A Bootstrap Test for the Existence of Moments for GARCH Processes

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titlepage\begin{center} {A Bootstrap Test for the Existence of Moments for GARCH Processes} \normalfont \end{center} \begin{center} Alexander Heinemann$^\dagger$ \today \end{center} \begin{abstract} This paper studies the joint inference on conditional volatility parameters and the innovation moments by means of bootstrap to test for the existence of moments for GARCH($p$,$q$) processes. We propose a residual bootstrap to mimic the joint distribution of the quasi-maximum likelihood estimators and the empirical moments of the residuals and also prove its validity. A bootstrap-based test for the existence of moments is proposed, which provides asymptotically correctly-sized tests without losing its consistency property. It is simple to implement and extends to other GARCH-type settings. A simulation study demonstrates the test's size and power properties in finite samples and an empirical application illustrates the testing approach.\\ \\ Keywords: Hypothesis Testing; GARCH; Residual bootstrap\\ JEL codes: C12; C14; C22; C58 \end{abstract} \begingroup \footnote{ $^{\dagger}$Department of Quantitative Economics, Maastricht University, Tongersestraat 53, 6211 LM Maastricht, Netherlands. E-mail address: [email removed] } \addtocounter{footnote}{-1} \endgroup

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Introduction

The existence of moments is key to statistical inference in financial time series. While researchers generally assume that returns are strictly stationary, there is a large dispute to which extend their corresponding moments are finite. In particular many econometricians question the existence of fourth-order moments of returns, whereas some even challenge the existence of second-order moments. In the absence of moments many statistical tools become unreliable such as the ordinary least squares (OLS) estimator, whose asymptotic distribution requires the existence of fourth-order moments. Frequently, returns are modeled as a product of a conditional volatility process and an sequence of innovations. In such case the existence of moments reduces to an inferential problem depending on the parameters of the conditional volatility model and on characteristics of the innovation process. ling1999probabilistic and ling2002necessary provide the necessary and sufficient condition for the existence of even-order moments in the well-known GARCH model. Similar results for other GARCH-type models are obtained by he1999properties, ling2002stationarity and francq2011garch (francq2011garch, Chapter 10). Recently, francq2018testing study the existence of moments for GARCH($1$,$1$) processes and derive the asymptotic distribution of the Wald statistic. Observing that the finite sample behavior is not always in par with the asymptotic results, they propose a bootstrap procedure, whose validity they prove for testing second-order stationarity. Unfortunately, neither for higher-order moments nor for higher-order GARCH models results are available. In particular the latter is a non-standard testing problem as the test-statistic is typically based on the spectral radius. In contrast, bootstrap methods are well-studied in conjunction with GARCH-type models hall2003inference,hidalgo2007goodness,corradi2008bootstrap,shimizu2009bootstrapping,cavaliere2018fixed,beutner2018residual,heinemann2018expected and are also proven to be suitable in non-standard testing problems cavaliere2018bootstrap. Therefore this paper studies the joint inference on conditional volatility parameters and the innovation moments by means of bootstrap. In particular, we prove the validity of the fixed design-residual bootstrap for a general class of volatility models and propose a bootstrap-based test for the existence of moments in the GARCH($p,q$) model. The testing procedure is simple to implement, provides asymptotically correctly-sized tests (without losing the consistency property) and can easily be extended to other GARCH-type settings.

The remainder of the paper is organized as follows. Section (ref) describes the model. The joint asymptotic distribution of the quasi-maximum likelihood (QML) estimators of the volatility parameters and the empirical moments of the residuals is derived in Section (ref). In Section (ref) we propose a fixed-design residual bootstrap method and prove its validity under mild assumptions. A bootstrap-based test for the existence of moments in the GARCH($p$,$q$) model is developed in Section (ref) and extended to other GARCH-type models. A simulation study is conducted in Section (ref) and an empirical application illustrates the bootstrap-based testing approach. Section (ref) concludes. Proofs and auxiliary results are collected in the Appendix.

Model

We consider conditional volatility models of the form

align[align omitted — 57 chars of source]

with $t\in \mathbb{Z}$, where $\epsilon_t$ denotes the log-return, $\{\sigma_t\}$ is a volatility process and $\{\eta_t\}$ is a sequence of independent and identically distributed (i.i.d.) variables. The volatility is assumed to be a measurable function of past observations

align[align omitted — 116 chars of source]

with $\sigma:\mathbb{R}^\infty\times \Theta\to(0,\infty)$ and $\theta_0$ denotes the true parameter vector belonging to the parameter space $\Theta \subset \mathbb{R}^r$, $r \in \mathbb{N}$. Various commonly used volatility models satisfy (ref)--(ref) such as GARCH($p$,$q$); for further examples see francq2015risk (francq2015risk, Table 1). Frequently we are not only interested in the parameter vector $\theta_0$, but also in characteristics of the innovation distribution. The following example illustrates.

exampleSuppose $\{\epsilon_t\}$ follows a GARCH$(1,1)$ process given by (ref) and $\sigma_{t}^2 = \omega_0 + \alpha_0 \epsilon_{t-1}^2+ \beta_0 \sigma_{t-1}^2$, where $\theta_0 = (\omega_0,\alpha_0,\beta_0)'\in (0,\infty)\times[0,\infty)\times[0,1)$. Writing $\mu_{k}= \mathbb{E}[\eta_t^{k}]$ for $k \in \mathbb{N}$, the necessary and sufficient condition for the existence of the fourth moment $\mathbb{E}[\epsilon_t^4]$ is \begin{align} \beta_0^2+2\alpha_0\beta_0\mu_2 +\alpha_0^2\mu_4<1. \end{align} francq2018testing propose a Wald statistic based on QML to test for (ref).

We collect the moment characteristics of the innovation distribution in a vector $\mu = \mathbb{E}[h(\eta_t)]$, where we confine ourselves here to the even moments, i.e.\

align[align omitted — 56 chars of source]

for some $m \in \mathbb{N}$. Generally, $\mu$ is unknown and needs to estimated just like $\theta_0$.

Estimation

For the estimation of the parameters $\theta_0$ and $\mu$ we use a two-step procedure, which is also employed by francq2015risk (2018). First, the vector of the conditional volatility parameters $\theta_0$ is estimated by QML. Since the conditional volatility $\sigma_{t}(\theta) = \sigma(\epsilon_{t-1},\dots,\epsilon_{1}, \epsilon_{0},\epsilon_{-1},\dots;\theta)$ can generally not be determined completely given a sample $\epsilon_1 ,\dots, \epsilon_n$, we replace the unknown presample observations by arbitrary values, say $\tilde{\epsilon}_t$, $t\leq 0$, yielding $\tilde{\sigma}_{t}(\theta) = \sigma(\epsilon_{t-1},\dots,\epsilon_{1}, \tilde{\epsilon}_{0},\tilde{\epsilon}_{-1},\dots;\theta)$. Then the QML estimator of $\theta_0$ is defined as a measurable solution $\hat{\theta}_n$ of

align[align omitted — 270 chars of source]

In the second step, the first-step residuals are obtained, i.e. $\hat{\eta}_t=\epsilon_t/\tilde{\sigma}_t(\hat{\theta}_n)$, and the moments estimated:

align[align omitted — 90 chars of source]

We first list several assumptions essential to the following analysis. Whereas in this paper we mainly focus on GARCH($p$,$q$) processes, the assumptions below are stated in a form that can readily applied to other GARCH-type processes (see Remark (ref)).

assumption{(Compactness)} $\Theta$ is a compact subset of $\mathbb{R}^r$.

assumption{(Stationarity & Ergodicity)} $\{\epsilon_t\}$ is a strictly stationary and ergodic solution of (ref) with (ref).

assumption{(Volatility process)} For any real sequence $\{x_i\}$, the function $\theta\to\sigma(x_1,x_2,\dots;\theta)$ is continuous. Almost surely, $\sigma_t(\theta)>\underline{\omega}$ for any $\theta \in \Theta$ and some $\underline{\omega}>0$ and $\mathbb{E}[\sigma_t^s(\theta_0)]<\infty$ for some $s>0$. Moreover, for any $\theta \in \Theta$, we assume $\sigma_t(\theta_0)/\sigma_t(\theta)=1$ almost surely (a.s.) if and only if $\theta=\theta_0$.

assumption{(Initial conditions)} There exists a constant $\rho \in (0,1)$ and a random variable $C_1$ measurable with respect to $\mathcal{F}_0$ and $\mathbb{E}[C_1^s]<\infty$ for some $s>0$ such that \begin{enumerate}[(i)] • $\sup_{\theta \in \Theta}|\sigma_t(\theta)-\tilde{\sigma}_t(\theta)|\leq C_1 \rho^t$; • $\theta\to \sigma(x_1, x_2, \dots;\theta)$ has continuous second-order derivatives satisfying \begin{align*} \sup_{\theta \in \Theta}\bigg|\bigg|\frac{\partial \sigma_t(\theta)}{\partial \theta}-\frac{\partial \tilde{\sigma}_t(\theta)}{\partial \theta}\bigg|\bigg|\leq C_1 \rho^t, \qquad \quad \sup_{\theta \in \Theta}\bigg|\bigg|\frac{\partial^2 \sigma_t(\theta)}{\partial \theta \partial \theta'}-\frac{\partial^2 \tilde{\sigma}_t(\theta)}{\partial \theta\partial \theta'}\bigg|\bigg|\leq C_1 \rho^t, \end{align*} where $||\cdot||$ denotes the Euclidean norm. \end{enumerate}

assumption{(Innovation process)} The innovations $\{\eta_t\}$ satisfy \begin{enumerate}[(i)] • $\eta_t\overset{iid}{\sim}F$ with $F$ being continuous, $\mu_2=1$, $\mu_4<\infty$ and $\eta_t$ is independent of $\{\epsilon_u:u<t\}$; • $\mathbb{E}\big[||h(\eta_t)||^2\big]<\infty$. \end{enumerate}

assumption{(Interior)} $\theta_0$ belongs to the interior of $\Theta$ denoted by $\mathring{\Theta}$.

assumption{(Non-degeneracy)} There does not exist a non-zero $\lambda\in \mathbb{R}^r$ such that $\lambda'\frac{\partial \sigma_t(\theta_0)}{\partial \theta}=0$ almost surely.

assumption{(Moments)} There exists a neighborhood $\mathscr{V}(\theta_0)$ of $\theta_0$ such that the following variables have finite expectation: \begin{align*} (i)\sup_{\theta \in \mathscr{V}(\theta_0)}\bigg|\frac{ \sigma_t(\theta_0)}{\sigma_t(\theta)}\bigg|^a, \qquad \;\;\: (ii) \sup_{\theta \in \mathscr{V}(\theta_0)}\bigg|\bigg|\frac{1}{\sigma_t(\theta)}\frac{\partial \sigma_t(\theta)}{\partial \theta}\bigg|\bigg|^{b}, \qquad \;\;\: (iii) \sup_{\theta \in \mathscr{V}(\theta_0)}\bigg|\bigg|\frac{1}{\sigma_t(\theta)}\frac{\partial^2 \sigma_t(\theta)}{\partial \theta \partial \theta'}\bigg|\bigg|^c \end{align*} for some $a$, $b$, $c$ (to be specified).

assumption{(Scaling Stability)} There exists a function $g$ such that for any $\theta \in \Theta$, for any $\lambda>0$, and any real sequence $\{x_i\}$ \begin{align*} \lambda \sigma(x_1,x_2,\dots;\theta)=\sigma(x_1,x_2,\dots;\theta_\lambda), \end{align*} where $\theta_\lambda=g(\theta,\lambda)$ and $g$ is differentiable in $\lambda$.

The assumptions are fairly standard in the literature; for a discussion we refer to francq2015risk and beutner2018residual. To lighten notation, we henceforth write $D_t(\theta) =\frac{1}{\sigma_t(\theta)}\frac{\partial\sigma_t(\theta)}{\partial \theta}$ and drop the argument when evaluated at the true parameter, i.e.\ $D_t=D_t(\theta_0)$. In addition, we define $d=deg(h)$, the highest polynomial degree of the function $h$, which reduces to $2m$ using (ref). The next result provides the joint asymptotic distribution of $\hat{\theta}_n$ and $\hat{\mu}_{n}$. A similar result for a GARCH($p$,$q$) model can be found in francq2018testing.

theorem(Asymptotic distribution) Suppose Assumptions (ref)--(ref) hold with $a=\max\{4,2d\}$, $b=4$ and $c=2$. Then, we have \begin{align} \begin{pmatrix} \sqrt{n}(\hat{\theta}_n-\theta_0)\\ \sqrt{n}(\hat{\mu}_{n} - \mu) \end{pmatrix} \overset{d}{\to}N\big(0, \Sigma \big) \qquad with\qquad \Sigma= \begin{pmatrix} \frac{\mu_4-1}{4}J^{-1} & -J^{-1}\Omega \nu'\\ -\nu \Omega'J^{-1} & \Xi \end{pmatrix}, \end{align} where $\Omega = \mathbb{E}[D_t]$, $J=\mathbb{E}[D_tD_t']$, $\nu = \mathbb{E}\big[\eta_t\frac{\partial h(\eta_t)}{\partial x}\big]$, $ \Xi = \frac{\mu_4-1}{4}\nu \nu' +\frac{1}{2}(\xi \nu'+\nu \xi')+\Upsilon$, $\Upsilon = \mathbb{V}\mbox{ar}[h(\eta_t)]$ and $\xi = \mathbb{C}\mbox{ov}[h(\eta_t),\eta_t^2]$.

The asymptotic distribution in Theorem (ref) can be used to perform inference on parameters after having obtained a consistent estimator for $\Sigma$. A powerful alternative to perform statistical inference provide bootstrap methods.

Bootstrap

We employ a fixed-design residual bootstrap scheme as in cavaliere2018fixed and beutner2018residual to approximate the distribution of the estimators in (ref)--(ref). We indicate the bootstrap quantities by a superscript $^*$ and use the usual bootstrap notation: “$\overset{p^*}{\to}$", “$\overset{d^*}{\to}$", “$O_{p^*}(1)$", “$o_{p^*}(1)$", $\mathbb{P}^*$ and $\mathbb{E}^*$ (cf.\ chang2003sieve, chang2003sieve).

algorithm[algorithm omitted — 851 chars of source]

The asymptotic validity of the bootstrap procedure is stated in the following theorem.

theorem(Boostrap consistency) If Assumptions (ref)--(ref) hold with $a=- 12, \max\{2d,12\}$, $b=12$ and $c=6$, then \begin{align} \begin{pmatrix} \sqrt{n}(\hat{\theta}_n^*-\hat{\theta}_n)\\ \sqrt{n}(\hat{\mu}_{n}^* - \hat{\mu}_{n}) \end{pmatrix} \overset{d^*}{\to} N(0, \Sigma), \end{align} in probability.

remarkThe estimator $\hat{\theta}_n$ in the first step of Algorithm (ref) can be replaced by any consistent estimator of $\theta_0$, say $\check{\theta}_n$. A close inspection of the proof of Theorem (ref) reveals that the bootstrap's consistency follows after an appropriate standardization, i.e.\ replace $\hat{\theta}_n$ by $\check{\theta}_n$ in (ref).

In the subsequent section we employ Theorem (ref) and Remark (ref) to derive a bootstrap-based test for the existence of moments in the GARCH model.

Bootstrap Test for the Existence of Moments

We consider a GARCH($p$,$q$) model, in which the recursive form of (ref) is given by

align[align omitted — 140 chars of source]

where $\theta_0 = (\omega_0,\alpha_{01},\dots,\alpha_{0q},\beta_{01},\dots, \beta_{0p})'\in \mathbb{R}_{>0} \times \mathbb{R}_{\geq 0}^{p+q}$. We are interested in testing whether for this GARCH process the moment $\mathbb{E}[\epsilon_t^{2m}]$ exists. ling1999probabilistic and ling2002necessary provide the necessary and sufficient condition for the existence of even moments of model (ref) and (ref). For any matrix $A$ we write $||A||_S$ to denote its spectral norm, i.e.\ $||A||_S=\sqrt{\lambda_{\max} (A'A)}$, and set $A^{\otimes m} = A \otimes A \otimes \dots \otimes A$ ($m$ factors), where $\otimes$ is the Kronecker product. Then the moment $\mathbb{E}[\epsilon_t^{2m}]$ of the GARCH process is finite if and only if $T=\big|\big|\mathbb{E}[A_t^{\otimes m}]\big|\big|_S<1$, where $A_t =A(\theta_0,\eta_t)$ and

align[align omitted — 418 chars of source]

We are interested in testing the null hypothesis $H_0$: $T < 1$ against the alternative hypothesis $H_1$: $T\geq 1$. As usual in hypothesis testing where the null hypothesis is characterized by an open set, the test is in fact constructed for the closure of $H_0$, i.e.\

align[align omitted — 112 chars of source]

Before proceeding with the test statistic, note that $T$ can be expressed in terms of $\theta_0$ and $\mu$. To illustrate this fact, we review the GARCH($1$,$1$) model from Example (ref).

Example1. (continued) We observe that the left-hand side of (ref) corresponds to $T$ for $m=2$. Further, for $p=q=1$ we find that $A_t$ reduces to $A_t = (\eta_t^2, 1)'(\alpha_{01}, \beta_{01})$, such that for general $m$ we have $\mathbb{E}[A_t^{\otimes m}]=\mathbb{E}\big[(\eta_t^2, 1)^{\prime \otimes m}\big](\alpha_{01}, \beta_{01})^{\otimes m}$. The latter possesses a single non-zero eigenvalue (c.f.\ francq2011garch, francq2011garch, p.\ 45) given by \begin{align} \big|\big|\mathbb{E}[A_t^{\otimes m}]\big|\big|_S=\sum_{k=0}^m \binom{m}{k} \alpha_{01}^k \beta_{01}^{m-k} \mu_{2k}. \end{align}

To appreciate why $T$ is a function of $\theta_0$ and $\mu$ also in higher order GARCH models, we state the following proposition.

propositionFor all $m \in \mathbb{N}$, we have $A(\theta,\eta)^{\otimes m}= \sum_{k=0}^m B_{k,m}(\theta) \eta^{2k}$ with $A(\theta,\eta)$ given in (ref) and $\big\{B_{k,m}(\theta): k=0,1,\dots,m\big\}$ is a sequence of matrices, where each matrix has dimension $(p+q)^m\times(p+q)^m$ and depends on $\theta$.

Employing Proposition (ref), one finds $\mathbb{E}\big[A_t^{\otimes m}\big]= \sum_{k=0}^m B_{k,m}(\theta_0) \mu_{2k}$ with $\mu_{0}=1$ and hence there exists a function $\tau:\Theta \times \mathbb{R}^{dim(\mu)} \to \mathbb{R}_+$ such that

align[align omitted — 121 chars of source]

With regard to Section (ref), a natural test statistic is given by

align[align omitted — 174 chars of source]

For $p=q=1$ one can rely on asymptotic theory to find critical values that control the size of the test.

corollarySuppose a GARCH(1,1) process $\{\epsilon_t\}$ with parameter $\theta_0$ and i.i.d. sequence $\{\eta_t\}$, which satisfies the assumptions of Theorem (ref). Then \begin{align} \sqrt{n}(\hat{T}_n-T) \overset{d}{\to}N(0,\varsigma^2), \end{align} where $\varsigma^2 = \frac{\partial \tau(\theta_0,\mu)}{\partial (\theta',\bar{\mu}')}\Sigma \frac{\partial \tau(\theta_0,\mu)}{\partial (\theta',\bar{\mu}')'}$.

The previous corollary is a direct consequence of Theorem (ref) and the delta-method. Hence, testing $\bar{H}_0$: $T\leq 1$ in the GARCH($1$,$1$) at the asymptotic level $\alpha \in (0,1)$ could be defined by the rejection region $\big\{\sqrt{n}(\hat{T}_n-1) >\hat{\varsigma}_n\Phi^{-1}(1-\alpha)\big\}$, where $\hat{\varsigma}_n$ is a consistent estimate for $\varsigma$ and $\Phi$ denotes the standard normal cumulative distribution function. However, as shown in francq2018testing, the finite sample distribution of $\hat{T}_n$ is not always in par with the asymptotic results. Moreover, for higher order GARCH models, this asymptotic approach is practically infeasible due to the complicated form of function $\tau$ (recall that $\tau$ is a composite function involving the spectral norm). Instead we propose to mimic the finite sample distribution of the test statistic by means of a bootstrap procedure similar to Section (ref). To construct such bootstrap scheme we re-estimate the parameter $\theta$ to impose the null hypothesis $\bar{H}_0$ for the “bootstrap world”. We denote the constrained estimator by $\hat{\theta}_n^c$, which satisfies

align[align omitted — 235 chars of source]

This estimator is strongly consistent for $\theta_0$ when $\tau(\theta_0,\mu)\leq 1$; for details we refer to Lemma (ref) in the Appendix. Note that, by construction, the corresponding constrained test statistic

align[align omitted — 158 chars of source]

satisfies $\hat{T}_n^c\leq 1$. Based on the constrained estimator $\hat{\theta}_n^c$ we propose a fixed-design residual bootstrap algorithm to mimic the distribution of the test statistic $\hat{T}_n$.

algorithm[algorithm omitted — 1,118 chars of source]

Since the bootstrap quantities are generated under the constrained estimator, a superscript $^\star$ is employed to distinguish them from the ones in Algorithm (ref). The corresponding bootstrap notation is given by: “$\overset{p^\star}{\to}$", “$\overset{d^\star}{\to}$", “$O_{p^\star}(1)$", “$o_{p^\star}(1)$", $\mathbb{P}^\star$ and $\mathbb{E}^\star$. The bootstrap procedure described in Algorithm (ref) is valid in the following sense.

corollarySuppose the assumptions of Theorem (ref) hold true. Under the null hypothesis $\bar{H}_0$: $T \leq 1$ we have \begin{align} \sup_x\Big|\mathbb{P}^\star\big[\sqrt{n}(\hat{T}_n^\star-\hat{T}_n^c)\leq x\big]-\mathbb{P}\big[\sqrt{n}(\hat{T}_n-T)\leq x\big]\Big|\overset{p}{\to}0. \end{align} Under the alternative $\bar{H}_1$: $T > 1$ we have $\sqrt{n}(\hat{T}_n^\star-\hat{T}_n^c)=O_{p^\star}(1)$ in probability.

The previous corollary legitimatizes the following bootstrap test to assess whether $\mathbb{E}[\epsilon_t^{2m}]$ is finite in the GARCH($p$,$q$) model. We acquire a set of $B$ bootstrap replicates, i.e.\ $\hat{T}_n^{\star (b)}$ for $b=1,\dots, B$, by repeating Algorithm (ref) and compute

align[align omitted — 155 chars of source]

which proxies the p-value of the null hypothesis $\bar{H}_0: T\leq 1$. Thus, one rejects the null hypothesis when (ref) is below the nominal level of the test (e.g.\ $5\%$ or $10\%$). To appreciate why the bootstrap test is consistent, we note that under the alternative $\bar{H}_1$: $T > 1$ we have $\sqrt{n}(\hat{T}_n^\star-\hat{T}_n^c)=O_{p^\star}(1)$ whereas

align[align omitted — 122 chars of source]

diverges in probability.

remarkIn case one is interested in the null hypothesis $\tilde{H}_0:T\geq 1$ against the alternative hypothesis $\tilde{H}_1:T< 1$, the outlined bootstrap testing procedure can be readily adapted: replace “$\leq$" in equations (ref) and (ref) by “$\geq$".

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landscape\begin{table}[] \resizebox{21cm}{!}{ \begin{tabular}{ccccc} & Moment & Volatility Recursion & $\bm{A(\theta,\eta)}$ & $\bm{h(x)}$ \\ \hline \hline & & & & \\ ARCH & $\mathbb{E}[\epsilon_t^{2m}]$ & $\sigma_{t}^2 = \omega_0+ \sum\limits_{i=1}^q\alpha_{0i} \epsilon_{t-i}^2$ & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}ccc@{}} \alpha_1 \eta^2 & \dots & \alpha_q \eta^2 \\ & I_{(q-1)\times(q-1)} & O_{(q-1)\times 1}\\ \end{array}\right)$\end{tabular} & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}c@{}} x^2 \\ \vdots \\ x^{2m}\\ \end{array}\right)$\end{tabular} \\ & & \multicolumn{1}{l} & \multicolumn{1}{l} & \multicolumn{1}{l} \\ GARCH & $\mathbb{E}[\epsilon_t^{2 m}]$ & \begin{tabular}[c]{@c@}$\sigma_{t}^2 = \omega_0 + \sum\limits_{i=1}^q\alpha_{0i}\epsilon_{t-i}^2$ \\ $\textcolor{white}{aaaaaaa} + \sum\limits_{j=1}^p\beta_{0j} \sigma_{t-j}^2 $\end{tabular} & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}ccc|ccc@{}} \alpha_1 \eta^2 & \dots & \alpha_q \eta^2 & \beta_1 \eta^2 & \dots & \beta_p \eta^2 \\ & I_{(q-1)\times(q-1)} & O_{(q-1)\times 1} & & O_{(q-1)\times p} & \\\hline \alpha_1 & \dots & \alpha_q & \beta_1 & \dots & \beta_p \\ & O_{(p-1)\times q} & & & I_{(p-1)\times(p-1)} & O_{(p-1)\times 1}\\ \end{array}\right)$\end{tabular} & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}c@{}} x^2 \\ \vdots \\ x^{2m}\\ \end{array}\right)$\end{tabular} \\ & & \multicolumn{1}{l} & \multicolumn{1}{l} & \multicolumn{1}{l} \\ T-GARCH & $\mathbb{E}[\epsilon_t^{m}]$ & \begin{tabular}[c]{@c@}$\sigma_{t} = \omega_0 + \sum\limits_{i=1}^q\big(\alpha_{0i}^+ \epsilon_{t-i}^+ + \alpha_{0i}^- \epsilon_{t-i}^-\big) $\\ $+ \sum\limits_{j=1}^p\beta_{0j} \sigma_{t-j}\textcolor{white}{aii}$\end{tabular} & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}ccccc|ccc@{}} \alpha_1^+ \eta^+ & \alpha_1^- \eta^+ & \dots & \alpha_q^+ \eta^+ & \alpha_q^- \eta^+ & \beta_1 \eta^+ & \dots & \beta_p \eta^+ \\ \alpha_1^+ \eta^- & \alpha_1^- \eta^- & \dots & \alpha_q^+ \eta^- & \alpha_q^- \eta^- & \beta_1 \eta^- & \dots & \beta_p \eta^- \\ & I_{(2q-2)\times(2q-2)} & & O_{(2q-2)\times 2} & & & O_{(2q-2)\times p} & \\\hline \alpha_1^+ & \alpha_1^- & \dots & \alpha_q^+ & \alpha_q^- & \beta_1 & \dots & \beta_p \\ & & O_{(p-1)\times 2q} & & & & I_{(p-1)\times(p-1)} & O_{(p-1)\times 1}\\ \end{array}\right)$\end{tabular} & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}c@{}} x^+ \\ \vdots \\ (x^+)^m \\ x^- \\ \vdots \\ (x^-)^m\\ \end{array}\right)$\end{tabular} \\ & & \multicolumn{1}{l} & \multicolumn{1}{l} & \multicolumn{1}{l} \\ AP-GARCH & $\mathbb{E}[\epsilon_t^{\delta m}]$ & \begin{tabular}[c]{@c@}$\sigma_{t}^\delta = \omega_0 + \sum\limits_{i=1}^q\alpha_{0i}\big(|\epsilon_{t-i}| - \gamma \epsilon_{t-i}\big)^\delta$ \\ $ + \sum\limits_{j=1}^p\beta_{0j} \sigma_{t-j}^\delta \textcolor{white}{aai}$\end{tabular} & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}ccc|ccc@{}} \alpha_1 (|\eta|-\gamma \eta)^\delta & \dots & \alpha_q (|\eta|-\gamma \eta)^\delta & \beta_1 (|\eta|-\gamma \eta)^\delta & \dots & \beta_p (|\eta|-\gamma \eta)^\delta \\ & I_{(q-1)\times(q-1)} & O_{(q-1)\times 1} & & O_{(q-1)\times p} & \\\hline \alpha_1 & \dots & \alpha_q & \beta_1 & \dots & \beta_p \\ & O_{(p-1)\times q} & & & I_{(p-1)\times(p-1)} & O_{(p-1)\times 1}\\ \end{array}\right)$\end{tabular} & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}c@{}} (x^+)^\delta \\ \vdots \\ (x^+)^{\delta m}\\ (x^-)^\delta \\ \vdots \\ (x^-)^{\delta m}\\ \end{array}\right)$\end{tabular} \\ & & \multicolumn{1}{l} & \multicolumn{1}{l} & \multicolumn{1}{l} \\ GJR-GARCH & $\mathbb{E}[\epsilon_t^{2m}]$ & \begin{tabular}[c]{@c@}$\sigma_{t}^2 = \omega_0 + \sum\limits_{i=1}^q\big(\alpha_{0i}^+ (\epsilon_{t-i}^+)^2 + \alpha_{0i}^- (\epsilon_{t-i}^-)^2\big)$\\ $ + \sum\limits_{j=1}^p\beta_{0j} \sigma_{t-j}^2\textcolor{white}{aaaaaai}$\end{tabular} & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}ccccc|ccc@{}} \alpha_1^+ (\eta^+)^2 & \alpha_1^- (\eta^+)^2 & \dots & \alpha_q^+ (\eta^+)^2 & \alpha_q^- (\eta^+)^2 & \beta_1 (\eta^+)^2 & \dots & \beta_p (\eta^+)^2 \\ \alpha_1^+ (\eta^-)^2 & \alpha_1^- (\eta^-)^2 & \dots & \alpha_q^+ (\eta^-)^2 & \alpha_q^- (\eta^-)^2 & \beta_1 (\eta^-)^2 & \dots & \beta_p (\eta^-)^2 \\ & I_{(2q-2)\times(2q-2)} & & O_{(2q-2)\times 2} & & & O_{(2q-2)\times p} & \\\hline \alpha_1^+ & \alpha_1^- & \dots & \alpha_q^+ & \alpha_q^- & \beta_1 & \dots & \beta_p \\ & & O_{(p-1)\times 2q} & & & & I_{(p-1)\times(p-1)} & O_{(p-1)\times 1}\\ \end{array}\right)$\end{tabular} & \begin{tabular}[c]{@c@}$\left(\begin{array}{@{}c@{}} (x^+)^2 \\ \vdots \\ (x^+)^{2m} \\ (x^-)^2 \\ \vdots \\ (x^-)^{2m}\\ \end{array}\right)$\end{tabular} \\ & & \multicolumn{1}{l} & \multicolumn{1}{l} & \multicolumn{1}{l} \\ \hline \hline \end{tabular}} \caption{Examples of GARCH-type models for which the proposed bootstrap-based test can be applied.} \end{table}

}

remarkThe bootstrap-based test for the existence of moments can be readily adapted to other GARCH-type processes such as the threshold GARCH (T-GARCH) of zakoian1994threshold, the asymmetric power GARCH (AP-GARCH) of ding1993long or the GARCH extension of glosten1993relation (GJR-GARCH). In fact, it only requires replacing the model-specific function $A(\theta,\eta)$ in (ref); see Table (ref) for details on the functional forms of the aforementioned GARCH-type models. The theoretical results presented for the GARCH carry over after a small adjustment of the moment function $h(x)$: e.g.\ in the T-GARCH case the corresponding function is given by $h(x)=\big(x^+,\dots,(x^+)^m,x^-,\dots,(x^-)^m\big)'$, where $x^+=\max(x,0)$ and $x^-=\max(-x,0)$.\footnote{Although $h(x)$ is not differentiable at $x=0$ in the T-GARCH case, it is worth mentioning that $\nu$ in Theorem (ref) is well defined as $h$ is differentiable almost everywhere.}

comment\begin{example} Suppose $\{\epsilon_t\}$ follows a T-GARCH$(1,1)$ process given by (ref) and $\sigma_{t+1} = \omega_0 + \alpha_0^+ \epsilon_t^+ + \alpha_0^- \epsilon_t^- + \beta_0 \sigma_{t}$, where $\theta_0 = (\omega_0,\alpha_0^+,\alpha_0^-,\beta_0)'\in (0,\infty)\times[0,\infty)^2\times[0,1)$. The necessary and sufficient condition for the existence of $\mathbb{E}[|\epsilon_t|^d]$ is \begin{align} \sum_{i=0}^d \binom{d}{i} \Big(\big(\alpha_0^+\big)^i \mu_i^+ + \big(\alpha_0^-\big)^i \mu_i^-\Big) \beta_{0}^{m-i} \textcolor{red}{\mu_{2i}}<1 \end{align} \end{example}

Numerical Illustration

Monte Carlo Experiment

A simulation study is conducted to gain further insights into the practical implications of the bootstrap-based test of Section (ref). In particular we focus on the GARCH($1$,$2$) model, which is motivated by the subsequent empirical application (see Section (ref)). The innovations are generated from a standard normal distribution, i.e.\ $\eta_t \overset{iid}{\sim}N(0,1)$, such that $(\mu_4,\mu_6,\mu_8,\mu_{10})=(3,15,105,945)$. Further, the GARCH parameters are set to $\omega_0 = 0.08$, $\alpha_{01} = 0.05$ and $\alpha_{02} = 0.10$ while $\beta_{01} \approx 0.80$ is chosen such that $T$ in (ref) is equal to unity when $m=3$. In other words, $m=3$ corresponds to the boundary case of the null hypothesis in which $\mathbb{E}[\epsilon_t^{2m}]$ is just evaluated infinite. We consider three estimation sample sizes, $n \in \{1{,}000; 5{,}000; 10{,}000\}$, whereas the number of bootstrap replicates is fixed and equal to $B=1{,}999$. For each model version we simulate $S=2{,}000$ independent Monte Carlo trajectories and investigate the proposed bootstrap test at two nominal levels: $5\%$ and $10\%$.

figure[figure omitted — 1,274 chars of source]

Figure (ref) displays the density of the distribution of $\sqrt{n}(\hat{T}_n-1)$ and the bootstrap distribution of $\sqrt{n}(\hat{T}_n^\star-\hat{T}_n^c)$ for varying $m$ and sample size $n=5{,}000$. For $m=1$ and $m=2$ one observes that the two densities have a similar shape. The key difference is that the bootstrap distribution is centered around zero, whereas the distribution of $\sqrt{n}(\hat{T}_n-1)$ is shifted to the left (as expected) with center $\sqrt{n}(T-1)$, i.e.\ $-3.00$ for $m=1$ and $-3.22$ for $m=2$. For the case $m=3$, which corresponds to the boundary of the null hypothesis, Figure (ref)(iii) shows that the bootstrap distribution of $\sqrt{n}(\hat{T}_n^\star-\hat{T}_n^c)$ mimics well the finite sample distribution of $\sqrt{n}(\hat{T}_n-1)$. For $m=4$ and $m=5$, the null hypothesis is violated and the bootstrap and the non-bootstrap distribution exhibit distinct behavior as visualized in Figures (ref)(d) and (ref)(e). Whereas the bootstrap distribution remains centered around the origin, the distribution of $\sqrt{n}(\hat{T}_n-1)$ is more disperse and starts to diverge with center $\sqrt{n}(T-1)$, i.e.\ $7.83$ for $m=4$ and $22.40$ for $m=5$.

Table (ref) reports the simulated rejection rates (in $\%$). For $m=1,2$ the null hypothesis of $\mathbb{E}[\epsilon_t^{2m}]<\infty$ is (almost) never rejected by the bootstrap test at the considered nominal values across sample sizes. For $m=3$, the relative rejection frequencies are below the corresponding nominal values, yet approach them with increasing sample size. This result suggests that the bootstrap test is rather conservative. For $m=4$ the the relative rejection frequency considerably increase (especially in larger samples) indicating that the null hypothesis is violated. For $m=5$ the results are more pronounced and the relative rejection rates are considerably higher reaching $100\%$ when the sample size is $n=10{,}000$.

table[table omitted — 1,581 chars of source]

Empirical Application

Next, we study the German stock market index DAX for the period January 2, 1990 until January 20, 2009. The information on the index price is retrieved from Yahoo Finance and daily (log-) returns (expressed in $\%$) are determined yielding $n=4{,}807$ observations.

figure[figure omitted — 658 chars of source]

Figure (ref)(i) displays the resulting series of returns. For this financial series francq2011garch (francq2011garch, p.\ 206) strongly reject the null hypothesis of a GARCH($1$,$1$) in favor for a GARCH($1$,$2$) model. Estimating the latter, we present the corresponding point estimates in Table (ref), where the reported standard errors are obtained by means of bootstrap.

table[table omitted — 714 chars of source]

Indeed we find a substantial point estimate for $\alpha_{0,2}$. Moreover, as documented in various studies we observe large volatility persistence in the data. The estimates of the fourth and sixth moments indicate that the innovation distribution is considerably more heavy-tailed than the standard normal distribution whose corresponding moments are $3$ and $15$, respectively. Although this can be hardly seen from the histogram of the residuals in Figure (ref)(ii), where a scaled normal distribution is superimposed, we find that a (normalized) Student-t distribution with $9$ degrees of freedom provides an improved fit. Next, we test to what extend the financial time series at hand has finite moments. In particular we focus on the second, fourth and sixth moment corresponding to $m=1,2,3$, respectively. Table (ref) presents the test-statistic and the corresponding p-value associated with the null hypothesis $\mathbb{E}[\epsilon_t^{2m}]<\infty$.

table[table omitted — 492 chars of source]

For $m=1$, we find a test statistic smaller than unity and henceforth the corresponding p-value is large. For $m=2$, the test statistic is slightly larger than unity, however there is not enough evidence to reject the null hypothesis that the fourth moment exists. In contrast, for $m=3$, the test statistic is substantial larger and the corresponding p-value indicates that it is unlikely that the sixth moment is finite. Summing up: while the series seems to admit moments of second-order, there is strong evidence against the existence of sixth-order moments. With regard to the fourth-order moment, the test is inconclusive.

Concluding Remarks

This paper studies the joint inference on conditional volatility parameters and the innovation moments by means of bootstrap to test for the existence of moments for GARCH processes. For a general class of volatility models we derive the joint asymptotic distribution of the QML estimators and the empirical moments of the residuals. Further, we propose a fixed-design residual bootstrap to mimic the estimators' finite sample distribution. The validity of the bootstrap method is proven under mild assumptions and a bootstrap-based test for the existence of moments in the GARCH($p$,$q$) model is proposed. This testing problem is non-standard as the test-statistic involves the spectral radius. Still the testing procedure is simple to implement and provides asymptotically correctly-sized tests without losing its consistency property. A simulation study demonstrates the test's size and power properties in finite samples. An empirical application illustrates the bootstrap-based testing approach, which can easily be extended to other GARCH-type settings.