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\begin{center}
\bigskip
{\Large INFERENCE UNDER RANDOM\ LIMIT\ BOOTSTRAP MEASURES{\LARGE \textsc{*}}}
\mbox{}
{\normalsize \ \vspace{-0.15cm}}
\textsc{Giuseppe Cavaliere}$^{a,b}$\textsc{\ and Iliyan Georgiev}$^{a,c}$
\textsc{\vspace{-0.15in}}
{\normalsize \vspace{0.45cm}}
{\small November 30, 2019}
\footnote{
\hspace{-7.2mm}
$^{*}
$ We thank Matias Cattaneo, Graham Elliott, Michael Jansson, S�ren Johansen, Ye Lu,
Adam McCloskey, Marcelo Moreira, Ulrich M�ller, Rasmus S�ndergaard Pedersen, Anders Rahbek,
Mervyn Silvapulle, Michael Wolf, for comments and suggestions. We also thank
seminar participants at Boston U, Cambridge, ETH, Exeter, FGV Sao Paulo, Geneve, Hitotsubashi, Kyoto, LSE, Melbourne,
Oxford, PUC Rio, Queen's, QUT, SMU, Tilburg, UCL, UPF, Verona, Vienna,
as well as participants to the $7^{th}$ Italian Congress of
Econometrics and Empirical Economics (ICEEE), the $14^{th}
$ SETA meeting and the
NBER/NSF 2018 Time Series Conference. Financial support from the
University of Bologna, ALMA IDEA 2017 grants, and from the Danish Council for Independent Research
(DSF Grant 7015-00028), is gratefully acknowledged. Address
correspondence to: Giuseppe Cavaliere, Department of Economics, University of Bologna,
Piazza Scaravilli 2, 40126 Bologna, Italy; email: [email removed].
\\
$^{a}$ Department of Economics, University of Bologna, Italy.
\\
$^{b}$ Department of Economics, Exeter Business School, UK.
\\
$^{c}
$ Nova School of Business and Economics, Universidade Nova de Lisboa, Portugal.}
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\end{center}
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\rightskip=1 cm
\small
\begin{center}
\textsc{Abstract\vspace{-0.15cm}}
\end{center}
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. \bigskip
\noindent\textsc{Keywords: }Bootstrap, random measures, weak convergence in
distribution, asymptotic inference.
\par\endgroup\normalsize
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\numberwithin{theorem}{section}
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\numberwithin{lemma}{section}
\newpage
\section{Introduction}
\textsc{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 \emph{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-\emph{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\textbf{\ }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
\emph{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 \emph{p}-values.
Specifically, let $p_{n}^{\ast}$ denote the bootstrap \emph{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
\begin{equation}
P(p_{n}^{\ast}\leq q)\rightarrow q\label{eq unconditional validity}
\end{equation}
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{eq unconditional validity}) 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 \emph{p}-value is uniformly distributed in large
sample \emph{conditionally }on $X_{n}$:
\begin{equation}
P(p_{n}^{\ast}\leq q|X_{n})\overset{p}{\rightarrow}
q\label{eq conditional validity}
\end{equation}
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{eq unconditional validity}). 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{eq conditional validity}) 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{eq conditional validity}) 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 \emph{exact }conditional inference and hence is also unconditionally
valid in the sense of (\ref{eq unconditional validity}). 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{eq unconditional validity}), we provide conditions for the validity of
alternative bootstrap schemes.\textbf{\ }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.
\subsection*{Structure of the paper}
The paper is organized as follows. In Section \ref{sec example} we outline the
main concepts and ideas using a simple linear regression model. Our main
theoretical results are presented in Section \ref{sec g}. Section
\ref{Section on Applications} contains the four applications of the theory,
whereas Section \ref{sec conclusion} concludes. The paper has two Appendices.
In Appendix \ref{sec itere} we collect some results on weak convergence in
distribution which are useful to prove our main theorems and develop the
applications. Appendix \ref{sec all proofs} contains the proofs of all theory
results given in the paper. The proofs of the results from Appendix
\ref{sec itere} and some additional material are collected in the accompanying
supplement, Cavaliere and Georgiev (2019).
\subsection*{Notation and definitions}
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),\textbf{\ }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 \emph{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 $P(\cdot|D_{n})$.
\section{A linear regression example}
\label{sec example}
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{Section on Applications}. 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{eq unconditional validity}). Finally, we illustrate the possibility that
bootstrap inference may have a conditional interpretation.
\subsection{Model, bootstrap and random limit bootstrap measures}
\label{sec example model and BS}
Assume that the data are given by $D_{n}:=\{y_{t},x_{t}\}_{t=1}^{n}$ and
consider the linear model
\begin{equation}
y_{t}=\beta x_{t}+\varepsilon_{t}\text{\hspace{1cm}(}t=1,2,...,n\text{)}
\label{eq:lm}
\end{equation}
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
\begin{equation}
y_{t}^{\ast}=\hat{\beta}x_{t}+\hat{\omega}_{\varepsilon}^{1/2}\varepsilon
_{t}^{\ast}\text{\hspace{1cm}(}t=1,2,...,n\text{)}\label{eq:blm}
\end{equation}
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 \emph{p}-value is given by
$p_{n}^{\ast}:=F_{n}^{\ast}\left( T_{n}\right) $.
\begin{remark}
\label{Remark on exact conditional inference}A special case where the ensuing
bootstrap inference is exact in finite samples, such that $p_{n}^{\ast}$ is
uniformly distributed for finite $n$, obtains when the original $\varepsilon
_{t}$'s are $N(0,\omega_{\varepsilon})$, independent of $X_{n}:=\{x_{t}
\}_{t=1}^{n}$, and $\omega_{\varepsilon}$ is known to the econometrician
(hence $\hat{\omega}_{\varepsilon}=\omega_{\varepsilon} $). Then the
conditional distribution of $T_{n}^{\ast}$ given the data $D_{n}$ equals the
distribution of the original statistic $T_{n}$ \emph{conditional }
on\emph{\ }the regressor $X_{n}$ (equivalently, on the ancillary statistic
$M_{n}$): $T_{n}^{\ast}|D_{n}\overset{d}{=}T_{n}|X_{n}\sim N\left(
0,\omega_{\varepsilon}M_{n}^{-1}\right) |M_{n}$. Put differently,
\[
F_{n}^{\ast}\left( u\right) :=P(T_{n}^{\ast}\leq u|D_{n})=P(T_{n}\leq
u|X_{n})=\Phi(\omega_{\varepsilon}^{-1/2}M_{n}^{1/2}u)\text{, }u\in
\mathbb{R}\text{.}
\]
Then, as $\omega_{\varepsilon}^{-1/2}M_{n}^{1/2}T_{n}|M_{n}\sim N(0,1)$, it is
straightforward that in this special case bootstrap inference is exact:
$p_{n}^{\ast}=F_{n}^{\ast}\left( T_{n}\right) =\Phi(\omega_{\varepsilon
}^{-1/2}M_{n}^{1/2}T_{n})\overset{d}{=}\Phi(N\left( 0,1\right) )\sim
U\left( 0,1\right) $, and that this result also holds conditionally on
$M_{n}$: $p_{n}^{\ast}|M_{n}\sim U(0,1)$.$\hfill\square$
\end{remark}
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.
\subsubsection{Classic bootstrap validity when the regressor is stationary}
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 \emph{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
\begin{equation}
P^{\ast}(\tau_{n}^{\ast}\leq u)=\Phi(n^{-1/2}\hat{\omega}_{\varepsilon}
^{-1/2}M_{n}^{1/2}u)\overset{p}{\rightarrow}\Phi(\omega_{\varepsilon}
^{-1/2}M^{1/2}u),\text{ }u\in\mathbb{R}.\label{eq lin model asy bootstrap cdf}
\end{equation}
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\textbf{\ }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 \emph{p}-value $p_{n}^{\ast}$
associated with $(\tau_{n},\tau_{n}^{\ast}) $ is asymptotically uniformly
distributed; i.e., (\ref{eq unconditional validity}) holds.
\subsubsection{Random limit bootstrap measures when the regressor is
non-stationary}
\label{sec Random limit bootstrap measures when the regressor is non-stationary}
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 \emph{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{{]},}
\begin{equation}
P^{\ast}(\tau_{n}^{\ast}\leq u)=\Phi(n^{-\alpha/2}\hat{\omega}_{\varepsilon
}^{-1/2}M_{n}^{1/2}u)\overset{w}{\rightarrow}\Phi(\omega_{\varepsilon}
^{-1/2}M^{1/2}u)\text{, }u\in\mathbb{R}
,\label{eq limit of BS measure - example}
\end{equation}
which is a random cdf. In terms of weak convergence in distribution, this
amounts to
\begin{equation}
\tau_{n}^{\ast}\overset{w^{\ast}}{\rightarrow}_{w}\left. N(0,\omega
_{\varepsilon}M^{-1})\right\vert M\text{.}\label{eq blim}
\end{equation}
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{Remark on exact conditional inference}). We proceed,
therefore, to identify in what sense bootstrap inference could remain meaningful.
\subsection{Bootstrap validity}
\label{sec example BS validity}
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.
\subsubsection{Unconditional bootstrap validity}
\label{sec intro to on average bs validity}\label{sec 2.2.1}
Under the assumption in Section
\ref{sec Random limit bootstrap measures when the regressor is non-stationary}
, 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{sec par on the boundary}), leading to no qualitative
differences from the case of diagonal $\Omega$.} Then, for $\beta\neq0$ eq.
(\ref{eq:lm}) 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{sec Random limit bootstrap measures when the regressor is non-stationary}
holds with $\alpha=2$ and
\begin{equation}
\tau_{n}:=n(\hat{\beta}-\beta)\overset{w}{\rightarrow}\left( \int B_{\eta
}^{2}\right) ^{-1}\int B_{\eta}dB_{\varepsilon}\sim N(0,\omega_{\varepsilon
}M^{-1})\text{,}\label{eq:ucl}
\end{equation}
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{eq blim}) and (\ref{eq:ucl}), 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{eq unconditional validity}). A direct argument is as
follows: the bootstrap \emph{p}-value $p_{n}^{\ast}:=P^{\ast}(\tau_{n}^{\ast
}\leq\tau_{n})$ satisfies, by the CMT,
\begin{align}
& p_{n}^{\ast}\overset{}{=}\Phi(\hat{\omega}_{\varepsilon}^{-1/2}M_{n}
^{1/2}(\hat{\beta}-\beta))\overset{w}{\rightarrow}\Phi((\omega_{\varepsilon}
{\textstyle\int}
B_{\eta}^{2})^{-1/2}
{\textstyle\int}
B_{\eta}dB_{\varepsilon})\label{eq:piv}\\
& \hspace{1.5in}\overset{d}{=}\Phi(N(0,1))\overset{}{\sim}U(0,1)\text{.}
\nonumber
\end{align}
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.
\subsubsection{Conditional bootstrap validity}
\label{sec intro to conditional bs validity}
In the case of unconditional bootstrap validity, it may be possible to find an
interpretation of bootstrap inference as also\textbf{\ }valid in the sense of
(\ref{eq conditional validity}), \textit{i.e.} conditionally on some $X_{n}$
defined on the probability space of the original data\textbf{\ }$D_{n}$ (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 \emph{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{sec intro to on average bs validity}, $\varepsilon_{t}$ 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{eq blim}) 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{Remark 3.5} where it is concluded that
$p_{n}^{\ast}|X_{n}\overset{w}{\rightarrow}_{p}U(0,1)$, \textit{i.e.}, the
bootstrap is valid conditionally on the regressor.
\subsubsection{A numerical illustration}
\label{sec uncond validity W/O conditional validity}
The result in Section \ref{sec intro to conditional bs validity} 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{Appendix MC}). In both cases, the
conditions of Section \ref{sec intro to conditional bs validity} 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.
\begin{figure}[tbh]
\centerline{\includegraphics[width=0.95\textwidth]{cattura.pdf}}
\caption{Fan chart of the simulated cdfs (conditional on $X_{n}$) of the
bootstrap $p$-values for the three DGPs (i)--(iii) and $n=10$ (upper panels),
$1000$ (lower panels).}
\end{figure}
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{sec 2.2.1} and
\ref{sec intro to conditional bs validity} 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{sec Proofs of the results in Section <ref>sec g</ref>}, eq.
(\ref{eq:phi}), that this endogeneity fact, not replicated in the bootstrap
world, induces the original statistic $\tau_{n}$ to satisfy
\begin{equation}
\text{$\tau_{n}|X_{n}$}\overset{w}{\rightarrow}_{w}\left. M^{-1/2}
(\omega_{\varepsilon|\eta}^{1/2}\xi_{1}+(1-\omega_{\varepsilon|\eta})^{1/2}
\xi_{2})\right\vert (M,\xi_{2})\text{,}\label{eq:xit}
\end{equation}
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{eq:xit}) contains more
randomness (through $\xi_{2}$) than the bootstrap limit in eq. (\ref{eq blim}
), thus resulting in a random limit for the distribution of the bootstrap
\emph{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{sec intro to on average bs validity}.
\begin{remark}
Although not valid conditionally on the regressor $X_{n}$, in the previous
example the bootstrap may be valid conditionally on a non-trivial function of
the regressor. See, in particular, Section \ref{subsec:id} and Remark
\ref{Remark 3.45} therein.$\hfill\square$
\end{remark}
\section{Main results}
\label{sec g}
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.
\subsection{Definitions}
\label{sec 3 definitions}
The following definition employs the bootstrap \emph{p}-value as a summary
indicator of the accuracy of bootstrap inferences (see also Remark
\ref{Remark 3.4} below). The original and the bootstrap statistic are denoted
by $\tau_{n}$ and $\tau_{n}^{\ast}$, respectively.
\begin{definition}
\label{def def}Let $\tau_{n}:=\tau_{n}(D_{n})$ and $\tau_{n}^{\ast}:=\tau
_{n}^{\ast}(D_{n},W_{n}^{\ast})$, $n\in\mathbb{N}$, where $D_{n}$ denotes the
data whereas $W_{n}^{\ast}$ are auxiliary variates defined jointly with
$D_{n}$ on a possibly extended probability space. Let $p_{n}^{\ast}:=P\left(
\left. \tau_{n}^{\ast}\leq\tau_{n}\right\vert D_{n}\right) $ be the
bootstrap p-value.
We say that the bootstrap based on $\tau_{n}$ and $\tau_{n}^{\ast}$ is valid
unconditionally if $p_{n}^{\ast}$ is asymptotically $U(0,1)$ distributed:
\begin{equation}
P\left( p_{n}^{\ast}\leq q\right) \rightarrow q,\text{ }q\in
(0,1),\label{eq BS unconditionally valid}
\end{equation}
where $P(\cdot)$ denotes probability w.r.t. the distribution of $D_{n}$.
Let further $X_{n}$ be a random element defined on the probability space of
$D_{n}$ and $W_{n}^{\ast}$. We say that the bootstrap based on $\tau_{n}$ and
$\tau_{n}^{\ast}$ is valid \textit{conditionally on} $X_{n}$ if $p_{n}^{\ast}$
is asymptotically $U(0,1)$ distributed conditionally on $X_{n}:$
\begin{equation}
P\left( \left. p_{n}^{\ast}\leq q\right\vert X_{n}\right) \overset
{p}{\rightarrow}q\text{, }q\in(0,1),\label{eq BS conditionally valid}
\end{equation}
where $P(\cdot|X_{n})$ is determined up to a.s. equivalence by the
distribution of $(D_{n},X_{n})$.
\end{definition}
\begin{remark}
\label{Remark 3.3}Bootstrap validity conditionally on some $X_{n}$ implies
unconditional validity, by the dominated convergence theorem. In applications,
therefore, the discussion of conditional validity may represent an
intermediate step to assess unconditional validity.
\end{remark}
\begin{remark}
\label{Remark 3.4 - part about right sided tests}The validity properties in
Definition \ref{def def} ensure correct asymptotic null rejection probability,
unconditionally or conditionally on some $X_{n}$, for bootstrap hypothesis
tests which reject the null when the bootstrap \emph{p}-value $p_{n}^{\ast}$
does not exceed a chosen nominal level, say $\alpha\in(0,1)$. If $P\left(
\tau_{n}^{\ast}\leq\cdot|D_{n}\right) $ converges weakly in
$\mathscr{D}(\mathbb{R})$ to a sample-path continuous random cdf, then correct
asymptotic null rejection probability is ensured also for bootstrap tests
rejecting the null hypothesis when $\tilde{p}_{n}^{\ast}:=P\left( \left.
\tau_{n}^{\ast}\geq\tau_{n}\right\vert D_{n}\right) \leq\alpha$ (for
applications, see Sections \ref{sec par on the boundary} and
\ref{Section on BS tests for parameter constancy}).
\end{remark}
\begin{remark}
\label{Remark 3.4} Validity as in Definition \ref{def def} has also
implications on the properties of bootstrap (percentile) confidence sets.
Suppose, for instance, that $T_{n}$ is an estimator of a population (scalar)
parameter, whose true value is denoted by $\theta_{0}$, and assume for
simplicity that $\tau_{n}$ is of the form $\tau_{n}=\rho(n)(T_{n}-\theta_{0}
)$, where $\rho(n)$ is a normalizing factor such that $\tau_{n}$ has a
non-degenerate limiting distribution (see Horowitz, 2001, p.3174). Its
bootstrap analog is denoted by $\tau_{n}^{\ast}$, and we assume that the
bootstrap is valid in the unconditional sense of
(\ref{eq BS unconditionally valid}). Interest is in constructing a right-sided
confidence interval for $\theta_{0}$, with (asymptotic) coverage $1-\alpha
\in\left( 0,1\right) $, using a simple bootstrap percentile method. With
$F_{n}^{\ast}(x):=P\left( \tau_{n}^{\ast}\leq\cdot|D_{n}\right) $, let
$q_{n}^{\ast}\left( 1-\alpha\right) :=\inf\{x\in\mathbb{R}:F_{n}^{\ast
}(x)\geq1-\alpha\}$ be the $(1-\alpha)$ quantile of the bootstrap distribution
$F_{n}^{\ast}$. Then, it is straightforward to show that, if $F_{n}^{\ast}$
converges weakly to a sample-path continuous random cdf, then
\[
P\left( \tau_{n}\leq q_{n}^{\ast}\left( 1-\alpha\right) \right) =P\left(
p_{n}^{\ast}\leq1-\alpha\right) +o\left( 1\right) \rightarrow1-\alpha
\]
This implies that a confidence interval of the form $[T_{n}-\rho(n)^{-1}
q_{n}^{\ast}\left( 1-\alpha\right) ,+\infty)$ has (unconditional) asymptotic
coverage probability of $1-\alpha$. If the bootstrap is valid conditionally on
some $X_{n}$, as in (\ref{eq BS conditionally valid}), then the (asymptotic)
coverage is $1-\alpha$ also conditionally on this $X_{n}$.$\hfill\square$
\end{remark}
Our main results make extensive use of \emph{joint }weak convergence in
distribution. Should the related notation not be self-explanatory, we refer
the reader to Appendix A for the formal definitions.
\subsection{Unconditional bootstrap validity}
\label{sec dis}\label{sec gen on unc val}
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{th2} do not
require a conditional analysis of $\tau_{n}$, in contrast to applications of
Theorem \ref{p2 copy(2)}.
\begin{theorem}
\label{th2}Let there exist a rv $\tau$ and a random element $X$, both defined
on the same probability space, such that $\left( \tau_{n},F_{n}^{\ast
}\right) \overset{w}{\rightarrow}\left( \tau,F\right) $ in
$\mathscr{\mathbb{R}}\times\mathscr{D}(\mathbb{R})$ for $F_{n}^{\ast
}(u):=P(\tau_{n}^{\ast}\leq u|D_{n})$ and $\text{\text{$F(u):=P(\tau\leq
u|X)$}, $u\in\mathbb{R}$.}$ If the (possibly)\textbf{\ }random cdf $F$ is
sample-path continuous, then the bootstrap based on $\tau_{n}$ and $\tau
_{n}^{\ast}$ is valid unconditionally.
\end{theorem}
\noindent Some remarks are in order.
\begin{remark}
\label{Remark 3.7} A trivial special case of Theorem \ref{th2} is obtained for
independent $\tau$ and $X$. In this case the bootstrap distribution of
$\tau_{n}^{\ast}$ estimates consistently the limiting unconditional
distribution of $\tau_{n}$ and the bootstrap is valid in the usual sense.
\end{remark}
\begin{remark}
\label{Remark 3.9}An important special case of Theorem \ref{th2} involves
stable convergence of the original statistic $\tau_{n}$ (see H\"{a}usler and
Luschgy, 2015, p.33, for a definition). With the notation of Theorem
\ref{th2}, let the data $D_{n}$ and the random element $X$ be defined on the
same probability space, whereas the rv $\tau$ be defined on an extension of
this probability space. Assume that $\tau_{n}\rightarrow\tau$ stably and
$F_{n}^{\ast}\overset{p}{\rightarrow}F$. Then $(\tau_{n},F_{n}^{\ast}
)\overset{w}{\rightarrow}(\tau,F)$ by Theorem 3.7(b) of H\"{a}usler and
Luschgy (2015). For instance, in the statistical literature on integrated
volatility, a result of the form $\tau_{n}\rightarrow\tau$ stably is contained
in Theorem 3.1 of Jacod, Mykland, Podolskij and Vetter (2009) for $\tau_{n}$
defined as a $t$-type statistic for integrated volatility, whereas the
corresponding $F_{n}^{\ast}\overset{p}{\rightarrow}F$ result is established in
Theorem 3.1 of Hounyo, Gon\c{c}alves and Meddahi (2017) for a combined wild
and blocks-of-blocks bootstrap introduced in the latter paper.
\end{remark}
\begin{remark}
\label{Remark conj}More generally, if $\tau_{n}^{\ast}\overset{w^{\ast}
}{\rightarrow}_{w}\tau|X$ and $\left( \tau_{n}^{\ast},\tau_{n},X_{n}\right)
\overset{w}{\rightarrow}\left( \tau^{\ast},\tau,X\right) $ with $D_{n}
$-measurable $X_{n}$ ($n\in\mathbb{N}$), then the joint convergence
$((\tau_{n}^{\ast}|D_{n}),\tau_{n},X_{n})\overset{w}{\rightarrow}_{w}
((\tau^{\ast}|X),\tau,X)$ follows (see Lemma \ref{le crpr}(b) in Appendix
\ref{sec itere}). If $\tau^{\ast}|X\overset{d}{=}\tau|X$ and $F$ is
sample-path continuous, then $\left( \tau_{n},F_{n}^{\ast}\right)
\overset{w}{\rightarrow}\left( \tau,F\right) $ by Lemma \ref{le kal}(b) in
Appendix \ref{sec itere}.
\end{remark}
\begin{remark}
\label{Remark redet}Alternatively, the convergence $\left( \tau_{n}
,F_{n}^{\ast}\right) \overset{w}{\rightarrow}\left( \tau,F\right) $ could
be obtained from (a) the convergence $(\tau_{n},X_{n})\overset{w}{\rightarrow
}(\tau,X)$ for some $D_{n}$-measurable random elements $X_{n}$ of the space of
$X$, and (b) the implication (were it to hold) from the strong version
$(\tau_{n},X_{n})\overset{a.s.}{\rightarrow}(\tau,X)$ to $\tau_{n}^{\ast
}\overset{w^{\ast}}{\rightarrow}_{p}\tau|X$. The idea is to choose $X_{n}$
such that $\tau_{n}^{\ast}$ depends on the data essentially through $X_{n}$.
Applications of Theorem \ref{th2} along these lines could proceed in two
steps: (i) prove that $(\tau_{n},X_{n})\overset{w}{\rightarrow}(\tau,X)$; (ii)
consider, by extended Skorokhod coupling (Corollary 5.12 of Kallenberg, 1997),
a representation of $D_{n}$ and $(\tau,X)$ such that, with an abuse of
notation, $(\tau_{n},X_{n})\overset{a.s.}{\rightarrow}(\tau,X)$ and, on a
product extension of the Skorokhod-representation space, prove that $\tau
_{n}^{\ast}\overset{w^{\ast}}{\rightarrow}_{p}\tau|X$. The latter conditional
assertion, due to the product structure of the probability space, reduces to a
collection of unconditional assertions by fixing the outcomes in the
factor-space of the data. It then holds that $\left( \tau_{n},F_{n}^{\ast
}\right) \overset{p}{\rightarrow}\left( \tau,F\right) $ on the
Skorokhod-representation space, whereas on a general probability space
$\left( \tau_{n},F_{n}^{\ast}\right) \overset{w}{\rightarrow}\left(
\tau,F\right) $. We proceed like this in the applications of Section
\ref{sec example K/S} (eq. (\ref{eq tri})) and Section
\ref{Section on BS tests for parameter constancy} (Theorem \ref{th fb} under
Assumption \textsc{$\mathcal{H}$}).$\hfill\square$
\end{remark}
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 $X_{n}$.
\begin{theorem}
\label{p2 copy(2)}With the notation of Definition \ref{def def}, let $X_{n}$
be $D_{n}$-measurable ($n\in\mathbb{N}$). Let it hold that
\begin{equation}
\left( P\left( \left. \tau_{n}\leq\cdot\right\vert X_{n}\right)
,\,P\left( \left. \tau_{n}^{\ast}\leq\cdot\right\vert D_{n}\right) \right)
\overset{w}{\rightarrow}\left( F,F^{\ast}\right) \label{eq joint conv}
\end{equation}
in $\mathscr{D}\left( \mathbb{R}\right) \times\mathscr{D}\left(
\mathbb{R}\right) $, where $F$ and $F^{\ast}$ are sample-path continuous
random cdf's, and let
\medskip
\noindent($\dagger$) there exist random elements $X^{\prime},X_{n}^{\prime}$
such that $F^{\ast}$ is $X^{\prime}$-measurable, $X_{n}^{\prime}$ are $X_{n}
$-measurable and $X_{n}^{\prime}\overset{w}{\rightarrow}X^{\prime}$ jointly
with (\ref{eq joint conv}).
\medskip
\noindent Then, if $E\{F(\cdot)|F^{\ast}\}=F^{\ast}(\cdot)$, the bootstrap
based on $\tau_{n}$ and $\tau_{n}^{\ast}$ is valid unconditionally.
\end{theorem}
\begin{remark}
\label{Remark exp}Condition ($\dagger$) of Theorem \ref{p2 copy(2)} implies
that $\tau_{n}^{\ast}$ depends on the data $D_{n}$ approximately through
$X_{n}$ alone, for under this condition $P\left( \left. \tau_{n}^{\ast}
\leq\cdot\right\vert X_{n}\right) \,$and $P\left( \left. \tau_{n}^{\ast
}\leq\cdot\right\vert D_{n}\right) $ are both close to $P\left( \left.
\tau_{n}^{\ast}\leq\cdot\right\vert X_{n}^{\prime}\right) $. Condition
($\dagger$) is trivially satisfied in the case $F=F^{\ast}$ with the choice
$X_{n}^{\prime}=P(\tau_{n}\leq\cdot|X_{n})$. It is also satisfied if $\tau
_{n}^{\ast}=\tilde{\tau}_{n}^{\ast}+o_{p}(1)$ for some $\tilde{\tau}_{n}
^{\ast}$ which is a measurable transformation of $X_{n}$ and $W_{n}^{\ast}$,
w.r.t. the probability measure on the space where $D_{n} $ and $W_{n}^{\ast}$
are jointly defined. In this case, $X_{n}^{\prime}=P(\tilde{\tau}_{n}^{\ast
}\leq\cdot|X_{n})$ satisfies condition ($\dagger$);\ see Appendix
\ref{sec prg}. An example of a pair $\tau_{n}^{\ast}$, $\tilde{\tau}_{n}
^{\ast}$ is given in eq. (\ref{ur zvez}) in Section \ref{sec example K/S}
.$\hfill\square$
\end{remark}
Convergence (\ref{eq joint conv}) in Theorem \ref{p2 copy(2)}\textbf{\ }could
be deduced from the weak convergence of the conditional distributions of
$\tau_{n}$ and $\tau_{n}^{\ast}$, as in the next corollary.
\begin{corollary}
\label{p2 part 2}Let $D_{n}$ and $X_{n}$ ($n\in\mathbb{N}$) be as in Theorem
\ref{p2 copy(2)}. Let the rv $\tau$ and the random elements $X$, $X^{\prime}$
be defined on a single probability space and
\begin{equation}
(\tau_{n}|X_{n},\tau_{n}^{\ast}|D_{n})\overset{w}{\rightarrow}_{w}(\tau
|X,\tau|X^{\prime})\label{eq coroll joint conv}
\end{equation}
in the sense of eq. (\ref{eq:wcrm}). Let further $F\left( u\right)
:=P(\tau\leq u|X)$ and $F^{\ast}\left( u\right) :=P(\tau\leq u|X^{\prime})$,
$u\in\mathbb{R}$, define sample-path continuous random cdf's. Then convergence
(\ref{eq joint conv}) holds. Moreover, the bootstrap based on $\tau_{n}$ and
$\tau_{n}^{\ast}$ is valid unconditionally provided that one of the following
extra conditions holds:
\medskip\noindent(a) $X^{\prime}=X$;
\medskip\noindent(b) $X=(X^{\prime},X^{\prime\prime})$ and $X_{n}^{\prime
}\overset{w}{\rightarrow}X^{\prime}$ jointly with (\ref{eq coroll joint conv})
for some $X_{n}$-measurable random elements $X_{n}^{\prime}$.
\end{corollary}
\begin{remark}
\label{Remark 3.6}An instance of (\ref{eq joint conv}) with $F\neq F^{\ast}$
is implied by the setup of Section
\ref{sec uncond validity W/O conditional validity}. There
(\ref{eq coroll joint conv}) holds with $\tau:=M^{-1/2}(\omega_{\varepsilon
|\eta}^{1/2}\xi_{1}+(1-\omega_{\varepsilon|\eta})^{1/2}\xi_{2})$ and
$X=(X^{\prime},X^{\prime\prime})=(M,(1-\omega_{\varepsilon|\eta})^{1/2}\xi
_{2})$. Moreover, (\ref{eq coroll joint conv}) is joint with the convergence
$X_{n}^{\prime}\overset{w}{\rightarrow}X^{\prime}$ for $X_{n}^{\prime}
=n^{-2}M_{n}$ (see Appendix \ref{sec prg}). Hence, Corollary \ref{p2 part 2}
(b) implies that the bootstrap is unconditionally valid, as was already
concluded in Section \ref{sec 2.2.1}.
\end{remark}
\begin{remark}
\label{Re renext}Convergence (\ref{eq coroll joint conv}) could be proved by
replacing in Remark \ref{Remark redet} the convergence $(\tau_{n}
,X_{n})\overset{}{\rightarrow}(\tau,X)$ (weakly and a.s.) by $((\tau_{n}
|X_{n}),X_{n}^{\prime})\rightarrow((\tau|X),X^{\prime})$ (weakly in
distribution and weakly a.s.) Other ways of proving
(\ref{eq coroll joint conv}), that could be relevant if conditional bootstrap
validity is of interest, are discussed in the next section.$\hfill\square$
\end{remark}
\subsection{Conditional bootstrap validity}
\label{subsec:id}Theorem \ref{p2} below states the asymptotic behavior of the
bootstrap \emph{p}-value conditional on an $X_{n}$ chosen to satisfy condition
($\dagger$) of Theorem \ref{p2 copy(2)}. 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{c1}(b) we provide a
result for validity conditional on a transformation of it.
\begin{theorem}
\label{p2}Under the conditions of Theorem \ref{p2 copy(2)}, the bootstrap
p-value $p_{n}^{\ast}$ satisfies
\begin{equation}
P\left( \left. p_{n}^{\ast}\leq q\right\vert X_{n}\right) \overset
{w}{\rightarrow}F(F^{\ast-1}(q))\label{eq:pis}
\end{equation}
for almost all $q\in(0,1)$, and the bootstrap based on $\tau_{n}$ and
$\tau_{n}^{\ast}$ is valid conditionally on $X_{n}$ if and only if $F=F^{\ast
}$ such that
\begin{equation}
\sup_{u\in\mathbb{R}}\left\vert P\left( \left. \tau_{n}\leq u\right\vert
X_{n}\right) -P\left( \left. \tau_{n}^{\ast}\leq u\right\vert D_{n}\right)
\right\vert \overset{p}{\rightarrow}0.\label{eq s}
\end{equation}
\end{theorem}
\begin{remark}
Under (\ref{eq s}), the bootstrap distribution of $\tau_{n}^{\ast}$
consistently estimates the limit of the conditional distribution of $\tau_{n}$
given $X_{n}$. Although under condition ($\dagger$) the proximity of $P\left(
\left. \tau_{n}\leq\cdot\right\vert X_{n}\right) $\ and $P\left( \left.
\tau_{n}^{\ast}\leq\cdot\right\vert D_{n}\right) $ is necessary for bootstrap
validity conditional on $X_{n}$, no such proximity is necessary for
conditional validity in the general case. In fact, validity conditional on
some $X_{n}$ implies validity conditional on any measurable transformation
$X_{n}^{\prime}=\psi_{n}(X_{n})$ and an analogue of (\ref{eq s}) with
$X_{n}^{\prime}$ in place of $X_{n}$ cannot generally hold for all $\psi_{n}$,
unless $F^{\ast}$ is non-random. This is similar to what happens with
unconditional bootstrap validity which, according to Theorem \ref{th2}, may
occur even if $P\left( \tau_{n}\leq\cdot\right) $\ and $P\left( \left.
\tau_{n}^{\ast}\leq\cdot\right\vert D_{n}\right) $ are not close.
$\hfill\square$
\end{remark}
A corollary in the terms of weak convergence in distribution is given next.
\begin{corollary}
\label{c1}Let $D_{n},X_{n}$ ($n\in\mathbb{N}$), $\tau,F, F^{\ast}$ be as in
Corollary \ref{p2 part 2}. Let (\ref{eq coroll joint conv}) hold and $F,
F^{\ast}$ be sample-path continuous random cdf's. Then:
(a) If $X^{\prime}=X$, the bootstrap based on $\tau_{n}$ and $\tau_{n}^{\ast}$
\ is valid conditionally on $X_{n}$ and (\ref{eq s}) holds.
(b) If $X=(X^{\prime},X^{\prime\prime})$, $(X_{n}^{\prime},X_{n}^{\prime
\prime})\overset{w}{\rightarrow}(X^{\prime},X^{\prime\prime})$ jointly with
(\ref{eq coroll joint conv}) for some $X_{n}$-measurable random elements
$(X_{n}^{\prime},X_{n}^{\prime\prime})$, and $X_{n}^{\prime\prime}
|X_{n}^{\prime}\overset{w}{\rightarrow}_{w}X^{\prime\prime}|X^{\prime}$, then
the bootstrap is valid conditionally on $X_{n}^{\prime} $ and (\ref{eq s})
holds with $X_{n}$ replaced by $X_{n}^{\prime}$.
\end{corollary}
Corollary \ref{c1} requires checking the joint convergence in
(\ref{eq coroll joint conv}); see Remark \ref{Re renext}. We provide further
strategies to establish this convergence, useful if interest is in conditional
bootstrap validity, in Remarks \ref{Remark remalt}--\ref{Remark joco} below.
\begin{remark}
\label{Remark 3.5}Consider the linear regression example under the extra
assumptions of Section \ref{sec intro to conditional bs validity} and
set\textbf{\ }$\tau=(\int B_{\eta}^{2})^{-1}\int B_{\eta}dB_{\varepsilon}$$, $
$X=M$. It then follows (by using Theorem 3 of Georgiev \textit{et al}., 2018)
that condition (\ref{eq coroll joint conv}) holds in the form
\begin{equation}
(\tau_{n}|X_{n},\tau_{n}^{\ast}|D_{n})\overset{w}{\rightarrow}_{w}
(\tau|B_{\eta},\tau|B_{\eta})\overset{d}{=}(1,1)N(0,\omega_{\varepsilon}
M^{-1})|M\text{,}\label{eq taun}
\end{equation}
where $X_{n}:=\{x_{t}\}_{t=1}^{n}$; equivalently, (\ref{eq joint conv}) holds
with $F=F^{\ast}=\Phi(\omega_{\varepsilon}^{-1/2}M^{1/2}(\cdot))$. Hence, the
bootstrap is consistent for the limit distribution of $\tau_{n} $ conditional
on the regressor and, by Corollary \ref{c1}(a), the bootstrap is valid
conditionally on the regressor.
\end{remark}
\begin{remark}
\label{Remark remalt}The joint convergence in (\ref{eq coroll joint conv})
would follow from the separate convergence facts $\tau_{n}|X_{n}\overset
{w}{\rightarrow}_{w}\tau|X$, $\tau_{n}^{\ast}\overset{w^{\ast}}{\rightarrow
}_{w}\tau^{\ast}|X^{\prime}$ and $(\tau_{n},\tau_{n}^{\ast},\phi_{n}
(X_{n}),\psi_{n}(D_{n}))\overset{w}{\rightarrow}\left( \tau,\tau^{\ast
},X,X^{\prime}\right) $ for some measurable $\phi_{n},\psi_{n}$, provided
that $\tau|X^{\prime}\overset{d}{=}\tau^{\ast}|X^{\prime}$; see Appendix
\ref{sec prg}. We use this approach in Section \ref{sec example K/S}, point
(ii). The convergence $\tau_{n}|X_{n}\overset{w}{\rightarrow}_{w}\tau|X$ is
the new ingredient compared to Remark \ref{Remark conj}.
\end{remark}
\begin{remark}
\label{Remark joco}Convergence (\ref{eq coroll joint conv}) would also follow
from $(\tau_{n},\phi_{n}(X_{n}),\psi_{n}(D_{n}))\overset{w}{\rightarrow}
(\tau,X,$ $X^{\prime})$ and $\tau_{n}|X_{n}\overset{w}{\rightarrow}_{w}\tau|X$
together with the implication (were it to hold) from $\psi_{n}(D_{n}
)\overset{a.s.}{\rightarrow}X^{\prime}$ to $\tau_{n}^{\ast}\overset{w^{\ast}
}{\rightarrow}_{p}\tau^{\ast}|X^{\prime}$, with $\tau|X^{\prime}\overset{d}
{=}\tau^{\ast}|X^{\prime}$. A possible implementation strategy is: (i) prove
that $\tau_{n}|X_{n}\overset{w}{\rightarrow}_{w}\tau|X$ and $(\tau_{n}
,\phi_{n}(X_{n}),\psi_{n}(D_{n}))\overset{w}{\rightarrow}\left(
\tau,X,X^{\prime}\right) $; (ii) consider a Skorokhod representation of
$D_{n}$ and $\left( \tau,X,X^{\prime}\right) $ such that, maintaining the
notation, $(\tau_{n},\phi_{n}(X_{n}),\psi_{n}(D_{n}))\overset{a.s.}
{\rightarrow}\left( \tau,X,X^{\prime}\right) \ $and, as a result, $\tau
_{n}|X_{n}\overset{w}{\rightarrow}_{w}\tau|X$ strengthens to $\tau_{n}
|X_{n}\overset{w}{\rightarrow}_{p}\tau|X$ (see Lemma \ref{le crpr} in Appendix
\ref{sec itere}); (iii) redefine the bootstrap variates $W_{n}^{\ast}$ on a
product extension of the Skorokhod-representation space and prove there that
$\tau_{n}^{\ast}\overset{w^{\ast}}{\rightarrow}_{p}\tau^{\ast}|X^{\prime}$.
Then (\ref{eq coroll joint conv}) holds on a general probability space. We
proceed like this in the proof of Theorem \ref{th fb} under Assumption
$\mathcal{C}$. The convergence $\tau_{n}|X_{n}\overset{w}{\rightarrow}_{w}
\tau|X$ is the extra ingredient compared to Remark \ref{Remark redet}. Notice
also that if $\phi_{n}(X_{n})=(X_{n}^{\prime},X_{n}^{\prime\prime})$ and
$\psi_{n}(D_{n})=X_{n}^{\prime}$, then the convergence $(X_{n}^{\prime}
,X_{n}^{\prime\prime})\overset{w}{\rightarrow}(X^{\prime},X^{\prime\prime})$
in Corollary \ref{c1}(b) would be joint with (\ref{eq coroll joint conv}).
\end{remark}
\begin{remark}
\label{Remark 3.45} In the setup of Section
\ref{sec uncond validity W/O conditional validity},
(\ref{eq coroll joint conv}) holds with $\tau$ and $X=(X^{\prime}
,X^{\prime\prime})\ $ given in Remark \ref{Remark 3.6}. Moreover,
(\ref{eq coroll joint conv}) is joint with the convergence $(X_{n}^{\prime
},X_{n}^{\prime\prime})\overset{w}{\rightarrow}(X^{\prime},X^{\prime\prime})$
for $X_{n}^{\prime}=n^{-2}M_{n}$ and $X_{n}^{\prime\prime}=M_{n}^{-1/2}
\sum_{t=1}^{n}x_{t}E(\varepsilon_{t}|\eta_{t})$ (see Appendix \ref{sec prg}).
By Corollary \ref{c1}(b), the bootstrap would be valid conditionally on
$M_{n}$ if it additionally holds that $X_{n}^{\prime\prime}|M_{n}\overset
{w}{\rightarrow}_{w}(1-\omega_{\varepsilon|\eta})^{1/2}\xi_{2}|M \overset
{d}{=}N(0,1-\omega_{\varepsilon|\eta})$ .$\hfill\square$
\end{remark}
\subsection{Local power of bootstrap tests}
\label{sec lop}
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{th2}.
\begin{theorem}
\label{th2a}Let there exist rv's $\tau,\tau^{\ast}$ and a random element $X$,
the three defined on the same probability space, such that $\left( \tau
_{n},F_{n}^{\ast}\right) \overset{w}{\rightarrow}\left( \tau,F^{\ast
}\right) $ in $\mathscr{\mathbb{R}}\times\mathscr{D}(\mathbb{R})$ for
$F_{n}^{\ast}(u):=P(\tau_{n}^{\ast}\leq u|D_{n})$ and $\text{\text{$F^{\ast
}(u):=P(\tau^{\ast}\leq u|X)$}, $u\in\mathbb{R}$.}$ If $F^{\ast}$ is
sample-path continuous, then the bootstrap p-value $p_{n}^{\ast}$ satisfies
$P(p_{n}^{\ast}\leq q)\rightarrow E\{F(F^{\ast-1}(q))\}$, $q\in(0,1)$ with
$F\left( \cdot\right) :=P\left( \tau\leq\cdot|X\right) $.
\end{theorem}
To illustrate, with $y_{t}$, $x_{t}$ and $\varepsilon_{t}$ as in Section
\ref{sec intro to conditional bs validity}, 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{eq:lm}). The original test statistic is
$\tau_{n}=n\hat{\beta}$.
Without recourse to the explicit expression in
(\ref{eq lin model asy bootstrap cdf})\ for the bootstrap \emph{p}-value
$p_{n}^{\ast}$ in terms of the Gaussian cdf, which in many applications may
have no analogue, we can instead\textbf{\ }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{le kal} in Appendix \ref{sec itere}, we can conclude that the
conditions of Theorem \ref{th2a} 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
\begin{equation}
P(p_{n}^{\ast}\leq q)\rightarrow E\{F(F^{\ast-1}(q))\}=E\{\Phi(\Phi
^{-1}(q)-\omega_{\varepsilon}^{-1/2}M^{1/2}b)\}\text{.}\label{eq uncond loc}
\end{equation}
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{eq uncond loc}) can also be derived
through a conditioning argument. This can be done using the results in Section
\ref{subsec:id} 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{eq taun}), which implies (\ref{eq joint conv}) with $F$ and
$F^{\ast}$ as defined above. Hence, by Theorem \ref{p2}, (\ref{eq:pis}) holds
and
\begin{equation}
P\left( \left. p_{n}^{\ast}\leq q\right\vert X_{n}\right) \overset
{w}{\rightarrow}\Phi(\omega_{\varepsilon}^{-1/2}M^{1/2}(F^{\ast}{}
^{-1}(q)-b))=\Phi(\Phi^{-1}(q)-\omega_{\varepsilon}^{-1/2}M^{1/2}
b)\label{eq conditional local power function}
\end{equation}
for $q\in(0,1)$. The latter expression is the (random) asymptotic local power,
conditional on\textbf{\ }$X_{n}$, of the one-sided, $q$-level test, bootstrap
test. By averaging the rhs of (\ref{eq conditional local power function}) over
$M$, the unconditional power function in (\ref{eq uncond loc}) follows.
\section{Applications}
\label{Section on Applications}
\subsection{A permutation CUSUM test under infinite variance}
\label{sec example CUSUM} 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 \emph{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{def def}, 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 \textit{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 \textit{et al}. (1997) in order to obtain the
conditional convergence of $\tau_{n}^{*}.$}
\smallskip{}
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{Remark on exact conditional inference}, 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
\emph{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{eq:wcrm})):
\begin{equation}
\left( \tau_{n}|X_{n},\tau_{n}^{\ast}|D_{n}\right) ^{\prime}\overset
{w}{\rightarrow}_{w}(\rho_{\alpha}(S)|S,\rho_{\alpha}(S)|S)\text{
.}\label{eq:cus}
\end{equation}
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{eq:cus}), 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
\begin{equation}
\left( \hat{\tau}_{n}|X_{n},\hat{\tau}_{n}^{\ast}|D_{n}\right) \overset
{w}{\rightarrow}_{w}(\rho_{\alpha}(S)|S,\rho_{\alpha}(S)|S)\label{eq:cusr}
\end{equation}
for $X_{n}:=\{\varepsilon_{(t)}\}_{t=1}^{n}$ again. It follows that:
\smallskip{}
(i) The permutation residual-based test is valid conditionally on $X_{n}$, by
Corollary \ref{c1}(a) with condition (\ref{eq coroll joint conv}) taking the
form (\ref{eq:cusr}).
\smallskip
(ii) This test is valid unconditionally, as a results of either the validity
conditional on $X_{n}$, or by Corollary \ref{p2 part 2}.
\subsection{A parametric bootstrap goodness-of-fit test}
\label{sec example K/S}
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{eq:lm}), 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.
\subsubsection{Set up and a random limit bootstrap measure}
Let the null hypothesis of interest, say $\mathsf{H}_{0}$, be that the
standardized errors $\omega_{\varepsilon}^{-1/2}\varepsilon_{t}$ in
(\ref{eq:lm}) 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 (\textit{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
\begin{equation}
\tilde{\tau}_{n}^{\ast}:=\sup_{u\in\lbrack0,1]}\left\vert n^{-1/2}\sum
_{t=1}^{n}(\mathbb{I}_{\{\varepsilon_{t}^{\ast}\leq q(u)\}}-u)+\Phi^{\prime
}(q(u))\hat{\beta}^{\ast}n^{-1/2}\sum_{t=1}^{n}x_{t}\right\vert \text{,}
\label{ur zvez}
\end{equation}
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{sec Proofs of the results in Section applications}
that, more strongly,
\begin{equation}
(n^{1/2}W_{n}^{\ast},\,n^{(\alpha+1)/2}\hat{\beta}^{\ast},\,n^{-\alpha/2-1}
\xi_{n})\overset{w^{\ast}}{\rightarrow}_{w}\left. (W,M^{-1/2}b,\xi
)\right\vert (M,\xi)\label{eq:jbb}
\end{equation}
on $\mathscr{D}\times\mathbb{R}^{2},$ where $(W,b$$)$ is a pair of a standard
Brownian bridge and a standard Gaussian rv individually independent of $X$
(and thus, of $M,\xi$), but with Gaussian joint conditional (on $X$)
distributions having covariance $\operatorname*{Cov}(W(u),b|X)=M^{-1/2}\xi
\psi(u),$ $u\in\lbrack0,1]$. Combining the expansion of $\tau_{n}^{\ast}$,
(\ref{ur zvez}) and (\ref{eq:jbb}) with the extended CMT (Theorem \ref{th cmt}
in Appendix \ref{sec itere}) yields
\begin{equation}
\tau_{n}^{\ast}\overset{w^{\ast}}{\rightarrow}_{w}\{\sup_{u\in\lbrack
0,1]}|W(u)+\Phi^{\prime}(q(u))M^{-1/2}b\xi|\}\big\vert(M,\xi)\overset{d}
{=}\tau\left\vert (M,\xi)\right. \text{,}\label{eq:tst}
\end{equation}
where $\tau:=\sup_{u\in\lbrack0,1]}|\tilde{W}(u)|$ for a process $\tilde{W}$
which conditionally on $X$ (and thus, on $M,\xi$), is a zero-mean Gaussian
process with $\tilde{W}(0)=\tilde{W}(1)=0$ a.s. and conditional covariance
function $K(u,v)=u(1-v)-M^{-1}\xi^{2}\psi(u)\psi(v)$ for $0\leq u\leq v\leq1$.
In summary, the limit bootstrap distribution is random because the latter
conditional covariance is random whenever $M$ or $\xi$ are such.
\subsubsection{Bootstrap validity}
We now discuss in what sense $\tau_{n}^{\ast}$ can provide a distributional
approximation of $\tau_{n}$ and whether the bootstrap can be valid in the
sense of Definition \ref{def def}.
\smallskip{}
\noindent(i)\ Under $\mathsf{H}_{0}$ that $\varepsilon_{t}\sim\text{i.i.d.}
$$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{th2}. More precisely, it is
proved in Appendix \ref{sec Proofs of the results in Section applications}
that
\begin{equation}
\left( \tau_{n},F_{n}^{\ast}\right) \overset{w}{\rightarrow}\left(
\tau,F\right) \text{, }F_{n}^{\ast}(\cdot):=P^{\ast}(\tau_{n}^{\ast}\leq
\cdot)\text{, }F(\cdot):=P(\tau\leq\cdot|M,\xi).\label{eq tri}
\end{equation}
As $F$ is sample-path continuous (e.g., by Proposition 3.2 of Linde, 1989,
applied conditionally on $M,\xi$), Theorem \ref{th2} guarantees the
unconditional validity of the bootstrap.\smallskip{}
\noindent(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{Remark remalt}, 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{eq tri}),
see Appendix \ref{sec Proofs of the results in Section applications}) are
sufficient for eq. (\ref{eq coroll joint conv}) 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{c1}(a).
\subsection{Parameters on the boundary in predictive regression}
\label{sec par on the boundary}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.
\subsubsection{General setup}
\label{sec par on the boundary setup}
Consider the predictive regression
\begin{equation}
y_{t}=\theta_{1}+\theta_{2}x_{n,t-1}+\varepsilon_{t}\text{ (}t=1,...,n;\text{
}n=1,2,...\text{)}\label{eq:prr}
\end{equation}
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 $\theta_{0}$:
\begin{description}
\item[$\mathscr{G}{}_{1}$.] $\mathsf{H}_{0}$ is the hypothesis that
$\theta_{0}$ belongs to the boundary: $\mathsf{H}_{0}:g(\theta_{0})=0$;
\item[$\mathscr{G}{}_{2}$.] $\mathsf{H}_{0}$ is a simple null hypothesis on
the boundary: $\mathsf{H}_{0}:\theta_{0}=\bar{\theta}$, $g(\bar{\theta})=0$;
\item[$\mathscr{G}{}_{3}$.] $\mathsf{H}_{0}:h(\theta_{0})=0$, where
$\{\theta\in\mathbb{R}^{2}:h\left( \theta\right) =0\}$ is not a subset of
the boundary $\partial\Theta$, but meets $\partial\Theta$ at a singleton set.
\end{description}
\noindent 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
\begin{equation}
y_{t}=\theta_{1}+\theta_{2}x_{n,t-1}+\delta\Delta x_{n,t}+e_{t}\label{eq:pr}
\end{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),
\begin{equation}
n^{1/2}(\hat{\theta}-\theta_{0})\overset{w}{\rightarrow}\ell(\theta_{0}
)=\ell:=\underset{\lambda\in\Lambda:=\{\lambda\in\mathbb{R}^{2}:\dot
{g}^{\prime}\lambda\geq0\}}{\arg\min}||\lambda-M^{-1/2}\xi||_{M}
\label{eq asy distribution}
\end{equation}
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
\begin{equation}
y_{t}^{\ast}=\hat{\theta}_{1}+\hat{\theta}_{2}x_{n,t-1}+\varepsilon_{t}^{\ast
}\text{,}\label{eq BS for PR}
\end{equation}
where $\varepsilon_{t}^{\ast}=\hat{e}_{t}w_{t}^{\ast}$, $t=1,...n$, with
$\hat{e}_{t}$ the residuals of (\ref{eq:pr}) 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:
\begin{equation}
n^{1/2}(\hat{\theta}^{\ast}-\hat{\theta})=n^{1/2}(\tilde{\theta}^{\ast}
-\hat{\theta})+o_{p}(1)\overset{w^{\ast}}{\rightarrow}_{w}\tilde{\ell
}|M\text{, }\label{eq random limit depending on M only}
\end{equation}
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 $n^{1/2}(\hat{\theta}-\theta_{0})$:
\begin{equation}
n^{1/2}(\hat{\theta}^{\ast}-\hat{\theta})\overset{w^{\ast}}{\rightarrow}
_{w}\ell^{\ast}|(M,\ell)\text{, }\ell^{\ast}:=\underset{\lambda\in
\Lambda_{\ell}^{\ast}:=\{\lambda\in\mathbb{R}^{2}:\dot{g}^{\prime}\lambda
\geq-\dot{g}^{\prime}\ell\}}{\arg\min}||\lambda-M^{-1/2}\xi^{\ast}
||_{M},\label{eq asy for standard BS with param on the boundary}
\end{equation}
where $\xi^{\ast}\sim N(0,1)$ is independent of $(M,\ell)$; see Theorem
\ref{Lemma bootstrap with boundary} below. In contrast with the case
$\theta_{0}\in\operatorname*{int}(\Theta)$, the limit in
(\ref{eq asy for standard BS with param on the boundary}) is not a conditional
version of the limit of $n^{1/2}(\hat{\theta}-\theta_{0})$, inasmuch as
$\Lambda_{\ell}^{\ast}$ in
(\ref{eq asy for standard BS with param on the boundary}) is a \emph{random}
half-plane, rather than the original set $\Lambda$ of
(\ref{eq asy distribution}). 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.
\subsubsection{Unconditionally valid bootstrap schemes}
\label{sec par on the boundary main}
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{eq BS for PR}) 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{sec par on the boundary setup} 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.
\par
{}}
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 $\dot{g}^{\ast}:=\frac{\partial}
{\partial\theta^{\prime}} g^{\ast}(\theta_{0})$.
\begin{theorem}
\label{Lemma bootstrap with boundary}Under the assumptions and the notation
introduced above, let a null hypothesis $\mathsf{H_{0}}$ as in
$\mathscr{G}{}_{1}$--$\,\mathscr{G}{}_{3}$ hold. Let also $\xi^{\ast}
|(M,\ell(\theta_{0}))\sim N(0,1)$. Then
\begin{equation}
(n^{1/2}(\hat{\theta}-\theta_{0}),(n^{1/2}(\hat{\theta}^{\ast}-\hat{\theta
})|D_{n}))\overset{w}{\rightarrow}_{w}\left( \ell(\theta_{0}),(\ell^{\ast
}(\theta_{0})|(M,\ell(\theta_{0})))\right) \text{, }
\label{eq asy distrib for general bootstraps}
\end{equation}
where in the case $g^{\ast}(\theta_{0})<g(\theta_{0})$,
\begin{equation}
\ell^{\ast}(\theta_{0})=\tilde{\ell}^{\ast}:=M^{-1/2}\xi^{\ast}\text{ with
}\tilde{\ell}^{\ast}|(M,\ell(\theta_{0}))\overset{d}{=}\tilde{\ell}|M\text{,
}\label{eq semno}
\end{equation}
whereas in the case $g^{\ast}(\theta_{0})=g(\theta_{0})$,
\begin{equation}
\ell^{\ast}(\theta_{0})=\ell^{\ast}:=\underset{\lambda\in\Lambda_{\ell}^{\ast
}}{\arg\min}||\lambda-M^{-1/2}\xi^{\ast}||_{M}\text{, }\Lambda_{\ell}^{\ast
}:=\{\lambda\in\mathbb{R}^{2}:\dot{g}^{\prime}\lambda\geq(\dot{g}^{\ast}
-\dot{g})^{\prime}\ell(\theta_{0})\}\text{. }\label{eq amans}
\end{equation}
\end{theorem}
\noindent The following conclusions could be drawn.
\medskip
\noindent(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{eq asy for standard BS with param on the boundary}) and (\ref{eq amans})
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{eq asy distribution}) and
(\ref{eq amans})). 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 $\kappa>0$.
\medskip
\noindent(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.
\medskip
\noindent(iii) In general, bootstrap unconditional validity can be evaluated
through the following corollary of Theorem \ref{Lemma bootstrap with boundary}.
\begin{corollary}
\label{corollary bootstrap with boundary}Under the assumptions of Theorem
\ref{Lemma bootstrap with boundary}, a necessary and sufficient condition for
the convergence
\begin{equation}
\left( n^{1/2}(\hat{\theta}-\theta_{0}),(n^{1/2}(\hat{\theta}^{\ast}
-\hat{\theta})|D_{n})\right) \overset{w}{\rightarrow}_{w}\left( \ell
(\theta_{0}),(\ell(\theta_{0})|M)\right) \text{ }\label{eq bound ave}
\end{equation}
is that: (i) under $\mathscr{G}{}_{1}$ and $\mathscr{G}{}_{2}$, $g(\theta
_{0})=g^{\ast}(\theta_{0})$ and $\dot{g}=\dot{g}^{\ast}$; (ii) under
$\mathscr{G}{}_{3}$, either $g(\theta_{0})=g^{\ast}(\theta_{0})$ and $\dot
{g}=\dot{g}^{\ast}$, or $g(\theta_{0})>\max\{0,g^{\ast}(\theta_{0})\}$.
Moreover, under (\ref{eq bound ave}) the bootstrap is unconditionally valid
for any pair of statistics $\tau=\phi(n^{1/2}(\hat{\theta}-\theta_{0}
))+o_{p}(1)$ and $\tau^{\ast}=\phi(n^{1/2}(\hat{\theta}^{\ast}-\hat{\theta
}))+o_{p}(1)$, where $\phi$ is a continuous real function such that the cdf of
$\phi(\ell(\theta_{0}))|M$ is continuous.
\end{corollary}
\noindent 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.
\subsubsection{Unconditional validity of one-sided standard bootstrap tests}
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
\emph{large} values of the bootstrap \emph{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{Remark 3.4 - part about right sided tests}). 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{eq asy distrib for general bootstraps}), which reduces to
(\ref{eq asy distribution}) and
(\ref{eq asy for standard BS with param on the boundary}), 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 \emph{p}-values $\tilde{p}_{n}^{\ast}$ are (asymptotically)
uniformly distributed below $\frac{1}{2}$. This follows rigorously from the
next generalization of Theorem \ref{th2} (the proof being analogous), where
conditions for unconditional bootstrap validity restricted to a subset of
nominal testing levels are formulated.
\bigskip
\noindent\textsc{Theorem \ref{th2}}\textbf{$^{\ast}$}. \textit{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 }$p$\textit{-value }$p_{n}^{\ast
}:=\mathit{F_{n}^{\ast}(\tau_{n})}$\textit{\ satisfies
\[
P(p_{n}^{\ast}\leq q)\rightarrow q
\]
for $q$ such that $q\in F(\mathbb{T})$ a.s.}\bigskip
\noindent By Theorem \ref{th2}\textbf{$^{\ast}$} 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
$(0,\frac{1}{2})$.
\subsection{Bootstrap tests of parameter constancy}
\label{Section on BS tests for parameter constancy}
\subsubsection{General set up}
Here we apply the results of Section \ref{sec g} 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:
\begin{equation}
y_{nt}=\beta_{t}^{\prime}x_{nt}+\varepsilon_{nt}\text{ \hspace{1cm}
(}t=1,2,...,n\text{).}\label{eq reg model}
\end{equation}
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).
\bigskip{}
\noindent\textsc{Assumption $\mathcal{H}$}. The following conditions on
$\{x_{nt},\varepsilon_{nt}\}$ hold:
\begin{description}
\item (i) \emph{(mda)}$\ \varepsilon_{nt}$\emph{\ is a martingale difference
array with respect to the current value of }$x_{nt}$\emph{\ and the lagged
values of }$\left( x_{nt},\varepsilon_{nt}\right) $\emph{;}
\item (ii) \emph{(wlln) }$\varepsilon_{nt}^{2}$\emph{\ satisfies the law of
large numbers }$n^{-1}\sum_{t=1}^{\left\lfloor nr\right\rfloor }
\varepsilon_{nt}^{2}\overset{p}{\rightarrow}r(E\varepsilon_{nt}^{2}
)=r\sigma^{2}>0$\emph{$,$ for all }$r\in(0,1]$\emph{;}
\item (iii) \emph{(non-stationarity) in }$\mathscr{D}{}_{m\times m}
\times\mathscr{D}{}_{m\times m}\times\mathscr{D}{}_{m}$\emph{:}
\[
\left( \tfrac{1}{n}\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor }x_{nt}
x_{nt}^{\prime},\tfrac{1}{n\sigma^{2}}\sum_{t=1}^{\left\lfloor n\cdot
\right\rfloor }x_{nt}x_{nt}^{\prime}\varepsilon_{nt}^{2},\tfrac{1}
{n^{1/2}\sigma}\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor }x_{nt}
\varepsilon_{nt}\right) \overset{w}{\rightarrow}(M,V,N),
\]
\emph{where }$M$\emph{\ and }$V$\emph{\ are a.s. continuous and (except at 0)
strictly positive-definite valued processes, whereas }$N$\emph{, conditionally
on }$\{V,M\}$\emph{,\ is a zero-mean Gaussian process with covariance kernel
}$E\{N\left( r_{1}\right) N\left( r_{2}\right) ^{\prime}\}=V\left(
r_{1}\right) $\emph{\ }$(0\leq r_{1}\leq r_{2}\leq1)$\emph{.}
\end{description}
\begin{remark}
\label{Remark 4.1}A special case of Assumption $\mathcal{H}$ is obtained when
the regressors satisfy the weak convergence $x_{n\left\lfloor n\cdot
\right\rfloor }\overset{w}{\rightarrow}U\left( \cdot\right) $ in
$\mathscr{D}{}_{m}$, such that $M\left( \cdot\right) =\int_{0}^{\cdot
}UU^{\prime}$. Under extra conditions (e.g., if $\sup_{n}\sup_{t=1,...,n}
E|E(\varepsilon_{nt}^{2}-\sigma^{2}|\mathcal{F}_{n,t-i})|\rightarrow0$ as
$i\rightarrow\infty$ for some filtrations $\mathcal{F}_{n,t}$, $n\in\mathbb{N}
$, to which $\{\varepsilon_{nt}^{2}$\} is adapted), also $V\left( \cdot\right)
=\int_{0}^{\cdot}UU^{\prime}$ (see Theorem A.1 of Cavaliere and Taylor, 2009).
$\hfill\square$ \medskip{}
\end{remark}
The null asymptotic distribution of $\mathscr{F}{}_{n}$ under Assumption
$\mathcal{H}$ is provided in Hansen (2000, Theorem 2):
\begin{equation}
\mathscr{F}{}_{n}\overset{w}{\rightarrow}\sup_{r\in\lbrack\underline
{r},\overline{r}]}\{\tilde{N}(r)^{\prime}\tilde{M}\left( r\right)
^{-1}\tilde{N}(r)\}\label{eq asy distr of supF}
\end{equation}
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{eq asy distr of supF}) depends on the joint
distribution of the limiting processes $M,N,V$, which is unspecified under
Assumption $\mathcal{H}$, asymptotic inference based on
(\ref{eq asy distr of supF}) is unfeasible. Simulation methods as the
bootstrap can therefore be appealing devices for computing \emph{p}-values
associated with $\mathscr{F}{}_{n}$.
\subsubsection{Bootstrap test and random limit bootstrap distribution}
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
\begin{equation}
y_{t}^{\ast}=\beta^{\ast\prime}x_{nt}+\theta^{\ast\prime}x_{nt}\mathbb{I}
_{\{t\geq\left\lfloor rn\right\rfloor \}}+\text{error}_{nt}^{\ast
},\label{eq br}
\end{equation}
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.
\begin{theorem}
\label{th h}Under Assumption $\mathcal{H}$ and under $\mathsf{H}_{0}$, it
holds that
\begin{equation}
\mathscr{F}{}_{n}^{\ast}\overset{w^{\ast}}{\rightarrow}_{w}\left. \sup
_{r\in\lbrack\underline{r},\overline{r}]}\{\tilde{N}(r)^{\prime}\tilde
{M}\left( r\right) ^{-1}\tilde{N}(r)\}\right\vert (M,V),\label{eq d}
\end{equation}
where $\tilde{M}\left( r\right) $, $\tilde{N}\left( r\right) $ are as in
(\ref{eq asy distr of supF}).
\end{theorem}
\begin{remark}
\label{Remark 4.4}Theorem \ref{th h} establishes that, in general, the weak
limit of the fixed-regressor bootstrap statistic is \emph{random}. In
particular, it is distinct from the limit in eq. (\ref{eq asy distr of supF})
and, as a result, the bootstrap does not estimate consistently the
unconditional limit distribution of the statistic $\mathscr{F}_{n}$ under
$\mathsf{H}_{0}$ (contrary to the claim in Theorem 6 of Hansen, 2000). To
illustrate the limiting randomness, consider the case $M=V$ with a scalar
regressor $x_{nt}\in\mathbb{R}$. By a change of variable (as in Theorem 3 of
Hansen, 2000), convergence (\ref{eq d}) reduces to
\[
\mathscr{F}{}_{n}^{\ast}\overset{w^{\ast}}{\rightarrow}_{w}\left. \sup_{u\in
I(M,\underline{r},\bar{r})}\left\{ \frac{W(u)^{2}}{u(1-u)}\right\}
\right\vert M\text{\ \ \ $\text{for\ \ \ }$}I(M,\underline{r},\bar
{r}):=\left[ \tfrac{M(\underline{r})}{M(1)},\tfrac{M(\bar{r})}{M(1)}\right]
,
\]
where $W$ is a standard Brownian bridge on $[0,1]$, independent of $M$. As the
maximization interval $I(M,\underline{r},\bar{r})$ depends on $M$, so does the
supremum itself.$\hfill\square$ \medskip{}
\end{remark}
\subsubsection{Bootstrap validity}
Although under Assumption $\mathcal{H}$ the bootstrap does not replicate the
asymptotic (unconditional) distribution in (\ref{eq asy distr of supF}),
unconditional bootstrap validity can be established under no further
assumptions than Assumption $\mathcal{H}$, by using the results in Section
\ref{sec gen on unc val}. 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 \textsc{$\mathcal{C}$}.
\bigskip
\noindent\textsc{Assumption $\mathcal{C}$}. \emph{Assumption }
\textsc{\emph{$\mathcal{H}$}}\emph{\ holds and, as random measures on
}$\mathscr{D}{}_{m\times m}\times\mathscr{D}{}_{m\times m}\times
\mathscr{D}{}_{m}$
\[
\left. \left( \tfrac{1}{n}\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor
}x_{nt}x_{nt}^{\prime},\tfrac{1}{n\sigma^{2}}\sum_{t=1}^{\left\lfloor
n\cdot\right\rfloor }x_{nt}x_{nt}^{\prime}\varepsilon_{nt}^{2},\tfrac
{1}{n^{1/2}\sigma}\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor }
x_{nt}\varepsilon_{nt}\right) \right\vert X_{n}\overset{w}{\rightarrow}
_{w}\left( M,V,N\right) |(M,V)
\]
\emph{jointly with the convergence in Assumption }\textsc{\emph{$\mathcal{H}$
}}\emph{(iii).}
\bigskip
The results on the validity of the bootstrap parameter constancy tests are
summarized in the following theorem.
\begin{theorem}
\label{th fb}Let the parameter constancy hypothesis $\mathsf{H}_{0}$ hold for
model (\ref{eq reg model}). Then, under Assumption $\mathcal{H}$, the
bootstrap based on $\tau_{n}=\mathscr{F}{}_{n}$ and $\tau_{n}^{\ast
}=\mathscr{F}{}_{n}^{\ast}$ is unconditionally valid. If Assumption
$\mathcal{C}$ holds, then the bootstrap based on $\mathscr{F}{}_{n}$ and
$\mathscr{F}{}_{n}^{\ast}$ is valid also conditionally on $X_{n}$.
\end{theorem}
\begin{remark}
\label{Remark 4.6}In the proof of Theorem \ref{th fb}, we refer to Theorem
\ref{th2} and Corollary \ref{c1}(a) for establishing respectively
unconditional and conditional bootstrap validity. Notice that Assumption
$\mathcal{C}$ is stronger than Assumption $\mathcal{H}$ due to the fact that
--\thinspace differently from the bootstrap variates $w_{t}^{\ast}$
--\thinspace the errors $\{\varepsilon_{nt}\}$ need not be independent of
$\{x_{nt}\}$. The third DGP of Section
\ref{sec uncond validity W/O conditional validity} could be used to construct
an example, with $x_{nt}:=n^{-1/2}x_{t}$ and $\varepsilon_{nt}:=\varepsilon
_{t}$, where Assumption $\mathcal{H}$(iii) holds but Assumption $\mathcal{C}$
does not.
\end{remark}
\begin{remark}
The meaning of `jointly' in Assumption $\mathcal{C}$ is given in eq.
(\ref{eq:wcrm1}). By Lemma \ref{le crpr}(b), the convergence will be
automatically joint if in $\mathscr{D}{}_{m\times m}$, $n^{-1}\sigma^{-2}
\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor }x_{nt}x_{nt}^{\prime
}(\varepsilon_{nt}^{2}-E(\varepsilon_{nt}^{2}|X_{n}))=o_{p}(1)$, such that
$n^{-1}\sigma^{-2}\sum_{t=1}^{\left\lfloor n\cdot\right\rfloor }x_{nt}
x_{nt}^{\prime}\varepsilon_{nt}^{2}$\emph{\ }is asymptotically equivalent to
an\emph{\ }$X_{n}$-measurable process.$\hfill\square$
\end{remark}
\section{Conclusions}
\label{sec conclusion}
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{Section on Applications}). 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{sec example CUSUM} 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.
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