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\begin{center}
\bigskip
{\Large {}PARAMETERS ON THE BOUNDARY IN PREDICTIVE REGRESSION}\textsc{
{\LARGE {}{* }}}
\mbox{}
\ \vspace{-0.15cm}
\textsc{Giuseppe Cavaliere}$^{a,b}$\textsc{, Iliyan Georgiev}$^{a}$ \textsc{
and Edoardo Zanelli}$^{a}$
\textsc{\vspace{-0.15in} }
\vspace{0.45cm}
{\small {}First draft: May 3, 2022; Revised: February 1, 2024 and July 1,
2024; \\This version: September 19, 2024}
\footnote{
\hspace{-7.2mm}
$^{*}$ We thank Rasmus S\o ndergaard Pedersen and Mervyn Silvapulle for comments.
Financial support from the Italian Ministry of University and Research (PRIN 2020 Grant 2020B2AKFW) 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, University of Exeter, UK.}
\addtocounter{footnote}{-2}
\end{center}
\par\begingroup\leftskip=1 cm
\rightskip=1 cm
\small
\begin{center}
\textsc{Abstract\vspace{-0.15cm}}
\end{center}
{\small We consider bootstrap inference in predictive (or Granger-causality)
regressions when the parameter of interest may lie on the boundary of the
parameter space, here defined by means of a smooth inequality constraint.
For instance, this situation occurs when the definition of the parameter
space allows for the cases of either no predictability or sign-restricted
predictability. We show that in this context constrained estimation gives
rise to bootstrap statistics whose limit distribution is, in general,
random, and thus distinct from the limit null distribution of the original
statistics of interest. This is due to both (i) the possible location of the
true parameter vector on the boundary of the parameter space, and (ii) the
possible non-stationarity of the posited predicting (resp. Granger-causing)
variable. We discuss a modification of the standard fixed-regressor wild
bootstrap scheme where the bootstrap parameter space is shifted by a
data-dependent function in order to eliminate the portion of limiting
bootstrap randomness attributable to the boundary, and prove validity of the
associated bootstrap inference under non-stationarity of the predicting
variable as the only remaining source of limiting bootstrap randomness. Our
approach, which is initially presented in a simple location model, has
bearing on inference in parameter-on-the-boundary situations beyond the
predictive regression problem. \bigskip {} }
\noindent \textsc{{\small {}Keywords: }}{\small {}}Parameter on the
boundary, random measures, weak convergence in distribution, asymptotic
inference, uniform inference.
\par\endgroup\normalsize
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\numberwithin{equation}{section}
\numberwithin{theorem}{section}
\numberwithin{corollary}{section}
\numberwithin{lemma}{section}
\newpage
\section{Introduction}
In this paper we revisit the well-known problem of bootstrap inference in
regressions with parameter space defined by means of smooth inequality
constraints. For instance, consider the setup of a regression $y_{t}=\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{$
{\Greekmath 010B}+$}{\Greekmath 010C} x_{t-1}+{\Greekmath 0122}_{t}$ where the parameter space for $
({\Greekmath 010B},{\Greekmath 010C})$ is defined by the constraint ${\Greekmath 010C}\geq0$. This framework
arises when only the possibilities ${\Greekmath 010C}=0$ of no predictability (or no
first-order Granger causality, generalizable to higher orders), and ${\Greekmath 010C}>0$
of sign-restricted predictability, are entertained, and the model is
estimated under the constraint ${\Greekmath 010C}\in\lbrack0,\infty)$. In applications,
economic theory is often informative about the direction of predictability,
and such information could be used to improve the efficiency of estimators
and increase the power of hypotheses tests. A prominent example is provided
by predictive regressions for financial returns; see, e.g., Phillips (2014)
and the references therein. Interest can then be in testing the very
hypothesis of no predictability (i.e., ${\Greekmath 010C}=0$) by means of a one-sided
test, or a special case of this hypothesis (e.g., ${\Greekmath 010B}={\Greekmath 010C}=0$), or a
hypothesis where the parameter vector may but need not lie on the boundary
of the parameter space (e.g., ${\Greekmath 010B}+{\Greekmath 010C}=0$).
While in this context the bootstrap is potentially useful, its application
is not straightforward if the parameter vector may lie on the boundary of
the parameter space; see Andrews (2000). In particular, as we discuss in the
following, even in a simple location model where the parameter space is a
closed half-line, the cumulative distribution function {[}cdf{]} of the
parametric bootstrap $t$-statistic, conditional on the original data,
converges weakly to a random cdf, rather than to the target asymptotic
distribution of the $t$-statistic computed from the original data.
Our first contribution is to show that in predictive regressions with
parameter values on the boundary, the distribution of fixed regressor
\footnote{
We focus on `fixed regressor' bootstrap schemes as they do not require
knowledge on the regressor generating process. For instance, and in contrast
to recursive-based schemes, they can be applied to both I(0) and I(1)
settings.} bootstrap statistics, like the $t$-statistic for ${\Greekmath 010C} =0$ in
the regression above, may be random in the limit. Limiting randomness may
arise in two ways. A first possible source of randomness in the limit
bootstrap measure is in the non-stationarity of the regressor, which
operates through the random limits of sample product moments. This is hardly
surprising, see e.g. Georgiev et al. (2019). A second potential source of
randomness is the location of the parameter vector on the boundary of the
parameter space. Invalidity of standard bootstrap schemes when a parameter
is on the boundary was initially discussed in Andrews (2000), where a simple
location-model example was given; see also Chatterjee and Lahiri (2011). In
the context of hypotheses tests in predictive regressions, we revisit
Andrews' result and show that, for a general bootstrap scheme, the
occurrence or non-occurrence of limiting bootstrap randomness due to the
possible location of a parameter on the boundary of the parameter space
depends on how well the bootstrap scheme approximates the mutual position of
three objects: (i)\ the boundary, (ii)\ the parameter set identified by the
null hypothesis, and (iii)\thinspace\ the true parameter value. Standard
bootstrap approximations of this mutual position may not be sufficiently
precise, giving rise to complex conditioning in the limit bootstrap
distribution, with ensuing bootstrap validity only for special types of
statistics.
Our second contribution is to show that certain non-standard bootstrap
schemes, designed to provide a better match with the geometric configuration
in the original parameter space, give rise to limit bootstrap distributions
where randomness, if present, is not attributable to the boundary value of
the parameter vector. This fact allows us to establish bootstrap validity in
an `unconditional' sense; see Cavaliere and Georgiev (2020). That is,
although randomness of the limiting bootstrap cdf prevents the possibility
that the bootstrap could mimic the asymptotic distribution of the original
statistic, we can show that in large samples bootstrap tests and asymptotic
tests are correctly sized for essentially the same set of nominal sizes.
Formally, we make use of the following definition, which generalizes the
definition of unconditional bootstrap validity given in Cavaliere and
Georgiev (2020, p.2555). Let $p_{n}$ and $p_{n}^{\ast }$ be respectively the
\emph{p}-value of an asymptotic test and of its bootstrap analogue. Let also
\begin{equation*}
C:=\{q\in (0,1):\lim_{n\rightarrow \infty }P(p_{n}\leq q)=q|\mathsf{H}_{0}\},
\end{equation*}
such that a test rejecting for $p_{n}\leq q$ (or for $p_{n}>q$) is correctly
sized for nominal significance levels $q$ (resp. $1-q$) with $q\in C$, as $
n\rightarrow \infty $.\footnote{{\small {}{}For $q\in \mathrm{int}C$ it
holds that $P(p_{n}=q)\rightarrow 0$ and rejections for $p_{n}\leq q$ (or $
p_{n}>q$) are asymptotically equivalent to rejections for $p_{n}<q$ (or $
p_{n}\geq q$).}} If, under the null hypothesis $\mathsf{H}_{0}$,
\begin{equation}
P(p_{n}^{\ast }\leq q)\rightarrow q\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for all $q\in \limfunc{int}C$,}
\label{eq unconditional validity}
\end{equation}
where $\limfunc{int}C$ denotes the interior of the set $C$, we say that the
bootstrap test based on $p_{n}^{\ast }$ is valid for $\mathsf{H}_{0}$.
\footnote{
Bootstrap unconditional validity as in Cavaliere and Georgiev (2020) is
obtained as the special case $\limfunc{int}$$C=(0,1)$.} The meaning is that
the bootstrap test and the asymptotic test are first-order asymptotically
equivalent in terms of correct size control. In particular, bootstrap
validity for simple hypotheses $\mathsf{H}_{0}$ characterizes pointwise size
control.
Notice that bootstrap validity as in (1.1) is implied by the classic
definition of bootstrap consistency, namely that $\sup_{x\in\mathbb{R}
}|F_{n}^{\ast}(x)-F(x)|\rightarrow_{p}0$ for a bootstrap statistic with cdf $
F_{n}^{\ast}$ conditionally on the data and an original test statistic with
continuous asymptotic cdf $F$. The converse does not hold; that is, (\ref{eq
unconditional validity}) does not imply classic bootstrap consistency, see
the discussion in Cavaliere and Georgiev (2020).
For test statistics whose asymptotic distribution is continuous, it holds
that $\limfunc{int}$$C=(0,1)$ and hence condition (\ref{eq unconditional
validity}) should hold for all $q\in (0,1)$ for the bootstrap to be valid.
Unfortunately, parameter values on the boundary of the parameter space may
induce discontinuities in the limiting cdf's, such that not even the exact
\emph{p}-values of the associated tests are asymptotically standard uniform
on $[0,1]$. This makes the above weaker version of the validity definition
unavoidable.
Finally, we turn to the special case of one-sided tests for the null
hypothesis that the parameter vector lies on the boundary of the parameter
space, such that the boundary coincides with the parameter set identified by
the null hypothesis. This case provides a transparent example of a limit
bootstrap cdf which is random only on a subset of its domain. Then, if
bootstrap validity is defined as in (\ref{eq unconditional validity}), in
this case also some standard bootstrap schemes can be proved to be valid.
This paper is related to recent work by Fang and Santos (2019) and Hong and
Li (2020). The latter two papers propose nonstandard bootstrap schemes --
involving a tuning tool -- which correct the inconsistency of `classic'
bootstrap methods in settings that cover parameters on the boundary as a
special case. The main difference from the present contribution is that our
theory applies to random limit bootstrap measures. Thus, Fang and Santos
(2019) consider bootstrap inference in settings where the target asymptotic
distribution, say that of a random element ${\Greekmath 011C} $, can be thought of as a
transformation ${\Greekmath 0127} $ of another random element ${\Greekmath 011C} ^{\prime }$, and
both the distribution of ${\Greekmath 011C} ^{\prime }$ and the transformation ${\Greekmath 0127} $
need to be estimated; see also the related works by D\"{u}mbgen (1993),
Hirano and Porter (2012), Fang (2014) and Chen and Fang (2019). Although
Fang and Santos (2019) consider deterministic ${\Greekmath 0127} $ and the
unconditional distribution of ${\Greekmath 011C} ^{\prime }$, such that their results are
not directly applicable here, their way of conceptualizing the problem
remains fruitful also in the case of random ${\Greekmath 0127} $ and random
conditional distributions $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{${\Greekmath 011C} ^{\prime }|{\Greekmath 011C} ^{\prime \prime }$
(for some random element }{\Greekmath 011C} ^{\prime \prime }$)$\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{. }$We discuss this
in Section \ref{subsec:Fang-and-Santos}.
Our contribution is also related to Hong and Li (2020), who propose a
`numerical bootstrap' which is valid in settings where a parameter space can
be approximated locally by a cone with vertex at the true value of the
parameters; see Geyer (1994) for a detailed discussion of the approximation.
Both the approaches in this paper and that by Hong and Li (2020) are
connected to the large body of literature considering estimation and
inference for constrained M-estimators; see, among others, Geyer (1994),
Andrews (1999, 2000), and the references therein. In Section \ref
{subsec:Fang-and-Santos} we argue that, when applied to a restricted
predictive regression, the `numerical bootstrap' of Hong and Li (2020)
performs a geometric approximation of the kind we propose, though at the
cost of a slower-than-standard convergence rate for the resulting bootstrap
estimator.
We present our main idea using first a simple location model for i.i.d.
scalar data whose location parameter is constrained to be positive. This is
done in Section \ref{sec example}. The predictive regression framework is
presented in Section \ref{sec par on the boundary setup}; in this section we
also show that the bootstrap limit measure associated with standard fixed
regressor wild bootstrap schemes is random. A new family of bootstrap
algorithms and their validity are discussed in Section \ref{sec par on the
boundary main}. Results on the validity of one-sided tests, connections to
the previous literature, and uniform size control for the bootstrap tests
are discussed in Section \ref{sec:Discussion}. Section \ref{sec:Numerical}
provides simulation evidence, whereas Section \ref{sec conclusion}
concludes. Proofs are collected in the Appendix.
\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, are denoted by $\mathscr{D}{}_{n}$,
$\mathscr{D}{}_{m\times n}$ and $\mathscr{D}(\mathbb{R})$, respectively; see
Kallenberg (1997, Appendix A2). For $n=1$, the subscript in $\mathscr{D}{}
_{n}$ is suppressed. $\mathscr{C}_{n}(\mathbb{R}^{n})$ is the space of
continuous functions from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ equipped with
the topology of uniform convergence on compacts. 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 cdf, possibly random, $F^{-1}$ 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 $.
With $(Z_{n},Y_{n})$ and $(Z,Y)$ being random elements of the metric spaces $
S_{Z}\times S_{Y_{n}}$ and $S_{Z}\times S_{Y}$ ($n\in \mathbb{N}$), and
defined on a common probability space, we denote by `$Z_{n}|Y_{n}\overset{w}{
\rightarrow }_{p}Z|Y$' (resp. `$Z_{n}|Y_{n}\overset{w}{\rightarrow }
_{a.s.}Z|Y$') the fact that $E\left\{ g\left( Z_{n}\right) |Y_{n}\right\} {
\rightarrow }E\left\{ g\left( Z\right) |Y\right\} $ in probability (resp.
a.s.) for all bounded continuous functions $g:S_{Z}\rightarrow \mathbb{R}$.
When $Z_{n}$ is a bootstrap statistic and $Y_{n}$ denotes the original data,
we write `$Z_{n}\overset{w^{\ast }}{\rightarrow }_{p}Z|Y$' (resp. `$Z_{n}
\overset{w^{\ast }}{\rightarrow }_{a.s.}Z|Y$'). Finally, with $(Z_{n},Y_{n})$
and $(Z,Y)$ possibly defined on different probability spaces, `$Z_{n}|Y_{n}
\overset{w}{\rightarrow }_{w}Z|Y$' means that $E(g(Z_{n})|Y_{n})\overset{w}{
\rightarrow }E(g(Z)|Y)$ for all bounded continuous functions $
g:S_{Z}\rightarrow \mathbb{R}$, see Kallenberg (1997, 2017); we label this
fact `weak convergence in distribution'. For the special case of scalar
random variables $Z_{n}$ and $Z$, if the conditional distribution $Z|Y$ is
diffuse (non-atomic), weak convergence in distribution is equivalent to the
following weak convergence in $\mathscr{D}(\mathbb{R})$:
\begin{equation}
F_{n}(\cdot |Y_{n}):=P(Z_{n}\leq \cdot |Y_{n})\overset{w}{\rightarrow }
P(Z\leq \cdot |Y)=:F(\cdot |Y).
\label{eq weak convergence of the conditional cdf}
\end{equation}
When $Z_{n}$ is a bootstrap statistic and conditioning is on the original
data, we use the notation `$\overset{w^{\ast }}{\rightarrow }_{w}$'. For
multivariate generalizations we refer to Cavaliere and Georgiev (2020,
Appendix A).
\section{Preview of the results in a location model}
\label{sec example}
To illustrate the main arguments that will be proposed in the predictive
regression framework later, consider as in Andrews (2000) and Cavaliere et
al. (2017) the location model
\begin{equation*}
y_{t}={\Greekmath 0112}+{\Greekmath 0122}_{t}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ (}t=1,...,n\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{)}
\end{equation*}
where the ${\Greekmath 0122}_{t}$'s are i.i.d.$\left(0,1\right)$ and the parameter
space is $\Theta:=\{{\Greekmath 0112}\in\mathbb{R}:{\Greekmath 0112}\geq0\}$. Interest is in
inference on the true value ${\Greekmath 0112}_{0}$ of ${\Greekmath 0112}$ by using the Gaussian
QMLE, $\hat{{\Greekmath 0112}}$. With $l_{n}\left({\Greekmath 0112}\right):=-\frac{1}{2}
\sum_{t=1}^{n}(y_{t}-{\Greekmath 0112})^{2}$, we find $\hat{{\Greekmath 0112}}:=\arg\max_{{\Greekmath 0112}
\in\Theta}l_{n}\left({\Greekmath 0112}\right)=\max\{0,\bar{y}_{n}\}$, $\bar{y}
_{n}:=n^{-1}\sum_{t=1}^{n}y_{t}$. If ${\Greekmath 0112}_{0}$ is an interior point of $
\Theta$, i.e. ${\Greekmath 0112}_{0}>0$, then $n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112}_{0})\overset{
w}{\rightarrow}{\Greekmath 0118}$, ${\Greekmath 0118}\sim N\left(0,1\right)$. In contrast, if $
{\Greekmath 0112}_{0} $ is on the boundary of $\Theta$, i.e. ${\Greekmath 0112}_{0}=0$, the
asymptotic distribution of $\hat{{\Greekmath 0112}}$ is
\begin{equation}
n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112}_{0})=n^{1/2}\hat{{\Greekmath 0112}}\overset{w}{\rightarrow}
\ell:=\max\{0,{\Greekmath 0118}\} \label{eq loc model asy distr}
\end{equation}
again with ${\Greekmath 0118}\sim N\left(0,1\right)$.
The first takeaway of this section is the fact that the location of a
parameter on the boundary of the parameter space may induce limiting
bootstrap randomness of a kind that invalidates bootstrap inference. To see
this, consider in the context of the location model a standard Gaussian
parametric bootstrap based on the bootstrap sample
\begin{equation*}
y_{t}^{\ast }=\hat{{\Greekmath 0112}}+{\Greekmath 0122} _{t}^{\ast },
\end{equation*}
where the ${\Greekmath 0122} _{t}^{\ast }$'s are i.i.d.$N\left( 0,1\right) $
independent of the original data. The bootstrap counterpart of $l_{n}\left(
{\Greekmath 0112} \right) $ is $l_{n}^{\ast }\left( {\Greekmath 0112} \right) :=-\frac{1}{2}
\sum_{t=1}^{n}(y_{t}^{\ast }-{\Greekmath 0112} )^{2}$, and the usual bootstrap QMLE is $
\hat{{\Greekmath 0112}}^{\ast }:=\arg \max_{{\Greekmath 0112} \in \Theta }l_{n}^{\ast }\left(
{\Greekmath 0112} \right) =\max \{0,\bar{y}_{n}^{\ast }\}$, $\bar{y}_{n}^{\ast }:=\hat{
{\Greekmath 0112}}+\bar{{\Greekmath 0122}}_{n}^{\ast }$, $\bar{{\Greekmath 0122}}_{n}^{\ast
}:=n^{-1}\sum_{t=1}^{n}{\Greekmath 0122} _{t}^{\ast }$. Conditionally on the
original sample, $\hat{{\Greekmath 0112}}^{\ast }$'s exact distribution is
\begin{equation}
n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})=n^{1/2}\max \{-\hat{{\Greekmath 0112}},\bar{
{\Greekmath 0122}}_{n}^{\ast }\}\sim \max \{-n^{1/2}\hat{{\Greekmath 0112}},{\Greekmath 0118} ^{\ast }\}|
\hat{{\Greekmath 0112}}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }{\Greekmath 0118} ^{\ast }|\hat{{\Greekmath 0112}}\sim {\Greekmath 0118} \sim N\left(
0,1\right) , \label{eq loc boot}
\end{equation}
with associated conditional cdf given by
\begin{equation}
P^{\ast }(n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})\leq x)=\Phi \left(
x\right) \mathbb{I}_{\{x\geq -n^{1/2}\hat{{\Greekmath 0112}}\}}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }x\in \mathbb{R}
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.} \label{eq bootstrap cdf loc mod}
\end{equation}
Now, when ${\Greekmath 0112} _{0}$ is an interior point of $\Theta $, $-n^{1/2}\hat{
{\Greekmath 0112}}$ diverges to $-\infty $ in probability and the distribution of $
n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})$ given the data converges weakly
in probability to the non-random distribution of ${\Greekmath 0118} ^{\ast }$; the
bootstrap therefore mimics the $N\left( 0,1\right) $ asymptotic distribution
of the original statistic, the bootstrap distributional approximation is
consistent and bootstrap inference is valid in the sense of (\ref{eq
unconditional validity}), with $\limfunc{int}$$C=(0,1)$. Conversely, when $
{\Greekmath 0112} _{0}$ is on the boundary of the parameter space, the cdf in (\ref{eq
bootstrap cdf loc mod}) converges weakly in $\mathscr{D}(\mathbb{R})$ to the
random cdf $\Phi \left( x\right) \mathbb{I}_{\{x\geq -\ell \}}$. In terms of
weak convergence in distribution,
\begin{equation}
n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})\overset{w^{\ast }}{\rightarrow }
_{w}\ell ^{\ast }|\ell \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }\ell ^{\ast }:=\max \{-\ell ,{\Greekmath 0118} ^{\ast }\}
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,} \label{eq:sy loc}
\end{equation}
where $\ell $ is distributed as in (\ref{eq loc model asy distr}) and is
independent of ${\Greekmath 0118} ^{\ast }$. The limit distribution in (\ref{eq:sy loc})
is random, since its cdf is a stochastic process depending on the
conditioning random variable $\ell $. Thus, it is distinct from the limit
distribution in (\ref{eq loc model asy distr}), which is unconditional and
hence characterized by a non-random cdf. Because the bootstrap limit
distribution is random, the bootstrap approximation is not consistent for
the limit in (\ref{eq loc model asy distr}).
As we shall see in Section \ref{sec par on the boundary main}, limiting
bootstrap randomness could be of two kinds: `benign', thus not compromising
the validity of bootstrap inference in the sense of (\ref{eq unconditional
validity}), or `malignant', thus invalidating bootstrap inference. In this
example, a bootstrap test employing a bootstrap statistic $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{${\Greekmath 011C}
_{n}^{\ast }:=$}{\Greekmath 011E} (n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}}))$ as the
analogue of a statistic ${\Greekmath 011C} _{n}:=$${\Greekmath 011E} (n^{1/2}\hat{{\Greekmath 0112}})$, where $
{\Greekmath 011E} $ is a real function, may not be valid in the sense of (\ref{eq
unconditional validity}) under the null hypothesis $\mathsf{H}_{0}:{\Greekmath 0112}
_{0}=0$ even if the function ${\Greekmath 011E} $ is continuous, thus implying
`malignant' randomness.
To get some further insight into the source of limiting bootstrap
randomness, which will be exploited in the next sections, it is useful to
notice that the asymptotic distributions in (\ref{eq loc model asy distr})
and (\ref{eq loc boot}) can be written as
\begin{eqnarray*}
\ell &=&\max \{0,{\Greekmath 0118} \}=\arg \min\nolimits_{{\Greekmath 0115} \in \Lambda }|{\Greekmath 0115}
-{\Greekmath 0118} |\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }\Lambda :=\{{\Greekmath 0115} \in \mathbb{R}:{\Greekmath 0115} \geq 0\} \\
\ell ^{\ast }|\ell &=&\max \{-\ell ,{\Greekmath 0118} ^{\ast }\}|\ell =\left( \arg
\min\nolimits_{{\Greekmath 0115} \in \Lambda (\ell )}|{\Greekmath 0115} -{\Greekmath 0118} ^{\ast }|\right)
\big|\ell \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }\Lambda (\ell ):=\{{\Greekmath 0115} \in \mathbb{R}:{\Greekmath 0115} \geq
-\ell \}=\Lambda -\ell \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}
\end{eqnarray*}
respectively. Hence, bootstrap randomness, and the implied bootstrap
invalidity, can be attributed to the fact that in the bootstrap world the
limit constraint set for the objective function $|{\Greekmath 0115} -{\Greekmath 0118} ^{\ast }|$ is
the \emph{random} half line $\Lambda (\ell )$ rather than the original fixed
half line $\Lambda =\Lambda (0)$. That is, the chosen bootstrap scheme
shifts the constraint set by the random variable $-\ell $, which is non-zero
with probability $1/2$.
The second takeaway of this section is the fact that bootstrap validity
could be restored by offsetting properly the previous shift of the limit
constraint set. Specifically, this requires an ad hoc construction of a
bootstrap parameter space intended to approximate well the mutual position
of the true parameter value and the boundary of the original parameter space.
Consider a bootstrap scheme where the boundary of the bootstrap parameter
space $\Theta ^{\ast }$ is chosen in a data-driven way such that the mutual
position of ${\Greekmath 0112} _{0}$ and the boundary of $\Theta $ is well approximated
irrespective of whether ${\Greekmath 0112} _{0}$ belongs to $\partial \Theta $ or not.
To this aim, introduce the half line $\Theta ^{\ast }:=\{{\Greekmath 0112} :{\Greekmath 0112} \geq
g^{\ast }(\hat{{\Greekmath 0112}})\}$, where $g^{\ast }({\Greekmath 0112} ):={\Greekmath 0112} -|{\Greekmath 0112}
|^{1+{\Greekmath 0114} }$, ${\Greekmath 0114} >0$, and the associated $\hat{{\Greekmath 0112}}^{\ast }:=\arg
\max_{{\Greekmath 0112} \in \Theta ^{\ast }}l_{n}^{\ast }\left( {\Greekmath 0112} \right) =\max
\{g^{\ast }(\hat{{\Greekmath 0112}}),\bar{y}_{n}^{\ast }\}$. The bootstrap QMLE
statistic is then given by $n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}}
)=n^{1/2}\max \{g^{\ast }(\hat{{\Greekmath 0112}})-\hat{{\Greekmath 0112}},\bar{{\Greekmath 0122}}
_{n}^{\ast }\}$. Conditionally on the data, it is distributed as $\max
\{n^{1/2}(g^{\ast }(\hat{{\Greekmath 0112}})-\hat{{\Greekmath 0112}}),{\Greekmath 0118} ^{\ast }\}|\hat{{\Greekmath 0112}}$
, with ${\Greekmath 0118} ^{\ast }|\hat{{\Greekmath 0112}}\sim N(0,1)$. If ${\Greekmath 0112} _{0}=0$, it then
follows that $n^{1/2}(g^{\ast }(\hat{{\Greekmath 0112}})-\hat{{\Greekmath 0112}})=-n^{1/2}\hat{
{\Greekmath 0112}}^{1+{\Greekmath 0114} }\overset{p}{\rightarrow }0$, and the bootstrap statistic
conditionally on the data converges weakly in probability to $\ell$ of (\ref
{eq loc model asy distr}). Conversely, if ${\Greekmath 0112} _{0}>0$ then $
n^{1/2}(g^{\ast }(\hat{{\Greekmath 0112}})-\hat{{\Greekmath 0112}})=-n^{1/2}\hat{{\Greekmath 0112}}^{1+{\Greekmath 0114}
}\overset{p}{\rightarrow }-\infty $ and the bootstrap statistic
conditionally on the data converges weakly in probability to the $N\left(
0,1\right) $ distribution. In both cases, the bootstrap mimics the
asymptotic distribution of $n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{0})$ and bootstrap
validity in the sense of (\ref{eq unconditional validity}) can be seen to be
successfully restored.\medskip {}
\noindent \textsc{Remark}. In the location model, an appropriate choice of $
\Theta ^{\ast }$ simultaneously restores bootstrap validity and removes all
the randomness from the limit bootstrap distribution. In the predictive
regression framework we shall conclude that, in order to achieve bootstrap
validity, it is essential to remove only the portion of limiting bootstrap
randomness that is due to the location of the parameter vector on the
boundary of the parameter space. As no other sources of limiting bootstrap
randomness exist in the context of the location model, in this section the
previous conclusion simplifies to eliminating all the limiting bootstrap
randomness.$\hfill \square $\medskip {}
Before moving on to predictive regressions, we notice that when a test of $
\mathsf{H}_{0}:{\Greekmath 0112} _{0}=0$ against $\mathsf{H}_{1}:{\Greekmath 0112} _{0}>0$ is
performed, employing ${\Greekmath 011C} _{n}^{\ast }:=n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{
{\Greekmath 0112}})$ as the bootstrap analogue of ${\Greekmath 011C} _{n}:=n^{1/2}\hat{{\Greekmath 0112}}$, the
standard parametric bootstrap with $\Theta ^{\ast }=\Theta $ is valid in the
sense of (\ref{eq unconditional validity}); see also Andrews (2000).
Specifically, the bootstrap test rejects $\mathsf{H}_{0}$ when the bootstrap
\emph{p}-value $\tilde{p}_{n}^{\ast }=1-p_{n}^{\ast }$ is small, with the
following convergence satisfied under the null hypothesis:
\begin{equation*}
p_{n}^{\ast }=P^{\ast }({\Greekmath 011C} _{n}^{\ast }\leq {\Greekmath 011C} _{n})=P^{\ast }({\Greekmath 0118}
^{\ast }\leq {\Greekmath 011C} _{n})\overset{w}{\rightarrow }\Phi (\ell ).
\end{equation*}
A similar convergence is satisfied by the \emph{p}-value $\tilde{p}
_{n}=1-p_{n}$ of the asymptotic test, with
\begin{equation*}
p_{n}=P(\ell\leq u)|_{u={\Greekmath 011C}_n}=\tfrac{1}{2}\mathbb{I}_{\{{\Greekmath 011C}_n=0\}}+\Phi(
{\Greekmath 011C}_n)\mathbb{I}_{\{{\Greekmath 011C}_n>0\}}=\Phi({\Greekmath 011C}_n)\overset{w}{\rightarrow}
\Phi(\ell).
\end{equation*}
As $\ell $ is distributed like $\Phi ^{-1}(U)\mathbb{I}_{\{U>1/2\}}$, $U\sim
U_{[0,1]}$, it follows that $\Phi(\ell)$ is distributed like $\Phi (\Phi
^{-1}(U)\mathbb{I}_{\{U>1/2\}})$. As a result, both the bootstrap and the
asymptotic test are correctly sized for nominal levels below $1/2$. This
phenomenon, whose extensions to predictive regression are discussed in
Section \ref{sec one sided tests}, does not generalize to hypotheses where
one-sided tests are not appropriate or straightforward. Therefore, a remedy
is necessary for the inference-invalidating limiting bootstrap randomness
induced by the location of a parameter on the boundary.
\section{The predictive regression setup}
\label{sec par on the boundary setup}
Consider the following predictive regression in a triangular array setup:
\begin{equation}
y_{t}={\Greekmath 0112} _{1}+{\Greekmath 0112} _{2}x_{n,t-1}+{\Greekmath 0122} _{t},\quad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ (}
t=1,...,n;\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }n=1,2,...\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{)}, \label{eq:prr}
\end{equation}
where ${\Greekmath 0122} _{t}$ is a martingale difference sequence {[}mds{]} and $
x_{n,t}$ is a non-stationary posited predicting variable satisfying the
following assumption; see, e.g. M\"{u}ller and Watson (2008) for references
to primitive conditions.
\begin{exercise}
Let $z_{n,t}:=n^{-1/2}\sum_{s=1}^t {\Greekmath 0122}_s$. Then:
\begin{itemize}
\item[(a)] $\{{\Greekmath 0122} _{t}\}$ is an mds w.r.t. some filtration to which $
(x_{n,t},z_{n,t})$ is adapted, with $E{\Greekmath 0122} _{t}^{2}={\Greekmath 0121} _{zz}\in
(0,\infty ).$
\item[(b)] a law of large numbers holds as $n \rightarrow \infty$:
\begin{equation*}
\sum_{t=1}^{n}\left(
\begin{array}{c}
\Delta x_{n,t} \\
\Delta z_{n,t}
\end{array}
\right) \left(
\begin{array}{cc}
\Delta x_{n,t} & \Delta z _{n,t}
\end{array}
\right) \overset{p}{\rightarrow }\Omega: =\left(
\begin{array}{cc}
{\Greekmath 0121} _{x x } & {\Greekmath 0121} _{x z } \\
{\Greekmath 0121} _{x z } & {\Greekmath 0121} _{z z }
\end{array}
\right)>0.
\end{equation*}
\item[(c)] an invariance principle holds in $\mathscr{D}_2$ as $n
\rightarrow \infty$:
\begin{equation*}
(x_{n,\left\lfloor n\cdot\right\rfloor }, z_{n,\left\lfloor
n\cdot\right\rfloor })^{\prime}\overset{w}{\rightarrow}(X,Z)^{\prime}\sim
BM(0,\Omega),
\end{equation*}
a bivariate Brownian motion on $[0,1]$.
\end{itemize}
\end{exercise}
Assumption 1 covers the specification $x_{n,t}=n^{-1/2}x_{t}$ for an $I(1)$
process $x_{t}$ driven by an mds that could be contemporaneously correlated
with ${\Greekmath 0122} _{t}$.\footnote{
As the bootstrap p-values discussed in the paper are invariant to rescaling
of the regressor, the normalization of $x_{t}$ by $n^{-1/2}$ has no
practical implication. It is equivalent to specifying a local-to-zero
regression coefficient, as is frequent in applications where $y_{t}$ is a
financial return and $x_{t}$ is non-stationary.}$^{,}$\footnote{
Results under two alternative stochastic specifications of $x_{n,t}$, as a
near-unit root and as a stationary process, are given in the accompanying
supplement, Section \ref{sec more DGPs}.{}}
Assumption 1 implies that $\sum_{t=1}^{n}x_{n,t-1}\Delta z_{n,t}\overset{w}{
\rightarrow }\int XdZ$, which need not have a mixed Gaussian distribution
because $X$ and $Z$ need not be independent. Nevertheless, it holds that $
\sum_{t=1}^{n}x_{n,t-1}(\Delta z_{n,t}-{\Greekmath 0121} _{xz}{\Greekmath 0121} _{xx}^{-1}\Delta
x_{n,t})\overset{w}{\rightarrow }\int Xd(Z-{\Greekmath 0121} _{xz}{\Greekmath 0121} _{xx}^{-1}X)$,
which is zero-mean mixed Gaussian with conditional variance ${\Greekmath 011B}
_{e}^{2}\int X^{2}$, where ${\Greekmath 011B} _{e}^{2}:={\Greekmath 0121} _{zz}-{\Greekmath 0121}
_{xz}^{2}{\Greekmath 0121} _{xx}^{-1}$ is the variance of ${\Greekmath 0122} _{t}$ corrected
for $\Delta x_{n,t}$. The bootstrap schemes discussed below all rely on the
independence of the processes $X$ and $Z-{\Greekmath 0121} _{xz}{\Greekmath 0121} _{xx}^{-1}X$.
Further, Assumption 1 imposes unconditional homoskedasticity for simplicity.
As all the bootstrap schemes below are based on `wild' bootstrap schemes,
unconditional heteroskedasticity can be accommodated at only a notational
cost.
The next assumption specifies the parameter space, say $\Theta$, by means of
a smooth inequality constraint.
\begin{exercise}
The parameter space is $\Theta:=\{{\Greekmath 0112}=({\Greekmath 0112}_{1},{\Greekmath 0112}_{2})^{\prime}\in
\mathbb{R}^{2}:g({\Greekmath 0112})\geq0\}$, with non-empty boundary $
\partial\Theta:=\{{\Greekmath 0112}\in\mathbb{R}^{2}:g({\Greekmath 0112})=0\}$, where $g:$ $
\mathbb{R}^{2}\rightarrow\mathbb{R}$ is continuously differentiable in some
neighborhood of the true parameter value ${\Greekmath 0112}_{0}:=({\Greekmath 0112}_{1,0},
{\Greekmath 0112}_{2,0})^{\prime}$ with gradient $\tfrac{\partial}{\partial{\Greekmath 0112}^{
\prime}}g({\Greekmath 0112})\neq0$ in that neighborhood.
\end{exercise}
In the following, $\dot{g}$ will denote the gradient of the function $g$
evaluated at ${\Greekmath 0112}_0$.
Assumption 2 generalizes the leading example of the parameter space $\Theta=
\mathbb{R}\times\lbrack0,\infty)$ obtained by setting $g({\Greekmath 0112})=(0,1)
{\Greekmath 0112}={\Greekmath 0112}_{2}$. The boundary of $\Theta$ then corresponds to the case $
{\Greekmath 0112}_{2}=0$ of no predictability of $y_{t}$ by $x_{n,t-1}$ whereas the
interior of $\Theta$ corresponds to the case of sign-restricted
predictability.
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 ${\Greekmath 0112}_{0}$:
\begin{description}
\item[$\mathscr{G}{}_{1}$.] $\mathsf{H}_{0}$ is the hypothesis that $
{\Greekmath 0112}_{0}$ belongs to the boundary: $\mathsf{H}_{0}:g({\Greekmath 0112}_{0})=0$;
\item[$\mathscr{G}{}_{2}$.] $\mathsf{H}_{0}$ is a simple null hypothesis on
the boundary: $\mathsf{H}_{0}:{\Greekmath 0112}_{0}=\bar{{\Greekmath 0112}}$, $g(\bar{{\Greekmath 0112}})=0$;
\item[$\mathscr{G}{}_{3}$.] $\mathsf{H}_{0}:h({\Greekmath 0112}_{0})=0$, where $
\{{\Greekmath 0112}\in\mathbb{R}^{2}:h\left({\Greekmath 0112}\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 $g({\Greekmath 0112})={\Greekmath 0112}_{2}$, such that the
parameter space is $\mathbb{R}\times\lbrack0,\infty)$ with boundary $
\partial\Theta=\mathbb{R}\times\{0\}$. Then the hypothesis of no
predictability $\mathsf{H}_{0}:{\Greekmath 0112}_{2,0}=0$ falls under $\mathscr{G}{}
_{1} $. The hypothesis $\mathsf{H}_{0}:{\Greekmath 0112}_{0}=\bar{{\Greekmath 0112}}
=(0,0)^{\prime} $ that $y_{t}$ is unpredictable with zero mean falls under $
\mathscr{G}{}_{2} $. Finally, the hypothesis $\mathsf{H}_{0}:(1,1)
{\Greekmath 0112}_{0}={\Greekmath 0112}_{1,0}+{\Greekmath 0112}_{2,0}=0$ falls under $\mathscr{G}{}_{3}$ by
setting $h\left({\Greekmath 0112}\right):=(1,1){\Greekmath 0112}$; in this case, the intersection
point of the boundary and the parameter set identified by $\mathsf{H}_{0}$
is $(0,0)^{\prime}$ which might, but need not, be the true value under $
\mathsf{H}_{0}$.
\subsection{Asymptotic distributions}
\label{subsec:Asymptotic-distributions}
Let $\hat{{\Greekmath 0112}}$ be the OLS estimator of $({\Greekmath 0112} _{1},{\Greekmath 0112}
_{2})^{\prime }$ in the equation
\begin{equation}
y_{t}={\Greekmath 0112} _{1}+{\Greekmath 0112} _{2}x_{n,t-1}+{\Greekmath 010E} \Delta x_{n,t}+e_{t}
\label{eq:pr}
\end{equation}
subject to the constraint $\hat{{\Greekmath 0112}}\in \Theta $, i.e. $g(\hat{{\Greekmath 0112}}
)\geq 0$, and where the role of the regressor $\Delta x_{n,t}$ is to ensure
that the residuals are asymptotically uncorrelated with the innovations
driving $x_{n,t}$, a convenient prerequisite for the bootstrap
implementations. The existence, with probability approaching one, of a
measurable minimizer of the residual sum of squares (\ref{eq:pr}) over the
set $\Theta $ can be established in a similar but simpler way than that of
its bootstrap counterpart in our detailed proof of Theorem \ref{Lemma
bootstrap with boundary}. Moreover, any two such minimizers are first-order
asymptotically equivalent, explaining our usage of `the' associated with the
constrained OLS estimator. Specifically, any such minimizer $\hat{{\Greekmath 0112}}$
satisfies $n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{0})\overset{w}{\rightarrow }\ell
({\Greekmath 0112} _{0})$, with $\ell ({\Greekmath 0112} _{0})$ depending on the position of $
{\Greekmath 0112} _{0}$ relative to the boundary $\partial \Theta $. Thus, $\ell
({\Greekmath 0112} _{0})=\tilde{\ell}:=M^{-1/2}{\Greekmath 0118} $ if ${\Greekmath 0112} _{0}\in $$\limfunc{int}$
$\Theta :=\Theta \setminus \partial \Theta $, where $M:=\int \tilde{X}\tilde{
X}^{\prime }$, $\tilde{X}:=(1,X)^{\prime }$, ${\Greekmath 0118} \sim N\left( 0,{\Greekmath 011B}
_{e}^{2}I_{2}\right) $ is independent of $X$, and ${\Greekmath 011B} _{e}^{2}>0$ is the
variance of ${\Greekmath 0122} _{t}$ corrected for $\Delta x_{n,t}$, whereas
\begin{equation}
n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{0})\overset{w}{\rightarrow }\ell ({\Greekmath 0112}
_{0})=\ell :=\underset{{\Greekmath 0115} \in \Lambda }{\arg \min }||{\Greekmath 0115}
-M^{-1/2}{\Greekmath 0118} ||_{M}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }\Lambda :=\{{\Greekmath 0115} \in \mathbb{R}^{2}:\dot{g}
^{\prime }{\Greekmath 0115} \geq 0\} \label{eq asy distribution}
\end{equation}
if $g({\Greekmath 0112} _{0})=0$, with $||x||_{M}:=(x^{\prime }Mx)^{1/2}$ for $x\in
\mathbb{R}^{2}$; see Section 12 in the working paper version of Andrews
(1999) or the proof of Theorem \ref{Lemma bootstrap with boundary} for the
bootstrap counterpart.
The previous asymptotic result is sufficient in order to see that the
possibility of having ${\Greekmath 0112}_{0}$ at the boundary of the parameter space $
\Theta$ induces a dichotomy in the limit distribution of $n^{1/2}(\hat{{\Greekmath 0112}
}-{\Greekmath 0112}_{0})$ similar to the dichotomy established in the introductory
location-model example. Replicating the constraint set in the limit
distribution by means of a bootstrap scheme will be our main concern in what
follows.
\subsection{Standard bootstrap invalidity}
\label{subsec:Standard-bootstrap-invalidity}
Consider first a fixed-regressor wild bootstrap sample generated as
\begin{equation}
y_{t}^{\ast }=\hat{{\Greekmath 0112}}_{1}+\hat{{\Greekmath 0112}}_{2}x_{n,t-1}+{\Greekmath 0122}
_{t}^{\ast }\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,} \label{eq BS for PR}
\end{equation}
where ${\Greekmath 0122} _{t}^{\ast }=\hat{e}_{t}w_{t}^{\ast }$, $t=1,...,n$, with
$\hat{e}_{t}$ the residuals of (\ref{eq:pr}) and $w_{t}^{\ast }$ i.i.d. $
N(0,1)$, independent of the original data.\footnote{
{}{}The conclusions do not change if another zero-mean unit-variance
distribution with a finite fourth moment is used instead of the standard
Gaussian distribution.} Then the distribution of $n^{1/2}(\hat{{\Greekmath 0112}}
-{\Greekmath 0112} _{0})$ could be tentatively approximated by the distribution of $
n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})$ conditional on the original
data, where $\hat{{\Greekmath 0112}}^{\ast }$ is obtained by regressing $y_{t}^{\ast }$
on $(1,x_{n,t-1})^{\prime }$ under the constraint $\hat{{\Greekmath 0112}}^{\ast }\in
\Theta ^{\ast }=\Theta $, i.e., $g(\hat{{\Greekmath 0112}}^{\ast })\geq 0$ as for the
original estimator; see Andrews (2000)\footnote{
Note that the term $\Delta x_{n,t}$ is no longer necessary because $
x_{n,t-1} $ and ${\Greekmath 0122} _{t}^{\ast }$ are independent conditionally on
the data.}.
To motivate the analysis in the next section, it is useful to anticipate
some asymptotic properties of $\hat{{\Greekmath 0112}}^{\ast}$ which obtain by
specializing Theorem \ref{Lemma bootstrap with boundary} below to the
fixed-regressor wild bootstrap scheme. For ${\Greekmath 0112}_{0}\in$$\limfunc{int}$$
\Theta$, it turns out that the bootstrap distribution converges to a
conditional version of the limit distribution of $n^{1/2}(\hat{{\Greekmath 0112}}
-{\Greekmath 0112}_{0})$ found earlier:
\begin{equation}
n^{1/2}(\hat{{\Greekmath 0112}}^{\ast}-\hat{{\Greekmath 0112}})=n^{1/2}(\tilde{{\Greekmath 0112}}^{\ast}-\hat{
{\Greekmath 0112}})+o_{p}(1)\overset{w^{\ast}}{\rightarrow}_{w}\tilde{\ell}|M\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }
\label{eq random limit depending on M only}
\end{equation}
where $\tilde{{\Greekmath 0112}}^{\ast}$ denotes the unconstrained OLS estimator from
the bootstrap sample. The limit bootstrap distribution is, therefore,
random. The vehicle of limiting bootstrap randomness is the random matrix $M$
, such that limiting bootstrap randomness is fully attributable to the
stochastic properties of the regressor. Due to the fact that the bootstrap
replicates a conditional version of the limit distribution of the original
estimator $\hat{{\Greekmath 0112}}$, bootstrap inference is not invalidated. Rigorous
statements in this sense will be provided in Corollary \ref{corollary
bootstrap with boundary}.
On the other hand, if ${\Greekmath 0112} _{0}\in \partial \Theta $ the bootstrap
statistic converges as follows:
\begin{eqnarray}
&&n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})\overset{w^{\ast }}{\rightarrow }
_{w}\ell ^{\ast }|(M,\ell )
\label{eq asy for standard BS with param on the boundary} \\
&&\ell ^{\ast }\overset{}{:=}\underset{{\Greekmath 0115} \in \Lambda _{\ell }^{\ast }}{
\arg \min }||{\Greekmath 0115} -M^{-1/2}{\Greekmath 0118} ^{\ast }||_{M}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }\Lambda _{\ell
}^{\ast }\overset{}{:=}\left. \{{\Greekmath 0115} \in \mathbb{R}^{2}:\dot{g}^{\prime
}{\Greekmath 0115} \overset{}{\geq }-\dot{g}^{\prime }\ell \}\right. \notag
\end{eqnarray}
where ${\Greekmath 0118} ^{\ast }\sim N(0,{\Greekmath 011B} _{e}^{2}I_{2})$ is independent of $
(M,\ell )$. In contrast with the case ${\Greekmath 0112} _{0}\in $$\limfunc{int}$$
\Theta $ and additionally to the random matrix $M$, in (\ref{eq asy for
standard BS with param on the boundary}) also the random vector $\ell $
appears as a vehicle of limiting bootstrap randomness. Moreover, 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{{\Greekmath 0112}}-{\Greekmath 0112} _{0})$,
inasmuch as $\Lambda _{\ell }^{\ast }$ in (\ref{eq asy for standard BS with
param on the boundary}) is a random half-plane, rather than the original
admissible set $\Lambda $ of (\ref{eq asy distribution}). The kind of
limiting bootstrap randomness introduced by $\ell $ is similar to the one
established in the introductory location model and, in general, it
invalidates bootstrap inference. The reason for the discrepancy between $
\Lambda $ and $\Lambda _{\ell }^{\ast }$ is that the parameter space of the
standard fixed-regressor wild bootstrap does not approximate well the
original mutual position of the true value ${\Greekmath 0112} _{0}$ and the boundary,
unless $g(\hat{{\Greekmath 0112}})=0$. Other, non-standard bootstrap schemes may be
designed in order to provide better approximations, at least under the null
hypothesis. Under these schemes the possible boundary position of ${\Greekmath 0112}
_{0}$ is no longer a vehicle of limiting bootstrap randomness, while the
role of the random matrix $M$ in the limit bootstrap distribution is
maintained. This topic is analyzed in the next section.
\section{Asymptotically 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{{\Greekmath 0112}}^{\ast }$ constrained to belong to a bootstrap
parameter space $\Theta^{\ast}$ satisfying the following assumption.
\begin{exercise}
\label{asstheta} The bootstrap parameter space is $\Theta ^{\ast }:=\{{\Greekmath 0112}
\in \mathbb{R}^{2}:g({\Greekmath 0112} )\geq g^{\ast } (\hat{{\Greekmath 0112}})\}$ for some
function $g^{\ast }:\mathbb{R}^{2}\rightarrow \mathbb{R}$ which is
continuously differentiable in a neighborhood of ${\Greekmath 0112} _{0}$ and satisfies
$g^{\ast }({\Greekmath 0112} )\leq g({\Greekmath 0112} )$ for ${\Greekmath 0112} \in \Theta $.
\end{exercise}
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{{\Greekmath 0112}}$ to lie on the boundary of
the bootstrap parameter space $\Theta ^{\ast }$.\footnote{{\small {}}{}As $
\hat{{\Greekmath 0112}}\overset{p}{\rightarrow }{\Greekmath 0112} _{0}$ and $\dot{g}({\Greekmath 0112}
_{0})\neq 0$, it follows by continuity that $P(\dot{g}(\hat{{\Greekmath 0112}})\neq
0)\rightarrow 1$, such that, with probability approaching one, $\hat{{\Greekmath 0112}}$
is not a stationary point of $g$. In particular, with probability
approaching one, $\hat{{\Greekmath 0112}}$ is not a local minimizer of $g$, implying
that $\hat{{\Greekmath 0112}}\in \partial $$\Theta ^{\ast }$ under $\Theta ^{\ast
}=\{{\Greekmath 0112} \in R^{2}:g({\Greekmath 0112} )\geq g(\hat{{\Greekmath 0112}})\}$.} Finally, setting $
g^{\ast }=g-|g|^{1+{\Greekmath 0114} }$ for some ${\Greekmath 0114} >0$ introduces a correction,
in the spirit of an alternative to the standard bootstrap mentioned in
Andrews (2000, p.403, Method two), Fang and Santos (2019, Example 2.1) and
Cavaliere et al. (2022), 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 ${\Greekmath 0112} _{0}$ belongs
to the original boundary $\partial \Theta $ or not. Other choices of $
g^{\ast }$ with the same implication are discussed in Sections \ref
{subsec:Fang-and-Santos} and \ref{sec tuning par}.
To formulate the next theorem, recall $M$ and $\ell({\Greekmath 0112}_{0})$ introduced
in Section \ref{subsec:Asymptotic-distributions}, and let $
{\Greekmath 0118}^{\ast}|(M,\ell({\Greekmath 0112}_{0}))\sim N(0,{\Greekmath 011B}_e^2 I_2)$ as in Section \ref
{subsec:Standard-bootstrap-invalidity}. Let also $D_{n}=\{y_{t},x_{n,t-1}
\}_{t=1}^{n}$ denote the original data. Finally, call a convergence in
distribution $Z_{n}\overset{w}{\rightarrow}Z$ and a weak convergence of
random distributions $Z_{n}^{*}|D_{n}\overset{w}{\rightarrow}_{w}Z^{*}|Y$
joint, denoted as $(Z_{n},(Z_{n}^{*}|D_{n}))$$\overset{w}{\rightarrow}
_{w}(Z,(Z^{*}|Y))$, if $(Z_{n},E\{g(Z_{n}^{*})|D_{n}\})\overset{w}{
\rightarrow}(Z,E\{g(Z)|Y\})$ for all continuous and bounded real functions $
g $ with matching domain.
\begin{theorem}
\label{Lemma bootstrap with boundary}Under a null hypothesis $\mathsf{H}_{0}$
as in $\mathscr{G}{}_{1}$--$\,\mathscr{G}{}_{3}$ and under Assumptions 1-3,
the bootstrap estimator $\hat{{\Greekmath 0112}}^{\ast}$ obtained by regressing $
y_{t}^{\ast}$ of (\ref{eq BS for PR}) on $(1,x_{n,t-1})^{\prime}$ under the
constraint $\hat{{\Greekmath 0112}}^{\ast}\in\Theta^{\ast}$, satisfies
\begin{equation*}
(n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112}_{0}),(n^{1/2}(\hat{{\Greekmath 0112}}^{\ast}-\hat{{\Greekmath 0112}}
)|D_{n}))\overset{w}{\rightarrow}_{w}\left(\ell({\Greekmath 0112}_{0}),(\ell^{\ast}(
{\Greekmath 0112}_{0})|(M,\ell({\Greekmath 0112}_{0})))\right)\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }
\label{eq asy distrib for general bootstraps}
\end{equation*}
where in the case $g^{\ast}({\Greekmath 0112}_{0})<g({\Greekmath 0112}_{0})$,
\begin{equation}
\ell^{\ast}({\Greekmath 0112}_{0})=\tilde{\ell}^{\ast}:=M^{-1/2}{\Greekmath 0118}^{\ast}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ with }
\tilde{\ell}^{\ast}|(M,\ell({\Greekmath 0112}_{0}))=\tilde{\ell}|M \label{eq semno}
\end{equation}
in the sense of a.s. equality of conditional distributions, whereas in the
case $g^{\ast}({\Greekmath 0112}_{0})=g({\Greekmath 0112}_{0})$,
\begin{equation}
\ell^{\ast}({\Greekmath 0112}_{0})=\ell^{\ast}:=\underset{{\Greekmath 0115}\in\Lambda_{\ell}^{
\ast}}{\arg\min}||{\Greekmath 0115}-M^{-1/2}{\Greekmath 0118}^{\ast}||_{M}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }
\Lambda_{\ell}^{\ast}:=\{{\Greekmath 0115}\in\mathbb{R}^{2}:\dot{g}^{\prime}{\Greekmath 0115}
\geq(\dot{g}^{\ast}-\dot{g})^{\prime}\ell({\Greekmath 0112}_{0})\}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{. }
\label{eq amans}
\end{equation}
\end{theorem}
\noindent The following conclusions could be drawn.
\begin{itemize}
\item[(i)] Consider first configurations $\mathscr{G}{}_{1}$ and $
\mathscr{G}{}_{2}$ under $\mathsf{H}_{0}$, such that $g({\Greekmath 0112} _{0})=0$.
Consider the magnitude order, in probability, of the distance between the
bootstrap `true' value $\hat{{\Greekmath 0112}}$ and the bootstrap boundary $\partial
\Theta ^{\ast }$ as a measure of how precisely the bootstrap approximates
the geometry of $\mathscr{G}_{1}$ and $\mathscr{G}_{2}$. As seen previously,
the standard bootstrap corresponds to $g^{\ast }=0$ and approximates the
geometry up to an exact magnitude order of $n^{-1/2}$, resulting in a
situation where the belonging of ${\Greekmath 0112} _{0}$ to the boundary contributes
to the randomness of the limit bootstrap distribution given by (\ref{eq asy
for standard BS with param on the boundary}) and (\ref{eq amans}) via
conditioning on the r.v. $\ell ({\Greekmath 0112} _{0})=\ell $. Conversely, bootstrap
schemes employing $g^{\ast }({\Greekmath 0112} _{0})=g({\Greekmath 0112} _{0})$ and $\dot{g}^{\ast
}=\dot{g}$, such that the bootstrap boundary is tangent to the original
boundary at ${\Greekmath 0112} _{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 random variable $M$, as now $
\ell ^{\ast }|(M,\ell )\overset{}{=}\ell |M$ in the sense of a.s. equality
of random distributions; 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{{\Greekmath 0112}}
^{\ast }-\hat{{\Greekmath 0112}})\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{{\Greekmath 0112}}$ on the bootstrap boundary, and the choices $g^{\ast
}=g-|g|^{1+{\Greekmath 0114} }$ for some ${\Greekmath 0114} >0$. In general, the limit
distribution of the resulting bootstrap estimator is random, with randomness
depending on both the stochastic regressor and the position of ${\Greekmath 0112} _{0}$
relative to the boundary.
\item[(ii)] Consider now the case in $\mathscr{G}{}_{3}$, such that $
g({\Greekmath 0112} _{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 limit distribution if $g({\Greekmath 0112} _{0})=0$
, while the `restricted' one would fail if $g({\Greekmath 0112} _{0})>0$. As an
alternative, consider the bootstrap based on $g^{\ast }=g-|g|^{1+{\Greekmath 0114} }$
for some ${\Greekmath 0114} >0$. If ${\Greekmath 0112} _{0}\in \partial \Theta $, then this choice
puts the bootstrap true value $\hat{{\Greekmath 0112}}$ at the asymptotically
negligible distance of $o_{p}(n^{-1/2})$ from the bootstrap boundary,
whereas if ${\Greekmath 0112} _{0}\in $$\limfunc{int}$$\Theta $, then $\hat{{\Greekmath 0112}}$ is
bounded away from the bootstrap boundary, in probability. This guarantees
bootstrap validity under some regularity conditions, see (ii)\ in Corollary
\ref{corollary bootstrap with boundary} below.
\end{itemize}
\medskip{}
In general, bootstrap validity in the sense of (\ref{eq unconditional
validity}) can be evaluated through the following corollary of Theorem \ref
{Lemma bootstrap with boundary} above.
\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}
(n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112}_{0}),(n^{1/2}(\hat{{\Greekmath 0112}}^{\ast}-\hat{{\Greekmath 0112}}
)|D_{n}))\overset{w}{\rightarrow}_{w}\left(\ell({\Greekmath 0112}_{0}),(\ell(
{\Greekmath 0112}_{0})|M)\right)\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ } \label{eq bound ave}
\end{equation}
is that: (i) under $\mathscr{G}{}_{1}$ and $\mathscr{G}{}_{2}$, $
g({\Greekmath 0112}_{0})=g^{\ast}({\Greekmath 0112}_{0})$ and $\dot{g}=\dot{g}^{\ast}$; (ii) under
$\mathscr{G}{}_{3}$, either $g({\Greekmath 0112}_{0})=g^{\ast}({\Greekmath 0112}_{0})$ and $\dot{g}
=\dot{g}^{\ast}$, or $g({\Greekmath 0112}_{0})>\max\{0,g^{\ast}({\Greekmath 0112}_{0})\}$.
Moreover, under (\ref{eq bound ave}) the bootstrap is valid in the sense of (
\ref{eq unconditional validity}) for any pair of statistics ${\Greekmath 011C} _{n},{\Greekmath 011C}
_{n}^{\ast }$ such that, under $\mathsf{H}_{0}$, ${\Greekmath 011C} _{n}={\Greekmath 011E} (n^{1/2}(
\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{0}))+o_{p}(1)$ and ${\Greekmath 011C} _{n}^{\ast }={\Greekmath 011E} (n^{1/2}(
\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}}))+o_{p}(1)$ for some continuous real
function ${\Greekmath 011E} $ such that ${\Greekmath 011E} (\ell ({\Greekmath 0112} _{0}))$ is well-defined a.s.
\end{corollary}
\noindent The class of functions $g^{\ast}=g-|g|^{1+{\Greekmath 0114}}$ for ${\Greekmath 0114}>0$
satisfies both conditions (i) and (ii) of Corollary \ref{corollary bootstrap
with boundary}; hence, the ensuing bootstrap inference is 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({\Greekmath 0112}_{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. Section \ref{sec one sided tests} focuses on this
setup.
\bigskip{}
\noindent \textsc{Remark}. The practical implications of Corollary \ref
{corollary bootstrap with boundary} depend on the choice of the statistic $
{\Greekmath 011C}_{n}$ and the respective function ${\Greekmath 011E}$, which will typically be a
linear ${\Greekmath 011E}(l)=l^{\prime}\frac{\partial r}{\partial{\Greekmath 0112}^{\prime}}
({\Greekmath 0112}_{0})$ arising from the delta method, with $l\in\mathbb{R}^{2},\frac{
\partial r}{\partial{\Greekmath 0112}^{\prime}}({\Greekmath 0112}_{0})\neq0$. For instance, if $
g({\Greekmath 0112}_{0})=0$ and ${\Greekmath 011E}(\ell)$ depends on $\ell$ only through $\dot{g}
^{\prime}\ell=\max\{0,\dot{g}^{\prime}M^{-1/2}{\Greekmath 0118}\}$, then the cdf of $
{\Greekmath 011E}(\ell)$ will not be continuous. Still, the bootstrap will be valid in
the sense of (\ref{eq unconditional validity}), meaning that the largest
open subset of $[0,1]$ on which the bootstrap test is correctly sized as $
n\rightarrow\infty$ coincides with the analogous set for the asymptotic
test. This set will be smaller than $(0,1)$, however. An example is $
{\Greekmath 011C}_{n}=n^{1/2}g(\hat{{\Greekmath 0112}})$, ${\Greekmath 011C}_{n}^{\ast}=n^{1/2}(g(\hat{{\Greekmath 0112}}
^{\ast})-g(\hat{{\Greekmath 0112}}))$ with ${\Greekmath 011E}(\ell)=\dot{g}^{\prime}\ell$,
corresponding to a right-sided test of $\mathsf{H}_{0}:g({\Greekmath 0112}_{0})=0$.
\medskip{}
\noindent \textsc{Remark}. Bootstrap validity extends readily to statistics
where\ $n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{0})$ is normalized by some $\hat{\Sigma
}=\Sigma (M_{n})+o_{p}(1)$ for a function $\Sigma :\mathbb{R}^{2\times
2}\rightarrow \mathbb{R}^{2\times 2}$ which is continuous on the set of
positive definite matrices. Specifically, bootstrap validity holds if, under
$\mathsf{H}_{0}$, ${\Greekmath 011C} _{n}={\Greekmath 011E} (n^{1/2}\hat{\Sigma}(\hat{{\Greekmath 0112}}-{\Greekmath 0112}
_{0}))+o_{p}(1)$ and ${\Greekmath 011C} _{n}^{\ast }={\Greekmath 011E} (n^{1/2}\hat{\Sigma}(\hat{{\Greekmath 0112}
}^{\ast }-\hat{{\Greekmath 0112}}))+o_{p}(1)$, where ${\Greekmath 011E} $ is a continuous real
function such that ${\Greekmath 011E} (\Sigma (M)\ell ({\Greekmath 0112} _{0}))$ is a.s.
well-defined.$\hfill \square $
\section{Discussion and extensions}
\label{sec:Discussion}
In this section we address the following three issues: (i) the validity of
one-sided bootstrap tests; (ii) a discussion of the bootstrap schemes from
Corollary \ref{corollary bootstrap with boundary} within the paradigm of
some previous works -- specifically, Fang and Santos (2019) and Hong and Li
(2020); and (iii) uniform bootstrap validity.
\subsection{Validity of one-sided standard bootstrap tests}
\label{sec one sided tests}
Under case $\mathscr{G}{}_{1}$, consider testing $\mathsf{H}_{0}:g({\Greekmath 0112}
_{0})=0$ against the alternative $H_{1}:g({\Greekmath 0112} _{0})>0$ using a one-sided
test and the standard bootstrap, i.e., with $g^{\ast }=0$. For a test
statistic of the form ${\Greekmath 011C} _{n}:=n^{1/2}g(\hat{{\Greekmath 0112}})$, a bootstrap
counterpart is given by ${\Greekmath 011C} _{n}^{\ast }:=n^{1/2}(g(\hat{{\Greekmath 0112}}^{\ast
})-g(\hat{{\Greekmath 0112}}))$ and the associated one-sided bootstrap test rejects for
large values of the bootstrap \emph{p}-value $p_{n}^{\ast }:=P^{\ast }({\Greekmath 011C}
_{n}^{\ast }\leq {\Greekmath 011C} _{n})$; equivalently, for small values of $\tilde{p}
_{n}^{\ast }:=1-p_{n}^{\ast }$. As for $\hat{{\Greekmath 0112}}^{\ast }$, also ${\Greekmath 011C}
_{n}^{\ast }$ is affected in the limit by extra randomness due to ${\Greekmath 0112}
_{0}$ 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
\begin{equation}
({\Greekmath 011C} _{n},({\Greekmath 011C} _{n}^{\ast }|D_{n}))\overset{w}{\rightarrow }_{w}\left(
\dot{g}^{\prime }\ell \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,}\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 ))),
\label{eq convergence after delta method}
\end{equation}
with $\ell $, $\ell ^{\ast }$ and $\tilde{\ell}^{\ast }$ as previously
defined. For ${\Greekmath 011C} _{n}^{\ast }$, however, the randomness induced by
conditioning on $\ell $ affects the sample paths of the associated random
cdf on the negative half-line alone, because $\dot{g}^{\prime }\ell \geq 0$,
and is thus irrelevant for bootstrap tests with nominal size 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 3.1 in Cavaliere and
Georgiev (2020), the proof being analogous, where conditions for bootstrap
validity restricted to a subset of nominal testing levels are formulated.
\begin{theorem}
\label{th2}Let there exist a random variable ${\Greekmath 011C} $ and a random element $X$
, both defined on the same probability space, such that the support of ${\Greekmath 011C}
_{n}$ is contained in a finite or infinite closed interval $T$, and $({\Greekmath 011C}
_{n},F_{n}^{\ast })\overset{w}{\rightarrow }({\Greekmath 011C} ,F)$ in $\mathbb{R}\times
\mathscr{D}(T)$ for $F_{n}^{\ast }(u):=P({\Greekmath 011C} _{n}^{\ast }\leq u|D_{n})$ and
$F(u):=P({\Greekmath 011C} \leq u|X)$, $u\in T$. If the possibly random cdf $F$ is
sample-path continuous on $T$, then the bootstrap p-value $p_{n}^{\ast
}:=F_{n}^{\ast }({\Greekmath 011C} _{n})$\ satisfies
\begin{equation*}
P(p_{n}^{\ast }\leq q)\rightarrow q
\end{equation*}
for $q$ such that $q\in F(T)$ a.s.
\end{theorem}
\noindent By Theorem \ref{th2} with $T=[0,\infty)$, which corresponds to the
support of ${\Greekmath 011C}_{n}$ and ${\Greekmath 011C}:=\dot{g}^{\prime}\ell$, it follows that the
standard bootstrap applied to the one-sided statistic ${\Greekmath 011C}_{n}$ is
asymptotically correctly sized for nominal test sizes in $(0,\frac{1}{2})$.
\subsection{Fang and Santos (2019) and Hong and Li (2020)}
\label{subsec:Fang-and-Santos}
In this section we put the geometric considerations of Section \ref{sec par
on the boundary main} in the perspective of Fang and Santos (2019), and of
the numerical bootstrap of Hong and Li (2020). The discussion is often
specialized to the case of an affine constraint.
Consider the constrained OLS estimator $\hat{{\Greekmath 0112}}$ of Section \ref
{subsec:Asymptotic-distributions}. Its limit distribution, see (\ref{eq asy
distribution}), is the distribution of $\ell ({\Greekmath 0112} _{0})={\Greekmath 0127} _{{\Greekmath 0112}
_{0}}(M^{-1/2}{\Greekmath 0118} )$ with
\begin{equation*}
{\Greekmath 0127} _{{\Greekmath 0112} _{0}}(u)=
\begin{cases}
\begin{array}{c}
u \\
\dot{g}_{\perp }(\dot{g}_{\perp }^{\prime }M\dot{g}_{\perp })^{-1}\dot{g}
_{\perp }^{\prime }Mu+M^{-1}\dot{g}(\dot{g}^{\prime }M^{-1}\dot{g})^{-1}\max
\{0,\dot{g}^{\prime }u\}
\end{array}
&
\begin{array}{c}
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ $g({\Greekmath 0112} _{0})>0$} \\
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{if}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ $g({\Greekmath 0112} _{0})=0$}
\end{array}
,
\end{cases}
\end{equation*}
with $u\in \mathbb{R}^{2}$. By a projection identity, the expression in the
second line of the previous display collapses to $u$ whenever $\dot{g}
^{\prime }u\geq 0$. Note that the distribution of $M^{-1/2}{\Greekmath 0118} $ conditional
on $M$ can be estimated consistently by the distribution of the
unconstrained bootstrap OLS estimator conditional on the data; that is,
\begin{equation*}
n^{1/2}\tilde{({\Greekmath 0112} ^{\ast }}-\hat{{\Greekmath 0112}})\overset{w^{\ast }}{\rightarrow
}_{w}M^{-1/2}{\Greekmath 0118} |M\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
One can then ask what properties of an estimator $\hat{{\Greekmath 0127}}_{n}$ of $
{\Greekmath 0127} _{{\Greekmath 0112} _{0}}$ are sufficient for $\hat{{\Greekmath 0127}}_{n}(n^{1/2}\tilde{
({\Greekmath 0112} ^{\ast }}-\hat{{\Greekmath 0112}}))\overset{w^{\ast }}{\rightarrow }_{w}{\Greekmath 0127}
_{{\Greekmath 0112} _{0}}(M^{-1/2}{\Greekmath 0118} )|M$ to hold. Fang and Santos (2019) address this
question in the setup of deterministic transformations of non-random limit
distributions, instead of the random transformation ${\Greekmath 0127} _{{\Greekmath 0112} _{0}}$
of the random distribution $M^{-1/2}{\Greekmath 0118} |M$. Although not directly
applicable here, Theorem 3.2 of Fang and Santos (2019) provides the key
insight: there should be sufficient uniformity in the convergence of $\hat{
{\Greekmath 0127}}_{n}$ to ${\Greekmath 0127} _{{\Greekmath 0112} _{0}}$. Consider for instance
\begin{eqnarray}
\hat{{\Greekmath 0127}}_{n}(u) &=&\hat{\dot{g}}_{\perp }(\hat{\dot{g}}_{\perp
}^{\prime }M_{n}\hat{\dot{g}}_{\perp })^{-1}\hat{\dot{g}}_{\perp }^{\prime
}M_{n}u \label{eq:phiest} \\
&&+M_{n}^{-1}\hat{\dot{g}}(\hat{\dot{g}}^{\prime }M_{n}^{-1}\hat{\dot{g}}
)^{-1}\max \{-n^{1/2}|g(\hat{{\Greekmath 0112}})|^{1+{\Greekmath 0114} },\hat{\dot{g}}^{\prime }u\}
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }u\overset{}{\in }\mathbb{R}^{2}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,} \notag
\end{eqnarray}
where $\hat{\dot{g}}=\frac{\partial }{\partial {\Greekmath 0112} ^{\prime }}g(\hat{
{\Greekmath 0112}})$, $M_{n}=n^{-1}\sum_{t=1}^{n}\tilde{x}_{t}\tilde{x}_{t}^{\prime }$
with $\tilde{x}_{t}=(1,x_{n,t-1})^{\prime }$, and ${\Greekmath 0114} >0$. Given that $
M_{n}\overset{w}{\rightarrow }M$, $\hat{\dot{g}}\overset{p}{\rightarrow }
\dot{g}$ and $n^{1/2}|g(\hat{{\Greekmath 0112}})|^{1+{\Greekmath 0114} }\overset{p}{\rightarrow }
\infty \mathbb{I}_{\{g({\Greekmath 0112} _{0})>0\}}$, it is easily checked that $\tilde{
{\Greekmath 0127}}_{n}\overset{w}{\rightarrow }{\Greekmath 0127} _{{\Greekmath 0112} _{0}}$ on $\mathscr{C}
_{2}(\mathbb{R}^{2})$, and the convergence of $\hat{{\Greekmath 0127}}_{n}$ is joint
with that of $n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{0})$ and $n^{1/2}\tilde{({\Greekmath 0112}
^{\ast }}-\hat{{\Greekmath 0112}})$, the latter one given the data. These facts are
sufficient to ensure that
\begin{equation*}
(n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{0}),(\hat{{\Greekmath 0127}}_{n}(n^{1/2}\tilde{({\Greekmath 0112}
^{\ast }}-\hat{{\Greekmath 0112}}))|D_{n}))\overset{w}{\rightarrow }_{w}\left( \ell
({\Greekmath 0112} _{0}),(\ell ({\Greekmath 0112} _{0})|M)\right) \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }
\end{equation*}
on $\mathbb{R}^{4}$, essentially as a consequence of the continuous mapping
theorem (CMT) and the continuity of the evaluation map from $\mathscr{C}_{2}(
\mathbb{R}^{2})\times \mathbb{R}^{2}$ to $\mathbb{R}^{2}$. As the previous
limit is the same as in Corollary \ref{corollary bootstrap with boundary},
it follows that bootstrap inference based on the distribution of $\hat{
{\Greekmath 0127}}_{n}(n^{1/2}\tilde{({\Greekmath 0112} ^{\ast }}-\hat{{\Greekmath 0112}}))$ conditional on
the data is valid. Moreover, for the valid bootstrap schemes obtained from
Corollary \ref{corollary bootstrap with boundary} with $g^{\ast
}=g-|g|^{1+{\Greekmath 0114} }$, ${\Greekmath 0114} >0$, the bootstrap estimator $\hat{{\Greekmath 0112}}
^{\ast }$ satisfies $n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})=\hat{{\Greekmath 0127}}
_{n}(n^{1/2}\tilde{({\Greekmath 0112} ^{\ast }}-\hat{{\Greekmath 0112}}))$ for affine functions $g$
. It can be concluded that $\hat{{\Greekmath 0127}}_{n}$ of (\ref{eq:phiest})
implicitly performs the geometric approximation proposed in Section \ref{sec
par on the boundary main}, and so does any other estimator of ${\Greekmath 0127}
_{{\Greekmath 0112} _{0}}$ that converges like $\hat{{\Greekmath 0127}}_{n}$.
We now argue that such an estimator of ${\Greekmath 0127}_{{\Greekmath 0112}_0}$ is embedded in
the numerical bootstrap of Hong and Li (2020). This ensures the validity of
the numerical bootstrap for the predictive regression of interest here,
though at the cost of a slower consistency rate of the bootstrap estimator
than in Corollary \ref{corollary bootstrap with boundary}. Let $
s_{n}\rightarrow \infty $ be a sequence such that $n^{-1/2}s_{n}\rightarrow
0 $. Hong and Li (2020) propose in their eq. (4.9) a bootstrap estimator $
\hat{{\Greekmath 0112}}_{nb}^{\ast }$ where the constraint set of our $\ell ({\Greekmath 0112}
_{0})$ (i.e., $\mathbb{R}^{2}$ if ${\Greekmath 0112} _{0}\in \mathrm{int}\Theta $ and
the half-plane $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{$\Lambda $}$ if ${\Greekmath 0112} _{0}\in \partial \Theta $),
would be estimated by $\Lambda _{nb}^{\ast }=\{{\Greekmath 0115} \in \mathbb{R}^{2}:g(
\hat{{\Greekmath 0112}}+s_{n}^{-1}{\Greekmath 0115} )\geq 0\}$, the implied bootstrap parameter
space being $\Theta _{nb}^{\ast }=\hat{{\Greekmath 0112}}+s_{n}^{-1}\Lambda _{nb}^{\ast
}=\Theta .$ The bootstrap estimator itself, adapted to our setup, could be
written as
\begin{equation*}
\hat{{\Greekmath 0112}}_{nb}^{\ast }=\underset{g({\Greekmath 0112} )\geq 0}{\arg \min }
||s_{n}({\Greekmath 0112} -\hat{{\Greekmath 0112}})-M_{n}^{-1/2}{\Greekmath 0118} _{n}^{\ast }||_{M_{n}},
\end{equation*}
where ${\Greekmath 0118} _{n}^{\ast }$ is a bootstrap variable such that ${\Greekmath 0118} _{n}^{\ast }
\overset{w^{\ast }}{\rightarrow }_{p}N(0,I_{2})$; e.g., ${\Greekmath 0118} _{n}^{\ast
}=n^{1/2}M_{n}^{1/2}(\tilde{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}})$. In the simple
case of an affine $g$ we find the explicit expression
\begin{equation*}
s_{n}(\hat{{\Greekmath 0112}}_{nb}^{\ast }-\hat{{\Greekmath 0112}})=\bar{{\Greekmath 0127}}
_{n}(M_{n}^{-1/2}{\Greekmath 0118} _{n}^{\ast })
\end{equation*}
for $\bar{{\Greekmath 0127}}_{n}$ defined similarly to $\hat{{\Greekmath 0127}}_n$, with the
only difference that in (\ref{eq:phiest}) the term $n^{1/2}|g(\hat{{\Greekmath 0112}}
)|^{1+{\Greekmath 0114}}$ is replaced by $s_{n}g(\hat{{\Greekmath 0112}})$. As $s_{n}g(\hat{{\Greekmath 0112}}
)\overset{p}{\rightarrow}\infty\mathbb{I}_{\{g({\Greekmath 0112}_0)>0\}}$ similarly to $
n^{1/2}|g(\hat{{\Greekmath 0112}})|^{1+{\Greekmath 0114} }$, ${\Greekmath 0114} >0$, it follows that $\bar{
{\Greekmath 0127}}_{n}$ converges similarly to $\hat{{\Greekmath 0127}}_n$. As a result,
\begin{equation*}
(n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{0}),(s_{n}(\hat{{\Greekmath 0112}}_{nb}^{\ast }-\hat{
{\Greekmath 0112}})|D_{n}))\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }\overset{w}{\rightarrow }_{w}\left( \ell ({\Greekmath 0112}
_{0}),(\ell ({\Greekmath 0112} _{0})|M)\right) ,
\end{equation*}
ensuring the validity of the numerical bootstrap, though the consistency
rate of $\hat{{\Greekmath 0112}}_{nb}^{\ast }$ is $s_{n}=o(n^{1/2})$ instead of $
n^{1/2} $. In contrast, the rate of $n^{1/2}$ would be achieved by our
proposed bootstrap estimator, with $n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}}
)=\bar{{\Greekmath 0127}}_{n}(M_{n}^{-1/2}{\Greekmath 0118} _{n}^{\ast })$, if $\Theta ^{\ast
}=\{{\Greekmath 0112} \in \mathbb{R}^{2}:g({\Greekmath 0112} )\geq g_{n}^{\ast }(\hat{{\Greekmath 0112}})\}$
with $g_{n}^{\ast }=g-n^{-1/2}s_{n}|g|$ is specified in Assumption 3.
\subsection{Uniformity considerations}
In agreement with Chatterjee and Lahiri (2011), Remark 3, the focus in this
paper is on pointwise bootstrap validity. For situations where uniform
bootstrap validity is of interest, our key takeaways are similar to the
literature on non-random limiting bootstrap measures. First, for the null
hypothesis $\mathscr{G}{}_{1}$ that the true parameter value lies on the
boundary of the parameter space, the pointwise-valid bootstrap schemes
outlined in Corollary \ref{corollary bootstrap with boundary} display
asymptotic rejection probabilities matching the local power of the bootstrap
test whenever the true parameter value varies along a sequence that is local
to the boundary at the $n^{-1/2}$ rate. This fact is associated with
rejection frequencies above the nominal test size along
local-to-the-boundary parameter sequences (cf. Fang and Santos, 2019, Remark
3.6). Second, if conservative bootstrap inference along such parameter
sequences is desired, it can be achieved for hypotheses $\mathscr{G}{}_{1}$--
$\mathscr{G}{}_{3}$ by adapting the approach of Doko Tchatoka and Wang
(2021), and Cavaliere et al. (2024), at the cost of a potential decrease in
power.
To illustrate these points, consider a sequence of true parameter values $
{\Greekmath 0112} _{n}={\Greekmath 0112} _{0}+n^{-1/2}{\Greekmath 0123} $ such that $g({\Greekmath 0112} _{0})=0$ and
$\dot{g}^{\prime }{\Greekmath 0123} =c>0$ with $g({\Greekmath 0112} _{n})=n^{-1/2}c+o(n^{-1/2})$
. Moreover, let
\begin{equation}
\ell ({\Greekmath 0123} ,c):={\Greekmath 0123} +\underset{{\Greekmath 0115} \in \Lambda ^{c}}{\arg
\min }||{\Greekmath 0115} -M^{-1/2}{\Greekmath 0118} ||_{M}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{, }\Lambda ^{c}=\{{\Greekmath 0115} \in
\mathbb{R}^{2}:\dot{g}^{\prime }{\Greekmath 0115} +c\geq 0\}, \label{eq:ltc}
\end{equation}
and $\ell (0,0)=\ell $ of eq. (\ref{eq asy distribution}). Then, the joint
convergence result
\begin{equation}
(n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{0}),(n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}}
)|D_{n}))\overset{w}{\rightarrow }_{w}(\ell ({\Greekmath 0123} ,c),(\ell (0,0)|M))
\label{eq:lct 2}
\end{equation}
holds for the bootstrap schemes satisfying conditions (i) and (ii) of
Corollary \ref{corollary bootstrap with boundary}. For a function $r:\mathbb{
R}^{2}\rightarrow \mathbb{R}$ which is continuously differentiable close to $
{\Greekmath 0112} _{0}$, consider the statistics ${\Greekmath 011C} _{n}=n^{1/2}r(\hat{{\Greekmath 0112}})$ and
${\Greekmath 011C} _{n}^{\ast }=n^{1/2}(r(\hat{{\Greekmath 0112}}^{\ast })-r(\hat{{\Greekmath 0112}}))$, and
distinguish among the extreme possibilities $\dot{r}={\Greekmath 010B} \dot{g}$ with $
{\Greekmath 010B} >0$, and $\dot{r}={\Greekmath 010B} \dot{g}_{\perp }$ with ${\Greekmath 010B} \neq 0$,
where $\dot{r}=\frac{\partial r}{\partial {\Greekmath 0112} ^{\prime }}({\Greekmath 0112} _{0})$.
The former possibility arises in testing the null hypothesis that ${\Greekmath 0112}
_{0}$ lies on the boundary (e.g., with $r=g$), whereas the latter one arises
when the null is orthogonal to the boundary (e.g., with $r({\Greekmath 0112} )={\Greekmath 0112}
_{1}$, $\mathsf{H}_{0}:{\Greekmath 0112} _{1}=0$ and $\Omega =\mathbb{R}\times \lbrack
0,\infty )$). If $\dot{r}=\dot{g}$ and, without loss of generality, ${\Greekmath 010B}
=1$, the delta method yields
\begin{equation*}
({\Greekmath 011C} _{n},({\Greekmath 011C} _{n}^{\ast }|D_{n}))\overset{w}{\rightarrow }_{w}(\max \{0,
\dot{g}^{\prime }M^{-1/2}{\Greekmath 0118} +c\},(\max \{0,\dot{g}^{\prime }M^{-1/2}{\Greekmath 0118}
\}|M)).
\end{equation*}
With ${\Greekmath 010D} _{M}:=(\dot{g}^{\prime }M^{-1}\dot{g})^{-1/2}$, it follows that
\begin{align*}
P^{\ast }({\Greekmath 011C} _{n}^{\ast }\leq {\Greekmath 011C} _{n})& \overset{w}{\rightarrow }{\Greekmath 0119}
(c;M,{\Greekmath 0118} ):=\tfrac{1}{2}\mathbb{I}_{\{\dot{g}^{\prime }M^{-1/2}{\Greekmath 0118}
+c<0\}}+\Phi ({\Greekmath 010D} _{M}(\dot{g}^{\prime }M^{-1/2}{\Greekmath 0118} +c))\mathbb{I}_{\{
\dot{g}^{\prime }M^{-1/2}{\Greekmath 0118} +c\geq 0\}} \\
& >{\Greekmath 0119} (0;M,{\Greekmath 0118} ),
\end{align*}
where ${\Greekmath 0119} (0;M,{\Greekmath 0118} )\overset{d}{=}\tfrac{1}{2}\mathbb{I}_{\{U<0.5\}}+U
\mathbb{I}_{\{U\geq 0.5\}}$, $U\sim U_{[0,1]}$, represents the limit
distribution of the bootstrap \emph{p}-value under the null. The inequality
above implies that bootstrap tests rejecting for large bootstrap \emph{p}
-values will exhibit rejection frequencies above the nominal test size.
On the other hand, if $\dot{r}={\Greekmath 010B} \dot{g}_{\perp }$, it holds that
\begin{equation*}
({\Greekmath 011C} _{n},({\Greekmath 011C} _{n}^{\ast }|D_{n}))\overset{w}{\rightarrow }_{w}(\dot{r}
^{\prime }{\Greekmath 010D} _{M}^{\perp }M{}^{1/2}{\Greekmath 0118} +\dot{r}^{\prime }{\Greekmath 010D}
_{M}^{\perp }M{\Greekmath 0123} ,(\dot{r}^{\prime }{\Greekmath 010D} _{M}^{\perp }M{}^{1/2}{\Greekmath 0118}
|M)),
\end{equation*}
with ${\Greekmath 010D} _{M}^{\perp }:=\dot{g}_{\perp }\dot{g}_{\perp }^{\prime }(\dot{g
}_{\perp }^{\prime }M\dot{g}_{\perp })^{-1}$, such that the boundary is
asymptotically irrelevant. Bootstrap tests of the null that $r({\Greekmath 0112}
_{0})=0 $ could be conservative or liberal according to the sign of $\dot{r}
^{\prime }\dot{g}_{\perp }\dot{g}_{\perp }^{\prime }M{\Greekmath 0123} $. Similar
considerations apply whenever $\dot{r}^{\prime }\dot{g}_{\perp }\neq 0$.
For situations where liberal tests are not desirable, a possible remedy is
suggested next. It involves a continuum of boundaries for the bootstrap
parameter space and its implementation requires a discretization of that
continuum.
Let $\tilde{{\Greekmath 0112}}$ be the unrestricted OLS estimator of ${\Greekmath 0112} $ in
regression (\ref{eq:pr}). For every $s\in I_{n}:=[-|g(\tilde{{\Greekmath 0112}}
)|^{1-{\Greekmath 0116} },g(\hat{{\Greekmath 0112}})]$, let $\hat{{\Greekmath 0112}}_{s}^{\ast }$ be the
bootstrap estimator over the parameter space $\Theta _{s}^{\ast }:=\{{\Greekmath 0112}
\in \mathbb{R}^{2}:g({\Greekmath 0112} )\geq s-g(\hat{{\Greekmath 0112}})^{1+{\Greekmath 0114} }\}$, where $
{\Greekmath 0116} \in (0,1)$ and ${\Greekmath 0114} >0$ are fixed. For a continuously differentiable
function $r$, let $p_{n}^{\ast }(s)$ be the \emph{p}-value of a test based
on ${\Greekmath 011C} _{n}=n^{1/2}r(\hat{{\Greekmath 0112}})$ and ${\Greekmath 011C} _{n}^{\ast }=n^{1/2}(r(\hat{
{\Greekmath 0112}}^{\ast })-r(\hat{{\Greekmath 0112}}))$. Then
\begin{equation*}
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{$\underset{n\rightarrow \infty }{\limsup }$}P(\sup_{s\in
I_{n}}p_{n}^{\ast }(s)\leq q)\leq q
\end{equation*}
for all $q\in \limfunc{int}C$, where $C$ is the set from display (\ref{eq
unconditional validity}) for the benchmark asymptotic test based on the
unfeasible statistic $n^{1/2}(r(\hat{{\Greekmath 0112}})-r({\Greekmath 0112} _{n}))$ and the
simple null hypothesis that ${\Greekmath 0112} _{n}$ is the true parameter value. This
conservative generalization of the validity property (\ref{eq unconditional
validity}) holds irrespective of the values of the drift parameter $c$.
Specifically, the role of $-|g(\tilde{{\Greekmath 0112}})|^{1-{\Greekmath 0116} }$ in the definition
of $I_{n}$ is to guarantee that $g(\hat{{\Greekmath 0112}})-cn^{-1/2}\in I_{n}$ with
probability approaching one. Conservative size control then follows from the
fact that $\hat{{\Greekmath 0112}}_{s}^{\ast }$ with $s=g(\hat{{\Greekmath 0112}})-cn^{-1/2}$
satisfies
\begin{equation*}
(n^{1/2}(\hat{{\Greekmath 0112}}-{\Greekmath 0112} _{n}),(n^{1/2}(\hat{{\Greekmath 0112}}_{s}^{\ast }-\hat{
{\Greekmath 0112}})|D_{n}))\overset{w}{\rightarrow }\left( \ell (0,c),(\ell
(0,c)|M)\right) ;
\end{equation*}
see eqs. (\ref{eq:ltc})--(\ref{eq:lct 2}).
\section{Numerical results and choice of the tuning parameters}
\label{sec:Numerical}
In this section we analyze the finite sample performance of the proposed
bootstrap methodology by means of numerical simulations. The purpose is
twofold: first, to investigate the practical advantage of our methodology
over standard bootstrap methods; second, to provide some practical guidance
on how to choose the functions $g^{\ast }$ and the tuning parameter ${\Greekmath 0114} $
in the definition of the bootstrap parameter space. Simulations are based on
setup $\mathcal{G}_{3}$ of Section \ref{sec par on the boundary setup}, as
it covers the general case of a true parameter value that could, but need
not, lie on the boundary of the parameter space under the null hypothesis.
This section is organized as follows. In Section \ref{sec MC design} we
describe the data generating processes, the null hypotheses and the adopted
bootstrap schemes. In Section \ref{sec MC comment} we discuss the
performance of the tests both under the null and under local alternatives.
Section \ref{sec tuning par} deals with the choice of $g^{\ast }$ and $
{\Greekmath 0114} $. Additional numerical results are provided in the accompanying
supplement, Section \ref{sec:additional MC}.
\subsection{Monte Carlo design}
\label{sec MC design}
We consider the same data generating process (DGP) as in (\ref{eq:prr}),
where $x_{n,t}=n^{-1/2}x_{t}$, $x_{t}:=\sum_{i=1}^{t}{\Greekmath 0122}_{x,i}$, $
{\Greekmath 0122}_{x,t}\sim iid$ $N(0,1)$, with the following specifications of $
{\Greekmath 0122}_{t}$:
\begin{enumerate}
\item ${\Greekmath 0122}_{t}\sim iid$ $N(0,1);$
\item ${\Greekmath 0122}_{t}={\Greekmath 011B}_{t}{\Greekmath 0117}_{t}$, where ${\Greekmath 011B}_{t}^{2}=0.7+0.3
{\Greekmath 0122}_{t-1}^{2}$ and ${\Greekmath 0117}_{t}\sim iid$ $N(0,1)$;
\item ${\Greekmath 0122}_{t}=\sqrt{0.5}{\Greekmath 0122}_{x,t}+\sqrt{0.5}{\Greekmath 0111}_{t}$,
where ${\Greekmath 0111}_{t}\sim iid$ $N(0,1)$.
\end{enumerate}
In each case, $\{{\Greekmath 0122}_{x,t}\}$ is independent of, respectively, $
\{{\Greekmath 0122}_{t}\},$ $\{{\Greekmath 0117}_{t}\}$ and $\{{\Greekmath 0111}_{t}\}$. In Case 1, the
regression errors are independent and Gaussian, while in Case 2 they exhibit
ARCH-type conditional heteroskedasticity. Case 3 allows for correlation
between ${\Greekmath 0122}_{t}$ and the regressor's innovation ${\Greekmath 0122}_{x,t}$.
The parameter space is specified as $\Theta :=\{{\Greekmath 0112} \in \mathbb{R}
^{2}:g({\Greekmath 0112} )\geq 0\}$ where $g({\Greekmath 0112} )={\Greekmath 0112} _{2}$. That is, $\Theta :=
\mathbb{R}\times \left[ 0,\infty \right) $ -- such that its boundary is
given by $\partial \Theta =\mathbb{R}\times \{0\}$. For all parameter
values, we test the null hypothesis $\mathsf{H}_{0}:h({\Greekmath 0112} _{0})=0$, with $
h({\Greekmath 0112} )={\Greekmath 0112} _{1}+{\Greekmath 0112} _{2}$, against the two-sided alternative $
h({\Greekmath 0112} _{0})\neq 0$. To do so, we employ the test statistics ${\Greekmath 011C}
_{n}={\Greekmath 011E} (\sqrt{n}h(\hat{{\Greekmath 0112}}))$ and ${\Greekmath 011C} _{n}^{\ast }={\Greekmath 011E} (\sqrt{n}(h(
\hat{{\Greekmath 0112}}^{\ast })-h(\hat{{\Greekmath 0112}})))$, where ${\Greekmath 011E} (x)=x^{2}$, while $
\hat{{\Greekmath 0112}}$ and $\hat{{\Greekmath 0112}}^{\ast }$ denote the original and bootstrap
constrained LS estimators, respectively. In order to analyze size control
and power of the proposed tests, we consider both empirical rejection
probabilities {[}ERPs{]} under the null and under local alternatives. For
tests performed under the null, we consider three different choices of the
true value ${\Greekmath 0112} _{0}$, one located on $\partial \Theta $ and two located
on $\Theta \backslash \partial \Theta $; specifically, ${\Greekmath 0112}_0 \in
\{(0,0)^{\prime },(-0.75,0.75)^{\prime },(-1.5,1.5)^{\prime }\}$. Under $
\mathsf{H}_{1}$, we employ a local alternative of the form ${\Greekmath 0112} _{0}={a}
_{0}n^{-1/2}$, ${a}_{0}\in \mathbb{R}^{2}$, such that $h({\Greekmath 0112} _{0})\neq 0$
unless $a=(0,0)^{\prime }$.
Tests are based on \emph{p}-values obtained using a `standard' -- i.e., with
$\Theta ^{\ast }=\Theta $ -- fixed-regressor Gaussian wild bootstrap and the
proposed `corrected' bootstrap scheme. For the latter, the bootstrap
parameter space is set to $\Theta ^{\ast }=\mathbb{R}\times \lbrack g^{\ast
}(\hat{{\Greekmath 0112}}_{2}),\infty )$, where the function $g^{\ast }$ satisfies the
assumptions of Corollary \ref{corollary bootstrap with boundary}, see also
Section \ref{sec tuning par}. In order to assess the impact of the tuning
parameter ${\Greekmath 0114} $, we consider a grid of possible values for ${\Greekmath 0114} $.
Numerical results are based on $50,000$ Monte Carlo simulations, each
involving $B=999$ bootstrap repetitions. Sample sizes are set to $n\in
\{100,200,400,800,1600\}$.
\subsection{Empirical rejection probabilities}
\label{sec MC comment}
We now discuss the ERPs of the bootstrap tests. Specifically, the Monte
Carlo results in Table 1 and 2 refer to the case in which data are generated
under the null and under local alternatives, respectively. The proposed
modified bootstrap parameter space is based on the function $
g^{\ast}=g-|g|^{1+{\Greekmath 0114}}$ for several values of ${\Greekmath 0114}>0$.
Table 1 shows that the `standard' bootstrap scheme typically under-rejects
the true null hypothesis when the parameter lies on the boundary of the
parameter space $\Theta $ whereas, as expected, its ERPs are closer to the
nominal level when ${\Greekmath 0112} _{0}$ is in the interior of $\Theta $. Our
proposed bootstrap performs similarly to the `standard' bootstrap for very
small values of ${\Greekmath 0114} $, with the impact of the correction becoming more
relevant as ${\Greekmath 0114} $ increases. If the parameter is on the boundary of the
parameter space (${\Greekmath 0112} _{0}\in \partial \Theta $), our proposed bootstrap
scheme gives rise to smaller absolute size distortions than the `standard'
bootstrap, for all the considered DGPs and all values of ${\Greekmath 0114} $. When $
{\Greekmath 0112} _{0}\in \mathrm{int}\Theta $, we observe very little variability in
the ERPs across the different bootstrap methods, at least for reasonably
small values of ${\Greekmath 0114} $.
Table 2 reports the ERPs of the tests when data are generated under local
alternatives ${\Greekmath 0112} _{0}=a_{0}n^{-1/2}$, $a_{0}\in \{(-3,0)^{\prime
},(3,0)^{\prime },(5,0)^{\prime }\}$, such that the true parameter values
lie on the boundary of the parameter space. Results show that both bootstrap
schemes have power under local alternatives, with the `corrected' bootstrap
generally showing higher ERPs than the `standard' bootstrap, in line with
the results obtained under the null. Finally, we notice that the sign of the
deviations from the null hypothesis matters, with positive deviations
showing higher ERPs. This finding can be explained by the fact that the
limit distribution of $n^{1/2}(h(\hat{{\Greekmath 0112}})-h({\Greekmath 0112} _{0}))$ is
asymmetric when ${\Greekmath 0112} _{0}$ lie on the boundary of $\Theta $. Results
about local alternatives such that ${\Greekmath 0112} _{0}$ are $n^{-1/2}$-local to the
boundary are substantially similar and are reported in Section S.2 of the
supplement.
\subsection{Choice of $g^{\ast}$ and $\protect{\Greekmath 0114}$}
\label{sec tuning par}
We now consider the practical issue of choosing the function $g^{*}$ and the
tuning parameter ${\Greekmath 0114}$ used to construct the modified bootstrap parameter
space $\Theta^{*}$.
Regarding $g^{\ast }$, in Section 4 we discussed the functions $
g_{(1)}^{\ast }:=g-|g|^{1+{\Greekmath 0114} }$, ${\Greekmath 0114} >0$, which satisfy the
assumptions of Corollary \ref{corollary bootstrap with boundary} and were
employed in the simulations so far, whereas in Section \ref
{subsec:Fang-and-Santos} we considered also $g_{(2)}^{\ast }:=g-n^{-{\Greekmath 0114}
}|g|$, ${\Greekmath 0114} \in (0,1/2)$, corresponding to $s_n=n^{1/2-{\Greekmath 0114}}$ in the
concluding paragraph of Section \ref{subsec:Fang-and-Santos}. Numerical
results in Table 1 and 2 and in the accompanying Supplement, Section S.2,
show that both choices of $g^{\ast }$ deliver good test performance, both
under the null and under local alternatives. The most salient difference
between $g_{(1)}^{\ast }$ and $g_{(2)}^{\ast }$ is that tests based on $
g_{(1)}^{\ast }$ tend to be more robust to the choice of ${\Greekmath 0114} $ when $
g({\Greekmath 0112} _{0})\geq 1$.
Concerning the choice of the tuning parameter ${\Greekmath 0114} $, we focus on $
g^{\ast }=g_{(1)}^{\ast }.$ Preliminary considerations point at a possible
trade-off between the cases of a boundary and an interior location of the
true parameter ${\Greekmath 0112} _{0}$. Thus, for ${\Greekmath 0112} _{0}\in \partial \Theta $,
larger values of ${\Greekmath 0114} $ accelerate the convergence of $g(\hat{{\Greekmath 0112}}
)^{1+{\Greekmath 0114} }$ to zero, which can be expected to favor bootstrap performance
as the bootstrap true value $\hat{{\Greekmath 0112}}$ is put at a smaller distance from
the bootstrap boundary. On the other hand, if ${\Greekmath 0112} _{0}\in \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{int}
(\Theta )$ and $g({\Greekmath 0112} _{0})\in (0,1)$, in small samples large values of $
{\Greekmath 0114} $ may put $\hat{{\Greekmath 0112}}$ too close to the bootstrap boundary,
yielding inferior bootstrap performance.
Our Monte Carlo study indeed confirms that small values of ${\Greekmath 0114} $ are
preferable when ${\Greekmath 0112} _{0}\in \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{int}(\Theta )$ and $g({\Greekmath 0112} _{0})\in
(0,1)$; however, it also shows that the proposed correction quickly provides
satisfactory size control for small values of ${\Greekmath 0114} $ even when ${\Greekmath 0112}
_{0}\in \partial \Theta $. Finally, we notice that when ${\Greekmath 0112} _{0}\in
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{int}(\Theta )$ and $g({\Greekmath 0112} _{0})\geq 1$ the choice of ${\Greekmath 0114} $ has
negligible impact on the ERPs. Overall, our numerical analysis suggests that
choices of ${\Greekmath 0114} $ close to $0.5$ provide quite satisfactory size control
across all the considered scenarios.
\medskip {}
\noindent \textsc{Remark}. The above guideline about the choice of ${\Greekmath 0114} $
is based on numerical evidence; it delivers a reasonable simple choice which
can be easily implemented. It is not optimal in any sense, and indeed
alternative methods could be employed to obtain data-driven choices of $
{\Greekmath 0114} $. For instance, the unrestricted parameter estimates could be used
to assess how far the true parameter value ${\Greekmath 0112} _{0}$ is from the
boundary of the parameter space, and then calibrate the choice of ${\Greekmath 0114} $
accordingly. This approach would be in the spirit of Romano, Shaikh and Wolf
(2014), who suggest to improve the power of tests of moment inequalities by
introducing a first step, where a confidence region for the moments is
constructed using their unrestricted estimates. Although this approach may
improve the finite sample properties of our tests, it would require a
preliminary choice of further tuning parameters, such as ${\Greekmath 010C} $ in Romano
et al. (2014),\ hence introducing an extra layer of complexity.$\hfill
\square $
\section{Conclusions}
\label{sec conclusion}
In this paper we analyzed the problem of bootstrap hypotheses tests on the
parameters $({\Greekmath 010B},{\Greekmath 010C})$ of a predictive regression $y_{t}={\Greekmath 010B}+{\Greekmath 010C}
x_{t-1}+{\Greekmath 0122}_{t}$, generalizable to higher dimensions, when the
parameter space is defined by means of a smooth constraint $
g({\Greekmath 010B},{\Greekmath 010C})\geq0$ and the true parameter vector under the null
hypothesis may lie on the boundary of the parameter space. In the framework
of constrained parameter estimation, implementation of the bootstrap is not
straightforward, as the presence of a parameter on the boundary of the
parameter space makes the bootstrap measure random in the limit.
We discussed possible solutions to this inference problem. Specifically, we
presented some modifications of standard bootstrap schemes where the
bootstrap parameter space is shifted by a data-dependent function, thus
allowing us to regain control over the boundary as a source of limiting
bootstrap randomness. We also proved validity of the associated bootstrap
inference in the cases where the posited predicting variable is I(1).
Our contribution is novel in the framework of predictive regression, in that
the existing literature has not analyzed the bootstrap in contexts combining
non-stationarity of the posited predictor with a priori knowledge about the
possible form of predictability, represented by a restricted parameter
space. The value of our work is to provide valid bootstrap implementations
in this setting.
\section*{Supplementary Material}
Cavaliere, G., Georgiev, I. \& Zanelli, E. (2024). Supplement to:
\textquotedblleft Parameters on the boundary in predictive
regression,\textquotedblright\ \emph{Econometric Theory }Supplementary
Material. To view, please visit [[doi to be inserted here by typesetter]]
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