Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
108,375 characters · 18 sections · 61 citation commands
\begingroup\leftskip=1 cm \rightskip=1 cm
In this paper we investigate to what extent the bootstrap can be applied to conditional mean models, such as regression or time series models, when the volatility of the innovations is random and possibly non-stationary. In fact, the volatility of many economic and financial time series displays persistent changes and possible non-stationarity. However, the theory of the bootstrap for such models has focused on deterministic changes of the unconditional variance and little is known about the performance and the validity of the bootstrap when the volatility is driven by a non-stationary stochastic process. This includes near-integrated exogenous volatility processes as well as near-integrated GARCH processes, where the conditional variance has a diffusion limit; a further important example is the case where volatility exhibits infrequent jumps. This paper fills this gap in the literature by developing conditions for bootstrap validity in time series and regression models with non-stationary, stochastic volatility. We show that in such cases the distribution of bootstrap statistics (conditional on the data) is random in the limit. Consequently, the conventional approaches to proofs of bootstrap consistency, based on the notion of weak convergence in probability of the bootstrap statistic, fail to deliver the required validity results. Instead, we use the concept of `weak convergence in distribution' to develop and establish novel conditions for validity of the wild bootstrap, conditional on the volatility process. We apply our results to several testing problems in the presence of non-stationary stochastic volatility, including testing in a location model, testing for structural change using CUSUM-type functionals, and testing for a unit root in autoregressive models. Importantly, we work under sufficient conditions for bootstrap validity that include the absence of statistical leverage effects, i.e., correlation between the error process and its future conditional variance. The results of the paper are illustrated using Monte Carlo simulations, which indicate that a wild bootstrap approach leads to size control even in small samples.
\noindentKeywords: Bootstrap; Non-stationary stochastic volatility; Random limit measures; Weak convergence in Distribution.
\noindentJEL Classification: C32.
\endgroup
In this paper we consider bootstrap and asymptotic inference on the conditional mean in econometric time series models when the (conditional) volatility is allowed to show a large degree of persistence due to possible permanent and stochastic changes, reflecting the well established fact that volatility in many economic and financial time series displays high persistence, and covariance non-$\frac{{}}{{}}$stationarity.
Earlier references in macroeconomics include KN99 and MP00, who find evidence of an (unanticipated) structural change in the volatility of US GDP growth rates. Evidence of changes in the unconditional volatility appear in many key time series, such as aggregate consumption and income, in interest rate data and in nominal and real price variables; see SvD04 . Evidence on changes in the long-run component of volatility in stock and currency markets are initially reported in LP94 and H95, who show that when stochastic volatility [SV] models are taken to the data, the largest autoregressive root in the SV process is so close to one that the assumption of stationary volatility seems to be at odds with the data. Similarly, it is a well-known stylized fact that GARCH models fit to stock market returns display parameter estimates which reflect high persistence as they (nearly) violate covariance stationarity conditions (often referred to as \textquotedblleft near-integrated GARCH\textquotedblright ), and that such parameters are smaller when a slowly-varying long run component is accounted for in the model, see ER08. HLST16 list a number of empirical studies that have found strong evidence of structural breaks in the unconditional variance of asset returns, with break dates driven by major financial and macroeconomic crises. Such (possibly random) volatility shifts are known to affect the asymptotic properties of estimators of the parameters of models for the conditional mean; see CT07, XP08 and, for multivariate models, Cavaliere, Rahbek and Taylor (2010a;b) and Boswijk, Cavaliere, Rahbek and Taylor (2016).
In the framework of a conditional mean, or a general (stationary, or non-stationary) regression type model, the wild bootstrap is an important tool to deliver consistent estimation of the asymptotic distributions of test statistics or parameter estimators. The wild bootstrap allows in particular to track changes in the quadratic variation of an econometric model by simply mimicking the (unknown) volatility dynamics through the squared model residuals, see GK04, GK07 for applications to stationary time series models and CRT10a,CRT10b for non-stationary multivariate models.
Consider the simple case where the volatility, say $\sigma _{t}$, can be approximated by a non-stochastic element of the space $ \mathscr{D}[0,1]$ of c\`{a}dl\`{a}g functions on $[0,1]$, such that $\sigma _{t}=\sigma (t/n)$ ($t=1,\ldots ,n$, $n$ denoting the sample size) with $ \sigma \in \mathscr{D}[0,1]$. Simple special cases are a single volatility break at time $\lfloor n\tau \rfloor $ (with $\lfloor \cdot \rfloor $ denoting the (floor) integer value), for some $\tau \in (0,1)$, as given by (with $\mathbb{I}_{A}(\cdot )$ denoting the indicator function of the set $A$ )
where $\sigma_A >0$ and $\sigma_B >0$; or the case of trending volatility,
where $\sigma_A >0$, $\sigma_B >0$ and $\delta >0$. A classic wild bootstrap, based on the resampling scheme $ \varepsilon _{t}^{\ast }=\hat{\varepsilon}_{t}w_{t}^{\ast }$, where the $ \hat{\varepsilon}_{t}$'s are the estimated residuals from the regression model and the $w_{t}^{\ast }$'s are i.i.d.\ $(0,1)$ bootstrap shocks, independent of the original sample, is in general able to track the volatility path (in terms of quadratic variation) of the original data, without any assumption on the initial values of the volatility process as, loosely speaking, under standard assumptions
Existing theory of the bootstrap mainly focuses on such deterministic changes of the unconditional variance and little is known about the performance and the validity\footnote{ Throughout the paper, with `validity' of the bootstrap we mean that the associated bootstrap tests control size asymptotically. With `consistency of the bootstrap test' we mean that the (bootstrap) test rejects with probability tending to one under the alternative.} of the bootstrap when the volatility is driven by a high-persistent, or (second order) non-stationary stochastic process. This includes leading key cases such as near-integrated exogenous volatility processes (as analyzed by H95), as well as near-integrated GARCH processes, where the conditional variance has a diffusion limit N90.
This paper fills this gap in the literature by developing conditions for bootstrap validity and consistency of the associated bootstrap tests in regression and time series models with persistent stochastic volatility. That is, we replace the deterministic volatility assumption by allowing that volatility is the realization of a (non-stationary) stochastic process $ \sigma _{t}$; specifically, we derive results under the general assumption that, for the c\`{a}dl\`{a}g version of the volatility, it holds that
where $\sigma $ is some random element in $\mathscr{D}[0,1]$.
The analysis of the bootstrap under a weak convergence assumption like ((ref)) is not straightforward. As we show in the paper, a key fact under non-stationary stochastic volatility is that the distribution of bootstrap statistics (conditional on the data), rather than converging to the unconditional distribution of the statistic of interest, converges weakly to a random limit. By this we mean that the distribution function of the bootstrap statistic (conditional on the data) is stochastic not only for finite sample sizes $n$, but also in the limit as $n\rightarrow \infty $. Consequently, the conventional approach, based on the notion of weak convergence in probability of the bootstrap statistic to the limiting distribution of the original statistic (which is obviously non-stochastic), fails to deliver the required result of validity of the bootstrap. This problem is not new in the bootstrap literature, as it appears in various areas of application of the bootstrap; for example, in models with infinite variance innovations K89 and in autoregressive models with unit roots B91,CNR15.
Specifically, in this paper we analyze the wild bootstrap under (non-stationary) stochastic volatility by adopting a new approach to assess bootstrap validity under random limit bootstrap measures. Thus, rather than focusing on the usual weak convergence in probability of the bootstrap conditional distribution, we make use of the concept of weak convergence in distribution (see CG20, for a general introduction) to develop novel conditions for validity of the wild bootstrap, conditional on the volatility process. This allows us to establish that, although the presence of a random limiting distribution for the bootstrap statistic makes the bootstrap unable to estimate the unconditional distribution of the statistic of interest, the bootstrap can still deliver hypothesis tests with the desired size. In particular, we do this by establishing that the high-level conditions for bootstrap validity in CG20 can be shown to hold for a large class of models with stochastic volatility, including the aforementioned near-integrated GARCH model and the non-stationary stochastic volatility model. We do so by showing new weak convergence results conditional on volatility paths.
To illustrate our new approach and its applicability, we apply our results to three leading testing problems in the presence of non-stationary stochastic volatility, including testing a hypothesis on the location of a time series, testing for a unit root and testing for stability of the conditional mean using CUSUM-type statistics. These illustrative examples can easily be extended to cover more general cases, such as cointegration (as in CRT10a, CNR15 and BCRT16) with multivariate stochastic volatility, or multivariate stability tests (see P06 and CP19). Importantly, for all examples we show that conditions for conditional wild bootstrap validity include the absence of statistical leverage effects, i.e.\ correlation between the error process and its future conditional variance. The results of the paper are illustrated using Monte Carlo simulations, which indicate that under the conditions developed in our paper, the wild bootstrap leads to excellent size control even in small samples.
The structure of the paper is the following. In Section (ref) we introduce the reference data generating process and our main assumptions, in particular on the volatility. Here we also introduce three examples which are used throughout the paper to illustrate the main results. We also derive the reference limit distribution for non-bootstrap statistics under non-stationary volatility. In Section (ref) we introduce the main (wild) bootstrap algorithm. We show that when volatility is non-stationary, the bootstrap fails to mimic the asymptotic distribution of the corresponding statistics, and hence it is not valid in the usual sense. In Section (ref) we discuss the wild bootstrap and prove, under proper assumptions, validity conditionally on the volatility path, as well as consistency of the bootstrap tests under the alternative. We first introduce in Section (ref) the concept of weak convergence in distribution and discuss how to prove validity of the bootstrap in the presence of random limit bootstrap distributions, as it happens here under non-stationary volatility. Then in Section (ref) we provide our main results under the required additional conditions on the original data. In Section (ref) we apply our results about validity of the bootstrap in our three applications. Finally, in Section (ref) we discuss consistency of the bootstrap tests under the alternative hypothesis. Results from a Monte Carlo study on the finite sample behavior of the bootstrap tests are reported in Section (ref). Section (ref) concludes. All proofs are reported in the online appendix.
The following (standard) notation is used throughout. With $x:=y$ ($y=:x$) we mean that $x$ is defined by $y$ ($y$ defined by $x$). For any $q\in \mathbb{R}$ ($\mathbb{R}$ denoting the set of real numbers), $\lfloor q\rfloor $ denotes the integer part of $q$. For random elements $(X_{n},X)$ of a metric space, weak convergence of $X_{n}$ to $X$ is denoted by $X_{n} \overset{w}{\rightarrow }X$. Also, $\overset{d}{=}$ denotes equality in distribution. We use $P^{\ast }$, $E^{\ast }$ and $V^{\ast }$ respectively to denote probability, expectation and variance, conditional on the original sample. With $\overset{w^{\ast }}{\rightarrow }_{p}$ we denote weak convergence in probability; in particular, for random varables $(X_{n}^{\ast },X)$, the notation $X_{n}^{\ast }\overset{w^{\ast }}{\rightarrow }_{p}X$ means that, as the sample size $n$ diverges, the cumulative distribution function [cdf] $G_{n}^{\ast }$ of $X_{n}$, conditional on the original data, converges in probability to the cdf $G$ of $X$, at all continuity points of $ G$. For a given sequence $X_{n}^{\ast }$ of random elements, computed from the bootstrap data, $X_{n}^{\ast }-X=o_{p}^{\ast }(1)$, in probability, or $ X_{n}^{\ast }\overset{p^{\ast }}{\rightarrow }_{p}X,$ means that for any $ \epsilon >0$, $P^{\ast }(||X_{n}^{\ast }-X||>\epsilon )\overset{p}{ \rightarrow }0$, as $n\rightarrow \infty $. Similarly, $X_{n}^{\ast }=O_{p}^{\ast }(1)$, in probability, means that, for every $\epsilon >0$, there exists a constant $M>0$ such that, for all large $n$, $P(P^{\ast }(||X_{n}^{\ast }||>M)<\epsilon )$ is arbitrarily close to one. Weak convergence in distribution and the related notation are introduced in Section (ref). The Skorokhod spaces of c\`{a}dl\`{a}g functions $[0,1]\rightarrow \mathbb{R} ^{m\times n}$ and $[0,1]\rightarrow \mathbb{R}^{n}$ are denoted by $ \mathscr{D}{}_{m\times n}[0,1]$ and $\mathscr{D}{}_{n}[0,1]$, respectively; for the latter, when $n=1$ the subscript is suppressed. The Skorokhod space of c\`{a}dl\`{a}g functions $\mathbb{R}\rightarrow \mathbb{R}$ is denoted by $\mathscr{D}{}(\mathbb{R})$.
In this section we introduce our reference class of models for the conditional mean under stochastic volatility as well as the (test) statistics of interest. In Section (ref) we focus on statistics which can be expressed (at least when the associated null hypothesis holds true) as functionals of the partial sum of the innovations and of the partial sum of the squared innovations. To illustrate ideas, we consider three simple univariate cases (which can easily be extended to multivariate cases) throughout: (i) testing a hypothesis on the mean in a simple location model; (ii) CUSUM\ testing for parameter constancy in a location model; (iii)\ testing for an autoregressive unit root in an AR(1) model. The main assumption on the volatility --- which, inter alia, allows for non-stationary stochastic volatility or near-integrated GARCH\ dynamics --- is discussed next in Section (ref). Under the assumptions in Sections (ref) and (ref) , the asymptotic (null) distributions can be derived. We do this in Section (ref), where we show that the limiting distribution can be expressed in terms of a continuous martingale and its quadratic variation process. The implications of these results on bootstrap inference and hypothesis testing are the focus of the main Sections (ref) and (ref).
We are concerned with inference and hypothesis testing on the regression parameters of a heteroskedastic time series regression model in a triangular array form:
where $\varepsilon _{n,t}$ is a martingale difference sequence (mds) relative to a suitable filtration $\mathcal{F}_{n,t}$, with conditional variance $\sigma _{n,t}^{2}=E(\varepsilon _{n,t}^{2}|\mathcal{F}_{n,t-1})$. To simplify notation, unless strictly required we simply write ((ref)) as $y_{t}=\beta ^{\prime }x_{t}+\varepsilon _{t}$, with $\sigma _{t}^{2}:=E(\varepsilon _{t}^{2}|\mathcal{F}_{t-1})$.
Inference focuses on test statistics, which we assume can be expressed (at least under the null hypothesis) as functionals of partial sum processes in terms of the innovations and squared innovations
as is the case for many testing problems, see also the discussion and examples below.
Defining $\mathcal{F}_{n}(u):=\mathcal{F}_{\lfloor nu\rfloor }$, the mds assumption implies that $\left\{ M_{n}(u),\mathcal{F}_{n}(u)\right\} _{u\in \lbrack 0,1]}$ is a martingale for all $n$, and $U_{n}(u)$ is its quadratic variation process, i.e.,
Throughout it will also be useful to define the predictable quadratic variation or angle bracket process (see JS03):
with the defining property that $\left\{ M_{n}^{2}(u)-\langle M_{n}\rangle (u),\mathcal{F}_{n}(u)\right\} _{u\in \lbrack 0,1]}$ is a martingale.
The following three testing problems are discussed in the paper. These are all special cases of ((ref)) where the statistic of interest is indeed a functional of $\left( M_{n},U_{n}\right) $.
Example 1 (testing in a location model). Consider the location model $y_{t}=\theta +\varepsilon _{t}$, which is trivially obtained from ((ref)) by setting $\beta =\theta $ and $x_{t}=1$. The true location parameter is denoted by $\theta _{0}$. Suppose that interest is in testing the simple null hypothesis $\theta =\bar{\theta}$. Then, one can consider the test statistic $S_{n}:=\sqrt{n}(\overline{y}_{n}-\bar{\theta})$ , where $\overline{y}_{n}=n^{-1}\sum_{t=1}^{n}y_{t}$, or, alternatively, its studentized version $T_{n}:=\sqrt{n}(\overline{y}_{n}-\bar{\theta})/s_{n}$, with $s_{n}^{2}=n^{-1}\sum_{t=1}^{n}(y_{t}-\bar{y}_{n})^{2}$. It is not difficult to see that, under the null hypothesis, it holds that $S_{n}$ and $ T_{n}$ can be expressed in terms of $M_{n}$ and $U_{n}$ defined in ((ref) ) as
If $s_{n}$ is constructed with the null imposed, i.e. $s_{n}^{2}=n^{-1} \sum_{t=1}^{n}(y_{t}-\bar{\theta})^{2}$, then under the null hypothesis $T_{n}$ simplifies to
Example 2 (CUSUM test in a location model). Consider the time-varying location model $y_{t}=\theta _{t}+\varepsilon _{t}$, and suppose that interest is in testing the null hypothesis of a constant location parameter, i.e. $\mathsf{H}_{0}:\theta _{t}=\theta _{1}$, $ t=2,\ldots ,n$. A standard CUSUM test can be constructed by considering the statistic (see e.g.\ DP08, and the references therein)
or its studentized version,
which, as in Example 1, reduce to
under $\mathsf{H}_{0}$.
Example 3 (Testing for a unit root) Consider the first-order autoregression $y_{t}=(1+\theta )y_{t-1}+\varepsilon _{t}$, with $y_{0}=0$ (which again follows from ((ref)) by setting $\beta =1+\theta $ and $x_{t}=y_{t-1}$). A test of the unit root hypothesis $\theta =0$ can be based on the Dickey-Fuller `coefficient' statistic $R_{n}:=n\hat{\theta } _{n}$, where $\hat{\theta }_{n}=\sum_{t=1}^{n}y_{t-1}\Delta y_{t}/\sum_{t=1}^{n}y_{t-1}^{2}$ is the least-squares estimator from the regression of $\Delta y_{t}$ on $y_{t-1}$. Under the null hypothesis, $\hat{ \theta }_{n}=\sum_{t=1}^{n}\varepsilon _{t}(\sum_{i=1}^{t-1}\varepsilon _{i})/\sum_{t=1}^{n}(\sum_{i=1}^{t-1}\varepsilon _{i})^{2}$ and the test statistic may be expressed as
If the test is based on the Dickey-Fuller `ratio' statistic $W_{n}:=\hat{ \theta }_{n}(s_{n}/(\sum_{t=1}^{n}y_{t-1}^{2}))^{-1/2}$, where $ s_{n}^{2}:=n^{-1}\sum_{t=1}^{n}(\Delta y_{t}-\hat{\theta}y_{t-1})^{2}$, then
under the null hypothesis.$\hfill \square $
Some remarks are in order.
We now introduce our basic hypotheses on the dynamic behavior of the conditional volatility $\sigma _{t}^{2}$ of the shocks $ \varepsilon _{t}$. More specifically, we will allow volatility to be a persistent stochastic process, with a stochastic volatility weak limit, as formulated in the next two assumptions. These are in the spirit of the seminal paper by H95, who considers conditional variances driven by nearly-integrated autoregressive shocks, although we do not constrain the behavior of the conditional variance to be of the autoregressive type.
Define now the $\mathscr{D}{}[0,1]$ version of the partial sum of the $z_{t}$'s as $B_{z,n}(u):=n^{-1/2}\sum_{t=1}^{\lfloor nu\rfloor }z_{t}$, $u\in \lbrack 0,1]$, and $\sigma _{n}(u)$, $u\in \lbrack 0,1]$, the $ \mathscr{D}{}[0,1]$ version of $\sigma _{t}$, as:
with $\sigma _{n}(1):=\sigma _{n}$. In the following, the process $\sigma _{n}(u)$, $u\in \lbrack 0,1]$, will be referred to as $\sigma _{n}$; this should not be confused with $\sigma _{t}$ for $t=n$, as also will be clear from the context where the notation is used.
While the convergence of the partial sum $B_{z,n}\in \mathscr{D}{}[0,1]$ is standard, the requirement on $\sigma _{n}\in \mathscr{D}{}[0,1]$ is not. More specifically, this assumption requires the conditional variance process $\{\sigma _{t}^{2}\}_{t=1}^{n}$ to be persistent enough such that its behavior can be approximated by an element of the space of c\`{a}dl\`{a}g functions $\mathscr{D}{}[0,1]$. No higher moment conditions are required on $\{\sigma _{t}^{2}\}_{t=1}^{n}$ or $ \{\varepsilon _{t}\}_{t=1}^{n}$. Some examples of processes satisfying Assumptions (ref) and (ref) are presented next. These will be analyzed in detail throughout the paper; for additional cases and discussions see e.g.\ CT09.
Assumptions (ref) and (ref) allow to analyze the asymptotic behavior of the functional $\left( M_{n},U_{n}\right) $, as is done in the following Lemma.
The implications for the testing problems in Section (ref) are given next.
Example 1 (cont'd). Consider the location model of Example 1. A straightforward application of Lemma (ref) along with the CMT yields that, under $\mathsf{H}_{0}$, $S_{n}=M_{n}(1)\overset{w}{ \rightarrow }M(1)$, which corresponds to the mixed normal distribution $ N(0,\int_{0}^{1}\sigma ^{2}(u)\mathsf{d}u)$ and hence is non-pivotal. For $ T_{n}$ it holds that
Notice that the limit distribution in ((ref)) is non-pivotal in cases where $\sigma $ and $B_{z}$ are not stochastically independent. In contrast, should independence hold, then ((ref)) corresponds to a standard Gaussian distribution.
Example 2 (cont'd). For the CUSUM test statistics of Example 2 it holds that, again by Lemma (ref) and the CMT, that under $\mathsf{H}_{0}$
Both statistics have a non-pivotal asymptotic null distribution, even in cases where the limit stochastic volatility process and the limit Brownian motions are stochastically independent.
Example 3 (cont'd). Finally, as shown in CT09, for the unit root testing problem the presence of non-stationary volatility renders the null distribution of the Dickey-Fuller coefficient and $t$ -statistics non-pivotal. More specifically, under the unit root null hypothesis it holds that
and
as the sample size diverges.$\hfill \square $
Consider the standardized sample mean statistic $S_{n}$ for the location model, see Example 1. Lemma (ref) implies that under the null hypothesis $S_{n}\overset{w}{\rightarrow }M(1)=\int_{0}^{1}\sigma (u)\mathsf{ d}B_{z}(u)$, see ((ref)). The distribution of $M(1)$ depends on the limit volatility process $\sigma $, implying that critical values cannot be tabulated without providing a complete specification of this process. When the Brownian motion $B_{z}$ and the limit volatility process $\sigma $ are independent, then in the location model of Example 1 this problem can be avoided by considering the studentized test statistic $T_{n}$, which has a standard normal limit distribution under the null. However, in other testing problems (such as those in Examples 2 and 3) it is generally not possible to find such asymptotically pivotal statistics.
This motivates the development of bootstrap tests. Following much of the literature (e.g. CT08,CT09), we consider the wild bootstrap, which replicates the volatility patterns in the original data. Let $w_{t}^{\ast }$ be an i.i.d.\ sequence with mean zero and variance one\footnote{ Further conditions on the moments of $w_{t}^{\ast }$ may be required in some specific applications.}, independent of $\{\sigma _{t},z_{t}\}_{t\geq 1}$, and define the bootstrap shocks as
Accordingly, we can define the bootstrap partial sum and the bootstrap partial sum of squares,
These processes are the bootstrap analogs of the processes $M_{n}$ and $ U_{n} $ of Section (ref). Notice that this implementation of the bootstrap assumes that $\varepsilon _{t}$ is observed under the null hypothesis, which is the case in Examples 1 and 3 (location and unit root test). In more general testing problems, including Example 2 (CUSUM test), $ \varepsilon _{t}$ will be replaced by some residuals $\hat{\varepsilon}_{t}$ (either restricted by the null hypothesis or unrestricted).
Classic validity of the bootstrap (usually denoted as `bootstrap consistency') is usually understood as the convergence in probability (or almost surely) of the conditional (on the original data) cdf of the bootstrap statistic to the limit cdf of the original statistic. We show here that, in the presence of stochastic volatility as in the previous section, in general classic validity of the bootstrap fails. This is essentially because the conditional cdf of the bootstrap statistic remains random in the limit. In this section we discuss this fact and its implications on bootstrap inference using, as the reference bootstrap algorithm, a wild bootstrap scheme as is typically applied when the data are heteroskedastic.
Focusing again on the location statistic $S_{n}$, its bootstrap counterpart is $S_{n}^{\ast }=n^{-1/2}\sum_{t=1}^{n}\varepsilon _{t}^{\ast }=M_{n}^{\ast }(1)$ where $M_{n}^{\ast }$ is as previously defined. Define $ P^{\ast }$ as the bootstrap measure conditional on the original data, $D_{n}$ . The bootstrap (conditional) cdf is
The classical condition for bootstrap validity is that, as $n\rightarrow \infty $, $S_{n}^{\ast }\overset{w^{\ast }}{\rightarrow }_{p}S:=M(1)$. If the limit cdf $F(x):=P(S\leq x)$ is continuous, then this weak convergence in probability corresponds to the property
where $F$ denotes the cdf of the asymptotic distribution of $S_{n}$:
where $\Phi (\cdot )$ is the standard normal cdf. Because $F(x)$ is the marginal cdf of $S=M(1)$, under independence of the processes $B_{z}$ and $ \sigma $ it corresponds to the cdf of the mixed normal random variable $ S=V(1)^{1/2}Z$, where $Z\sim N(0,1)$, independent of $V(1)$.
However, under Assumption (ref), condition ((ref)) fails to hold, which is seen as follows. Choosing $w_{t}^{\ast }\sim \mathrm{i.i.d.}~N(0,1)$ for convenience, it is seen that
This follows because, conditional on the original data,
In terms of the conditional distribution $F_{n}^{\ast }$ of $S_{n}^{\ast }$ given the data $D_{n}$, ((ref)) corresponds to
Letting $n\rightarrow \infty $, the limit distribution of the bootstrap statistic given the data follows from Lemma (ref) and the CMT. Specifically, we have that
for all $x\in \mathbb{R}$; eq.\ ((ref)) implies that the limit distribution of the bootstrap cdf is in fact random. That is, the bootstrap cdf $F_{n}^{\ast }$ does not converge in probability but weakly, and the limiting cdf is random, as it depends on the random variable $V(1)$. Therefore, there is no reason to expect that the difference between the random function $F_{n}^{\ast }$ and the non-random function $F$ converges in probability to $0$, as required for standard bootstrap validity to apply.
The fact that the conditional cdf $F_{n}^{\ast }$ converges weakly (in $ \mathscr{D}{}(\mathbb{R})$), rather than in probability, to a random cdf, will be referred to as `weak convergence in distribution', and denoted as `$ \overset{w}{\rightarrow }_{w}$'. More specifically, for sequences of random variables $(Z_{n},Y_{n})$ and $(Z,Y)$ (possibly defined on different probability spaces), the notation $Z_{n}|Y_{n}\overset{w}{\rightarrow } _{w}Z|Y$, when the conditional distribution of $Z|Y$ is diffuse (non-atomic), means that
A more general definition of $Z_{n}|Y_{n}\overset{w}{\rightarrow }_{w}Z|Y$, which includes non-diffuse conditional distributions and that is applicable to the case where $(Z_{n},Y_{n})$ and $(Z,Y)$ are random elements of the metric spaces $\mathcal{S}_{Z}\times \mathcal{S}_{Y_{n}}$ and $\mathcal{S} _{Z}\times \mathcal{S}_{Y}$, respectively (and hence to stochastic processes), is that $E(g(Z_{n})|Y_{n})\overset{w}{\rightarrow }E(g(Z)|Y)$ for all bounded continuous functions $g:\mathcal{S}_{Z}\rightarrow \mathbb{R} $, see CG20 and the references therein. We generalize this definition to multivariate (joint) convergence in Section (ref) below and employ it to prove bootstrap validity conditionally on the volatility path.
When $Z_{n}$ represents a bootstrap statistic and the conditioning set $ Y_{n} $ is the original data $D_{n}$, we use the notation `$\overset{w^{\ast }}{\rightarrow }_{w}$'. Hence, eq.\ ((ref)) corresponds to the weak convergence in distribution
Unless $V(1)$ is non-random (which is not the case under stochastic volatility as considered here), this convergence shows that the limit bootstrap measure is indeed a random measure. Hence, the bootstrap cannot be valid in the usual sense of weak convergence in probability of $F_{n}^{\ast } $ to $F$.
The result in Section (ref) applies to the other examples considered, except for the asymptotically pivotal statistic $ T_{n}$.
Example 1 (cont'd). Consider the location model example and assume that the bootstrap data are generated as $y_{t}^{\ast }=\varepsilon _{t}^{\ast }$, with $\varepsilon _{t}^{\ast }$ as defined above. The bootstrap test statistics are $S_{n}^{\ast }:=\sqrt{n}\bar{ \varepsilon}_{n}^{\ast }$ and $T_{n}^{\ast }:=\sqrt{n}\bar{\varepsilon} _{n}^{\ast }/s_{n}^{\ast }$, $s_{n}^{\ast }=(n^{-1}\sum_{t=1}^{n}(\varepsilon _{t}^{\ast }-\bar{\varepsilon}_{n}^{\ast })^{2})^{1/2}$. Using the argument discussed above, we have that $ S_{n}^{\ast }\overset{w^{\ast }}{\rightarrow }_{w}M^{\ast }(1)|V(1)$, where $ M^{\ast }(u):=\int_{0}^{u}\sigma (s)\mathsf{d}B_{z}^{\ast }(s)$ with $ B_{z}^{\ast }$ a standard Brownian motion, stochastically independent of $ \sigma $, and $V(1):=\int_{0}^{1}\sigma (u)^{2}\mathsf{d}u$. The conditional distribution of $M^{\ast }(1)|V(1)$ is equal (a.s.) to the conditional normal distribution $N(0,V(1))|V(1)$. In contrast, for $T_{n}^{\ast }$ it holds that
where the limit corresponds to a conditional $N(0,1)$ distribution (independent of $V(1)$), following from the conditional $N(0,V(1))$ distribution of $M^{\ast}(1)\vert V(1)$. Since weak convergence of a conditional distribution to a non-random cdf corresponds to weak convergence in probability, in this special case $T_{n}^{\ast }\overset{w^{\ast }}{ \rightarrow }_{p}N(0,1)$.
Two facts are worth stressing. First, in the above representations of the limit conditional distribution of the bootstrap statistic, $\sigma $ and $ B_{z}^{\ast }$ are independent, even if the original processes $\sigma $ and $B_{z}$ are not. This result stems from the assumption that the wild bootstrap shocks $w_{t}^{\ast }$ are independent of the original data. Second, if $\sigma $ and $B_{z}$ are stochastically independent, then the conditional distributions of $M^{\ast }(1)|V(1)$ and $M(1)|V(1)$ are equal (a.s). This distributional equality is crucial to determine validity of the bootstrap.
Example 2 (cont'd). For the bootstrap CUSUM statistics, suppose that the bootstrap data are generated as $\varepsilon _{t}^{\ast }= \hat{\varepsilon}_{t}w_{t}^{\ast }$ where $\hat{\varepsilon}_{t}:=y_{t}-\bar{ y}_{n}$, such that when the null hypothesis is true $\hat{\varepsilon} _{t}=\varepsilon _{t}-\bar{\varepsilon}_{n}$. The bootstrap statistics are defined as
with $M_{n}^{\ast }(u):=n^{-1/2}\sum_{t=1}^{\lfloor n\cdot \rfloor }\varepsilon _{t}^{\ast }$ as above and $U_{n}^{\ast }(1):=n^{-1}\sum_{t=1}^{n}(\varepsilon _{t}^{\ast })^{2}$. Under the null hypothesis,
uniformly in $u\in \lbrack 0,1]$, and it holds that
where again $M^{\ast }(u):=\int_{0}^{u}\sigma (s)\mathsf{d}B_{z}^{\ast }(s)$ with $B_{z}^{\ast }$ a standard Brownian motion, stochastically independent of $\sigma $, and $V(1):=\int_{0}^{1}\sigma (u)^{2}\mathsf{d}u$. Both bootstrap statistics have a random non-pivotal asymptotic null distribution.
Example 3 (cont'd). Finally, consider the unit root example. To avoid the problems described in B91, the bootstrap data are generated with the unit root imposed, i.e.\ $y_{t}^{\ast }=y_{t-1}^{\ast }+\varepsilon _{t}^{\ast }$, with $y_{0}^{\ast }=0$ and $\varepsilon _{t}^{\ast }:=(\Delta y_{t})w_{t}^{\ast }$; see e.g.\ CT08. Under the null, clearly $\varepsilon _{t}^{\ast }:=\varepsilon _{t}w_{t}^{\ast }$. As discussed earlier for the non-bootstrap case, we have that
and, up to a negligible term,
In this case it holds that
where $M^{\ast }(u):=\int_{0}^{u}\sigma (s)\mathsf{d}B_{z}^{\ast }(s)$ with $ B_{z}^{\ast }$ a standard Brownian motion, stochastically independent of $ \sigma $, and $V(u):=\int_{0}^{u}\sigma (s)^{2}\mathsf{d}s$. The asymptotic distributions in ((ref)) are random, except in the special case where $\sigma $ is non-stochastic.$ \hfill \square $
CG20 provide a number of other examples where the bootstrap validity condition ((ref)) fails for any non-random cdf $F$, and develop an alternative criterion for conditional bootstrap validity. We will apply this concept to the present situation and extend it to the analysis of consistency of the bootstrap tests in the next section.
Despite the fact that under non-stationary stochastic volatility the bootstrap is unable to consistently estimate the limiting distribution of the original statistic, it can still be valid, in the sense that it delivers control over type one error probabilities as $n$ diverges. This can be seen by focusing on the bootstrap p-value. Taking the statistic $S_{n}$ and associated bootstrap analog $S_{n}^{\ast }$ to illustrate, the bootstrap p-value is defined as
where $F_{n}^{\ast }(\cdot )$ is the cdf of $S_{n}^{\ast }$, conditional on the data (this definition of the $p$-value assumes a left-tailed test, which will be assumed below unless indicated otherwise). As in CG20, we say that the bootstrap based on $S_{n},S_{n}^{\ast }$ is valid conditionally on the volatility process $\{\sigma _{t}\}_{t=1}^{n}$ if $p_{n}^{\ast }$ is asymptotically $U(0,1)$ distributed conditionally on $\{\sigma _{t}\}_{t=1}^{n}$, i.e.
If this is the case, even if (as shown in the previous section) the limiting conditional distribution of $S_{n}^{\ast }$ is random, the bootstrap test can still be correctly sized in large samples. Moreover, proofs of validity in the form of ((ref)) also imply\footnote{ For fixed $q\in (0,1)$, $Q_{n}:=P(p_{n}^{\ast }\leq q|\left\{ \sigma _{t}\right\} _{t=1}^{n})$, is a random sequence in $[0,1]$, and hence uniformly integrable. Result ((ref)), i.e.\ $Q_{n} \overset{p}{\rightarrow }q$, together with uniform integrability of $Q_{n}$, implies $L^{1}$ convergence and hence $P(p_{n}^{\ast }\leq q)=E(Q_{n})\rightarrow q$ (see Kallenberg, 1997, Lemma 3.11).} that, unconditionally,
In the next subsections we discuss a set of sufficient conditions for ((ref)) to hold. These are new in the literature on bootstrapping conditional mean models when the volatility can be stochastic. First, in Section (ref) we provide our strategy to assess bootstrap validity. Our main results are given in Section (ref). Application to our examples are provided in Section (ref). Finally, the behavior under the alternative is analyzed and applied to our examples in Section (ref).
In this section, we summarize the approach developed by CG20, applied here to establish conditional bootstrap validity in the presence of non-stationary stochastic volatility. Before turning to bootstrap validity, we introduce a multivariate version of the concept of weak convergence in distribution that we anticipated in Section (ref).
Precisely, let $Z_{n}:=(Z_{n}^{(1)},Z_{n}^{(2)})$ and $Z:=(Z^{(1)},Z^{(2)})$ be random elements of the (complete and separable)\ metric space $\mathcal{S} _{Z^{(1)}}\times \mathcal{S}_{Z^{(2)}}$, and $ Y_{n}:=(Y_{n}^{(1)},Y_{n}^{(2)})$ and $Y:=(Y^{(1)},Y^{(2)})$ be random elements of the (complete and separable)\ metric spaces $\mathcal{S} _{Y^{(1)}}^{\prime }\times \mathcal{S}_{Y^{(2)}}^{\prime }$ and $\mathcal{S} _{Y^{(1)}}\times \mathcal{S}_{Y^{(2)}}$, respectively. We say that
jointly if, for all bounded continuous $g:\mathcal{S}_{Z^{(1)}}\rightarrow \mathbb{R}$ and $h:\mathcal{S}_{Z^{(2)}}\rightarrow \mathbb{R}$,
see K97 and CG20. If $Z_{n}^{(i)},Z^{(i)}$, $i=1,2$ are random variables and the conditional distributions $Z^{(1)}|Y^{(1)}$, $ Z^{(2)}|Y^{(2)}$ are diffuse, then the above convergence is equivalent to
Finally, if $Z^{(1)}$ and $Z^{(2)}$ are the same random element (and similary for $Y^{(1)}$ and $Y^{(2)}$), then we use the following alternative notation for ((ref)):
and for $Z_{n}=(Z_{n}^{(1)},Z_{n}^{(2)})$ a bivariate random variable, ((ref)) becomes
Given this definition, we turn the attention to a statistic $\tau _{n}$ which is a function of the data $D_{n}$, which in our general set-up may be represented by $(M_{n},U_{n})$; that is, $\tau _{n}=\tau (M_{n},U_{n})$. Its bootstrap equivalent is $\tau _{n}^{\ast }=\tau (M_{n}^{\ast },U_{n}^{\ast }) $, and we let $\tau =\tau (M,V)$, with $(M,V)$ denoting the weak limit of $\left( M_{n},U_{n}\right) $, see Section 2.3.
Recall that, with $\sigma _{n}\in \mathscr{D}{}[0,1]$ the volatility process defined in ((ref)), conditioning on $\sigma _{n}$ is equivalent to conditioning on $\{\sigma _{t}\}_{t=1}^{n}$, and moreover that $\sigma \in \mathscr{D}[0,1]$ is the weak limit of $\sigma _{n}$ by Assumption (ref). It then follows by CG20 that if the condition
is satisfied, with the random cdf of $\tau |\sigma $ being sample-path continuous, then
This means that the bootstrap consistently estimates the distribution of the original statistic conditional on the volatility process, which in turn implies that the bootstrap is conditionally valid, i.e.,
for all $q\in (0,1)$, where $p_{n}^{\ast }=P(\tau _{n}^{\ast }\leq \tau _{n}|D_{n})$ is the bootstrap p-value. The key condition to verify is therefore the one given in ((ref)), along with continuity of the limiting (random) cdf.
Because $\tau _{n}=\tau (M_{n},U_{n})$ with $(M_{n},U_{n})\in \mathscr{D}{} _{2}[0,1]$, proving ((ref)) involves proving conditional functional limit theorems. It is known, see G94 and CP05, that joint weak convergence of e.g.\ $(\left( M_{n},U_{n}\right) ,\sigma _{n})$ is not sufficient for conditional weak convergence. For example, G94 shows that a sufficient condition is that $\left( M_{n},U_{n}\right) $ is independent of $\sigma _{n}$, or that a change of measure can be found (with weakly convergent Radon-Nikodym derivative) under which this independence holds. These conditions do not seem to be directly applicable to the present case.
Recall that the main assumption used to derive the limiting distribution of the original statistic and the limiting (conditional) distribution of the bootstrap statistic is that the errors form a mds with respect to the past information set. This condition, however, is not sufficient for conditional bootstrap validity, unless the volatility is deterministic or stationary. In the presence of non-stationary stochastic volatility, further conditions are required. A sufficient set of conditions is provided in the following assumption.
A few remarks are in order.
The main result in this section is given in Theorem (ref) and Corollary (ref) below.
The key result of Theorem (ref) is that the bootstrap processes $ M_{n}^{\ast }$ and $U_{n}^{\ast }$, conditionally on the data, replicate in the limit the distribution of the original processes $M_{n}$ and $U_{n}$, conditionally on the volatility process $\sigma _{n}$ (or, equivalently, on $ \{\sigma _{t}\}_{t=1}^{n}$). The implication of Theorem (ref) on the behaviour of the bootstrap p-values is provided in the following corollary, which applies to a statistic $\tau _{n}=\tau (M_{n},U_{n})$ (which, under the null, converges weakly to $\tau =\tau (M,V)$) and its bootstrap equivalent $\tau _{n}^{\ast }=\tau (M_{n}^{\ast },U_{n}^{\ast })$.
In this section we check whether the conditions for bootstrap validity hold for the examples. We assume throughout that Assumptions (ref) and (ref), strengthened by (ref), hold.
Example 1 (cont'd). As earlier, the bootstrap statistics are given by $S_{n}^{\ast }:=\sqrt{n}\bar{\varepsilon}_{n}^{\ast }$ and $ T_{n}^{\ast }:=\sqrt{n}\bar{\varepsilon}_{n}^{\ast }/s_{n}^{\ast }$. Under the null hypothesis, the original statistics are given by $S_{n}^{\ast }= \sqrt{n}\bar{\varepsilon}_{n}$ and $T_{n}^{\ast }=\sqrt{n}\bar{\varepsilon} _{n}/\hat{s}_{n}$. The original (bootstrap) statistics obtain as a continuous transformation of $(M_{n},U_{n})$ (of $(M_{n}^{\ast },U_{n}^{\ast })$). Hence by Theorem (ref) and a version of the CMT (see CG20, Theorem A.1) it holds that
with $M(1)=\int_{0}^{1}\sigma (u)\mathsf{d}B_{z}(u)$. With $ V(1)=\int_{0}^{1}\sigma (u)^{2}\mathsf{d}u$, the cdf of $M(1)|\sigma $ is given by $\Phi (uV(1)^{-1/2})$, which is sample-path continuous with probability 1. Hence, by Corollary (ref), the bootstrap is valid conditionally on the volatility path $\sigma $. For the studentized statistic it holds that
where $Z\sim N(0,1)$, independent of $\sigma $; this implies that the bootstrap is conditionally valid. Notice also that, see the discussion in Section (ref), ((ref)) and continuity of the limiting Gaussian cdf $\Phi $ imply that $\sup_{x\in \mathbb{R}}|P^{\ast }(T_{n}^{\ast }\leq x)-\Phi (x)|=o_{p}\left( 1\right) $. As $T_{n}\overset{w}{\rightarrow }Z$, the bootstrap test based on the studentized statistics $T_{n},T_{n}^{\ast }$ is also valid in the classic sense, i.e.\ the distribution of the bootstrap statistic $T_{n}^{\ast }$ conditional on the data consistenty estimates the (unconditional)\ distribution of the original statistic $T_{n}$. The same type of result does not hold for the bootstrap based on $S_{n},S_{n}^{\ast }$; however, bootstrap conditional validity is guaranteed by Corollary (ref).
Example 2 (cont'd). As for the previous example, since the CUSUM (bootstrap) statistics are continuous transformations of $ (M_{n},U_{n})$ (of $(M_{n}^{\ast },U_{n}^{\ast })$), from Theorem (ref) and the CMT in CG20 we have that, for $\tau _{S}:=\sup_{u\in \lbrack 0,1]}|M(u)-uM(1)|$
similarly, for $\tau _{T}:=V(1)^{-1/2}\sup_{u\in \lbrack 0,1]}|M(u)-uM(1)|,$
Both conditional asymptotic distributions are continuous with probability 1. As discussed in A97, this holds using the results in L82 because the limiting random distribution corresponds (up to an almost surely strictly positive term) to the supremum of a conditionally Gaussian process with conditional covariance function which is nonsingular almost surely. Hence, by Corollary (ref) the bootstrap is valid conditionally on the volatility process $\sigma $.
Example 3 (cont'd). Finally, in the unit root example we have that, under the stated assumption and if the null hypothesis holds, with $\tau _{R}:=(\int_{0}^{1}M^{2}(u)\mathsf{d}u)^{-1}\int_{0}^{1}M(u) \mathsf{d}M(u)$,
Similarly, for the $t$ ratio test, with $\tau _{W}:=(\int_{0}^{1}M^{2}(u) \mathsf{d}u)^{-1/2}\int_{0}^{1}M(u)\mathsf{d}M(u)$,
As proved in Lemma A.1 in the online appendix, the limiting conditional cdfs have almost surely continuous sample paths, and hence by Corollary (ref), the bootstrap is valid conditionally on $\sigma $.$\hfill \square $
We now briefly discuss the behavior of the bootstrap tests under the alternative hypothesis when the stochastic volatility process induces randomness of the limiting distribution of the bootstrap statistic. As before, consider a left-sided test based on the statistic $\tau _{n}=\tau (M_{n},U_{n})$ and its bootstrap equivalent $\tau _{n}^{\ast }=\tau (M_{n}^{\ast },U_{n}^{\ast })$. Suppose that under the alternative the original statistic diverges, say to $-\infty $, while the bootstrap statistic satisfies
for some random element $\tau ^{\ast} $. Then, the following lemma holds.
Lemma (ref) shows that the fact that the limit distribution of the bootstrap statistic is random and depends on the volatility path does not affect the consistency of the bootstrap test. Essentially, weak convergence in distribution of $\tau _{n}^{\ast }$ given the data implies that the bootstrap statistic is $O_{p}^{\ast }(1) $, in probability. If the original statistic diverges to $-\infty $, it then holds that the bootstrap test rejects with probability converging to 1.\footnote{ Notice that a consistent right-sided test can be obtained by focusing on the bootstrap $p$-value $\tilde{p}_{n}^{\ast }:=1-p_{n}^{\ast }$.} We now apply this result to the three leading examples.
Example 1 (cont'd). Consider the location model example, where the econometrician is interested in testing the simple null hypothesis $\theta =\bar{\theta}$ when $\bar{\theta}>\theta _{0}$, $\theta _{0}$ being the true parameter value. A wild bootstrap with the null imposed generates bootstrap data as $\varepsilon _{t}^{\ast }:=\hat{\varepsilon} _{t}w_{t}^{\ast }$ with $w_{t}^{\ast }$ i.i.d.\ $N(0,1)$ and $\hat{ \varepsilon}_{t}:=y_{t}-\bar{\theta}=\varepsilon _{t}+\delta $, $\delta :=\theta _{0}-\bar{\theta}<0$. It follows that, conditionally on the data, $ M_{n}^{\ast }(\cdot ):=n^{-1/2}\sum_{t=1}^{\lfloor n\cdot \rfloor }\varepsilon _{t}^{\ast }$ is a zero-mean Gaussian process with independent increments and with conditional variance function $\hat{U}_{n}(\cdot )$, where
Hence, under Assumptions (ref) and (ref), we have, as $ n\rightarrow \infty $, that\footnote{ The proof reduces to the standard characterization of weak convergence to a Gaussian process by means of a Skorokhod representation.} $M_{n}^{\ast } \overset{w^{\ast }}{\rightarrow }_{w}\tilde{M}^{\ast }|\sigma $, where $ \tilde{M}^{\ast }(u):=\int_{0}^{u}\tilde{\sigma}(s)\mathsf{d}B_{z}^{\ast }(s) $, with $\tilde{\sigma}(s):=(\sigma (s)^{2}+\delta ^{2})^{1/2}$. This implies that $S_{n}^{\ast }\overset{w^{\ast }}{\rightarrow }_{w}\tilde{M} ^{\ast }(1)|\sigma $. As $S_{n}\rightarrow -\infty $ as $n\rightarrow \infty $, the conditions of Lemma (ref) are satisfied and $p_{n}^{\ast } \overset{p}{\rightarrow }0$. Consistency of the test based on $T_{n}$ follows by standard arguments as $T_{n}^{\ast }\overset{w^{\ast }}{ \rightarrow }_{p}N(0,1)$.
Example 2 (cont'd). For the bootstrap CUSUM statistics, consider the alternative $\theta _{t}=\theta _{1}+g(t/n)$, where $ g:[0,1]\rightarrow \mathbb{R}$ is an arbitrary function satisfying $ 0<\int_{0}^{1}g^{2}(u)\mathsf{d}u<\infty $, see PK92. The wild bootstrap partial sum process is $M_{n}^{\ast }(\cdot):=n^{-1/2}\sum_{t=1}^{\lfloor n\cdot\rfloor }\varepsilon _{t}^{\ast }$, a zero-mean Gaussian process with independent increments and with conditional variance function $\hat{U}_{n}(\cdot):=n^{-1}\sum _{t=1}^{\lfloor n\cdot\rfloor }\hat{\varepsilon}_{t}^{2}=U_{n}( \cdot)+G_{n}(\cdot)+O_{p}(n^{-1/2})$, with
in $\mathscr{D}$. This implies that
with $\tilde{\sigma}(s):=(\sigma (s)^{2}+(g(s)-\int_{0}^{1}g(r)\mathsf{d} r)^2)^{1/2}$ and $B_{z}^{\ast }$ a standard Brownian motion, stochastically independent of $\sigma $. The limiting distribution of the bootstrap statistic $CS_{n}^{\ast }$ is then given by
In order to analyze the $CT_{n}^{\ast }$ statistic, notice that its denominator satisfies, in probability,
which implies that
As both $S_{n}$ and $T_{n}$ diverge under the alternative considered, Lemma (ref) applies and for both tests $p_{n}^{\ast }\overset{p}{ \rightarrow }0$.
Example 3 (cont'd). Consider the unit root example with wild bootstrap shocks generated with the null hypothesis, i.e. $\varepsilon _{t}^{\ast }:=(\Delta y_{t})w_{t}^{\ast }$. The bootstrap $R_{n}^{\ast }$ statistic is given as in ((ref)) with $ M_{n}^{\ast }(u):=n^{-1/2}\sum_{t=1}^{\lfloor nu\rfloor }(\Delta y_{t})w_{t}^{\ast }$ and $U_{n}^{\ast }(u):=n^{-1/2}\sum_{t=1}^{\lfloor nu\rfloor }(\Delta y_{t})^{2}w_{t}^{\ast }{}^{2}$. Conditionally on the data, $M_{n}^{\ast }$ is a zero-mean Gaussian process with independent increments and conditional variance function $\hat{U}_{n}(\cdot):=n^{-1} \sum_{t=1}^{\lfloor n\cdot\rfloor }(\Delta y_{t})^{2}$. Under the alternative that $y_{t}=(1+\theta )y_{t-1}+\varepsilon _{t}$ with $\theta \in (-2,0)$, $\Delta y_{t}$ can be written as the linear process with exponentially decaying coefficients $\Delta y_{t}=\sum_{i=0}^{t-1}\psi _{i}\varepsilon _{t-i}$ with $\psi _{0}=0$ and $\psi _{i}=\theta (1+\theta )^{i-1}$, $i=1,2,\ldots $. Hence, by standard decompositions for squared stationary autoregressions it holds that (the proof is reported in the online appendix)
where the $o_{p}(1)$ term is uniform in $\cdot \in \lbrack 0,1]$, which implies that $\hat{U}_{n}\overset{w}{\rightarrow }\overline{\psi }^{2} V$. Hence, $M_{n}^{\ast }\overset{w^{\ast }}{\rightarrow }_{w} \overline{ \psi } M^{\ast }|\sigma $. Finally, using the fact that $U_{n}^{\ast }= \overline{\psi }^{2}\hat{U}_{n}+o_{p}^{\ast }(1)$, in probability, where $ \hat{U}_{n}\overset{w}{\rightarrow }V$, it holds that
Hence, under the alternative the bootstrap replicates the null distribution of the original statistic conditional on the volatility process and consistency of the bootstrap test follows from Lemma (ref). An identical result holds for the $t$-ratio test based on $W_{n}$.$\hfill \square $
In this section we analyze finite sample size and power properties of bootstrap tests under non-stationary stochastic volatility using Monte Carlo simulations. To study the behavior of the tests from Examples 1--3 under the null hypothesis, we report the Monte Carlo (empirical) cdfs of bootstrap $p$-values, both unconditionally over all Monte Carlo replications and conditionally on specific simulated volatility paths. Following CG20, we report the results in the form of fan charts of the conditional cdfs, displayed together with the unconditional cdf and the theoretical cdf of the $U(0,1)$ distribution. Similarly, we display conditional power curves of the tests under local alternatives in fan charts.
In all experiments, we draw observations $\{\varepsilon _{t}\}_{t=1}^{n}$ from the GARCH(1,1) process from Example V.2, with
corresponding to a limit process $\sigma ^{2}(u)$ with unit unconditional variance $\bar{\sigma}^{2}=1$, mean-reversion parameter $\kappa $, and volatility-of-volatility parameter $\sigma _{\eta }$. The conditional variance sequence is initialized at the unconditional variance, i.e., $ \sigma _{1}^{2}=1$. We set $\kappa =5$ and $\sigma _{\eta }=\sqrt{10}$, corresponding to a rather persistent volatility process with a reasonable amount of short-run variability of the volatility, which we know from earlier simulation studies to lead to substantial size distortions in tests based on standard (constant-volatility) asymptotic critical values. We expect similar results from stochastic volatility processes (Example V.1) with the same type of persistence and volatility-of-volatility properties. We report results for two sample sizes, $n\in \{100,500\}$. The standardized errors $ z_{t}$ are drawn from three different distributions, discussed below.
For each distribution and sample size, we first simulate $100$ different realizations of the volatility path $\{\sigma _{t}\}_{t=1}^{n}$. For each of these paths, we draw $50,000$ replications from the conditional distribution of $\{\varepsilon _{t}=\sigma _{t}z_{t}\}_{t=1}^{n}$ given $\{\sigma _{t}\}_{t=1}^{n}$. As discussed in Remark (ref), this is equivalent to drawing $v_{nt}=\mathrm{sgn}(z_{t})$ conditional on $\psi _{nt}=|z_{t}|$ for $t=1,\ldots ,n-1$, and drawing $z_{n}$ from its unconditional distribution (independent of $\{\sigma _{t}\}_{t=1}^{n}$). For each choice of the distribution of $\{z_{t}\}_{t=1}^{n}$, we can check the conditions of Assumption (ref) for conditional validity of the bootstrap.
The first data-generating process, labelled DGP 1, is defined by $z_{t}\sim N(0,1)$. In that case the conditional distribution of the signs $v_{nt}$ is discrete uniform over $\{-1,1\}$, independent of $|z_{t}|$. This in turn implies that the mda condition of Assumption (ref) is satisfied (as well as the independence, measurability and Lindeberg conditions), such that the bootstrap is conditionally valid.
In DGP 2, $z_{t}$ is drawn from the following mixed normal density
where $\phi (z;\mu ,\sigma )$ is the pdf of the $N(\mu ,\sigma ^{2})$ distribution, and where $\mu _{1}=-2a,\sigma _{1}=a,\mu _{2}=a,\sigma _{2}=a \sqrt{2}$, with $a=\sqrt{3/11}$. This distribution was constructed by M00 to be asymmetric but with skewness $0$ (and with mean zero and unit variance). Because $v_{nt}=\mathrm{sgn}(z_{t})$ in this case has a conditional distribution depending on $\psi _{nt}=|z_{t}|$, with $ P(v_{nt}=1|\psi _{nt})=f(\psi _{nt})/(f(\psi _{nt})+f(-\psi _{nt}))\neq \frac{1}{2}$, it follows that Assumption (ref) is violated, and conditional validity of the bootstrap is not guaranteed. On the other hand, the zero skewness implies that the limit result of Example V.2 still applies, with $B_{z}$ independent of $B_{\eta }$ and hence $\sigma $. As conjectured in Remark (ref), we may expect unconditional bootstrap validity in this case.
In DGP 3, $z_{t}$ is drawn from another version of ((ref)), but now with $\mu _{1}=-2b,\sigma _{1}=b\sqrt{2},\mu _{2}=b,\sigma _{2}=b$, with $b= \sqrt{3/10}$. This is a distribution with mean zero, unit variance and negative skewness, so that $B_{z}$ and $B_{\eta }$ in Example V.2 have a negative correlation, corresponding to long-run leverage effects. This implies that the wild bootstrap is invalid in this case, both conditionally and unconditionally.
Figures 1--3 display the results for the behavior of bootstrap $p$-values (based on $199$ bootstrap replications) under the null hypothesis, for the studentized tests based on $T_{n}$, $CT_{n}$ and $W_{n}$, respectively. Unreported results for the other three test statistics $S_{n}$, $CS_{n}$ and $R_{n}$ are very similar to the results for the corresponding studentized tests.
From Figure 1, we observe that when the standardized errors $z_{t}$ are standard normal (DGP 1, left panels), then the conditional distribution of the bootstrap $p$-values is very close to uniform, and appears to be independent of $\sigma $ for both sample sizes considered. Thus the theoretical conditional validity of the bootstrap in this case is clearly reflected in finite-sample behavior. When the distribution of $z_{t}$ is asymmetric with zero skewness (DGP 2, centre panels), then the bootstrap appears to be valid on average (indicated by the solid line almost coinciding with the $U(0,1)$ cdf, especially for $n=500$), but the conditional cdfs of bootstrap $p$-values do depend on the volatility path and deviate from the uniform cdf, illustrating the conjectured violation of conditional bootstrap validity. Finally, for DGP 3 (right panels, skewed $ z_{t}$), we observe more extreme dependence of bootstrap $p$-values on the volatility path. In this case the bootstrap does not appear to be valid on average either, as predicted by the dependence between $B_{z}$ and $\sigma $ implied by this DGP, which is not replicated by the wild bootstrap.
Figure 2 displays the results for the studentized CUSUM test based on $CT_n$ . For this test, the finite-sample size distortion (indicated by the difference between the solid and dashed line) is more pronounced than for the location test, in particular for the smaller sample size ($n=100$). Unreported additional simulations show that these size distortions are even stronger for the test based on $CS_n$. The results improve when the sample size increases, and it should be noted that the rejection frequencies at the $5\%$ significance level are still fairly close to $0.05$; the deviations are larger at the centre of the distribution. For this test, the dependence of the conditional cdf of $p$-values on $\sigma$ is much weaker than for the location test. For DGP 2, we do not observe any deviation of conditional cdfs from their average; in case of DGP 3, there is a clear violation of conditional bootstrap validity, but the deviations are less pronounced than for $T_n$.
The results for the unit root test based on $W_{n}$ are given in Figure 3. For DGP 1 and 2, the size distortions appear to be negligible for both sample sizes. Similarly to the CUSUM test, the dependence of bootstrap $p$ -values on the volatility path for DGP 1 and 2 appears to be very weak. On the other hand, for DGP 3 we find this dependence to be clearly present, illustrating again a violation of conditional bootstrap validity. On average, the bootstrap appears to be valid even for DGP 3, although theoretically we would not expect this to be the case because of the dependence between $B_{z}$ and $\sigma $. Unreported simulations have shown that in case of stronger leverage effects (i.e., a stronger correlation between $B_{z}$ and $B_{\eta }$), the unconditional cdf of bootstrap $p$ -values does differ from the $U(0,1)$ cdf, as predicted by the theoretical results.
Next, we investigate the local power of the bootstrap tests, again conditional on the same realizations of the volatility path as considered for the size of the tests. As in Remark (ref), for the location tests we evaluate the rejection frequency of the test for $ \mathsf{H}_{0}:\theta =0$ against local alternatives $\theta _{n}=-n^{-1/2}c$ , with $c\in \lbrack 0,8]$. For the CUSUM tests, the local alternative is a break in the mean of the series, at $t=n/2$, from $\theta _{n,t}=0$ to $ \theta _{n,t}=n^{-1/2}c$, with $c\in \lbrack 0,15]$. For the unit root tests, we consider local alternatives $\theta _{n}=-n^{-1}c$, with $c\in \lbrack 0,20]$. We provide results for the tests based on the studentized statistics $T_{n}$, $CT_{n}$ and $W_{n}$, and for the sample size $n=100$.
Figure 4 displays the rejection frequencies of the bootstrap tests, based on $10,000$ replications of the test for each volatility path, plotted against $ c$. We observe that the conditional rejection probabilities under the alternative hypothesis depend on both the non-centrality parameter $c$ and the volatility process $\sigma $, for each test and DGP 1--3. While the dependence on the volatility is as expected for DGP 2 and 3, we note that for DGP 1, where the rejections probabilities under the null hypothesis are conditionally independent of the volatility process, the power of the tests clearly depends on the volatility (as discussed in Remark (ref)).
To gain some insight in the sensitivity of the simulation results to the chosen parameter values, we have repeated the Monte Carlo experiments with $ \kappa =0$, implying an integrated GARCH process ($\alpha _{n}+\beta _{n}=1$ ) with infinite unconditional variance. As in the original simulations, the initial condition is $\sigma _{1}^{2}=1$, the volatility of volatility is $ \sigma _{\eta }=\sqrt{10}$, and the same three DGPs for $z_{t}$ have been used. The corresponding figures (available from the authors upon request) show qualitatively very similar results as those displayed in Figures 1--4. The increased persistence and variation in the realizations of $\{\sigma _{t}^{2}\}_{t=1}^{n}$ leads to a bigger bias in DGP 3 for the $W_{n}$ test statistic, and a bit more variation in the conditional distributions for DGP 3 for the $W_{n}$ and $CT_{n}$ test statistics. Furthermore, we find more variation in conditional power functions and a slightly higher average (unconditional) power.
In this paper we have analyzed the properties of wild bootstrap inference in time series models for the conditional mean under non-stationary stochastic volatility. In our setting, we do not make any specific assumption on the volatility process, rather than assuming that it admits a weak limit in $ \mathscr{D}{}[0,1]$. The additional advantage of this semi-parametric nature of our analysis is that we do not need any assumptions on the existence of higher moments, or on the initial condition of the conditional variance process. On the other hand, if the parametric form of the volatility process was known to the econometrician, then other model-based estimators and tests (such as S99, for the unit root testing problem under GARCH(1,1) errors) and related bootstrap methods could be implemented. These approaches, although particularly interesting and likely to deliver power improvements over our wild bootstrap methods, are beyond the scope of this paper.
A central ingredient in our analysis is that the bootstrap distribution (the distribution of a bootstrap statistic conditional on the data) has a random limit. In our case, this random limit can be characterized as the conditional distribution of the limiting statistic conditional on a continuous-time volatility process (which is itself the weak limit of the finite-sample volatility process). The random nature of the limit of bootstrap distributions is not new: a well-known example in the unit-root literature is B91, and SBW10 provide another example from non-parametric statistics involving cube-root asymptotics. CG20 analyze a number of other examples of this phenomenon. In the examples encountered so far, this random limit distribution can always be expressed as a conditional distribution, but its specific form is determined by a combination of model assumptions, statistical methods and bootstrap implementation details. Some of these combinations lead to bootstrap invalidity, which we may try to resolve by another bootstrap implementation. For example, the randomness of the bootstrap limit distribution in the unit-root testing problem, and the associated invalidity of the standard recursive-design bootstrap, is resolved by the use of restricted residuals (with the unit root imposed) in the bootstrap algorithm. The challenging part of our work is to show that the bootstrap distribution matches a particular conditional distribution of the original statistic, in our case conditional on the volatility process $\{\sigma _{t}\}_{t=1}^{n}$. If we were to change the bootstrap scheme (e.g., by using an i.i.d. or `$m$ out of $n$' bootstrap), then such a match would not occur, and the bootstrap would be invalid.
Our results can be generalized in several directions. First, our applications deal with univariate time series models and it is naturally of interest to apply our results to multivariate (time series)\ models, where volatilities and correlations are time-varying, stochastic and non-stationary. In particular, in BCRT16 the bootstrap was considered for multivariate cointegrated vector autoregressions in the presence of stationary volatility, in combination with possible deterministic changes in the volatility; we conjecture that our results obtained here also apply to the case of non-stationary multivariate stochastic volatility. Second, it would be important to understand how to bootstrap conditional mean time series models in the presence of leverage. Although, as we have shown, the wild bootstrap is not valid in this context, our theory may be useful for assessing validity of other bootstrap methods when the volatility displays leverage effects.
We thank the guest Co-Editor and two anonymous referees for important suggestions on an earlier draft of the paper. We also thank participants of the conference in honour of Pierre Perron (Boston University, 14--15 March 2019), of the 12th World Congress of the Econometric Society (Bocconi University, 17--21 August, 2020), and of workshops and seminars in Amsterdam, Canterbury, \'{I}lhavo, Nottingham, Oxford, Rotterdam and Vienna for useful feedbacks on earlier versions of the paper. This research was supported by the Danish Council for Independent Research (DSF Grant 015-00028B), by the University of Bologna (ALMA IDEA 2017 Grant), and by the Italian Ministry of University and Research (PRIN 2017 Grant 2017TA7TYC).