Allowing for weak identication when testing GARCH-X type models Philipp Ketz

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Allowing for weak identiﬁcation when testing

GARCH-X type models

Philipp Ketz∗

October 21, 2022

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.

∗Paris School of Economics - CNRS. E-mail: philipp.ketz@psemail.eu.

arXiv:2210.11398v1 [econ.EM] 20 Oct 2022

1 Introduction

In a GARCH-X type model, the variance of a generalized autoregressive conditional het-

eroskedasticity (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 pres-

ence 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) pro-

cedure, 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 dis-

tribution 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 P¨otscher, 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.1

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,2to 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

1The 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.

2Here, 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.

control asymptotic size (at the 5% nominal level).3Furthermore, 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 approxi-

mations 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 overrejec-

tion 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 empiri-

cally 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→

∞”. eidenotes a vector of zeros (of suitable dimension) with a one in the ith position. For

any matrix A,Aij denotes the entry with row index iand column index j. Furthermore,

Xn(π) = opπ(1) means that supπ∈ΠkXn(π)k=op(1), where k · k denotes the Euclidean

norm. Lastly, “for all δn→0” abbreviates “for all sequences of positive scalar constants

{δn:n≥1}for which δn→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

yt=ht(θ)1/2zt,(1)

3These 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.

where θ= (ψ0, π)0= (β0, ζ, π)0and

ht(θ) = ht(ψ, π) = ht(β, ζ, π) = ζ(1 −π) + β1y2

t−1+πht−1(ψ, π) + β2x2

t−1(2)

with h0(θ) = ζ.4Here, {yt, xt}n

t=0 is observed and {zt}n

t=0 is unobserved. The true parameter

space for θ, i.e., the space of all possible true values of θ, is given by Θ∗= Ψ∗×Π∗, where

Ψ∗={ψ: 0 ≤β1≤β∗

1,0≤β2≤β∗

2, ζ∗≤ζ≤ζ∗}and Π∗={π: 0 ≤π≤π∗}

for some 0 < β∗

1<∞, 0 < β∗

2<∞, 0 < ζ∗< ζ∗<∞, and 0 < π∗<1.

The model is estimated by quasi-maximum likelihood. In particular, the objective func-

tion is given by (- 1

ntimes) the Gaussian-based conditional quasi log-likelihood function, i.e.,

Qn(θ) = 1

nPn

t=1 lt(θ), where

lt(θ) = 1

2log(2˜π) + 1

2log(ht(θ)) + y2

2ht(θ)

and where ˜π= 3.14.... The quasi-maximum likelihood estimator is given by

θn= arg min

θ∈Θ

Qn(θ),

where Θ = Ψ ×Π denotes the optimization parameter space with

Ψ = {ψ: 0 ≤β1≤β1,0≤β2≤β2, ζ ≤ζ≤ζ}and Π = {π: 0 ≤π≤π}

for some β∗

1< β1<∞,β∗

2< β2<∞, 0 < ζ < ζ∗,ζ∗< ζ < ∞, and π∗< π < 1. Note that,

given the deﬁnitions of Θ∗and Θ, (the true values of) β1,β2, and πare 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

H0:β2= 0 vs. H1:β2>0.(3)

As pointed out in PR, this testing problem is non-standard in that, under H0, there are two

nuisance parameters that may be at the boundary of the (optimization) parameter space,

β1and π. Furthermore, when β1is at the boundary (β1= 0) then πis not identiﬁed under

4For ease of reference, we adopt the notation in AC, where βgoverns the identiﬁcation strength of π(see

(4) below) and ψis always identiﬁed.

H0. To see this, note that, given h0(θ) = ζ, we have

ht(θ) = ζ+β1

t−1

i=0

πiy2

t−i−1+β2

t−1

i=0

πix2

t−i−1(4)

and, thus, ht(0, ζ, π) = ζ∀θ= (β, ζ, π) = (0, ζ, π)∈Θ,∀n≥1. In words, under H0,β1= 0

implies that the distribution of the data does not depend on π, i.e., π1and π2(with π16=π2)

are observationally equivalent.

Let Tndenote a generic test statistic for testing (3) and let cvn,1−αdenote the correspond-

ing nominal level αcritical value, which may depend on n. The size of the test that rejects

H0when Tn>cvn,1−αis given by SzT= supγ∈Γ:β2=0 Pγ(Tn>cvn,1−α), where Γ denotes the

true parameter space for γ= (θ, φ) and where φdenotes the distribution of {xt, zt}. We say

that a test controls size if SzT≤α. We note that “uniformity” (over Γ) is built into the

deﬁnition of SzTand that whether a test controls size crucially depends on Γ. 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

AsySzT= lim sup

n→∞

sup

γ∈Γ:β2=0

Pγ(Tv>cvn,1−α).

While AsySzT“still” depends on Γ, it generally only does so through a ﬁnite-dimensional

parameter, making its evaluation “easier”. We say that a test controls asymptotic size if

AsySzT≤α. In large samples, AsySzTprovides a good approximation of SzTso 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 pro-

cedure to deal with the presence of nuisance parameters on the boundary of the parameter

space as well as the lack of identiﬁcation of πwhen β1= 0. In the ﬁrst step, PR propose to

test H†

0:β1=β2= 0 using the corresponding (rescaled) quasi-likelihood ratio statistic

LR†

n= 2n(Qn(ˆ

θ†

n,0)−Qn(ˆ

θn))/ˆc†

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AllowingforweakidenticationwhentestingGARCH-XtypemodelsPhilippKetz*October21,2022AbstractInthispaper,weusetheresultsinAndrewsandCheng(2012),extendedtoallowforparameterstobenearorattheboundaryoftheparameterspace,toderivetheasymptoticdistributionsofthetwoteststatisticsthatareusedinthetwo-step(testing...

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Allowing for weak identication when testing GARCH-X type models Philipp Ketz

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