = OP (p/T ) , (A.6)
where we have used Assumptions A.5–A.7.
284 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
For the first term in Eq. (A.4), we haveT−1−j=1
k2 (j/p) Tj
∫∫ B11j (u, v)2 dW (u) dW (v)
≤
√T (θ − θ0)
2 T−1−j=1
k2 (j/p)
×
∫∫ T−1j
T−t=j+1
∂
∂θϕ (u, t|It−1, θ0) ψt−j (v)
2
× dW (u) dW (v)= OP (1) , (A.7)
as is shown below: Put ηj(u, v) ≡ E[∂∂θϕ (u, t|It−1, θ0) ψt−j(v)]
= cov[ ∂∂θϕ (u, t|It−1, θ0) , ψt−j(v)]. Then we have supu,v∈R2N Σ∞
j=1ηj(u, v) ≤ C by Assumption A.4. Next, expressing the momentsby cumulants via well-known formulas (e.g., Hannan (1970, (5.1),p. 23), for real-valued processes), we can obtain
TjE
T−1j
T−t=j+1
∂
∂θϕ (u, t|It−1, θ0) ψt−j(v)− ηj(u, v)
2
⩽
Tj−τ=−Tj
cov ∂∂θϕ u,max(0, τ )+ 2|Imax(0,τ )+1, θ0,
∂
∂θϕu,max(0, τ )+ 2 − τ |Imax(0,τ )+1−τ , θ0
′× |Ωτ (v,−v)| +
Tj−τ=−Tj
ηj+|τ |(u,−v) ηj−|τ |(u, v)
+
Tj−τ=−Tj
κj,|τ |,j+|τ |(u, v)
⩽ C, (A.8)
given Assumption A.4, where κj,l,τ (v) is the fourth order cumulantof the joint distribution of theprocess
∂∂θϕ (u, t|It−1, θ0) , ψt−j(v),
∂∂θϕ (u, t|It−1, θ0) , ψ
∗
t−j(v). See also (A.7) ofHong (1999, p. 1212).Consequently, from (A.5), (A.8), |k(·)| ≤ 1, and p/T → 0, we have
T−1−j=1
k2(j/p)E∫∫ T−1
j
T−t=j+1
∂
∂θϕ (u, t|It−1, θ0) ψt−j(v)
2
× dW (u)dW (v)
≤ CT−1−j=1
∫∫ ηj(u, v)2 dW (u)dW (v)+ CT−1−j=1
aT (j)
= OP(1)+ OP(p/T ) = OP(1).
Hence (A.7) is OP(1). The desired result of Lemma A.1 follows from(A.5)–(A.7).
Proof of Lemma A.2. We first decompose B2j (u, v) to
B2j(u, v) =ϕ (v)− ϕj (v)
T−1j
×
T−t=j+1
ϕ (u, t|It−1, θ0)− ϕ
u, t|It−1, θ
+ϕ (v)− ϕj (v)
T−1j
T−t=j+1
ϕu, t|It−1, θ
− ϕ
u, t|IĎ
t−1, θ
= B21j (u, v)+ B22j (u, v) , say.
For the first term, we haveT−1−j=1
k2 (j/p) Tj
∫∫ B21j (u, v)2 dW (u) dW (v)
≤
T−1−j=1
k2 (j/p) Tj
∫∫ ϕ (v)− ϕj (v)2 T−1
j
×
T−t=j+1
ϕ (u, t|It−1, θ0)− ϕu, t|It−1, θ
2× dW (u) dW (v)
= OP (p/T ) , (A.9)
where we made use of the fact that Eϕ (v)− ϕj (v)
2 ≤ CT−1j
given Assumption A.4 andT−
t=j+1
ϕ (u, t|It−1, θ0)− ϕu, t|It−1, θ
2 ≤ Tjθ − θ0
2 T−1j
×
T−t=1
supu∈RN
supθ∈2
∂∂θϕ (u, t|It−1, θ)
2 = OP(1).
For the second term, we haveT−1−j=1
k2 (j/p) Tj
∫∫ B22j (u, v)2 dW (u) dW (v)
≤
T−1−j=1
k2 (j/p) Tj
∫∫ ϕ (v)− ϕj (v)2 T−1
j
×
T−t=j+1
supu∈RN
supθ∈2
ϕ (u, t|It−1, θ)− ϕu, t|IĎ
t−1, θ2
× dW (u) dW (v)= OP (p/T ) , (A.10)
where we made use of the fact that Eϕ (v)− ϕj (v)
2 ≤ CT−1j
given Assumptions A.4–A.7. The desired result of Lemma A.2follows from (A.9) and (A.10).
Proof of Proposition A.2. Given the decomposition in (A.3), wehave[Γj(u, v)− Γj(u, v)]Γj(u, v)∗
≤
2−a=1
|Baj(u, v)||Γj(u, v)|,
where the Baj(u, v) are defined in (A.3), a = 1, 2.We first consider the term with a = 2. By the Cauchy–Schwarz
inequality, we haveT−1−j=1
k2(j/p)Tj
∫∫|B2j(u, v)||Γj(u, v)|dW (u)dW (v)
≤
T−1−j=1
k2(j/p)Tj
∫∫|B2j(u, v)|2dW (u)dW (v)
12
×
T−1−j=1
k2(j/p)Tj
∫∫|Γj(u, v)|2dW (u)dW (v)
12
= OP(p12 /T
12 )OP(p
12 ) = oP(p
12 ), (A.11)
given Lemma A.2, and p/T → 0, where p−1∑T−1j=1 k2(j/p)Tj
|Γj(u, v)|2dW (u)dW (v) = OP(1) by Markov’s inequality, them.d.s. hypothesis of Zt (u, θ0), and (A.5).
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 285
For a = 1, by (A.4) and the triangular inequality, we haveT−1−j=1
k2(j/p)Tj
∫∫|B1j(u, v)‖Γj(u, v)|dW (u)dW (v)
≤
T−1−j=1
k2(j/p)Tj
∫∫|B11j(u, v)‖Γj(u, v)|dW (u)dW (v)
+
T−1−j=1
k2(j/p)Tj
∫∫|B12j(u, v)‖Γj(u, v)|
× dW (u)dW (v)+
T−1−j=1
k2(j/p)Tj
×
∫∫|B13j(u, v)‖Γj(u, v)|dW (u)dW (v) . (A.12)
For the first term in (A.12), we haveT−1−j=1
k2(j/p)Tj
∫∫ B11j(u, v) Γj(u, v)
dW (u)dW (v)
≤
θ − θ0
T−1−j=1
k2(j/p)Tj
×
∫∫ T−1j
T−t=j+1
∂
∂θϕ (u, t|It−1, θ0) ψt−j(v)
×Γj(u, v)
dW (u)dW (v)
= OP(1 + p/T12 ) = oP(p
12 ),
given p → ∞, p/T → 0, Assumptions A.2, A.3, A.6 and A.7, andTjE|Γj(u, v)|2 ≤ C . Note that we have made use of the fact that
E
T−1j
T−t=j+1
∂
∂θϕ (u, t|It−1, θ0) ψt−j(v)
Γj(u, v)
≤
E
T−1j
T−t=j+1
∂
∂θϕ (u, t|It−1, θ0) ψt−j(v)
2 1
2
×
EΓj(u, v)
2 12
≤ C[ηj(u, v)+ CT
−12
j
]T
−12
j ,
by (A.7), and consequently,θ − θ0
T−1−j=1
k2(j/p)TjE∫∫ T−1
j
T−t=j+1
∂
∂θϕ
× (u, t|It−1, θ0) ψt−j(v)
Γj(u, v) dW (u)dW (v)
≤ CT−1−j=1
∫∫ ηj(u, v) dW (u)dW (v)
+ CT−12
T−1−j=1
k2(j/p) = O(1 + p/T12 ),
given |k(·)| ≤ 1 and Assumption A.7.For the second term in (A.12), we have
T−1−j=1
k2(j/p)Tj
∫∫ B12j(u, v) Γj(u, v)
dW (u)dW (v)
≤ CTθ − θ0
2 T−1T−
t=j+1
supu∈RN
supθ∈2
∂2
∂θ∂θ′ϕ (u, t|It−1, θ)
×
T−1−j=1
k2(j/p)∫∫ Γj(u, v)
dW (u)dW (v)
= OP
p/T
12
,
by the Cauchy–Schwarz inequality, Markov’s inequality, Assump-tions A.2, A.3, A.6 and A.7, and E|Γj (u, v) |2 ≤ CT−1
j .For the third term in (A.12), we have
T−1−j=1
k2(j/p)Tj
∫∫ B13j(u, v) Γj(u, v)
dW (u)dW (v)
≤ CT−
t=1
supu∈RN
supθ∈2
ϕ (u, t|It−1, θ)− ϕu, t|IĎ
t−1, θ
×
T−1−j=1
k2(j/p)∫∫ Γj(u, v)
dW (u)dW (v)
= OP(p/T12 ),
by the Cauchy–Schwarz inequality, Markov’s inequality, Assump-tions A.2, A.3 and A.5–A.7, and E|Γj (u, v) |2 ≤ CT−1
j . Hence, wehaveT−1−j=1
k2(j/p)Tj
∫∫|B1j(u, v)||Γj(u, v)|dW (u)dW (v)
= OP
1 + p/T
12
+ OP
p/T
12
+ OP
p/T
12
= oP
p
12
. (A.13)
Combining (A.11) and (A.13) then yields the result of thisproposition.
Proof of Theorem A.2. We first state a lemma, which will be usedin the proof.
Lemma A.3. Suppose that Inm are the σ -fields generated by a
stationary β-mixing process Xi with mixing coefficient β (i). For somepositive integers m let yi ∈ I
tisi where s1 < t1 < s2 < t2 < · · · < tm
and suppose ti − si > τ for all i. Assume that
‖yi‖pipi = E |yi|pi < ∞,
for some pi > 1 for which Q =∑l
i=11pi< 1. ThenE
l∏
i=1
yi
−
l∏i=1
E (yi)
≤ 10 (l − 1) β(1−Q ) (τ )
l∏i=1
‖yi‖pi .
Proof of Lemma A.3. See Theorem 5.4 of Roussas and Ioannides(1987).
Let q = p1+1
4b−2 (ln2 T )1
2b−1 . We shall show Propositions A.3 andA.4 below.
Proposition A.3. Under the conditions of Theorem 1,
p−12
T−1−j=1
k2(j/p)Tj
∫∫ Γj(u, v)2 dW (u) dW (v)
= p−1/2C(0, 0)+ p−1/2Vq + oP(1),
where Vq =∑T
t=2q+2
Zt (u, θ0)
∑qj=1 aT (j)ψt−j(v)[
∑t−2q−1s=1 Z∗
s(u, θ0) ψ
∗
s−j(v)]dW (u) dW (v).
286 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
Proposition A.4. Under the conditions of Theorem 1, D−1/2(0, 0)Vq
d→N(0, 1).
Proof of Proposition A.3. We first decompose
T−1−j=1
k2(j/p)Tj
∫∫ Γj(u, v)2 dW (u) dW (v)
=
T−1−j=1
aT (j)∫∫ T−
t=j+1
Zt (u, θ0) ψt−j(v)
2
dW (u)dW (v)
+
T−1−j=1
aT (j)∫∫ T−
t=j+1
Zt (u, θ0)
2 ϕ (v)− ϕj (v)
2× dW (u) dW (v)
+ 2 ReT−1−j=1
αT (j)∫∫
T−t=j+1
Zt (u, θ0) ψt−j(v)
×
T−
t=j+1
Zt (u, θ0)ϕ (v)− ϕj (v)
∗
dW (u) dW (v)
≡ M + R1 + 2 Re(R2). (A.14)
Next we write
Mq =
T−1−j=1
aT (j)∫∫ T−
t=j+1
|Zt (u, θ0)|2ψt−j(v)
2× dW (u)dW (v)+ 2 Re
T−1−j=1
aT (j)
×
∫∫ T−t=j+2
t−1−s=j+1
Zt (u, θ0) Z∗
s (u, θ0) ψt−j(v)ψ∗
s−j(v)
× dW (u) dW (v) ≡ C1(0, 0)+ 2 Re(U), (A.15)
where we further decompose
U =
T−t=2q+2
∫∫Zt (u, θ0)
t−1−j=1
aT (j)ψt−j(v)
×
t−2q−1−s=j+1
Z∗
s (u, θ0) ψ∗
s−j(v)dW (u) dW (v)
+
T−t=2
∫∫Zt (u, θ0)
t−1−j=1
aT (j)ψt−j(v)
×
t−1−s=max(j+1,t−2q)
Z∗
s (u, θ0) ψ∗
s−j(v)dW (u)dW (v)
≡ U1 + R3, (A.16)
where in the first term U1, we have t − s > 2q so that we canbound it with the mixing inequality. In the second term R3q, wehave 0 < t − s ≤ 2q. Finally, we write
U1 =
T−t=2q+2
∫∫Zt (u, θ0)
q−j=1
aT (j)ψt−j(v)
×
t−2q−1−s=j+1
Z∗
s (u, θ0) ψ∗
s−j(v)dW (u) dW (v)
+
T−t=2q+2
∫∫Zt (u, θ0)
t−1−j=q+1
aT (j)ψt−j(v)
×
t−2q−1−s=j+1
Z∗
s (u, θ0) ψ∗
s−j(v)dW (u)dW (v)
≡ Vq + R4, (A.17)
where the first term Vq is contributed by the lag orders j from 1 toq; and the second term R4 is contributed by the lag orders j > q. Itfollows from (A.14)–(A.17) that
T−1−j=1
k2(j/p)Tj
∫∫ Γj(u, v)2 dW (u)dW (v)
= C1(0, 0)+ 2 Re(Vq)+ R1 − 2 Re(R2 − R3 − R4).
It suffices to show Lemmas A.4–A.8 below, which imply p−12 [C1
(0, 0) − C(0, 0)] = oP(1) and p−12 Ra = oP(1) given q =
p1+1
4b−2 (ln2 T )1
2b−1 and p = cTλ for 0 < λ <3 +
14b−2
−1.
Lemma A.4. Let C1(0, 0) be defined as in (A.15). Then C1(0, 0) −
C(0, 0) = OP(p/T12 ).
Lemma A.5. Let R1 be defined as in (A.14). Then R1 = OP(p/T ).
Lemma A.6. Let R2 be defined as in (A.14). Then R2 = OP(p/T12 ).
Lemma A.7. Let R3 be defined as in (A.16). Then R3 = OP(qp/T12 ).
Lemma A.8. Let R4 be defined as in (A.17). Then R4 = OP(p2bln T/q2b−1).
Proof of Lemma A.4. By Markov’s inequality and E|C1(0, 0) −
C(0, 0)| ≤ Cp/T 1/2 given∑T−1
j=1 (j/p)aT (j) = O(p/T ) and E|ϕ(v)−
ϕj(v)|4 = O(T−2), which are shown in Hong (1999) and belowrespectively.
Let t1, . . . , t4 be distinct integers and |j| + 1 ≤ ti ≤ T , let|j| + 1 ≤ r1 < · · · < r4 ≤ T be the permutation of t1, . . . , t4in ascending order and let dc be the cth largest difference amongrl+1 − rl, l = 1, 2, 3.
By Lemma 1 of Yoshihara (1976), we have−|j|+1≤r1<···<r4≤T
r2−r1=d1
E eiv′Xr1−j − ϕ (v)
×
eiv
′Xr2−j − ϕ (v)
eiv′Xr3−j − ϕ (v)
eiv
′Xr4−j − ϕ (v)
≤
T−3−r1=|j|+1
T−2−r2=r1+max
j≥3(rj−rj−1)
T−1−r3=r2+1
T−r4=r3+1
4C1
1+δ βδ
1+δ (r2 − r1)
≤ 4C1
1+δ
T−3−r1=|j|+1
T−2−r2=r1+1
(r2 − r1)2 βδ
1+δ (r2 − r1)
≤ 4TC1
1+δ
T−r=|j|+1
r2βδ
1+δ (r) = O (T ) .
Similarly,−|j|+1≤r1<···<r4≤T
r4−r3=d1
E eiv′Xr1−j − ϕ (v)
×
eiv
′Xr2−j − ϕ (v)
eiv′Xr3−j − ϕ (v)
eiv
′Xr4−j − ϕ (v)
= O (T ) .
B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293 287
Further, it can be shown in a similar way that−|j|+1≤r1<···<r4≤T
r3−r2=d1
E eiv′Xr1−j − ϕ (v)
×
eiv
′Xr2−j − ϕ (v)
eiv′Xr3−j − ϕ (v)
eiv
′Xr4−j − ϕ (v)
= OT 2 .
Similar to the above, we can show that if rs are not distinct to eachother, we have−|j|+1≤t1,t2,t3≤Tt1,t2,t3 different
E eiv′Xt1−j − ϕ (v)2
×
eiv
′Xt2−j − ϕ (v)
eiv′Xt3−j − ϕ (v)
= O
T 2 ,
and−|j|+1≤t1,t2≤Tt1,t2 different
E eiv′Xt1−j − ϕ (v)2
eiv′Xt2−j − ϕ (v)
2= O
T 2 .
Therefore, Eϕ (v)− ϕj (v)
4 = OT−2
.
Proof of Lemma A.5.
E|R1| ≤
T−1−j=1
aT (j)∫∫ E
T−t=j+1
Zt (u, θ0)
4 1
2
×
Eϕ (v)− ϕj (v)
4 12dW (u) dW (v) = O (p/T ) ,
where we have used the fact E∑T
t=j+1 Zt (u, θ0)
4 ≤ CT 2j by
Rosenthal’s inequality and Eϕ (v)− ϕj (v)
4 = OT−2
.
Proof of Lemma A.6. By the m.d.s. property of Zt (u, θ0), theCauchy–Schwarz inequality, we have
E|R2| ≤ 2T−1−j=1
aT (j)∫∫ E
T−t=j+1
Zt (u, θ0) ψt−j (v)
2 1
2
×
E
T−t=j+1
Zt (u, θ0)ϕ (v)− ϕj (v)
2
12
dW (u) dW (v)
≤ 2T−1−j=1
aT (j)∫∫
ET−
t=j+1
Z2t (u, θ0) ψ
2t−j (v)
12
×
E
T−t=j+1
Zt (u, θ0)
4 1
4
×
Eϕ (v)− ϕj (v)
4 14dW (u) dW (v)
= Op/T
12
,
given Eϕ (v)− ϕj (v)
4 = OT−2
.
Proof of Lemma A.7. By them.d.s. property of Zt (u, θ0), Minkow-ski’s inequality and (A.5), we have
ER3
2 =
T−t=2
E
t−1−j=1
aT (j)∫∫
Zt (u, θ0) ψt−j(v)
×
t−1−s=max(j+1,t−2q)
Z∗
s (u, θ0) ψ∗
s−j(v)dW (u) dW (v)
2
⩽
T−t=2
t−1−j=1
aT (j)∫∫ E
Zt (u, θ0) ψt−j(v)
×
t−1−s=max(j+1,t−2q)
Z∗
s (u, θ0) ψ∗
s−j(v)
21/2
dW (u) dW (v)
2
≤ CTq2
T−1−j=1
aT (j)
2
= O(p2q2/T ).
Proof of Lemma A.8. By the m.d.s. property of Zt (u, θ0) andMinkowski’s inequality, we have
ER4
2 =
T−t=2q+2
E
t−1−j=q+1
aT (j)∫∫
Zt (u, θ0) ψt−j(v)
×
t−2q−1−s=j+1
Z∗
s (u, θ0) ψ∗
s−j(v)dW (u) dW (v)
2
⩽
T−t=2q+2
t−1−
j=q+1
aT (j)∫∫
EZt (u, θ0) ψt−j(v)
41/4
×
E
t−2q−1−s=j+1
Z∗
s (u, θ0) ψ∗
s−j(v)
41/4
dW (u) dW (v)
2
⩽ CT 2
T−1−
j=q+1
aT (j)
2
⩽ CT 2
T−1−
j=q+1
(j/p)−2bT−1j
2
= O(p4b ln2 T/q4b−2),
given Assumption A.6 (i.e., k(z) ⩽ C |z|−bas z → ∞).
Proof of Proposition A.4. We rewrite Vq =∑T
t=2q+2 Vq(t), where
Vq(t) =
∫∫Zt (u, θ0)
q−j=1
aT (j)ψt−j(v)Hj,t−2q−1(u, v)
× dW (u) dW (v),
andHj,t−2q−1(u, v) =∑t−2q−1
s=j+1 Z∗s (u, θ0)ψ
∗
s−j(v).We apply Brown’s
(1971) martingale limit theorem, which states var(2 Re Vq)−
12 2 Re
Vqd
→N(0, 1) if
var(2 Re Vq)−1
T−t=2q+2
2 Re Vq(t)
2 1×
2 Re Vq(t) > η · var(2 Re Vq)
12
→ 0 ∀η > 0, (A.18)
var(2 Re Vq)−1
T−t=2q+2
E
2 Re Vq(t)2
|It−1
p
→ 1. (A.19)
288 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
First, we compute var(2 Re Vq). By the m.d.s. property of Zt(u, θ0)under H0, we have
Proof of Theorem 2. The proof of Theorem2 consists of the proofsof Theorems A.3 and A.4 below.
Theorem A.3. Under the conditions of Theorem 2, (p12 /T )[Q (0, 0)−
Q (0, 0)]p
→ 0.
Theorem A.4. Under the conditions of Theorem 2,
(p12 /T )Q (0, 0)
p→(D (0, 0))−
12
∫∫∫ π
−π
|F(ω,u, v)
− F0(ω,u, v)|2dωdW (u)dW (v) .
Proof of Theorem A.3. It suffices to show that
T−1∫∫ T−1−
j=1
k2(j/p)Tj|Γj(u, v)|2 − |Γj(u, v)|2
× dW (u)dW (v)
p→ 0, (A.31)
292 B. Chen, Y. Hong / Journal of Econometrics 164 (2011) 268–293
p−1[C(0, 0) − C(0, 0)] = OP(1), and p−1
[D(0, 0) − D(0, 0)]p
→ 0,where C(0, 0) and D(0, 0) are defined in the same way as C(0, 0)and D(0, 0) in (3.8), with Zt(u) replaced by Zt(u, θ∗). Since theproofs for p−1
[C(0, 0) − C(0, 0)] = OP(1) and p−1[D(0, 0) −
D(0, 0)]p
→ 0 are straightforward, we focus on the proof of (A.31).From (A.5), the Cauchy–Schwarz inequality, and the fact thatT−1
∑T−1j=1 k2(j/p)Tj|Γj(u, v)|2dW (u)dW (v) = OP(1) as is
implied by Theorem A.4 (the proof of Theorem A.4 does notdepend on Theorem A.3), it suffices to show that T−1A1
p→ 0,
where A1 is defined as in (A.2). Given (A.3), we shall show thatT−1
∑T−1j=1 k2(j/p)Tj|Baj(u, v)|2dW (u)dW (v)
p→ 0, a = 1, 2.
We first consider a = 1. By the Cauchy–Schwarz inequality and|ψt−j(v)| ≤ 2, we have
|B1j(u, v)|2 ≤ CT−1j
T−t=j+1
ϕ u, t|It−1, θ∗− ϕ
u, t|It−1, θ
2+ CT−1
j
T−t=j+1
ϕ u, t|It−1, θ
− ϕu, t|IĎ
t−1, θ2
≤ CT−1j T
θ − θ0
2 T−1T−
t=1
supu∈RN
supθ∈8
∂∂θϕ (u, t|It−1, θ)
2+ CT−1
j
T−t=j+1
supu∈RN
supθ∈8
ϕ (u, t|It−1, θ)− ϕu, t|IĎ
t−1, θ2
= OPT−1j
.
It follows from (A.4) and Assumption A.7 that
T−1∫∫ T−1−
j=1
k2(j/p)Tj|B1j(u, v)|2dW (u)dW (v)
= OP(p/T ).
The proof for a = 2 is similar. This completes the proof forTheorem A.3.
Proof of Theorem A.4. The proof is very similar to Hong (1999,Proof of Thm. 5), for the case (m, l) = (0, 0). The consistency resultfollows from (a) p−1∑T
j=1 k4 (j/p) →
∞
0 k4 (ω) dω; (b) π
−π
|F (ω,u, v)− F (ω,u, v) |2 ×dωdW (u)dW (v) → 0; (c) C (0, 0) =
OP(p); (d) D (0, 0)→p D (0, 0).
Proof of Theorem 3. The proof is very similar to Theorem 1,with Z (m)t (0, θ), Z (m)t (0, θ0), Γ
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