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{ 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. }
{ Keywords: }{ }Parameter on the boundary, random measures, weak convergence in distribution, asymptotic inference, uniform inference.
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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 p-value of an asymptotic test and of its bootstrap analogue. Let also
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{{ 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 $\mathsf{H}_{0}$,
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}$$C=(0,1)$.} The meaning is that the bootstrap test and the asymptotic test are first-order asymptotically equivalent in terms of correct size control. In particular, bootstrap validity for simple hypotheses $\mathsf{H}_{0}$ characterizes pointwise size control.
Notice that bootstrap validity as in (1.1) is implied by the classic definition of bootstrap consistency, namely that $\sup_{x\in\mathbb{R} }|F_{n}^{\ast}(x)-F(x)|\rightarrow_{p}0$ for a bootstrap statistic with cdf $ F_{n}^{\ast}$ conditionally on the data and an original test statistic with continuous asymptotic cdf $F$. The converse does not hold; that is, ((ref)) does not imply classic bootstrap consistency, see the discussion in Cavaliere and Georgiev (2020).
For test statistics whose asymptotic distribution is continuous, it holds that $\limfunc{int}$$C=(0,1)$ and hence condition ((ref)) 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 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)), 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).
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) 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). The predictive regression framework is presented in Section (ref); 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). 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). Section (ref) provides simulation evidence, whereas Section (ref) concludes. Proofs are collected in the Appendix.
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 $\mathscr{D}(\mathbb{R})$:
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).
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
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
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
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
with associated conditional cdf given by
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)), with $\limfunc{int}$$C=(0,1)$. Conversely, when $ {\Greekmath 0112} _{0}$ is on the boundary of the parameter space, the cdf in ((ref)) converges weakly in $\mathscr{D}(\mathbb{R})$ to the random cdf $\Phi \left( x\right) \mathbb{I}_{\{x\geq -\ell \}}$. In terms of weak convergence in distribution,
where $\ell $ is distributed as in ((ref)) and is independent of ${\Greekmath 0118} ^{\ast }$. The limit distribution in ((ref)) is random, since its cdf is a stochastic process depending on the conditioning random variable $\ell $. Thus, it is distinct from the limit distribution in ((ref)), which is unconditional and hence characterized by a non-random cdf. Because the bootstrap limit distribution is random, the bootstrap approximation is not consistent for the limit in ((ref)).
As we shall see in Section (ref), limiting bootstrap randomness could be of two kinds: `benign', thus not compromising the validity of bootstrap inference in the sense of ((ref)), or `malignant', thus invalidating bootstrap inference. In this example, a bootstrap test employing a bootstrap statistic $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{${\Greekmath 011C} _{n}^{\ast }:=$}{\Greekmath 011E} (n^{1/2}(\hat{{\Greekmath 0112}}^{\ast }-\hat{{\Greekmath 0112}}))$ as the analogue of a statistic ${\Greekmath 011C} _{n}:=$${\Greekmath 011E} (n^{1/2}\hat{{\Greekmath 0112}})$, where $ {\Greekmath 011E} $ is a real function, may not be valid in the sense of ((ref)) 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)) and ((ref)) can be written as
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 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)). 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)) can be seen to be successfully restored.
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 $
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)); see also Andrews (2000). Specifically, the bootstrap test rejects $\mathsf{H}_{0}$ when the bootstrap p-value $\tilde{p}_{n}^{\ast }=1-p_{n}^{\ast }$ is small, with the following convergence satisfied under the null hypothesis:
A similar convergence is satisfied by the p-value $\tilde{p} _{n}=1-p_{n}$ of the asymptotic test, with
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), 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.
Consider the following predictive regression in a triangular array setup:
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.
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).}
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.
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 ${\Greekmath 0112}_{0}$:
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}$.
Let $\hat{{\Greekmath 0112}}$ be the OLS estimator of $({\Greekmath 0112} _{1},{\Greekmath 0112} _{2})^{\prime }$ in the 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)) 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). 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
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) 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.
Consider first a fixed-regressor wild bootstrap sample generated as
where ${\Greekmath 0122} _{t}^{\ast }=\hat{e}_{t}w_{t}^{\ast }$, $t=1,...,n$, with $\hat{e}_{t}$ the residuals of ((ref)) 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) below to the fixed-regressor wild bootstrap scheme. For ${\Greekmath 0112}_{0}\in$$\limfunc{int}$$ \Theta$, it turns out that the bootstrap distribution converges to a conditional version of the limit distribution of $n^{1/2}(\hat{{\Greekmath 0112}} -{\Greekmath 0112}_{0})$ found earlier:
where $\tilde{{\Greekmath 0112}}^{\ast}$ denotes the unconstrained OLS estimator from the bootstrap sample. The limit bootstrap distribution is, therefore, random. The vehicle of limiting bootstrap randomness is the random matrix $M$ , such that limiting bootstrap randomness is fully attributable to the stochastic properties of the regressor. Due to the fact that the bootstrap replicates a conditional version of the limit distribution of the original estimator $\hat{{\Greekmath 0112}}$, bootstrap inference is not invalidated. Rigorous statements in this sense will be provided in Corollary (ref).
On the other hand, if ${\Greekmath 0112} _{0}\in \partial \Theta $ the bootstrap statistic converges as follows:
where ${\Greekmath 0118} ^{\ast }\sim N(0,{\Greekmath 011B} _{e}^{2}I_{2})$ is independent of $ (M,\ell )$. In contrast with the case ${\Greekmath 0112} _{0}\in $$\limfunc{int}$$ \Theta $ and additionally to the random matrix $M$, in ((ref)) also the random vector $\ell $ appears as a vehicle of limiting bootstrap randomness. Moreover, the limit in ((ref)) 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)) is a random half-plane, rather than the original admissible set $\Lambda $ of ((ref)). 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.
In order to unify the discussion of several bootstrap schemes for inference on $\mathsf{H}_{0}$ under the three cases $\mathscr{G}{}_{1}$, $\mathscr{G}{} _{2}$ and $\mathscr{G}{}_{3}$, consider a bootstrap sample generated as in ( (ref)) and, more generally than before, a bootstrap OLS estimator $\hat{{\Greekmath 0112}}^{\ast }$ constrained to belong to a bootstrap parameter space $\Theta^{\ast}$ satisfying the following assumption.
The standard bootstrap considered in Section (ref) obtains by setting $g^{\ast }=0$, such that $\Theta ^{\ast }=\Theta $ , the original parameter space. Alternatively, setting $g^{\ast }=g$ restricts the bootstrap true value $\hat{{\Greekmath 0112}}$ to lie on the boundary of the bootstrap parameter space $\Theta ^{\ast }$.\footnote{{ }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) and (ref).
To formulate the next theorem, recall $M$ and $\ell({\Greekmath 0112}_{0})$ introduced in Section (ref), and let $ {\Greekmath 0118}^{\ast}|(M,\ell({\Greekmath 0112}_{0}))\sim N(0,{\Greekmath 011B}_e^2 I_2)$ as in Section (ref). 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.
The following conclusions could be drawn.
In general, bootstrap validity in the sense of ((ref)) can be evaluated through the following corollary of Theorem (ref) above.
The class of functions $g^{\ast}=g-|g|^{1+{\Greekmath 0114}}$ for ${\Greekmath 0114}>0$ satisfies both conditions (i) and (ii) of Corollary (ref); 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) focuses on this setup.
Remark. The practical implications of Corollary (ref) 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)), 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 $\mathsf{H}_{0}:g({\Greekmath 0112}_{0})=0$.
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 $
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) within the paradigm of some previous works -- specifically, Fang and Santos (2019) and Hong and Li (2020); and (iii) uniform bootstrap validity.
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 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)), which reduces to ((ref)) and ((ref)), it follows by the delta method that
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 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.
By Theorem (ref) 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 $(0,\frac{1}{2})$.
In this section we put the geometric considerations of Section (ref) 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). Its limit distribution, see ((ref)), is the distribution of $\ell ({\Greekmath 0112} _{0})={\Greekmath 0127} _{{\Greekmath 0112} _{0}}(M^{-1/2}{\Greekmath 0118} )$ with
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,
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
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
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), 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) 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)) implicitly performs the geometric approximation proposed in Section (ref), 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). 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
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
for $\bar{{\Greekmath 0127}}_{n}$ defined similarly to $\hat{{\Greekmath 0127}}_n$, with the only difference that in ((ref)) 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,
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.
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) 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
and $\ell (0,0)=\ell $ of eq. ((ref)). Then, the joint convergence result
holds for the bootstrap schemes satisfying conditions (i) and (ii) of Corollary (ref). 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
With ${\Greekmath 010D} _{M}:=(\dot{g}^{\prime }M^{-1}\dot{g})^{-1/2}$, it follows that
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 p-value under the null. The inequality above implies that bootstrap tests rejecting for large bootstrap 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
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)). 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 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
for all $q\in \limfunc{int}C$, where $C$ is the set from display ((ref)) 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)) 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
see eqs. ((ref))--((ref)).
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), 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) we describe the data generating processes, the null hypotheses and the adopted bootstrap schemes. In Section (ref) we discuss the performance of the tests both under the null and under local alternatives. Section (ref) deals with the choice of $g^{\ast }$ and $ {\Greekmath 0114} $. Additional numerical results are provided in the accompanying supplement, Section (ref).
We consider the same data generating process (DGP) as in ((ref)), 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}$:
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 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), see also Section (ref). 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 $n\in \{100,200,400,800,1600\}$.
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.
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) and were employed in the simulations so far, whereas in Section (ref) 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). 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.
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 $
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.
Cavaliere, G., Georgiev, I. & Zanelli, E. (2024). Supplement to: \textquotedblleft Parameters on the boundary in predictive regression,\textquotedblright\ Econometric Theory Supplementary Material. To view, please visit [[doi to be inserted here by typesetter]]