# Allowing for weak identification when testing GARCH-X type models

Philipp Ketz111Paris School of Economics - CNRS. E-mail:
<EMAIL_ADDRESS>
###### Abstract

In this paper, we use the results in Andrews and Cheng (2012), extended to
allow for parameters to be near or at the boundary of the parameter space, to
derive the asymptotic distributions of the two test statistics that are used
in the two-step (testing) procedure proposed by Pedersen and Rahbek (2019).
The latter aims at testing the null hypothesis that a GARCH-X type model, with
exogenous covariates (X), reduces to a standard GARCH type model, while
allowing the “GARCH parameter” to be unidentified. We then provide a
characterization result for the asymptotic size of any test for testing this
null hypothesis before numerically establishing a lower bound on the
asymptotic size of the two-step procedure at the 5% nominal level. This lower
bound exceeds the nominal level, revealing that the two-step procedure does
not control asymptotic size. In a simulation study, we show that this finding
is relevant for finite samples, in that the two-step procedure can suffer from
overrejection in finite samples. We also propose a new test that, by
construction, controls asymptotic size and is found to be more powerful than
the two-step procedure when the “ARCH parameter” is “very small” (in which
case the two-step procedure underrejects).

Keywords: Boundary, weak identification, testing.

##  1 Introduction

In a GARCH-X type model, the variance of a generalized autoregressive
conditional heteroskedasticity (GARCH) type model is augmented by a set of
exogenous regressors (X). Naturally, the question arises if the more general
GARCH-X type model can be reduced to the simpler GARCH type model.
Statistically speaking, the problem reduces to testing whether the
coefficients on the exogenous regressors are equal to zero. As noted in
Pedersen and Rahbek (2019) (PR hereinafter), the testing problem is non-
standard due to the presence of two nuisance parameters that could possibly be
at the boundary of the parameter space. In addition, under the null
hypothesis, one of the nuisance parameters, the “GARCH parameter”, is not
identified when the other, the “ARCH parameter”, is at the boundary. In order
to address this possible lack of identification, PR suggest a two-step
(testing) procedure, where rejection in the first step is taken as “evidence”
that the model is identified. In the second step, the authors then impose an
“additional assumption”, which implies that a specific entry of the inverse
information equals zero, to obtain an asymptotic null distribution of their
second-step test statistic that is nuisance parameter free. There are two
potential problems with this two-step procedure. First, it may not control
(asymptotic) size, i.e., its (asymptotic) size may exceed the nominal level,
for reasons similar to those that invalidate “naive” post-model-selection
inference (see e.g., Leeb and Pötscher, 2005, 2008). In addition, the
aforementioned “additional assumption” may not be satisfied, which may
possibly aggravate the problem. Second, the two-step procedure may, due to its
two-step nature and despite the possible lack of (asymptotic) size control,
have poor power in certain parts of the parameter space, as suggested by
simulations in PR.222The simulation results in Appendix D of PR show that the
two-step procedure has a null rejection frequency below the nominal level for
“very small” values of the ARCH parameter; see also Section 5.

In this paper, we use the results in Andrews and Cheng (2012) (AC
hereinafter), extended to allow for parameters to be near or at the boundary
of the parameter space,333Here, the parameter space is equal to a product
space; see Cox (2022) for related results in the context of more general
shapes of the parameter space. to derive the asymptotic distributions of the
two test statistics used in PR under weak, semi-strong, and strong
identification (using the terminology in AC). These asymptotic distribution
results, in turn, allow us characterize the asymptotic size of any test for
testing the null hypothesis that the coefficients on the exogenous regressors
are equal to zero. We numerically establish lower bounds on the asymptotic
sizes of the two-step procedure proposed by PR as well as a second testing
procedure proposed by PR that assumes that the ARCH parameter is known to be
in the interior of the parameter space. These bounds are given by 6.65% and
9.48%, respectively, for a 5% nominal level, which implies that the two
testing procedures do not control asymptotic size (at the 5% nominal
level).444These lower bounds (also) apply if the “additional assumption” and,
in case of the second testing procedure, the assumption that the ARCH
parameter is in the interior of the parameter space are satisfied; see Remark
1 for details. Furthermore, we propose a new test based on the second-step
test statistic of PR that uses plug-in least favorable configuration critical
values and, thus by construction, controls asymptotic size.

In a small simulation study, we find that our asymptotic theory provides good
approximations to the finite-sample behaviors of the tests, or testing
procedures, that we consider. In particular, we find that the testing
procedures proposed by PR can suffer from overrejection in finite samples.
Furthermore, we find that our new test has greater power than the two-step
procedure for “very small” values of the ARCH parameter, a presumably
empirically important region of the parameter space. This finding is line with
the intuition that the two-step procedure, in some sense, “sacrifices” power
for such parameter constellations due to its two-step nature.

The remainder of this paper is organized as follows. In Section 2, we
introduce the testing problem as well as the two testing procedures proposed
by PR. In Section 3, we present the asymptotic distribution results. Section 4
presents the characterization result for asymptotic size and obtains the lower
bounds on the asymptotic sizes of the two testing procedures proposed by PR.
It also introduces our new test. The result of our simulation study are
presented in Section 5. Additional material, including proofs, is relegated to
the Appendix.

Throughout this paper, we use the following conventions. All limits are taken
“as $n\to\infty$”. $e_{i}$ denotes a vector of zeros (of suitable dimension)
with a one in the $i^{\text{th}}$ position. For any matrix $A$, $A_{ij}$
denotes the entry with row index $i$ and column index $j$. Furthermore,
$X_{n}(\pi)=o_{p\pi}(1)$ means that $\sup_{\pi\in\Pi}\|X_{n}(\pi)\|=o_{p}(1)$,
where $\|\cdot\|$ denotes the Euclidean norm. Lastly, “for all $\delta_{n}\to
0$” abbreviates “for all sequences of positive scalar constants
$\\{\delta_{n}:n\geq 1\\}$ for which $\delta_{n}\to 0$”.

##  2 Testing problem

For ease of exposition, we consider a simple version of the GARCH-X(1,1) model
with a single exogenous variable (as in PR). In particular, the model is given
by

$y_{t}=h_{t}(\theta)^{1/2}z_{t},$ (1)

where
$\theta=(\psi^{\prime},\pi)^{\prime}=(\beta^{\prime},\zeta,\pi)^{\prime}$ and

$h_{t}(\theta)=h_{t}(\psi,\pi)=h_{t}(\beta,\zeta,\pi)=\zeta(1-\pi)+\beta_{1}y_{t-1}^{2}+\pi
h_{t-1}(\psi,\pi)+\beta_{2}x^{2}_{t-1}$ (2)

with $h_{0}(\theta)=\zeta$.555For ease of reference, we adopt the notation in
AC, where $\beta$ governs the identification strength of $\pi$ (see (4) below)
and $\psi$ is always identified. Here, $\\{y_{t},x_{t}\\}_{t=0}^{n}$ is
observed and $\\{z_{t}\\}_{t=0}^{n}$ is unobserved. The true parameter space
for $\theta$, i.e., the space of all possible true values of $\theta$, is
given by $\Theta^{*}=\Psi^{*}\times\Pi^{*}$, where

$\Psi^{*}=\\{\psi:0\leq\beta_{1}\leq\overline{\beta}^{*}_{1},0\leq\beta_{2}\leq\overline{\beta}^{*}_{2},\underline{\zeta}^{*}\leq\zeta\leq\overline{\zeta}^{*}\\}\text{
and }\Pi^{*}=\\{\pi:0\leq\pi\leq\overline{\pi}^{*}\\}$

for some $0<\overline{\beta}^{*}_{1}<\infty$,
$0<\overline{\beta}^{*}_{2}<\infty$,
$0<\underline{\zeta}^{*}<\overline{\zeta}^{*}<\infty$, and
$0<\overline{\pi}^{*}<1$.

The model is estimated by quasi-maximum likelihood. In particular, the
objective function is given by (-$\frac{1}{n}$ times) the Gaussian-based
conditional quasi log-likelihood function, i.e.,
$Q_{n}(\theta)=\frac{1}{n}\sum_{t=1}^{n}l_{t}(\theta)$, where

$l_{t}(\theta)=\frac{1}{2}\log(2\tilde{\pi})+\frac{1}{2}\log(h_{t}(\theta))+\frac{y_{t}^{2}}{2h_{t}(\theta)}$

and where $\tilde{\pi}=3.14...$. The quasi-maximum likelihood estimator is
given by

$\hat{\theta}_{n}=\operatornamewithlimits{arg\
min}_{\theta\in\Theta}Q_{n}(\theta),$

where $\Theta=\Psi\times\Pi$ denotes the optimization parameter space with

$\Psi=\\{\psi:0\leq\beta_{1}\leq\overline{\beta}_{1},0\leq\beta_{2}\leq\overline{\beta}_{2},\underline{\zeta}\leq\zeta\leq\overline{\zeta}\\}\text{
and }\Pi=\\{\pi:0\leq\pi\leq\overline{\pi}\\}$

for some $\overline{\beta}^{*}_{1}<\overline{\beta}_{1}<\infty$,
$\overline{\beta}^{*}_{2}<\overline{\beta}_{2}<\infty$,
$0<\underline{\zeta}<\underline{\zeta}^{*}$,
$\overline{\zeta}^{*}<\overline{\zeta}<\infty$, and
$\overline{\pi}^{*}<\overline{\pi}<1$. Note that, given the definitions of
$\Theta^{*}$ and $\Theta$, (the true values of) $\beta_{1}$, $\beta_{2}$, and
$\pi$ are allowed to be at the boundary of the optimization parameter space.

While our asymptotic distribution results are useful for analyzing a wide
range of testing problems, we are mainly interested in testing

$H_{0}:\beta_{2}=0\text{\ vs.\ }H_{1}:\beta_{2}>0.$ (3)

As pointed out in PR, this testing problem is non-standard in that, under
$H_{0}$, there are two nuisance parameters that may be at the boundary of the
(optimization) parameter space, $\beta_{1}$ and $\pi$. Furthermore, when
$\beta_{1}$ is at the boundary ($\beta_{1}=0$) then $\pi$ is not identified
under $H_{0}$. To see this, note that, given $h_{0}(\theta)=\zeta$, we have

$h_{t}(\theta)=\zeta+\beta_{1}\sum_{i=0}^{t-1}\pi^{i}y_{t-i-1}^{2}+\beta_{2}\sum_{i=0}^{t-1}\pi^{i}x_{t-i-1}^{2}$
(4)

and, thus, $h_{t}(0,\zeta,\pi)=\zeta$
$\forall\theta=(\beta,\zeta,\pi)=(0,\zeta,\pi)\in\Theta,\forall n\geq 1$. In
words, under $H_{0}$, $\beta_{1}=0$ implies that the distribution of the data
does not depend on $\pi$, i.e., $\pi_{1}$ and $\pi_{2}$ (with
$\pi_{1}\neq\pi_{2}$) are observationally equivalent.

Let $\mathcal{T}_{n}$ denote a generic test statistic for testing (3) and let
cvn,1-α denote the corresponding nominal level $\alpha$ critical value, which
may depend on $n$. The size of the test that rejects $H_{0}$ when
$\mathcal{T}_{n}>\text{cv}_{n,1-\alpha}$ is given by
Sz${}_{\mathcal{T}}=\sup_{\gamma\in\Gamma:\beta_{2}=0}P_{\gamma}(\mathcal{T}_{n}>\text{cv}_{n,1-\alpha})$,
where $\Gamma$ denotes the true parameter space for $\gamma=(\theta,\phi)$ and
where $\phi$ denotes the distribution of $\\{x_{t},z_{t}\\}$. We say that a
test controls size if Sz${}_{\mathcal{T}}\leq\alpha$. We note that
“uniformity” (over $\Gamma$) is built into the definition of SzT and that
whether a test controls size crucially depends on $\Gamma$. Typically, it is
infeasible to compute SzT. Therefore, we rely on asymptotic approximations. In
particular, we approximate the (“finite-sample”) size of a test by its
asymptotic size, which is given by

$\text{AsySz}_{\mathcal{T}}=\limsup_{n\to\infty}\sup_{\gamma\in\Gamma:\beta_{2}=0}P_{\gamma}(\mathcal{T}_{v}>\text{cv}_{n,1-\alpha}).$

While $\text{AsySz}_{\mathcal{T}}$ “still” depends on $\Gamma$, it generally
only does so through a finite-dimensional parameter, making its evaluation
“easier”. We say that a test controls asymptotic size if
AsySz${}_{\mathcal{T}}\leq\alpha$. In large samples, AsySzT provides a good
approximation of SzT so that a test that controls asymptotic size can be
expected to “approximately” control size. Therefore, in what follows we focus
on whether or not a given test, or testing procedure, controls asymptotic
size.

###  2.1 Testing procedures proposed by PR

Given the non-standard nature of the testing problem in (3), PR propose a two-
step procedure to deal with the presence of nuisance parameters on the
boundary of the parameter space as well as the lack of identification of $\pi$
when $\beta_{1}=0$. In the first step, PR propose to test
$H_{0}^{\dagger}:\beta_{1}=\beta_{2}=0$ using the corresponding (rescaled)
quasi-likelihood ratio statistic

$LR^{\dagger}_{n}=2n(Q_{n}(\hat{\theta}^{\dagger}_{n,0})-Q_{n}(\hat{\theta}_{n}))/\hat{c}_{n}^{\dagger},$

where $\hat{\theta}^{\dagger}_{n,0}=\operatornamewithlimits{arg\
min}_{\theta\in\Theta^{\dagger}_{0}}Q_{n}({\theta})$ with
$\Theta^{\dagger}_{0}=\\{\theta\in\Theta:\beta_{1}=\beta_{2}=0\\}$ and
where666Note that other estimators of
$c_{0}=c(\gamma_{0})=E_{\gamma_{0}}(z_{t}^{2}-1)^{2}/2$ are available (see
Section 3 for the “definition” of $\gamma_{0}$): For example,
$\hat{c}_{\text{alt}}(\theta)=\left(\frac{1}{n}\sum_{t=1}^{n}\frac{y_{t}^{4}}{h_{t}(\theta)^{2}}-1\right)/2$
evaluated at $\hat{\theta}^{\dagger}_{n,0}$ or $\hat{\theta}_{n}$; similarly,
one could evaluate $\hat{c}(\theta)$ at $\hat{\theta}_{n}$. Note that Andrews
(2001) also considers (rescaled) quasi-likelihood ratio statistics of the form
$2n(\min_{\pi\in\Pi}Q_{n}(\hat{\psi}^{\dagger}_{n,0}(\pi),\pi)/c(\hat{\psi}_{n}(\pi),\pi)-\min_{\pi\in\Pi}Q_{n}(\hat{\psi}_{n}(\pi),\pi)/c(\hat{\psi}_{n}(\pi),\pi)),$
where
$\hat{\theta}^{\dagger}_{n,0}=((\hat{\psi}^{\dagger}_{n,0}(\hat{\pi}^{\dagger}_{n,0}))^{\prime},\hat{\pi}^{\dagger}_{n,0})^{\prime}$.
We note that $Q_{n}(\hat{\psi}^{\dagger}_{n,0}(\pi),\pi)$ does not depend on
$\pi$.

$\hat{c}^{\dagger}_{n}=\hat{c}(\hat{\theta}^{\dagger}_{n,0})=\left(\frac{1}{n}\sum_{t=1}^{n}\left(\frac{y_{t}^{2}}{h_{t}(\hat{\theta}^{\dagger}_{n,0})}-1\right)^{2}\right)/2=\left(\frac{1}{n}\sum_{t=1}^{n}\left(\frac{y_{t}^{2}}{\hat{\zeta}^{\dagger}_{n,0}}-1\right)^{2}\right)/2.$

Here,
$\hat{\theta}^{\dagger}_{n,0}=(\hat{\zeta}^{\dagger}_{n,0},0,0,\hat{\pi}^{\dagger}_{n,0})^{\prime}$
with $\hat{\pi}^{\dagger}_{n,0}=\Pi$. The asymptotic null distribution of
$LR^{\dagger}_{n}$ can be derived using the results in Andrews (2001) (see
e.g., Theorem 2.1 in PR) and, although not available in closed form, can
easily be simulated from. If $H_{0}^{*}$ is not rejected, then $H_{0}$ is not
rejected. If $H_{0}^{*}$ is rejected, then PR conclude that $\beta_{1}>0$.
Given that $\beta_{1}>0$, the nuisance parameter $\pi$ is identified and the
only remaining “problem” is that $\pi$ may still be at the boundary ($\pi=0$).
In the second step, PR then suggest to test (3), under the maintained
assumption that $\beta_{1}>0$, using the corresponding (rescaled) quasi-
likelihood ratio statistic

$LR_{n}=2n(Q_{n}({\hat{\theta}_{n,0}})-Q_{n}(\hat{\theta}_{n}))/\hat{c},$

where $\hat{\theta}_{n,0}=\operatornamewithlimits{arg\
min}_{\theta\in\Theta_{0}}Q_{n}({\theta})$ with
$\Theta_{0}=\\{\theta\in\Theta:\beta_{2}=0\\}$ and where
$\hat{c}_{n}=\hat{c}(\hat{\theta}_{n,0})$.777Alternatively, we could evaluate
$\hat{c}(\theta)$ at $\hat{\theta}_{n}$ or use $\hat{c}_{\text{alt}}(\theta)$
(see footnote 6) evaluated at $\hat{\theta}_{n,0}$ or $\hat{\theta}_{n}$. In
this context, PR make an “additional assumption” on the dependence between
$\\{x_{t}\\}$ and $\\{y_{t}\\}$ that implies that a certain entry of the
inverse of the information matrix equals zero when $\pi=0$. This, in turn,
implies that the asymptotic null distribution of $LR_{n}$ simplifies to
$\max(0,Z)^{2}$, where $Z\sim N(0,1)$. Formally, the two-step procedure (TS)
is defined as follows: Reject $H_{0}$ if

${TS}_{n}=\mathbbm{1}(LR_{n}^{\dagger}>\widetilde{LR}^{\dagger}_{n,1-\alpha})\times
LR_{n}>\text{cv}_{1-\alpha},$

where $\text{cv}_{1-\alpha}$ and $\widetilde{LR}^{\dagger}_{1-\alpha}$ denote
the $1-\alpha$ quantiles of $\max(0,Z)^{2}$ and the asymptotic distribution
given in (14) with $b=0$ and unknown quantities replaced by consistent
estimators (see e.g., Appendix B), respectively.888PR do not formally define
their two-step procedure in that they do not define the nominal levels that
ought to be used in the two steps, say $\alpha_{1}$ and $\alpha_{2}$. Here, we
take $\alpha_{1}=\alpha_{2}=\alpha$. Of course, given the results in this
paper, it is in principle possible to choose $\alpha_{1}$ and $\alpha_{2}$ in
order to ensure that AsySz${}_{TS}\leq\alpha$. We refrain from doing so,
however, because, given the results in this paper, the motivation for using a
two-step procedure is rendered obsolete; see e.g., the new test. While it is
possible to derive the asymptotic null distribution of $LR_{n}$ without this
“additional assumption”, PR refrain from doing so.999If $\beta_{1}>0$, or
rather $\beta_{1}\geq c>0$ for some $c\in\mathbb{R}$, then it is
straightforward to test (3) using the approach in Ketz (2018), without any
assumptions on the dependence between $\\{x_{t}\\}$ and $\\{y_{t}\\}$.

PR also mention that if it is a priori known that $\beta_{1}>0$ then one may
directly test (3) using $LR_{n}$ and their suggested critical value,
$\text{cv}_{1-\alpha}$. We refer to this test as the “second testing procedure
(of PR)”. Formally, the second testing procedure (S) is defined as follows: If
$\beta_{1}>0$, reject $H_{0}$ if

$S_{n}=LR_{n}>\text{cv}_{1-\alpha}.$

There are several potential issues with the above testing procedures. First,
the intuition underlying the two-step procedure is based on the assumption
that the first-step test never makes a type I error. This assumption, however,
cannot hold and a type I error in the first step may very well propagate to
the type I error of the overall (two-step) procedure.101010The only way for
the first-step test to never make a type I error is to take the nominal level
of the corresponding test equal to zero. This, however, would lead the first-
step test (as well as the two-step procedure) to have zero power. Relatedly,
$\beta_{1}$ may be close to zero relative to the sample size; see Section 3
for details. In that case, the probability of rejecting $H_{0}^{\dagger}$
exceeds the first-step nominal level and $\pi$ is only weakly identified such
that the asymptotic distribution result for $LR_{n}$ that takes $\pi$ to be
(strongly) identified may only provide a very poor approximation to its actual
finite-sample distribution. The latter may also be an issue for the second
testing procedure. In both cases, one may be worried that the testing
procedure does not control asymptotic size. As mentioned above, asymptotic
size depends on $\Gamma$. For sake of brevity, the definition of $\Gamma$ is
given in Appendix A.

##  3 Asymptotic distribution results

As shown in the recent literature (see e.g., Andrews and Guggenberger, 2010,
AC), asymptotic size is intrinsically linked to the asymptotic distribution of
the test statistic under (drifting) sequences of true parameters. Let
$\gamma_{n}=(\theta_{n},\phi_{n})=(\beta_{n},\zeta_{n},\pi_{n},\phi_{n})\in\Gamma$
denote the true parameter for $n\geq 1$. In the context at hand, the following
(sets of) sequences are key:

$\displaystyle\Gamma(\gamma_{0})=$
$\displaystyle\\{\\{\gamma_{n}\in\Gamma:n\geq
1\\}:\gamma_{n}\to\gamma_{0}=(\theta_{0},\phi_{0})=(\beta_{0},\zeta_{0},\pi_{0},\phi_{0})\in\Gamma\\},$
$\displaystyle\Gamma(\gamma_{0},0,b)=$
$\displaystyle\\{\\{\gamma_{n}\\}\in\Gamma(\gamma_{0}):\beta_{0}=0\text{ and
}\sqrt{n}\beta_{n}\to b\in\mathbb{R}^{2}_{+,\infty}\\},\text{ and }$
$\displaystyle\Gamma(\gamma_{0},\infty,b,\omega_{0},p)=$
$\displaystyle\\{\\{\gamma_{n}\\}\in\Gamma(\gamma_{0}):\sqrt{n}\beta_{n}\to
b\in\mathbb{R}^{2}_{+,\infty}\text{ with }\|b\|=\infty,$ $\displaystyle\
\beta_{n}/\|\beta_{n}\|\to\omega_{0}\in\mathbb{R}_{+}^{2},\text{ and
}\sqrt{n}\|\beta_{n}\|\pi_{n}\to p\in[0,\infty]\\}.$

In what follows, we use the terminology “under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0})$” to mean “when the true parameters are
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0})$ for any $\gamma_{0}\in\Gamma$, “under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$” to mean “when the true parameters
are $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ for any $\gamma_{0}\in\Gamma$
with $\beta_{0}=0$ and any $b\in\mathbb{R}^{2}_{+,\infty}$”, and “under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$” to mean “when
the true parameters are
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ for any
$\gamma_{0}\in\Gamma$, any $b\in\mathbb{R}^{2}_{+,\infty}\text{ with
}\|b\|=\infty$, any $\omega_{0}\in\mathbb{R}_{+}^{2}$ with $\|\omega_{0}\|=1$
if $b_{1}=b_{2}=\infty$ or $\omega_{0}=e_{j}\text{ if }b_{j}=\infty\text{ and
}b_{j^{\prime}}<\infty\text{ for }j,j^{\prime}\in\\{1,2\\}$ and
$j^{\prime}\neq j$, and any $p\in[0,\infty]$ if $\pi_{0}=0$ or $p=\infty$ if
$\pi_{0}>0$”. We note that under sequences of true parameters for which
$\sqrt{n}\beta_{n}\to b$ with $\|b\|<\infty$, identification of $\pi$ is weak,
while under sequences of true parameters for which $\beta_{n}\to 0$ but
$\sqrt{n}\beta_{n}\to\infty$, identification of $\pi$ is semi-strong.

All claims in this section, including the following, are verified in Appendix
A. Under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0})$, we have
$\sup_{\pi\in\Pi}\|\hat{\psi}_{n}(\pi)-\psi_{n}\|\to_{p}0$ and
$\|\hat{\psi}_{n}-\psi_{n}\|\to_{p}0$ when $\beta_{0}=0$, while
$\hat{\theta}_{n}-\theta_{n}\to_{p}0$ when $\beta_{0}\neq 0$.

###  3.1 Results for $\beta_{n}\to 0$ including asymptotic distribution for
$\|b\|<\infty$

First, we derive results for $\beta$ close to zero, i.e., $\beta_{n}\to 0$. To
that end, let
$\hat{\theta}_{n}=(\hat{\psi}_{n}(\hat{\pi}_{n}),\hat{\pi}_{n})$, where
$\hat{\psi}_{n}(\pi)=(\hat{\beta}_{n}(\pi),\hat{\zeta}_{n}(\pi))\in\Psi$ is
the concentrated extremum estimator of $\psi$ for given $\pi\in\Pi$, i.e.,

$\hat{\psi}_{n}(\pi)=\operatornamewithlimits{arg\
min}_{\psi\in\Psi}Q_{n}(\psi,\pi),$

and where $\hat{\pi}_{n}$ is the minimizer of the concentrated objective
function $Q_{n}^{c}(\pi)=Q_{n}(\hat{\psi}_{n}(\pi),\pi)$, i.e.,

$\hat{\pi}_{n}=\operatornamewithlimits{arg\ min}_{\pi\in\Pi}Q_{n}^{c}(\pi).$

With a slight abuse of notation, we let $\hat{\pi}_{n}=1$ whenever
$\hat{\pi}_{n}=\Pi$, which occurs when $\hat{\beta}_{n}(\pi)=0$ for all
$\pi\in\Pi$. Under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$,
$Q_{n}(\psi,\pi)$ admits the following quadratic expansion in $\psi$ around
$\psi_{0,n}=(0,\zeta_{n})$ for given $\pi$:

$\displaystyle Q_{n}(\psi,\pi)$
$\displaystyle=Q_{n}(\psi_{0,n},\pi)+D_{\psi}Q_{n}(\psi_{0,n},\pi)^{\prime}(\psi-\psi_{0,n})$
$\displaystyle+\frac{1}{2}(\psi-\psi_{0,n})^{\prime}H(\pi;\gamma_{0})(\psi-\psi_{0,n})+R_{n}(\psi,\pi),$
(5)

where the remainder $R_{n}(\psi,\pi)$ satisfies

$\sup_{\psi\in\Psi:\|\psi-\psi_{0,n}\|\leq\delta_{n}}|a_{n}^{2}(\gamma_{n})R_{n}(\psi,\pi)|=o_{p\pi}(1)$

for all $\delta_{n}\to 0$ and where

$a_{n}(\gamma_{n})=\left\\{\begin{array}[]{ll}n^{1/2}&\text{if
}\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)\text{ and }\|b\|<\infty\\\
\|\beta_{n}\|^{-1}&\text{if }\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)\text{
and }\|b\|=\infty\end{array}\right..$

Here, $D_{\psi}Q_{n}(\theta)$ denotes the left/right partial derivatives of
$Q^{\infty}_{n}(\theta)=\frac{1}{n}\sum_{t=1}^{n}l^{\infty}_{t}(\theta)$ with
respect to $\psi$, where

$l^{\infty}_{t}(\theta)=\frac{1}{2}\log(2\tilde{\pi})+\frac{1}{2}\log(h_{t}^{\infty}(\theta))+\frac{y_{t}^{2}}{2h^{\infty}_{t}(\theta)}$

and where

$h_{t}^{\infty}(\theta)=\zeta+\beta_{1}\sum_{i=0}^{\infty}\pi^{i}y_{t-i-1}^{2}+\beta_{2}\sum_{i=0}^{\infty}\pi^{i}x_{t-i-1}^{2}.$

In particular,

$D_{\psi}Q_{n}(\theta)=\frac{1}{n}\sum_{t=1}^{n}l^{\infty}_{\psi,t}(\theta),$

where

$l^{\infty}_{\psi,t}(\theta)=\frac{\partial}{\partial\psi}l^{\infty}_{t}(\theta)=\frac{1}{2h^{\infty}_{t}(\theta)}\left(1-\frac{y_{t}^{2}}{h^{\infty}_{t}(\theta)}\right)\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}$

and where

$\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}=\left(\sum_{i=0}^{\infty}\pi^{i}y_{t-i-1}^{2},\sum_{i=0}^{\infty}\pi^{i}x_{t-i-1}^{2},1\right)^{\prime}.$

$H(\pi;\gamma_{0})$ is defined below. Next, define the empirical process
$\\{G_{n}(\pi):\pi\in\Pi\\}$ by

$G_{n}(\pi)=\sqrt{n}\frac{1}{n}\sum_{t=1}^{n}(l^{\infty}_{\psi,t}(\psi_{0,n},\pi)-E_{\gamma_{n}}l^{\infty}_{\psi,t}(\psi_{0,n},\pi)).$

Under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$, $G_{n}(\cdot)\Rightarrow
G(\cdot;\gamma_{0})$, where $\Rightarrow$ denotes weak convergence and where
$G(\cdot;\gamma_{0})$ is a mean zero Gaussian process with bounded continuous
sample paths and covariance Kernel given by

$\displaystyle\Omega(\pi_{1},\pi_{2};\gamma_{0})=$
$\displaystyle\frac{c_{0}}{2}\left[\begin{array}[]{ccc}\frac{2c_{0}}{1-\pi_{1}\pi_{2}}+\frac{1}{(1-\pi_{1})(1-\pi_{2})}&\frac{1}{\zeta_{0}}E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi_{1}^{i}z^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi_{2}^{j}x^{2}_{t-j-1}&\frac{1}{\zeta_{0}}\frac{1}{1-\pi_{1}}\\\
\frac{1}{\zeta_{0}}E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi_{1}^{i}x^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi_{2}^{j}z^{2}_{t-j-1}&\frac{1}{\zeta_{0}^{2}}E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi_{1}^{i}x^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi_{2}^{j}x^{2}_{t-j-1}&\frac{1}{\zeta_{0}^{2}}E_{\gamma_{0}}\sum_{j=0}^{\infty}\pi_{1}^{j}x^{2}_{t-j-1}\\\
\frac{1}{\zeta_{0}}\frac{1}{1-\pi_{2}}&\frac{1}{\zeta_{0}^{2}}E_{\gamma_{0}}\sum_{j=0}^{\infty}\pi_{2}^{j}x^{2}_{t-j-1}&\frac{1}{\zeta_{0}^{2}}\end{array}\right]$
(9)

for $\pi_{1},\pi_{2}\in\Pi$, where
$c_{0}=c(\gamma_{0})=\frac{E_{\gamma_{0}}(z_{t}^{2}-1)^{2}}{2}=\frac{E_{\gamma_{0}}z_{t}^{4}-1}{2}$.111111The
last equality follows by the definition of $\Gamma$. Furthermore, we have that

$H(\pi;\gamma_{0})=\Omega(\pi,\pi;\gamma_{0})/c_{0}$

for $\pi\in\Pi$. Let

$Z_{n}(\pi;\gamma_{0})=-a_{n}(\gamma_{n})H^{-1}(\pi;\gamma_{0})D_{\psi}Q_{n}(\psi_{0,n},\pi).$

Then, under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$, we have

$Z_{n}(\pi;\gamma_{0})\overset{d}{\to}\begin{cases}Z(\pi;\gamma_{0},b)&\text{if
}\|b\|<\infty\\\ -H^{-1}(\pi;\gamma_{0})K(\pi;\gamma_{0})\omega_{0}&\text{if
}\|b\|=\infty\text{ and }\beta_{n}/\|\beta_{n}\|\to\omega_{0}\end{cases},$
(10)

where

$Z(\pi;\gamma_{0},b)=-H^{-1}(\pi;\gamma_{0})\\{G(\pi;\gamma_{0})+K(\pi;\gamma_{0})b\\}$

and where

$K(\pi;\gamma_{0})=-\frac{1}{2}\left[\begin{array}[]{cc}\frac{2c}{1-\pi\pi_{0}}+\frac{1}{(1-\pi)(1-\pi_{0})}&\frac{1}{\zeta_{0}}E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi^{i}z^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi_{0}^{j}x^{2}_{t-j-1}\\\
\frac{1}{\zeta_{0}}E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi^{i}x^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi_{0}^{j}z^{2}_{t-j-1}&\frac{1}{\zeta_{0}^{2}}E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi^{i}x^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi^{j}_{0}x^{2}_{t-j-1}\\\
\frac{1}{\zeta_{0}}\frac{1}{1-\pi_{0}}&\frac{1}{\zeta_{0}^{2}}E_{\gamma_{0}}\sum_{j=0}^{\infty}\pi_{0}^{j}x^{2}_{t-j-1}\end{array}\right].$

Next, let

$q_{n}(\lambda,\pi;\gamma_{0})=(\lambda-
Z_{n}(\pi;\gamma_{0}))^{\prime}H(\pi;\gamma_{0})(\lambda-
Z_{n}(\pi;\gamma_{0}))$

such that the quadratic expansion in (5) can be written as

$\displaystyle Q_{n}(\psi,\pi)$
$\displaystyle=Q_{n}(\psi_{0,n},\pi)-\frac{1}{2a^{2}_{n}(\gamma_{n})}Z_{n}(\pi;\gamma_{0})^{\prime}H(\pi;\gamma_{0})Z_{n}(\pi;\gamma_{0})$
$\displaystyle+\frac{1}{2a^{2}_{n}(\gamma_{n})}q_{n}(a_{n}(\gamma_{n})(\psi-\psi_{0,n}),\pi;\gamma_{0})+R^{*}_{n}(\psi,\pi).$

Note that the first two terms do not depend on $\psi$. As a result, following
Andrews (1999, 2001), it can then be shown that the (scaled and demeaned)
minimizer of $Q_{n}(\psi,\pi)$ with respect to $\psi$ for given $\pi$
asymptotically behaves like the minimizer of the “asymptotic version” of
$q_{n}(\cdot)$ over an appropriately defined parameter space for $\lambda$.

####  3.1.1 Asymptotic distribution for $\|b\|<\infty$

In particular, under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with
$\|b\|<\infty$, we have

$\sqrt{n}(\hat{\psi}_{n}(\pi)-\psi_{0,n})\overset{d}{\to}\hat{\lambda}(\pi;\gamma_{0},b),$
(11)

where

$\hat{\lambda}(\pi;\gamma_{0},b)=\operatornamewithlimits{arg\
min}_{\lambda\in\Lambda}q(\lambda,\pi;\gamma_{0},b).$

Here,

$q(\lambda,\pi;\gamma_{0},b)=(\lambda-Z(\pi;\gamma_{0},b))^{\prime}H(\pi;\gamma_{0})(\lambda-Z(\pi;\gamma_{0},b))$

and

$\Lambda=[0,\infty]^{2}\times[-\infty,\infty].$

Next, let

$\hat{\pi}(\gamma_{0},b)=\operatornamewithlimits{arg\
min}_{\pi\in\Pi}q(\hat{\lambda}(\pi;\gamma_{0},b),\pi;\gamma_{0},b)$

and, with a slight abuse of notation, let $\hat{\pi}(\gamma_{0},b)=1$ whenever
$\hat{\pi}(\gamma_{0},b)=\Pi$. Then, we have the following first main result
of this paper. Namely, under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with
$\|b\|<\infty$, we have

$\left(\begin{array}[]{c}\sqrt{n}(\hat{\psi}_{n}-\psi_{0,n})\\\
\hat{\pi}_{n}\end{array}\right)\overset{d}{\to}\left(\begin{array}[]{c}\hat{\lambda}(\hat{\pi}(\gamma_{0},b);\gamma_{0},b)\\\
\hat{\pi}(\gamma_{0},b)\end{array}\right)$ (12)

and121212Note that
$\min_{\pi\in\Pi}-\hat{\lambda}(\pi;\gamma_{0},b)^{\prime}H(\pi;\gamma_{0})\hat{\lambda}(\pi;\gamma_{0},b)=-\hat{\lambda}(\hat{\pi}(\gamma_{0},b);\gamma_{0},b)^{\prime}H(\pi;\gamma_{0})\hat{\lambda}(\hat{\pi}(\gamma_{0},b);\gamma_{0},b)$.

$2n(Q_{n}(\hat{\theta}_{n})-Q_{0,n})\overset{d}{\to}\min_{\pi\in\Pi}-\hat{\lambda}(\pi;\gamma_{0},b)^{\prime}H(\pi;\gamma_{0})\hat{\lambda}(\pi;\gamma_{0},b).$
(13)

The corresponding asymptotic distribution results for
$\hat{\theta}^{\dagger}_{n,0}$ and $\hat{\theta}_{n,0}$ are obtained
similarly, by replacing $\Lambda$ with $\\{0\\}^{2}\times[-\infty,\infty]$ and
$[0,\infty]\times\\{0\\}\times[-\infty,\infty]$, respectively.

Figure 1: Asymptotic and finite-sample ($n=500$) densities of $\hat{\pi}_{n}$
with $\sqrt{n}\beta_{1}=b_{1}=0,2,$ and 4 from left to right and
$\beta_{2}=b_{2}=0$. Here, $\pi=0.2$, $\varphi=0.5$, and $\kappa=0$.

Figure 1 shows the asymptotic and finite-sample ($n=500$) densities of
$\hat{\pi}_{n}$ for several values of $b_{1}$ and $b_{2}=0$; see Section 4 for
details on the data generating process (dgp) and Appendix B for details on how
the asymptotic distribution is simulated.131313Figure 4 in Appendix C plots
the corresponding distributions for the elements of $\hat{\psi}_{n}$. We
observe that the asymptotic distribution provides a good approximation to the
finite-sample distribution. Furthermore, we observe that identification
strength strongly impacts the distribution of $\hat{\pi}_{n}$, with large
deviations from normality. When $b=0$, the distribution of $\hat{\pi}_{n}$ is
(almost) completely flat with point masses at the boundaries of the
optimization parameter space as well as at 1; recall that $\hat{\pi}_{n}=1$
whenever $\hat{\pi}_{n}=\Pi$. And, as $b_{1}$ increases, the distribution of
$\hat{\pi}_{n}$ starts to resemble the normal distribution (centered at the
true value, $\pi=0.2$, and truncated to the support $\Pi$). This observation
is in line with the asymptotic distribution results for $\|b\|=\infty$; see
Section 3.2.

The asymptotic distribution result in (13) allows us to derive the asymptotic
distributions of $LR_{n}^{\dagger}$ and $LR_{n}$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $\|b\|<\infty$. Let
$S_{\beta}=[I_{2}\ 0_{2}]$ denote the selection matrix that (among $\psi$)
selects the entries pertaining to $\beta$ and let
$\hat{\lambda}_{\beta}(\pi;\gamma_{0},b)=S_{\beta}\hat{\lambda}(\pi;\gamma_{0},b)$.
Then, the asymptotic distribution of $LR^{\dagger}_{n}$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $\|b\|<\infty$ is given by

$LR^{\dagger}(\gamma_{0},b)=\max_{\pi\in\Pi}\hat{\lambda}_{\beta}(\pi;\gamma_{0},b)^{\prime}(c_{0}S_{\beta}H^{-1}(\pi;\gamma_{0})S_{\beta}^{\prime})^{-1}\hat{\lambda}_{\beta}(\pi;\gamma_{0},b).$
(14)

We note that for $b=0$ we recover the asymptotic distribution of
$LR^{\dagger}_{n}$ under $H_{0}^{\dagger}$, see also Theorem 2.1 in PR. The
asymptotic distribution of $LR_{n}$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $\|b\|<\infty$ is given
by141414If $\\{z_{t}\\}$ and $\\{x_{t}\\}$ are independent, in which case PR’s
“additional assumption” is also satisfied, then
$S_{\beta}H^{-1}(\pi;\gamma_{0})S_{\beta}^{\prime}$ is diagonal and the
asymptotic distributions of $LR_{n}^{\dagger}$ and $LR_{n}$ simplify. Letting
$S_{\beta_{1}}=[1\ 0\ 0]$ and $S_{\beta_{2}}=[0\ 1\ 0]$ as well as
$\hat{\lambda}_{\beta_{1}}(\pi;\gamma_{0},b)=S_{\beta_{1}}\hat{\lambda}(\pi;\gamma_{0},b)$
and
$\hat{\lambda}_{\beta_{2}}(\pi;\gamma_{0},b)=S_{\beta_{2}}\hat{\lambda}(\pi;\gamma_{0},b)$,
we have, for $i\in\\{1,2\\}$,
$\hat{\lambda}_{\beta_{i}}(\pi;\gamma_{0},b)\sim\max(Z_{\beta_{i}}(\pi;\gamma_{0},b),0),$
with $Z_{\beta_{i}}(\pi;\gamma_{0},b)=S_{\beta_{i}}Z(\pi;\gamma_{0},b)$.
Furthermore, we have
$S_{\beta_{1}}\hat{\lambda}^{r}(\pi;\gamma_{0},b)=S_{\beta_{1}}\hat{\lambda}(\pi;\gamma_{0},b)$.
Letting
$\Delta(\pi)=(c_{0}S_{\beta}H^{-1}(\pi;\gamma_{0})S_{\beta}^{\prime})^{-1}$,
the asymptotic distribution of $LR_{n}$, for example, is then given by
$\max_{\pi\in\Pi}\left[\hat{\lambda}_{\beta_{1}}(\pi;\gamma_{0},b)^{2}\Delta_{11}(\pi)+\hat{\lambda}_{\beta_{2}}(\pi;\gamma_{0},b)^{2}\Delta_{22}(\pi)\right]-\max_{\pi\in\Pi}\hat{\lambda}_{\beta_{1}}(\pi;\gamma_{0},b)^{2}\Delta_{11}(\pi).$

$\displaystyle LR(\gamma_{0},b)=$
$\displaystyle\max_{\pi\in\Pi}\hat{\lambda}_{\beta}(\pi;\gamma_{0},b)^{\prime}(c_{0}S_{\beta}H^{-1}(\pi;\gamma_{0})S_{\beta}^{\prime})^{-1}\hat{\lambda}_{\beta}(\pi;\gamma_{0},b)$
$\displaystyle-$
$\displaystyle\max_{\pi\in\Pi}\hat{\lambda}^{r}_{\beta}(\pi;\gamma_{0},b)^{\prime}(c_{0}S_{\beta}H^{-1}(\pi;\gamma_{0})S_{\beta}^{\prime})^{-1}\hat{\lambda}^{r}_{\beta}(\pi;\gamma_{0},b),$
(15)

where
$\hat{\lambda}^{r}_{\beta}(\pi;\gamma_{0},b)=S_{\beta}\hat{\lambda}^{r}(\pi;\gamma_{0},b)$,

$\hat{\lambda}^{r}(\pi;\gamma_{0},b)=\operatornamewithlimits{arg\
min}_{\lambda\in\Lambda^{r}}q(\lambda,\pi;\gamma_{0},b),$

and

$\Lambda^{r}=[0,\infty]\times\\{0\\}\times[-\infty,\infty].$

The results in (14) and (15) provide us with the asymptotic distributions of
$LR^{\dagger}_{n}$ and $LR_{n}$, respectively, for true values of $\beta_{1}$
that are close to the boundary relative to the sample size, $b_{1}<\infty$,
which are permitted under $H_{0}$. Together, these results allow us to
determine the asymptotic null rejection frequencies of the testing procedures
proposed by PR under empirically relevant values of $\beta_{1}$. These
asymptotic null rejection frequencies play a crucial role in determining
whether the testing procedures control asymptotic size; see Section 4.

Figure 2: Asymptotic and finite-sample ($n=500$) densities of
$LR_{n}^{\dagger}$ (top row) and $LR_{n}$ (bottom row) with
$\sqrt{n}\beta_{1}=b_{1}=0,2,$ and 4 from left to right and
$\beta_{2}=b_{2}=0$. Here, $\pi=0.2$, $\varphi=0.5$, and $\kappa=0$.

Figure 2 shows the asymptotic and the finite-sample ($n=500$) densities of
$LR_{n}^{\dagger}$ and $LR_{n}$ for the same dgp that underlies Figure 1.
While the asymptotic distribution of $LR_{n}$ provides a very good
approximation to its finite-sample distribution, the asymptotic distribution
of $LR_{n}^{\dagger}$ seems to be first-order stochastically dominated by its
finite-sample distribution. However, looking at Figure 3, which reproduces the
top row of Figure 2 with $n=$ 10,000, this seems to be a “small sample”
phenomenon.

Figure 3: Asymptotic and finite-sample ($n=$ 10,000) densities of
$LR_{n}^{\dagger}$ with $\sqrt{n}\beta_{1}=b_{1}=0,2,$ and 4 from left to
right and $\beta_{2}=b_{2}=0$. Here, $\pi=0.2$, $\varphi=0.5$, and $\kappa=0$.

####  3.1.2 Results for $\|b\|=\infty$ (and $\beta_{n}\to 0$)

Under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$\beta_{0}=0$, we have that

$\|\beta_{n}\|^{-2}(Q_{n}^{c}(\pi)-Q_{0,n})\to\eta(\pi;\gamma_{0},\omega_{0})$
(16)

uniformly over $\pi\in\Pi$, where

$\eta(\pi;\gamma_{0},\omega_{0})=-\frac{1}{2}\omega_{0}^{\prime}K(\pi;\gamma_{0})^{\prime}H^{-1}(\pi;\gamma_{0})K(\pi;\gamma_{0})\omega_{0}.$

Furthermore, we have that $\eta(\pi;\gamma_{0},\omega_{0})$ is uniquely
minimized at $\pi=\pi_{0}\ \forall\gamma_{0}\in\Gamma$ with $\beta_{0}=0$. It
can then be shown that $\hat{\pi}_{n}-\pi_{n}\to_{p}0$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ and
$\|\beta_{n}\|^{-1}(\hat{\psi}_{n}-\psi_{n})=o_{p}(1)$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$\beta_{0}=0$, where the latter result ensures that (17) holds; see the
discussion below Assumption D1 in AC.

###  3.2 Asymptotic distribution for $\|b\|=\infty$

Under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$, we have

$Q_{n}(\theta)=Q_{n}(\theta_{n})+DQ_{n}(\theta_{n})^{\prime}(\theta-\theta_{n})+\frac{1}{2}(\theta-\theta_{n})^{\prime}D^{2}Q_{n}(\theta_{n})(\theta-\theta_{n})+R^{*}_{n}(\theta),$
(17)

where the remainder $R^{*}_{n}(\theta)$ satisfies

$\sup_{\theta\in\Theta_{n}(\delta_{n})}|nR_{n}^{*}(\theta)|=o_{p}(1)$

for all $\delta_{n}\to 0$, where
$\Theta_{n}(\delta_{n})=\\{\theta\in\Theta:\|\psi-\psi_{n}\|\leq\delta_{n}\|\beta_{n}\|\text{
and }\|\pi-\pi_{n}\|\leq\delta_{n}\\}$. Here, $DQ_{n}(\theta)$ and
$D^{2}Q_{n}(\theta)$ denote the vector of first-order left/right partial
derivatives and the matrix of second-order left/right partial derivaties of
$Q^{\infty}_{n}(\theta)$ with respect to $\theta$, respectively. Define

$B(\beta)=\left[\begin{array}[]{cc}I_{3}&0_{3\times 1}\\\ 0_{1\times
3}&\|\beta\|\end{array}\right],$
$J_{n}=B^{-1}(\beta_{n})D^{2}Q_{n}(\theta_{n})B^{-1}(\beta_{n}),$

and

$Z_{n}^{\infty}=-\sqrt{n}J_{n}^{-1}B^{-1}(\beta_{n})DQ_{n}(\theta_{n}).$

Then (17) can be rewritten as

$Q_{n}(\theta)=Q_{n}(\theta_{n})-\frac{1}{2n}{Z_{n}^{\infty}}^{\prime}J_{n}Z_{n}^{\infty}+\frac{1}{2n}q_{n}^{\infty}(\sqrt{n}B(\beta_{n})(\theta-\theta_{n}))+R^{*}_{n}(\theta),$

where

$q_{n}^{\infty}(\lambda)=(\lambda-Z_{n}^{\infty})^{\prime}J_{n}(\lambda-
Z_{n}^{\infty}).$

Note that

$B^{-1}(\beta_{n})DQ_{n}(\theta_{n})=\frac{1}{n}\sum_{t=1}^{n}\frac{1}{2h^{\infty}_{t}(\theta_{n})}\left(1-\frac{y_{t}^{2}}{h^{\infty}_{t}(\theta_{n})}\right)\tau(\beta_{n}/\|\beta_{n}\|,\pi_{n}),$

where

$\displaystyle\tau(\beta/\|\beta\|,\pi)$
$\displaystyle=B^{-1}(\beta)\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\theta}$
$\displaystyle=\left(\sum_{i=0}^{\infty}\pi^{i}y_{t-i-1}^{2},\sum_{i=0}^{\infty}\pi^{i}x_{t-i-1}^{2},1,\sum_{i=1}^{\infty}i\pi^{i-1}\left(\frac{\beta_{1}}{\|\beta\|}y_{t-i-1}^{2}+\frac{\beta_{2}}{\|\beta\|}x_{t-i-1}^{2}\right)\right)^{\prime}.$

Then, under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$, we
have
$\sqrt{n}B^{-1}(\beta_{n})DQ_{n}(\theta_{n})\overset{d}{\to}N(0,V(\gamma_{0},\omega_{0}))$,
where

$V(\gamma_{0},\omega_{0})=\frac{c_{0}}{2}E_{\gamma_{0}}\frac{\tau(\omega_{0},\pi_{0})}{h^{\infty}_{t}(\theta_{0})}\frac{\tau^{\prime}(\omega_{0},\pi_{0})}{h^{\infty}_{t}(\theta_{0})},$

and $J_{n}\to_{p}J(\gamma_{0},\omega_{0})$, where
$J(\gamma_{0},\omega_{0})=V(\gamma_{0},\omega_{0})/c_{0}$, such that

$Z_{n}^{\infty}\overset{d}{\to}Z^{\infty}(\gamma_{0},\omega_{0})\sim
N(0,c_{0}J^{-1}(\gamma_{0},\omega_{0})).$ (18)

Let

$q^{\infty}(\lambda;\gamma_{0},\omega_{0})=(\lambda-Z^{\infty}(\gamma_{0},\omega_{0}))^{\prime}J(\gamma_{0},\omega_{0})(\lambda-Z^{\infty}(\gamma_{0},\omega_{0})).$

Then, under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$, we
have

$\sqrt{n}B(\beta_{n})(\hat{\theta}_{n}-\theta_{n})\overset{d}{\to}\hat{\lambda}^{\infty}(\gamma_{0},\omega_{0},b,p),$
(19)

where
$\hat{\lambda}^{\infty}(\gamma_{0},\omega_{0},b,p)=\operatornamewithlimits{arg\
min}_{\lambda\in\Lambda^{\infty}}q^{\infty}(\lambda;\gamma_{0},\omega_{0})$
and where

$\Lambda^{\infty}=[-b_{1},\infty]\times[-b_{2},\infty]\times[-\infty,\infty]\times[-p,\infty].$

Furthermore, under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$, we have

$2n(Q_{n}(\hat{\theta}_{n})-Q_{n}(\theta_{n}))\overset{d}{\to}-\hat{\lambda}^{\infty}(\gamma_{0},\omega_{0},b,p)^{\prime}J(\gamma_{0},\omega_{0})\hat{\lambda}^{\infty}(\gamma_{0},\omega_{0},b,p).$
(20)

Let $S_{\beta,\pi}=[I_{2}\ 0_{2\times 2};0_{3}^{\prime}\ 1]$ be the selection
matrix that selects (among $\theta$) the entries pertaining to $\beta$ and
$\pi$ and define
$\hat{\lambda}^{\infty}_{\beta,\pi}(\gamma_{0},\omega_{0},b,p)=S_{\beta,\pi}\hat{\lambda}^{\infty}(\gamma_{0},\omega_{0},b,p)$.
Also, let

$\lambda^{\infty,r}(\gamma_{0},\omega_{0},b,p)=\operatornamewithlimits{arg\
min}_{\lambda\in\Lambda^{\infty,r}}q^{\infty}(\lambda;\gamma_{0},\omega_{0}),$

where

$\Lambda^{\infty,r}=[-b_{1},\infty]\times[-b_{2},-b_{2}]\times[-\infty,\infty]\times[-p,\infty],$

and define
$\hat{\lambda}^{\infty,r}_{\beta,\pi}(\gamma_{0},\omega_{0},b,p)=S_{\beta,\pi}\hat{\lambda}^{\infty,r}(\gamma_{0},\omega_{0},b,p)$.
Then, the asymptotic distribution of $LR_{n}$, under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$, is given by

$\displaystyle LR^{\infty}(\gamma_{0},\omega_{0},b,p)=$
$\displaystyle\hat{\lambda}^{\infty}_{\beta,\pi}(\gamma_{0},\omega_{0},b,p)^{\prime}(cS_{\beta,\pi}J^{-1}(\gamma_{0},\omega_{0})S_{\beta,\pi}^{\prime})^{-1}\hat{\lambda}^{\infty}_{\beta,\pi}(\gamma_{0},\omega_{0},b,p)$
$\displaystyle-$
$\displaystyle\hat{\lambda}^{\infty,r}_{\beta,\pi}(\gamma_{0},\omega_{0},b,p)^{\prime}(cS_{\beta,\pi}J^{-1}(\gamma_{0},\omega_{0})S_{\beta,\pi}^{\prime})^{-1}\hat{\lambda}^{\infty,r}_{\beta,\pi}(\gamma_{0},\omega_{0},b,p).$
(21)

In the following section, we use the foregoing asymptotic distribution results
to analyze the asymptotic size of the two testing procedures proposed by PR.

##  4 Asymptotic size and a new test

First, we provide a characterization result for asymptotic size that holds for
any test for testing (3). To that end, let

$H=\\{h=(b_{1},\gamma_{0}):0\leq b_{1}<\infty\text{ and
}\gamma_{0}\in\text{cl}(\Gamma)\text{ with }\beta_{0}=0\\}$

and

$H^{\infty}=\\{h=(p,\gamma_{0}):0\leq p<\infty\text{ and
}\gamma_{0}\in\text{cl}(\Gamma)\text{ with }\pi_{0}=0\\}.$

Furthermore, let $RP_{\mathcal{T}}(h)$ denote the asymptotic rejection
probability of the test (that uses $\mathcal{T}_{n}$) under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $b_{1}<\infty$ and $b_{2}=0$,
where $h=(b_{1},\gamma_{0})$. Similarly, let $RP_{\mathcal{T}}^{\infty}(h)$
denote the asymptotic rejection probability of the test under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$\pi_{0}=0$, $b_{1}=\infty$, $b_{2}=0$, and $\omega_{0}=e_{1}$, where
$h=(p,\gamma_{0})$. Lastly, let $RP_{\mathcal{T}}^{\infty,\infty}\in[0,1]$ be
such that $\limsup_{n\to\infty}RP_{\mathcal{T},n}(\gamma_{n})\geq
RP_{\mathcal{T}}^{\infty,\infty}$ for any
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},\infty)$ with
$\pi_{0}=0$, $b_{1}=\infty$, $b_{2}=0$, and $\omega_{0}=e_{1}$ and
$RP_{\mathcal{T},n}(\gamma_{n})\to RP_{\mathcal{T}}^{\infty,\infty}$ for some
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},\infty)$ with
$\pi_{0}=0$, $b_{1}=\infty$, $b_{2}=0$, and $\omega_{0}=e_{1}$, where
$RP_{\mathcal{T},n}(\gamma_{n})$ denotes the finite-sample rejection
probability of the test under $\gamma_{n}$. We can now state our
characterization result for AsySzT.

######  Proposition 1.

Given $\Gamma$ as defined in Appendix A, we have

$\textup{AsySz}_{\mathcal{T}}=\max\\{\sup_{h\in
H}RP_{\mathcal{T}}(h),\sup_{h\in
H^{\infty}}RP_{\mathcal{T}}^{\infty}(h),RP_{\mathcal{T}}^{\infty,\infty}\\}$

for any test for testing (3).

The proof follows along the lines of the proof of Lemma 2.1 in AC and is given
in Appendix A, along with the verification of all claims made in this section.

Next, we consider the two testing procedures proposed by PR. We have

$RP_{TS}(h)=P\left(\mathbbm{1}(LR^{\dagger}(\gamma_{0},b)>LR^{\dagger}_{1-\alpha}(\gamma_{0},0))\times
LR(\gamma_{0},b)>\text{cv}_{1-\alpha}\right),$

where $LR^{\dagger}_{1-\alpha}(\gamma_{0},0)$ denotes the $1-\alpha$ quantile
of $LR^{\dagger}(\gamma_{0},0)$, and

$RP_{S}(h)=P\left(LR(\gamma_{0},b)>\text{cv}_{1-\alpha}\right).$

We note that $RP_{TS}(h)$ and $RP_{S}(h)$ depend on $\gamma_{0}$ through
$H(\pi;\gamma_{0})$, $\Omega(\pi_{1},\pi_{2};\gamma_{0})$, and
$K(\pi;\gamma_{0})$.

Since $LR^{\dagger}_{n}\to\infty$ and
$P(\widetilde{LR}^{\dagger}_{n,1-\alpha}<\infty)=1$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$\pi_{0}=0$, $b_{1}=\infty$, $b_{2}=0$, and $\omega_{0}=e_{1}$, we have that

$RP_{TS}^{\infty}(h)=RP_{S}^{\infty}(h)=P\left(LR^{\infty}(\gamma_{0},p)>\text{cv}_{1-\alpha}\right),$

where
$LR^{\infty}(\gamma_{0},p)=LR^{\infty}(\gamma_{0},e_{1},(\infty,0)^{\prime},p)$
with $\pi_{0}=0$. Now, let

$\rho(\gamma_{0})=\frac{\left(J^{-1}(\gamma_{0},e_{1})\right)_{\beta_{2},\pi}}{\sqrt{\left(J^{-1}(\gamma_{0},e_{1})\right)_{\beta_{2},\beta_{2}}\left(J^{-1}(\gamma_{0},e_{1})\right)_{\pi,\pi}}}$

and
$q(\gamma_{0})=q(\rho(\gamma_{0}))=\sin^{-1}(\rho(\gamma_{0}))/2\tilde{\pi}$.
Furthermore, let ${LR}^{\infty}_{1-\alpha}(\gamma_{0},0)$ denote the
$1-\alpha$ quantile of ${LR}^{\infty}(\gamma_{0},p)$ and note that
$LR_{1-\alpha}^{\infty}(\gamma_{0},\infty)=\text{cv}_{1-\alpha}$. It can be
shown that $LR_{1-\alpha}^{\infty}(\gamma_{0},p)$ is strictly monotonically
increasing (decreasing) in $p$ if $\rho(\gamma_{0})<0$ ($\rho(\gamma_{0})>0$),
while $LR_{1-\alpha}^{\infty}(\gamma_{0},p)=\text{cv}_{1-\alpha}$ if
$\rho(\gamma_{0})=0$; this result is in line with and extends the result in
PR. It then follows, from Theorem 2.1 in Kopylev and Sinha (2011), that

$\displaystyle\sup_{h\in
H^{\infty}}P\left(LR^{\infty}(\gamma_{0},b)>\text{cv}_{1-\alpha}\right)$
$\displaystyle=$ $\displaystyle\max\\{\sup_{\gamma_{0}:(0,\gamma_{0})\in
H^{\infty},\rho(\gamma_{0})>0}1-((1/2-q(\gamma_{0}))+1/2F_{\chi^{2}}(\text{cv}_{1-\alpha};1)+q(\gamma_{0})F_{\chi^{2}}(\text{cv}_{1-\alpha};2)),\alpha\\},$
(22)

where $F_{\chi^{2}}(\cdot;k)$ denotes the cumulative distribution function of
a $\chi^{2}$ random variable with degree of freedom $k$. Noting that
$1-((1/2-q(\gamma_{0}))+1/2F_{\chi^{2}}(\text{cv}_{1-\alpha};1)+q(\gamma_{0})F_{\chi^{2}}(\text{cv}_{1-\alpha};2))$
is strictly increasing in $\rho(\gamma_{0})$, we have that $\sup_{h\in
H^{\infty}}P\left(LR^{\infty}(\gamma_{0},b)>\text{cv}_{1-\alpha}\right)$ is
bounded from above by $11.46\%$ for $\alpha=0.05$; this number is obtained by
evaluating (22) at $\rho(\gamma_{0})=1$.

In what follows, we determine lower bounds on $\sup_{h\in H}RP_{TS}(h)$ and
$\sup_{h\in H}RP_{S}(h)$ by numerically evaluating $RP_{TS}(h)$ and
$RP_{S}(h)$ at certain choices of $h\in H$. Similarly, we determine a lower
bound on (22) by numerically evaluating $\rho(\gamma_{0})$ at certain choices
of $\gamma_{0}$ such that $(0,\gamma_{0})\in H^{\infty}$ (and by plugging the
resulting value into
$1-((1/2-q(\gamma_{0}))+1/2F_{\chi^{2}}(\text{cv}_{1-\alpha};1)+q(\gamma_{0})F_{\chi^{2}}(\text{cv}_{1-\alpha};2))$).
Combining the above lower bounds then provides us with lower bounds on AsySzTS
and AsySzS.

The numerical evaluation is based on the following dgp, which is inspired by
PR (see their Appendix D). $\phi$ is such that $x_{t}$ follows an AR(1), i.e.,

$x_{t}=\varphi x_{t-1}+\epsilon_{t},$ (23)

where

$\left(\begin{array}[]{c}z_{t}\\\ \epsilon_{t}\end{array}\right)\sim
N\left(\left(\begin{array}[]{c}z_{t}\\\
\epsilon_{t}\end{array}\right),\left(\begin{array}[]{cc}1&\kappa\\\
\kappa&1\end{array}\right)\right).$

We note that, given the above $\phi$, the “additional assumption” of PR is
satisfied if and only if $\kappa=0$. In what follows, we take $\bar{\pi}=0.9$
and obtain results for $\alpha=5\%$; details on the numerical evaluation can
be found in Appendix B. We find that $RP_{TS}(h)=6\%$ where $h$ is given by
$b_{1}=2.5$, $\zeta_{0}=1$, $\pi_{0}=0.64$, $\varphi_{0}=0.5$, and
$\kappa_{0}=0$. Furthermore, we find that $RP_{S}(h)=9.48\%$ where $h$ is as
before, except that $b_{1}=0$ (and $\pi_{0}=\Pi^{*}$). Lastly, we find that
$RP^{\infty}_{TS}(h)=RP^{\infty}_{S}(h)=6.65\%$ where $h$ is given by $p=0$,
$\beta_{1}=0.3$, $\zeta_{0}=1$, $\pi_{0}=0$, $\varphi_{0}=0$, and
$\kappa_{0}=0.99$ so that $\rho(\gamma_{0})=0.39$.151515This lower bound is
increasing in $\kappa_{0}$, e.g., we have
$RP^{\infty}_{TS}(h)=RP^{\infty}_{S}(h)=7.11\%$ where $h$ is as before except
that $\beta_{1}=0.25$ and $\kappa_{0}=0.999$ so that $\rho(\gamma_{0})=0.49$.
We refrain from reporting results for larger $\kappa_{0}$ due to the
decreasing accuracy in the numerical evaluation (as $\kappa_{0}$ approaches
1). Noting that $RP_{TS}^{\infty,\infty}=RP_{S}^{\infty,\infty}=\alpha$, we
obtain the following Corollary to Proposition 1.

######  Corollary 1.

Given $\Gamma$ as defined in Appendix A with $\overline{\pi}=0.9$,
AsySz${}_{TS}\geq 6.65\%$ and AsySz${}_{S}\geq 9.48\%$ at $\alpha=5\%$.

######  Remark 1.

We note that the two lower bounds obtained in Corollary 1 also apply if
$\Gamma$ is restricted to satisfy the “additional assumption” in PR and, in
case of the second testing procedure, the assumption that $\beta_{1}>0$. This
is due to the point-wise nature of the assumptions and the “suprema” in the
definitions of SzT and AsySzT. For example, the “additional assumption” in PR
is only imposed at $\pi=0$ and does not exclude sequences of true parameters
that are such that $\rho(\gamma_{n})\to\rho(\gamma_{0})>0$ and $\pi_{n}>0$ for
all $n\geq 1$, while $\pi_{n}\to\pi_{0}=0$.

Figure 2 shows that the (asymptotic) distribution of $LR_{n}$ does not vary a
lot with $\beta_{1}$, at least for the particular $\gamma$ under
consideration. This motivates our suggestion to test (3) using $LR_{n}$
combined with a plug-in least favorable configuration critical value (PI-LF);
in what follows, the resulting test is abbreviated as $LR$-$LF$. In the
context at hand, we have to consider two identification scenarios
($b_{1}<\infty$ and $b_{1}=\infty$) that result in two different asymptotic
null distributions of $LR_{n}$. The idea is simple: in each scenario, all
unknown quantities/parameters of the asymptotic distribution of $LR_{n}$ that
are consistently estimable are replaced by estimators. For the remaining
parameters, we determine the least favorable configuration, i.e., we determine
under which values of these parameters the $1-\alpha$ quantile of the
asymptotic distribution is maximized. The corresponding $1-\alpha$ quantile
then serves as critical value for the scenario under consideration. Finally,
the maximum of the thus obtained critical values and $\text{cv}_{1-\alpha}$
constitutes our proposed PI-LF. We note that, under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $b_{1}<\infty$ and $b_{2}=0$,
$b_{1}$ and $\pi_{0}$ are not consistently estimable, while, under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$\pi_{0}=0$, $b_{1}=\infty$, $b_{2}=0$, and $\omega_{0}=e_{1}$, $p$ is not
consistently estimable. Our final implementation follows the above (“general”)
approach with two twists. First, under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $b_{1}<\infty$ and $b_{2}=0$,
we restrict the search of the least favorable configuration to values of
$b_{1}$ and $\pi_{0}$ that satisfy
$3(\sqrt{n}b_{1})^{2}+\pi_{0}^{2}+2(\sqrt{n}b_{1})\pi_{0}<1$; this condition
is motivated by the condition $E_{\gamma}y_{t}^{2}<\infty$ (see e.g., Mikosch
and Starica, 2000) that is also imposed by $\Gamma$. Second, if
$\hat{\beta}_{n,1}=0$, then we only consider the critical value obtained from
the $b_{1}<\infty$ scenario, because the probability of observing
$\hat{\beta}_{n,1}=0$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$\pi_{0}=0$, $b_{1}=\infty$, $b_{2}=0$, and $\omega_{0}=e_{1}$ approaches 0.
See Appendix B for more details on the construction of critical values. By
design, we have the following result.

######  Corollary 2.

Given $\Gamma$ as defined in Appendix A, AsySz${}_{LR\text{-}LF}=\alpha$ for
$\alpha\in(0,1/2)$.

##  5 Monte Carlo

In this section, we use simulations to assess (i) how well the finite-sample
null rejection frequencies of the different tests, or testing procedures, that
we consider are approximated by the foregoing asymptotic theory and (ii) how
these tests compare in terms of (finite-sample) power. We generate data from
the model given in (1), (2), and (23). We use a burn-in phase of 100
observations and set the starting values for the $y_{t}$ and $x_{t}$ series
equal to zero. The sample size $n$ (after discarding the burn-in observations)
is equal to 500. The number of simulations is 1,000. Table 1 shows the finite-
sample rejection frequencies (in %) of the likelihood ratio test for testing
$H_{0}^{\dagger}$ ($LR^{\dagger}$), the two-step procedure ($TS$), the second
testing procedure ($S$), and the test that uses $LR_{n}$ together with PI-LF
($LR$-$LF$) at the 5% nominal level for different true parameter
constellations. Throughout, $\zeta$ is set equal to 1, while $\pi$, $\kappa$,
$\varphi$, $\beta_{1}$, and $\beta_{2}$ are varied as indicated in the table.

Table 1: Finite-sample rejection frequencies (in %) at 5% nominal level | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
---|---|---|---|---|---|---|---|---|---|---|---
$\pi$ | 0.2 | 0.2 | 0.2 | 0.64 | 0
$\varphi$ | 0.5 | 0.5 | 0.5 | 0.5 | 0
$\kappa$ | 0 | 0 | 0 | 0 | 0.99
$\beta_{1}$ | 0 | 0.05 | 0.1 | 0.11 | 0.3
$\beta_{2}$ | 0 | 0.05 | 0.1 | 0 | 0.05 | 0.1 | 0 | 0.05 | 0.1 | 0 | 0
$LR^{\dagger}$ | 3.1 | 22.9 | 57.5 | 14.0 | 36.0 | 65.2 | 40.6 | 57.5 | 79.1 | 70.5 | 99.0
$TS$ | 2.0 | 21.9 | 56.6 | 3.4 | 25.8 | 59.7 | 5.6 | 29.9 | 62.9 | 6.2 | 8.2
$S$ | 11.0 | 45.6 | 77.9 |  9.0 | 41.1 | 73.3 | 8.3 | 37.2 | 68.9 | 6.9 | 8.2
$LR$-$LF$ | 5.3 | 34.5 | 67.7 | 4.8 | 30.9 | 61.9 | 4.0 | 26.8 | 57.1 | 3.9 | 0.0

Columns (1), (4), (7), (10), and (11) of Table 1 report null rejection
frequencies for $TS$, $S$, and $LR$-$LF$ ($\beta_{2}=0$). We see that the
asymptotic null rejection probabilities that underlie the lower bounds on
asymptotic size obtained in Section 4 provide good approximations to the
corresponding finite-sample rejection frequencies. In particular, column (1)
shows an 11% rejection frequency of $S$, which is close to the corresponding
asymptotic rejection probability of 9.48%. Similarly, columns (10) and (11)
show rejection frequencies of 6.2% and 8.2% for $TS$, which are close to the
corresponding asymptotic rejection probabilities of 6% and 6.65%,
respectively; note that $\sqrt{500}\times 0.11\approx 2.5$ and
$\sqrt{500}\times 0.3\approx 6.7$.161616Here, 6.7 appears to be large enough
for the asymptotic theory obtained for $b_{1}=\infty$ to provide a good
approximation, which is corroborated by a rejection frequency of
$LR^{\dagger}$ close to 100%. Furthermore, Table 1 reveals that $TS$ has null
rejection frequencies below the nominal level for “very small” values of
$\beta_{1}$ (see columns (1) and (4)); this is not surprising given the nature
of the two-step procedure and the low rejection frequency of $LR^{\dagger}$
for such values of $\beta_{1}$. For the dgps considered in columns (1)–(9) our
numerical evaluations show that $b_{1}=0$ is the (unique) least favorable
configuration for $LR_{n}$. As a result, the $LR$-$LF$ offers sizeable power
gains over $TS$ for “very small” values of $\beta_{1}$ (see columns (2), (3),
(5), and (6)), where the latter, by continuity of the power curve, in some
sense “sacrifices” power. Not surprisingly, for large(r) values of $\beta_{1}$
the power ranking is reversed (see columns (8) and (9)).

## References

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  * Andrews (2001) Andrews, D. W. K. (2001).  Testing When a Parameter Is on the Boundary of the Maintained Hypothesis.  Econometrica 69(3), 683–734.
  * Andrews and Cheng (2012) Andrews, D. W. K. and X. Cheng (2012).  Estimation and Inference With Weak, Semi-Strong, and Strong Identification.  Econometrica 80(5), 2153–2211.
  * Andrews and Guggenberger (2010) Andrews, D. W. K. and P. Guggenberger (2010).  Asymptotic Size and a Problem With Subsampling and With the m out of n Bootstrap.  Econometric Theory 26(2), 426–468.
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  * Ketz (2018) Ketz, P. (2018).  Subvector inference when the true parameter vector may be near or at the boundary.  Journal of Econometrics 207(2), 285–306.
  * Kopylev and Sinha (2011) Kopylev, L. and B. Sinha (2011).  On the asymptotic distribution of likelihood ratio test when parameters lie on the boundary.  Sankhya B 73, 20–41.
  * Leeb and Pötscher (2005) Leeb, H. and B. M. Pötscher (2005).  Model selection and inference: Facts and fiction.  Econometric Theory 21(1), 21–59.
  * Leeb and Pötscher (2008) Leeb, H. and B. M. Pötscher (2008).  Can one estimate the unconditional distribution of post-model-selection estimators?  Econometric Theory 24(2), 338–376.
  * Mikosch and Starica (2000) Mikosch, T. and C. Starica (2000).  Limit theory for the sample autocorrelations and extremes of a GARCH (1, 1) process.  The Annals of Statistics 28(5), 1427–1451.
  * Pedersen and Rahbek (2019) Pedersen, R. S. and A. Rahbek (2019).  Testing GARCH-X Type Models.  Econometric Theory 35(5), 1012–1047.

##  Appendix A Definition of $\Gamma$, verification of claims in Sections 3
and 4, and proof of Proposition 1

First, we define $\Gamma$. [Details to be added.]

The claim in the last paragraph before Section 3.1 follows from Lemma 3.1 in
AC by verifying Assumptions A and B3 in AC. [Details to be added.]

Next, we show that equation (5) holds, which amounts to verifying Assumptions
C1 and C4 in AC. [Details to be added.]

Next, we verify the weak convergence result for $G_{n}(\cdot)$, which amounts
to verifying Assumptions C2 and C3 in AC. [Details to be added.] Note that

$\Omega(\pi_{1},\pi_{2};\gamma_{0})=E_{\gamma_{0}}l^{\infty}_{\psi,t}(\psi_{0},\pi_{1})l^{\infty}_{\psi,t}(\psi_{0},\pi_{2})^{\prime}=\frac{c_{0}}{2\zeta_{0}^{2}}E_{\gamma_{0}}\frac{\partial
h^{\infty}_{t}(\psi_{0},\pi_{1})}{\partial\psi}\frac{\partial
h^{\infty}_{t}(\psi_{0},\pi_{2})}{\partial\psi^{\prime}}.$

Then, equation (9) is obtained by noting (i)

$\displaystyle\frac{1}{\zeta_{0}^{2}}E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi_{1}^{i}y^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi_{2}^{j}y^{2}_{t-j-1}$
$\displaystyle=E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi_{1}^{i}z^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi_{2}^{j}z^{2}_{t-j-1}$
$\displaystyle=\sum_{i=0}^{\infty}\pi_{1}^{i}\pi_{2}^{i}E_{\gamma_{0}}z^{4}_{t-i-1}+\sum_{i=0}^{\infty}\sum_{j\neq
i}\pi_{1}^{i}\pi_{2}^{j}E_{\gamma_{0}}z^{2}_{t-i-1}z^{2}_{t-j-1}$
$\displaystyle=\sum_{i=0}^{\infty}\pi_{1}^{i}\pi_{2}^{i}E_{\gamma_{0}}z^{4}_{t-i-1}-\sum_{i=0}^{\infty}\pi_{1}^{i}\pi_{2}^{i}+\sum_{i=0}^{\infty}\pi_{1}^{i}\sum_{j=0}^{\infty}\pi_{2}^{j}$
$\displaystyle=\frac{E_{\gamma_{0}}z_{t}^{4}-1}{1-\pi_{1}\pi_{2}}+\frac{1}{(1-\pi_{1})(1-\pi_{2})}$
$\displaystyle=\frac{2c}{1-\pi_{1}\pi_{2}}+\frac{1}{(1-\pi_{1})(1-\pi_{2})},$

where the second to last equality uses
$E_{\gamma_{0}}z^{2}_{t-i-1}z^{2}_{t-j-1}=1$ for $i\neq j$, (ii)

$\frac{1}{\zeta_{0}}E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi_{1}^{i}y^{2}_{t-i-1}=E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi_{1}^{i}z^{2}_{t-i-1}=\sum_{i=0}^{\infty}\pi_{1}^{i}E_{\gamma_{0}}z^{2}_{t-i-1}=\frac{1}{1-\pi_{1}},$

and (iii)

$\frac{1}{\zeta_{0}}E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi_{1}^{i}y^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi_{2}^{j}x^{2}_{t-j-1}=E_{\gamma_{0}}\sum_{i=0}^{\infty}\pi_{1}^{i}z^{2}_{t-i-1}\sum_{j=0}^{\infty}\pi_{2}^{j}x^{2}_{t-j-1}.$

To show that $H(\pi;\gamma_{0})=\Omega(\pi,\pi;\gamma_{0})/c_{0}$, note that,
by definition, we have

$H(\pi;\gamma_{0})=E_{\gamma_{0}}\frac{\partial}{\partial\psi^{\prime}}\rho_{\psi,t}(\psi_{0},\pi),$

where

$\frac{\partial}{\partial\psi^{\prime}}\rho_{\psi,t}(\theta)=\frac{\partial}{\partial\psi^{\prime}}\left[\frac{1}{2h^{\infty}_{t}(\theta)}\left(1-\frac{y_{t}^{2}}{h^{\infty}_{t}(\theta)}\right)\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}\right]=\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}\frac{\partial}{\partial\psi^{\prime}}\left[\frac{1}{2h^{\infty}_{t}(\theta)}\left(1-\frac{y_{t}^{2}}{h^{\infty}_{t}(\theta)}\right)\right],$

since $\frac{\partial h^{\infty}_{t}(\theta)}{\partial\psi}$ is not a function
of $\psi$. Furthermore, we have

$\frac{\partial}{\partial\psi^{\prime}}\left[\frac{1}{2h^{\infty}_{t}(\theta)}\left(1-\frac{y_{t}^{2}}{h^{\infty}_{t}(\theta)}\right)\right]=\left[-\frac{1}{2(h^{\infty}_{t}(\theta))^{2}}\left(1-\frac{y_{t}^{2}}{h^{\infty}_{t}(\theta)}\right)+\frac{1}{2h^{\infty}_{t}(\theta)}\frac{y_{t}^{2}}{(h^{\infty}_{t}(\theta))^{2}}\right]\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi^{\prime}}.$

Now, given $\Gamma$, we have

$E_{\gamma_{0}}-\frac{1}{2\zeta_{0}^{2}}\left(1-z_{t}^{2}\right)\frac{\partial
h^{\infty}_{t}(\psi_{0},\pi)}{\partial\psi}\frac{\partial
h^{\infty}_{t}(\psi_{0},\pi)}{\partial\psi^{\prime}}=0$

and the desired result follows.

Next, we derive $K(\pi;\gamma_{0})=K(\psi_{0},\pi;\gamma_{0})$. The expected
value entering $K_{n}(\theta,\gamma^{*})$, see equation (3.6) in AC, is given
by

$\displaystyle
E_{\gamma^{*}}\frac{1}{2h^{\infty}_{t}(\theta)}\left(1-\frac{y_{t}^{2}}{h^{\infty}_{t}(\theta)}\right)\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}=E_{\gamma^{*}}\frac{1}{2h^{\infty}_{t}(\theta)}\left(1-\frac{h^{\infty}_{t}(\theta^{*})}{h^{\infty}_{t}(\theta)}\right)\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}$ $\displaystyle=$ $\displaystyle
E_{\gamma^{*}}\frac{1}{2h^{\infty}_{t}(\theta)}\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}-E_{\gamma^{*}}\frac{1}{2h^{\infty}_{t}(\theta)}\frac{h^{\infty}_{t}(\theta^{*})}{h^{\infty}_{t}(\theta)}\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}$ $\displaystyle=$ $\displaystyle
E_{\gamma^{*}}\frac{1}{2h^{\infty}_{t}(\theta)}\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}-E_{\gamma^{*}}\frac{1}{2h^{\infty}_{t}(\theta)}\frac{\zeta^{*}+\beta^{*}_{1}\sum_{i=0}^{\infty}\pi^{*i}y_{t-i-1}^{2}+\beta^{*}_{2}\sum_{i=0}^{\infty}\pi^{*i}x_{t-i-1}^{2}}{h^{\infty}_{t}(\theta)}\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}$

such that

$K(\psi_{0},\pi;\gamma_{0})=-E_{\gamma_{0}}\frac{1}{2\zeta_{0}^{2}}\left.\frac{\partial
h^{\infty}_{t}(\theta)}{\partial\psi}\right|_{\theta=(\psi_{0},\pi)}\left[\sum_{j=0}^{\infty}\pi^{j}_{0}y_{t-j-1}^{2};\sum_{j=0}^{\infty}\pi^{j}_{0}x_{t-j-1}^{2}\right],$

cf. Assumption C5 in AC. The desired result then follows using similar
observations to those used in the derivation of
$\Omega(\pi_{1},\pi_{2};\gamma_{0})$.

Equation (10) follows from Lemma 9.1 in AC, upon verification of Assumptions
C2, C3, and C5 in AC. [Details to be added.]

Equation (11) follows from arguments analogous to those underlying Theorem
1(a) in Andrews (2001). [Details to be added.] Similarly, equations (12) and
(13) follow from combining the arguments underlying Theorem 3.1 in AC and
Theorem 1(b) and (c) in Andrews (2001). [Details to be added.] Furthermore,
equations (14) and (15) follow from arguments analogous to those underlying
Theorem 2(b) in Andrews (2001). [Details to be added.]

Equation (16) follows from Lemma 3.2(b) in AC. [Details to be added.] To gain
intuition for this result, note that, under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$\beta_{0}=0$, the equivalent of the above $q(\cdot)$ function is given by

$(\lambda-\\{-H^{-1}(\pi;\gamma_{0})K(\pi;\gamma_{0})\omega_{0}\\})^{\prime}H(\pi;\gamma_{0})(\lambda-\\{-H^{-1}(\pi;\gamma_{0})K(\pi;\gamma_{0})\omega_{0}\\}),$
(24)

which is minimized over $\Lambda$, since

$\frac{1}{\|\beta_{n}\|}(\Psi-\psi_{0,n})\to\Lambda.$

Then, noting that $-H^{-1}(\pi;\gamma_{0})K(\pi;\gamma_{0})\omega_{0}\geq 0$
(since all elements of $K(\pi;\gamma_{0})$ are nonpositive), we have that the
minimum of (24) over $\Lambda$ is equal to zero.

Next, we show that $\eta(\pi;\gamma_{0},\omega_{0})$ is uniquely minimized at
$\pi=\pi_{0}\ \forall\gamma_{0}\in\Gamma$ with $\beta_{0}=0$. [Details to be
added.]

The two claims in the last sentence of Section 3.1.2 follow from Lemmas 3.3
and 3.4 in AC, respectively. [Details to be added.]

Next, we show that equation (17) holds. [Details to be added.]

Equation (18) holds given the definition of $\Gamma$. [Details to be added.]

Equations (19) and (20) follow from combining the arguments underlying Theorem
3.2 in AC and Theorem 1(b) and (c) in Andrews (2001). [Details to be added.]
Similarly, equation (21) follows from arguments analogous to those underlying
Theorem 2(b) in Andrews (2001). [Details to be added.]

###### Proof of Proposition 1.

[Details to be added.] ∎

Next, we show that $LR^{\dagger}_{n}\to\infty$ and
$P(\widetilde{LR}^{\dagger}_{n,1-\alpha}<\infty)=1$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$\pi_{0}=0$. [Details to be added.]

Lastly, we show that $LR_{1-\alpha}^{\infty}(\gamma_{0},p)$ is strictly
monotonically increasing (decreasing) in $p$ if $\rho(\gamma_{0})<0$
($\rho(\gamma_{0})>0$). [Details to be added.]

##  Appendix B Computational details

In what follows, we let $\bar{\Pi}$ and $\bar{\Pi}^{*}$ denote discretized
versions of $\Pi$ and $\Pi^{*}$, respectively.

###  B.1 Evaluation of asymptotic distribution for $\|b\|<\infty$

The asymptotic distributions displayed in Figures 1–4 as well as the
asymptotic rejection probabilities $RP_{TS}(h)$ and $RP_{S}(h)$ are evaluated
using simulation. The underlying dgp is given in (1), (2), and (23). To
simulate from $G(\cdot;\gamma_{0})$, we use the method of Hansen (1996). In
particular, we obtain the $j^{\text{th}}$ draw from $G(\pi;\gamma_{0})$ by
computing

$\frac{1}{\sqrt{N}}\sum_{k=1}^{N}Z^{j}_{k}\left(\sum_{i=0}^{99}\pi^{i}z^{2}_{k,t-i-1},\sum_{i=0}^{99}\pi^{i}x^{2}_{k,t-i-1}/\zeta_{0},1/\zeta_{0}\right)^{\prime}{\sqrt{c_{0}/2}},$

where $\\{Z^{j}_{k}\\}_{k=1}^{N}$ are iid $N(0,1)$, independent across $j$,
and where $\\{z_{k,t-i-1},x_{k,t-i-1}\\}_{i=0}^{99}$ are generated according
to (23) (with $z_{k,t-i-1}=z_{t-i-1}$ and $x_{k,t-i-1}=x_{t-i-1}$) using a
burn-in phase of 100 observations. Here and in what follows, $c_{0}=1$.

We approximate $H(\pi;\gamma_{0})$ and $K(\pi;\gamma_{0})$ by

$\frac{1}{2}\left[\begin{array}[]{ccc}\frac{2{c_{0}}}{1-\pi}+\frac{1}{(1-\pi)^{2}}&\frac{1}{\zeta_{0}}\frac{1}{N^{*}}\sum_{l=1}^{N^{*}}\sum_{i=0}^{99}\pi^{i}z^{2}_{l,t-i-1}\sum_{j=0}^{99}\pi^{j}x^{2}_{l,t-j-1}&\frac{1}{\zeta_{0}}\frac{1}{1-\pi}\\\
\frac{1}{\zeta_{0}}\frac{1}{N^{*}}\sum_{l=1}^{N^{*}}\sum_{i=0}^{99}\pi^{i}z^{2}_{l,t-i-1}\sum_{j=0}^{99}\pi^{j}x^{2}_{l,t-j-1}&\frac{1}{\zeta_{0}^{2}}\frac{1}{N^{*}}\sum_{l=1}^{N^{*}}(\sum_{i=0}^{99}\pi^{i}x^{2}_{l,t-i-1})^{2}&\frac{1}{\zeta_{0}^{2}}\frac{1}{N^{*}}\sum_{l=1}^{N^{*}}\sum_{i=0}^{99}\pi^{i}x^{2}_{l,t-i-1}\\\
\frac{1}{\zeta_{0}}\frac{1}{1-\pi}&\frac{1}{\zeta_{0}^{2}}\frac{1}{N^{*}}\sum_{l=1}^{N^{*}}\sum_{i=0}^{99}\pi^{i}x^{2}_{l,t-i-1}&\frac{1}{\zeta_{0}^{2}}\end{array}\right]$

and

$-\frac{1}{2}\left[\begin{array}[]{cc}\frac{2{c_{0}}}{1-\pi\pi_{0}}+\frac{1}{(1-\pi)(1-\pi_{0})}&\frac{1}{\zeta_{0}}\frac{1}{N^{*}}\sum_{l=1}^{N^{*}}\sum_{i=0}^{99}\pi^{i}z^{2}_{l,t-i-1}\sum_{j=0}^{99}\pi_{0}^{j}x^{2}_{l,t-j-1}\\\
\frac{1}{\zeta_{0}}\frac{1}{N^{*}}\sum_{l=1}^{N^{*}}\sum_{i=0}^{99}\pi_{0}^{i}z^{2}_{l,t-i-1}\sum_{j=0}^{99}\pi^{j}x^{2}_{l,t-j-1}&\frac{1}{\zeta_{0}^{2}}\frac{1}{N^{*}}\sum_{l=1}^{N^{*}}\sum_{i=0}^{99}\pi^{i}x^{2}_{l,t-i-1}\sum_{j=0}^{99}\pi_{0}^{j}x^{2}_{l,t-j-1}\\\
\frac{1}{\zeta_{0}}\frac{1}{1-\pi_{0}}&\frac{1}{\zeta_{0}^{2}}\frac{1}{N^{*}}\sum_{l=1}^{N^{*}}\sum_{i=0}^{99}\pi_{0}^{i}x^{2}_{l,t-i-1}\end{array}\right],$

respectively, where $\\{z_{l,t-i-1},x_{l,t-i-1}\\}_{i=0}^{99}$ are generated
similarly to $\\{z_{k,t-i-1},x_{k,t-i-1}\\}_{i=0}^{99}$.

Together, this allows us to obtain the $j^{\text{th}}$ draw from
$Z(\pi;\gamma_{0},b)$. The corresponding draws of the “asymptotic versions” of
the relevant estimators and test statistics can then be obtained using the
formulae in Section 3.1.1 with $\bar{\Pi}$ in place of $\Pi$ for a total of
$J$ draws.

We use $J=N=$ 10,000, $N^{*}=$ 100,000, and
$\bar{\Pi}=\\{0,0.01,0.02,\dots,\overline{\pi}\\}$ with $\overline{\pi}=0.9$.

###  B.2 Evaluation of $\rho(\gamma_{0})$

In order to evaluate $\rho(\gamma_{0})$, it suffices to evaluate
$J(\gamma_{0},e_{1})$ with $\beta_{0,2}=\pi_{0}=0$. As before, the underlying
dgp is given in (1), (2), and (23). We approximate $J(\gamma_{0},e_{1})$ with
$\beta_{0,2}=\pi_{0}=0$ by

$\frac{1}{2N^{*}}\sum_{l=1}^{N^{*}}\frac{\tau_{l}(e_{1},0)}{h^{\infty}_{l,t}(\theta_{0})}\frac{\tau^{\prime}_{l}(e_{1},0)}{h^{\infty}_{l,t}(\theta_{0})},$

where
$\tau_{l}(e_{1},0)=\left(y_{l,t-1}^{2},x_{l,t-1}^{2},1,y_{l,t-2}^{2}\right)^{\prime}$
and $h_{l,t}^{\infty}(\theta_{0})=\zeta_{0}+\beta_{0,1}y_{l,t-1}^{2}$. Here,
$\\{y_{l,t-i-1},x_{l,t-i-1}\\}_{i=0}^{1}$ is generated according to (1), (2),
and (23) (with $y_{l,t-i-1}=y_{t-i-1}$ and $x_{l,t-i-1}=x_{t-i-1}$) using a
burn-in phase of 100 observations.

We use $N^{*}=$ 100,000.

###  B.3 Computation of critical values

####  B.3.1 $b_{1}<\infty$ and critical value for testing $H_{0}^{\dagger}$
using $LR_{n}^{\dagger}$

Under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $b_{1}<\infty$ and
$b_{2}=0$, $\theta_{0}$ is consistently estimated by
$\hat{\theta}^{\dagger}_{n,0}$ (as $\beta_{0}=0_{2}$). Let $\tilde{G}(\cdot)$
denote $G(\cdot;\gamma_{0})$ with unknown quantities replaced by consistent
estimators, i.e., by estimators that are consistent under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $b_{1}<\infty$ and $b_{2}=0$.
To simulate from $\tilde{G}(\cdot)$, we use the method of Hansen (1996). In
particular, we obtain the $j^{\text{th}}$ draw from $\tilde{G}(\pi)$ by
computing

$\tilde{G}_{j}(\pi)=\frac{1}{\sqrt{n}}\sum_{t=1}^{n}Z^{j}_{t}\left(\sum_{i=0}^{t-1}\pi^{i}y^{2}_{t-i-1},\sum_{i=0}^{t-1}\pi^{i}x^{2}_{t-i-1},1\right)^{\prime}{\sqrt{\hat{c}^{\dagger}_{n}/2}}/\hat{\zeta}^{\dagger}_{n,0},$

where $\\{Z^{j}_{t}\\}_{t=1}^{n}$ are iid $N(0,1)$, independent across $j$.
Here, we use the fact that
$h_{t}(\hat{\theta}^{\dagger}_{n,0})=\hat{\zeta}^{\dagger}_{n,0}$.
Furthermore, we estimate $H(\pi;\gamma_{0})$ and $K(\pi;\gamma_{0})$ by

$\displaystyle\tilde{H}(\pi)=$
$\displaystyle\frac{1}{2}\left[\begin{array}[]{ccc}\frac{2{\hat{c}^{\dagger}_{n}}}{1-\pi}+\frac{1}{(1-\pi)^{2}}&\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\frac{1}{n}\sum_{t=1}^{n}\sum_{i=0}^{t-1}\pi^{i}y^{2}_{t-i-1}\sum_{j=0}^{t-1}\pi^{j}x^{2}_{t-j-1}&\frac{1}{\hat{\zeta}^{\dagger}_{n,0}}\frac{1}{1-\pi}\\\
\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\frac{1}{n}\sum_{t=1}^{n}\sum_{i=0}^{t-1}\pi^{i}y^{2}_{t-i-1}\sum_{j=0}^{t-1}\pi^{j}x^{2}_{t-j-1}&\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\frac{1}{n}\sum_{t=1}^{n}(\sum_{i=0}^{t-1}\pi^{i}x^{2}_{t-i-1})^{2}&\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\frac{1}{n}\sum_{t=1}^{n}\sum_{i=0}^{t-1}\pi^{i}x^{2}_{t-i-1}\\\
\frac{1}{\hat{\zeta}^{\dagger}_{n,0}}\frac{1}{1-\pi}&\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\frac{1}{n}\sum_{t=1}^{n}\sum_{i=0}^{t-1}\pi^{i}x^{2}_{t-i-1}&\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\end{array}\right]$

and

$\displaystyle\tilde{K}(\pi,\pi_{0})=$
$\displaystyle-\frac{1}{2}\left[\begin{array}[]{cc}\frac{2{\hat{c}^{\dagger}_{n}}}{1-\pi\pi_{0}}+\frac{1}{(1-\pi)(1-\pi_{0})}&\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\frac{1}{n}\sum_{t=1}^{n}\sum_{i=0}^{t-1}\pi^{i}y^{2}_{t-i-1}\sum_{j=0}^{t-1}\pi_{0}^{j}x^{2}_{t-j-1}\\\
\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\frac{1}{n}\sum_{t=1}^{n}\sum_{i=0}^{t-1}\pi_{0}^{i}y^{2}_{t-i-1}\sum_{j=0}^{t-1}\pi^{j}x^{2}_{t-j-1}&\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\frac{1}{n}\sum_{t=1}^{n}\sum_{i=0}^{t-1}\pi^{i}x^{2}_{t-i-1}\sum_{j=0}^{t-1}\pi_{0}^{j}x^{2}_{t-j-1}\\\
\frac{1}{\hat{\zeta}^{\dagger}_{n,0}}\frac{1}{1-\pi_{0}}&\frac{1}{(\hat{\zeta}^{\dagger}_{n,0})^{2}}\frac{1}{n}\sum_{t=1}^{n}\sum_{i=0}^{t-1}\pi_{0}^{i}x^{2}_{t-i-1}\end{array}\right],$

respectively. Then, the $j^{\text{th}}$ draw from
$\tilde{Z}(\pi;\pi_{0},(b_{1},0)^{\prime})$, the “estimated counterpart” to
$Z(\pi;\gamma_{0},(b_{1},0)^{\prime})$, is given by

$\tilde{Z}_{j}(\pi;\pi_{0},(b_{1},0)^{\prime})=-\tilde{H}^{-1}(\pi)\\{\tilde{G}_{j}(\pi)+\tilde{K}(\pi,\pi_{0})(b_{1},0)^{\prime}\\}.$

The $j^{\text{th}}$ draw of $\widetilde{LR}^{\dagger}$, the “estimated
counterpart” to $LR^{\dagger}(\gamma_{0},0_{2})$, can then be approximated by
evaluating $\tilde{Z}_{j}(\pi;\pi_{0},0_{2})$ over $\bar{\Pi}$ using the
corresponding formulae in Section 3.1.1. We note that
$\tilde{Z}_{j}(\pi;\pi_{0},0_{2})$ and thus the $j^{\text{th}}$ draw of
$\widetilde{LR}^{\dagger}$ do not depend on $\pi_{0}$. Finally, the $1-\alpha$
quantile of the empirical distribution of $J$ draws from
$\widetilde{LR}^{\dagger}$ provides us with an estimate of the $1-\alpha$
quantile of the asymptotic distribution of $LR^{\dagger}_{n}$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $b=0_{2}$. We denote this
quantile by $\widetilde{LR}^{\dagger}_{n,1-\alpha}$.

The $j^{\text{th}}$ draw of $\widetilde{LR}(\pi_{0},(b_{1},0)^{\prime})$, the
“estimated counterpart” to $LR(\gamma_{0},(b_{1},0)^{\prime})$, is
approximated in a similar fashion. And we take the $1-\alpha$ quantile of the
empirical distribution of $J$ draws from
$\widetilde{LR}(\pi_{0},(b_{1},0)^{\prime})$ as estimator of the $1-\alpha$
quantile of the asymptotic distribution of $LR_{n}$ under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},0,b)$ with $b_{1}<\infty$ and $b_{2}=0$,
which we denote by $\widetilde{LR}_{n,1-\alpha}(\pi_{0},b_{1})$.

####  B.3.2 $b_{1}=\infty$

Under $\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$b_{1}=\infty$, $b_{2}=0$, and $\omega_{0}=e_{1}$, $\theta_{0}$ is
consistently estimated by $\hat{\theta}_{n,0}$. Here, we do not impose
$\pi_{0}=0$ and estimate $J(\gamma_{0},e_{1})$ by

$\tilde{J}=\frac{1}{2n}\sum_{t=1}^{n}\tilde{\tau}_{t}\tilde{\tau}^{\prime}_{t}/h^{2}_{t}(\hat{\theta}_{n,0}),$

where

$\tilde{\tau}_{t}=\left(\sum_{i=0}^{t-1}\hat{\pi}_{n,0}^{i}y_{t-i-1}^{2},\sum_{i=0}^{t-1}\hat{\pi}_{n,0}^{i}x_{t-i-1}^{2},1,\sum_{i=1}^{t-1}i\hat{\pi}_{n,0}^{i-1}y_{t-i-1}^{2}\right)^{\prime}.$

Then, we estimate $\rho(\gamma_{0})$ by

$\tilde{\rho}=\frac{\left(\tilde{J}^{-1}\right)_{\beta_{2},\pi}}{\sqrt{\left(\tilde{J}^{-1}\right)_{\beta_{2},\beta_{2}}\left(\tilde{J}^{-1}\right)_{\pi,\pi}}}.$

Finally, let $\tilde{q}=q(\tilde{\rho})$ and define

$\widetilde{LR}^{\infty}_{n,1-\alpha}=\begin{cases}\text{cv}_{1-\alpha}&\text{if
}\tilde{\rho}<0\\\
\left(\sum_{i=0}^{2}\mathbbm{1}(W=i)\times\chi^{2}_{i}\right)_{1-\alpha}&\text{if
}\tilde{\rho}\geq 0\end{cases},$

where $\\{\chi^{2}_{i}\\}_{i=0}^{2}$ are independent $\chi^{2}$ random
variables with degree of freedom $i=0,1,$ and 2 that are independent of the
random variable $W$ that is equal to 0, 1, and 2 with probabilities
$1/2-\tilde{q}$, 1/2, and $\tilde{q}$, respectively. Here, the subscript
${1-\alpha}$ indicates the $1-\alpha$ quantile of the random variable in
parentheses. Note that under
$\\{\gamma_{n}\\}\in\Gamma(\gamma_{0},\infty,b,\omega_{0},p)$ with
$\pi_{0}=0$, $b_{1}=\infty$, $b_{2}=0$, and $\omega_{0}=e_{1}$ such that
$\rho(\gamma_{0})\geq 0$, $\widetilde{LR}^{\infty}_{n,1-\alpha}$ consistently
estimates the $1-\alpha$ quantile of
$LR^{\infty}(\gamma_{0},e_{1},(\infty,0)^{\prime},0)$.

####  B.3.3 Plug-in least favorable configuration critical value

The plug-in least favorable configuration critical value that we propose is
given by

$\text{cv}^{\text{PI-
LF}}_{n,1-\alpha}=\max\\{\max_{(\pi_{0},b_{1})\in{PB}}\widetilde{LR}_{n,1-\alpha}(\pi_{0},b_{1}),\mathbbm{1}(\hat{\beta}_{n,1}>0)\cdot\widetilde{LR}^{\infty}_{n,1-\alpha},\mathbbm{1}(\hat{\beta}_{n,1}>0)\cdot\text{cv}_{1-\alpha}\\},$

where

$PB=\\{(\pi_{0},b_{1})\in\bar{\Pi}^{*}\times
B_{1}:3(\sqrt{n}b_{1})^{2}+\pi_{0}^{2}+2(\sqrt{n}b_{1})\pi_{0}<1\\}$

and where $B_{1}$ is a finite grid.

In our simulations, we use $J=$ 10,000, $\bar{\Pi}=\\{0,0.1,0.2,\dots,0.9\\}$,
$\bar{\Pi}^{*}=\\{0,0.1,0.2,\dots,0.8\\}$, and

$B_{1}=\\{0,0.1,0.2,0.3,0.4,0.5,1,1.5,2,3,4,5,6,8,10,12\\}.$

We chose the grids relatively coarsely to keep the computation cost of the
Monte Carlo simulations low. For empirical applications, we recommend the use
of finer grids. We also note that there is no need to include values of
$b_{1}$ greater than $\sqrt{n/3}$ in $B_{1}$.

##  Appendix C Additional graphs

Figure 4: Asymptotic and finite-sample ($n=500$) densities of
$\sqrt{n}\hat{\beta}_{n,1}$, $\sqrt{n}\hat{\beta}_{n,2}$, and
$\sqrt{n}(\hat{\zeta}_{n}-\zeta_{n})$ from top to bottom with
$\sqrt{n}\beta_{1}=b_{1}=0,2,$ and 4 from left to right and
$\beta_{2}=b_{2}=0$. Here, $\pi=0.2$, $\varphi=0.5$, and $\kappa=0$.