Consistent nonparametric specification tests for stochastic volatility models based on the return distribution

Transcription

1 Consistent nonparametric specification tests for stocastic volatility models based on te return distribution Yang Zu City University London H. Peter Boswijk 2 University of Amsterdam April 29, Yang.Zu@city.ac.uk. Department of Economics, City University London, Nortampton Square, ECV 0HB, London, United Kingdom. Telepone: +44 ( Fax: +44 ( H.P.Boswijk@uva.nl. Amsterdam Scool of Economics, University of Amsterdam, Valckenierstraat 65-67, 08 XE Amsterdam, Te Neterlands. Telepone: +3 (

2 Abstract Tis paper develops nonparametric specification tests for stocastic volatility models by comparing te nonparametically estimated return density and distribution functions wit teir parametric counterparts. Asymptotic null distributions of te tests are derived and te tests are sown to be consistent. Extensive Monte Carlo experiments are performed to study te finite sample properties of te tests. Te tests are applied to an empirical dataset and we find te estimated stocastic volatility model is misspecified.

3 JEL Classification: C58, C2, C4. Keywords: nonparametric test, stocastic volatility models.

4 Introduction In tis paper we consider specification tests for a class of parametric stocastic volatility models, given by dy t = σ t db t, ( dσt 2 = b(σt 2 ; θdt + a(σt 2 ; θdw t, were (B t, W t t 0 is a bivariate standard Brownian motion process, were b and a are known functions, and were θ is an unknown parameter vector. Te model is tested witin a larger class of nonparametric stocastic volatility models dy t = σ t db t, (2 were (σ t t 0 is a stocastic process satisfying certain regularity conditions. Te model (2 is nonparametric in te sense tat tere is no parametric structure specified for te volatility process. Model ( is often used in financial econometrics to describe a logaritmic stock price process (Y t t 0, were (σ t t 0 is an unobserved spot volatility process. It includes popular models suc as te Hull-Wite model, te Heston model and te GARCH diffusion model, wic motivates te development of specification tests for tis class of models. Let Y be observed discretely at times t i = i, i = 0,,..., n. Consider te re-scaled -period return sequence X i = (Y ti Y ti = ti t i σ s db s, i =,..., n. (3 Let (X i n i= aving a stationary density, denoted by q(x, and let q(x; θ be its specification implied by te parametric model (. In tis paper we propose to test te specification ( by comparing te estimated parametric return density to its nonparametrically estimated counterpart. Stated formally, we are testing H 0 : q(x = q 0 (x {q(x; θ, θ Θ}, were Θ R k is te parameter space, and define θ 0 to be te true parameter under te null ypotesis: tat is, it satisfies q(x; θ 0 = q 0 (x. Specification tests based on te stationary marginal return distribution ave teir empirical justifications as discussed in Section 3.3 of Aït-Saalia, Hansen, and Sceinkman (200, reproducing te stationary distribution is an important aspect of structural economic modeling. Return distribution is also widely used as te basis to formulate specification tests for continuous-time diffusion processes, see e.g. Aït-Saalia (996 and Gao and King (2004. Admittedly, formulating te test based on q(.; θ will limit its power in 2

5 detecting certain deviations in te functions {b(.; θ, a(.; θ} ; owever, tests constructed tis way would still be an important first ceck because of its empirical significance in any structural modelling. Te problem could be solved by defining test statistics based on te transition distribution of te observed returns, we discuss tis issue in Section 7. To formulate te test statistic, one can compare eiter te density functions or te cumulative distribution functions. It is known from te literature tat generally speaking, density-based tests are more sensitive to local deviations, wereas distribution-based tests are more sensitive to global deviations (see e.g. Eubank and LaRiccia (992, Escanciano (2009 and Aït-Saalia, Fan, and Peng (2009, we tus consider bot in tis paper. A long-span asymptotic sceme is used in tis paper. Tat is, we consider te asymptotics wen n wit fixed. Tis is because model ( is often used to describe price processes observed at relatively low frequencies (usually daily, prominent microstructure noise effects in prices observed at iger frequencies make te model unsuitable for suc data. Trougout, we assume tat (X i n i= is a stationary and ergodic sequence, and tat it is β-mixing wit exponentially decaying coefficients. In Appendix A, ceckable sufficient conditions for tese properties to old in te parametric model are given. Te stocastic volatility model we consider ere is essentially a (partially observed two dimensional diffusion process, so our test is related to te vast literature of nonparametric test for diffusion models, suc as Aït-Saalia (996, Hong and Li (2005, Corradi and Swanson (2005, Li (2007, Cen et al. (2008, Aït-Saalia et al. (200, Kristensen (20 and Aït-Saalia and Park (202, among oters. However, te unobservability of te volatility process in Model ( makes te aforementioned researc not applicable. Closely related to tis paper is Corradi and Swanson (20, wo consider a conditional distribution based nonparametric test for stocastic volatility models. Oter tan te test statistics we consider are different from tat of Corradi and Swanson, te model we consider is also different. Corradi and Swanson consider te stocastic volatility model were te observed series is assumed to be strictly stationary; wile in our model we assume te observed return series (first difference of te observed series to be stationary and we allow te observed series to exibit say unit-root type of dynamics. From a practical perspective, te stocastic volatility model considered in Corradi and Swanson in more appropriate to be used wit interest rate data, were mean-reversion is often observed; wile our model is more appropriate for equity and excange rate price data, were unit-root beaviour is often observed. Zu (205 analyzes an alternative approac to a similar testing problem, by comparing te nonparametric kernel deconvolution estimator of te volatility density wit its parametric counterpart. Te structure of tis paper is as follows. Sections 2 and 3 discuss nonparametric and parametric estimation of te return density and distribution functions, respectively. Tat is, tere migt exist two different specifications {b(.; θ, a(.; θ} and { b(.; θ, ã(.; θ} leading to te same return distribution wit density q(x. 3

6 Section 4 defines te test statistics, derives teir asymptotic null distributions and consistency, and discusses using te bootstrap to approximate te null distribution. In section 5 Monte Carlo evidence for te size and power properties of te tests are given. In Section 6 we study an empirical application. Section 7 discusses possible extensions and concludes. Tecnical assumptions are collected in Appendix A. Te proofs for te teorems are collected in Appendix B. 2 Nonparametric estimation We now discuss te nonparametric estimation of density and distribution functions. In te nonparametric model (2, estimation of te stationary marginal return density and distribution functions is considered under te direct assumption tat te sequence (X i n i= is stationary, ergodic and β-mixing wit exponentially decaying coefficients. To estimate q(x, te stationary return density function of (X i n i=. Let n be a bandwidt, K(. be a kernel function. It is well known tat te density function q(x can be estimated by te kernel density estimator ˆq(x = n n n ( x Xi K. i= Under appropriate conditions on te bandwidt parameter and te kernel function, te consistency and asymptotic distribution of te kernel density estimator are classical results, we refer te readers to e.g. Pagan and Ulla (999. Denote te distribution function of te sequence (X i n i= to be Q(x. n Letting I(. denote te indicator function, te distribution function Q(x can be estimated by te empirical distribution function ˆQ(x = n n I(X i x. i= Te consistency and asymptotic normality of te empirical distribution function are classical results in statistics, see e.g. Capter 9 of Van der Vaart (2000 for te results wit independent and identically distributed data. For stationary dependent data, suc properties still old wit te application of Ergodic Law of Large Numbers and Central Limit Teorem for dependent data, see Appendix A.5 in Pagan and Ulla (999 for a summary. 4

7 3 Parametric estimation Given a parameterization {b(x; θ, a(x; θ}, to obtain te parametric estimate of te functions q(x; θ and Q(x; θ, we first need an estimate for te parameter vector, denoted as ˆθ, ten evaluate te two functions given ˆθ. On te one and, parametric estimation of stocastic volatility model is by no means an easy task; substantial researc efforts were devoted to it in te past decades. Here we first briefly review te existing metods and just assume we ave a parametric estimator satisfying certain conditions. On te oter and, evaluating te two functions given ˆθ is also not trivial, because te density and distribution functions of te observed returns usually do not ave closed-form expressions and one needs to resort to approximation metods to evaluate tem. 3. Parametric estimation of stocastic volatility models Many efforts ave been devoted to te estimation of stocastic volatility models in te past decades. For a review, see e.g. Renault (2009. Here we do not confine to any particular parametric estimation metod, but only give conditions tat a parametric estimator sould satisfy. We will need different assumptions for te density function based test and te distribution function based test. For te density function based test, we only need to assume te parametric estimator ˆθ n is n-consistent. We will also need te parametrization to be smoot. (Pa Under te null ypotesis, ˆθ θ 0 = O p (n /2, and q(x, θ is Lipscitz in te parameter θ wit te Lipscitz constant L(x to be square integrable. For te distribution function based test, owever, stronger assumptions are needed te estimator as to satisfy a certain first order asymptotic expansion, wic will be a non-vanising part of te asymptotic distribution. We also need te parameterization to be differentiable. (Pb Under te null ypotesis, n(ˆθ θ0 = n n ψ θ0 (X i + o p (, i= wit P θ0 ψ θ0 = 0 and P θ0 ψ θ0 2 <, and Q(x; θ is differentiable wit respect to θ. 5

8 3.2 Approximating parametric density and distribution function Wen no closed-form expressions for te density and distribution functions exist, we can in principle use an Euler sceme to simulate te process and ence evaluate intractable functionals of te process. Given an estimate ˆθ, te parameterization b(., θ and a(., θ, te observation interval, and te objective variables X i = ti t i σ s db s, i =,..., n, to be approximated, we first coose an integer m as te steps to simulate witin te interval, and anoter integer M as te number of -interval returns, suc tat we simulate te process Y wit step size δ = /m for m M steps. Ten take first differences to get δ-returns, and aggregate and rescale over every m returns to get M simulated -returns, X i, i =,..., M. Using a kernel density estimator we can approximate q(x; ˆθ from te simulated sample wit q (x; ˆθ = M M M ( x X K i i= M, were K(. is a kernel function, and M is te bandwidt parameter. Standard consistency results for te kernel density estimator and convergence teorems for te Euler sceme simulation of stocastic differential equations imply tat wen M, M 0 and m, q (x; ˆθ q(x; ˆθ pointwise in x R. Te convergence sould be understood as in te probability space of Monte Carlo simulation. For te tecnical conditions on te kernel function K(., bandwidt M and te consistency result for te kernel density estimator, we refer to, e.g. Section of Pagan and Ulla (999. For te convergence result of te Euler simulation metod, we refer to Capter 9 of Kloeden and Platen (992. Te accuracy of tis approximation is determined by te number M and m tat we coose. Because tese numbers do not ave to be bounded by te sample size n, tey can be cosen very large to make te approximation error arbitrarily small. Te parametric distribution function ˆQ(x, ˆθ can be approximated analogously using te empirical distribution function wit te simulated data. In tis reason, in te following we take te approximated q (x; θ and Q (x; θ to be equal to te corresponding exact ones q(x; θ and Q(x; θ to avoid complication. Remark Wit te purpose to approximate te conditional distribution in a stocastic volatility model, Bardwaj, Corradi, and Swanson (2008 propose to simulate multiple pats of fixed lengt starting from a common initial value and take average across tese pats. We remark tat wen te simulated process is stationary, our simulation of a long pat can be understood as multiple pats of sorter lengt and tus te two approaces can bot simulate te intended observations. One difference is tat our pats ave different starting values wile tey ave a common one. Tey impose tis restriction in 6

9 order to evaluate te conditional distribution given a certain value. However, we remark tat for te purpose of conditional distribution function approximation given a certain value, simulating pats conditional on tat value is not necessary. Remark 2 Bardwaj, Corradi, and Swanson (2008 notice tat te Milestein sceme simulation for te stocastic volatility is not convergent if te commutative condition is not satisfied, wic is te case for most of te stocastic volatility models wit leverage effects. Tey use a generalized Milestein sceme from Kloeden and Platen (992 to deal wit tis problem. We empasize tat te Euler s sceme is valid bot in univariate and multivariate diffusions, and it is convergent bot in te weak and in te strong sense wen te simulating interval goes to 0. Wen used wit stocastic volatility model, usually te Eulers Sceme is applied to a finer grid witin te needed sampling interval, as in te metod used in tis paper. Tis will not cause te stocastic integral problem as discussed in Bardwaj, Corradi, and Swanson (2008. Actually te Euler s sceme is widely used in te literature to simulate stocastic volatility models wit leverage effects. Tis include Andersen and Lund (997, Bollerslev and Zou (2002, Aït-Saalia and Kimmel (2007 and Barndorff-Nielsen, Hansen, Lunde, and Separd (2008, to name just a few. 4 Test statistics and asymptotic properties 4. Asymptotic null distribution and te consistency Define T = n /2 R (ˆq(x K q(x; ˆθ 2 dx, were K q(x; ˆθ = R K (x yq(y; ˆθdy is te convolution of K (x = K(x// wit q(x; ˆθ, te function K(. and bandwidt are te same as used in te definition of ˆq(x. Using te convoluted return density in te formulation of te test statistic corrects te bias of te test statistic and deliver better asymptotic properties of te test statistic, we refer to Fan (994 for a discussion of tis issue in te general density test problem wit i.i.d. data. Teorem Under te null ypotesis, and if (SV0 (SV5, (N (N4, and (Pa are satisfied, ten d (T /2 N R ( 0, 2 q0(xdx 2 R K 2 (udu R ( K (2 (v 2 dv, 7

10 were K (2 (v denote te convolution of te kernel function K wit itself. Let ˆσ 2 = 2 n n i= ( ˆq(X i K (2 (v 2 dv, R wic is a consistent estimator of te variance of te asymptotic distribution, ten T 2 = T /2 R K2 (udu ˆσ d N(0,. Te test T is not pivotal as it depends on te unknown density q 0 (x. However te corresponding studentized test T 2 is pivotal. A Cramer-von Mises type statistic can be formulated by comparing distribution estimates: T 3 = n ( Q(x Q(x; ˆθ n 2 dq(x; ˆθ. R Teorem 2 Under te null ypotesis, and if (SV0 (SV5, (N2 and (Pb are satisfied, ten as n T 3 d R ( 2 G Q I( x G Q ψθ T Q(x, θ 0 ( θ=θ0 dq(x, (4 θ were G Q is a Q-Brownian bridge indexed by F = {I( x, x R} {ψ θ ( }, wit zero mean and te covariance function Γ(f, g = lim k i= Cov(f(X k, g(x i wit f, g F. Te above limiting distribution of T 3 is a functional of a Brownian bridge process, and it depends on te model structure (tus not model-free as well as te unknown parameter values. For tis reasons, tis limiting teorem cannot be used directly to define critical values of te test. We discuss te approximation metod to obtain test critical values in te next Section 4.2. We ten look at te asymptotic power of tese tests under fixed alternatives. To be specific, we consider H : {q(x = q (x q(x; θ, θ Θ}. We will need assumptions on te parametric estimator under te alternative model. (Pa Under te alternative ypotesis H, ˆθ θ = O p (n /2, were θ is te pseudo true value of te model corresponding to q (x. 8

11 Teorem 3 Assume Conditions (SV0 (SV5 and Assumptions (N (N4, and (Pa in Appendix A; let α (0, be a level of significance, and Z α be te α quantile of te standard normal distribution. Ten under H, P (T /2 K 2 (udu > ˆσZ α, R and P (T 2 > Z α. For te distribution based test, we assume tat under te alternative ypotesis te parametric estimator satisfy (Pb Under te alternative ypotesis H, n(ˆθ θ = n n ψ θ (X i + o p (, i= wit P θ ψ θ θ. = 0 and P θ ψ θ 2 <, and Q(x; θ is differentiable wit respect to Teorem 4 Under te null ypotesis, and if (SV0 (SV5, (N2 and (Pb are satisfied; let α (0, be a level of significance, and c α be te α quantile of te limiting distribution in (4, ten as n, P (T 3 > c α. As wit most nonparametric tests, bot tests are consistent. Tat is, tey can detect any fixed deviation to te true model as long as te sample size is sufficiently large. 4.2 Bootstrap null distribution In te literature of nonparametric goodness of fit test, it is a usual problem tat te asymptotic distribution of te density based nonparametric test statistic will provide a poor approximation for te null distribution in finite sample. For te distribution function based test, as we noted before tat te asymptotic distribution of te test statistic is not feasible. Tese problems lead us to develop a bootstrap approximation to te null distribution, wic is a common practice in nonparametric tests. Different bootstrap metods can be considered in nonparametric test. Parametric-type of bootstrap as been considered in e.g. Fan (995, Andrews (997, Franke, Kreiss, and Mammen (2002, Andrews (2005, Gao and Gijbels (2008 and Aït-Saalia, Fan, and Peng (2009among oters. A nonparametric-type block bootstrap as been considered in Bardwaj, Corradi, and Swanson (2008, Corradi and Swanson (20 and several of teir oter works. 9

12 We use a parametric bootstrap procedure to approximate te distributions of te tests under te null ypotesis. Parametric bootstrap is also called model based bootstrap. As contrary to classical bootstrap, were one generate bootstrap samples by resampling te available dataset, parametric bootstrap involves generating bootstrap samples by first estimating a parametric model and ten simulating data from te estimated parametric model (see Section 6.5 of Efron and Tibsirani (994. Te dependence of te bootstrap sample on te original data is only troug te estimated parameters. Te parametric bootstrap is in particular useful in approximating te null distribution in a testing context because it always simulates data based on te null model: it will mimic te null model bot under te null ypotesis and under te alternative ypotesis. In contrast, bootstrap procedures tat do not exploit te model structures will usually mimic te data generating process (alternative model under te alternative ypotesis. For example, in testing diffusion models, Corradi and Swanson (20 use a block bootstrap procedure. Since te block bootstrap procedure mimics te data generating process under te alternative ypotesis, te bootstrapped statistic cannot reproduce te null distribution under a misspecified model, and tey furter define a re-centered test statistic to make te block bootstrap to work. Te parametric bootstrap procedure is as follows (use T as an example: Step Given a parametric estimate ˆθ, and step size, simulate n (original sample size discretely observed returns {X i } n i=, wic is called one bootstrap sample. Notice tis step as to be done using te metod in Section 3.2 over a finer grid. Step 2 Wit tis bootstrap sample, compute te nonparametric estimator ˆq (x and te parametric estimator ˆθ, ten compute te test statistic T analogous to T. Step 3 Repeat step and 2 B times to get a bootstrap sample T,..., T B for te statistic T. Wen B is large, te empirical distribution of T,..., T B approximates te finite sample null distribution. Teoretical justification of te proposed parametric procedure is missing in tis paper. Tis is a igly non-trivial problem, altoug it can probably be solved using te metodology developed in Fan (994 and Andrews (997. In absence of suc results, we use extensive Monte Carlo simulations to study te power properties of te tests under various realistic scenarios and across different sample sizes in te next section. 5 Monte Carlo simulations In tis section, we study te finite sample performance of te density-based tests T and T 2 and te distribution-based test T 3. We use te bootstrap metod described in te previous 0

13 section to determine te null distribution of te test statistic. Te empirical quantiles of te null distribution are used to determine te critical values. Te cross-validation metod (e. g. Wasserman (2004, Section 20.3 is used to determine te bandwidt, and we use te Gaussian kernel in all te nonparametric kernel density estimators. Te GMM metod of Meddai (2002 is used to estimate te parametric model. Te GMM metod is less efficient tan te likeliood based metod, but it acieves a good compromise between te estimation efficiency and computation time. To save space, for all te simulated size and power, we only report te results at 5% significance level Size of te tests We simulate 000 sample pats of daily observations ( = /252 from te Heston model, dy t = σ t dw t, dσ 2 t = 5(0. σ 2 t dt σ 2 t db t, (5 were te two Brownian motions W and B are independent. Witin one day, 0 steps are simulated to reduce discretization error. We consider te sample sizes 000, 2000 and 3000, rougly corresponds to 4 years, 8 years and 2 years of daily observations. Wit te parameters and sample sizes, te test statistics T, T 2 and T 3 are simulated 000 times. Te distribution of tese realized test statistics are taken as te true distribution (except for te Monte Carlo errors. For eac of te realized 000 sample pat, we obtain 5 bootstrap samples and compute teir resulting test statistics T, T2 and T3. Aggregating tem togeter across 000 sample yields 5000 bootstrap statistics. Teir sampling distribution is taken as te distribution of te bootstrap metod. Table summarizes te simulated 5% level size of all te tests for te tree sample sizes. Te bootstrap test statistics seem to ave a reasonable size property, especially wen te sample size is large. For T and T 3, te size property becomes better as te sample size increases, toug tis is not te case for T 2. [Table about ere.] 5.2 Power of te tests We study te power performance of te tests under four different sample sizes 500, 000, 2000 and 3000, and we still use 000 simulations. We take te Heston model (5 as te null ypotesis. We evaluate te power functions of te tree test statistics under te 2 Te computations in tis section are conducted wit Matlab R 202b on te Lisa computing cluster at te University of Amsterdam. Te default random number generator in Matlab (Te Mersenne twister algoritm is used wit seed 2345.

14 tree families of alternative models. Eac family of models is indexed by τ, wit τ = 0 corresponding to te null model (5. In te first family of alternative models, te drift functions of te volatility processes are deviated from te Heston model. In te second family of models, te diffusion functions are deviated from te Heston model. In te tird family of models, jumps are included in te volatility process Misspecification in te drift function We evaluate te power function of te tree test statistics under te following sequence of alternative models, dσ 2 t = {( τ(α(β σ 2 t + τµ(σ 2 t }dt + γ σ 2 t dw t, (6 for τ = 0, 0.,...,, were µ(σ 2 t = σ 2 t [a(b ln σ 2 t ] wit a = 9, b = 3.5. Te functional form of te drift part is motivated by te log SARV model dy t = σ t db t, d ln σ 2 t = κ(θ ln σ 2 t dt + γdw t. By Ito s lemma, te volatility process of te log SARV model is dσ 2 t = σ 2 t [κ(θ ln σ 2t + 2 γ2 ] dt + γσ 2 t dw t, were te drift function is a igly nonlinear function of σt 2 and we use tis to determine te specification of µ(σt 2. Figure sows te differences of te drift functions between te null model and te alternative models. It also gives te 5% level power functions of te tree tests at te 4 different sample sizes. All te tree tests sow increasing power as te sample size goes large, confirming te consistency result of te tests. Te performance of te tree tests seems to be similar for tis type of deviations in te drift function. [Figure about ere.] Misspecification in te diffusion function Here we consider te misspecification in te diffusion function of te volatility process. Te null model is still te Heston model (5. In te alternative model, te drift function remains te same, but te diffusion function is deviating away to a GARCH diffusion 2

15 process. dy t = σ t db t, dσ 2 t = α(β σ 2 t dt + {( τγ σ 2 t + τρ(σ 2 t }dw t (7 for τ = 0, 0.,...,, were ρ(σt 2 = cσt 2 wit c = 5. Wen τ =, te alternative model is a GARCH diffusion process. Figure 2 sows te differences of te diffusion functions between te null model and te alternative models. It also gives te 5% level power functions of te tree tests at te 4 different sample sizes. Again all te tree tests sow increasing power (to as te sample size goes large. Te power of T seems to be sligtly better tan T 3 wen te deviation is small, wile wen te deviation is large, te power seems to be similar. Te power of T 2 seems to be lower tan te oter two tests; wen te sample size is small (n = 500, te test T 2 seems to ave no power at all. [Figure 2 about ere.] Jumps in te volatility process We now consider te power of te tree tests against a sequence of models were te volatility process contains jumps dσt 2 = α(β σt 2 dt + γ σt 2 db t + J t dn t, (8 were N t is a Poisson process wit intensity λ. J t is te jump size tat is independent of (W t and (B t. We consider te following 5 jump intensities: 52, 04, 252, 252.5, 252 2, tese can be understood as te average number of jumps in a year. We consider a jump size tat is normally distributed wit mean 0 and standard deviation equal to.5%. Figure 3 gives te power functions of te tree tests under different sample sizes. We observe again te consistency of te tree tests for tis type of deviations to te null ypotesis. Also we see tat as te jump intensity of te volatility process increases, te tests are more powerful in detecting te deviations. Te relative performance of te tree tests seems to be similar to tis type of deviations. [Figure 3 about ere.] 6 Empirical application In tis section, we apply our tests to a daily Apple stock price dataset. Te dataset contains te adjusted close prices from Jan 3rd 2000 to 3rd Feb 204, making

16 observations in total. Figure 4 gives te plot of te series, te log returns, te nonparametrically estimated density function, as well as te empirical distribution function of te dataset. [Figure 4 about ere.] Te dataset is fitted to te Heston model (5. Te estimated parameter is ˆα = 7.59, ˆβ = 0.793, ˆγ = We ten apply te nonparametric specification tests proposed in tis paper to study te validity of te Heston model estimated. We still use te cross-validation metod to determine te bandwidt and use te Gaussian kernel. Based on 000 bootstrap samples, te estimated p-values for T, T 2 and T 3 are 0.009, 0.04 and respectively. Te p-values of all te tests provide strong evidence of rejection of te Heston model. Te p-value of te distribution-based test is smaller tan te p-values of te oter two tests. From te Monte Carlo evidence, under te misspecification of te diffusion function, te distribution-based test seems to be sligtly more powerful tan te density-based tests. Tis give us te int tat te misspecification of te current model may come from te diffusion function. Te results of tis empirical application provides evidences tat te model may not provide a good approximation of te marginal distribution of empirical data, altoug it is widely used in pricing options in real world applications. 7 Discussion and conclusion We propose tree tests for stocastic volatility model specification by comparing te parametrically and nonparametrically estimated stationary marginal density functions and distribution functions. Our approac can be adapted to discrete-time stocastic volatility models easily, as long as te volatility process is stationary. For example,consider te classical discrete-time stocastic volatility model y ti = σ ti ε ti, log σt 2 i = ω + γ log σt 2 i + σ η η t, (ε t, η t i. i. d. N(0, I 2. Wen γ <, and te volatility process log σt 2 is initiated from te stationary distribution N(ω/( γ, ση/( 2 γ 2, te volatility process is strictly stationary. It is also β-mixing wit exponentially decaying coefficients, see Pam and Tran (985, and tus ergodic. Ten te conditions for nonparametric estimation and parametric estimation are satisfied, and one can compare te estimated density functions and distribution functions as discussed for continuous-time model analogously. 4

17 As discussed in te introduction section, te stationary marginal return distribution does not contain te information of dynamics in te data, suc tat tests defined on te marginal distribution will not be able to detect te misspecification in te dynamic structure of a model. To exploit te dependence structure in te model, we could consider to te one-step conditional distribution function and density function of X i X i, i = 2,..., n, to formulate te test statistics. Denote te density function of X i X i by p(y, x and te corresponding conditional distribution function by P (y, x. For te nonparametric estimation of te two functions, we can proceed as follows. A simple kernel type density estimator for p(y, x is ˆp(y, x = ( ( n n 2 n i= K x X i n K y Xi+ n (, n n n i= K x X i n and te estimator for te conditional distribution function P (y, x is ˆP (y, x = n i= I(X i xi(x i+ y n i= I(X. i x For parametric estimation of te above functions, we again need approximations and we can again resort to simulation based approximation. Analogously to te univariate density based test, conditional density based test statistics can be formulated as: T 4 = R 2 ( ˆp(x, y K p(x, y; ˆθ n 2 dxdy, were K (x, y = K(x/ K(y// 2 is te two dimensional kernel used in te definition of ˆp(x, y. And te conditional distribution function based test is ( T 5 = n P (x, y; ˆθ n ˆP 2 (x, y dp (x, y; ˆθn. R 2 A similar parametric bootstrap is used to obtained te null distribution of te tests. Appendix A: Basic setup and probabilistic properties (N (kernel function Te kernel function K is a bounded, symmetric, nonnegative function on R, satisfying K(xdx =, xk(xdx = 0, x 2 K(xdx = 2k <, 5

18 were k > 0 is a constant, and K 2 (xdx <. (N2 (density function q(x and its second order derivative are bounded and uniformly continuous on R. Te above assumptions on te kernel function, and te smootness assumption on te density function are not te weakest possible. However, Assumptions (N and (N2 are sufficient for te present purpose and simplify te argument in te proof. (N3 For te process (X i n i=, all four dimensional joint densities of (y i,..., y i exist, are bounded and Lipscitz continuous. Tis implies tat te corresponding distribution functions satisfy te same conditions. (N4 As n, n 0 and n n. Tese set of conditions will be used to derive te asymptotic properties of te test statistics, but togeter wit te mixing conditions we assume trougout, tey are also sufficient to make ˆq(x a (pointwise consistent estimator of q(x for all x R. (N3 is stronger tan necessary for consistency, but will be required for te asymptotic distribution of te test based on te bivariate distribution of (X i, X i+ n i=. Te tests developed in tis paper require te observed return sequence to be stationary, ergodic and β-mixing wit exponentially decaying coefficients. In te nonparametric model, it is sufficient to assume te observed return sequence (X i n i= to satisfy te above conditions directly. However, in te parametric model, it is non-trivial to ceck tat tese conditions are satisfied for particular coices of te functions b(x; θ and a(x; θ. In te parametric stocastic volatility model (, we first assume (SV0 (B, W is a standard Brownian motion in R 2, defined on te probability space (Ω, F, P, and σ 2 0 is random variable defined on te same probability space, independent of (B, W. Te following are standard assumptions from Genon-Catalot et al. (998. (SV For all θ Θ, te function b(x = b(x; θ is continuous on (0, +, and te function a(x = a(x; θ is continuously differentiable on (0, +, suc tat K > 0, x > 0, b 2 (x + a 2 (x K( + x 2, and x > 0, a(x; θ > 0. 6

19 Tis assumption ensures te existence and uniqueness of an almost surely positive strong solution to te stocastic differential equation ( generating te volatility process. Define, for v 0 > 0, te scale measure ( x b(v; θ s(x; θ = exp 2 v 0 a 2 (v; θ dv, and te speed measure Ten te assumption m(x; θ = a 2 (x; θs(x; θ. (SV2 s(x; θdx =, 0 s(x; θdx =, 0 m(x; θdx = M <, were te in te integral means a arbitrary point in te domain of s(x; θ, ensures a unique and positive recurrent solution on (0,, see Genon-Catalot et al. (998. Te last condition in (SV2 guarantees te existence of a stationary distribution (for te volatility process, wit density defined as π(x; θ = m(x; θ I(x > 0. M If te process is initiated from tis stationary distribution, i.e., under assumption (SV3 Te initial random variable σ0 2 as density π(x; θ, te solution is strictly stationary and ergodic. Now we give sufficient conditions to ensure tat te volatility process is β-mixing wit exponentially decaying coefficients. From Teorem 3.6 of Cen et al. (200, a sufficient condition (togeter wit (SV and (SV2 for exponential decay of te β-mixing coefficients is tat te process is ρ-mixing, so in te following we give te conditions for te process to be ρ-mixing. Also, we note te result tat if a diffusion process is ρ-mixing, its ρ-mixing coefficients decay at an exponential rate (Bradley (2005, Teorem 3.3, or Genon-Catalot, Jeanteau, and Laredo (2000, Proposition 2.5. Furtermore, β-mixing and ρ-mixing wit exponential decay are almost equivalent concepts for scalar diffusions, as discussed in Cen et al. (200. (SV4 lim a(x; θm(x; θ = 0, x 0 lim a(x; θm(x; θ = 0. x (SV5 Let γ(x; θ = a (x; θ 2b(x; θ a(x; θ ; te limits of /γ(x; θ, as x 0 and as x, exist. 7

20 Appendix B: Lemmas and proofs Conditions in Appendix A ensure strict stationarity, ergodicity and β-mixing of te volatility process. Notice tat te return sequence (X i n i= is a sequence of stocastic integrals of te volatility process wit respect to an independent Brownian motion B over small fixed intervals. By te following lemma from Zu (205, te return series inerit te stationarity, ergodicity and te β-mixing properties from te volatility process. Lemma In te model (, if te volatility process (σt 2 t 0 is stationary, ergodic and β-mixing wit a certain decay rate, ten te normalized return sequence (X i n i= is also stationary, ergodic and β-mixing, wit a mixing decay rate at least as fast as tat of (σt 2 t 0. In all te proofs in tis appendix, we take te above mentioned probabilistic properties for te return series as given to avoid repetition. Wen we use a integral witout te range of integration, te integration is over te full real axis R. Proof (of Teorem We first derive te asymptotic distribution of T. Notice tat T = n /2 (ˆq(x K q(x; ˆθ n 2 dx = n /2 (ˆq(x K q(x 2 dx + n /2 (K q(x K q(x; ˆθ n 2 dx +n /2 (ˆq(x K q(x(k q(x K q(x; ˆθ n dx. Define T = n /2 (ˆq(x K q(x 2 dx. It will be sown later tat T = O p (; te second term n /2 (K q(x K q(x; ˆθ n 2 dx = O p ( /2 because ˆθ n is a n-consistent estimator for θ 0 and te square integrable assumption on te Lipscitz constant L(x as in Assumption (Pa, so tis term is dominated by T ; te crossproduct term is clearly dominated by T by te Caucy- Scuwarz inequality. Tus we ave tat T = T ( + o p (, and we derive te asymptotic distribution of T in te following. 8

21 First notice ( n i= 2 n (K (x X i K q(x dx i= = n (K n 2 (x X i K q(x 2 dx + 2 (K n 2 (x X i K q(x(k (x X j K q(xdx i<j := n ϕ n 2 n (X i, X i + 2 ϕ n 2 n (X i, X j, i= i<j were ϕ n (u, v := (K (x u K q(x(k (x v K q(xdx. Next, we sow tat. Te sum of te diagonal terms n 2 n ϕ n (X i, X i p (n i= K 2 (udu. 2. Te sum of te off-diagonal terms ( n /2 2 ( d ϕ n 2 n (X i, X j N 0, 2 i<j q0(udu 2 (K (2 (u 2 du. 3. Ten we sow tat T ( d N 0, 2 q0(udu 2 (K (2 (u 2 du, and te asymptotic distribution of T follows easily. 9

22 Step We first compute te order of te mean, ( E n 2 n ϕ n (X i, X i i= = n Eϕ n(x, X = (K (x X K q(x 2 dxq(x dx n = ( ( 2 x X K q(x n 2 dxdx ( + o( = (K(u 2 q(x dudx ( + o( n = (n K 2 (udu( + o(. (9 Ten we compute te order of te variance ( n Var ϕ n 2 n (X i, X i i= ( 2 n E ϕ n 2 n (X i, X i i= = n 3 Eϕ2 n(x, X + 2 Eϕ n 4 n (X i, X i ϕ n (X j, X j. (0 We look at te two terms separately. For te first term i<j = Eϕ 2 n(x, X ( 2 K(x 2 X dx q(x dx ( + o( = ( ( 2 x K 2 X dx q(x 4 dx ( + o( = ( 2 K (udu 2 q(x 2 dx ( + o( ( = O. ( 2 20

23 For te second term, = Eϕ n (X i, X i ϕ n (X j, X j K(x 2 X i dx = 2 ( = O (. 2 K 2 (x X j dxq(x i, X j dx i dx j ( + o( K 2 (udu 2 q(x i, X j dx i dx j ( + o( (2 Use te result in ( and (2 in (0, we get ( Var n 2 n ϕ n (X i, X i i= ( n O n 4 n2 O ( = O n 2 2 ( 2. (3 Use te results in (9 and (3 and apply Markov s inequality we ave ( P n 2 n ϕ n (X i, X i (n i= K 2 (udu > ε E n n 2 i= ϕ n(x i, X i (n K 2 (udu 2 ( ε 2 = O = o(, n 2 2 because n. Tus we ave proved tat n 2 n ϕ n (X i, X i (n i= K 2 (udu = o p (. Step 2 sow Now we use Teorem A, Appendix in Hjellvik, Yao, and Tjøsteim (998 to ( n /2 2 ( d ϕ n 2 n (X i, X j N 0, 2 i<j q0(udu 2 (K (2 (u 2 du. Notice tat ϕ n (x, y is a degenerate symmetric kernel, and te mixing condition is satisfied. First we calculate te asymptotic variance. Let X i, Xj be independent variables wit 2

24 te same distribution as X i. First we compute = Eϕ 2 n( X i, X j ( 2 K (x X i K (x X j dx q(x i q(x j dx i dx j ( + o( = ( ( K(uK u + X 2 i X j dx q(x 2 i q(x j dx i dx j ( + o( = ( ( 2 K (2 Xi X j q(x 2 i q(x j dx i dx j ( + o( = (K (2 (u 2 q(x j + uq(x j dudx j ( + o( = (K (2 (u 2 du q 2 (xdx( + o(. ten te asymptotic variance σn 2 = n2 2 Eϕ2 n( X i, X j = n2 2 (K (2 (u 2 du q 2 (xdx( + o(. (4 Ten we ceck te conditions related to te 6 quantities M ni, i =,..., 6, as defined in Teorem A, Appendix in Hjellvik, Yao, and Tjøsteim (998. For M n, notice tat for > δ > 0, = E ϕ n (X, X j ϕ n (X i, X j +δ K (x X K (x X j dx K (x X i K (x X j dx q(x, X i, X j dx dx i dx j ( + o( ( ( ( ( +δ x X x Xj x Xi x Xj = K K dx K K dx 4 q(x, X i, X j dx dx i dx j ( + o( ( ( X X j = 2 K(2 K (2 Xi X i +δ q(x, X i, X j dx dx i dx j ( + o( +δ = 2 2 K(2 (uk (2 (v q(x j + u, X j + v, X j dudvdx j ( + o( = ( 2 K (2 (u +δ du q(x 2δ j + u, X j + v, X j dx j ( + o( ( = O. 2δ Using te same strategy it can be sown tat M n also as tis upper bound and we tus ave n 2 M +δ n ( /σn 2 = O 2δ/(+δ = O ( δ +δ = o(. +δ 22

25 Similarly, we can sow tat ( E ϕ n (X, X j ϕ n (X i, X j 2(+δ 2 = O ( 4(+δ E ϕ n (X, X j ϕ n (X i, X j 2 = O, ( 2 E ϕ n (X, X i ϕ n (X j, Y k 2(+δ = O, 4δ+2 wic furter imply tat n 3 2 M 2(+δ n2 n 3 2 M ( /σn 2 = O ( n 3 2 M 2 n3/σn 2 = O 2(+δ n4 ( /σn 2 = O n /2 δ/(+δ n /2 = o(. n /2 δ/(+δ by noticing again tat δ/( + δ < /2 and n. For M n5, we first calculate = o(, = o(, ( = O, 4δ+2 = E ϕ n (X, X i ϕ n (X, X j dp (X ( K (x X K (x X i dx 2(+δ 2(+δ K (x X K (x X j dx q(x dx = = = = q(x i, X j dx i dx j ( + o( ( ( x X 4 K q(x i, X j dx i dx j ( + o( ( X X i 2 K(2 K(2 (uk (2 ( K (2 (uk (2 2(+δ 2(+δ C = O 2(+δ ( 2δ+. ( K ( x Xi dx ( K (2 X X j ( u + X i X j u + X i X j ( x X K K ( x Xj 2(+δ dx q(x dx q(x dx 2(+δ q(x i, X j dx i dx j ( + o( q(x i + udu 2(+δ q(x i, X j dx i dx j ( + o( 2(+δ q(x i + uduq(x i, X j dx i dx j ( + o( K (2 (uk (2 (u + v 2(+δ q(x i + uduq(x i, X i vdx i dv( + o( 2 K (2 (u 2(+δ du Using te same metod, we can sow tat te oter quantities in te definition of M n5 23

26 also ave tis upper bound and M n5 = O ( 2δ+. We tus ave n 2 M 2(+δ n5 ( /σn 2 = O = O ( 2δ+ 2(+δ = o(. 2(+δ Similarly we ave E ϕ n (X, X i ϕ n (X, X j dp (X 2 = O (, and ( n 2 M 2 n6 /σn 2 = O 2 = o(. We ave ten cecked te condition { } { {n 2 M +δ 2(+δ n + Mn5 + M 2 n6, n 3 2 M σ 2 n 2(+δ n2 + M 2 n3 + M 2(+δ n4 }} 0, and te CLT for te U-statistic is proved and we ave finised proving Step 2. Step 3 Te asymptotic distribution of T is an easy consequence of Step and 2. Te asymptotic distribution of T is obtained by noticing T = T ( + o p (. For te asymptotic distribution of T 2. First use te result in Fan and Ulla (999 Teorem 4. we ave n n ˆq(X i p i= q 2 (xdx, ten te CLT for T 2 follows easily from Slutsky s lemma. In proving Teorem 2, we will need te following weak convergence results for empirical process of β-mixing sequences: Lemma 2 (Kosorok (2008, Teorem.24 Let X, X 2,... be stationary wit marginal distribution P, and β-mixing wit for some 2 < p <. k 2/(p 2 β(k <, k= condition, were te bracketing number satisfies Let F be a class of functions in L 2 (P satisfying te entropy J [ ] (, F, L p (P <. (5 ten G n f = n /2 n (f(x i P f d G P f, i= 24

27 in l (F, were f G P f is a tigt, mean zero Gaussian process wit covariance function for all f, g F. Proof Γ(f, g = lim k Cov(f(X k, g(x i, i= (of Teorem 2 We prove te teorem using empirical processes tecniques. We are working wit dependent data, so we need te empirical process result for β-mixing sequences in Lemma 2, wic is Teorem.24 in Kosorok (2008. Te exponential decay of β-mixing coefficients is sufficient for te above lemma to work. We first prove n( Q(x Q(x; ˆθn l (F x G Q I(u x G Q ψθ T Q(x, θ 0 θ=θ0, (6 θ using te strategy discussed in Van der Vaart (2000, pp Here te notation l denote weak convergence of stocastic process. Notice tat n( Q(x Q(x; ˆθn = n( Q(x Q(x; θ n(q(x; ˆθ n Q(x; θ = n( Q(x Q(x; θ Q(x, θ n(ˆθ n θ, θ = n( Q(x Q(x; θ n Q(x, θ ψ θ (X i n θ i= + o p (, were we use te differentiability of te parameterization and te assumption (Pb about te expansion of te parametric estimator. Wit tis, te above limiting distribution is determined by te joint distribution of ( n( Q(x Q(x; θ, n n ψ θ0 (X i. Notice tat our F is te class of indicator functions F = {I(, x}, wic satisfies te entropy condition (5 and tus is a Donsker class. Adding te k components of ψ θ to F will make a larger class wic we call G, wic is again Donsker (a finite class is Donsker; tis is because te union of Donsker classes is also Donsker. Terefore ( n( Q(x Q(x; θ, n i= n ψ θ0 (X i l (G g G Q g, i= and using te continuous mapping teorem we get n( Q(x Q(x; θ n n i= Q(x, θ ψ θ0 (X i θ l x G Q I(u x G Q ψ T θ Q(x, θ. θ 25

28 Notice tat te process G Q I(u x and te variable G Q ψθ T are dependent because tey can be viewed as marginals of te process g G Q g. Terefore, under te null ypotesis, (6 is proved. Ten, under te null ypotesis, T 2n = n ( Q(x Q(x; ˆθ n 2 dq(x; ˆθ = = ( n( 2 Q(x Q(x; ˆθn dq(x; θ0 ( + O p (n /2 ( n( 2 Q(x Q(x; ˆθn dq(x; θ( + op (, and te result of te teorem follows easily from continuous mapping, because te map z z 2 (tdq(t from D[, + ] into R is continuous wit respect to te supremum norm. Proof (of Teorem 3 Under te alternative ypotesis, we can make te following decomposition of te test statistic, T = n /2 (ˆq(x K q(x; ˆθ n 2 dx = n /2 (ˆq(x K q (x 2 dx + n /2 (K q (x K q(x; ˆθ n 2 dx +2n /2 (ˆq(x K q (x(k q (x K q(x; ˆθ n dx Using te same approac as in Teorem, it can be sown tat te first term satisfies d n /2 ( N ( 0, 2 (ˆq(x K q (x 2 dx (n q0(udu 2 (K K 2 (udu. K 2 (udu For te second term, by definition (K q (x K q(x; ˆθ n 2 dx p (q (x q(x; ˆθ n 2 dx = O p (, as 0, because tis is te L 2 distance between te alternative model and te pesudotrue model. Te limit exists because we are considering functions in te L 2 space. Tus we ave n /2 (K q (x K q(x; ˆθ n 2 dx. 26

29 For te tird term, by Caucy-Scuwarz inequality ( (ˆq(x K q (x(k q (x K q(x; ˆθ n dx /2 ( (ˆq(x K q (x 2 dx = O p (n /2 /4. /2 (K q (x K q(x; ˆθ n 2 dx Ten it is obvious tat T under H and te test is consistent. Te consistency of te test T 2 can be sown analogously. Proof (of Teorem 4 Let F (x = Q(x, θ be te projection of Q (x onto te space of parametric models. Tat is, Q(x, θ is te pseudotrue model. Let X, X 2,... be te observations generated from Q(x, θ, and denote ˆF (x be te empirical distribution function of te sample {X i } n i=. Ten under te alternative ypotesis, we can make te following decomposition T 3 = n ( ˆQ(x Q(x; ˆθ n 2 dq(x; ˆθ n = n ( ˆQ(x ˆF (x 2 dq(x; ˆθ n + n ( ˆF (x Q(x; ˆθ n 2 dq(x; ˆθ n +2n ( ˆQ(x ˆF (x( ˆF (x Q(x; ˆθ n ddq(x; ˆθ n. Wen n, te first term n ( ˆQ(x ˆF (x 2 dq(x; ˆθ n p n (Q (x F (x 2 df (x = O p (n. Using te same metod as in te proof of Teorem 2, it can be sown tat te second term n ( ˆF (x Q(x; ˆθ n 2 dq(x; ˆθ n satisfy te same convergence in distribution result as in tat teorem, suc tat n ( ˆF (x Q(x; ˆθ n 2 dq(x; ˆθ n = O p (. By Caucy-Scuwarz inequality te cross product term is O p (n /2. Ten it is obvious tat T 3 under H wen n and te test is consistent. 27

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