Uniform Quasi ML based inference for the panel AR(1) model

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Uniform Quasi ML based inference for the panel AR(1) model.



\title{Uniform Quasi ML based inference for the panel AR(1) model.}
\author{Hugo Kruiniger\thanks{
Address: [email removed]; Department of Economics, 1 Mill Hill
Lane, Durham DH1 3LB, England. I thank N. Peyerimhoff for helpful
discussions. } \\
Durham University}
\date{This version: 12 December 2025}
\maketitle

\vspace{6.2cm}

\bigskip

\bigskip

\bigskip

\noindent JEL\ classification: C12, C13, C23.\bigskip

\noindent Keywords: dynamic panel data model, identification robust
inference, quasi Lagrange multiplier test, score test, second-order
identification, singular information matrix, uniform inference, Wald test.

\setcounter{page}{0} \thispagestyle{empty}

\newpage

\baselineskip=20pt


\begin{center}
\textbf{Abstract}
\end{center}

\vspace{1cm}

Maximum Likelihood (ML) offers attractive alternatives to Generalized Method
of Moments (GMM)\ estimators for dynamic panel data models. However, to date
no\linebreak identification robust inference methods exist that can be used
in conjunction with the ML estimators for these models. In this paper we
propose ML based inference methods for panel AR(1) models with arbitrary
initial conditions and heteroskedasticity that are\linebreak robust to the
strength of identification. We show that (Quasi) Lagrange Multiplier (LM)
tests and confidence sets (CSs) that use the expected Hessian rather than
the observed Hessian of the log-likelihood function have correct asymptotic
size and coverage\linebreak probability in a uniform sense, respectively.
Such Quasi LM\ tests and CSs are also robust to misspecification of the
distribution of the data and to heterogeneity, including heteroskedasticity.
We derive the power envelope of a Fixed Effects version of such an LM test
for hypotheses involving the autoregressive parameter when the average
information matrix is estimated by a centered OPG\ estimator and the model
is only second-order identified, and show that it coincides with the maximal
attainable power curve in the worst case setting. We also study the
empirical size and power properties of these (Quasi) LM tests and find that
the hypothesis that the (Quasi) LM test has correct size cannot be rejected.

\setcounter{page}{0} \thispagestyle{empty}\newpage

\section{Introduction}

This paper proposes new inference methods for panel AR models with arbitrary
initial conditions and heteroskedasticity and possibly additional regressors
that are robust to the strength of identification. Specifically, we consider
several Maximum Likelihood based methods of constructing tests and
confidence sets (CSs) and show that (Quasi) LM tests and CSs that use the
expected Hessian rather than the observed Hessian of the log-likelihood have
correct asymptotic size in a uniform sense.

There exists a vast literature devoted to estimation of versions of the
panel AR(1) model or, more generally, dynamic panel data models. The
estimators in this literature can roughly be classified into two groups:
Instrumental Variables/Generalized Method of Moments (IV/GMM) type
estimators and Maximum Likelihood (ML) type estimators. Seminal
contributions to the first group include Anderson and Hsiao (1981, 1982),
Holtz-Eakin et al. (1988), Arellano and Bond (AB, 1991), Arellano and Bover
(ABov, 1995) and Ahn and Schmidt (AS, 1995), while seminal contributions to
the second group include Chamberlain (1980), Anderson and Hsiao (1981,
1982), MaCurdy (1982) and Hsiao et al. (2002). The last two papers
considered Fixed Effects (FE) ML\ estimators, that is, estimators that are
based on data in first differences and are consistent under minimal
assumptions, whereas the other two papers in the second group considered
(correlated) Random Effects (RE) ML estimators.\footnote{
The RE approach assumes finite second moments of the data in levels while
the FE approach only assumes finite second moments of the data in
differences, cf. Kruiniger (2001, 2022).} \footnote{
The Transformed MLE of Hsiao et al. (2002) and the FEMLE in Kruiniger (2001,
2013, 2022) are the same estimator and are the FE counterpart of the REMLE.}

Kruiniger (2001) has shown that when the data are i.i.d. and normal, then
the RE and FE MLEs of the autoregressive parameter ${\Greekmath 011A} $ in a panel AR(1)
model with homoskedastic errors and $\left\vert {\Greekmath 011A} \right\vert <1$ are
asymptotically equivalent to optimal AS-type GMM\ estimators,\linebreak
which in this case also exploit moment conditions that rely on time-series
homoskedasticity, and that when ${\Greekmath 011A} =1,$ the information matrix associated
with the FEMLE for the panel AR(1) model with homoskedastic errors is
singular. Kruiniger (2013) and Alvarez and Arellano (2022) have developed
RE\ and FE MLEs for the panel AR(1) model that allows for time-series
heteroskedasticity. These estimators of ${\Greekmath 011A} $ remain large $N$, fixed $T$
consistent when the data are non-normal or i.h.d., where $N$ and $T$ are the
dimensions of the panel. Kruiniger (2013) and Ahn and Thomas (2023)\ have
shown that when ${\Greekmath 011A} =1$ and the errors bar possibly the error of the last
period are homoskedastic over time, then RE and FE MLEs of ${\Greekmath 011A} $ that
allow for time-series heteroskedasticity are $N^{1/4}$-consistent and have
non-normal\ limiting distributions. The slower rate of convergence is
related to the fact that in this case the information matrix is singular so
that ${\Greekmath 011A} $ is only second-order locally identified, cf. Sargan (1983) and
Rotnitzky et al. (2000).

Monte Carlo studies in Hsiao et al. (2002), Kruiniger (2013) and Hayakawa
and Pesaran (2015) have found that the (Quasi) MLEs for the panel AR(1) and
ARX(1) models have very good finite sample properties but that when ${\Greekmath 011A} $
is close to one, Wald tests for hypotheses about ${\Greekmath 011A} $ are oversized and
asymptotic confidence intervals based on the QMLEs of ${\Greekmath 011A} $ are too narrow
because ${\Greekmath 011A} $ is weakly identified. However, to date no iden-\linebreak
tification robust inference methods have been proposed that are related to
these QMLEs.

Weak identification including second- rather than first-order local
identification affect the rate of convergence and the limiting distributions
of the estimators and pose a challenge to conducting inference both in the
case of GMM and the ML method. For reliable inference it is important that
tests and/or confidence sets (CSs) have correct asymptotic size in a uniform
sense. As Andrews et al. (2020) explain, pointwise asymptotics often provide
very poor approximations to the finite-sample size when the test statistic
of interest has a discontinuous pointwise asymptotic distribution. For this
reason Wald-type tests and CSs for (hypotheses/parameter vectors that
include) ${\Greekmath 011A} $ will generally not have correct asymptotic size in a
uniform sense. Also likelihood based (Quasi) LR-type tests and CSs for
(hypotheses/parameter vectors that include) ${\Greekmath 011A} $ will in many cases not
have correct (locally) uniform asymptotic size, cf. Rotnitzky et al. (2000)
and Bottai (2003). In this paper we will instead propose tests and CSs for
(hypotheses/parameter vectors that include) ${\Greekmath 011A} $ that are based on
(Quasi) LM test statistics that use the expected Hessian of a RE or FE
log-likelihood and show that they have correct uniform asymptotic size.

Our approach generalizes that of Bottai (2003), who considered CSs based on
similar LM test statistics in the context of identifiable one-dimensional
parametric models with a smooth likelihood function and information equal to
zero at a critical point, in at least two ways. Firstly, we consider
multi-parameter models and hypotheses. To show that our (Quasi) LM tests and
CSs have correct uniform asymptotic size, we make use of the fact that in
suitably reparametrized versions of the panel AR(1) model, the parameters
other than ${\Greekmath 011A} $ are still strongly identified when ${\Greekmath 011A} =1.$ Secondly,
by using Quasi LM test statistics, we also allow for misspecification of the
distribution of the data and heterogeneity.

Bottai (2003) explains for the one-parameter case why an LM\ test that is
based on the observed Hessian rather than the expected Hessian will not have
correct uniform asymptotic size when the parameter, say ${\Greekmath 0112} $, is
second-order identified at a critical point ${\Greekmath 0112} ^{\ast }$. In this case
the LM\ test statistic does not converge in distribution to a ${\Greekmath 011F} ^{2}(1)$
random variable under the sequence ${\Greekmath 0112} _{n}={\Greekmath 0112} ^{\ast }-cn^{-1/4}$
as the sample size $n$ tends to infinity although it does converge to a $
{\Greekmath 011F} ^{2}(1)$ random variable under sequences ${\Greekmath 0112} _{n}={\Greekmath 0112} ^{\ast
}-cn^{-b}$ with $b>1/4.$ The same situation arises in the multi-parameter
case.

We will now discuss alternative, GMM-based methods. When the autoregressive
parameter ${\Greekmath 011A} $ is local to unity, then the AB GMM\ estimator has poor
finite sample properties and is inconsistent due to weak instruments, cf.
ABov, Blundell and Bond (BB, 1998) and Kruiniger (K, 2009). If in addition
the individual time series are covariance stationary or the number of
pre-sample realizations is large, then the System estimator of Abov and BB
may have a non-normal limiting distribution and its rate of convergence may
depend on the choice of the weight matrix, cf. K2009. Furthermore, when
mean-stationarity holds but the ratio of the variance of the individual
effects to the variance of the idiosyncratic errors is large, then the AB
GMM and the System estimator can also suffer from a weak instruments problem
and be biased, cf. Hayakawa (2007), Bun and Windmeijer (2010) and Kruiniger
(2001, 2022). Incidentally, Bun and Kleibergen (2022) and Kruiniger (2022)
have shown that provided that $T$ is not too small, ${\Greekmath 011A} $ is identified,
albeit possibly only second-order identified when ${\Greekmath 011A} =1$, by a set of
linear and quadratic AS-type moment conditions that only depend on a lack of
serial correlation assumption for the errors and on differenced data and
hence do not require mean-stationarity to hold.

GMM-based tests (and CSs) for ${\Greekmath 011A} $ that have correct uniform asymptotic
size when ${\Greekmath 011A} =1$ include (CSs based on) the Newey and West (1987) type
GMM\ LM\ test(-statistic)s that exploit ABov and SYS moment conditions,
respectively, see Madsen (2003) and K2009, and identification-robust
test(-statistics)s such as the GMM AR test of Stock and Wright (2000) and
the KLM and GMM-CLR tests of Kleibergen (2005) that exploit AB, ABov, SYS
and AS moments conditions, cf. Bun and Kleibergen (2014).\footnote{
These GMM LM\ tests have correct uniform asymptotic size when ${\Greekmath 011A} =1$
because the scaled sample moment conditions they depend on (evaluated at $
{\Greekmath 011A} =1$) and their first derivates are asymptotically independent. As a
result these GMM\ LM test statistics have the same limiting distribution as
they would have under strong identification, i.e., $N(0,1)$ (or ${\Greekmath 011F}
^{2}(1) $ when squared), see Kruiniger (2009). However, in general GMM LM
tests are not robust to lack of identification. For instance, if ${\Greekmath 011A} =1,$
the GMM LM test that only exploits the moment conditions of AB does not have
correct uniform asymptotic size and shouldn't be used,\thinspace
cf.\thinspace Bond and Windmeijer (2005).} Results in Newey and Windmeijer
(NW, 2009) imply that most of these identification robust test statistics
also have correct asymptotic size under many weak moment conditions
asymptotics with a restriction on the relative rate at which the number of
moment conditions and $N$ tend to infinity. The only possible exceptions are
the tests that exploit the non-linear AS moment conditions when NW's second
assumption, in particular global identification, does not hold. However,
this only happens when $T\geq 3$ and the variances of the errors change over
time at a constant rate, cf. Alvarez and Arellano (2022). Andrews and
Guggenberger (2017) have shown that under rather general conditions the KLM
and GMM-CLR tests for nonlinear moment condition models have correct
asymptotic size as $N\rightarrow \infty $ although they note that for
GMM-CLR tests this result depends in the multi-parameter case on how the
conditioning statistic, upon which the GMM-CLR test depends, is weighted.

Bun and Kleibergen (2014) have shown for $T\geq 4$ that in a worst case
scenario, where the variance of the initial observations goes to infinity,
the true value of ${\Greekmath 011A} $ is equal to one and the errors are homoskedastic
so that ${\Greekmath 011A} $ is only second-order locally identified, the power envelope
of the KLM test based on the AS\ moment conditions and a centered optimal
weight matrix, viz. the AS\ KLM\ test, coincides with the maximal attainable
power curve for testing $H_{0}:{\Greekmath 011A} =1-cN^{-1/4},$ and that in such a
scenario the AS GMM\ AR\ and the AS\ GMM LM tests have less power than the
AS KLM test when $T\geq 5$.\footnote{
In the worst case scenario effectively only moment conditions based on
second moments of differences of the data are exploited, which do not
require mean-stationarity when $\left\vert {\Greekmath 011A} \right\vert <1$.}\ The
analysis of local power of these tests that is provided in Bun and
Kleibergen (2014) is different from that in Dovonon et al. (2020). The
latter assumes that the true value of ${\Greekmath 011A} $ equals $1-cN^{-1/4}$ and tests
$H_{0}:{\Greekmath 011A} =1$, and finds in a simulation study that the AS\ GMM\ AR test
has better power properties than the AS\ KLM\ and AS\ GMM-CLR tests.

Andrews et al. (2019) list some limitations and weaknesses of inference
procedures that are based on identification robust test statistics. Firstly,
the KLM and GMM-CLR tests for subvectors of ${\Greekmath 0112} $ generally no longer
have correct asymptotic size when the nuisance parameters are not strongly
identified, although the GMM AR\ test for subvectors of ${\Greekmath 0112} $ still has
correct asymptotic size in this case when the errors are homoskedastic, cf.
Guggenberger et al. (2012, 2019) and Kleibergen (2021). Secondly, a large
number of inference procedures have been proposed for overidentified models
with nonhomoskedastic errors, some of which are based on analogs or
generalizations of the GMM-CLR test, but there is no consensus on what
procedures to use in practice, beyond the recommendation to use methods that
are efficient when the instruments are strong. Recently, Andrews (2018)
introduced a widely applicable approach to detecting weak identification and
constructing two-step confidence sets that controls coverage distortions
under weak identification. In cases where the model is well identified, his
method indicates this and reports nonrobust confidence sets with probability
tending to 1. Finally, it has been found that increases in the number of
moment conditions can lead to size distortions of the identification robust
tests in finite samples, cf. Kleibergen and Mavroeidis (2009), Kleibergen
(2019) and Bun et al. (2020). The GMM LM test is also susceptible to this
problem. In their Monte Carlo studies based on various panel AR(1) and panel
ARX(1) models with parameter values that correspond to different degrees of
identification strength both Hayakawa and Pesaran (2015) and Bun and
Poldermans (2015) find evidence for size distortions for the GMM\ LM, GMM
AR, KLM and GMM-CLR tests that exploit different sets of moment conditions
(e.g. AB, AS, SYS and subsets and collapsed versions of these sets) and that
the distortions get worse as $T$ or the number of moment conditions
increases. The size distortions become smaller when $N$ gets larger. Bun et
al. (2020) show that when the number of moment conditions is moderately
large, then the GMM AR\ test that is based on a weighting matrix that uses
centered moment conditions is oversized, whereas the uncentered version of
the GMM AR test is conservative. Bun et al. (2020) also propose a
degrees-of-freedom corrected centered GMM\ AR test that has good size
properties.

The paper proceeds as follows. In section 2 we present the panel AR(1)
model, the assumptions, the (Q)ML estimators and their asymptotic properties
including their local-to-unit root limiting distributions. In section 3 we
discuss the asymptotic size properties of various ML\ based tests including
Quasi LM tests that use the expected Hessian of the RE\ or the FE
log-likelihood function. We also derive the power envelope of the FE version
of such an QLM test for $H_{0}:{\Greekmath 011A} =1-cN^{-1/4}$ when the average
information matrix is estimated by a centered OPG\ estimator and the model
is only second-order identified, and the maximal attainable power curve for
testing $H_{0}:{\Greekmath 011A} =1-cN^{-1/4}$ in the worst case scenario. In section 4
we conduct a Monte Carlo study into the empirical size and power properties
of the QLM tests. Section 5 concludes and the appendix contains the proofs.$
\vspace*{-0.22in}$

\section{The panel AR(1) model$\protect\vspace*{-0.1in}$}

Consider the panel AR(1) model with individual effects:$\vspace*{-0.12in}$

\begin{equation}
y_{i,t}={\Greekmath 011A} y_{i,t-1}+{\Greekmath 0111} _{i}+{\Greekmath 0122} _{i,t},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ where }{\Greekmath 0111}
_{i}=(1-{\Greekmath 011A} ){\Greekmath 0116} _{i},\vspace*{-0.18in}  \label{mdl}
\end{equation}
for $i=1,...,N$ and $t=2,...,T.$ When deriving the asymptotics results, we
let $N$ become large while the number of observations per `individual', $T,$
remains fixed. The analysis below can be extended straightforwardly for
models that also include strictly exogenous covariates; we have omitted them
from the model to keep the presentation simple.

We assume that $-1<{\Greekmath 011A} \leq 1.$ Note that the model can be rewritten as $
y_{i,t}-{\Greekmath 0116} _{i}={\Greekmath 011A} (y_{i,t-1}-{\Greekmath 0116} _{i})+{\Greekmath 0122} _{i,t}$ and that the $
{\Greekmath 0111} _{i}$ disappear (the ${\Greekmath 0116} _{i}$ drop out) when ${\Greekmath 011A} =1$. The parame-
trization ${\Greekmath 0111} _{i}=(1-{\Greekmath 011A} ){\Greekmath 0116} _{i}$ prevents the individual effects from
turning into individual trends at ${\Greekmath 011A} =1$ and thereby avoids a
discontinuity in the data generating process at ${\Greekmath 011A} =1$.

The vectors of idiosyncratic errors ${\Greekmath 0122} _{i}=({\Greekmath 0122} _{i,2}$ $
...$ ${\Greekmath 0122} _{i,T})^{\prime }$ are independently distributed across
individuals and satisfy the following Standard Assumptions, SA$k$, for $k=2$
or $k=4$:$\vspace{-0.12in}$
\begin{equation}
E({\Greekmath 0122} _{i,t})=0\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }E\left\vert {\Greekmath 0122} _{i,t}\right\vert
^{k+{\Greekmath 0126} }<\infty \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }i=1,...,N\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }t=2,...,T,
\label{b4}
\end{equation}
where ${\Greekmath 0126} \geq 0$ is an arbitrarily small constant. In the sequel SA2
is denoted by SA.

The individual effects can often be treated as random effects. In this case
we make the following Random Effects Assumptions, REA$k$, for $k=2$ or $k=4$:
$\vspace{-0.12in}$
\begin{eqnarray}
&&(y_{i,1}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }{\Greekmath 0111} _{i})^{\prime },\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }i=1,...,N,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ are i.i.d.
with }{\Greekmath 011B} _{y}^{2}=Var(y_{i,1}),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }E\left\vert y_{i,1}\right\vert
^{k+{\Greekmath 0126} }<\infty ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and\qquad }  \label{b1} \\
&&E({\Greekmath 0116} _{i})=0,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }{\Greekmath 011B} _{{\Greekmath 0116} }^{2}=E({\Greekmath 0116} _{i}^{2}),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }
E\left\vert {\Greekmath 0116} _{i}\right\vert ^{k+{\Greekmath 0126} }<\infty \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }E({\Greekmath 0116}
_{i}y_{i,1})={\Greekmath 011B} _{{\Greekmath 0116} y}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ when }\left\vert {\Greekmath 011A} \right\vert <1.
\vspace{-0.12in}  \notag
\end{eqnarray}
In addition, we let ${\Greekmath 011B} _{{\Greekmath 0111} }^{2}=E({\Greekmath 0111} _{i}^{2})$ and ${\Greekmath 011B}
_{{\Greekmath 0111} y}=E({\Greekmath 0111} _{i}y_{i,1}).$ The i.i.d. assumption in (\ref{b1}) is only
made for presentational convenience. It can easily be relaxed. The
assumption $E\left\vert {\Greekmath 0116} _{i}\right\vert ^{k+{\Greekmath 0126} }<\infty $ (for $
k=2$ or $k=4$) ensures that under covariance stationarity the means of the
data, i.e., ${\Greekmath 0111} _{i}/(1-{\Greekmath 011A} )={\Greekmath 0116} _{i},$ $i=1,...,N$, are drawn from a
distribution with a finite variance rather than a variance that tends to
infinity when ${\Greekmath 011A} $ approaches one.

Unlike the RE ML\ estimators, the FE ML\ estimators only exploit data in
first differences. This reflects the fact that the FE approach entails
making minimal assumptions about the ${\Greekmath 0116} _{i}$ and the $y_{i,1}.$ In the FE
case we assume that $v_{i,1}\equiv y_{i,1}-{\Greekmath 0116} _{i},$ $i=1,...,N$, satisfy a
Fixed Effects Assumption, FEA$k$, for $k=2$ or $k=4$, cf. Kruiniger (2013):$
\vspace{-0.12in}$
\begin{eqnarray}
&&v_{i,1},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }i=1,...,N,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ are i.i.d. with }{\Greekmath 011B}
_{v_{1}}^{2}=Var(v_{i,1})\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }  \label{b2} \\
&&E\left\vert v_{i,1}\right\vert ^{k+{\Greekmath 0126} }<\infty \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ when }
\left\vert {\Greekmath 011A} \right\vert <1.  \notag
\end{eqnarray}
The i.i.d. assumption in (\ref{b2}) is made for presentational convenience.
It can be relaxed.

Suppose that $y_{i,1}$ depends on ${\Greekmath 0116} _{i}$ in a linear fashion, i.e. $
y_{i,1}=E(y_{i,1})+{\Greekmath 010B} _{1}{\Greekmath 0116} _{i}+{\Greekmath 0122} _{i,1}$ with $
E({\Greekmath 0122} _{i,1})=0,$ and ${\Greekmath 0116} _{i}\perp {\Greekmath 0122} _{i,1}$. In the
important case that ${\Greekmath 010B} _{1}=1,$ we have ${\Greekmath 0116} _{i}\perp v_{i,1}$ and FEA$
k$ does not impose any restrictions on ${\Greekmath 0116} _{i}$ and $y_{i,1}$ other than
those on $y_{i,1}-{\Greekmath 0116} _{i}$. However when ${\Greekmath 010B} _{1}\neq 1,$ FEA$k$
implies restrictions on the ${\Greekmath 0116} _{i}$ themselves$.$

In the sequel REA2 and FEA2 are denoted by REA and FEA, respectively.

We add assumption B (Basic), which in the RE\ case amounts to\vspace{-0.1in}
\begin{equation}
{\Greekmath 0122} _{i,s}\perp {\Greekmath 0122} _{i,t}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }i=1,...,N\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }
t\neq s.\vspace{-0.1in}  \label{b3}
\end{equation}
and\vspace{-0.1in}
\begin{equation}
{\Greekmath 0122} _{i,t}\perp y_{i,1}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }{\Greekmath 0122} _{i,t}\perp {\Greekmath 0111} _{i}
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }i=1,...,N\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }t=2,...,T,  \label{b5}
\end{equation}
and in the FE case amounts to (\ref{b3}) and\vspace{-0.1in}
\begin{equation}
{\Greekmath 0122} _{i,t}\perp v_{i,1}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }i=1,...,N\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }t=2,...,T
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ when }\left\vert {\Greekmath 011A} \right\vert <1.  \label{b7}
\end{equation}
We sometimes use an augmented version of assumption B, i.e., assumption B$
^{\prime }$\ that in addi-\linebreak tion to (\ref{b3}) and (\ref{b5})
assumes that ${\Greekmath 0122} _{i,l}{\Greekmath 0122} _{i,s}{\Greekmath 0122} _{i,t}\perp
y_{i,1}$ for all $l,s,t\in \{2,...,T\}$ and $i=1,...,N$.

In the paper we will also make reference to the assumption of
mean-stationarity:\vspace{-0.1in}
\begin{equation}
E(y_{i,1}-{\Greekmath 0116} _{i})=0\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad and\quad }E({\Greekmath 0116} _{i}(y_{i,1}-{\Greekmath 0116} _{i}))=0.
\label{mstat}
\end{equation}

We can allow for heteroskedasticity of the ${\Greekmath 0122} _{i,t}$ in the
time-series dimension:\vspace{-0.1in}
\begin{equation}
E({\Greekmath 0122} _{i,t}^{2})={\Greekmath 0115} _{t}^{2}<\infty ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }i=1,...,N
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }t=2,...,T.\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }  \label{b8}
\end{equation}
In some cases we make the stronger assumption of Time Series
Homoskedasticity (TSH):\vspace{-0.1in}
\begin{equation}
E({\Greekmath 0122} _{i,t}^{2})={\Greekmath 011B} ^{2}<\infty ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ for }i=1,...,N\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{
and }t=2,...,T\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{.}  \label{b6}
\end{equation}
Let $\overline{{\Greekmath 011B} }_{s}^{2}=N^{-1}\sum_{i=1}^{N}E({\Greekmath 0122}
_{i,s}^{2}) $, then we say that assumption TSH$_{t}^{\ast }$ holds if and
only if $\overline{{\Greekmath 011B} }_{s}^{2}=\overline{{\Greekmath 011B} }_{2}^{2},$ for $
s=3,...,t.$ We will often make a `homoskedasticity' assumption that is
weaker than TSH, namely TSH$^{\ast }$, which is short for TSH$_{T}^{\ast }.$
Similarly we can use a weaker version of time-series heteroskedasticity,
namely $\overline{{\Greekmath 011B} }_{t}^{2}={\Greekmath 0115} _{t}^{2}<\infty ,$ for $
t=2,...,T. $\vspace{-0.14in}

\subsection{ML estimators}

Direct application of the Maximum Likelihood method to the nonstationary
panel AR(1) model with RE will generally yield an inconsistent estimator for
${\Greekmath 011A} $ due to correlation between the individual effects (${\Greekmath 0111} _{i}$) and
the regressors ($y_{i,t-1},$ $t=2,...,T$). However, a consistent ML
estimator for ${\Greekmath 011A} $ can be obtained after reformulating the model.
Following Chamberlain (1980) we can decompose the ${\Greekmath 0111} _{i}$ into a term
that depends on the initial observation, $y_{i,1},$ and a term that does
not: \footnote{
For the sake of a simple exposition we assume that $E(y_{i,1})=0$. A
situation where $E(y_{i,1})\neq 0$ can be handled by including an intercept
term in (\ref{re1}) and in (\ref{cm1}).\smallskip \medskip} \vspace{-0.14in}
\begin{equation}
{\Greekmath 0111} _{i}={\Greekmath 0119} (1-{\Greekmath 011A} )y_{i,1}+(1-{\Greekmath 011A} )v_{i},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }i=1,...,N,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }
\label{re1}
\end{equation}
where $v_{i}$ is a new individual effect with $E(v_{i})=0$, ${\Greekmath 0119} (1-{\Greekmath 011A} )=$
plim$_{N\rightarrow \infty }\sum_{i=1}^{N}({\Greekmath 0111} _{i}y_{i,1})/$ $
\sum_{i=1}^{N}y_{i,1}^{2}$ and plim$_{N\rightarrow \infty
}N^{-1}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}(y_{i,1}v_{i})=0.$

Let $y_{i}=(y_{i,2}$ $...$ $y_{i,T})^{\prime }$ and $y_{i,-1}=(y_{i,1}$ $...$
$y_{i,T-1})^{\prime }$ and let ${\Greekmath 0113} $ denote a vector of ones. Then using
the decomposition of the `correlated effects' ${\Greekmath 0111} _{i}$ given in (\ref{re1}
), we can rewrite the panel AR(1) model with RE\ as\vspace{-0.14in}
\begin{eqnarray}
&&y_{i}={\Greekmath 011A} y_{i,-1}+{\Greekmath 0119} (1-{\Greekmath 011A} )y_{i,1}{\Greekmath 0113} +u_{i},\quad \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{where}
\label{cm1} \\
&&u_{i}=(1-{\Greekmath 011A} )v_{i}{\Greekmath 0113} +{\Greekmath 0122} _{i}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad with\quad }
E({\Greekmath 0122} _{i}{\Greekmath 0122} _{i}^{\prime })=\Psi ({\Greekmath 0110} )=diag({\Greekmath 0115}
_{2}^{2},{\Greekmath 0115} _{3}^{2},...,{\Greekmath 0115} _{T}^{2}).  \notag
\end{eqnarray}

Let plim$_{N\rightarrow \infty
}N^{-1}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}v_{i}^{2}={\Greekmath 011B} _{v}^{2}$ and $\Phi =(1-{\Greekmath 011A}
)^{2}{\Greekmath 011B} _{v}^{2}{\Greekmath 0113} {\Greekmath 0113} ^{\prime }+\Psi ({\Greekmath 0110} )$. Then it is easily
verified that plim$_{N\rightarrow \infty
}N^{-1}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}(y_{i,0}{\Greekmath 0113} ^{\prime }\Phi ^{-1}u_{i})=0$
and plim$_{N\rightarrow \infty
}N^{-1}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}(y_{i,-1}^{\prime }\Phi ^{-1}u_{i})=0,$ cf.
similar results under homoskedasticity in Blundell and Bond (1998). After
imposing the assumption that the error components are i.i.d. and normal,
i.e., $u_{i}\sim i.i.d.$ $N(0,\Phi )$, application of the ML method to (\ref
{cm1}) yields the RE ML\ estimator of ${\Greekmath 011A} $, ${\Greekmath 0119} $, ${\Greekmath 011B} _{v}^{2}$ and
${\Greekmath 0110} =({\Greekmath 0115} _{2}^{2}$ ${\Greekmath 0115} _{3}^{2}$ $...$ ${\Greekmath 0115}
_{T}^{2})^{\prime }$. This estimator will still be consistent when the $
{\Greekmath 0122} _{i,t}$ are heteroskedastic across both dimensions of the panel.$
\,$

When calculating the REMLE it is convenient to use the reparameterization $
\tilde{{\Greekmath 0119}}={\Greekmath 0119} (1-{\Greekmath 011A} )$ and $\widetilde{{\Greekmath 011B} }_{v}^{2}=(1-{\Greekmath 011A}
)^{2}{\Greekmath 011B} _{v}^{2}.$ Let ${\Greekmath 0112} _{0}=({\Greekmath 011A} ,\tilde{{\Greekmath 0119}},\widetilde{{\Greekmath 011B}
}_{v}^{2},{\Greekmath 0110} ^{\prime })^{\prime }.$ Then the log-likelihood function for
the above model will be denoted by $l_{RE}({\Greekmath 0112}
)=\mathop{\textstyle \sum }\nolimits_{i=1}^{N}l_{RE,i}({\Greekmath 0112} )$ where ${\Greekmath 0112} =(r,\widetilde{p},
\widetilde{s}_{v}^{2},z^{\prime })^{\prime }$ and $l_{RE,i}({\Greekmath 0112} )$ is the
contribution to $l_{RE}({\Greekmath 0112} )$ from `individual' $i$. The REMLE of $
{\Greekmath 0112} _{0}$ will be denoted by $\widehat{{\Greekmath 0112} }_{RE}$ or simply by $
\widehat{{\Greekmath 0112} }.$

Hsiao et al. (2002) has proposed the Transformed MLE for the panel AR(1)
model, which can be viewed as the FE counterpart of the REMLE, cf. Kruiniger
(2001). One can obtain the FEMLE for the model that allows for
heteroskedasticity by replacing ${\Greekmath 0116} _{i}$ in (\ref{mdl}) by $y_{i,1}+v_{i},$
and imposing that the $(v_{i}$ ${\Greekmath 0122} _{i}^{\prime })$ are i.i.d. and
normal with $v_{i}\perp {\Greekmath 0122} _{i}$ when $\left\vert {\Greekmath 011A} \right\vert
<1$.\footnote{
For the sake of a simple exposition we assume that $E(v_{i})=0$. A situation
where $E(v_{i})\neq 0$ can be handled by including an intercept term in (\ref
{cm2}).\smallskip \medskip} This amounts to imposing the restriction ${\Greekmath 0119} =1
$ on the model in (\ref{cm1}) and leads to the following formulation of the
nonstationary panel AR(1) model with FE:\vspace{-0.14in}
\begin{eqnarray}
&&y_{i}={\Greekmath 011A} y_{i,-1}+(1-{\Greekmath 011A} )y_{i,1}{\Greekmath 0113} +u_{i}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{,\quad where }
\label{cm2} \\
&&u_{i}=(1-{\Greekmath 011A} )v_{i}{\Greekmath 0113} +{\Greekmath 0122} _{i}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad with\quad }
E({\Greekmath 0122} _{i}{\Greekmath 0122} _{i}^{\prime })={\Greekmath 011B} _{i}^{2}\Psi ({\Greekmath 0110} ),
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }  \notag
\end{eqnarray}
and where $v_{i}=-v_{i,1}$ satisfy assumption (\ref{b7}). After imposing $
u_{i}\sim i.i.d.$ $N(0,\Phi )$ with $\Phi =\widetilde{{\Greekmath 011B} }_{v}^{2}{\Greekmath 0113}
{\Greekmath 0113} ^{\prime }+\Psi ({\Greekmath 0110} )$, application of the ML method to (\ref{cm2})
yields the FE MLE of ${\Greekmath 011A} $, $\widetilde{{\Greekmath 011B} }_{v}^{2}$, ${\Greekmath 0110}
=({\Greekmath 0115} _{2}^{2}$ ${\Greekmath 0115} _{3}^{2}$ $...$ ${\Greekmath 0115} _{T}^{2})^{\prime }$.
This estimator will also be consistent when the ${\Greekmath 0122} _{i,t}$ are
heteroskedastic across both dimensions of the panel.$\,$\ The log-likelihood
function for the above model will be denoted by $l_{FE}({\Greekmath 0112} )$ where in
this case ${\Greekmath 0112} =(r,\widetilde{s}_{v}^{2},z^{\prime })^{\prime }$. \vspace{
-0.14in}

\subsection{QML estimation}

In the previous subsection we allowed for cross-sectional heteroskedasticity
but otherwise assumed homogeneity of the distributions of the (standardized)
idiosyncratic errors. However, such strong distributional assumptions with
respect to the errors are almost never satisfied by panel data. K2013 has
shown that when the data exhibit heterogeneity the ML method still yields
consistent RE and FE Quasi ML\ estimators for ${\Greekmath 011A} $ if SA, REA (or FEA)
and B hold, and $T\geq 3$ when $\left\vert {\Greekmath 011A} \right\vert <1$ and $T\geq 4$
when ${\Greekmath 011A} =1$ (cf. Theorem 2 in K2013). Under stronger conditions on the $
{\Greekmath 0122} _{i},$ $y_{i,1}$ and ${\Greekmath 0116} _{i}$ (or $v_{i,1}$), namely under
SA4, REA4 (or FEA4), B and an appropriate Lindeberg condition, K2013 has
also shown that the first-order fixed parameter asymptotic distributions of
the RE and FE QMLEs of ${\Greekmath 0112} _{0}$ are normal when either ${\Greekmath 011A} =1$ and
the average information matrix and the Expected Hessian of the
log-likelihood function are nonsingular or when $\left\vert {\Greekmath 011A} \right\vert
<1$, that is, $\sqrt{N}(\widehat{{\Greekmath 0112} }-{\Greekmath 0112} _{0})\overset{d}{
\rightarrow }N(0,H({\Greekmath 0112} _{0})^{-1}G({\Greekmath 0112} _{0})H({\Greekmath 0112} _{0})^{-1})$ in
these cases, where $H({\Greekmath 0112} _{0})$ is the asymptotic Hessian of the
log-likelihood function and $G({\Greekmath 0112} _{0})$ is the asymptotic information
matrix (cf. Theorem 3 in K2013).\vspace{-0.08in}

\subsubsection{Asymptotic properties of the QMLEs when $\protect{\Greekmath 011A} =1$ and
$TSH^{\ast }$ holds}

K2001 has shown that if ${\Greekmath 011A} =1,$ the ${\Greekmath 0122} _{i}$ are i.i.d. and
normal, and assumption TSH holds and has been imposed on the likelihood
function, then the Expected Hessian of $N^{-1}l_{FE}({\Greekmath 0112} )$ (and thus
also the information matrix) is singular. Ahn and Thomas (2023) and Bond et
al. (2005) have established an analogous result for the Expected Hessian of $
N^{-1}l_{RE}({\Greekmath 0112} ).$ In these cases standard ML theory cannot be used to
derive the limiting distribution of the MLE. Applying the asymptotic theory
recently developed by Rotnitzky, Cox, Bottai and Robins (2000, henceforth
RCBR) for cases with an information matrix of rank one less than full, Ahn
and Thomas (2023) have derived the non-normal limiting distribution of the
REMLE. The theory of RCBR is based on using suitable Taylor expansions of
the possibly reparametrized log-likelihood and score functions around the
value of the parameter vector ${\Greekmath 0112} $\ at which the information matrix is
singular, viz. ${\Greekmath 0112} _{\ast }.$

Unsurprisingly, when ${\Greekmath 011A} =1$ and TSH$^{\ast }$ holds but TSH\ has not been
imposed, the Expected Hessians of $N^{-1}l_{RE}({\Greekmath 0112} )$ and $
N^{-1}l_{FE}({\Greekmath 0112} )$ are singular as well. By extending the theory of RCBR
to allow for i.h.d. data, K2013 has generalized the result of Ahn and Thomas
(2023) as follows.

Application of the asymptotic results of RCBR or their extension for i.h.d.
data to the (Q)MLEs requires that one reparametrizes the RE and FE models
described in section 2.1 so that three conditions, viz. (B1)-(B3), which are
stated just above Theorem 1, are satisfied. They have been verified in K2013
and are similar to conditions given in RCBR. The reparametrizations are such
that the scores at ${\Greekmath 0112} _{\ast }$ for all the new parameters but one are
linearly independent, the QMLEs of these parameters converge at rate $
O_{p}(N^{-1/2})$ and the score at ${\Greekmath 0112} _{\ast }$ for the remaining new
parameter, which is ${\Greekmath 011A} $, is equal to zero w.p.1.

In the case of the RE(Q)MLE we need to use the following new parametrization
(indicated by the subscript $n$): ${\Greekmath 0112} _{0,n}=({\Greekmath 011A} _{n},\widetilde{
{\Greekmath 011B} }_{v,n}^{2},{\Greekmath 0110} _{n}^{\prime },\tilde{{\Greekmath 0119}}_{n})^{\prime }$ where $
{\Greekmath 011A} _{n}={\Greekmath 011A} ,$ $\widetilde{{\Greekmath 011B} }_{v,n}^{2}=\widetilde{{\Greekmath 011B} }
_{v}^{2}/{\Greekmath 011B} _{2}^{2}-(1-{\Greekmath 011A} ),$ ${\Greekmath 0110} _{n}=({\Greekmath 011B} _{2,n}^{2},{\Greekmath 011B}
_{3,n}^{2},...,{\Greekmath 011B} _{T,n}^{2})^{\prime }$ with ${\Greekmath 011B} _{t,n}^{2}={\Greekmath 011B}
_{t}^{2}/{\Greekmath 011A} ,$ $t=2,...,T,$ so that ${\Greekmath 0110} _{n}={\Greekmath 0110} /{\Greekmath 011A} $, and $\tilde{
{\Greekmath 0119}}_{n}=\tilde{{\Greekmath 0119}}-(1-{\Greekmath 011A} ).$ Under this reparametrization the regression
equation becomes $y_{i}-y_{i,1}{\Greekmath 0113} ={\Greekmath 011A} _{n}(y_{i,-1}-y_{i,1}{\Greekmath 0113} )+
\tilde{{\Greekmath 0119}}_{n}y_{i,1}{\Greekmath 0113} +u_{i}.$ Noting that we can express the elements
of ${\Greekmath 0112} $ as functions of the elements of ${\Greekmath 0112} _{n}=(r_{n},\widetilde{s
}_{v,n}^{2},z_{n}^{\prime },\widetilde{p}_{n})^{\prime },$ viz. ${\Greekmath 0112}
({\Greekmath 0112} _{n})=(r_{n},r_{n}s_{2,n}^{2}(\widetilde{s}
_{v,n}^{2}+(1-r_{n})),r_{n}z_{n}^{\prime },\widetilde{p}_{n}+(1-r_{n}))^{
\prime },$ the reparameterized log-likelihood function is given by $
l_{n,RE}({\Greekmath 0112} _{n})=\sum_{i=1}^{N}l_{n,RE,i}({\Greekmath 0112} _{n})$ where $
l_{n,RE,i}({\Greekmath 0112} _{n})=l_{RE,i}({\Greekmath 0112} ({\Greekmath 0112} _{n})).\,$To obtain results
for the FE(Q)MLE one leaves out the last elements of ${\Greekmath 0112} _{0},$ ${\Greekmath 0112}
, $ ${\Greekmath 0112} _{0,n}$ and ${\Greekmath 0112} _{n},$ i.e., the elements that correpond to $
\widetilde{{\Greekmath 0119} },$ $\widetilde{p},$ $\widetilde{{\Greekmath 0119} }_{n}$\ and $\widetilde{p
}_{n}$, respectively. Hence it suffices to focus on the RE(Q)MLE.

Note that if ${\Greekmath 011A} =1$ and TSH$^{\ast }$ holds, then ${\Greekmath 0112} _{0}={\Greekmath 0112}
_{0,n}=(1,0,{\Greekmath 011B} ^{2}{\Greekmath 0113} ^{\prime },0)^{\prime }$ for some ${\Greekmath 011B} ^{2}$
. Thus ${\Greekmath 0112} _{\ast }=(1,0,{\Greekmath 011B} ^{2}{\Greekmath 0113} ^{\prime },0)^{\prime }$ for
some ${\Greekmath 011B} ^{2}$. We may partition ${\Greekmath 0112} _{0}$ as ${\Greekmath 0112} _{0}=({\Greekmath 011A}
,{\Greekmath 010E} ^{\prime })^{\prime }$, partition ${\Greekmath 0112} _{\ast }$ as ${\Greekmath 0112}
_{\ast }=({\Greekmath 011A} _{\ast },{\Greekmath 010E} _{\ast }^{\prime })^{\prime }=(1,{\Greekmath 010E}
_{\ast }^{\prime })^{\prime }$, partition ${\Greekmath 0112} _{0,n}$ as ${\Greekmath 0112}
_{0,n}=({\Greekmath 011A} _{n},{\Greekmath 010E} _{n}^{\prime })^{\prime }$ and partition ${\Greekmath 0112}
_{n}$ as ${\Greekmath 0112} _{n}=(r_{n},d_{n}^{\prime })^{\prime }.$ We will use $
l_{i}, $ $l_{n}$ and $l_{n,i}$ as short for $l_{RE,i}({\Greekmath 0112} ),$ $
l_{n,RE}({\Greekmath 0112} _{n})$ and $l_{n,RE,i}({\Greekmath 0112} _{n}),$ respectively. We also
define:
\begin{eqnarray*}
S_{i,1} &=&\frac{1}{2}\frac{\partial ^{2}l_{n,i}}{\partial r_{n}^{2}}
|_{{\Greekmath 0112} _{\ast }},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }S_{i,2}=\frac{\partial l_{n,i}}{\partial
d_{n}}|_{{\Greekmath 0112} _{\ast }},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }\medskip
S_{i}=(S_{i,1},S_{i,2}^{\prime })^{\prime }, \\
\mathcal{I} &=&\mathcal{I}_{{\Greekmath 0112} _{\ast }{\Greekmath 0112} _{\ast }}=\medskip
\lim_{N\rightarrow \infty }N^{-1}\sum\nolimits_{i=1}^{N}E_{{\Greekmath 0112} _{\ast
}}(S_{i}S_{i}^{\prime }), \\
\mathcal{H}_{1,1} &=&\lim_{N\rightarrow \infty }\frac{2}{4!}N^{-1}E_{{\Greekmath 0112}
_{\ast }}\frac{\partial ^{4}l_{n}}{\partial r_{n}^{4}}|_{{\Greekmath 0112} _{\ast }},
\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }\vspace*{0.35in}\mathcal{H}_{1,2}^{\prime }=\mathcal{H}
_{2,1}=\lim_{N\rightarrow \infty }\frac{1}{2!}N^{-1}E_{{\Greekmath 0112} _{\ast }}\frac{
\partial ^{3}l_{n}}{\partial r_{n}^{2}\partial d_{n}}|_{{\Greekmath 0112} _{\ast }}, \\
\mathcal{H}_{2,2} &=&\lim_{N\rightarrow \infty }\frac{2}{2!}N^{-1}E_{{\Greekmath 0112}
_{\ast }}\frac{\partial ^{2}l_{n}}{\partial d_{n}\partial d_{n}^{\prime }}
|_{{\Greekmath 0112} _{\ast }},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }\vspace*{0.35in}\mathcal{H}=\mathcal{H}
_{{\Greekmath 0112} _{\ast }{\Greekmath 0112} _{\ast }}=\left[
\begin{array}{cc}
\mathcal{H}_{1,1} & \mathcal{H}_{1,2} \\
\mathcal{H}_{2,1} & \mathcal{H}_{2,2}
\end{array}
\right] , \\
\mathcal{H}^{-1} &=&[\mathcal{H}^{k,l}]\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ with }\dim (\mathcal{H}
^{1,1})=1\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }\vspace*{0.2in}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{and }\dim (\mathcal{H}^{2,2})=\dim
(d_{n}), \\
C_{1,1} &=&-\lim_{N\rightarrow \infty }\frac{1}{5!}N^{-1}E_{{\Greekmath 0112} _{\ast }}
\frac{\partial ^{5}l_{n}}{\partial r_{n}^{5}}|_{{\Greekmath 0112} _{\ast }},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad
}C_{1,2}^{\prime }=C_{2,1}=\medskip -\lim_{N\rightarrow \infty }\frac{1}{
2\times (3!)}N^{-1}E_{{\Greekmath 0112} _{\ast }}\frac{\partial ^{4}l_{n}}{\partial
r_{n}^{3}\partial d_{n}}|_{{\Greekmath 0112} _{\ast }},\quad \\
C_{2,2} &=&-\lim_{N\rightarrow \infty }\frac{1}{2!}N^{-1}E_{{\Greekmath 0112} _{\ast }}
\frac{\partial ^{3}l_{n}}{\partial r_{n}\partial d_{n}\partial d_{n}^{\prime
}}|_{{\Greekmath 0112} _{\ast }},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }C=\medskip \left[
\begin{array}{cc}
C_{1,1} & C_{1,2} \\
C_{2,1} & C_{2,2}
\end{array}
\right] , \\
U_{1,N} &=&(\frac{1}{3!}N^{-1/2}\frac{\partial ^{3}l_{n}}{\partial r_{n}^{3}}
|_{{\Greekmath 0112} _{\ast }},N^{-1/2}\frac{\partial ^{2}l_{n}}{\partial r_{n}\partial
d_{n}^{\prime }}|_{{\Greekmath 0112} _{\ast }})^{\prime },\quad U_{2,N}=(\frac{1}{2!}
N^{-1/2}\frac{\partial ^{2}l_{n}}{\partial r_{n}^{2}}|_{{\Greekmath 0112} _{\ast
}},N^{-1/2}\frac{\partial l_{n}}{\partial d_{n}^{\prime }}|_{{\Greekmath 0112} _{\ast
}})^{\prime }, \\
U_{N} &=&U_{1,N}+C\mathcal{H}^{-1}U_{2,N},\quad U\sim N(0,\Sigma _{U}), \\
\breve{Z}_{N} &=&(\breve{Z}_{1,N},\breve{Z}_{2,N}^{\prime })^{\prime }=-
\mathcal{H}^{-1}U_{2,N}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ with }\dim (\breve{Z}_{1,N})=1, \\
\breve{Z} &=&(\breve{Z}_{1},\breve{Z}_{2}^{\prime })^{\prime }\sim N(0,
\mathcal{H}^{-1}\mathcal{IH}^{-1})\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ with }\dim (\breve{Z}_{1})=1,
\end{eqnarray*}
\begin{equation*}
\breve{B}\sim Bernoulli\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ with }\Pr (\breve{B}=0)=\Pr (U^{\prime }\breve{
Z}>0).
\end{equation*}
Note that $U_{2,N}=(N^{-1/2}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}S_{i,1},$ $
N^{-1/2}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}S_{i,2}^{\prime })^{\prime }$ and that
condition (B2), which requires that for all $K\in
\mathbb{R}
^{T+1}$ $\frac{\partial ^{2}l_{n}}{\partial {\Greekmath 011A} _{n}^{2}}|_{{\Greekmath 0112} _{\ast
}}\neq K^{\prime }\frac{\partial l_{n}}{\partial {\Greekmath 010E} _{n}}|_{{\Greekmath 0112}
_{\ast }}$ with positive probability, implies that $\mathcal{I}$ is
nonsingular.

Condition (B3), which requires that for all $\overline{K}=(k$ $K^{\prime
})^{\prime }\in
\mathbb{R}
^{T+2}$\ $\frac{\partial ^{3}l_{n}}{\partial {\Greekmath 011A} _{n}^{3}}|_{{\Greekmath 0112} _{\ast
}}\neq k\frac{\partial ^{2}l_{n}}{\partial {\Greekmath 011A} _{n}^{2}}|_{{\Greekmath 0112} _{\ast
}}+K^{\prime }\frac{\partial l_{n}}{\partial {\Greekmath 010E} _{n}}|_{{\Greekmath 0112} _{\ast }}$
with positive probability, ensures that $U_{N}$ is not identically equal to
zero.

Condition (B1) requires that $\frac{\partial l_{n}}{\partial {\Greekmath 011A} _{n}}
|_{{\Greekmath 0112} _{\ast }}=0$ w.p.1.

In order to obtain the parametrization for which conditions (B1)-(B3) are
met\linebreak K2013 has used the formula ${\Greekmath 0112} _{n}={\Greekmath 0112} +[0,\widetilde{K
}^{\prime }]^{\prime }(r-1),$ where $\widetilde{K}=\mathop{\textstyle \sum }_{i}E((\frac{
\partial l_{i}}{\partial {\Greekmath 011A} }|_{{\Greekmath 0112} _{\ast }})(\frac{\partial l_{i}}{
\partial {\Greekmath 010E} }|_{{\Greekmath 0112} _{\ast }}))\times $\linebreak\ $[\mathop{\textstyle \sum }_{i}E((
\frac{\partial l_{i}}{\partial {\Greekmath 010E} }|_{{\Greekmath 0112} _{\ast }})(\frac{\partial
l_{i}}{\partial {\Greekmath 010E} }|_{{\Greekmath 0112} _{\ast }})^{\prime })]^{-1}$, cf. RCBR p.
264.

Let $\mathbf{1}\{\breve{Z}_{1}>0\}=1$ if $\breve{Z}_{1}>0,$ let $\mathbf{1}\{
\breve{Z}_{1}>0\}=0$ if $\breve{Z}_{1}\leq 0,$ let $\mathbf{1}\{\breve{Z}
_{1}\leq 0\}=1-\mathbf{1}\{\breve{Z}_{1}>0\},$ and let ${\Greekmath 0112} _{a,n}=({\Greekmath 011A}
_{n},\widetilde{{\Greekmath 011B} }_{v,n}^{2},{\Greekmath 0110} _{n}^{\prime })^{\prime }.$ Then
K2013 obtained the following result when the ${\Greekmath 0122} _{i}$ (or,
equivalently, the data) are i.h.d.:\pagebreak

\begin{theorem}[K2013]
Let assumptions SA4, REA4 (FEA4) and B hold and let $T\geq 4$. Let ${\Greekmath 011A} =1$
and TSH$^{\ast }$ hold, that is, let ${\Greekmath 0112} _{0,n}={\Greekmath 0112} _{\ast }$ for
some ${\Greekmath 011B} ^{2}.$ Finally, let $\widehat{{\Greekmath 0112} }_{n}\equiv (\widehat{{\Greekmath 011A}
}_{n},\widehat{{\Greekmath 010E} }_{n})^{\prime }$ be the REQMLE (FEQMLE) that
maximizes the value of $l_{n}({\Greekmath 0112} _{n})=l_{n,RE}({\Greekmath 0112} _{n})$ $
(l_{n,FE}({\Greekmath 0112} _{n}))$. If the regularity conditions (A1)-(A7) given in
K2013 hold and $\mathcal{H}$ is nonsingular, then \vspace{0.08in}\newline
a) $\frac{\partial l_{n}}{\partial r_{n}}|_{{\Greekmath 0112} _{\ast }}=0$ and $\frac{
\partial l_{n}}{\partial d_{n}}|_{{\Greekmath 0112} _{\ast }}\neq 0$ a.s. Furthermore, $
rank(I_{{\Greekmath 010E} _{n}{\Greekmath 010E} _{n}})=\dim (d_{n})$ a.s. \vspace{0.14in}\newline
b) $\left[
\begin{array}{c}
N^{1/4}(\widehat{{\Greekmath 011A} }_{n}-1) \\
N^{1/2}(\widehat{{\Greekmath 010E} }_{n}-{\Greekmath 010E} _{n})
\end{array}
\right] \overset{d}{\rightarrow }\left[
\begin{array}{c}
(-1)^{\breve{B}}\breve{Z}_{1}^{1/2} \\
\breve{Z}_{2}
\end{array}
\right] \mathbf{1}\{\breve{Z}_{1}>0\}+\left[
\begin{array}{c}
0 \\
\breve{Z}_{2}-(\mathcal{H}^{2,1}/\mathcal{H}^{1,1})\breve{Z}_{1}
\end{array}
\right] \mathbf{1}\{\breve{Z}_{1}\leq 0\}$\vspace{0.14in}\newline
$U_{N}\overset{d}{\rightarrow }U$ and $\breve{Z}_{N}\overset{d}{\rightarrow }
\breve{Z},$ and if in addition assumption B$^{\prime }$ holds, then$
\smallskip $\newline
c1) $\mathcal{I}_{RE}=diag(\mathcal{I}_{RE,{\Greekmath 0112} _{a,n}{\Greekmath 0112} _{a,n}},
\mathcal{I}_{RE,\tilde{{\Greekmath 0119}}_{n}\tilde{{\Greekmath 0119}}_{n}})$ where $\dim (\mathcal{I}
_{RE,\tilde{{\Greekmath 0119}}_{n}\tilde{{\Greekmath 0119}}_{n}})=1;$ $\mathcal{I}_{RE,{\Greekmath 0112}
_{a,n}{\Greekmath 0112} _{a,n}}=\mathcal{I}_{FE}.\smallskip $\newline
c2) $\mathcal{H}_{RE}=diag(\mathcal{H}_{RE,{\Greekmath 0112} _{a,n}{\Greekmath 0112} _{a,n}},
\mathcal{H}_{RE,\tilde{{\Greekmath 0119}}_{n}\tilde{{\Greekmath 0119}}_{n}})$ where $\dim (\mathcal{H}
_{RE,\tilde{{\Greekmath 0119}}_{n}\tilde{{\Greekmath 0119}}_{n}})=1;$ $\mathcal{H}_{RE,{\Greekmath 0112}
_{a,n}{\Greekmath 0112} _{a,n}}=\mathcal{H}_{FE}.\smallskip \newline
$d) in general $\widehat{{\Greekmath 011A} }_{RE}\overset{asy}{\nsim }\widehat{{\Greekmath 011A} }
_{FE} $ but if the ${\Greekmath 0122} _{i}$ are i.i.d. and normal, then $\widehat{
{\Greekmath 011A} }_{RE}\overset{asy}{\sim }\widehat{{\Greekmath 011A} }_{FE}.$
\end{theorem}

The QMLEs of ${\Greekmath 0112} $ are given by $\widehat{{\Greekmath 0112} }={\Greekmath 0112} (\widehat{
{\Greekmath 0112} }_{n})$. The parameter ${\Greekmath 011A} $ is second-order identified (cf.
Sargan, 1983) because $\frac{\partial l_{n}}{\partial r_{n}}|_{{\Greekmath 0112} _{\ast
}}=0,$ while $\frac{\partial ^{2}l_{n,i}}{\partial r_{n}^{2}}|_{{\Greekmath 0112}
_{\ast }}\neq 0$, and because for all $K\in
\mathbb{R}
^{T+1}$ $\frac{\partial ^{2}l_{n}}{\partial r_{n}^{2}}|_{{\Greekmath 0112} _{\ast
}}\neq K^{\prime }\frac{\partial l_{n}}{\partial d_{n}}|_{{\Greekmath 0112} _{\ast }}$
with positive probability, that is, (B.2) holds. As a result the expansion
of the log-likelihood involves $N^{1/4}(\widetilde{{\Greekmath 011A} }-1)$ rather than $
N^{1/2}(\widetilde{{\Greekmath 011A} }-1)$ and the QMLEs of $\left\vert {\Greekmath 011A}
-1\right\vert $ converge at rate $O_{p}(N^{-1/4}).$ Furthermore, when $
{\Greekmath 0112} _{0,n}={\Greekmath 0112} _{\ast }$ and $\breve{Z}_{1,N}>0,$ $l_{n}({\Greekmath 0112} _{n})$
is bimodal. In this case the QMLE of ${\Greekmath 011A} $ is determined by higher-order
terms in the expansion of $l_{n}({\Greekmath 0112} _{n})$ around ${\Greekmath 0112} _{n}={\Greekmath 0112}
_{\ast }$, i.e., $U_{N}^{\prime }\breve{Z}_{N}$, cf. K2013. Although $\frac{
\partial l_{n}}{\partial d_{n}}|_{{\Greekmath 0112} _{\ast }}\neq 0$ a.s.\ and $
\widehat{{\Greekmath 010E} }_{n}-{\Greekmath 010E} _{n}=O_{p}(N^{-1/2}),$ whereas $\frac{\partial
l_{n}}{\partial r_{n}}|_{{\Greekmath 0112} _{\ast }}=0$ a.s.\ and $\widehat{{\Greekmath 011A} }
_{n}-1=O_{p}(N^{-1/4}),$ all elements of $\widehat{{\Greekmath 0112} }_{a,n}$ have a
non-normal limiting distribution. All these findings would still hold if the
model also included exogenous regressors.

Theorem 1 holds for i.h.d. data. In the special case where the ${\Greekmath 0122}
_{i}$ are i.i.d. and normal, $\mathcal{H}=-\mathcal{I}$, $\breve{Z}\sim N(0,
\mathcal{I}^{-1}),$ $U\perp \breve{Z}$ and hence $\Pr (\breve{B}=0|\breve{Z}
_{1})=1/2,$ $\widehat{{\Greekmath 011A} }$ has a symmetric limiting distribution, and $
\widehat{{\Greekmath 011A} }_{RE}$ is asymptotically equivalent to $\widehat{{\Greekmath 011A} }_{FE}$
. However, in general (i.e., except for some special cases) if the $
{\Greekmath 0122} _{i}$ are i.i.d and non-normal or i.h.d., then $\mathcal{H}\neq -
\mathcal{I}$, $E(U\breve{Z}^{\prime })\neq 0$ and hence $\Pr (\breve{B}=0|
\breve{Z}_{1})\neq \Pr (\breve{B}=0)$, and $\widehat{{\Greekmath 011A} }$ has an
asymmetric limiting distribution. Moreover, $\widehat{{\Greekmath 011A} }_{RE}$ is not
asymptotically equivalent to $\widehat{{\Greekmath 011A} }_{FE}$.

\subsection{The distributional properties of QML estimators when $\protect
{\Greekmath 011A} $ is close to unity}

It is well-known that (Quasi) ML estimators can be reinterpreted as GMM
estimators, cf. e.g. Newey and McFadden (NMcF, 1994). The underlying moment
conditions can be obtained by setting the expected score vector equal to
zero. It follows that the Expected Hessian of the (quasi) log-likelihood
function equals the first derivative of the vector of moment conditions
exploited by the (Q)MLE with respect to the parameters. Therefore when the
Expected Hessian of the (quasi) log-likelihood function is almost singular,
the (Q)MLE suffers from a `weak moment conditions problem.' K2013 gives
necessary and sufficient conditions for this situation to arise when $
Var(y_{i,1}-{\Greekmath 0116} _{i})\propto (1-{\Greekmath 011A} )^{0}=1$ $\forall i\in \{1,2,...,N\}$:

\begin{theorem}[K2013]
Let assumptions SA, REA (or FEA) and B hold and let\linebreak $
Var(y_{i,1}-{\Greekmath 0116} _{i})\propto (1-{\Greekmath 011A} )^{0}=1$ $\forall i\in \{1,2,...,N\}$.
Furthermore let ${\Greekmath 011A} $ be local to unity. Then the Expected Hessian of $
l_{RE}({\Greekmath 0112} )$ $(l_{FE}({\Greekmath 0112} ))$ is almost singular if and only if
either assumption TSH$^{\ast }$ holds, assumption TSH has been imposed on $
l_{RE}({\Greekmath 0112} )$ $(l_{FE}({\Greekmath 0112} ))$ and $T\geq 3,$ or assumption TSH$
_{T-1}^{\ast }$ (almost) holds and $T\geq 4$.
\end{theorem}

If the Expected Hessian of the (quasi) log-likelihood function is almost
singular, one can obtain a better approximation to the finite sample
distribution of the (Q)MLE than the usual approximation by using local
asymptotics. K2013 obtained the following result:

\begin{theorem}[K2013]
Let assumptions SA4, FEA4 (or REA4) and B hold, let $T\geq 4$, let ${\Greekmath 010E}
_{\ast }$ be such that $(1,{\Greekmath 010E} _{\ast }^{\prime })^{\prime }={\Greekmath 0112}
_{\ast },$ and let $\{{\Greekmath 0112} _{n,N}\}$ be such that $N^{1/4}({\Greekmath 011A}
_{n,N}-1)=o(1)$ and $N^{1/2}({\Greekmath 010E} _{n,N}-{\Greekmath 010E} _{\ast })=o(1)$ for some $
{\Greekmath 011B} ^{2}.$ Let $\{{\Greekmath 0112} _{i,n,N}\},$ $i=1,...,N,$ be such that ${\Greekmath 011A}
_{i,n,N}={\Greekmath 011A} _{n,N},$ $N^{-1}\sum_{i=1}^{N}\widetilde{{\Greekmath 011B} }
_{v,i,n,N}^{2}=\widetilde{{\Greekmath 011B} }_{v,n,N}^{2},$ $N^{-1}\sum_{i=1}^{N}{\Greekmath 0110}
_{i,n,N}={\Greekmath 0110} _{n,N}$ $($and $\tilde{{\Greekmath 0119}}_{i,n,N}=\tilde{{\Greekmath 0119}}_{n,N}).$ Let $
\widehat{{\Greekmath 0112} }_{n}=(\widehat{{\Greekmath 011A} }_{n},\widehat{{\Greekmath 010E} }_{n})^{\prime }$
be the FEQMLE (REQMLE) that maximizes the value of $l_{n}({\Greekmath 0112}
_{n})=l_{n,FE}({\Greekmath 0112} _{n})$\linebreak $(l_{n,RE}({\Greekmath 0112} _{n}))$. Let $
\mathcal{H},$ $\breve{B}$ and $\breve{Z}=(\breve{Z}_{1},\breve{Z}
_{2}^{\prime })^{\prime }$ satisfy the definitions given just above theorem
1. If $\mathcal{H}$ is nonsingular and $(y_{i,1}$ $y_{i}^{\prime })^{\prime
} $ is generated under ${\Greekmath 0112} _{i,n,N},$ $i=1,...,N,$ then\vspace{0.06in}
\newline
a) $\widehat{{\Greekmath 0112} }_{n}-{\Greekmath 0112} _{\ast }=o_{p}(1).$\vspace{0.06in}\newline
b) $\left[
\begin{array}{c}
N^{1/4}(\widehat{{\Greekmath 011A} }_{n}-1) \\
N^{1/2}(\widehat{{\Greekmath 010E} }_{n}-{\Greekmath 010E} _{\ast })
\end{array}
\right] \overset{d}{\rightarrow }\left[
\begin{array}{c}
(-1)^{\breve{B}}\breve{Z}_{1}^{1/2} \\
\breve{Z}_{2}
\end{array}
\right] \mathbf{1}\{\breve{Z}_{1}>0\}+\left[
\begin{array}{c}
0 \\
\breve{Z}_{2}-(\mathcal{H}^{2,1}/\mathcal{H}^{1,1})\breve{Z}_{1}
\end{array}
\right] \mathbf{1}\{\breve{Z}_{1}\leq 0\}$\vspace{0.14in}\newline
c) in general $\widehat{{\Greekmath 011A} }_{RE}\overset{asy}{\nsim }\widehat{{\Greekmath 011A} }_{FE}$
but if assumption B$^{\prime }$ also holds and the $u_{i}$ are i.i.d. and
normal, then $\widehat{{\Greekmath 011A} }_{RE}\overset{asy}{\sim }\widehat{{\Greekmath 011A} }_{FE}.$
\end{theorem}

Note that since $\widetilde{{\Greekmath 011B} }_{v,n}^{2}=\widetilde{{\Greekmath 011B} }
_{v}^{2}/{\Greekmath 011B} _{2}^{2}-(1-{\Greekmath 011A} )=(1-{\Greekmath 011A} )((1-{\Greekmath 011A} ){\Greekmath 011B} _{v}^{2}/{\Greekmath 011B}
_{2}^{2}-1)$ and $\tilde{{\Greekmath 0119}}_{n}=({\Greekmath 0119} -1)(1-{\Greekmath 011A} )$, both condition $
N^{1/2}(\widetilde{{\Greekmath 011B} }_{v,n,N}^{2}-0)=o(1)$ and condition $N^{1/2}(
\tilde{{\Greekmath 0119}}_{n,N}-0)=o(1)$ normally require that $N^{1/2}({\Greekmath 011A}
_{n,N}-1)=o(1) $ rather than $N^{1/4}({\Greekmath 011A} _{n,N}-1)=o(1).$ However, when
for instance ${\Greekmath 0119} =1$ and ${\Greekmath 011B} _{v}^{2}=2{\Greekmath 011B} _{2}^{2}/(1-{\Greekmath 011A} ^{2}),$
then the results of theorem 3 hold under any $\{{\Greekmath 011A} _{n,N}\}$ such that $
N^{1/4}({\Greekmath 011A} _{n,N}-1)=o(1).$ In any case, when both $N^{1/4}({\Greekmath 011A}
_{n,N}-1)=o(1)$ and $N^{1/2}({\Greekmath 010E} _{n,N}-{\Greekmath 010E} _{\ast })=o(1)$ for some $
{\Greekmath 011B} _{2}^{2},$ the RE-\ and the FEQMLE of ${\Greekmath 011A} $ are $N^{1/4}-$
consistent.

Results in RCBR suggest that in principle under $({\Greekmath 011A} _{n,N}-1)=-{\Greekmath 0115}
N^{-b}$ different local asymptotic distributions and rates of convergence
are obtained for the QMLE of ${\Greekmath 011A} $ when $b=1/4,$ $1/6<b<1/4$, $b=1/6$ or $
0<b<1/6,$ respectively.

\section{Likelihood based tests}

Wald test statistics, some versions of (Quasi) LM test statistics, and
(Quasi) LR test statistics that are used for testing an hypothesis about $
{\Greekmath 011A} $ and are based on the reparametrized RE or FE\ likelihood do not
uniformly converge to their fixed parameter first-order limiting
distributions near the singularity point, cf. Bottai (2003). As a
consequence these tests do not have correct asymptotic size in a uniform
sense. (Q)LM test statistics that are standardised by (using a sandwich
formula involving) the \emph{expected} rather than the observed average
Hessian are exceptions to this rule. Bottai (2003) explains why this is the
case in the context of models with a single parameter. As we will show in
the proof of theorem 4, when testing a hypothesis that includes a
restriction on the parameter that has zero score at the singularity point,
these test statistics still converge uniformly to their fixed parameter
first-order limiting distribution, which is a central ${\Greekmath 011F} ^{2}$
-distribution, near the singularity point and these tests have correct
asymptotic size in a uniform sense. Crucially, the expected average Hessian
of $l_{n}({\Greekmath 0112} _{n})$, i.e., $\overline{H}(\mathcal{\breve{{\Greekmath 0112}}}
_{n})\equiv E_{\mathcal{\breve{{\Greekmath 0112}}}_{n}}(\frac{1}{N}\frac{\partial
^{2}l_{n}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}\partial {\Greekmath 0112} _{n}^{\prime }}
|_{\mathcal{\breve{{\Greekmath 0112}}}_{n}})$, is\linebreak negative definite and hence
nonsingular for any value of $\mathcal{\breve{{\Greekmath 0112}}}_{n}$ that differs
from the singu- larity point ${\Greekmath 0112} _{\ast }$. Note that the values of the
elements of $\overline{H}(\mathcal{\breve{{\Greekmath 0112}}}_{n})$ do not depend on
the true distribution of the data. Alternative testing approaches that are
based on a QLR test statistic but use a so-called type 2 robust critical
value, which is determined by a statistic that indicates the closeness to
the singularity point, or a least favourable critical value (cf. Andrews and
Cheng, 2013) either do not work due to non-monotonicity of the limiting
distributions of the QLR test statistic near the singularity point or are
conservative and have low power.

We will now introduce the QLM test statistic $QLM(\widetilde{\mathcal{{\Greekmath 0112}
}}_{n})$ for testing $H_{0}:$ $A\mathcal{{\Greekmath 0112} }_{0,n}=a$, which includes a
restriction on ${\Greekmath 011A} $, where $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}$ is a
restricted QML\ estimate of $\mathcal{{\Greekmath 0112} }_{0,n}$ such that $A\widetilde{
\mathcal{{\Greekmath 0112} }}_{n}=a,$ $A$ is a $J\times \dim (\mathcal{{\Greekmath 0112} })$
constant matrix of rank $J,$ and $a\ $is a constant vector. Define the
average information matrix $\overline{\mathcal{J}}\mathcal{(\mathcal{\breve{
{\Greekmath 0112}}}}_{n}\mathcal{)}=N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}\mathcal{J}_{i}\mathcal{(
\mathcal{\breve{{\Greekmath 0112}}}}_{n}\mathcal{)}$ with $\mathcal{J}_{i}\mathcal{(
\breve{{\Greekmath 0112}}}_{n}\mathcal{)}=\left( \frac{\partial l_{n,i}({\Greekmath 0112} _{n})}{
\partial {\Greekmath 0112} _{n}}|_{\mathcal{\breve{{\Greekmath 0112}}}_{n}}\right) \left( \frac{
\partial l_{n,i}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}^{\prime }}|_{\mathcal{
\breve{{\Greekmath 0112}}}_{n}}\right) ,$ where $l_{n,i}({\Greekmath 0112} _{n})$ is the
contribution to the reparametrized log-likelihood function, $l_{n}({\Greekmath 0112}
_{n})$, by individual $i$. If $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112}
_{\ast }$ for all ${\Greekmath 011B} ^{2}>0$, then $QLM(\widetilde{\mathcal{{\Greekmath 0112} }}
_{n})$ is given by, cf. White (1994, p. 173):\vspace{-0.1in}
\begin{eqnarray}
QLM(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}) &=&N^{-1}\times \frac{\partial l_{n}(
\widetilde{\mathcal{{\Greekmath 0112} }}_{n})}{\partial {\Greekmath 0112} _{n}^{\prime }}\overline{
H}^{-1}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})A^{\prime }\times  \label{qlm} \\
&&(A\overline{H}^{-1}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})\overline{\mathcal{J}
}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})\overline{H}^{-1}(\widetilde{\mathcal{
{\Greekmath 0112} }}_{n})A^{\prime })^{-1}A\overline{H}^{-1}(\widetilde{\mathcal{{\Greekmath 0112}
}}_{n})\frac{\partial l_{n}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})}{\partial
{\Greekmath 0112} _{n}}.  \notag
\end{eqnarray}
If $H_{0}$ is true and $\mathcal{{\Greekmath 0112} }_{0,n}\neq {\Greekmath 0112} _{\ast }$ (for
all ${\Greekmath 011B} ^{2}>0$), then $QLM(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})\overset{d}
{\rightarrow }{\Greekmath 011F} ^{2}(J).$ To test $H_{0}:$ ${\Greekmath 011A} =a$ when $\widetilde{
\mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112} _{\ast }$ (for all ${\Greekmath 011B} ^{2}>0$,) for
some known value of $a\in (-1,1]$, one can use $QLM(\widetilde{\mathcal{
{\Greekmath 0112} }}_{n})$ in (\ref{qlm}) with $A=(1$ $\mathbf{0}^{\prime })$ and $
\frac{\partial l_{n}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})}{\partial {\Greekmath 0112} _{n}
}=A^{\prime }\frac{\partial l_{n}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})}{
\partial {\Greekmath 011A} }$. Note also that, unlike the values of the\linebreak (Quasi)
Wald and Hausman test-statistics that use $\overline{H}^{-1}(\widehat{
\mathcal{{\Greekmath 0112} }}_{n})$ (cf. White, 1994, p. 173), the value of $QLM(
\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ remains the same when $\overline{H}^{-1}(
\mathcal{\breve{{\Greekmath 0112}}}_{n})$ for some $\mathcal{\breve{{\Greekmath 0112}}}_{n}$ is
replaced by $adj(\overline{H}(\mathcal{\breve{{\Greekmath 0112}}}_{n}))$. Furthermore,
if $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112} _{\ast }$, then $\overline{
\mathcal{J}}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ and p$\lim_{N\rightarrow
\infty }\overline{\mathcal{J}}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ are
positive definite.

If $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$ for some ${\Greekmath 011B}
^{2}>0,$ $rk(\overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}))=\dim (
\overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}))-1$ and hence $rk(adj(
\overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})))=1.$ Furthermore, $
\overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})_{i,j}=0$ iff $i=1$ and/or $
j=1,$ $adj(\overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}))_{i,j}\neq 0$ iff
$i=j=1,$ and $adj(\overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}))(1$ $
\mathbf{0}^{\prime })^{\prime }\propto (1$ $\mathbf{0}^{\prime })^{\prime }.$
Hence if $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$ (for some $
{\Greekmath 011B} ^{2}>0$), the QLM test statistic in (\ref{qlm}) for $H_{0}:$ ${\Greekmath 011A} =a$
would be equal to $(\frac{\partial l_{n}(\mathcal{{\Greekmath 0112} }_{n})}{\partial r}
|_{{\Greekmath 0112} _{\ast }})^{2}/(\mathop{\textstyle \sum }_{i=1}^{N}(\frac{\partial l_{n,i}(\mathcal{
{\Greekmath 0112} }_{n})}{\partial r}|_{{\Greekmath 0112} _{\ast }})^{2})$. However, this test
statistic cannot be used, because $\frac{\partial l_{n,i}(\mathcal{{\Greekmath 0112} }
_{n})}{\partial r}|_{{\Greekmath 0112} _{\ast }}=0$ a.s.\thinspace and p$
\lim_{N\rightarrow \infty }N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}(\frac{\partial l_{n,i}(
\mathcal{{\Greekmath 0112} }_{n})}{\partial r}|_{{\Greekmath 0112} _{\ast }})^{2}=0$. More
generally, $N^{-1/2}A\,adj(\overline{H}({\Greekmath 0112} _{\ast }))\frac{\partial
l_{n}(\mathcal{{\Greekmath 0112} }_{n})}{\partial \mathcal{{\Greekmath 0112} }_{n}}|_{{\Greekmath 0112}
_{\ast }}=0$ a.s. and p$\lim_{N\rightarrow \infty }[A\,adj(\overline{H}
({\Greekmath 0112} _{\ast }))\times $ $\overline{\mathcal{J}}({\Greekmath 0112} _{\ast })adj(
\overline{H}({\Greekmath 0112} _{\ast }))A^{\prime }]=\mathbf{0.}$ To test $
H_{0}:\nolinebreak $ $\nolinebreak A\mathcal{{\Greekmath 0112} }_{0,n}=a$ when $
\widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$ (for some ${\Greekmath 011B} ^{2}>0$
), one can use the following QLM test statistic, cf. Bottai (2003):\vspace{
-0.1in}
\begin{eqnarray}
QLM(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}) &=&N^{-1}\times \widetilde{S}^{\prime
}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})\widetilde{\mathcal{H}}^{-1}(\widetilde{
\mathcal{{\Greekmath 0112} }}_{n})A^{\prime }\times  \label{qlm1} \\
&&(A\widetilde{\mathcal{H}}^{-1}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})
\widetilde{\mathcal{J}}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})\widetilde{
\mathcal{H}}^{-1}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})A^{\prime })^{-1}A
\widetilde{\mathcal{H}}^{-1}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})\widetilde{S}(
\widetilde{\mathcal{{\Greekmath 0112} }}_{n}),\vspace{-0.12in}  \notag
\end{eqnarray}
with\pagebreak
\begin{eqnarray*}
\widetilde{S}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})
&=&\sum\nolimits_{i=1}^{N}S_{i},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }\widetilde{\mathcal{J}}(
\widetilde{\mathcal{{\Greekmath 0112} }}_{n})=N^{-1}\sum
\nolimits_{i=1}^{N}(S_{i}S_{i}^{\prime }),\medskip \\
S_{i} &=&(S_{i,1},S_{i,2}^{\prime })^{\prime },\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }S_{i,1}=\frac{1}{
2}\frac{\partial ^{2}l_{n,i}}{\partial r_{n}^{2}}|_{\widetilde{\mathcal{
{\Greekmath 0112} }}_{n}},\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }S_{i,2}=\frac{\partial l_{n,i}}{\partial d_{n}}
|_{\widetilde{\mathcal{{\Greekmath 0112} }}_{n}},\vspace*{-0.35in}
\end{eqnarray*}
$\vspace*{-0.25in}$and\vspace{-0.07in}
\begin{eqnarray*}
\widetilde{\mathcal{H}}_{1,1} &=&\frac{2}{4!}E_{\widetilde{\mathcal{{\Greekmath 0112} }}
_{n}}(\frac{\partial ^{4}l_{n}}{\partial r_{n}^{4}}|_{\widetilde{\mathcal{
{\Greekmath 0112} }}_{n}}),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }\widetilde{\mathcal{H}}_{1,2}^{\prime }=
\widetilde{\mathcal{H}}_{2,1}=\medskip \frac{1}{2!}E_{\widetilde{\mathcal{
{\Greekmath 0112} }}_{n}}(\frac{\partial ^{3}l_{n}}{\partial r_{n}^{2}\partial d_{n}}|_{
\widetilde{\mathcal{{\Greekmath 0112} }}_{n}}), \\
\widetilde{\mathcal{H}}_{2,2} &=&\frac{2}{2!}E_{\widetilde{\mathcal{{\Greekmath 0112} }}
_{n}}(\frac{\partial ^{2}l_{n}}{\partial d_{n}\partial d_{n}^{\prime }}|_{
\widetilde{\mathcal{{\Greekmath 0112} }}_{n}}),\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{\quad }\widetilde{\mathcal{H}}(
\widetilde{\mathcal{{\Greekmath 0112} }}_{n})=\left[
\begin{array}{cc}
\widetilde{\mathcal{H}}_{1,1} & \widetilde{\mathcal{H}}_{1,2} \\
\widetilde{\mathcal{H}}_{2,1} & \widetilde{\mathcal{H}}_{2,2}
\end{array}
\right] ,\vspace{-0.1in}
\end{eqnarray*}
where we have used $l_{n}$ and $l_{n,i}$ as short for $l_{n}({\Greekmath 0112} _{n})$
and $l_{n,i}({\Greekmath 0112} _{n})$. In contrast, the Wald and Hausman
test-statistics that use $\overline{H}^{-1}(\widehat{\mathcal{{\Greekmath 0112} }}_{n})$
when $\widehat{\mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112} _{\ast }$ cannot be defined
when $\widehat{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }.$ If $H_{0}$ is true
and $\mathcal{{\Greekmath 0112} }_{0,n}={\Greekmath 0112} _{\ast }$ for some ${\Greekmath 011B} ^{2}>0$, then
$QLM({\Greekmath 0112} _{\ast })\overset{d}{\rightarrow }{\Greekmath 011F} ^{2}(J).$ To test $H_{0}:$
${\Greekmath 011A} =a$ when $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$, one can
use $QLM(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ in (\ref{qlm1}) with $A=(1$ $
\mathbf{0}^{\prime })$ and $\widetilde{S}(\widetilde{\mathcal{{\Greekmath 0112} }}
_{n})=A^{\prime }(\sum\nolimits_{i=1}^{N}S_{i,1})$. It can easily be shown
that the test statistic $QLM(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ for testing
$H_{0}:$ ${\Greekmath 011A} =a$ given by (\ref{qlm}) and (\ref{qlm1}) is continuous at $
\widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$ (for any ${\Greekmath 011B} ^{2}>0$)
by using de l'H\^{o}pital's rule twice.\vspace{-0.06in}

\begin{theorem}
Under regularity conditions (A1)-(A7) given in the appendix, the Quasi LM
test for $H_{0}:$ $A\mathcal{{\Greekmath 0112} }_{0,n}=a,$ where $A_{1,.}=(1$ $\mathbf{0
}^{\prime }),$ that is based on (\ref{qlm}) if $\widetilde{\mathcal{{\Greekmath 0112} }}
_{n}\neq {\Greekmath 0112} _{\ast }$ for all ${\Greekmath 011B} ^{2}>0,$ and on (\ref{qlm1}) if $
\widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$ for some ${\Greekmath 011B} ^{2}>0,$
has correct asymptotic size in a uniform sense.\vspace{-0.04in}
\end{theorem}

Confidence Sets (CSs) that have correct uniform asymptotic size can be
obtained by "inverting" the QLM test, i.e., $QLM(\widetilde{\mathcal{{\Greekmath 0112} }
}_{n})$ given by (\ref{qlm}) and (\ref{qlm1}), or any of the other
aforementioned tests that have correct uniform asymptotic size. For
instance, a CS for ${\Greekmath 011A} $ of level $(1-{\Greekmath 010B} )\ast 100\%$ constructed in
this way is the set of points $a\in {\mathbb{R}}$ for which the test fails
to reject $H_{0}:{\Greekmath 011A} =a$ at significance level ${\Greekmath 010B} .$

When the data are i.i.d. and normal and ${\Greekmath 011A} $ is not local or equal to
one, the (Q)LM test that uses the expected Hessian has optimal local power
properties. Next we derive the power envelope of a centered version of $QLM(
\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ for testing $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /
\sqrt[4]{N}$, where ${\Greekmath 0114} $ is a constant, viz. $QLM^{c}(\widetilde{
\mathcal{{\Greekmath 0112} }}_{n}),$ when the data are i.i.d. and normal and ${\Greekmath 0112}
_{0}={\Greekmath 0112} _{\ast }.$ Let $\overline{\mathcal{J}}^{c}\mathcal{({\Greekmath 0112} }_{n}
\mathcal{)}=N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}\mathcal{J}_{i}^{c}\mathcal{({\Greekmath 0112} }_{n}
\mathcal{)}$ with $\mathcal{J}_{i}^{c}\mathcal{(\mathcal{\mathcal{\breve{
{\Greekmath 0112}}}}}_{n}\mathcal{)}=(\frac{\partial l_{n,i}({\Greekmath 0112} _{n})}{\partial
{\Greekmath 0112} _{n}}|_{\mathcal{\breve{{\Greekmath 0112}}}_{n}}-N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}(\frac{
\partial l_{n,i}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}}|_{\mathcal{\breve{{\Greekmath 0112}
}}_{n}}))(\frac{\partial l_{n,i}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}^{\prime }
}|_{\mathcal{\breve{{\Greekmath 0112}}}_{n}}-N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}(\frac{\partial
l_{n,i}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}^{\prime }}|_{\mathcal{\breve{
{\Greekmath 0112}}}_{n}}))$. If $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112} _{\ast },$
we have\vspace{-0.04in}
\begin{eqnarray*}
QLM^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}) &=&N^{-1}\times \frac{\partial
l_{n}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})}{\partial {\Greekmath 011A} }A\overline{H}^{-1}(
\widetilde{\mathcal{{\Greekmath 0112} }}_{n})A^{\prime }\times \\
&&(A\overline{H}^{-1}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})\overline{\mathcal{J}
}^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})\overline{H}^{-1}(\widetilde{
\mathcal{{\Greekmath 0112} }}_{n})A^{\prime })^{-1}A\overline{H}^{-1}(\widetilde{
\mathcal{{\Greekmath 0112} }}_{n})A^{\prime }\frac{\partial l_{n}(\widetilde{\mathcal{
{\Greekmath 0112} }}_{n})}{\partial {\Greekmath 011A} },\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }\vspace{-0.12in}\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }
\end{eqnarray*}
where $A=(1$ $\mathbf{0}^{\prime }).$ If $\widetilde{\mathcal{{\Greekmath 0112} }}
_{n}={\Greekmath 0112} _{\ast },$ $QLM^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ is
similar to (\ref{qlm1}) with $\widetilde{S}(\widetilde{\mathcal{{\Greekmath 0112} }}
_{n})=A^{\prime }(\sum\nolimits_{i=1}^{N}S_{i,1})$ and $\widetilde{\mathcal{J
}}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ replaced by $\widetilde{\mathcal{J}}
^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})=N^{-1}\sum\nolimits_{i=1}^{N}(S_{i}-
\overline{S})(S_{i}-\overline{S})^{\prime },$ where $\overline{S}
=N^{-1}\sum\nolimits_{i=1}^{N}S_{i}$.

If mean stationarity (\ref{mstat}) holds when $\left\vert {\Greekmath 011A} \right\vert
<1 $ and $Var(\Delta y_{i,t})/Var(y_{i,1})=o(N^{-1}),$ $i=1,2,...,N,$ then
only the full set of moment conditions based on differenced\linebreak data
matters asymptotically for identification of ${\Greekmath 011A} $ and there is no loss in
efficiency\linebreak when exploiting only the latter for estimation or
testing purposes; additional moment conditions involving levels of the data
are redundant (cf. Kruiniger, 2022, and Bun and Kleibergen, 2022). Note that
$Var(\Delta y_{i,t})=O(1)$ for any ${\Greekmath 011A} \in (-1,1].$ We will investigate
the power of the $QLM^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ tests under a
worst case scenario where ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast }$ and $
1/Var(y_{i,1})=o(N^{-1})$ $i=1,2,...,N$ and consider a sequence of null
hypotheses that is local-to-unity: $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$
(cf. Bun and Kleibergen, 2022). In this scenario it is sufficient to focus
on the FE version of $QLM^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}),$ viz. $
QLM_{FE}^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}).$

\begin{theorem}
When the data are i.i.d. and normal and ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast },$ (so
that TSH\ holds,) then the large sample distribution of $QLM_{FE}^{c}(
\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ for testing $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /
\sqrt[4]{N}$ is given by ${\Greekmath 011F} ^{2}($p$\lim_{N\rightarrow \infty
}c_{1}^{\prime }S^{-1}\overline{H}^{-1}A^{\prime }(A\overline{H}^{-1}
\overline{\mathcal{J}}^{c}\overline{H}^{-1}A^{\prime })^{-1}A\overline{H}
^{-1}S^{-1}c_{1}{\Greekmath 0114} ^{4},1)$ where $c_{1}$ and $S$ are defined in the
proof in the appendix.
\end{theorem}

The quartic root rate in the sequence of hypotheses is related to the fact
that ${\Greekmath 011A} $ is only second-order identified when ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast
}$. When we impose TSH on $l_{n}({\Greekmath 0112} _{n}),$ we obtain the following
result:

\begin{theorem}
When the data are i.i.d. and normal, ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast }$ and TSH
has been imposed on $l_{n}({\Greekmath 0112} _{n})$, then the large sample distribution
of $QLM_{FE}^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ for testing $H_{0}:$ $
{\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ is given by ${\Greekmath 011F} ^{2}(\frac{(2T-3)T(T-1)(T-2)
}{72}{\Greekmath 0114} ^{4},1).$
\end{theorem}

We will obtain the maximal attainable power (MAP) curve for testing $H_{0}:$
${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ by first considering the asymptotic
distribution of an GMM-Anderson-Rubin statistic which tests $H_{0}$ by using
all the moment conditions based on first-differences of the data whilst (the
true value of) ${\Greekmath 011A} =1.$ This GMM-AR statistic is given by
\begin{equation*}
GMM-AR({\Greekmath 011A} )=Nm({\Greekmath 011A} )^{\prime }[\widehat{V}_{mm}({\Greekmath 011A} )]^{-1}m({\Greekmath 011A} )
\vspace{-0.12in}
\end{equation*}
with$\vspace{-0.12in}$
\begin{equation}
m(r)=N^{-1}\sum\nolimits_{i=1}^{N}\left[ P\times vech\left( D_{r}(\Delta
y_{i}(\Delta y_{i})^{\prime })D_{r}^{\prime }\right) \right]  \label{momc}
\end{equation}
where $D_{r}$ is a $(T-1)\times (T-1)$ band matrix with $(D_{r})_{i,i}=1$
and $(D_{r})_{i+1,i}=-r$ for $i=1,2,...,T-2,($and $T-1)$ and $
(D_{r})_{i,j}=0 $ elsewhere; $P=(0$ $g$ $I_{p})$ is a $p\times \frac{1}{2}
(T-1)T$ matrix where $p=\frac{1}{2}T(T-1)-2$ and $g$ is a $p$-vector such
that $(1,-1,$ $g^{\prime })^{\prime }=vech(D_{r}D_{r}^{\prime })$; and $
\widehat{V}_{mm}({\Greekmath 011A} )$ is the Eicker-White estimator of the covariance
matrix of $m({\Greekmath 011A} ).$

\begin{theorem}
When the data are i.i.d. and normal, ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast }$ and TSH\
is exploited by the testing procedure, then the large sample distribution of
$GMM-AR({\Greekmath 011A} )$ for testing\linebreak $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}
$ is given by ${\Greekmath 011F} ^{2}(\frac{(2T-3)T(T-1)(T-2)}{72}{\Greekmath 0114} ^{4},\frac{1}{2}
T(T-1)-2).$
\end{theorem}

We obtain the MAP curve for testing $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$
by deriving the asymptotic distribution of an GMM-Anderson-Rubin statistic
which tests $H_{0}$ by using the (infeasible) weighted average of the moment
conditions in $E(m({\Greekmath 011A} ))=0_{p}$ (with $m(r)$ given in (\ref{momc})) that
leads to the largest value of the non-centrality parameter of this
distribution whilst (the true value of) ${\Greekmath 011A} =1$.

\begin{theorem}
When the data are i.i.d. and normal, ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast }$ and TSH\
is exploited by the testing procedure, then the maximal attainable power
curve for testing $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ is given by ${\Greekmath 011F}
^{2}(\frac{(2T-3)T(T-1)(T-2)}{72}{\Greekmath 0114} ^{4},1).$
\end{theorem}

\begin{corollary}
When the data are i.i.d. and normal, ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast }$ and TSH\
has been imposed on $l_{n}({\Greekmath 0112} _{n})$, then the large sample distribution
of $QLM_{FE}^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ for testing $H_{0}:$ $
{\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ attains the maximal attainable power curve
for testing $H_{0}.$
\end{corollary}

This result implies that, like the KLM-statistic, the centered LM-statistic $
QLM^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ is efficient both when ${\Greekmath 011A} $
is less than one and when ${\Greekmath 011A} $ is equal to one.

To test $H_{0}:$ ${\Greekmath 011A} =1$ one could also use a Wald test based on $\sqrt{N}(
\widehat{{\Greekmath 011A} }-1)^{2}$ where $\widehat{{\Greekmath 011A} }$ is the REQMLE or FEQMLE of $
{\Greekmath 011A} .$ Under $H_{0}\ $we have $\sqrt{N}(\widehat{{\Greekmath 011A} }-1)^{2}\overset{d}{
\rightarrow }\breve{Z}_{1}\mathbf{1}\{\breve{Z}_{1}>0\},$ cf. Theorem 1.
Recall that $\breve{Z}_{N}=(\breve{Z}_{1,N},\breve{Z}_{2,N}^{\prime
})^{\prime }=-\mathcal{H}^{-1}U_{2,N}\overset{d}{\rightarrow }\breve{Z}$
with $U_{2,N}=(\frac{1}{2!}N^{-1/2}\frac{\partial ^{2}l_{n}}{\partial
r_{n}^{2}}|_{{\Greekmath 0112} _{\ast }},N^{-1/2}\frac{\partial l_{n}}{\partial
d_{n}^{\prime }}|_{{\Greekmath 0112} _{\ast }})^{\prime }$ and $\breve{Z}=(\breve{Z}
_{1},\breve{Z}_{2}^{\prime })^{\prime }\sim N(0,\mathcal{H}^{-1}\mathcal{IH}
^{-1}).$ When the data are i.i.d. non-normal or i.h.d., one can bootstrap
the distribution of $U_{2,N}$. To do this, one can make use of the fact that
under $H_{0}$, ${\Greekmath 0122} _{i}=y_{i}-y_{i,-1}$ for $i=1,...,N.$\vspace{
-0.15in}

\section{The finite sample performance of the QLM\ tests\protect\vspace{
-0.1in}}

In this section we investigate through Monte Carlo simulations the empirical
size and power properties of QLM-tests for testing a simple hypothesis of
the type $H_{0}:$ ${\Greekmath 011A} =a$, namely $QLM(\mathcal{{\Greekmath 011A} }),$ that are based
on the RE and FE\ likelihood functions for various panel AR(1) models
without covariates. The data were generated using the panel AR(1) model
given in (\ref{mdl}). We study how the properties of these tests are
affected if we change (1) the value of ${\Greekmath 011A} ,$ (2) the distributions of the
initial conditions $v_{i,1}=y_{i,1}-{\Greekmath 0116} _{i}$, (3) the distributions of the
idiosyncratic errors (the ${\Greekmath 0122} _{i,t}$) and/or (4) the ratio of the
variances of the error components. We conducted the simulation experiments
for $(T,N)=(4,100),$ $(9,100),$ $(4,250)$ or $(9,250).$ The nominal size of
the tests was $0.05$ and for each scenario the number of replications was
2500.

We calculated the empirical size of the tests for ${\Greekmath 011A} =0.2,$ $0.5,$ $0.8,$
$0.9,$ $0.95,$ $0.98$ or $0.99$ and we calculated the empirical power of the
tests for $H_{0}:{\Greekmath 011A} =0.8$ when ${\Greekmath 011A} =0.5,$ $0.6,$ $0.7,$ $0.9,$ $0.95$ or
$0.99.$

In all simulation experiments the individual effects, the ${\Greekmath 0116} _{i}$, were
i.i.d. $N(0,{\Greekmath 011B} _{{\Greekmath 0116} }^{2})$ with ${\Greekmath 011B} _{{\Greekmath 0116} }^{2}=1$ or $25.$ The
errors, the ${\Greekmath 0122} _{i,t}$, were either i.i.d. $N(0,1)$ or i.i.d. $
({\Greekmath 011F} ^{2}(1)-1)/\sqrt{2}.$ Note that in both cases $Var({\Greekmath 0122}
_{i,t})=1$ for $i=1,...,N$ and $t=2,...,T$.

In order to assess how the assumptions with respect to $y_{i,1}-{\Greekmath 0116} _{i}$, $
i=1,...,N,$ affect the finite sample properties of the tests, we conducted
three different kinds of experiments: in one set, labeled NS-Normal, the
initial observations are non-stationary, i.e., $y_{i,1}-{\Greekmath 0116} _{i}=0$, $
i=1,...,N,$ and the ${\Greekmath 0122} _{i,t}\sim N(0,1)$, whereas in the other
two sets, labeled S-Normal and S-ChiSq., respectively, the initial
observations are drawn from stationary distributions, i.e., either $
(y_{i,1}-{\Greekmath 0116} _{i})\sim N(0,1/(1-{\Greekmath 011A} ^{2}))$ when the ${\Greekmath 0122}
_{i,t}\sim N(0,1)$ or $(y_{i,1}-{\Greekmath 0116} _{i})\sim ({\Greekmath 011F} ^{2}(1)-1)/\sqrt{
2(1-{\Greekmath 011A} ^{2})}$ when the ${\Greekmath 0122} _{i,t}\sim ({\Greekmath 011F} ^{2}(1)-1)/\sqrt{2}$
. Note that in design NS-Normal the data are still mean stationary, i.e., $
E(y_{i,1}-{\Greekmath 0116} _{i})=0$ and $E({\Greekmath 0116} _{i}(y_{i,1}-{\Greekmath 0116} _{i}))=0$, and that in
all designs $E(y_{i,t}-y_{i,t-1})=0$.

In the cases of the RE and FE models, $(1-{\Greekmath 011A} ){\Greekmath 0116} _{i}+{\Greekmath 0122} _{i}$
is decomposed as $(1-{\Greekmath 011A} ){\Greekmath 0119} y_{i,1}+(1-{\Greekmath 011A} )v_{i}+{\Greekmath 0122}
_{i}=(1-{\Greekmath 011A} ){\Greekmath 0119} y_{i,1}+u_{i}$ with ${\Greekmath 0119} =1$ for the FE case. In the
experiments we imposed homoskedasticity on the likelihood functions and
added the restrictions ${\Greekmath 011B} ^{2}>0$ and $\widetilde{{\Greekmath 011B} }_{v}^{2}\geq
0 $ or the restrictions ${\Greekmath 011B} ^{2}>0$ and $(T-1)\widetilde{{\Greekmath 011B} }
_{v}^{2}+{\Greekmath 011B} ^{2}>0$ in case the restriction $\widetilde{{\Greekmath 011B} }
_{v}^{2}\geq 0$ was binding to ensure that the estimates of $
E(u_{i}u_{i}^{\prime })$ were positive definite.

We allowed for time effects by subtracting cross-sectional averages from the
data.

Note that the QML estimators suffer from a weak moment conditions problem
when ${\Greekmath 011A} $ is close to one.

Tables 1-12 report the simulation results in terms of the relative rejection
frequenties of the tests. Tables 1-4, 9, 10 report results concerning the
empirical size of the tests whereas tables 5-8, 11, 12 report results
concerning the empirical power of the tests. The tables also differ with
respect to the the value of ${\Greekmath 011B} _{{\Greekmath 0116} }^{2}$: tables 1-8 correspond to $
{\Greekmath 011B} _{{\Greekmath 0116} }^{2}=1$ and report results for both RE and FE versions of the
QLM tests whereas tables 9-12 correspond to ${\Greekmath 011B} _{{\Greekmath 0116} }^{2}=25$ and
report results for the RE version of the QLM\ test only because changes in
the value of ${\Greekmath 011B} _{{\Greekmath 0116} }^{2}$ do not affect the FE affects version of
the QLM test by construction. If the tests have correct size (i.e., 0.05),
then the standard errors of the estimates of the empirical size are $\sqrt{
0.05\ast (1-0.05)}/\sqrt{2500}\approx 0.0044.$

Inspection of the results in tables 1-4, 9 and 10 leads to the following
conclusions regarding the empirical size of the QLM tests:

\begin{enumerate}
\item Using a 5\% significance level, we would not be able to reject the
hypothesis that the tests have correct size for the various scenarios with $
{\Greekmath 011B} _{{\Greekmath 0116} }^{2}=1.$ Only 8 out of 168 estimates of the empirical size lie
outside the acceptance region $(0.0412,0.0588).$ The most extreme estimates
of the empirical sizes are $0.0628$ and $\ 0.0380.$ If we restrict attention
to $N=250$ and ${\Greekmath 011A} =0.95,$ $0.98$ or $0.99,$ then 2 out of 36 estimates of
the empirical size lie outside the acceptance region $(0.0412,0.0588).$

\item Using a 5\% significance level, we would not be able to reject the
hypothesis that the tests have correct size for the various scenarios with $
{\Greekmath 011B} _{{\Greekmath 0116} }^{2}=25.$ Only 5 out of 84 estimates of the empirical size lie
outside the acceptance region $(0.0412,0.0588).$ The most extreme estimates
of the empirical sizes are $0.0664$ and $\ 0.0644.$ If we restrict attention
to $N=250$ and ${\Greekmath 011A} =0.95,$ $0.98$ or $0.99,$ then 1 out of 18 estimates of
the empirical size lies outside the acceptance region $(0.0412,0.0588).$
\end{enumerate}

Inspection of the results in tables 5-8, 11 and 12 leads to the following
conclusions regarding the empirical power of the QLM tests:

\begin{enumerate}
\item[3.] The power of the tests increases in $N$ and $T.$

\item[4.] The power of the RE version of the QLM\ test is higher than the
power of the FE version of the QLM test unless the initial conditions $
v_{i,1}=y_{i,1}-{\Greekmath 0116} _{i}$ are zero (NS) in which case the power is the same
for both versions of the QLM tests.

\item[5.] The power curves for testing $H_{0}:{\Greekmath 011A} =0.8$ are asymmetric
around ${\Greekmath 011A} =0.8$, i.e., the QLM\ tests have more power against an
alternative below ${\Greekmath 011A} =0.8$ than against an equidistant alternative above $
{\Greekmath 011A} =0.8.$\vspace{-0.16in}
\end{enumerate}

\section{Concluding remarks}

In this paper we proposed new ML based inference methods for panel AR models
with arbitrary initial conditions and heteroskedasticity and possibly
additional regressors that are robust to the strength of identification.
Specifically, we showed that (Quasi) LM tests and CSs that use the expected
Hessian rather than the observed Hessian of the RE or the FE\ log-likelihood
function have correct asymptotic size in a uniform sense. We also derived
the power envelope of a FE version of such an LM test for testing $H_{0}:$ $
{\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ when the average information matrix is
estimated by a centered OPG\ estimator and the model is only second-order
identified, and showed that it coincides with the maximal attainable power
curve for testing $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ in the worst case
setting. In a Monte Carlo study that included a variety of experiments we
found that these (Quasi) LM tests have correct empirical size and good
empirical power properties.

None of the existing ML based inference methods for dynamic panel data
models have correct uniform asymptotic size close to the point in the
parameter space at which such a model is only second-order identified and
therefore the methods proposed in this paper will result in more reliable
inference. The proposed methods can be adapted to generalizations of the
panel AR(1) model that was considered in this paper including dynamic panel
models with a factor structure and panel VAR models. Similarly, Quasi LM\
test that have correct uniform asymptotic size can also be developed for
other multi-parameter models that are only second-order identified at some
point in the parameter space, such as the examples mentioned in e.g.
Rotnitzky et. al. (2000) and Dovonon and Hall (2018).

An issue that requires additional study is the effect of the choice of the
estimator for the average information matrix on the power of the QLM tests.
\newpage