Specification tests for GARCH processes

Specification tests for GARCH processes

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{ \setcounter{footnote}{1} \footnotetext{This research was supported by the Danish Council for Independent Research (DSF Grant 015-00028B) and by the Italian Ministry of University and Research (PRIN 2017 Grant 2017TA7TYC).} \setcounter{footnote}{0} }

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Keywords: GARCH model; Bootstrap; Specification test; Kolmogorov-Smirnov test; Cram\'{e}r-von Mises test; Marked empirical process; Nuisance parameters on the boundary.

Introduction

Generalized autoregressive conditionally heteroskedastic (GARCH) models introduced by Bollerslev:86 are widely used for modelling various financial time series processes. The data generation mechanism of a GARCH model requires the conditional variance to be always strictly positive, which is generally obtained by imposing a strictly positive intercept and non-negative GARCH coefficients in the conditional variance equation. Consequently, in GARCH models, the admissible parameter space typically needs to be inequality restricted. This represents an important difference between GARCH and other popular time series models, such as AR and ARMA models. Although omnibus specification testing in GARCH type models against unspecified alternatives has attracted considerable attention in the recent literature, a crucial weakness in the current theory remains the exclusion of the presence of nuisance parameters on the boundary. This paper contributes towards addressing this issue by developing new statistical methodology for specification testing in GARCH models.

There are a number of different GARCH models available in the literature and many of them are nonnested models (see Francq:2010). Therefore, in many cases, a sensible way to proceed when testing a specification of a GARCH model is to leave the alternative model unspecified, or to test the lack-of-fit. This type of tests, also known as omnibus tests, have their roots in the seminal work of Kolmogorov:33 on testing for a specific probability distribution function, and Grenander:Rosenblatt:57 on testing the hypothesis of white noise dependence. Several omnibus specification tests in GARCH type models have been proposed in the literature. These include tests based on weighted empirical processes of standardized residuals (Koul:06,Escanciano:2010), spectral distributions based tests in the frequency domain (Hidalgo:Zaffaroni:07,Escanciano:2008), residual based tests for nonnegative valued processes (Fernandes:Gramg:05,Koul:Indee:12,Perera:Silvapulle:14), and Khmaladze type (khm:81) martingale transformations based tests (Bai:03,Indee:Hira:15), amongst others.

A key regularity condition imposed by the aforementioned specification tests is to restrict the true parameter to the interior of the null parameter space. Since the parameter space of a GARCH-type model is inequality restricted, this condition is not typically satisfied if some ARCH or GARCH coefficients are zero, because then the true parameter may lie on the boundary of the parameter space. Therefore, for the theory developed in the above cited papers, the true parameter being an interior point is essential; for example, the limiting process obtained in Theorem 2.1 in Hidalgo:Zaffaroni:07 would not be Gaussian if, for instance, a GARCH($p, q$) model is estimated when the underlying true process is a GARCH($p-1, q$), or a GARCH($p, q-1$) process. Similarly, the asymptotic properties of the other aforementioned papers would also not hold when some nuisance parameters lie on the boundary.

In this paper we contribute towards the literature of specification testing in GARCH models by developing a new class of tests for the correct specification of the conditional variance function while allowing the null model to have an unknown number of nuisance parameters on the boundary of the parameter space. Our test statistics are functionals of an empirical process marked by centered squared residuals and are easy to compute. The limiting distributions of the test statistics are not free from (unknown) nuisance parameters, and hence critical values cannot be tabulated for general use. We propose a bootstrap method to implement the tests and show that it is asymptotically valid under general conditions, irrespective of the presence of nuisance parameters on the boundary. The proposed bootstrap approach is simple to implement, and is based on a method of shrinkage of the parameter estimates used to generate the bootstrap sample toward the boundary of the parameter space at an appropriate rate. This approach is similar to the related bootstrap scheme advocated in Cavaliere:19, in a different context, for bootstrapping likelihood ratio statistics, and it also has its roots in the modified bootstrap approach considered in Chatterjee:Lahiri:11 for bootstrapping Lasso-type estimators. Our bootstrap tests are shown to be consistent against fixed alternatives. We also separately consider the case the nuisance parameters lie in the interior of the parameter space for Kolmogorov-Smirnov and Cram\'{e}r-von Mises type tests based on the aforementioned marked empirical process, and show that the bootstrap implementations of these tests under standard residual based bootstrap are asymptotically valid and consistent. Our tests can be implemented easily because the test statistics have simple closed form expressions. A simulation study shows that the proposed tests have desirable finite sample properties. We illustrate the testing procedure by considering two real data examples.

The rest of this paper is structured as follows. Section (ref) formulates the problem, defines the estimators and test statistics. Section (ref) provides the results relating to the asymptotic validity and consistency of the bootstrap tests when the parameters are in the interior of the parameter space. Section (ref) considers inference when some components of the true parameter lie on the boundary of the parameter space. Section (ref) describes a simulation study. Two empirical illustrations are discussed in Section (ref). Section (ref) concludes the paper. The proofs and some assumptions are relegated to Appendix (ref).

Formulation of the Problem

Let $(Y_1, Y_2, \ldots, Y_n)$ be a realization of an observable stationary process $\{Y_i\}$ satisfying

where the errors $\varepsilon_i,\, i\in{\mathbb Z},$ are independent and identically distributed (i.i.d.) random variables (r.v.'s) having zero mean and unit variance with common cumulative distribution function (c.d.f.) $F_0$, and ${\it h}_i = {\mathrm{E}}[Y_i^2 \mid {\mathcal{H}}_{i-1}]$, where ${\mathcal{H}}_{i-1}$ denotes the information available up to time $i-1$ for forecasting $Y_i$, $i \in {\mathbb Z}$.

As is well-known, a GARCH($p_1,p_2$) model for ${\it h}_i$ takes the form

the vector of parameters ${\boldsymbol{\phi}} = (\phi_1, \ldots, \phi_{p_1+p_2+1})^\prime = (\omega, \alpha_1, \ldots, \alpha_{p_1}, \beta_1, \ldots, \beta_{p_2})^\prime$, usually belongs to a compact parameter space

with $\omega > 0$, $\alpha_{k} \ge 0 \, (k=1,\ldots, p_1), \beta_{k} \ge 0 \, (k=1, \ldots, p_2)$, and in order to avoid well-known identification issues (see also Assumption (ref) below) one typically imposes $\sum_{k=1}^{p_1} \alpha_{k} \ne 0$.

Suppose we wish to test the adequacy of the above GARCH($p_1,p_2$) model for ${\it h}_i$, i.e., to test the null hypothesis

against the alternative $\mathsf{H}_1 : \mathsf{H}_0$ is not true.

Since some ARCH or GARCH coefficients may be zero, the null model (ref) allows some components of ${\boldsymbol{\phi}} _0$ to be on the boundary of the parameter space $\Phi$.

Let $\hat {\boldsymbol{\phi}} $ denote the Gaussian quasi maximum likelihood estimator [QMLE] defined by

with ${\it h}_i({\boldsymbol{\phi}} )$ being defined recursively by (ref) for $i=1, 2, \ldots, n$. To simplify the exposition, the vector of initial values, $\varsigma_0=(Y_0, \ldots, Y_{1-p_1}, {\it h}_0, \ldots, {\it h}_{1-p_2})^\prime \in {\mathbb R}^{p_1} \times [0,\infty)^{p_2}$, is assumed to be fixed for the statistical analysis. The asymptotic results do not change if $\varsigma _{0}$ is replaced by an arbitrarily chosen vector (e.g., by setting $Y_{t}=0$ and ${\it h}_{t}=0$, all $t\leq 0$); see, for example, the discussions in Straumann:Mikosch:06, Perera:Silvapulle:20 and Jensen:Rahbek:04.

Let $ \widehat{\varepsilon}_i := Y_i/\{{\it h}_i(\hat{{\boldsymbol{\phi}} })\}^{1/2}, \ i =1, \ldots, n, $ denote the estimated residuals. With $\boldsymbol{\phi } \in \Phi,$ we propose an omnibus test statistic based on the marked empirical process:

where $\mathbb{I}$ denotes the indicator function. We allow the domain of ${\mathcal{U}}_n(\cdot, {\boldsymbol{\phi}} )$ to extend over the whole real line by letting ${\mathcal{U}}_n(-\infty, {\boldsymbol{\phi}} ):=0$ and ${\mathcal{U}}_n(\infty, {\boldsymbol{\phi}} ):= n^{-1/2} \sum_{i=1}^n \{Y_i^2/{\it h}_i({\boldsymbol{\phi}} ) - 1\}$. Hence, ${\mathcal{U}}_n(\cdot, {\boldsymbol{\phi}} )$ in (ref) can be viewed as a process in the space of cadlag functions on $[-\infty,\infty]$, equipped with the uniform metric, which we denote by ${\mathcal{D}}({\mathbb R})$. This process is an extension of the so-called cumulative sum process for the one sample setting to the current set up. Under $\mathsf{H}_0$, ${\mathrm{E}}{\mathcal{U}}_n(y,{\boldsymbol{\phi}} _0) = 0$, for all $y$, but not under $\mathsf{H}_1$. Hence, if $\mathsf{H}_0$ is true, then we would expect ${\mathcal{U}}_n(y, \hat {\boldsymbol{\phi}} \,)$ to be close to zero for all $y$, but not otherwise. Therefore, a suitable functional of ${\mathcal{U}}_n(\cdot, \hat {\boldsymbol{\phi}} \,)$ can potentially be used as a test statistic for testing $\mathsf{H}_0$ against $\mathsf{H}_1$.

The use of cumulative sum processes for specification testing similar to ${\mathcal{U}}_n(\cdot, \hat {\boldsymbol{\phi}} \,)$ goes back to Neumann:41, who proposed a test of constant regression based on an analog of this process. A motivation for basing inference in nonnegative valued processes on an analog of the process $\mathcal{U}_n(\cdot,{\boldsymbol{\phi}} ), \, {\boldsymbol{\phi}} \in \Phi,$ also appears in Koul:Indee:12. Similar tests have also been considered by Stute:97 and Koul:Stute:99 for certain regression and additive time series models. More recently, analogs of ${\mathcal{U}}_n(\cdot, \hat {\boldsymbol{\phi}} \,)$ have been used by several authors to propose asymptotically distribution free specification tests in related time series models; see, for example, Indee:Hira:15 and Bala:Koul:19. In the econometric analyses presented in these papers certain tests based on analogs of ${\mathcal{U}}_n(\cdot, \hat {\boldsymbol{\phi}} \,)$ have demonstrated desirable finite sample and asymptotic properties. Therefore, we find it of interest to develop specification tests based on similar statistics involving the process ${\mathcal{U}}_n(\cdot, \hat {\boldsymbol{\phi}} \,)$ for testing $\mathsf{H}_0$ against $\mathsf{H}_1$ in the current setup. In particular, we consider the Kolomogorov-Smirnov (KS) and Cram\'{e}r-von Mises (CvM) type statistics which can be defined in terms of ${\mathcal{U}}_n(\cdot, \hat {\boldsymbol{\phi}} )$ as:

where $G_n(y) := n^{-1}\sum_{i=1}^n \mathbb{I}(Y_{i-1}\le y)$. Other suitable functionals of ${\mathcal{U}}_n(\cdot, \hat {\boldsymbol{\phi}} \,)$ may also be considered as possible test statistics (see Stephens:86).

Inference when the parameters are in the interior of the parameter space

Before moving to the general case which includes possible parameters on the boundary of the parameter space, we here consider the case the true parameter ${\boldsymbol{\phi}} _0$ is in the interior of $\Phi$.

The asymptotic distribution of ${\mathcal{U}}_{n}(\cdot ,{\boldsymbol{\phi }}_{0})$ under the null hypothesis $\mathsf{H}_{0}$ can be derived by standard arguments, under the assumptions on the GARCH process discussed in the next subsection. Specifically, from a martingale central limit theorem [for example Hall:Heyde:80, Corollary 3.1] and the Cram\'{e}r-Wold device it follows that all finite dimensional distributions of ${\mathcal{U}}_{n}(\cdot ,{ \boldsymbol{\phi }}_{0})$ converge weakly to a multivariate normal distribution with mean vector zero and covariance matrix given by the covariance function

where $G$ denotes the (unconditional) distribution function (d.f.)\thinspace of $Y_{0}$, $\kappa _{\varepsilon }:={\mathrm{E}}\varepsilon _{i}^{4}<\infty $ and $x\wedge y=\min (x,y)$. Under $\mathsf{H}_{0}$, $G$ may depend on ${\boldsymbol{\phi }}_{0}$, but we do not exhibit this dependence. Then, since the function $\pi (x):=K(x,x)= (\kappa _{\varepsilon }-1)G(x)$ is nondecreasing and nonnegative, tightness of the process ${\mathcal{U}}_{n}(\cdot ,{\boldsymbol{\phi }}_{0})$ follows by e.g. Theorem 15.7 in Billingsley:68, and therefore, under $\mathsf{H}_{0}$, ${\mathcal{U}}_{n}(\cdot ,{\boldsymbol{\phi }}_{0})$ converges weakly to the time-transformed Brownian motion $B\circ \pi $, in the space ${\mathcal{D}}({\mathbb R})$ equipped with the uniform metric.

However, since ${\boldsymbol{\phi }}_{0}$ is replaced by $\hat{{\boldsymbol{ \phi }}}$, the weak limit of ${\mathcal{U}}_{n}(\cdot ,\hat{{\boldsymbol{ \phi }}}\,)$ will not be of the form $B\circ \pi $; rather, it depends on $({ \boldsymbol{\phi }}_{0},G)$. We derive this result in the next subsection, where weak convergence of ${\mathcal{U}}_{n}(\cdot ,\hat{{\boldsymbol{\phi }}}\,)$ is derived for the case where the true value ${\boldsymbol{\phi }} _{0}$ lies in the interior of the parameter space.

Asymptotics for the original test statistics

First we introduce some notation to facilitate the presentation of the underlying assumptions for the asymptotic results. Let ${\mathcal{A}}_{{\boldsymbol{\phi}} }(z) = \sum_{i=1}^{p_1} \alpha_{i} z^i$ and ${\mathcal{B}}_{{\boldsymbol{\phi}} }(z) = 1 - \sum_{i=1}^{p_2} \beta_{i} z^i$ with ${\mathcal{A}}_{{\boldsymbol{\phi}} }(z) = 0$ if $p_1=0$ and ${\mathcal{B}}_{{\boldsymbol{\phi}} }(z) = 1$ if $p_2=0$. Furthermore, let

with ${\boldsymbol{I}}_k$ denoting the $k \times k$ identity matrix.

In order to study the limiting behaviour of ${\mathcal{U}}_n(\cdot, \hat {\boldsymbol{\phi}} )$ we make the following assumptions on the process $\{Y_i\}_{ i \in \mathbb{Z}}$ which satisfies (ref)--(ref).

The condition $\gamma ({\boldsymbol{A}}_0) < 0$ in (ref) ensures the existence of a unique strictly stationary solution $\{Y_i\}_{ i \in \mathbb{Z}}$ to Model (ref)--(ref); see, e.g. Bougerol:Picard:92. Note that, in (ref), the strict stationarity condition $\gamma ({\boldsymbol{A}}_0) < 0$ is imposed only on the true value ${\boldsymbol{\phi}} _0$, but for ${\boldsymbol{\phi}} \neq {\boldsymbol{\phi}} _0$ we only impose the weaker restriction $\sum_{j=1}^{p_2} \beta_j < 1$. In Assumption (ref), the condition ${\mathcal{A}}_{{\boldsymbol{\phi}} _0}(1) \neq 0$ ensures that all the $\alpha_{0i}$ are not zero when $p_1 \neq 0$, and hence we do not allow the strictly stationary solution of (ref)--(ref) to be a strong white noise process. This in turn allows us to avoid certain identifiability issues when estimating the GARCH parameters with $p_2 \neq 0$ (see Francq:2010). Note that, in the ARCH case (i.e.\,\,when $p_2 =0$), the Assumption (ref) is not required. In the general GARCH case when $p_2 > 0$, the Assumption (ref) allows for an overidentification of either the order of the ARCH parameters $p_1$ or the order of the GARCH parameters $p_2$, but not both. The condition ${\mathrm{E}} |\varepsilon_i|^{4+d} < \infty$ in Assumption (ref) is only required for the existence of the variance of the score vector $\partial \ell_i({\boldsymbol{\phi}} _0)/\partial {\boldsymbol{\phi}} $; this is necessary for establishing the limiting distribution of the QMLE. Note that we do not assume that the true parameter ${\boldsymbol{\phi}} _0$ is in the interior of $\Phi$. Thus, the assumptions do not exclude the cases where some $\alpha_i$ or $\beta_j$ are zero. Assumptions similar to (ref)--(ref) have previously been discussed in the literature for establishing asymptotic properties of the QMLE; see, e.g., Francq:2010 and Cavaliere:19.

Let

The next lemma provides an asymptotic uniform expansion for ${\mathcal{U}}_n(y,\hat {\boldsymbol{\phi}} )$. We make use of this expansion in the proof of establishing the weak convergence of ${\mathcal{U}}_n(\cdot,\hat {\boldsymbol{\phi}} )$.

Unlike the process ${\mathcal{U}}_n(y,{\boldsymbol{\phi}} _0)$, the estimated process ${\mathcal{U}}_n(y,\hat {\boldsymbol{\phi}} )$ does not converge weakly to a time transformed Brownian motion, because the term $n^{1/2}(\hat {\boldsymbol{\phi}} -{\boldsymbol{\phi}} _0)^\prime J(y,{\boldsymbol{\phi}} _0)$ in (ref), is of order $O_p(1)$ and hence is not asymptotically negligible. In fact, if Assumptions (ref)--(ref) are satisfied, then $\hat {\boldsymbol{\phi}} $ converges to ${\boldsymbol{\phi}} _0$ almost surely (a.s.), and additionally, if Assumption (ref) also holds and ${\boldsymbol{\phi}} _0$ is an interior point in $\Phi$, then $\hat {\boldsymbol{\phi}} $ is asymptotically linear and satisfies

where

see, for example, Berkes:etal:03.

By using Lemma (ref) and (ref), when ${\boldsymbol{\phi}} _0$ is an interior point in $\Phi$, one can show that ${\mathcal{U}}_n(\cdot,\hat {\boldsymbol{\phi}} )$ converges weakly to a centred Gaussian process. This result is stated in the next theorem.

In view of Theorem (ref), the limiting distributions of KS and CvM statistics defined in (ref) depend on the unknown $({\boldsymbol{\phi}} _0, G)$ in a non-trivial way, despite the fact the true parameter is in the interior of $\Phi$. Consequently, it does not appear that it would be possible to find a transformation that would lead to an asymptotically distribution free test, for example as in Bai:03,Koul:Indee:12,Indee:Hira:15,Escanciano:etal:18. Hence, we proceed by considering bootstrap implementations of the tests.

Bootstrap implementation

In this section, we propose a bootstrap procedure for computing the critical values for the KS and CvM statistics in (ref). We perform the resampling scheme under the null hypothesis and derive the asymptotic properties of the bootstrap statistics, irrespective of whether or not the data generating process satisfies the null hypothesis. To this end, we initially standardize the residuals $ \widehat{\varepsilon}_i := Y_i/\{{\it h}_i(\hat{{\boldsymbol{\phi}} })\}^{1/2}, \ i =1, \ldots, n, $ as

and define the associated empirical distribution function of $\{\check{\varepsilon}_1,\ldots, \check{\varepsilon}_n\}$ as

By construction, $\int_{\mathbb{R}}u\check{F}_{n}(u)du=0$ and $\int_{\mathbb{R}}u^{2}\check{F}_{n}(u)du=1$, hence a random variable with distribution function $\check{F}_{n}$ has zero mean and unit variance, therefore matching the first and second order moments of the error distribution $F_{0}$. From Lemma (ref) in Appendix (ref) we obtain that $\check{F}_{n}$ converges to $F_{0}$ with probability one under the null hypothesis.

We next outline the bootstrap algorithm.

Bootstrap algorithm 1

{Step 1:} Compute $\{\hat {\boldsymbol{\phi}} , T_j\}$ on the original sample $\{Y_1, \ldots, Y_n\}$, where $T_j$ is the test statistic defined in (ref) ($j=1, 2$); \newline { Step 2:} Compute $\check{\varepsilon}_{i},\,i=1,\ldots ,n $ as in (ref) and draw a random sample (with replacement) of size $n$, say $\{\varepsilon _{1}^{\ast },\ldots ,\varepsilon _{n}^{\ast }\}$, independent of the original data, from the empirical distribution function $\check F_n(\cdot)$ in (ref); \newline { Step 3:} Generate the bootstrap sample $\{{Y}_{1}^{*}, \ldots, Y_n^{*}\}$ with bootstrap true values $(\hat {\boldsymbol{\phi}} , \check F_n)$ by

initialized with $(Y_0^*, \ldots, Y_{1-q}^*, {\it h}_0^{*}(\hat {\boldsymbol{\phi}} ), \ldots, {\it h}_{1-p}^{*}(\hat {\boldsymbol{\phi}} ))^\prime = \varsigma_0$, where $\varsigma _{0}$ is an arbitrarily chosen vector (e.g. $Y_{t}^*=0$ and ${\it h}_{t}^*=0$, all $t\leq 0$); \newline { Step 4:} Using $\{Y_1^{*}, \ldots, Y_n^{*}\},$ compute $\hat{{\boldsymbol{\phi}} }^{*}$, the bootstrap analog of $\hat {\boldsymbol{\phi}} $; \newline { Step 5:} Compute the bootstrap test statistic $T_j^{*}$ as

where $G_n^*(y)$ and ${\mathcal{U}}_n^*(y,{\boldsymbol{\phi}} )$ are the bootstrap analogs of $G_n(y)$ and ${\mathcal{U}}_n(y,{\boldsymbol{\phi}} )$, respectively. The bootstrap p-value is then defined as

where $P_{n}^{\ast }$ denotes the probability measure induced by the bootstrap (i.e., conditional on the original data). The bootstrap test corresponds to the decision rule:

As is standard, $p_{n}^{\ast }$ of ((ref)) is unknown. It can be approximated with arbitrary accuracy by repeating steps 2--5 a large number of times, say $B$, and then setting $p_{n}^{\ast (B)}$ to be the fraction of times $T_{j}^{\ast }$ exceeds $T_{j}$.

The above bootstrap algorithm is designed to mimic the null data generating process by replacing the unknown $({\boldsymbol{\phi}} _0, F_0)$ by the estimators $(\hat {\boldsymbol{\phi}} , \check F_n)$. Therefore, to establish the validity of the bootstrap test in (ref), we need to generalize the regularity assumptions of Lemma (ref) and Theorem (ref) allowing an arbitrary true value $({\boldsymbol{\phi}} , F)$ in a neighbourhood of $({\boldsymbol{\phi}} _0, F_0)$. The required conditions are introduced as Assumptions (ref)--(ref) in Appendix (ref). Theorem (ref) below establishes the asymptotic validity of the bootstrap test (ref) under (ref)--(ref).

In the bootstrap setup, we define $G_n^*(y) := n^{-1}\sum_{i=1}^n \mathbb{I}(Y_{i-1}^*\le y),$ $y \in {\mathbb R}$. Similarly, the bootstrap analogue of the marked empirical process ${\mathcal{U}}_n(y,{\boldsymbol{\phi}} )$ in (ref) is defined by

Let $O_{p}^{\ast }(1)$, in probability, $o_{p}^{\ast }(1)$, in probability, and ${\mathrm{E}}^{\ast }$ denote the usual stochastic orders of magnitude and expectation, respectively, with respect to $P_{n}^{\ast }$ defined above. We denote convergence in distribution of bootstrap statistics as `$\overset{d^{\ast }}{ \longrightarrow }$'. That is, `$T_{j}^{\ast }\overset{d^{\ast }}{\rightarrow }g_{j}({\mathcal{U}}_{0})$ in probability' means that $ P_{n}^{\ast }(T_{j}^{\ast }\leq \cdot)\overset{p}{\rightarrow }P\{g _{j}({\mathcal{U}}_{0})\leq \cdot\},$ at every continuity point of $P\{ g_{j}({\mathcal{U}}_{0})\leq \cdot\}$.

Next theorem establishes the asymptotic validity of the bootstrap test (ref) under $\mathsf{H}_0$.

In view of Theorem (ref), the bootstrap test ((ref)) based on $T_j$ is asymptotically valid under $\mathsf{H}_0$ ($j = 1, 2$). The next theorem shows that the bootstrap tests are consistent under $\mathsf{H}_1$.

First we need to introduce some notation. Let $({\boldsymbol{\phi}} _0^*,F_0^*)$ be the probability limit of $(\hat {\boldsymbol{\phi}} , \check F_n)$, such that $\hat {\boldsymbol{\phi}} \overset{p}{\rightarrow } {\boldsymbol{\phi}} _0^*$ and $d_2(\check F_n,F_0^*) \overset{p}{\rightarrow } 0$ as $n \to \infty$, where $d_2(F_X, F_Y)$ is the Mallows metric for the distance between two probability distributions $F_X$ and $F_Y$ (see also Lemma (ref) in Appendix (ref)). Clearly, under the null hypothesis $\mathsf{H}_0$, we have that $({\boldsymbol{\phi}} _0^*,F_0^*) = ({\boldsymbol{\phi}} _0,F_0)$.

In view of Theorem (ref), our tests have asymptotic power against a given alternative as long as Assumptions (ref)--(ref) hold, (ref) holds if some components of ${\boldsymbol{\phi}} _0^*$ are zero, and $n^{1/2}(\hat {\boldsymbol{\phi}} - {\boldsymbol{\phi}} _0^*) = O_p(1)$. Assumption (ref) introduces some regularity conditions in order to ensure the stationarity of the bootstrap data generating process under $\mathsf{H}_1$. Assumptions (ref)--(ref) are the same as in Theorem (ref) except that now $({\boldsymbol{\phi}} _0^*,F_0^*) \neq ({\boldsymbol{\phi}} _0,F_0)$, and Assumption (ref) claims that there exists a $y \in {\mathbb R}$ such that ${\mathrm{E}}[\{{\it h}_1/{\it h}_1({\boldsymbol{\phi}} _0^*) - 1\}\mathbb{I}(Y_0 \le y)] \neq 0$, where ${\it h}_i = {\mathrm{E}}(Y_i^2 \mid {\mathcal{H}}_{i-1})$, $i \in {\mathbb Z}$. Since ${\it h}_i$ is not of the form ${\it h}_i({\boldsymbol{\phi}} )$ under $\mathsf{H}_1$ and $({\boldsymbol{\phi}} _0^*,F_0^*)$ is the pseudo-true value, the requirement ${\mathrm{E}}[\{{\it h}_1/{\it h}_1({\boldsymbol{\phi}} _0^*) - 1\}\mathbb{I}(Y_0 \le y)] \neq 0$ is not very restrictive under $\mathsf{H}_1$. However, in finite samples, the power of the tests can be sensitive to the form of the discrepancy between ${\it h}_i$ and ${\it h}_i(\hat {\boldsymbol{\phi}} )$. More precisely, if ${\it h}_i(\hat {\boldsymbol{\phi}} )$ is significantly different from ${\it h}_i$ such that the magnitude of the process $n^{-1/2}\sum_{i=1}^n \{Y_i^2/{\it h}_i(\hat{\boldsymbol{\phi}} ) - 1\}\mathbb{I}(Y_{i-1} \le y)$ is `large' for some $y$, then the KS and CvM functionals of $n^{-1/2}\sum_{i=1}^n \{Y_i^2/{\it h}_i(\hat{\boldsymbol{\phi}} ) - 1\}\mathbb{I}(Y_{i-1} \le y)$ are likely to be significantly large compared to realizations from the empirical distributions of KS$^*$ and CvM$^*$, respectively, leading to finite sample power of the bootstrap tests. Importantly, if the true conditional variance ${\it h}_i$ is non-linear while the null parametric form ${\it h}_i({\boldsymbol{\phi}} )$ is linear, then our tests are likely to have better finite sample power compared to the case where ${\it h}_i$ and ${\it h}_i({\boldsymbol{\phi}} )$ are both linear and the misspecification is only in terms of some missing lags, because in the latter case the KS and CvM functionals of $n^{-1/2}\sum_{i=1}^n \{Y_i^2/{\it h}_i(\hat{\boldsymbol{\phi}} ) - 1\}\mathbb{I}(Y_{i-1} \le y)$ are likely to be smaller compared to the former.

For the validity of our bootstrap tests we have so far required the true value ${ \boldsymbol{\phi }}_{0}$ to be an interior point of $\Phi $ under $\mathsf{H}_0$. It is of interest to see whether the bootstrap implementation of $T_{j}(j=1,2)$ can be modified to obtain a consistent bootstrap test for the case ${\boldsymbol{ \phi }}_{0}$ lies on the boundary of $\Phi $ under $\mathsf{H}_0$. We consider this in the next section.

Inference when the true value is on the boundary

Heuristic arguments suggest that $T_1$ and $T_2$ in (ref) could serve as possible test statistics for testing $\mathsf{H}_0$ against $\mathsf{H}_1$ regardless of whether ${\boldsymbol{\phi}} _0$ lies in the interior or on the boundary of the parameter space. In fact, from Lemma (ref), under assumptions (ref) and (ref), we have

uniformly in $y \in {\mathbb R}$, irrespective of whether ${\boldsymbol{\phi}} _0$ is in the interior or on the boundary of $\Phi$, with ${\mathcal{U}}_n(\cdot,{\boldsymbol{\phi}} _0)$ converging weakly to a time transformed Brownian motion. Therefore, the weak limit of ${\mathcal{U}}_n(\cdot, \hat {\boldsymbol{\phi}} \,)$, and hence the limiting distributions of $T_1$ and $T_2$, depend on the asymptotic behaviour of $n^{1/2}(\hat {\boldsymbol{\phi}} - {\boldsymbol{\phi}} _0)^\prime J(\cdot,{\boldsymbol{\phi}} _0)$. Hence, to establish the limiting distributions of the test statistics it is essential to study the large sample properties of $\hat {\boldsymbol{\phi}} $ when ${\boldsymbol{\phi}} _0$ lies on the boundary of the parameter space. Several important results on this have already been obtained by Andrews:01 and Francq:Zakoian:07. For the ease of reference, in the next subsection, we summarize some of these results in the notation used in this paper.

Limiting distributions of the estimators

In this section, we summarize several technical results regarding the asymptotic behaviour of the QMLE $\hat {\boldsymbol{\phi}} $ in (ref) when some components of ${\boldsymbol{\phi}} _0$ are allowed to be zero, and hence ${\boldsymbol{\phi}} _0$ could be on the boundary of $\Phi$. First, we introduce the following additional regularity condition.

Condition (ref) is equivalent to assuming that the ARCH coefficients being nonzero up to the order of the first GARCH coefficient that is zero. As shown in Berkes:etal:03, a recursive formula may be obtained to compute $b_j({\boldsymbol{\phi}} )$ for any given $j$. Further, we have that $b_j({\boldsymbol{\phi}} ) \to 0$ exponentially fast as $j \to \infty$, uniformly in ${\boldsymbol{\phi}} \in \Phi$. This means that there exists some $0 < \nu < 1$ such that $\nu^{-j} \sup_{{\boldsymbol{\phi}} \in \Phi} b_j({\boldsymbol{\phi}} ) \rightarrow 0$ as $j \to \infty$.

Since the parameter ${\boldsymbol{\phi}} _0$ is allowed to contain zero components, by the assumption that $\Phi$ contains a hypercube (see (ref)), the space $n^{1/2}(\Phi - {\boldsymbol{\phi}} _0)$ increases to the convex cone

where $\Lambda_1 = {\mathbb R}$, and for each $i = 2, \ldots, p_1+p_2+1$, denoting ${\boldsymbol{\phi}} _0 = ({\boldsymbol{\phi}} _{01}, \ldots, {\boldsymbol{\phi}} _{0(1+p_1+p_2)})^\prime$, $\Lambda_i = {\mathbb R}$ if ${\boldsymbol{\phi}} _{0i} \neq 0$ and $\Lambda_i = [0,\infty)$ if ${\boldsymbol{\phi}} _{0i} = 0$. Next lemma shows that, under (ref)--(ref), the asymptotic distribution of $n^{1/2}(\hat {\boldsymbol{\phi}} - {\boldsymbol{\phi}} _0)$ can be represented as the projection of a normal vector distribution onto $\Lambda$; for further details on the nature of this projection, see Section 4 in Francq:Zakoian:07.

The proof of Lemma (ref) follows from Francq:Zakoian:07. If ${\boldsymbol{\phi}} _0$ is an interior point, then $\Lambda = {\mathbb R}^{p_1+p_2+1}$ and $\lambda^\Lambda = Z \sim {\mathcal{N}} (0, (\kappa_{\varepsilon}-1)\Sigma^{-1}({\boldsymbol{\phi}} _0))$, which is the same as the classical case (e.g., see Berkes:Horvath:2004) as we also considered in the previous section.

Inconsistency of the standard bootstrap test with parameters on the boundary

The bootstrap true parameter value, say ${\boldsymbol{\phi}} _n^*$, plays a crucial role in defining the properties of any bootstrap test. For the standard bootstrap test in Section (ref) we set ${\boldsymbol{\phi}} _n^*$ equal to $\hat {\boldsymbol{\phi}} $. In the proof of Theorem (ref), under Assumptions (ref)--(ref) and (ref)--(ref), we obtain that the limiting behaviour of $n^{1/2}(\hat {\boldsymbol{\phi}} ^* - {\boldsymbol{\phi}} _n^*)$, conditional on $(Y_1, \ldots, Y_n)$, is the same as that of $n^{1/2}(\hat {\boldsymbol{\phi}} - {\boldsymbol{\phi}} _0)$, under $\mathsf{H}_0$, since ${\boldsymbol{\phi}} _0$ is an interior point and ${\boldsymbol{\phi}} _n^* = \hat {\boldsymbol{\phi}} $. This result plays a key role in the proof of establishing the validity of the bootstrap tests for the case the true parameter lies in the interior of the parameter space. Convergence results of this type have also been used in establishing the asymptotic validity of other bootstrap methods in similar contexts (see Hidalgo:Zaffaroni:07,Perera:Silvapulle:14).

However, in the current setup, the parameter ${\boldsymbol{\phi}} _0$ is allowed to contain zero components, and hence we require additional conditions to ensure that the bootstrap tests are consistent. In particular, a crucial requirement for the validity of the bootstrap tests is to have the following rate of consistency for the bootstrap true value ${\boldsymbol{\phi}} _n^* = ({\boldsymbol{\phi}} _{n1}^*, \ldots, {\boldsymbol{\phi}} _{n(1+p_1+p_2)}^*)^\prime$:

This requirement has previously been introduced in Cavaliere:19 for establishing the validity of a bootstrap based inference procedure in a different context. In the current setup, the requirement (ref) ensures that the bootstrap method based on ${\boldsymbol{\phi}} _n^*$ replicates the unknown limiting distribution of $T_j$ under the null, while being of order $O^*_p(1)$, in probability, under the alternative ($j=1,2$), as is established in Theorems (ref) and (ref) below. If we set ${\boldsymbol{\phi}} _n^* = \hat {\boldsymbol{\phi}} $, then it only holds that $n^{1/2}({\boldsymbol{\phi}} _{ni}^* - {\boldsymbol{\phi}} _{0i}) = O_p(1)$ for $i= 1,2, \ldots, 1+p_1+p_2$, and hence (ref) is not satisfied. Therefore, the standard bootstrap test outlined in Section (ref) is not consistent when some parameters lie on the boundary of $\Phi$. Hence, in the next subsection, instead of the standard bootstrap, we propose a new bootstrap method based on using a different mechanism in choosing the bootstrap true values ${\boldsymbol{\phi}} _{ni}^*$, $i= 1,2, \ldots, 1+p_1+p_2$.

Consistent bootstrap implementations

In this section we propose a modified bootstrap testing procedure based on shrinking the parameter estimators in the bootstrap data generation. The main idea is that, instead of using $\hat{\boldsymbol{\phi }}=\hat{ \boldsymbol{\phi }}_{n}=(\hat{\boldsymbol{\phi }}_{n1},\ldots ,\hat{ \boldsymbol{\phi }}_{n(1+p_{1}+p_{2})})^{\prime }$ as the true value in the bootstrap data generation, we make use of a transformed version of $\hat{ \boldsymbol{\phi }}$, denoted $\hat{\boldsymbol{\phi }}^{\dag }=\hat{ \boldsymbol{\phi }}_{n}^{\dag }=(\hat{\boldsymbol{\phi }}_{n1}^{\dag },\ldots ,\hat{\boldsymbol{\phi }}_{n(1+p_{1}+p_{2})}^{\dag })^{\prime }$ defined by

where $c_n$ is a scalar sequence converging to zero at an appropriate rate:

This approach has its roots in the Hodges-Le Cam super-efficient type estimators, see e.g. Bickel:etal:98, Chatterjee:Lahiri:11 and Cavaliere:19.

In view of the parameter restrictions in (ref), denoting ${\boldsymbol{\phi}} _0 = ({\boldsymbol{\phi}} _{01}, \ldots, {\boldsymbol{\phi}} _{0(1+p_1+p_2)})^\prime$, we have that ${\boldsymbol{\phi}} _{01} = \omega_0 > 0$, ${\boldsymbol{\phi}} _{0i}=\alpha_{0(i-1)} \ge 0$ $(i=2,\ldots, 1+p_1),$ and ${\boldsymbol{\phi}} _{0i}=\beta_{0(i-1-p_1)} \ge 0 $ $(i=2+p_1, \ldots, 1+p_1+p_2).$ Thus, ${\boldsymbol{\phi}} _{01}$ is always in the interior, and ${\boldsymbol{\phi}} _{0j}$ is on the boundary of the parameter space only if ${\boldsymbol{\phi}} _{0j}=0$ for some $j \in \{2, 3, \ldots, 1+p_1+p_2\}$; i.e.\,\,some ARCH or GARCH coefficient is zero. Since $\hat {\boldsymbol{\phi}} $ is root-$n$ consistent, the proposed shrinkage in terms of the $c_n$ sequence ensures that $P(\hat {\boldsymbol{\phi}} ^\dag_{nj} = 0) \to 1$ as $n \to \infty$ whenever $\mathop {\rm plim} \hat {\boldsymbol{\phi}} _{nj}=0,$ $j \in \{2, 3, \ldots, 1+p_1+p_2\}$, where `$\mathop {\rm plim}$' is the probability limit as $n \to \infty$. Hence, unlike $\hat {\boldsymbol{\phi}} _{nj}$, in large samples, the transformed estimator $\hat {\boldsymbol{\phi}} ^\dag_{nj}$ lies on the boundary of the parameter space with large probability whenever ${\boldsymbol{\phi}} _{0j}$ is on the boundary; i.e.\,\,${\boldsymbol{\phi}} _{0j}=0$. Since $n^{1/2}(\hat {\boldsymbol{\phi}} - {\boldsymbol{\phi}} _0)=O_p(1)$ and $c_n$ converges at a rate slower than $n^{-1/2}$, this ensures that the requirement (ref) is satisfied by the parameter $\hat {\boldsymbol{\phi}} ^\dag_n$ defined by (ref)--(ref). Hence, as established in Theorems (ref) and (ref) below, the bootstrap based on ${\boldsymbol{\phi}} _n^*=\hat {\boldsymbol{\phi}} ^\dag_n$ allows us to replicate the unknown limiting distributions of $T_1$ and $T_2$ under $\mathsf{H}_0$, while being of order $O_p^*(1)$, in probability, under the alternative.

We next provide a step-by-step guide of the proposed modified bootstrap approach.

Bootstrap algorithm 2 (shrinking parameter estimators approach)

{Step 1:} Compute $\{\hat {\boldsymbol{\phi}} , T_j\}$ on the original sample $\{Y_1, \ldots, Y_n\}$; \newline { Step 2:} Compute $\check{\varepsilon}_{i},\,i=1,\ldots ,n $ as in (ref) and draw a random sample (with replacement) of size $n$, say $\{\varepsilon _{1}^{\ast },\ldots ,\varepsilon _{n}^{\ast }\}$, independent of the original data, from the empirical distribution function $\check F_n(\cdot):= n^{-1} \sum_{i=1}^n \mathbb{I}(\check{\varepsilon}_i \leq \cdot)$; \newline { Step 3:} Generate the bootstrap sample $\{{Y}_{1}^{*}, \ldots, Y_n^{*}\}$ with bootstrap true values $(\hat {\boldsymbol{\phi}} ^\dag, \check F_n)$ as

initialized with $(Y_0^*, \ldots, Y_{1-p_1}^*, {\it h}_0^{*}(\hat {\boldsymbol{\phi}} ^\dag), \ldots, {\it h}_{1-p_2}^{*}(\hat {\boldsymbol{\phi}} ^\dag))^\prime = \varsigma_0$, where $\varsigma_0$ is an arbitrarily chosen vector (e.g., set $Y_{t}^*=0$ and ${\it h}_{t}^*=0$, all $t\leq 0$); \newline { Step 4:} Using $\{Y_1^{*}, \ldots, Y_n^{*}\},$ compute $\{\hat{{\boldsymbol{\phi}} }^{*}, T_j^{*} \}$ the bootstrap analogs of $\{\hat {\boldsymbol{\phi}} , T_j\}$.

The bootstrap test then corresponds to the decision rule:

The bootstrap p-value $p_{n}^{\ast }$ is defined as in ((ref)), and it can be approximated with arbitrary accuracy by repeating steps 2--4 a large number of times, say $B$, and then setting $p_{n}^{\ast (B) }$ to be the fraction of times $T_{j}^{\ast }$ exceeds $T_{j}$ ($j=1, 2$).

Note that, the limiting distribution of $n^{1/2}(\hat {\boldsymbol{\phi}} ^\dag - {\boldsymbol{\phi}} _0)$ is the same as that of $n^{1/2}(\hat {\boldsymbol{\phi}} - {\boldsymbol{\phi}} _0)$ whenever ${\boldsymbol{\phi}} _0$ is in the interior of $\Phi$. Hence, the bootstrap test (ref) collapses into the bootstrap method outlined in Section (ref) as $n \to \infty$, whenever ${\boldsymbol{\phi}} _0$ is in the interior of $\Phi$.

Asymptotic validity

In this section we establish the asymptotic validity of the bootstrap based on the shrinking parameter estimators approach introduced in the previous subsection.

Note that the bootstrap analogue of the marked empirical process ${\mathcal{U}}_n(y,{\boldsymbol{\phi}} )$ for the bootstrap test (ref) is defined as in (ref), with

except that $Y_i^*$ and ${\it h}_i^*({\boldsymbol{\phi}} )$ are now based on the bootstrap method outlined in Section (ref). The next theorem establishes the asymptotic validity of the bootstrap test (ref). First, we introduce the following additional assumption. Recall that $F_0$ is the c.d.f\,\,of $\varepsilon_i$ in (ref).

Since Assumptions (ref)--(ref) correspond to the original data generating process, the underlying true parameter value $\zeta_0=({\boldsymbol{\phi}} _0, F_0)$ is fixed. However, in the bootstrap data generation the true parameter $(\hat {\boldsymbol{\phi}} ^\dag, \check F_n)$ is not fixed but converges to $({\boldsymbol{\phi}} _0^*, F_0^*)$ as $n \to \infty$. Therefore, it is not adequate to assume only (ref)--(ref) in order to establish the validity of the bootstrap tests. Assumption (ref) ensures that (ref)--(ref) hold for triangular arrays, and hence allows us to extend the arguments in the proof of Lemma (ref) to a triangular array setup, which in turn is essential for establishing that the limiting distribution of $n^{1/2}(\hat {\boldsymbol{\phi}} ^* - \hat {\boldsymbol{\phi}} ^\dag_n)$, conditional on $(Y_1, \ldots, Y_n)$, is the same as that of $n^{1/2}(\hat {\boldsymbol{\phi}} - {\boldsymbol{\phi}} _0)$ under $\mathsf{H}_0$, while being of order $O_p^*(1)$, in probability, under $\mathsf{H}_1$. This result plays a key role in the proofs of the asymptotic validity and consistency of the bootstrap tests obtained in the next two theorems.

The next theorem establishes the consistency of the bootstrap test (ref).

Theorem (ref) shows that the proposed shrinkage in terms of the $c_n$ sequence, or more generally, the requirement (ref) ensures that the bootstrap test statistics $T_1^*$ and $T_2^*$ based on ((ref)) replicate the unknown limiting distributions of $T_1$ and $T_2$ under the null hypothesis. Theorem (ref) establishes that $T_1^*$ and $T_2^*$ are of order $O_p^*(1)$, in probability, under the alternative; that is, the proposed bootstrap method is also consistent even if it is unknown whether any of the nuisance parameters are on the boundary or not.

Numerical Study

In this section we carry out a Monte Carlo simulation study to evaluate the finite sample performance of the KS and CvM tests based on the bootstrap method (ref) in Section (ref). Our main focus is the case where the true parameter value ${\boldsymbol{\phi }}_{0}$ of the data generating process lies on the boundary of the parameter space. For comparison, we also consider the case where ${\boldsymbol{\phi }}_{0}$ is an interior point. Several data generating processes under the alternative hypothesis are also considered in order to investigate the finite sample power properties of the tests. Although there are several other tests that can be applied for testing the conditional variance specification in GARCH-type models, as mentioned in the introduction, the theory for their validity does not hold when the true parameter is on the boundary. Hence, in these simulations, we compare the proposed tests with the general purpose Ljung-Box $Q$ test which tests the significance of the serial dependence of the squared residuals estimated from the fitted model. We denote the Ljung-Box $Q$ test for a lag length $\ell$ by $\text{LBQ}(\ell)$.

Design of the simulation study

Tests are evaluated when the parametric form ${\it h}_i({\boldsymbol{\phi}} )$ under $\mathsf{H}_0$ is

For the error distribution, we consider the standard normal distribution.

For the conditional variance ${\it h}_i$ of the true data generating process [DGP] we consider the following 9 cases:

DGP1 [ARCH(1)]: ${\it h}_i = 0.20 + 0.7 Y_{i-1}^2,$ {[$\mathsf{H}_0^A$ and $\mathsf{H}_0^B$ both true]}

DGP2 [GARCH(1,1)]: ${\it h}_i = 0.10 + 0.20 Y_{i-1}^2 + 0.70 {\it h}_{i-1},$ {[$\mathsf{H}_0^A$ and $\mathsf{H}_0^B$ both true]}

DGP3 [GARCH(1,2)]: ${\it h}_i = 0.10 + 0.10 Y_{i-1}^2 +0.15 {\it h}_{i-1} + 0.70 {\it h}_{i-2},$ {[$\mathsf{H}_0^A$ false and $\mathsf{H}_0^B$ true]}

DGP4 [GJR-GARCH(1,1)]\footnote{The Glosten-Jagannathan-Runkle GARCH model of Glosten:GJR:93.}: {[$\mathsf{H}_0^A$ and $\mathsf{H}_0^B$ both false]} \newline ${\it h}_i = 0.10 + 0.1 Y_{i-1}^2 +0.5 {\it h}_{i-1} + 0.3 Y_{i-1}^2 \mathbb{I}(Y_{i-1}<0)$,

DGP5 [GARCH(2,2)]: \newline ${\it h}_i = 0.10 + 0.05 Y_{i-1}^2 + 0.6 Y_{i-2}^2 +0.1 {\it h}_{i-1} + 0.2 {\it h}_{i-2},$ {[$\mathsf{H}_0^A$ and $\mathsf{H}_0^B$ both false]}

DGP6 [EGARCH(1,1)]: \newline $\ln {\it h}_i = 0.1 + 0.4 \ln {\it h}_{i-1} + 0.2 ( |\varepsilon_{i-1}| - {\mathrm{E}} |\varepsilon_{i-1}| ) - 0.2\varepsilon_{i-1},$ {[$\mathsf{H}_0^A$ and $\mathsf{H}_0^B$ both false]}

DGP7 [i.i.d.]: ${\it h}_i = 1,$ {[$\mathsf{H}_0^A$ and $\mathsf{H}_0^B$ both true, but Assumption (ref) false]}

DGP8 [Threshold GARCH(1,1)]: \newline ${\it h}_i = 0.10 + 0.1 Y_{i-1}^2 +0.5 {\it h}_{i-1} + 0.3 {\it h}_{i-1} \mathbb{I}(Y_{i-1}<0),$ {[$\mathsf{H}_0^A$ and $\mathsf{H}_0^B$ both false]}

DGP9 [T-CHARM]\footnote{T-CHARM refers to the conditionally heteroscedastic AR models proposed by Chan:Tong:14.}: ${\it h}_i = \mathbb{I}(Y_{i-1}\le 0) + 1.2\mathbb{I}(Y_{i-1}>0).$ {[$\mathsf{H}_0^A$ and $\mathsf{H}_0^B$ both false.]}

DGPs 1, 2, 3, 5 and 7 are linear models. We use these DGPs to evaluate the size and power properties of the tests, focusing particularly on the cases where the true parameter lies on the boundary. The DGPs 4, 6, 8 and 9 are nonlinear models, and hence the linear models specified under the null hypotheses $\mathsf{H}_0^A$ and $\mathsf{H}_0^B$ are misspecified for each of DGPs 4, 6, 8 and 9. We use these four DGPs to evaluate the finite sample power properties of the tests.

In the next two subsections we consider testing for the two null models $\mathsf{H}_0^A$ [GARCH(1,1)] and $\mathsf{H}_0^B$ [GARCH(1,2)] separately. The results are based on $2000$ Monte Carlo replications. For each replication and data generating process, we first compute the QMLE $\hat {\boldsymbol{\phi}} $ and compute the test statistics KS, CvM, and LBQ($\ell$), $\ell = 3,5,10,15,20$. To implement the proposed KS and CvM tests we use the bootstrap method (ref) outlined in Section (ref) with $c_n=n^{-1/3}/50$, while adopting the `Warp-Speed' Monte Carlo method of Giacomini:etal:07 in order to reduce the computational burden. The results are presented in Figures (ref)--(ref). In these figures the results of the LBQ($\ell$) tests are presented for only $\ell = 5, 10$ and $15$; the patterns of the results for $\ell = 3$ and $\ell = 20$ are similar to those for $\ell = 5, 10$ and $15$ and hence are omitted.

Testing for $\mathsf{H}_0^A$ [GARCH(1,1)]

The DGP1 [ARCH(1)] and DGP2 [GARCH(1,1)] are members of the null family under $\mathsf{H}_0^A$. For the DGP1, the GARCH coefficient is zero, and hence the true parameter lies on the boundary of the parameter space. For the DGP2, the true parameter is an interior point. The DGP7 [i.i.d.] is a member of the null family $\mathsf{H}_0^A$ with the true value on the boundary. However, since there are no ARCH or GARCH coefficients, the DGP 7 does not satisfy Assumption (ref) and hence the theory developed in this paper does not apply in this case. The DGPs 3, 4, 5, 6, 8 and 9 are considered in order to evaluate the empirical power properties of the tests; these DGPs are part of the alternative hypothesis. A summary of the results are given in Figures (ref), (ref), (ref) and (ref), and are discussed in Section (ref).

Testing for $\mathsf{H}_0^B$ [GARCH(1,2)]

The DGPs 1, 2 and 3 are members of the null family $\mathsf{H}_0^B$ [GARCH(1,2)]. For each of DGP1 and DGP2, one of ARCH or GARCH coefficients is zero, and hence the true parameter lies on the boundary of the parameter space. The true parameter of the DGP3 [GARCH(1,2)] is an interior point. Since there are no ARCH or GARCH coefficients, when testing for $\mathsf{H}_0^B$, the DGP7 does not satisfy Assumption (ref) and hence as in the previous case, the theory developed in this paper does not apply. The DGPs 4, 5, 6, 8 and 9 continue to be under the alternative hypothesis when testing for GARCH(1,2).

The results are given in Figures (ref), (ref), (ref) and (ref), and are discussed in the next subsection.

Summary of the results

The main observations of the simulation results are the following:

Empirical illustrations

To illustrate the bootstrap testing procedure, we briefly discuss two real data examples.

Example 1

We first consider a data example based on the daily log returns of the SPDR exchange-traded fund (ETF) for the S&P 500 index. This ETF is usually denoted by the tick symbol SPY. The data spans the period January 3, 2007 to June 30, 2017 and contains 2640 observations. A shorter version of this data set was previously studied by tsay2018, and by using some preliminary diagnostics, they concluded that a GJR-GARCH(1,1) model provides a good fit. In their empirical analysis, tsay2018 concluded that the leverage effect of the fitted GJR-GARCH(1,1) model is statistically significant at the 5% level. This indicates that if one specifies a GARCH(1,2) model for the conditional variance then that may not provide a good fit for the data. In order to investigate this, in this empirical illustration, we employ the proposed KS and CvM bootstrap tests to test the adequacy of the GARCH(1,2) specification, expecting that the proposed tests would be able to detect a misspecification. For comparison the LBQ test considered in the simulations in the previous section is also considered.

For the GARCH(1,2) specification, the $p$-values of the KS and CvM tests are both zero up to 3 decimal places, whereas the $p$-value for the LBQ(20) turns out to be 0.123, and those for LBQ(15) and LBQ(5) are 0.032 and 0.033, respectively. Thus, the KS and CvM tests proposed in this paper clearly reject the GARCH(1,2) specification, but the LBQ(20) fails to reject the GARCH(1,2) specification at the 10% level of significance. Note that the Ljung-Box $Q$ test is designed to check the significance of the autocorrelations of the squared residuals at multiple lags jointly. Figure (ref) shows the sample autocorrelations for both the squared values of the observed time series and the squared residuals estimated from the fitted GARCH(1,2). As expected, squared SPY log returns are significantly serially correlated, but the correlogram of squared residuals suggests no significant serial correlations except for some minor ones at lags 1 and 10. This explains the relatively large $p$-values of the LBQ test. However, squared residuals can be serially uncorrelated, but dependent, and hence it appears that the tests proposed in this paper are better suited than the Ljung-Box $Q$ test in detecting the misspecification of the conditional variance specification in this case.

Example 2

In this illustrative example we consider a data set from the Caterpillar stock traded on the New York Stock Exchange. The variable of interest is the daily log return of the Caterpillar stock, defined by $Y_{t}=100(\log P_{t}-\log P_{t-1})$ where $P_t$ is the stock price at time $t$. The sample contains $2515$ observations and spans the period Jan 02, 2001 to Dec 31, 2010. Tsay:12 analyzed this data set by applying several diagnostic methods, and fitted a GARCH(1,1) model (see Table 5.1 in Tsay:12). When we fit a GARCH(1,2) model to this data set, the estimated GARCH(2) coefficient turns out to be statistically insignificant, practically at any level of significance. This indicates that, when testing the GARCH(1,2) specification, one component of the true parameter could potentially be a boundary point of the parameter space, whereas when the null model is GARCH(1,1) the true parameter could potentially be an interior point. Of course we do not have any certainty that this is actually true. But, as an illustration, we employ the proposed KS and CvM bootstrap tests to test GARCH(1,1) and GARCH(1,2) specifications. For comparison the LBQ test is also considered. The $p$-values of the tests are given in Table (ref). As expected the tests support both GARCH(1,1) and GARCH(1,2) specifications with large $p$-values. In the simulations in the previous section, the LBQ test was undersized when testing for the correct specification. Thus, the large $p$-values of the LBQ test in Table (ref) are consistent with the simulation results reported in the previous section.

Conclusion

This paper contributes to advance the current statistical methodology for inference in GARCH models by developing bootstrap based omnibus specification tests while allowing parameters on the boundary of the parameter space. In particular, Kolmogorov-Smirnov and Cram\'{e}r-von Mises type test statistics are proposed based on a certain empirical process marked by centered squared residuals. We first derive the asymptotic null distributions of the proposed test statistics when the true parameter is in the interior of the parameter space. Since the limiting distributions of the test statistics are not free from (unknown) nuisance parameters, we propose a bootstrap method to implement the tests and establish that the proposed bootstrap method is asymptotically valid and consistent. However, when some components of the nuisance parameters lie on the boundary of the parameter space, this bootstrap testing procedure is not consistent. Hence, as an alternative, we also propose a modified version of the bootstrap by employing a method of shrinkage of the parameter estimates in the bootstrap data generation. We show that the modified bootstrap procedure is asymptotically valid and consistent, regardless of the presence of nuisance parameters on the boundary. Our bootstrap methods can be implemented easily under fairly general and easily verifiable assumptions and have desirable finite sample properties in terms of empirical size and power.

Our results can be extended in several directions. For instance, it is of interest to see if the methods we propose in this paper can be extended to models beyond the standard GARCH($p_1,p_2$). To this end, consider the model ${\mathcal{M}}$ defined by

for some ${\boldsymbol{\phi}} \in \Phi \subset \mathbb{R}^{p_1+p_2}$, where $\{g_{\boldsymbol{\phi}} ; {\boldsymbol{\phi}} \in \Phi\}$ is a parametric family of nonnegative functions on ${\mathbb R}^{p_1+p_2}$, and the error terms $\{\varepsilon_i\}_{i \in \mathbb{Z}}$ are i.i.d.\,\,with zero mean and unit variance. Thus, $ {\it h}_i={\it h}_i({\boldsymbol{\phi}} )=\hbox{Var}(Y_i\mid {\mathcal{H}}_{i-1}), \ i \in {\mathbb Z}. $ Consider the hypothesis testing problem

The $\textrm{GARCH($p_1,p_2$)}$ model is a special case of ${\mathcal{M}}$. Another example is the asymmetric AGARCH($p_1,p_2$) model defined by $ {\it h}_i = {\it h}_i({\boldsymbol{\phi}} ) = \alpha_0 + \sum_{j=1}^{p_1} \alpha_j \big(|Y_{i-j}| - \gamma Y_{i-j} \big)^2 + \sum_{k=1}^q \beta_{k} {\it h}_{i-k}, $ where ${\boldsymbol{\phi}} =(\alpha_0, \ldots, \alpha_p, \beta_1, \ldots, \beta_q, \gamma)^\prime$, $\alpha_0 >0, \alpha_j \geq 0$, $\beta_{k} \geq 0$ ($i \in \mathbb{Z}$, $1 \leq j \leq p$, $1 \leq k \leq q$). Similarly, several other extensions of the standard GARCH model can also be written in the general form (ref).

Heuristic arguments suggest that the bootstrap tests proposed in this paper for ARCH($p$) and GARCH($p_1,p_2$) models can also be extended to this general setup. In fact, the bootstrap algorithm outlined in Section (ref) can be readily applied to any model of the form (ref), based on a suitable estimator for ${\boldsymbol{\phi}} $. However, parameter estimation, when the true value is on the boundary of the parameter space, in the family of models in (ref), has not yet been studied, and therefore it is not a trivial task to extend the methods developed in this paper to a general setup of the form (ref); this would provide a potential direction for a possible extension of the paper. Furthermore, our testing procedures can also be potentially extended to Poisson autoregressions with exogenous covariates as considered in AGOSTO:16.