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Asymptotic bootstrap validity is usually understood as consistency of the distribution of a bootstrap statistic, conditional on the data, for the unconditional limit distribution of a statistic of interest. From this perspective, randomness of the limit bootstrap measure is regarded as a failure of the bootstrap. We show that such limiting randomness does not necessarily invalidate bootstrap inference if validity is understood as control over the frequency of correct inferences in large samples. We first establish sufficient conditions for asymptotic bootstrap validity in cases where the unconditional limit distribution of a statistic can be obtained by averaging a (random) limiting bootstrap distribution. Further, we provide results ensuring the asymptotic validity of the bootstrap as a tool for conditional inference, the leading case being that where a bootstrap distribution estimates consistently a conditional (and thus, random) limit distribution of a statistic. We apply our framework to several inference problems in econometrics, including linear models with possibly non-stationary regressors, functional CUSUM statistics, conditional Kolmogorov-Smirnov specification tests, the `parameter on the boundary' problem and tests for constancy of parameters in dynamic econometric models.
\noindentKeywords: Bootstrap, random measures, weak convergence in distribution, asymptotic inference.
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Consider a data sample $D_{n}$ of size $n$ and a statistic $\tau _{n}:=\tau_{n}\left( D_{n}\right) $, say a test statistic or a parameter estimator, possibly normalized. Interest is in a distributional approximation of $\tau_{n}$. Let a bootstrap procedure generate a bootstrap analogue $\tau_{n}^{\ast}$ of $\tau_{n}$; i.e., computed on a bootstrap sample. Assume that $\tau_{n}$ converges in distribution to a non-degenerate random variable [rv], say $\tau$. In classic bootstrap inference, asymptotic bootstrap validity is usually understood and established as convergence in probability (or almost surely) of the cumulative distribution function {[}cdf{]} of the bootstrap statistic $\tau_{n}^{\ast}$ conditional on the data $D_{n}$, say $F_{n}^{\ast}$, to the unconditional cdf of $\tau$, say $F$. This convergence, along with continuity of $F$, implies by Polya's theorem that $\sup _{x\in\mathbb{R}}|F_{n}^{\ast}\left( x\right) -F\left( x\right) |\rightarrow0$, in probability (or almost surely).
In many applications, however, the bootstrap statistic $\tau_{n}^{\ast}$ may possess, conditionally on the data, a random limit distribution. Cases of random bootstrap limit distributions appear in various areas of econometrics and statistics; for instance, they are documented for infinite-variance processes (Athreya, 1987; Knight, 1989; Aue, Berkes and Horv\'{a}th, 2008; Cavaliere, Georgiev and Taylor, 2016), time series with unit roots (Basawa, Mallik, McCormick, Reeves and Taylor, 1991; Cavaliere, Nielsen and Rahbek, 2015), parameters on the boundary of the parameter space (Andrews, 2000), subsample inference based on fixed-b asymptotics (Shao and Politis, 2013), cube-root consistent estimators (Sen, Banerjee and Woodroofe, 2010; Cattaneo, Jansson and Nagasawa, 2017), Hodges-LeCam superefficient estimators (Beran, 1997). In most of these cases, the occurrence of a random limit distribution for the bootstrap statistic $\tau_{n}^{\ast}$ given the data -- in contrast to a non-random limit of the unconditional distribution of the original statistic $\tau_{n}$ -- is taken as evidence of failure of the bootstrap.
In this paper we show that randomness in the limiting distribution of a bootstrap statistic need not invalidate bootstrap inference. On the contrary, although the bootstrap no longer estimates the limiting unconditional distribution of the statistic of interest, it may\ still deliver hypothesis tests (or confidence intervals) with the desired null rejection probability (or coverage probability) when the sample size diverges. This happens because asymptotic control over the frequency of wrong inferences can be guaranteed by the asymptotic distributional uniformity of the bootstrap p-values, which in its turn can occur without the convergence in probability (or almost surely) of the bootstrap cdf $F_{n}^{\ast}$ of $\tau_{n}^{\ast}$ to the asymptotic cdf $F$ of $\tau$.
Therefore, instead of assessing bootstrap validity in terms of the convergence of $F_{n}^{\ast}$ to $F$, in cases where the limit of the bootstrap distribution is random we study bootstrap validity in terms of the property of asymptotic distributional uniformity of bootstrap p-values. Specifically, let $p_{n}^{\ast}$ denote the bootstrap p-value, usually defined as $p_{n}^{\ast}:=F_{n}^{\ast}\left( \tau_{n}\right) $. We define `bootstrap validity' or `unconditional bootstrap validity' the fact that
for $q\in\left( 0,1\right) $. The focus on this property is not new in the literature on bootstrap and simulation-based inference (see, e.g., Hansen, 1996, and Lockhart, 2012, among others).
Our first set of results provides sufficient conditions for bootstrap validity in the sense of ((ref)) in situations where the bootstrap distribution is random in the limit. Classic results for bootstrap validity when the limit bootstrap measure is not random can be obtained as special cases. The main requirement in our results is that the unconditional limit distribution of $\tau_{n}$ should be an average of the random limit distribution of $\tau_{n}^{\ast}$ given the data.
It is often the case that bootstrap\ validity can be addressed through the lens of a conditioning argument. In this regard, our second set of results concerns the possibility that, for a sequence of random elements $X_{n}$, it holds that the bootstrap p-value is uniformly distributed in large sample conditionally on $X_{n}$:
for $q\in\left( 0,1\right) $. This property, that we define `bootstrap validity conditional on $X_{n}$', implies unconditional validity in the sense of ((ref)). Moreover, conditional bootstrap validity given $X_{n}$ implies that the bootstrap replicates asymptotically the property of conditional tests and confidence intervals to have, conditionally on $X_{n}$, constant null rejection probability and coverage probability, respectively (for further roles of conditioning in inference, like the relevance of the drawn inferences and information recovery, see Reid, 1995, and the references therein). The leading case where we show ((ref)) to hold --\thinspace under regularity conditions that will be discussed in the paper --\thinspace is that where the (random) limit of the conditional distribution of $\tau_{n}$ given $X_{n}$ matches the (random) limit distribution of the bootstrap statistic. A property like ((ref)) was initially established by LePage and Podgorski (1996) for permutation tests in location models. Their approach has not been developed further in the bootstrap literature, in particular because it requires probabilistic tools that are not widely popular in this field.
When dealing with random limit distributions, the usual convergence concept employed to establish bootstrap validity, i.e. weak convergence in probability, can only be employed in some very special cases. Instead, our formal discussion makes extensive use of the probabilistic concept of weak convergence of random measures; see e.g. Kallenberg (2017, Ch.4). To our knowledge, in the bootstrap context this concept has so far been mostly used to obtain negative results of lack of validity for specific bootstrap procedures (as e.g. in Knight, 1989, and Basawa et al., 1991), rather than positive validity results, as we do here. As an ingredient of our analysis, we also present some novel results on the weak convergence of conditional expectations.
To illustrate the practical relevance of our results, we initially present them by using a simple linear model with either stationary or non-stationary regressors, and later we analyze four well-known cases in the econometric literature where the bootstrap features a random limit distribution. The first is a standard CUSUM-type test of the i.i.d. property for a random sequence with infinite variance. This is a case where the limit distribution of the CUSUM statistic depends on unknown nuisance parameters (e.g., the tail index) and bootstrap or permutation tests fail to estimate this distribution consistently. We argue that a simple bootstrap based on permutations, albeit having a random limit distribution and hence being invalid in the usual sense, provides exact conditional inference and hence is also unconditionally valid in the sense of ((ref)). The second application considers a Kolmogorov-Smirnov-type test for correct specification of the conditional distribution of a response variable given a vector of covariates. Andrews (1997) considers a parametric bootstrap implementation where the covariates are kept fixed across bootstrap samples. While in the independent case the limit of the bootstrap distribution is non-random, this is not the case in general. Using our theory we discuss conditions for validity of the bootstrap within this framework. The third application is the implementation of the bootstrap in `parameter on the boundary' problems (Andrews, 1999,2000). Taking hypothesis testing in a predictive regression framework as an illustration, we show that the kind of randomness in the limit distribution of the bootstrap statistics of interest depends, when the true parameter lies on the boundary of the parameter space, on how well the mutual position of the boundary, the set identified by the null hypothesis and the true parameter value, is approximated in the bootstrap world. Although the standard bootstrap may fail to be valid in the sense of ((ref)), we provide conditions for the validity of alternative bootstrap schemes.\ The fourth application includes an analysis of the much applied bootstrap `$\sup F$' tests of parameter constancy in regression models where the design matrix could be random but be conditioned upon; see Hall (1992, p.170). In the resampling process forming the bootstrap sample, it appears natural to take the design matrix as fixed across the bootstrap repetitions. Under a set of assumptions proposed by Hansen (2000), we argue that the fixed-regressor bootstrap `$\sup F$' statistic has a random limit distribution, thus invalidating previous claims in the literature that the bootstrap is consistent for the unconditional limit distribution of the original `$\sup F$' test statistic. We then provide conditions under which the fixed-regressor bootstrap is unconditionally valid and, additionally, valid conditionally on the chosen set of regressors.
The paper is organized as follows. In Section (ref) we outline the main concepts and ideas using a simple linear regression model. Our main theoretical results are presented in Section (ref). Section (ref) contains the four applications of the theory, whereas Section (ref) concludes. The paper has two Appendices. In Appendix (ref) we collect some results on weak convergence in distribution which are useful to prove our main theorems and develop the applications. Appendix (ref) contains the proofs of all theory results given in the paper. The proofs of the results from Appendix (ref) and some additional material are collected in the accompanying supplement, Cavaliere and Georgiev (2019).
We use the following notation throughout. The spaces of c\`{a}dl\`{a}g functions $[0,1]\rightarrow\mathbb{R}^{n}$, $[0,1]\rightarrow\mathbb{R} ^{m\times n}$ and $\mathbb{R}\rightarrow\mathbb{R}$ (all equipped with the respective Skorokhod $J_{1}$-topologies; see Kallenberg, 1997, Appendix A2),\ are denoted by $\mathscr{D}{}_{n}$, $\mathscr{D}{}_{m\times n}$ and $\mathscr{D}({\mathbb{R}})$, respectively; for the first one, when $n=1$ the subscript is suppressed. Integrals are over $[0,1]$ unless otherwise stated, $\Phi$ is the standard Gaussian cdf, $U(0,1)$ is the uniform distribution on $[0,1]$ and $\mathbb{I}_{\{\cdot\}}$ is the indicator function. If $F$ is a (random) cdf, $F^{-1}$ stands for the right-continuous generalized inverse, i.e., $F^{-1}(u):=\sup\{v\in\mathbb{R}:F\left( v\right) \leq u\}$, $u\in\mathbb{R}$. Unless differently specified, limits are for $n\rightarrow\infty$.
Polish (i.e., complete and separable metric) spaces are always equipped with their Borel $\sigma$-algebras. Throughout, we assume that all random elements are Polish-space valued and that well-defined conditional distributions exist. For random elements of a Polish space, the existence of regular conditional distributions is guaranteed and we assume without loss of generality that conditional probabilities are regular (Kallenberg, 1997, Theorem 5.3). Equality of conditional distributions is understood in the almost sure [a.s.] sense and, for random cdf's as random elements of $\mathscr{D}({\mathbb{R}})$, equalities are up to indistinguishability.
Let $\mathcal{C}_{b}(\mathcal{S})$ be the set of all continuous and bounded real-valued functions on a metric space $\mathcal{S}$. For random elements $Z,Z_{n}$ ($n\in\mathbb{N}$) of a metric space $\mathcal{S}_{Z}$, we employ the usual notation $Z_{n}\overset{w}{\rightarrow}Z$ for the property that the distribution of $Z_{n}$ weakly converges to the distribution of $Z$, defined by the convergence $E\left\{ g\left( Z_{n}\right) \right\} {\rightarrow }E\left\{ g\left( Z\right) \right\} $ for all $g\in\mathcal{C} _{b}(\mathcal{S}_{Z})$. For random elements $(Z,X)$, $(Z_{n},X_{n})$ of the metric spaces $\mathcal{S}_{Z}\times\mathcal{S}$ and $\mathcal{S}_{Z} \times\mathcal{S}_{n} $ ($n\in\mathbb{N}$), and defined on a common probability space, we denote by $Z_{n}|X_{n} \overset{w}{\rightarrow}_{p}Z|X$ (resp. $Z_{n}|X_{n}\overset{w}{\rightarrow}_{a.s.}Z|X$) the fact that $E\left\{ g\left( Z_{n}\right) |X_{n}\right\} {\rightarrow}E\left\{ g\left( Z\right) |X\right\} $ in probability (resp. a.s.) for all $g\in\mathcal{C}_{b}(\mathcal{S}_{Z})$. In the special case where $E\left\{ g\left( Z_{n}\right) |X_{n}\right\} \overset{w}{\rightarrow}E\left\{ g\left( Z\right) \right\} $ in probability (resp. a.s.) for all $g\in\mathcal{C}_{b}(\mathcal{S}_{Z})$, we write $Z_{n}|X_{n} \overset {w}{\rightarrow}_{p}Z$ (resp. $Z_{n}|X_{n}\overset{w}{\rightarrow}_{a.s.}Z$). In such a case the weak limit (in probability or a.s.) of the random conditional distribution $Z_{n}|X_{n}$ is the non-random distribution of $Z$, thus reducing our definition to the one of weak convergence in probability (resp. a.s.) usually employed in the bootstrap literature.
In order to deal with random limit measures, we need a further convergence concept. For $(Z,X)$,$\,(Z_{n},X_{n})$ ($n\in\mathbb{N}$) defined on possibly different probability spaces, we denote by $Z_{n}|X_{n}\overset{w} {\rightarrow}_{w}Z|X$ the fact that $E\{g(Z_{n})|X_{n}\}\overset {w}{\rightarrow}E\{g\left( Z\right) |X\}$ for all $g\in\mathcal{C} _{b}(\mathcal{S}_{Z})$ and label it `weak convergence in distribution'. It coincides with the probabilistic concept of weak convergence of random measures (here, of the random conditional distributions $Z_{n}|X_{n}$; see Kallenberg, 2017, Ch.4). Whenever $Z_{n}$ and $Z$ are rv's and the conditional distribution of $Z$ given $X$ is diffuse (non-atomic), this is equivalent to the weak convergence $P\left( Z_{n}\leq\cdot|X_{n}\right) \overset{w}{\rightarrow}P\left( Z\leq\cdot|X\right) $ of the random cdf's as random elements of $\mathscr{D}(\mathbb{R})$ (see Kallenberg, 2017, Theorem 4.20). Finally, on probability spaces where both the data $D_{n}$ and the auxiliary variates used in the construction of the bootstrap data are defined, we use $Z_{n}\overset{w^{\ast}}{\rightarrow}_{p}Z|X$ (resp. $\overset{w^{\ast }}{\rightarrow}_{a.s}$, $\overset{w^{\ast}}{\rightarrow}_{w}$) interchangeably with $Z_{n}|D_{n}\overset{w}{\rightarrow}_{p}Z|X$ (resp. $\overset {w}{\rightarrow}_{a.s}$, $\overset{w}{\rightarrow}_{w}$), and write $P^{\ast }(\cdot)$ for $P(\cdot|D_{n})$.
In this section we provide an overview of the main results established in the sections below, and the concepts employed, by using a simple linear regression model. Further applications will be given in Section (ref). We observe that even for this basic model bootstrap statistics may have a random limit distribution. Then, we show that convergence of the bootstrap statistic to a random limit may imply (asymptotic) bootstrap validity in the unconditional sense of eq. ((ref)). Finally, we illustrate the possibility that bootstrap inference may have a conditional interpretation.
Assume that the data are given by $D_{n}:=\{y_{t},x_{t}\}_{t=1}^{n}$ and consider the linear model
where $x_{t},y_{t}$ are scalar rv's and $\varepsilon_{t}$ are unobservable zero-mean errors with $\omega_{\varepsilon}:=\operatorname*{Var} (\varepsilon_{t})\in(0,\infty)$, $t=1,...,n$. Assume that $M_{n} :=\sum\nolimits_{t=1}^{n}x_{t}^{2}>0$ a.s. for all $n$; further assumptions will be introduced gradually. Interest is in inference on $\beta$ based on $T_{n}:=\hat{\beta}-\beta$, with $\hat{\beta}$ the OLS estimator of $\beta$; for instance, a confidence interval or a test of a null hypothesis of the form $\mathsf{H}_{0}:\beta=0$.
The classic (parametric) fixed-design bootstrap, see e.g. Hall (1992), entails generating a bootstrap sample $\{y_{t}^{\ast},x_{t}\}_{t=1}^{n}$ as
where $\{\varepsilon_{t}^{\ast}\}_{t=1}^{n}$ are i.i.d. $N\left( 0,1\right) $, independent of the original data, and $\hat{\omega}_{\varepsilon}$ is an estimator of $\omega_{\varepsilon}$, e.g., the residual variance $n^{-1} \sum_{t=1}^{n}(y_{t}-\hat{\beta}x_{t})^{2}$. The OLS estimator of $\beta$ from the bootstrap sample is denoted by $\hat{\beta}^{\ast}$ and, conditionally on the original data, $T_{n}^{\ast}:=\hat{\beta}^{\ast}-\hat{\beta}\sim N\left( 0,\hat{\omega}_{\varepsilon}M_{n}^{-1}\right) $. As is standard, the distribution of $T_{n}$ is approximated by the distribution of $T_{n}^{\ast}$ conditional on the data. With $F_{n}^{\ast}$ denoting the cdf of $T_{n}^{\ast }$ under $P^{\ast}$ (the probability measure induced by the bootstrap; i.e., conditional on the original data), the bootstrap p-value is given by $p_{n}^{\ast}:=F_{n}^{\ast}\left( T_{n}\right) $.
Although bootstrap inference is not exact in general, it may be still be asymptotically valid. To show this, we distinguish between the cases of a stationary and a non-stationary regressor $x_{t}$. It is the second case that anticipates the main results of the paper. We assume $\hat{\omega }_{\varepsilon}\overset{p}{\rightarrow}\omega_{\varepsilon}$ throughout.
Suppose initially that $\{x_{t}\}_{t\in\mathbb{N}}$ is weakly stationary and $n^{-1}M_{n}\overset{p}{\rightarrow}M:=Ex_{1}^{2}>0$. Define $\tau _{n}:=n^{1/2}(\hat{\beta}-\beta)$ and $\tau_{n}^{\ast}:=n^{1/2}(\hat {\beta^{\ast}}-\hat{\beta})$; the bootstrap p-values based on $(\tau_{n},\tau_{n}^{\ast})$ and $(T_{n},T_{n}^{\ast})$ are identical. The distribution of the bootstrap statistic $\tau_{n}^{\ast}$ conditional on the original data $D_{n}$ satisfies
Hence, $\tau_{n}^{\ast}\overset{w^{\ast}}{\rightarrow}_{p}\tau\sim N(0,\omega_{\varepsilon}M^{-1})$ and the limit distribution is non-random.
If the initial assumptions are strengthened such that a central limit theorem [CLT]\ holds for $\{x_{t}\varepsilon_{t}\}_{t\in\mathbb{N}}$; that is, $n^{-1/2}\sum_{t=1}^{n}x_{t}\varepsilon_{t}\overset{w}{\rightarrow}N\left( 0,\omega_{\varepsilon}M\right) $, then it also holds that $\tau_{n} \overset{w}{\rightarrow}\tau\sim N(0,\omega_{\varepsilon}M^{-1})$. Hence, the bootstrap distribution of $\tau_{n}^{\ast}$ consistently estimates the unconditional limit distribution of $\tau_{n}$ in the usual\ sense that $\sup_{u\in\mathbb{R}}|P^{\ast}\left( \tau_{n}^{\ast}\leq u\right) -P\left( \tau\leq u\right) |$$\overset{p}{\rightarrow}0$, by Polya's theorem. As the limit cdf is continuous, the p-value $p_{n}^{\ast}$ associated with $(\tau_{n},\tau_{n}^{\ast}) $ is asymptotically uniformly distributed; i.e., ((ref)) holds.
Suppose now that $\{x_{t}\}_{t\in\mathbb{N}}$ is such that, for some constant $\alpha$, $n^{-\alpha}M_{n}\overset{w}{\rightarrow}M$, with $M>0$ a.s. having a non-degenerate distribution. A well-known special case is that where $x_{t}$ is a finite-variance random walk and $\alpha=2$. Redefine $\tau_{n} :=n^{\alpha/2}(\hat{\beta}-\beta)$ and $\tau_{n}^{\ast}:=n^{\alpha/2} (\hat{\beta^{\ast}}-\hat{\beta})$; bootstrap p-values remain unchanged. Now the bootstrap distribution of $\tau_{n}^{\ast}$, conditional on the data, remains random in the limit. Specifically, by the continuous mapping theorem {[}CMT{{]},}
which is a random cdf. In terms of weak convergence in distribution, this amounts to
As a result, with $\tau_{n}^{\ast}$ and $M$ generally defined on different probability spaces, weak convergence in probability of $\tau_{n}^{\ast}$ does not occur. Moreover, whatever the (unconditional) limit distribution of $\tau_{n}$ is, provided that it exists, $P\left( \tau_{n}\leq u\right) $, $u\in\mathbb{R}$, will tend to a deterministic cdf. Therefore, the bootstrap cannot estimate consistently the limit distribution of $\tau_{n}$ and it cannot hold that $\sup_{u\in\mathbb{R}}|P^{\ast}\left( \tau_{n}^{\ast}\leq u\right) -P\left( \tau_{{}}\leq u\right) |$$\overset{p}{\rightarrow}0 $. Nevertheless, bootstrap inference need not become meaningless, as it may even be exact (see Remark (ref)). We proceed, therefore, to identify in what sense bootstrap inference could remain meaningful.
Within the framework of the linear regression model, we discuss two concepts of bootstrap validity in the case of a random limit bootstrap measure. These are employed to interpret the bootstrap as a tool for unconditional or conditional inference.
Under the assumption in Section (ref) , consider the random-walk special case, where $x_{t}:=\sum_{s=1}^{t}\eta_{s}$ with $e_{t}:=(\varepsilon_{t},\eta_{t})^{\prime}$ forming a stationary, ergodic and conditionally homoskedastic martingale difference sequence {[}mds{]} with p.d.\thinspace variance matrix $\Omega:=\operatorname*{diag} \{\omega_{\varepsilon},\omega_{\eta}\}$.\footnote{Non-diagonal $\Omega$ could be handled by augmenting the estimated regression with $\Delta x_{t}$ (as we do in section (ref)), leading to no qualitative differences from the case of diagonal $\Omega$.} Then, for $\beta\neq0$ eq. ((ref)) is an instance of a cointegration regression. It holds that $(n^{-1/2}\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor }e_{t}^{\prime} ,n^{-1}\sum_{t=1}^{n}x_{t-1}\varepsilon_{t})\overset{w}{\rightarrow }(B_{\varepsilon},B_{\eta},\int B_{\eta}dB_{\varepsilon})$ in $\mathscr{D}{}_{2}\times\mathbb{R}$, where $(B_{\varepsilon},B_{\eta} )^{\prime}$ is a bivariate Brownian motion with covariance matrix $\Omega$; see Theorem 2.4 of Chan and Wei (1988). Moreover, $n^{-2}M_{n}\overset {w}{\rightarrow}M:=\int B_{\eta}^{2}$ by the CMT, jointly with the convergence to a stochastic integral above, so that the assumption in Section (ref) holds with $\alpha=2$ and
the limit being (by independence of $B_{\eta}$ and $B_{\varepsilon}$) a variance mixture of normals, with mixing variable $M^{-1}$ and cdf $\int_{\mathbb{R}}\Phi(\omega_{\varepsilon}^{-1/2}M^{1/2}u)dP\left( M\right) $.
A comparison between the limit distributions of $\tau_{n}^{\ast}$ and $\tau_{n}$, resp. in ((ref)) and ((ref)), shows that the bootstrap mimics a component of the mixture limit distribution of $\tau_{n}$, since the limit distribution of $\tau_{n}$ can be recovered by integrating over $M$ the conditional limit distribution of $\tau_{n}^{\ast} $ given the data. This turns out to be sufficient for bootstrap unconditional validity in the sense of eq. ((ref)). A direct argument is as follows: the bootstrap p-value $p_{n}^{\ast}:=P^{\ast}(\tau_{n}^{\ast }\leq\tau_{n})$ satisfies, by the CMT,
Thus, when inference on $\beta$ is based on the distribution of $\tau _{n}^{\ast}$ conditional on the data, the large-sample frequency of wrong inferences can be controlled.
In the case of unconditional bootstrap validity, it may be possible to find an interpretation of bootstrap inference as also\ valid in the sense of ((ref)), i.e. conditionally on some $X_{n}$ defined on the probability space of the original data\ $D_{n}$ (for instance, but not necessarily, the regressor $X_{n}:=\{x_{t}\}_{t=1}^{n}$).
In the linear regression case considered here, conditional bootstrap validity with respect to the regressor $X_{n}$ can be obtained under a tightening of our previous assumptions such that the invariance principle $n^{-1/2} \sum_{t=1}^{\left\lfloor n\cdot\right\rfloor }e_{t}\overset{w}{\rightarrow }(B_{\varepsilon},B_{\eta})^{\prime}$ holds conditionally (on $X_{n}$ for finite $n$ and on $B_{\eta}$ in the limit, in the sense of weak convergence in distribution). A sufficient condition for the conditional invariance principle is that, additionally to the assumptions on $e_{t}$ in Section (ref), $\varepsilon_{t}$ is an mds with respect to $\mathcal{G}_{t}=\sigma(\{\varepsilon\}_{s=-\infty}^{t} \cup\{\eta_{s}\}_{s\in\mathbb{Z}})$, and that $n^{-1}\sum_{t=1}^{n} E(\varepsilon_{t}^{2}|\{\eta_{s}\}_{s\in\mathbb{Z}})\rightarrow\omega _{\varepsilon}$ a.s. (see the proof of Theorem 2 in Rubshtein, 1996). Then, by using Theorem 3 of Georgiev, Harvey, Leybourne and Taylor (2018), it follows that \[ \tau_{n}|X_{n}\overset{w}{\rightarrow}_{w}\left. N(0,\omega_{\varepsilon }M^{-1})\right\vert M\text{,} \] which compared to ((ref)) shows that the distribution of $\tau _{n}^{\ast}$ conditional on the data estimates consistently the random limit distribution of $\tau_{n}$ conditional on the regressor $X_{n}$. This fact is stated more precisely in Remark (ref) where it is concluded that $p_{n}^{\ast}|X_{n}\overset{w}{\rightarrow}_{p}U(0,1)$, i.e., the bootstrap is valid conditionally on the regressor.
The result in Section (ref) implies that unconditional bootstrap validity can sometimes be established by means of a conditioning argument; for example, by showing validity conditional on the regressor $X_{n}$. To illustrate, in Figure 1, panels (a) and (b), we summarize for two different data generating processes [DGPs] the cdf's of $p_{n}^{\ast}|X_{n}$ across $M=1,000$ independent realizations of $X_{n}$ for samples of size $n=10$ (upper panels) and $n=1,000$ (lower panels). Specifically, the DGP used for panel (i) is based on i.i.d. shocks, while the one for (ii) features ARCH-type shocks (details are reported in the accompanying Supplement, Section (ref)). In both cases, the conditions of Section (ref) are satisfied. For both DGPs, the conditional distributions of $p_{n}^{\ast}$ given $X_{n}$ are, as expected, close to the $45 {{}^\circ} $ line, which corresponds to the implied asymptotic $U\left( 0,1\right) $ distribution. Unconditional validity follows accordingly.
Nevertheless, unconditional validity may also hold without validity conditional on an apparently `natural' conditioning variable $X_{n}$, like the regressor in a fixed-regressor bootstrap design. For instance, suppose that for the DGP\ in Sections (ref) and (ref) it holds that $\eta_{t}=\xi _{t}(1+\mathbb{I}_{\{\varepsilon_{t}<0\}})$, with $\{\varepsilon_{t}\}$ and $\{\xi_{t}\}$ two independent i.i.d. sequences of zero-mean, unit-variance rv's. Since $\eta_{t}$ is informative about the sign of $\varepsilon_{t}$, the $\varepsilon_{t}$'s conditionally on their own past and the regressor $X_{n}$ do not form an mds. It is shown in Appendix (ref), eq. ((ref)), that this endogeneity fact, not replicated in the bootstrap world, induces the original statistic $\tau_{n}$ to satisfy
where $\omega_{\varepsilon|\eta}:=E\{\operatorname*{Var}(\varepsilon_{s} |\eta_{s})\}\in(0,1)$, and $M$, $\xi_{1}$, $\xi_{2}$ are jointly independent with $\xi_{i}\sim N(0,1)$,$\,i=1,2$. The limit in ((ref)) contains more randomness (through $\xi_{2}$) than the bootstrap limit in eq. ((ref) ), thus resulting in a random limit for the distribution of the bootstrap p-value $p_{n}^{\ast}$ conditional on the regressor $X_{n}$; see panel (iii) of Figure 1, where for this DGP\ the cdf's of $p_{n}^{\ast}|X_{n}$ are reported for 1,000 realizations of $X_{n}$. These cdf's display substantial dispersion around the $45^{ {{}^\circ} }$ line, and this feature does not vanish as $n$\ increases. However, and in agreement with the earlier discussion, their unconditional average (plotted in black) is very close to the $45 {{}^\circ} $ line, showing indeed unconditional validity of the bootstrap. This follows because $e_{t}:=(\varepsilon_{t},\eta_{t})^{\prime}$ is a zero-mean i.i.d. sequence with a diagonal covariance matrix and $p_{n}^{\ast}\overset {w}{\rightarrow}U(0,1)$ as derived in Section (ref).
We provide general conditions for bootstrap validity in cases where a bootstrap statistic conditionally on the data possesses a random limit distribution. Before all else, we formally distinguish between two concepts of bootstrap validity.
The following definition employs the bootstrap p-value as a summary indicator of the accuracy of bootstrap inferences (see also Remark (ref) below). The original and the bootstrap statistic are denoted by $\tau_{n}$ and $\tau_{n}^{\ast}$, respectively.
Our main results make extensive use of joint weak convergence in distribution. Should the related notation not be self-explanatory, we refer the reader to Appendix A for the formal definitions.
The unconditional validity results in this section have in common the requirement, explicit or implicit, that the unconditional limit distribution of $\tau_{n}$ should be an average of the random limit distribution of $\tau_{n}^{\ast}$ given the data. Applications of Theorem (ref) do not require a conditional analysis of $\tau_{n}$, in contrast to applications of Theorem (ref).
Some remarks are in order.
Unconditional bootstrap validity could also be established by means of an auxiliary conditional analysis of the original statistic $\tau_{n}$. In the next theorem the conditioning sequence $X_{n}$ is chosen such that the bootstrap statistic $\tau_{n}^{\ast}$ depends on the data $D_{n}$ approximately through $X_{n}$ (condition (\dag)). Then, the main requirement for bootstrap validity is that the limit bootstrap distribution should be a conditional average of the limit distribution of $\tau_{n}$ given $X_{n}$.
Convergence ((ref)) in Theorem (ref)\ could be deduced from the weak convergence of the conditional distributions of $\tau_{n}$ and $\tau_{n}^{\ast}$, as in the next corollary.
Theorem (ref) below states the asymptotic behavior of the bootstrap p-value conditional on an $X_{n}$ chosen to satisfy condition ($\dagger$) of Theorem (ref). It also characterizes the cases where the bootstrap is valid conditionally on such an $X_{n}$. Should validity conditional on such an $X_{n}$ fail, in Corollary (ref)(b) we provide a result for validity conditional on a transformation of it.
A corollary in the terms of weak convergence in distribution is given next.
Corollary (ref) requires checking the joint convergence in ((ref)); see Remark (ref). We provide further strategies to establish this convergence, useful if interest is in conditional bootstrap validity, in Remarks (ref)--(ref) below.
When the limit bootstrap measure is random, the power function of the bootstrap test, conditionally on the data, is also random, even asymptotically. Its unconditional power function can be investigated using the following generalization of Theorem (ref).
To illustrate, with $y_{t}$, $x_{t}$ and $\varepsilon_{t}$ as in Section (ref), let interest be in the large-sample behavior of the bootstrap test for the hypothesis $\mathsf{H} _{0}:\beta=0$ against $\mathsf{H}_{1}:\beta<0$, under the local alternative $\beta=\beta_{n}:=b/n$ in ((ref)). The original test statistic is $\tau_{n}=n\hat{\beta}$.
Without recourse to the explicit expression in ((ref))\ for the bootstrap p-value $p_{n}^{\ast}$ in terms of the Gaussian cdf, which in many applications may have no analogue, we can instead\ use, for $\tau_{n}=b+n(\hat{\beta }-\beta)$ and $\tau_{n}^{\ast}$, the joint convergence\footnote{The conditional analysis of $\tau_{n}^{\ast}$, needed to show that under local alternatives it behaves asymptotically as under $\mathsf{H}_{0}$, is straightforward and is omitted.} \[ (\tau_{n},(\tau_{n}^{\ast}|D_{n}))\overset{w}{\rightarrow}_{w}(\tau ,(\tau^{\ast}|M)), \] with $\tau:=b+\tau^{\ast}$ and $\tau^{\ast}:=\omega_{\varepsilon} ^{1/2}M^{-1/2}\xi$ for $\xi\sim N\left( 0,1\right) $ independent of $M$. By using Lemma (ref) in Appendix (ref), we can conclude that the conditions of Theorem (ref) hold with $X=M$, $F\left( u\right) =P\left( \tau\leq u|M\right) =\Phi(\omega_{\varepsilon}^{-1/2}M^{1/2}(u-b))$ and $F^{\ast}(u)=\Phi(\omega_{\varepsilon}^{-1/2}M^{1/2}u)$. The unconditional asymptotic local power function of the one-sided, $q$-level, bootstrap test then follows as
Notice that this power function is distinct from the asymptotic local power of the unconditional test based on critical values from the null asymptotic (unconditional) distribution of $\tau_{n}$, which is that of $\omega _{\varepsilon}^{1/2}M^{-1/2}\xi$. Hence, when the limit bootstrap measure is random, the bootstrap test in general does not replicate, in terms of (unconditional) power, the standard asymptotic test (in this specific case, numerical evidence shows that for small $b$, where both local powers are relatively low, the bootstrap test is more powerful, whereas for large negative $b$, where the local power of both tests is high, the asymptotic test\ is preferable).
The unconditional power function in ((ref)) can also be derived through a conditioning argument. This can be done using the results in Section (ref) by considering the joint convergence \[ (\tau_{n}|X_{n},\tau_{n}^{\ast}|D_{n})\overset{w}{\rightarrow}_{w} (b,0)+\left. (1,1)N(0,\omega_{\varepsilon}M^{-1})\right\vert M\text{,} \] see ((ref)), which implies ((ref)) with $F$ and $F^{\ast}$ as defined above. Hence, by Theorem (ref), ((ref)) holds and
for $q\in(0,1)$. The latter expression is the (random) asymptotic local power, conditional on\ $X_{n}$, of the one-sided, $q$-level test, bootstrap test. By averaging the rhs of ((ref)) over $M$, the unconditional power function in ((ref)) follows.
Consider a standard CUSUM\ test for the null hypothesis (say, $\mathsf{H}_{0}$) that $\{\varepsilon_{t}\}_{t=1}^{n}$ is a sequence of i.i.d. random variables. The test statistic is of the form \[ \tau_{n}:=\nu_{n}^{-1}\max_{t=1,...,n}\left\vert \sum\nolimits_{i=1} ^{t}(\varepsilon_{i}-\overline{\varepsilon}_{n})\right\vert \text{, } \overline{\varepsilon}_{n}:=n^{-1}\sum\nolimits_{t=1}^{n}\varepsilon_{t}, \] where $\nu_{n}$ is a permutation-invariant normalization sequence. Standard choices are $\nu_{n}^{2}=\sum_{t=1}^{n}(\varepsilon_{t}-\overline{\varepsilon }_{n})^{2}$ in the case where $E\varepsilon_{t}^{2}<\infty$, and $\nu_{n} =\max_{t=1,...,n}|\varepsilon_{t}|$ when $E\varepsilon_{t}^{2}=\infty$. If $\varepsilon_{t}$ is in the domain of attraction of a strictly $\alpha$-stable law with $\alpha\in(0,2)$, such that $E\varepsilon_{t}^{2}=\infty$, the asymptotic distribution of $\tau_{n}$ depends on unknown parameters (e.g., the characteristic exponent $\alpha$), which makes the test difficult to apply (see also Politis, Romano and Wolf, 1999, and the references therein). To overcome this problem, Aue et al. (2008) consider a permutation analogue of $\tau_{n}$, defined as \[ \text{$\tau_{n}^{\ast}:=\nu_{n}^{-1}\max_{t=1,...,n}\left\vert \sum \nolimits_{i=1}^{t}(\varepsilon_{\pi(i)}-\overline{\varepsilon}_{n} )\right\vert $} \] where $\pi$ is a (uniformly distributed) random permutation of $\{1,2,...,n\}$ , independent of the data.\footnote{The normalization of $\nu_{n}$ is only of theoretical importance for obtaining non-degenerate limit distributions. In practice, any bootstrap procedure comparing $\tau_{n}$ to the quantiles of $\tau_{n}^{\ast}$ is invariant to the choice of $\nu_{n}$ and can be implemented by setting $\nu_{n}=1$.} In terms of Definition (ref), the data is $D_{n}:=\{\varepsilon_{t}\}_{t=1}^{n}$ and the auxiliary `bootstrap' variate is $W_{n}^{\ast}:=\pi$. With $X_{n}:=\{\varepsilon_{(t)}\}_{t=1}^{n}$ denoting the vector of order statistics of $\{\varepsilon_{t}\}_{t=1}^{n}$, there exists a random permutation $\varpi$ of $\{1,...,n\}$ (under $\mathsf{H}_{0}$, uniformly distributed conditionally on $X_{n}$) for which it holds that $\varepsilon_{t}=\varepsilon_{(\varpi(t))}$ ($t=1,...,n$), whereas the `bootstrap' sample is $\{\varepsilon_{\pi(t)}\}_{t=1}^{n}$ . The results in Aue et al. (2008, Corollary 2.1, Theorem 2.4) imply that, if $\mathsf{H} _{0}$ holds and $\varepsilon_{t}$ is in the domain of attraction of a strictly $\alpha$-stable law with $\alpha\in(0,2)$, then $\tau_{n}$$\overset {w}{\rightarrow}\rho_{\alpha}(S)$ and $\tau_{n}^{\ast}\overset{w^{\ast} }{\rightarrow}_{w}\rho_{\alpha}(S)|S$ for a certain random function $\rho_{\alpha}$ and $S=(S_{1},S_{2})^{\prime}$, with $S_{i}=\{S_{ij} \}_{j=1}^{\infty}$ ($i=1,2$) being partial sums of sequences of i.i.d. standard exponential rv's, and with $\rho_{\alpha}$ independent of $S$.\footnote{To avoid centering terms, Aue et al. (2008) assume additionally that the location parameter of the limit stable law is zero when $\alpha\in\lbrack1,2).$ Moreover, although they provide conditional convergence results only for the finite-dimensional distributions of the CUSUM process, these could be strengthened to conditional functional convergence as in Proposition 1 of LePage et al. (1997) in order to obtain the conditional convergence of $\tau_{n}^{*}.$}
Aue et al. (2008) do not report the fact that statistical inferences are not invalidated by the failure of the permutation procedure to estimate consistently the distribution of $\rho_{\alpha}(S)$. In fact, the situation is similar to that of Remark (ref), as $\tau_{n}|X_{n}\overset{d}{=}\tau_{n}^{\ast}|D_{n}$ under $\mathsf{H}_{0}$. As a consequence, under $\mathsf{H}_{0}$ the permutation test implements exact\footnote{Here by `exact' we mean that bootstrap inference replicates the finite-sample (conditional) distribution of the test statistic for any sample size with no error. } finite-sample inference conditional on $X_{n}$ and, additionally, the distribution of $\tau_{n}^{\ast}$ given the data estimates consistently the limit of the conditional distribution $\tau_{n}|X_{n}$, in the sense of joint weak convergence in distribution (see eq. ((ref))):
CUSUM tests can also be applied to residuals from an estimated model in order to test for correct model specification or stability of the parameters (see e.g., Ploberger and Kr\"{a}mer, 1992). Consider thus the case where $\{\varepsilon_{t}\}_{t=1}^{n}$ are the disturbances in a statistical model (e.g., the regression model of Section 2), and we observe residuals $\hat{\varepsilon}_{t}$ obtained upon estimation of the model using a sample $D_{n} $ not containing the unobservable $\{\varepsilon_{t}\}_{t=1}^{n}$. The residual-based CUSUM\ statistic is $\hat{\tau}_{n}:=$$\hat{\nu}_{n}^{-1} \max_{t=1,...,n}|\sum_{i=1}^{t}(\hat{\varepsilon}_{i}-\overline{\hat {\varepsilon}}_{n})|$, where $\hat{\nu}_{n}$ and $\overline{\hat{\varepsilon} }_{n}$ are the analogues of $\nu_{n}$ and $\bar{\varepsilon}_{n} $ computed from $\hat{\varepsilon}_{t}$ instead of $\varepsilon_{t}$. The bootstrap statistic could be defined as $\hat{\tau}_{n}^{\ast}:=$$\hat{\nu}_{n}^{-1} \max_{t=1,...,n}|\sum_{i=1}^{t}(\hat{\varepsilon}_{\pi(i)}-\overline {\hat{\varepsilon}}_{n})|$. If $\hat{\tau}_{n}-\tau_{n}\overset{p} {\rightarrow}0$ and $(\hat{\tau}_{n}^{\ast}-\tau_{n}^{\ast})|D_{n}\overset {w}{\rightarrow}_{p}0$ under $\mathsf{H}_{0}$ (e.g., due to consistent parameter estimation), then also $(\hat{\tau}_{n}-\tau_{n})|X_{n}\overset {w}{\rightarrow}_{p}0$, such that the (L\'{e}vy) distances between the pairs of conditional distributions $\hat{\tau}_{n}|X_{n}$ and $\tau_{n}|X_{n}$ on the one hand, and $\hat{\tau}_{n}^{\ast}|D_{n}\ $and $\tau_{n}^{\ast}|D_{n}$ on the other hand, converge in probability to zero. Hence, in view of ((ref)), and under the conjecture that $P(\rho_{\alpha}(S)\leq\cdot|S)$ defines a sample-path continuous cdf, the residual-based permutation procedure is consistent in the sense that
for $X_{n}:=\{\varepsilon_{(t)}\}_{t=1}^{n}$ again. It follows that:
(i) The permutation residual-based test is valid conditionally on $X_{n}$, by Corollary (ref)(a) with condition ((ref)) taking the form ((ref)).
(ii) This test is valid unconditionally, as a results of either the validity conditional on $X_{n}$, or by Corollary (ref).
The parametric bootstrap is a standard technique for the approximation of a conditional distribution of goodness-of-fit test statistics (Andrews, 1997; Lockhart, 2012). When these are discussed in the i.i.d. finite-variance setting, the limit of the bootstrap distribution is non-random. However, if we return to the relation ((ref)), there exist relevant settings where a random limit of the normalized $M_{n}$ implies that parametrically bootstrapped goodness-of-fit test statistics have random limit distributions.
Let the null hypothesis of interest, say $\mathsf{H}_{0}$, be that the standardized errors $\omega_{\varepsilon}^{-1/2}\varepsilon_{t}$ in ((ref)) have a certain known density $f$ with mean 0 and variance 1. For expositional ease we assume that $\omega_{\varepsilon}=1$ and is known to the econometrician. Then, the Kolmogorov-Smirnov statistic based on OLS residuals $\hat{\varepsilon}_{t}$ is \[ \tau_{n}:=n^{1/2}\sup_{u\in\mathbb{R}}\left\vert n^{-1}\sum_{t=1} ^{n}\mathbb{I}_{\{\hat{\varepsilon}_{t}\leq u\}}-\int_{-\infty}^{u} f\right\vert \text{.} \] A (parametric) bootstrap counterpart, $\tau_{n}^{\ast}$, of $\tau_{n}$ could be constructed under $\mathsf{H}_{0}$ by (i) drawing $\{\varepsilon_{t}^{\ast }\}_{t=1}^{n}$ as i.i.d. from $f$, independent of the data; (ii), regressing them on $x_{t}$, thus obtaining an estimator $\hat{\beta}^{\ast}$ and associated residuals $\hat{\varepsilon}_{t}^{\ast}$; and (iii) calculating $\tau_{n}^{\ast}$ as $\tau_{n}^{\ast}:=n^{1/2}\sup_{u\in\mathbb{R}}|n^{-1} \sum_{t=1}^{n}\mathbb{I}_{\{\hat{\varepsilon}_{t}^{\ast}\leq u\}} -\int_{-\infty}^{u}f|$.
To see that the distribution of the bootstrap statistic $\tau_{n}^{\ast}$ conditional on the data $D_{n}:=\{x_{t},y_{t}\}_{t=1}^{n}$ may have a random limit, consider the Gaussian case, $f=\Phi^{\prime}$. Under the assumptions of Johansen and Nielsen (2016, Sec. 4.1-4.2), it holds (ibidem) that $\tau_{n}^{\ast}=\tilde{\tau}_{n}^{\ast}+o_{p}(1)$ under the product probability on the product probability space where the data and $\{\varepsilon _{t}^{\ast}\}$ are jointly defined, with
where $q(u)=\Phi^{-1}(u)$ is the $u$-th quantile of $\Phi$. The expansion of $\tau_{n}^{\ast}$ holds also conditionally on the data, i.e., $\tau_{n}^{\ast }-\tilde{\tau}_{n}^{\ast}\overset{w^{\ast}}{\rightarrow}_{p}0$, since convergence in probability to a constant is preserved upon such conditioning. Hence, if $\left. \tilde{\tau}_{n}^{\ast}\right\vert D_{n}$ converges to a random limit, so does $\left. \tau_{n}^{\ast}\right\vert D_{n}$ for the same limit. Assume that $X_{n}:=n^{-\alpha/2}x_{\left\lfloor n\cdot\right\rfloor } $$\overset{w}{\rightarrow}X$ in $\mathscr{D}$ for some $\alpha>0$ and that $M:=\int X^{2}>0$ a.s. (e.g., $X=B_{\eta}$ if $x_{t}=\sum_{s=1}^{t-1}\eta_{s}$ with $\{\eta_{t}\}$ introduced in Section 2.2). Then $(\text{$M_{n},\xi_{n})$ }:=(\sum_{t=1}^{n}x_{t}^{2},\sum_{t=1}^{n}x_{t})$ satisfies $(\text{$n^{-\alpha-1}M_{n},n^{-\alpha/2-1}\xi_{n})$}\overset{w}{\rightarrow }(M,\xi)$, $\xi:=\int X$.\ Furthermore, if $W_{n}^{\ast}(u):=n^{-1}\sum _{t=1}^{n}(\mathbb{I}_{\{\varepsilon_{t}^{\ast}\leq q(u)\}}-u)$, $u\in \lbrack0,1],$ is the bootstrap empirical process in probability scale, then $W_{n}^{\ast}$ and $M_{n}^{1/2}\hat{\beta}^{\ast}$ are independent of the data individually (the second one being conditionally standard Gaussian), but not jointly independent of the data, because \[ \operatorname*{Cov}\nolimits^{\ast}(n^{1/2}W_{n}^{\ast}(u),M_{n}^{1/2} \hat{\beta}^{\ast})=(n^{-\alpha-1}M_{n})^{-1/2}n^{-\alpha/2-1}\xi_{n} \psi(u)\overset{w}{\rightarrow}M^{-1/2}\xi\psi(u)\text{,} \] $u\in\lbrack0,1]$, where $\psi(\cdot):=E^{\ast}[\varepsilon_{1}^{\ast }\mathbb{I}_{\{\varepsilon_{1}^{\ast}\leq q(\cdot)\}}]=-\Phi^{\prime} (q(\cdot))$ is a trimmed mean function, with $\operatorname*{Cov} \nolimits^{\ast}(\cdot)$ and $E^{\ast}(\cdot)$ calculated under $P^{\ast}$. It is shown in Appendix (ref) that, more strongly,
on $\mathscr{D}\times\mathbb{R}^{2},$ where $(W,b$$)$ is a pair of a standard Brownian bridge and a standard Gaussian rv individually independent of $X$ (and thus, of $M,\xi$), but with Gaussian joint conditional (on $X$) distributions having covariance $\operatorname*{Cov}(W(u),b|X)=M^{-1/2}\xi \psi(u),$ $u\in\lbrack0,1]$. Combining the expansion of $\tau_{n}^{\ast}$, ((ref)) and ((ref)) with the extended CMT (Theorem (ref) in Appendix (ref)) yields
where $\tau:=\sup_{u\in\lbrack0,1]}|\tilde{W}(u)|$ for a process $\tilde{W}$ which conditionally on $X$ (and thus, on $M,\xi$), is a zero-mean Gaussian process with $\tilde{W}(0)=\tilde{W}(1)=0$ a.s. and conditional covariance function $K(u,v)=u(1-v)-M^{-1}\xi^{2}\psi(u)\psi(v)$ for $0\leq u\leq v\leq1$. In summary, the limit bootstrap distribution is random because the latter conditional covariance is random whenever $M$ or $\xi$ are such.
We now discuss in what sense $\tau_{n}^{\ast}$ can provide a distributional approximation of $\tau_{n}$ and whether the bootstrap can be valid in the sense of Definition (ref).
(i)\ Under $\mathsf{H}_{0}$ that $\varepsilon_{t}\sim\text{i.i.d.} $$N(0,1)$, the bootstrap could be shown to be unconditionally valid using Theorem \ref{th2}. Specifically, under $\mathsf{H}_{0}$, the assumptions and results of Johansen and Nielsen (2016, Sec. 4.1-4.2) guarantee that $\tau_{n}$ has the expansion $\tau_{n}=\tilde{\tau}_{n}+o_{p}(1)$, with $\tilde{\tau} _{n}:=\sup_{u\in\lbrack0,1]}|n^{-1/2}\sum_{t=1}^{n}(\mathbb{I}_{\{\varepsilon _{t}\leq q(u)\}}-u)+\Phi^{\prime}(q(u))(\hat{\beta}-\beta)n^{-1/2}\sum _{t=1}^{n}x_{t}|$ defined similarly to $\tilde{\tau}_{n}^{\ast} $. Assume that $\hat{\beta}$ is asymptotically mixed Gaussian, such that jointly with $n^{-\alpha/2}x_{\left\lfloor n\cdot\right\rfloor }$$\overset{w}{\rightarrow }X$ it holds that \[ (n^{-1/2}\sum_{t=1}^{n}(\mathbb{I}_{\{\varepsilon_{t}\leq q(u)\}} -u),\,n^{(\alpha+1)/2}(\hat{\beta}-\beta),\,n^{-\alpha/2-1}\xi_{n})\overset {w}{\rightarrow}(W,M^{-1/2}b,\xi)\text{ ;} \] then $\tau_{n}=\tilde{\tau}_{n}+o_{p}(1)\overset{w}{\rightarrow}\tau =\sup_{u\in\lbrack0,1]}|\tilde{W}(u)|$. Thus, the unconditional limit of $\tau_{n}$ obtains by averaging (over $M,\xi$) the conditional limit of $\tau_{n}^{\ast}$. This is the main prerequisite for establishing unconditional bootstrap validity via Theorem (ref). More precisely, it is proved in Appendix (ref) that
As $F$ is sample-path continuous (e.g., by Proposition 3.2 of Linde, 1989, applied conditionally on $M,\xi$), Theorem (ref) guarantees the unconditional validity of the bootstrap.
(ii) As $\tau_{n}=\tilde{\tau}_{n}+o_{p}(1)$ under $\mathsf{H}_{0} $, with $\tilde{\tau}_{n}$ related to $\left( M_{n},\xi_{n}\right) $ through the same functional form as $\tilde{\tau}_{n}^{\ast}$, it is possible for $\tau_{n}|X_{n}$ to have the same random limit distribution under $\mathsf{H}_{0}$ as $\tau_{n}^{\ast}$ given the data, i.e., $\tau_{n} |X_{n}\overset{w}{\rightarrow}_{w}\tau\left\vert (M,\xi)\right. $. For instance, this occurs if $\{\varepsilon_{t}\}$ is an i.i.d. sequence independent of $X_{n}$, by the same argument as for $\tilde{\tau}_{n}^{\ast} $. According to Remark (ref), the convergence $\tau_{n} |X_{n}\overset{w}{\rightarrow}_{w}\tau|(M,\xi)$ and the convergence $(\tau _{n},\tau_{n}^{\ast},n^{-\alpha-1}M_{n},n^{-\alpha/2-1}\xi_{n})\overset {w}{\rightarrow}\left( \tau,\tau^{\ast},M,\xi\right) $ with $\tau^{\ast }|(M,\xi)\overset{d}{=}\tau|(M,\xi)$ (shown in the proof of ((ref)), see Appendix (ref)) are sufficient for eq. ((ref)) to hold in the form \[ (\tau_{n}|X_{n},\tau_{n}^{\ast}|D_{n})\overset{w}{\rightarrow}_{w}(\tau |(M,\xi),\tau|(M,\xi))\text{.} \] As $F$ is sample-path continuous, the bootstrap is valid conditionally on $X_{n}$ by Corollary (ref)(a).
Here we consider an instance of the `parameter on the boundary' problem in the framework of predictive regressions for financial returns; see e.g. Phillips (2014) and the references therein. While in this context the bootstrap is potentially useful (e.g., when there is uncertainty about the degree of persistence of the posited predicting variable), its application is not straightforward if some of the parameters may lie on the boundary of the parameter space; see Andrews (2000).
We show that in the presence of parameters on the boundary, the distribution of the bootstrap statistic may be random in the limit. Moreover, the type of randomness induced by parameters on the boundary depends on how well the bootstrap scheme approximates the mutual position of three objects, namely (i)\ the boundary, (ii)\ the set identified by the null hypothesis, and (iii)\, the true parameter value. Standard bootstrap approximations may not be sufficiently precise, giving rise to complex conditioning in the limit bootstrap distribution, with ensuing unconditional bootstrap validity only for special statistics. Conversely, non-standard, or ad hoc, bootstrap schemes, designed to provide a better match with the original geometry, may feature limit bootstrap distributions where no randomness attributable to the possibly boundary value of a parameter is present.
Consider the predictive regression
under Assumption 1 of Georgiev et al. (2018), specialized for simplicity to unconditionally homoskedastic errors. The posited predicting variable $x_{n,t}$ is such that, in $\mathscr{D}{}$, $x_{n,\left\lfloor n\cdot \right\rfloor }\overset{w}{\rightarrow}X$, e.g. a Brownian motion or an Ornstein-Uhlenbeck process, and hence features low frequency variability in the sense of M\"{u}ller and Watson (2008). We assume that the parameter space, say $\Theta$, is defined by an inequality constraint and that the true value of the parameter $\theta:=(\theta_{1},\theta_{2})^{\prime}$, say $\theta _{0}:=(\theta_{1,0},\theta_{2,0})^{\prime}$, may lie on the boundary of $\Theta$. An important example is when $\theta$ is assumed to belong to the set $\mathbb{R}\times\lbrack0,\infty)$, with the boundary corresponding to the case $\theta_{2}=0$ of no predictability of $y_{t}$ by $x_{n,t-1}$ and the interior corresponding to (sign-restricted) predictability.
More specifically, assume that $\Theta:=\{\theta\in\mathbb{R}^{2} :g(\theta)\geq0\}$, where $g:$ $\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a real function, continuously differentiable on some neighborhood of $\theta_{0}$ and with gradient $\tfrac{\partial}{\partial\theta^{\prime}}g(\theta)\neq0$ on that neighborhood, with $\dot{g}:=\tfrac{\partial}{\partial\theta^{\prime} }g(\theta_{0})$. The boundary of $\Theta$ is denoted by $\partial \Theta:=\{\theta\in\mathbb{R}^{2}:g(\theta)=0\}$. The aforementioned example $\theta_{2}\geq0$ is obtained by setting $g(\theta)=(0,1)\theta=\theta_{2}$.
Interest is in bootstrap inference on a null hypothesis $\mathsf{H}_{0}$ identifying a set of parameter values that has a non-empty intersection with the boundary of the parameter space. In particular, we consider the following mutual positions of the boundary, the parameter set identified by $\mathsf{H}_{0}$ and the true value $\theta_{0}$:
For example, let again $g(\theta)=\theta_{2}\text{.}$ Then the hypothesis of no predictability $\mathsf{H}_{0}:\theta_{2,0}=0$ falls under $\mathscr{G}{}_{1}$; the hypothesis $\mathsf{H}_{0}:\theta_{0}=(0,0)^{\prime}$ that $y_{t}$ is unpredictable with zero mean falls under $\mathscr{G}{}_{2}$; the hypothesis $\mathsf{H}_{0}:(1,1)^{\prime}\theta_{0}=\theta_{1,0} +\theta_{2,0}=0$ falls under $\mathscr{G}{}_{3}$. In the latter case, the intersection point of the boundary and $\mathsf{H}_{0}$ is $(0,0)^{\prime}$ which might, but need not, be the true value under $\mathsf{H}_{0}$.
Let $\hat{\theta}$ be the OLS estimator of the first two coefficients in the equation
subject to the constraint $\hat{\theta}\in\Theta$, i.e. $g(\hat{\theta})\geq0$ (here $\Delta x_{n,t}$ is included in order to obtain residuals asymptotically uncorrelated with the innovations driving $x_{n,t}$). It holds that $n^{1/2}(\hat{\theta}-\theta_{0})\overset{w}{\rightarrow}\ell(\theta_{0})$, with $\ell(\theta_{0})$ depending on the position of $\theta_{0}$ relative to the boundary $\partial\Theta$. Thus, $\ell(\theta_{0})=\tilde{\ell} :=M^{-1/2}\xi$ if $\theta_{0}\in\operatorname*{int}(\Theta):=\Theta \setminus\partial\Theta$, where $M:=\int\tilde{X}\tilde{X}^{\prime}$, $\tilde{X}:=(1,X)^{\prime}$, $\xi\sim N\left( 0,\sigma_{e}^{2}I_{2}\right) $ is independent of $X$, and $\sigma_{e}>0$, whereas (see Section 12 in the working paper version of Andrews, 1999),
if $g(\theta_{0})=0$, where we use the notation $||x||_{M}:=(x^{\prime }Mx)^{1/2}$ for $x\in\mathbb{R}^{2}$.
Consider now a bootstrap sample generated as
where $\varepsilon_{t}^{\ast}=\hat{e}_{t}w_{t}^{\ast}$, $t=1,...n$, with $\hat{e}_{t}$ the residuals of ((ref)) and $w_{t}$ i.i.d. $N(0,1)$, independent of the original data.\footnote{The conclusions do not change if, instead of this wild (fixed regressor) bootstrap, a standard residual-based i.i.d. bootstrap or a parametric bootstrap is used.} Then the distribution of $n^{1/2}(\hat{\theta}-\theta_{0})$ could be tentatively approximated by the distribution of $n^{1/2}(\hat{\theta}^{\ast}-\hat{\theta})$ conditional on the original data, where $\hat{\theta}^{\ast}$ is obtained by regressing $y_{t}^{\ast}$ on $(1,x_{n,t-1})^{\prime}$ (the term $\Delta x_{n,t}$ is no longer necessary) under the constraint $\hat{\theta}^{\ast}\in\Theta^{\ast }=\Theta$ (as for the original estimator), i.e., $g(\hat{\theta}^{\ast})\geq 0$; see Andrews (2000).
For $\theta_{0}\in\operatorname*{int}(\Theta)$, it turns out that the bootstrap statistic converges to a conditional version of the limit of $n^{1/2}(\hat{\theta}-\theta_{0})$ found earlier:
where $\tilde{\theta}^{\ast}$ denotes the unconstrained OLS estimator from the bootstrap sample.
On the other hand, if $\theta_{0}\in\partial\Theta$ the bootstrap statistic converges as follows, jointly with $n^{1/2}(\hat{\theta}-\theta_{0})$:
where $\xi^{\ast}\sim N(0,1)$ is independent of $(M,\ell)$; see Theorem (ref) below. In contrast with the case $\theta_{0}\in\operatorname*{int}(\Theta)$, the limit in ((ref)) is not a conditional version of the limit of $n^{1/2}(\hat{\theta}-\theta_{0})$, inasmuch as $\Lambda_{\ell}^{\ast}$ in ((ref)) is a random half-plane, rather than the original set $\Lambda$ of ((ref)). The reason is that the standard bootstrap does not approximate well the original mutual position of the true value and the boundary, unless $g(\hat{\theta})=0$. Other, non-standard bootstraps may be designed in order to provide better approximations, at least under the null hypothesis. This is analyzed next.
In order to unify the discussion of several bootstrap schemes for inference on $\mathsf{H}_{0}$ under the three cases $\mathscr{G}{}_{1}$, $\mathscr{G}{}_{2} $ and $\mathscr{G}{}_{3}$, consider a bootstrap sample generated as in ((ref)) and, more generally than before, a bootstrap OLS estimator $\hat{\theta}^{\ast}$ constrained to belong to the (random) set \[ \Theta^{\ast}:=\{\theta\in\mathbb{R}^{2}:g(\theta)\geq g^{\ast}(\hat{\theta })\} \] where the function $g^{\ast}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is continuously differentiable on some neighborhood of $\theta_{0}$ and satisfies $g^{\ast}(\theta)\leq g(\theta)$ for $\theta\in\Theta$. The standard bootstrap considered in Section (ref) obtains by setting $g^{\ast}=0$ (such that $\Theta^{\ast}=\Theta$, the original parameter space). Alternatively, setting $g^{\ast}=g$ restricts the bootstrap true value $\hat{\theta}$ to lie on the boundary of the bootstrap parameter space $\Theta^{\ast}$ (as, in this case, $\Theta^{\ast}=\{\theta\in\mathbb{R} ^{2}:g(\theta)\geq g(\hat{\theta})\}$); see Cavaliere, Nielsen and Rahbek (2017) for an application of this `restricted' bootstrap to the location model. Finally, setting $g^{\ast}=g-|g|^{1+\kappa}$ for some $\kappa>0$ introduces a correction, in the spirit of an alternative to the standard bootstrap mentioned in Andrews (2000,p.403, Method two) and Fang and Santos (2019, Example 2.1), where the bootstrap true value either shrinks to the boundary of the bootstrap parameter space at a proper rate or remains bounded away from this boundary, according to whether $\theta_{0}$ belongs to the original boundary $\partial\Theta$ or not.\footnote{Instead of setting $g^{\ast}=g-|g|^{1+\kappa}$, one could alternatively set $g^{\ast }:=g-n^{-\kappa}|g|$ for $\kappa\in(0,\tfrac{1}{2})$. The results would be unchanged. }
In general, the limit distribution of the resulting bootstrap estimator is random, with randomness depending on both the stochastic regressor and the position of $\theta_{0}$ relative to the boundary. This distribution is given in the following theorem, where $\dot{g}^{\ast}:=\frac{\partial} {\partial\theta^{\prime}} g^{\ast}(\theta_{0})$.
The following conclusions could be drawn.
(i) Consider first configurations $\mathscr{G}{}_{1}$ and $\mathscr{G}{}_{2}$ under $\mathsf{H}_{0}$, such that $g(\theta_{0})=0$. Consider the magnitude order, in probability, of the distance between the bootstrap `true' value $\hat{\theta}$ and the bootstrap boundary $\partial\Theta^{\ast}$ as a precision measure for a bootstrap approximation to the geometry of $\mathscr{G}_{1}$ and $\mathscr{G}_{2}$. As seen above, the standard bootstrap (corresponding to $g^{\ast}=0$) approximates the geometry up to an exact magnitude order of $n^{-1/2}$, resulting in a situation where the belonging of $\theta_{0}$ to the boundary contributes to the randomness of limit bootstrap distribution given by ((ref)) and ((ref)) via conditioning on the rv $\ell(\theta_{0})=\ell$. Conversely, bootstrap schemes employing $g^{\ast}(\theta_{0})=g(\theta_{0})$ and $\dot{g}^{\ast }=\dot{g}$, such that the bootstrap boundary is tangent to the original boundary at $\theta_{0}$, give rise to approximations of order $o_{p} (n^{-1/2})$ and all the randomness in the bootstrap limit is due to the properties of the stochastic regressor (via the rv $M$, as now $\ell^{\ast }|(M,\ell)\overset{d}{=}\ell|M$; see ((ref)) and ((ref))). Moreover, for such schemes the bootstrap mimics a conditional version of the asymptotic distribution of the original estimator: $n^{1/2}(\hat{\theta}^{\ast}-\hat{\theta})\overset{w^{\ast}}{\rightarrow} _{w}\ell|M$. Examples are the `restricted' bootstrap based on $g^{\ast}=g$, which replicates the geometry of the original data under $\mathsf{H}_{0}$ by putting $\hat{\theta}$ on the bootstrap boundary, and the choices $g^{\ast }=g-|g|^{1+\kappa}$ for some $\kappa>0$.
(ii) Consider now the case in $\mathscr{G}{}_{3}$, such that $g(\theta_{0})=0$ need not, but may hold under $\mathsf{H}_{0}$. Among the bootstraps considered in (i), the standard one would fail to mimic a conditional version of the original distribution if $g(\theta_{0})=0$, while the `restricted' one would fail if $g(\theta_{0})>0$. As an alternative, consider the bootstrap based on $g^{\ast}=g-|g|^{1+\kappa}$ for some $\kappa>0$, see above. If $\theta_{0}\in\partial\Theta$, then this choice puts the bootstrap true value $\hat{\theta}$ at an (asymptotically negligible) distance of $o_{p}(n^{-1/2})$ from the bootstrap boundary, whereas if $\theta_{0}\in\operatorname*{int}(\Theta)$, then $\hat{\theta}$ is bounded away from the bootstrap boundary, in probability. This guarantees bootstrap unconditional validity, see (iii)\ below.
(iii) In general, bootstrap unconditional validity can be evaluated through the following corollary of Theorem (ref).
The class of functions $g^{\ast}=g-|g|^{1+\kappa}$ for $\kappa>0 $ satisfies both conditions (i) and (ii) of the previous corollary; hence, the ensuing bootstrap inference is unconditionally valid under all of $\mathscr{G}{}_{1}$-$\mathscr{G}{}_{3}$. In contrast, the standard bootstrap violates condition (i) and, in general, is asymptotically invalid if $g(\theta_{0})=0$. An exception is when the discrepancy between the original and the bootstrap geometry is offset by the use of a test statistic that takes into account the geometric position of the null hypothesis in the original parameter space. The next section focuses on this setup.
Under case $\mathscr{G}{}_{1}$, consider testing $\mathsf{H}_{0}:g(\theta _{0})=0$ against the alternative $\mathsf{H}_{1}:g(\theta_{0})>0$ using the standard bootstrap (i.e., with $g^{\ast}=0$). For a test statistic of the form $\tau_{n}:=n^{1/2}g(\hat{\theta})$,\footnote{What follows easily generalizes to statistics of the form $\tau_{n}:=\phi(n^{1/2}g(\hat{\theta}))$ with $\phi(\cdot)$ strictly increasing and normalized by $\phi(0)=0$.} its bootstrap counterpart is given by $\tau_{n}^{\ast}:=n^{1/2}(g(\hat{\theta }^{\ast})-g(\hat{\theta}))$ and the (one-sided) bootstrap test rejects for large values of the bootstrap p-value $p_{n}^{\ast}:=P^{\ast }(\tau_{n}^{\ast}\leq\tau_{n})$; equivalently, for small values of $\tilde {p}_{n}^{\ast}:=1-p_{n}^{\ast}$ (see Remark (ref)). As for $\hat{\theta}^{\ast} $, also $\tau_{n}^{\ast}$ is affected in the limit by extra randomness due to $\theta_{0}$ being on the boundary. From ((ref)), which reduces to ((ref)) and ((ref)), it follows by the Delta method that \[ (\tau_{n},(\tau_{n}^{\ast}|D_{n}))\overset{w}{\rightarrow}_{w}\left( \dot {g}^{\prime}\ell\text{,}\left( \dot{g}^{\prime}\ell^{\ast}|(M,\ell)\right) \right) =(\dot{g}^{\prime}\ell,(\max\{-\dot{g}^{\prime}\ell,\dot{g}^{\prime }\tilde{\ell}^{\ast}\}|(M,\ell)))\text{.} \] For $\tau_{n}^{\ast}$, however, the randomness induced by conditioning on $\ell$ affects the sample paths of the associated random cdf on the negative half-line alone (because $\dot{g}^{\prime}\ell\geq0$), and is thus irrelevant for bootstrap tests with nominal levels in $(0,\frac{1}{2})$. Put differently, the bootstrap p-values $\tilde{p}_{n}^{\ast}$ are (asymptotically) uniformly distributed below $\frac{1}{2}$. This follows rigorously from the next generalization of Theorem (ref) (the proof being analogous), where conditions for unconditional bootstrap validity restricted to a subset of nominal testing levels are formulated.
\noindentTheorem (ref)$^{\ast}$. Let there exist a rv $\tau$ and a random element $X$, both defined on the same probability space, such that the support of $\tau_{n}$ is contained in a closed interval $\mathbb{T}$ (finite or infinite), and $(\tau_{n},F_{n}^{\ast})\overset {w}{\rightarrow}(\tau,F)$ in $\mathbb{R}\times D(\mathbb{T})$ for $F_{n} ^{\ast}(u):=P(\tau_{n}^{\ast}\leq u|D_{n})$ and $F(u):=P(\tau\leq u|X)$, $u\in\mathbb{T}$. If the (possibly random) cdf $F$ is sample-path continuous on $\mathbb{T}$, then the bootstrap $p$-value $p_{n}^{\ast }:=\mathit{F_{n}^{\ast}(\tau_{n})}$\ satisfies \[ P(p_{n}^{\ast}\leq q)\rightarrow q \] for $q$ such that $q\in F(\mathbb{T})$ a.s.
By Theorem (ref)$^{\ast}$ with $\mathbb{T}=[0,\infty)$ (which corresponds to the support of $\tau_{n}$ and $\tau:=\dot{g}^{\prime }\ell$), it follows that the standard bootstrap applied to the one-sided statistic $\tau_{n}$ is unconditionally valid for nominal levels in $(0,\frac{1}{2})$.
Here we apply the results of Section (ref) to the classic problem of parameter constancy testing in regression models (Chow, 1960; Quandt, 1960; Nyblom, 1989; Andrews, 1993; Andrews and Ploberger, 1994). Specifically, we deal with bootstrap implementations when the moments of the regressors may be unstable over time; see Hansen (2000) and Zhang and Wu (2012), inter alia.
Consider a linear regression model for $y_{nt}\in\mathbb{R}$ given $x_{nt} \in\mathbb{R}^{m}$, in triangular array notation:
The null hypothesis of parameter constancy is $\mathsf{H}_{0}:\beta_{t} =\beta_{1}\,(t=2,...,n)$, which is tested here against the alternative $\mathsf{H}_{1}:\beta_{t}=\beta_{1}+\theta\mathbb{I}_{\{t\geq n^{\star}\}} $ ($t=2,...,n$), where $n^{\star}:=\lfloor r^{\star}n\rfloor$ and $\theta\neq0$ respectively denote the timing and the magnitude of the possible break,\footnote{We suppress the possible dependence of $\beta_{t}=\beta_{nt}$ on $n$ with no risk of ambiguities.} both assumed unknown to the econometrician. The so-called break fraction $r^{\star}$ belongs to a known closed interval $[\underline{r},\overline{r}]\ $in $(0,1)$. In order to test $\mathsf{H}_{0}$ against $\mathsf{H}_{1}$, it is customary to consider the `$\sup F$' (or `$\sup$ Wald') test (Quandt, 1960; Andrews, 1993), based on the statistic $\mathscr{F}{}_{n}:=\max_{r\in\lbrack\underline{r},\overline{r} ]}F_{\left\lfloor nr\right\rfloor },$ where $F_{\left\lfloor nr\right\rfloor }$ is the usual $F$ statistic for testing the auxiliary null hypothesis that $\theta=0$ in the regression \[ y_{nt}=\beta^{\prime}x_{nt}+\theta^{\prime}x_{nt}\mathbb{I}_{\{t\geq \left\lfloor rn\right\rfloor \}}+\varepsilon_{nt}\text{.} \]
We make the following assumption, allowing for non-stationarity in the regressors (see also Hansen, 2000, Assumptions 1 and 2).
\noindentAssumption $\mathcal{H}$. The following conditions on $\{x_{nt},\varepsilon_{nt}\}$ hold:
The null asymptotic distribution of $\mathscr{F}{}_{n}$ under Assumption $\mathcal{H}$ is provided in Hansen (2000, Theorem 2):
with $\tilde{N}\left( u\right) :=N\left( u\right) -M\left( u\right) M\left( 1\right) ^{-1}N\left( 1\right) $ and $\tilde{M}\left( r\right) :=M\left( r\right) -M\left( r\right) M\left( 1\right) ^{-1}M\left( r\right) $. In the case of (asymptotically)\ stationary regressors, $\mathscr{F}{}_{n}$ converges to the supremum of a squared tied-down Bessell process; see Andrews (1993). In the general case, however, since the asymptotic distribution in ((ref)) depends on the joint distribution of the limiting processes $M,N,V$, which is unspecified under Assumption $\mathcal{H}$, asymptotic inference based on ((ref)) is unfeasible. Simulation methods as the bootstrap can therefore be appealing devices for computing p-values associated with $\mathscr{F}{}_{n}$.
Following Hansen (2000), we consider here a fixed-regressor wild bootstrap introduced to accommodate possible conditional heteroskedasticity of $\varepsilon_{nt}$. It is based on the residuals $\tilde{e}_{nt}$ from the OLS regression of $y_{nt}$ on $x_{nt}$ and $x_{nt}\mathbb{I}_{\{t\geq\left\lfloor \tilde{r}n\right\rfloor \}}$, where $\tilde{r}:=\arg\max_{r\in\lbrack \underline{r},\overline{r}]}F_{\left\lfloor nr\right\rfloor }$ is the estimated break fraction for the original sample. The bootstrap statistic is \[ \mathscr{F}{}_{n}^{\ast}:=\max_{r\in\lbrack\underline{r},\overline{r} ]}F_{\left\lfloor nr\right\rfloor }^{\ast}\text{,} \] where $F_{\left\lfloor nr\right\rfloor }^{\ast}$ is the $F$ statistic for the auxiliary null hypothesis that $\theta^{\ast}=0$ in the regression
with bootstrap data $y_{t}^{\ast}:=\tilde{e}_{nt}w_{t}^{\ast}$ for an i.i.d. N(0,1) sequence of bootstrap multipliers $w_{t}^{\ast}$ independent of the data.
The weak limit of the bootstrap statistic $\mathscr{F}{}_{n}^{\ast}$ is stated in the next theorem.
Although under Assumption $\mathcal{H}$ the bootstrap does not replicate the asymptotic (unconditional) distribution in ((ref)), unconditional bootstrap validity can be established under no further assumptions than Assumption $\mathcal{H}$, by using the results in Section (ref). In contrast, despite fixing the regressors across bootstrap samples, if interest is in achieving bootstrap validity conditional on the regressors $X_{n}:=\{x_{nt}\}_{t=1}^{n}$, further conditions are required; e.g., the following Assumption $\mathcal{C}$.
\noindentAssumption $\mathcal{C}$. Assumption $\mathcal{H}$\ holds and, as random measures on $\mathscr{D}{}_{m\times m}\times\mathscr{D}{}_{m\times m}\times \mathscr{D}{}_{m}$ \[ \left. \left( \tfrac{1}{n}\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor }x_{nt}x_{nt}^{\prime},\tfrac{1}{n\sigma^{2}}\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor }x_{nt}x_{nt}^{\prime}\varepsilon_{nt}^{2},\tfrac {1}{n^{1/2}\sigma}\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor } x_{nt}\varepsilon_{nt}\right) \right\vert X_{n}\overset{w}{\rightarrow} _{w}\left( M,V,N\right) |(M,V) \] jointly with the convergence in Assumption \textsc{\emph{$\mathcal{H}$ }}\emph{(iii).}
The results on the validity of the bootstrap parameter constancy tests are summarized in the following theorem.
When the distribution of a bootstrap statistic conditional on the data is random in the limit, the bootstrap fails to estimate consistently the asymptotic distribution of the original statistic. In this case, the bootstrap is usually regarded as invalid. Renormalization of the statistic of interest cannot always be used to eliminate the limiting bootstrap randomness (e.g., it cannot be used in any of the four applications discussed in Section (ref)). We have shown, however, that if (asymptotic)\ bootstrap validity is defined as (large sample)\ control over the frequency of correct inferences, then randomness of the limit bootstrap distribution does not imply invalidity of the bootstrap, even without renormalizing the original statistic.
For the asymptotic validity of bootstrap inference, in an unconditional or a conditional sense, we have established sufficient conditions and strategies to verify these conditions in specific applications. The conditions differ mainly in their demands on the dependence structure of the data, and are more restrictive for conditional validity to hold.
We have provided four applications to well-known econometric inference problems which feature randomness of the limit bootstrap distribution. Among the further applications where randomness of the limit bootstrap distribution is likely to appear, and that could be analyzed using our approach, are bootstrap inference in weakly or partially identified models, inference in time series models with time-varying (stochastic) volatility, inference after model selection, and the bootstrap in high-dimensional models. In addition, the methods we provide for establishing conditional bootstrap validity could be useful in problems involving nuisance parameters that are not consistently estimable under the null hypothesis but where sufficient statistics are available (with the bootstrap being potentially valid conditionally on such statistics).
An important issue not analyzed in the paper is whether the bootstrap can deliver refinements over standard asymptotics in cases where the limit bootstrap measure is random. We have seen in Sections 2 and (ref) that bootstrap inference in such cases could be exact or close to exact. This seems to suggest that a potential for refinements exists. Moreover, there is also a potential for the bootstrap to inherit the finite-sample refinements offered by conditional asymptotic expansions (in line with Barndorff-Nielsen's p{*}-formula, see Barndorff-Nielsen and Cox, 1994, Sec. 6.2), as has been established for some bootstrap procedures (DiCiccio and Young, 2008) in the special case of correctly specified parametric models. The study of such questions requires mathematical tools different from those employed here, and is therefore left for further research.