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Large sample properties of GMM estimators under second-order identification.
\title{Large sample properties of GMM estimators under second-order
identification.}
\author{Hugo Kruiniger\thanks{
Address: [email removed]; Department of Economics, 1 Mill Hill
Lane, Durham DH1 3LB, England.} \\
Durham University}
\date{This version: 5 April 2026\\
Previous version: 24 December 2022}
\maketitle
\vspace{6.2cm}
\bigskip
\bigskip
\bigskip
\noindent JEL\ classification: C12, C13, C23.\bigskip
\noindent Keywords: bias, Generalized Method of Moments (GMM), moment
conditions, optimal weight matrix, rank deficiency, rate of convergence,
second-order local identification, underidentification.
\setcounter{page}{0} \thispagestyle{empty}
\newpage
\baselineskip=20pt
\begin{center}
\textbf{Abstract}
\end{center}
\vspace{1cm}
Dovonon and Hall (Journal of Econometrics, 2018) proposed a limiting
distribution theory for GMM\ estimators for a $p$ - dimensional globally
identified parameter vector ${\Greekmath 011E} $ when local identification conditions
fail at first-order but hold at second-order. They assumed that the
first-order underidentification is due to the expected Jacobian having rank $
p-1$ at the true value ${\Greekmath 011E} _{0}$, i.e., having a rank deficiency of one.
After reparametrizing the model such that the last column of the Jacobian
vanishes, they showed that the GMM\ estimator of the vector comprising the
first $p-1$ parameters, $\widehat{{\Greekmath 011E} }_{1},$ converges at rate $T^{-1/2}$
and the GMM estimator of the remaining parameter, $\widehat{{\Greekmath 011E} }_{p},$
converges at rate $T^{-1/4}$. They also provided a limiting distribution of $
T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ subject to a (non-transparent)
condition which they claimed to be not restrictive in general. However, as
we show in this paper, their condition is in fact only satisfied when ${\Greekmath 011E} $
is overidentified and the limiting distribution of $T^{1/4}(\widehat{{\Greekmath 011E} }
_{p}-{\Greekmath 011E} _{0,p}),$ which is non-standard, depends on whether ${\Greekmath 011E} $ is
exactly identified or overidentified. In particular, the limiting
distributions of the sign of $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ for
the cases of exact and overidentification, respectively, are different and
are obtained by using expansions of the GMM objective function of different
orders. Unsurprisingly, we find that the limiting distribution theories of
Dovonon and Hall (2018) for Indirect Inference (II) estimation under two
different scenarios with second-order identification where the target
function is a GMM\ estimator of the auxiliary parameter vector, are
incomplete for similar reasons. We discuss how our results for GMM
estimation can be used to complete both theories and in particular how they
can be used to obtain the limiting distributions of the II estimators in the
case of exact identification under either scenario. We also derive the
optimal, in the sense of limiting Mean Squared Error minimising, weight
matrices for $\widehat{{\Greekmath 011E} }_{1}$ and $\widehat{{\Greekmath 011E} }_{p},$ respectively.
\setcounter{page}{0} \thispagestyle{empty}\newpage
\section{Introduction}
Global identification is a necessary condition for consistency of an
estimator. In models that are linear in the parameters, global
identification is equivalent to first-order local identification. However,
in models that are nonlinear in the parameters, global identication of the
parameter vector may hold even when some of the parameters are not
first-order but higher order locally identified although in this case the
rate of convergence of the estimators of these parameters is slower than the
usual rate.
For the situation where one of the elements of ${\Greekmath 011E} ,$ say ${\Greekmath 011E} _{p},$ is
not first-order but only second-order locally identified, Sargan (1983),
Rotnitzky et al. (2000) and Kruiniger (2013) developed asymptotic theory for
IV estimators, MLEs and Quasi MLEs, respectively. A common finding is that
the estimator of the parameter that is only second-order locally identified
converges at a quartic root rate, i.e., at rate $T^{-1/4}$ and has a
non-normal asymptotic distribution, while the estimators of the parameters
that are first-order locally identified converge at the usual square root
rate, i.e., at rate $T^{-1/2}$ and have asymptotic distributions that are
mixtures of normal distributions. Furthermore, the limiting distribution of $
T^{1/2}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})^{2}$ is a mixture of a half-normal
distribution and $0$.
Dovonon and Renault (2009) give a formal definition of second-order local
identification in the context of GMM\ estimation. Dovonon and Hall (2018),
henceforth DH, present an asymptotic theory for GMM estimators under
second-order identification. The limiting distribution they give for the
estimator of the second-order locally identified parameter, i.e., ${\Greekmath 011E} _{p}$
, holds if a certain condition is satisfied. However, as we show in this
paper, their condition is only satisfied when ${\Greekmath 011E} $ is overidentified.
Furthermore, we show that the limiting distribution of $\widehat{{\Greekmath 011E} }_{p}$
depends on whether ${\Greekmath 011E} $ is exactly identified or overidentified. In
particular, the limiting distributions of the sign of $T^{1/4}(\widehat{{\Greekmath 011E}
}_{p}-{\Greekmath 011E} _{0,p})$ for the cases of exact and overidentification,
respectively, are different and are obtained by using expansions of the GMM
objective function of different orders. In fact, in the case of exact
identification the limiting distribution of the sign of $T^{1/4}(\widehat{
{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ may not even exist. The reason for these
differences is that in the case of exact identification some terms in the
expansion vanish. On the other hand, we find that the formula for the
limiting distribution of the GMM\ estimator of the vector with the other
elements of ${\Greekmath 011E} $, viz. $\widehat{{\Greekmath 011E} }_{1}$, is the same for both cases.
When the parameters are overidentified, the GMM estimator $\widehat{{\Greekmath 011E} }$
depends on a weight matrix. We show that the asymptotically optimal weight
matrix for $\widehat{{\Greekmath 011E} }_{1}$ is different from the asymptotically
optimal weight matrix for $\widehat{{\Greekmath 011E} }_{p}$, and that only the latter
weight matrix is the same as the one that is used when all the parameters
are first-order identified.
Kruiniger (2018a) derived the limiting distributions of two Modified MLEs
for a panel ARX(1) model with homoskedastic errors when the autoregressive
parameter equals one by viewing them as GMM\ estimators. In the unit root
case the parameter vector is only second-order locally identified by the
objective functions of the Modified MLEs due to the nonlinear terms in the
modified score vector. Alvarez and Arellano (2021) found that in the same
case (i.e., the case of a unit root and homoskedastic errors) the
autoregressive parameter of the panel AR(1) model is only second-order
locally identified by certain nonlinear moment conditions due to Ahn and
Schmidt (1995). DH showed that a set of moment conditions that are related
to a conditionally heteroskedastic factor model for asset returns has a rank
deficient Jacobian matrix and that the vector of parameters in these moment
conditions is second-order locally identified. Sargan (1983) discussed IV
and FIML estimation of dynamic simultaneous equation models that are linear
in the variables and nonlinear in the parameters and where the parameter
vector is only second-order locally identified. Finally, Rotnitzky et al.
(2000) give additional examples of models where the parameter vector is only
second-order locally identified.
The paper is organized as follows. Section 2 briefly reviews GMM estimation
under first-order local identification. Section 3 defines second-order
identification. Section 4 presents the limiting distribution theory for GMM
estimators under second-order local identification and discusses its
implications for Indirect Inference (II) estimation under two scenarios with
second-order local identification where the target function is a GMM\
estimator of the auxiliary parameter vector. Section 5 offers some
concluding remarks. The appendix contains the proofs.
\section{GMM under first-order identification$\protect\vspace{-0.09in}$}
In this section we briefly review the basic GMM framework based on
first-order asymptotics, paying special attention to the role of first-order
local identification. We first define the GMM estimator and then discuss
some first-order asymptotic theory for this estimator. To this end, we
introduce the following notation. The model involves the random vector $X$
which is assumed strictly stationary with distribution $P({\Greekmath 011E} _{0})$ which
is indexed by the parameter vector ${\Greekmath 011E} \in \Phi \subset \mathcal{
\mathbb{R}
}^{p}$. ${\Greekmath 011E} _{0}$\ is the true value of ${\Greekmath 011E} .$
GMM is a semi-parametric method in the sense that its implementation does
not require complete knowledge of $P($\textperiodcentered $)$ but only
population moment conditions implied by this distribution. In view of this,
we suppose that the model implies:
\begin{equation}
E[g(X,{\Greekmath 011E} _{0})]=0, \label{f1}
\end{equation}
where $g($\textperiodcentered $)$ is a $q\times 1$ vector of continuous
functions. The GMM estimator of ${\Greekmath 011E} _{0}$ based on\ (1) is defined as:
\begin{equation}
\widehat{{\Greekmath 011E} }=\underset{{\Greekmath 011E} \in \Phi }{argmin}Q_{T}({\Greekmath 011E} ),
\end{equation}
where
\begin{equation*}
Q_{T}({\Greekmath 011E} )=m_{T}^{\prime }({\Greekmath 011E} )W_{T}m_{T}({\Greekmath 011E} )\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ with }m_{T}({\Greekmath 011E}
)=T^{-1}\mathop{\textstyle \sum }\limits_{t=1}^{T}g(x_{t},{\Greekmath 011E} ),
\end{equation*}
$W_{T}$ is a positive definite matrix, and $\{x_{t}\}_{t=1}^{T}$ represents
the sample observations on $X$.
We will assume that $q\geq p$ and that $m_{T}({\Greekmath 011E} )$ satisfies
\begin{assumption}
(i) $m_{T}({\Greekmath 011E} )=O_{p}(1)$ for all ${\Greekmath 011E} \in \Phi $; (ii) $
T^{1/2}m_{T}({\Greekmath 011E} _{0})\overset{d}{\rightarrow }N(0,V_{m})$, where $V_{m}$
is a positive definite matrix of finite constants.
\end{assumption}
To consider the first-order asymptotic properties of GMM estimators, we
introduce a number of high level assumptions.
\begin{assumption}
(i) $W_{T}\overset{p}{\rightarrow }W$, a positive definite matrix of
constants; (ii) $\Phi $ is a compact set; (iii) $Q_{T}({\Greekmath 011E} )\overset{p}{
\rightarrow }Q({\Greekmath 011E} )=m({\Greekmath 011E} )^{\prime }Wm({\Greekmath 011E} )$ uniformly in ${\Greekmath 011E} $;
(iv) $Q({\Greekmath 011E} )$ is continuous on $\Phi $; (v) $Q({\Greekmath 011E} _{0})<Q({\Greekmath 011E} )$ $
\forall {\Greekmath 011E} \neq {\Greekmath 011E} _{0},$ ${\Greekmath 011E} \in \Phi $.
\end{assumption}
Assumption 2(v) serves as a global identification condition. These
conditions are sufficient to establish consistency, see, for example, Newey
and McFadden (1994).
\begin{proposition}
If Assumption 2 holds, then $\widehat{{\Greekmath 011E} }\overset{p}{\rightarrow }{\Greekmath 011E}
_{0}$.
\end{proposition}
Let $M_{T}(\tilde{{\Greekmath 011E}})=\partial m_{T}({\Greekmath 011E} )/\partial {\Greekmath 011E} ^{\prime
}|_{{\Greekmath 011E} =\tilde{{\Greekmath 011E}}}$ and let $N_{{\Greekmath 011E} ,{\Greekmath 010F} }$ be an ${\Greekmath 010F} $
-neighbourhood of ${\Greekmath 011E} _{0}$, that is,\linebreak $N_{{\Greekmath 011E} ,{\Greekmath 010F}
}=\{{\Greekmath 011E} :\parallel {\Greekmath 011E} -{\Greekmath 011E} _{0}\parallel <{\Greekmath 010F} \}$. We can derive
the first-order asymptotic distribution of $\widehat{{\Greekmath 011E} }$ after adding
the following assumption, cf. Newey and McFadden (1994).
\begin{assumption}
(i) ${\Greekmath 011E} _{0}$ is an interior point of $\Phi $; (ii) $m_{T}({\Greekmath 011E} )$ is
continuously differentiable on $N_{{\Greekmath 011E} ,{\Greekmath 010F} }$; (iii) $M_{T}({\Greekmath 011E} )
\overset{p}{\rightarrow }M({\Greekmath 011E} )$ uniformly on $N_{{\Greekmath 011E} ,{\Greekmath 010F} }$; (iv) $
M({\Greekmath 011E} )$ is continuous at ${\Greekmath 011E} _{0}$; (v) $M({\Greekmath 011E} _{0})$ has rank $p$.
\end{assumption}
Assumption 3(v) is the condition for first-order local identification. It is
sufficient but not necessary for local identification of ${\Greekmath 011E} _{0}$ on $
N_{{\Greekmath 011E} ,{\Greekmath 010F} }$, but it is necessary for the development of the
standard first-order asymptotic theory.
\begin{proposition}
If Assumptions 1--3 hold, then $T^{1/2}(\widehat{{\Greekmath 011E} }_{MD}-{\Greekmath 011E} _{0})
\overset{d}{\rightarrow }N(0,V_{{\Greekmath 011E} })$, where
$V_{{\Greekmath 011E} }=[M({\Greekmath 011E} _{0})^{\prime }WM({\Greekmath 011E} _{0})]^{-1}M({\Greekmath 011E} _{0})^{\prime
}WV_{m}WM({\Greekmath 011E} _{0})[M({\Greekmath 011E} _{0})^{\prime }WM({\Greekmath 011E} _{0})]^{-1}$.
\end{proposition}
Global identification is crucial for consistency; global and first-order
local identification are needed for the preceding asymptotic distribution
theory.
Given Assumption 2(i), the global identification condition for GMM can be
equivalently stated as $E[g(X,{\Greekmath 011E} )]=0$ has a unique solution at ${\Greekmath 011E}
={\Greekmath 011E} _{0}$. The first-order local identification condition can also be
stated as $E[\partial g(X,{\Greekmath 011E} )/\partial {\Greekmath 011E} ^{\prime }|_{{\Greekmath 011E} ={\Greekmath 011E}
_{0}}] $ has full column rank.
\section{Second-order local identification}
For our analysis of GMM, we adopt the definition of second-order local
identification originally introduced by Dovonon and Renault (2009). To
present this definition, we introduce the following notations. Let $m({\Greekmath 011E}
)=E[g(X,{\Greekmath 011E} )]$ and
\begin{equation*}
M_{k}^{(2)}({\Greekmath 011E} _{0})=E\left[ \frac{\partial ^{2}g_{k}(X,{\Greekmath 011E} )}{\partial
{\Greekmath 011E} \partial {\Greekmath 011E} ^{\prime }}|_{{\Greekmath 011E} ={\Greekmath 011E} _{0}}\right] ,\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ }
k=1,2,\ldots ,q,
\end{equation*}
where $g_{k}(X,{\Greekmath 011E} )$ is the $k$th element of $g(X,{\Greekmath 011E} )$ and $g($
\textperiodcentered $)$ is defined in (\ref{f1}). Second-order local
identification is defined as follows.
\begin{definition}
The moment condition $m({\Greekmath 011E} )=0$ locally identifies ${\Greekmath 011E} _{0}\in \Phi $ up
to the second order if:
(a) $m({\Greekmath 011E} _{0})=0.$
(b) For all $u$ in the range of $M({\Greekmath 011E} _{0})^{\prime }$ and all $v$ in the
nullspace of $M({\Greekmath 011E} _{0})$, we have:$\medskip $
$\left( M({\Greekmath 011E} _{0})u+\left( v^{\prime }M_{k}^{(2)}({\Greekmath 011E} _{0})v\right)
_{1\leq k\leq q}=0\right) \Rightarrow (u=v=0).$
\end{definition}
The latter condition is derived using a second-order expansion of $m({\Greekmath 011E} )$
around $m({\Greekmath 011E} _{0})$ and can be motivated as follows. For any non-zero $
{\Greekmath 011E} -{\Greekmath 011E} _{0}$ with ${\Greekmath 011E} \in N_{{\Greekmath 011E} ,{\Greekmath 010F} },$ we have ${\Greekmath 011E} -{\Greekmath 011E}
_{0}=c_{1}u+c_{2}v$ where $c_{1},$ $c_{2}$ are constants such that $
c_{1}\neq 0$ and/or $c_{2}\neq 0$. For those directions for which $c_{1}$ is
non-zero, the first-order term is non-zero and dominates, and for those
directions in which $c_{1}=0$, the second-order term is non-zero. Thus,
without requiring the expected Jacobian matrix $M({\Greekmath 011E} _{0})$ to have full
rank, conditions (a) and (b) in Definition 1 guarantee local identification
in the sense that there is no sequence of points $\{{\Greekmath 011E} _{n}\}$ different
from ${\Greekmath 011E} _{0}$ but converging to ${\Greekmath 011E} _{0}$ such that $m({\Greekmath 011E} _{n})=0$
for all $n$. The difference between first-order local identification and
second-order local identification (with $M({\Greekmath 011E} _{0})$ rank deficient) is
how sharply $m({\Greekmath 011E} )$ moves away from $0$ in the neighbourhood of ${\Greekmath 011E}
_{0} $.
\noindent \textbf{Example}\textit{.} Consider the following panel AR(1)
model with individual effects:
\begin{eqnarray}
y_{i,t} &=&{\Greekmath 011A} y_{i,t-1}+w_{i,t}, \label{mdl} \\
w_{i,t} &=&{\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}, \notag
\end{eqnarray}
for $i=1,...,N$ and $t=1,2,3.$ The number of individuals, $N,$ may be large.
Note that\ when ${\Greekmath 011A} =1,$ then ${\Greekmath 0111} _{i}=0$. Adding the term $
x_{i,t}^{\prime }\breve{{\Greekmath 010C}}$ to (\ref{mdl}) with $x_{i,t}$ exogenous and $
\breve{{\Greekmath 010C}}={\Greekmath 010C} (1-{\Greekmath 011A} )$ does not affect the essence of the analysis
below except that $p$ and $q$ increase by $\dim (\breve{{\Greekmath 010C}}$)\ and $
\breve{{\Greekmath 010C}}$ is first-order locally identified in all cases. We make the
following assumption.\smallskip
\textbf{Assumption A} $\left\{ {\Greekmath 0111}
_{i},y_{i,0},y_{i,1},y_{i,2},y_{i,3}\right\} _{i=1}^{N}$ \textit{is a random
sample from a joint distribution with finite fourth-order moments that
satisfies} $E({\Greekmath 0122} _{i,t}|{\Greekmath 0111} _{i},y_{i,0},\ldots ,y_{i,t-1})=0$
\textit{for} $t=1,2,3.$\medskip
Let the unconditional variances of the errors be denoted as $E({\Greekmath 0122}
_{i,t}^{2})={\Greekmath 011B} _{t}^{2}$ for $t=1,2,3.$ We are interested in GMM
estimation of ${\Greekmath 011A} $. Assumption A implies the following three linear
moment conditions for the model in (\ref{mdl}), cf. Arellano and Bond (1991):
\begin{equation}
m_{AB,s,t}({\Greekmath 011A} ):=E[y_{i,t-s}(\Delta y_{i,t}-{\Greekmath 011A} \Delta y_{i,t-1})]=0\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{
for }s=2,t\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{ and }t=2,3, \label{bega}
\end{equation}
where $\Delta y_{i,t}=y_{i,t}-y_{i,t-1}$. Assumption A also implies one
nonlinear moment condition\ for the model in (\ref{mdl}), cf. Ahn and
Schmidt (1995):
\begin{equation}
m_{AS,3}({\Greekmath 011A} ):=E[(y_{i,3}-{\Greekmath 011A} y_{i,2})(\Delta y_{i,2}-{\Greekmath 011A} \Delta
y_{i,1})]=0. \label{asm}
\end{equation}
\pagebreak
If ${\Greekmath 011A} \neq 1,$ then ${\Greekmath 011A} $ is both globally and first-order locally
identified by each of the above four moment conditions, while if ${\Greekmath 011A} =1,$
then the first three moment conditions do not help to identify ${\Greekmath 011A} $ at
all, because they are linear in ${\Greekmath 011A} $ and $dm_{AB,s,t}({\Greekmath 011A} )/d{\Greekmath 011A} =0$
for $s=2,t$ and $t=2,3.$ When ${\Greekmath 011A} =1$ and ${\Greekmath 011B} _{1}^{2}\neq {\Greekmath 011B}
_{2}^{2}$, then ${\Greekmath 011A} $ is still first-order locally identified by $
m_{AS,3}({\Greekmath 011A} )=0,$ because $dm_{AS,3}({\Greekmath 011A} )/d{\Greekmath 011A} =-{\Greekmath 011B} _{2}^{2}+{\Greekmath 011B}
_{1}^{2}\neq 0,$ but \textit{technically speaking} ${\Greekmath 011A} $ is no longer
globally identified because $m_{AS,3}({\Greekmath 011A} )=0$ now has two solutions, i.e.,
${\Greekmath 011A} =1$ and ${\Greekmath 011A} ={\Greekmath 011B} _{2}^{2}/{\Greekmath 011B} _{1}^{2}$. However, the fact
that $m_{AS,3}({\Greekmath 011A} )=0$ has multiple solutions only in this case (and not
when ${\Greekmath 011A} \neq 1$) means that \textit{in practice} ${\Greekmath 011A} $ is globally
identified in this case (because the occurance of two roots implies that $
{\Greekmath 011A} =1$) and that the GMM\ estimator that exploits $m_{AS,3}({\Greekmath 011A} )=0$ is
also consistent in this case, see Kruiniger (2013) for details. If ${\Greekmath 011A} =1$
and ${\Greekmath 011B} _{1}^{2}={\Greekmath 011B} _{2}^{2}$, then ${\Greekmath 011A} $ is second-order rather
than first-order locally identified by $m_{AS,3}({\Greekmath 011A} )=0,$ because $
dm_{AS,3}({\Greekmath 011A} )/d{\Greekmath 011A} =0$ and $d^{2}m_{AS,3}({\Greekmath 011A} )/d{\Greekmath 011A} ^{2}=2{\Greekmath 011B}
_{1}^{2}\neq 0$, and ${\Greekmath 011A} $ is also globally identified because the two
solutions of $m_{AS,3}({\Greekmath 011A} )=0$ are now both equal to $1$, that is, $1$ is
a double root of $m_{AS,3}({\Greekmath 011A} )=0$, cf. Alvarez and Arellano (2021).
Finally, as we have just seen, when ${\Greekmath 011A} =1,$ then ${\Greekmath 011A} $ is only
identified by $m_{AS,3}({\Greekmath 011A} )=0$ and not by $m_{AB,s,t}({\Greekmath 011A} )=0$ for $
s=2,t $ and $t=2,3,$ so in this case we really have $q=1$ rather than $q=4,$
that is, ${\Greekmath 011A} $ is exactly identified rather than overidentified because $
q=p=1.$
\section{The limiting distribution of the GMM estimator}
In this section we consider the moment condition model (1) and study the
asymptotic behaviour of the GMM estimator when ${\Greekmath 011E} _{0}$ is second-order
locally identified because the moment condition exhibits the properties in
Definition 1 but the standard local identification condition (Assumption
3(v)) fails.
\subsection{Main results}
We study the asymptotic behaviour of the GMM estimator by restricting
ourselves to the case of a rank deficiency of one, i.e., the rank of $M({\Greekmath 011E}
_{0})$ is equal to $p-1$, since this case is relatively easy to analyse
compared to the general case. W.l.o.g. we consider the case where the rank
deficiency of $M({\Greekmath 011E} _{0})$ is due to its last column being a null vector.
\footnote{
As mentioned by Sargan (1983), any model with an expected Jacobian that has
a rank deficiency of one can be brought into this configuration by
reparametrizing the model as needed.} To\ this end, we partition ${\Greekmath 011E} $
into (${\Greekmath 011E} _{1:p-1}^{\prime },{\Greekmath 011E} _{p})^{\prime }$ where ${\Greekmath 011E} _{1:p-1}$
is the vector consisting of the first $p-1$ elements of ${\Greekmath 011E} $ and ${\Greekmath 011E}
_{p}$ is the $p-$th element of ${\Greekmath 011E} $. For ease of presentation below, we
shorten the subscript and write ${\Greekmath 011E} _{1}$ for ${\Greekmath 011E} _{1:p-1}$. Thus ${\Greekmath 011E}
_{0}=({\Greekmath 011E} _{0,1}^{\prime },{\Greekmath 011E} _{0,p})^{\prime }$ where ${\Greekmath 011E} _{0,1}$ is a
$(p-1)\times 1$ vector containing the true value of ${\Greekmath 011E} _{1:p-1}$ and $
{\Greekmath 011E} _{0,p}$ is the true value of ${\Greekmath 011E} _{p}$. If $M({\Greekmath 011E} _{0})$ has rank $
p-1$ with $\frac{\partial m}{\partial {\Greekmath 011E} _{p}}({\Greekmath 011E} _{0})=0$, then
second-order local identification is equivalent to:
\begin{equation*}
Rank\left( \frac{\partial m}{\partial {\Greekmath 011E} _{1}^{\prime }}({\Greekmath 011E} _{0})\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{
\quad }\frac{\partial ^{2}m}{\partial {\Greekmath 011E} _{p}^{2}}({\Greekmath 011E} _{0})\right) =p
\end{equation*}
This is the setting studied by Sargan (1983) for the instrumental variables
estimator for a nonlinear in parameters model. We now present the regularity
conditions under which we derive the asymptotic distribution of the GMM
estimator. Define $D=\frac{\partial m}{\partial {\Greekmath 011E} _{1}^{\prime }}({\Greekmath 011E}
_{0})$ and $G=$\linebreak $\frac{\partial ^{2}m}{\partial {\Greekmath 011E} _{p}^{2}}
({\Greekmath 011E} _{0})$. The next assumption states formally the identification pattern
described above.
\begin{assumption}
(i) $m({\Greekmath 011E} )=0$ $\Leftrightarrow $ ${\Greekmath 011E} ={\Greekmath 011E} _{0}$; (ii) $\frac{\partial m
}{\partial {\Greekmath 011E} _{p}}({\Greekmath 011E} _{0})=0$; (iii) $Rank(D$ $G)=p$.
\end{assumption}
As mentioned by Sargan (1983), any model with an expected Jacobian that has
a rank deficiency of one can be brought into this configuration by rotating
the parameter space, see DH for details. We also require the following
regularity conditions to hold.
\begin{assumption}
(i) $m_{T}({\Greekmath 011E} )$ has partial derivatives up to order 3 if $q>p$ and up to
order 5 if $q=p$ in a neighbourhood $N_{{\Greekmath 011E} ,{\Greekmath 010F} }$ of ${\Greekmath 011E} _{0}$ and
the derivatives of $m_{T}({\Greekmath 011E} )$ converge in probability uniformly on $
N_{{\Greekmath 011E} ,{\Greekmath 010F} }$ to those of $m({\Greekmath 011E} )$.$\smallskip \smallskip
\smallskip \smallskip $
(ii) $\sqrt{T}\left(
\begin{array}{c}
m_{T}({\Greekmath 011E} _{0}) \\
\frac{\partial m_{T}}{\partial {\Greekmath 011E} _{p}}({\Greekmath 011E} _{0})
\end{array}
\right) \overset{d}{\rightarrow }\left(
\begin{array}{c}
\mathcal{Z}_{0} \\
\mathcal{Z}_{1}
\end{array}
\right) .\smallskip \smallskip \smallskip \smallskip $
(iii) $W_{T}-W=o_{p}(T^{-1/4}),$ $\frac{\partial m_{T}}{\partial {\Greekmath 011E}
_{1}^{\prime }}({\Greekmath 011E} _{0})-D=O_{p}(T^{-1/2}),$ $\frac{\partial ^{2}m_{T}}{
\partial {\Greekmath 011E} _{p}^{2}}({\Greekmath 011E} _{0})-G=O_{p}(T^{-1/2}),$ $\frac{\partial
^{2}m_{T}}{\partial {\Greekmath 011E} _{1}^{\prime }\partial {\Greekmath 011E} _{p}}({\Greekmath 011E}
_{0})-G_{1p}=o_{p}(1)$ and $\frac{\partial ^{3}m_{T}}{\partial {\Greekmath 011E} _{p}^{3}}
({\Greekmath 011E} _{0})-L=o_{p}(1),$ and if $q=p,$ $\frac{\partial ^{3}m_{T}}{\partial
{\Greekmath 011E} _{1}^{\prime }\partial {\Greekmath 011E} _{p}^{2}}({\Greekmath 011E} _{0})-G_{1pp}=o_{p}(1),$ $
\frac{\partial ^{4}m_{T}}{\partial {\Greekmath 011E} _{1}^{\prime }\partial {\Greekmath 011E} _{p}^{3}}
({\Greekmath 011E} _{0})-G_{1ppp}=o_{p}(1),$ $\frac{\partial ^{4}m_{T}}{\partial {\Greekmath 011E}
_{p}^{4}}({\Greekmath 011E} _{0})-F=o_{p}(1)$ and $\frac{\partial ^{2}m_{k,T}}{\partial
{\Greekmath 011E} _{1}\partial {\Greekmath 011E} _{1}^{\prime }}({\Greekmath 011E} _{0})-K_{k}=o_{p}(1)$ for $
k=1,2,\ldots ,q,$ with $G_{1p}=\frac{\partial ^{2}m}{\partial {\Greekmath 011E}
_{1}^{\prime }\partial {\Greekmath 011E} _{p}}({\Greekmath 011E} _{0}),$ $L=\frac{\partial ^{3}m}{
\partial {\Greekmath 011E} _{p}^{3}}({\Greekmath 011E} _{0}),$ $G_{1pp}=\frac{\partial ^{3}m}{\partial
{\Greekmath 011E} _{1}^{\prime }\partial {\Greekmath 011E} _{p}^{2}}({\Greekmath 011E} _{0}),$ $G_{1ppp}=\frac{
\partial ^{4}m}{\partial {\Greekmath 011E} _{1}^{\prime }\partial {\Greekmath 011E} _{p}^{3}}({\Greekmath 011E}
_{0}),$ $F=\frac{\partial ^{4}m}{\partial {\Greekmath 011E} _{p}^{4}}({\Greekmath 011E} _{0})$ and $
K_{k}=\frac{\partial ^{2}m_{k}}{\partial {\Greekmath 011E} _{1}\partial {\Greekmath 011E} _{1}^{\prime
}}({\Greekmath 011E} _{0})$ for $k=1,2,\ldots ,q,$ where $m_{k,T}({\Greekmath 011E} )$\ $(m_{k}({\Greekmath 011E}
)) $ is the kth element of $m_{T}({\Greekmath 011E} )$\ $($of $m({\Greekmath 011E} )).\vspace{-0.05in}$
\end{assumption}
These conditions are stronger than those imposed in the standard first-order
asymptotic analysis. The derivation of the asymptotic distribution of the
GMM estimator requires an expansion of $Q_{T}({\Greekmath 011E} )$ involving derivatives
of $m_{T}({\Greekmath 011E} )$ up to the third order when $q>p$ and up to the fifth order
when $q=p,$ and the uniform convergence guaranteed by Assumption 5(i) is
useful to control the remainder of our expansions. The orders of our
expansions of $Q_{T}({\Greekmath 011E} )$ for the cases $q>p$ and $q=p$ are different
because in the case of exact identification some terms in the expansion
vanish. Assumption 5(ii) states that $\sqrt{T}(m_{T}({\Greekmath 011E} _{0})^{\prime
},\partial m_{T}({\Greekmath 011E} _{0})^{\prime }/\partial {\Greekmath 011E} _{p})^{\prime }$
converges in distribution. Under Assumption 4 and additional mild conditions
on $g(X,{\Greekmath 011E} _{0})$ and $\frac{\partial g}{\partial {\Greekmath 011E} _{p}}(X,{\Greekmath 011E} _{0})$
, the central limit theorem guarantees that $(\mathcal{Z}_{0},\mathcal{Z}
_{1})^{\prime }\sim N(0,v)$, with $v=lim_{T\rightarrow \infty }Var[\sqrt{T}
(m_{T}({\Greekmath 011E} _{0})^{\prime },\partial m_{T}({\Greekmath 011E} _{0})^{\prime }/\partial
{\Greekmath 011E} _{p})^{\prime }]$. Assumption 5(iii) imposes the asymptotic order of
magnitude on the differences between some sample dependent quantities and
their probability limits. These orders of magnitude are enough to make these
differences negligible in the expansions. Assumption 5(iii) is not
particularly restrictive since most of the orders of magnitude imposed are
guaranteed by the central limit theorem.
To facilitate the presentation of our main result in this section, we
introduce the following definitions. Let $M_{d}$ be the matrix of the
orthogonal projection on the orthogonal complement of $W^{1/2}D:\vspace{
-0.14in}$
\begin{equation*}
M_{d}=I_{q}-W^{1/2}D(D^{\prime }WD)^{-1}D^{\prime }W^{1/2},\vspace{-0.14in}
\end{equation*}
where $I_{q}$ is the identity matrix of size $q$, let $P_{g}$ be the matrix
of the orthogonal projection\ on $M_{d}W^{1/2}G:\vspace{-0.06in}$
\begin{equation*}
P_{g}=M_{d}W^{1/2}G(G^{\prime }W^{1/2}M_{d}W^{1/2}G)^{-1}G^{\prime
}W^{1/2}M_{d},\vspace{-0.1in}
\end{equation*}
and let $M_{dg}$ be the matrix of the orthogonal projection on the
orthogonal complement of $(W^{1/2}D$\quad $W^{1/2}G):\vspace{-0.06in}$
\begin{equation*}
M_{dg}=M_{d}-P_{g}.\vspace{-0.1in}
\end{equation*}
Let$\vspace{-0.14in}$
\begin{eqnarray}
\mathcal{R}_{1} &=&(\mathcal{Z}_{0}^{\prime }W^{1/2}P_{g}W^{1/2}\mathcal{Z}
_{0}G^{\prime }-G^{\prime }W^{1/2}P_{g}W^{1/2}\mathcal{Z}_{0}\mathcal{Z}
_{0}^{\prime })W^{1/2}M_{d}W^{1/2}\times \vspace{-0.14in} \notag \\
&&(\frac{1}{3}L+G_{1p}HG)/{\Greekmath 011B} _{G}+\mathcal{Z}_{0}^{\prime
}W^{1/2}M_{dg}W^{1/2}(\mathcal{Z}_{1}+G_{1p}H\mathcal{Z}_{0}), \label{f7}
\end{eqnarray}
with ${\Greekmath 011B} _{G}=G^{\prime }W^{1/2}M_{d}W^{1/2}G$ and $H=-(D^{\prime
}WD)^{-1}D^{\prime }W$. In addition, let $V=$ $-2\mathcal{Z}\mathbf{1}(
\mathcal{Z}<0)/{\Greekmath 011B} _{G},$ where $\mathcal{Z}=G^{\prime
}W^{1/2}M_{d}W^{1/2}\mathcal{Z}_{0}$ and $\mathbf{1}($\textperiodcentered $)$
is the usual indicator function.
The following lemma, which is based on Theorem 1 in DH, and theorem give the
asymptotic properties of the GMM estimator $\widehat{{\Greekmath 011E} }$ under
Assumptions 2, 4 and 5.\pagebreak
\begin{lemma}[Dovonon and Hall (2018)]
Under Assumptions 2, 4 and 5, we have:
$(a)$ $\widehat{{\Greekmath 011E} }_{1}-{\Greekmath 011E} _{0,1}=O_{p}(T^{-1/2})$ and $\widehat{{\Greekmath 011E} }
_{p}-{\Greekmath 011E} _{0,p}=O_{p}(T^{-1/4});$
$(b)$ if in addition ${\Greekmath 011E} _{0}\in interior(\Phi )$, then$\smallskip
\smallskip $
$\qquad \left(
\begin{array}{c}
\sqrt{T}(\widehat{{\Greekmath 011E} }_{1}-{\Greekmath 011E} _{0,1}) \\
\sqrt{T}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})^{2}
\end{array}
\right) \overset{d}{\rightarrow }\left(
\begin{array}{c}
H\mathcal{Z}_{0}+HGV/2 \\
V
\end{array}
\right) .$
\end{lemma}
\begin{theorem}
Under Assumptions 2, 4 and 5, and if ${\Greekmath 011E} _{0}\in interior(\Phi )$, we have:
$(a)$ if in addition $q>p$, then$\smallskip \smallskip $
$\qquad T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})\overset{d}{\rightarrow }
(-1)^{B_{1}}\sqrt{V},\smallskip \smallskip $
with $B_{1}=\mathbf{1}(\mathcal{R}_{1}\geq 0);$
$(b)$ if in addition $q=p$ and $\Pr (\mathcal{R}_{2}=0)=0$, where $\mathcal{R
}_{2}$ is defined in the proof below equation $(\ref{h4})$, then$\smallskip
\smallskip $
$\qquad T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})\overset{d}{\rightarrow }
(-1)^{B_{2}}\sqrt{V},\smallskip \smallskip $
with $B_{2}=\mathbf{1}(\mathcal{R}_{2}\geq 0).$
\end{theorem}
Parts (a) and (b) of Lemma 1 are the same as parts (a) and (b) of Theorem 1
in DH. In the Appendix we provide an alternative, self-contained proof for
part (b) of Lemma 1. There we also provide a proof for our Theorem 1.$
\vspace{-0.14in}$
\subsection{Discussion$\protect\vspace{-0.06in}$}
Our theorem is different from part (c) of Theorem 1 of DH. The latter only
states that $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ converges in
distribution to the limiting distribution given in part (a) of our Theorem 1
under the condition that $\Pr (\mathcal{R}_{1}=0)=0.$ DH claim in their
Remark 1 that this condition \textquotedblleft is not expected to be
restrictive in general ...\textquotedblright , although they also add the
following caveat: \textquotedblleft However, when $q=p=1$ (one moment
restriction with one non first-order locally identified parameter), we can
see that $\mathcal{R}_{1}=0$.\textquotedblright\ However, as we show in the
proof of Lemma 2 in the Appendix, the condition $\Pr (\mathcal{R}_{1}=0)=0$
is actually only satisfied when ${\Greekmath 011E} $ is overidentified, i.e., when $q>p;$
when ${\Greekmath 011E} $ is exactly identified, i.e., when $q=p,$ then $\Pr (\mathcal{R}
_{1}=0|\mathcal{Z}<0)=1$ and hence $\Pr (\mathcal{R}_{1}=0)\neq 0.$ In other
words, the theory of DH only provides the limiting distribution of $T^{1/4}(
\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ for the case where ${\Greekmath 011E} $ is
overidentified.
In part (b) of Theorem 1 we provide the limiting distribution of $T^{1/4}(
\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ for the case where ${\Greekmath 011E} $ is exactly
identified and $\Pr (\mathcal{R}_{2}=0)=0$. The difference between the
limiting distributions of $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ given
in parts (a) and (b) is related to the difference between the distributions
of the Bernoulli r.v.'s $B_{1}$ and $B_{2}$ that determine the sign of $
T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ when $\mathcal{Z}<0.$ The
condition $\Pr (\mathcal{R}_{2}=0)=0$ holds if $F+3!G_{1pp}HG+\frac{4!}{2}
\tilde{{\Greekmath 0115}}_{3}\neq 0,$ where $\tilde{{\Greekmath 0115}}_{3}=({\Greekmath 0115} _{3,1}\ldots
{\Greekmath 0115} _{3,q})^{\prime }$ with ${\Greekmath 0115} _{3,k}=\frac{1}{4}G^{\prime
}H^{\prime }K_{k}HG$ for $k=1,2,\ldots ,q,$ cf. Lemma 2(b). When $q=p>1$ and
$F+3!G_{1pp}HG+\frac{4!}{2}\tilde{{\Greekmath 0115}}_{3}=0,$ then the condition $\Pr (
\mathcal{R}_{2}=0)=0$ may still hold, but $\mathcal{R}_{2}=0$ when $F=0,$ $
G_{1pp}=0$ and $K_{k}=0$ for $k=1,2,\ldots ,q$. When $q=p=1,$ then $\Pr (
\mathcal{R}_{2}=0)=0$ if only if $F\neq 0$. If $q=p$ and $\Pr (\mathcal{R}
_{2}=0)>0,$ then one may still be able to describe the limiting distribution
(of the sign) of $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p}),$ see later in
this subsection.
Part (a) of Lemma 1 gives the rates of convergence of $\widehat{{\Greekmath 011E} }_{1}$
and $\widehat{{\Greekmath 011E} }_{p}$. Because ${\Greekmath 011E} _{1}$ is first-order identified and
${\Greekmath 011E} _{p}$ is second-order identified, $\widehat{{\Greekmath 011E} }_{1}-{\Greekmath 011E} _{0,1}$
converges at the usual rate $T^{-1/2}$ while $\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E}
_{0,p} $ converges at the slower rate $T^{-1/4}.$
Part (b) of Lemma 1 gives the limiting distribution of $(\sqrt{T}(\widehat{
{\Greekmath 011E} }_{1}-{\Greekmath 011E} _{0,1}),$ $\sqrt{T}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})^{2})$.
This result is obtained by minimizing the sum of the leading $O_{p}(T^{-1})$
terms of an expansion of $m_{T}^{\prime }(\widehat{{\Greekmath 011E} })W_{T}m_{T}(
\widehat{{\Greekmath 011E} })$ around ${\Greekmath 011E} _{0}$ which are collected into $\underline{K}
_{T}({\Greekmath 011E} _{0})$ as given by (\ref{g4}) in the Appendix. As $\underline{K}
_{T}({\Greekmath 011E} _{0})$ is a quadratic function of $(\widehat{{\Greekmath 011E} }_{1}-{\Greekmath 011E}
_{0,1})$ and $(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})^{2}$ only, it only allows
one to obtain the limiting distribution of $T^{1/4}|\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E}
_{0,p}|.$ To obtain the limiting distribution of the sign of $T^{1/4}(
\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ one needs to employ a higher order
expansion of $m_{T}^{\prime }(\widehat{{\Greekmath 011E} })W_{T}m_{T}(\widehat{{\Greekmath 011E} })$
which includes an odd power of $(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p}).$
When $q>p,$ the limiting distribution of the sign of $T^{1/4}(\widehat{{\Greekmath 011E} }
_{p}-{\Greekmath 011E} _{0,p})$ can be obtained from the $O_{p}(T^{-5/4})$ terms in the
expansion of $m_{T}^{\prime }(\widehat{{\Greekmath 011E} })W_{T}m_{T}(\widehat{{\Greekmath 011E} }).$
In that case we have:$\vspace{-0.13in}$
\begin{equation*}
m_{T}^{\prime }(\widehat{{\Greekmath 011E} })W_{T}m_{T}(\widehat{{\Greekmath 011E} })=\underline{K}
_{T}({\Greekmath 011E} _{0,p})+(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})\times
2R_{1T}+o_{p}(T^{-5/4}),\vspace*{-0.11in}
\end{equation*}
where $\underline{K}_{T}({\Greekmath 011E} _{0,p})$ and $R_{1T}$ are quadratic functions
of $(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})^{2}.$ We show in the Appendix that $
TR_{1T}\overset{d}{\rightarrow }\mathcal{R}_{1}$ and that $\Pr (\mathcal{R}
_{1}=0)=0$ if $q>p.$ When $Z_{T}\equiv G^{\prime
}W^{1/2}M_{d}W^{1/2}m_{T}({\Greekmath 011E} _{0})<0,$ $q>p$ and $T$ is large, the minimum
of $m_{T}^{\prime }({\Greekmath 011E} )W_{T}m_{T}({\Greekmath 011E} )$ is reached when $({\Greekmath 011E}
_{p}-{\Greekmath 011E} _{0,p})$ has the opposite sign to $R_{1T}$. This suggests that
when $q>p,$ then the limiting distribution of the sign of $T^{1/4}(\widehat{
{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ can be described by $(-1)^{B_{1}}$ with $B_{1}=
\mathbf{1}(\mathcal{R}_{1}\geq 0).$
When $q=p,$ then $\Pr (\mathcal{R}_{1}=0|\mathcal{Z}<0\mathbf{)}=1$. In
fact, when $q=p,$ then $\mathcal{R}_{1}=0$, see the end of the proof of
Lemma 2(a1). However, if $q=p$ and $\Pr (\mathcal{R}_{2}=0)=0,$ then the
limiting distribution of the sign of $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E}
_{0,p})$ can be obtained from the $O_{p}(T^{-7/4})$ terms in the expansion
of $m_{T}^{\prime }(\widehat{{\Greekmath 011E} })W_{T}m_{T}(\widehat{{\Greekmath 011E} }).$ In that
case we have:$\vspace{-0.12in}$
\begin{equation*}
m_{T}^{\prime }(\widehat{{\Greekmath 011E} })W_{T}m_{T}(\widehat{{\Greekmath 011E} })=\underline{K}
_{T}({\Greekmath 011E} _{0,p})+(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})\times
2R_{1T}+O_{p}(T^{-6/4})+(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})\times
2R_{2T}+o_{p}(T^{-7/4}),\vspace*{-0.1in}
\end{equation*}
where the $O_{p}(T^{-6/4})$ term and $R_{2T}$ are cubic functions of $(
\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})^{2}.$ We show in the Appendix that $
T^{6/4}R_{2T}\overset{d}{\rightarrow }\mathcal{R}_{2}$ if $q=p.$ When $
Z_{T}<0$, $q=p,$ $\Pr (\mathcal{R}_{2}=0)=0$ and $T$ is large, then the
minimum of $m_{T}^{\prime }({\Greekmath 011E} )W_{T}m_{T}({\Greekmath 011E} )$ is reached when $({\Greekmath 011E}
_{p}-{\Greekmath 011E} _{0,p})$ has the opposite sign to $R_{2T}$. This suggests that if $
q=p$ and $\Pr (\mathcal{R}_{2}=0)=0$, then the limiting distribution of the
sign of $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ can be described by $
(-1)^{B_{2}}$ with $B_{2}=\mathbf{1}(\mathcal{R}_{2}\geq 0).$ If $q=p$ and $
\Pr (\mathcal{R}_{2}=0)>0,$ then we may still be able to characterize the
sign of $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ by using an expansion of $
m_{T}^{\prime }(\widehat{{\Greekmath 011E} })W_{T}m_{T}(\widehat{{\Greekmath 011E} })$ of an even
higher order. However, if, for instance, $q=p=1$ and $\frac{\partial ^{k}m}{
\partial {\Greekmath 011E} _{p}^{k}}({\Greekmath 011E} _{0})=0$ for $k\geq 4,$ then we cannot
characterize the sign of $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ and the
latter does not have a proper limiting distribution, whereas $\sqrt{T}(
\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})^{2}$ has one, which is given in Lemma 1(b).
Generally, the limiting distribution of\ $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E}
_{0,p})$ given in Theorem 1 is asymmetric around $0$ and $\widehat{{\Greekmath 011E} }
_{p} $ has an asymptotic bias, unless $\Pr (B_{1}=1|\mathcal{Z}_{0})=\frac{1
}{2}$ when $q>p$ or $\Pr (B_{2}=1|\mathcal{Z}_{0})=\frac{1}{2}$ when $q=p.$
Furthermore, $\widehat{{\Greekmath 011E} }_{1}$ has an asymptotic bias.\vspace{-0.12in}
\subsubsection{Optimal weight matrices for $\protect\widehat{\protect{\Greekmath 011E} }
_{1}$ and $\protect\widehat{\protect{\Greekmath 011E} }_{p}$}
When $q>p,$ the limiting Mean Squared Errors (MSEs) of $\widehat{{\Greekmath 011E} }_{1}$
and $\widehat{{\Greekmath 011E} }_{p}$ depend on the choice of the limiting weight matrix
$W$. Recall that $V$ depends on $M_{d}$ and that $M_{d}$ depends on $W$.
Hence $V$ depends on $W.$ Let $M_{d}(W)=M_{d},$ $V(W)=V,$ $\Psi _{1}(W)=E((
\mathcal{Z}_{0}+GV(W)/2)(\mathcal{Z}_{0}+GV(W)/2)^{\prime })$ and $\Psi
_{p}=E(\mathcal{Z}_{0}\mathcal{Z}_{0}^{\prime }).$ Furthermore, let $\hat{
\Psi}_{p}$ be a consistent estimate of $\Psi _{p}.$ At the end of the
appendix we show that the optimal weight matrix for $\widehat{{\Greekmath 011E} }_{p}$
that minimizes its limiting MSE is $W_{p,opt}=\Psi _{p}^{-1}$ and that the
limiting MSE of $T^{1/4}(\widehat{{\Greekmath 011E} }_{p,opt}-{\Greekmath 011E} _{0,p})$ is given by $
(G^{\prime }\Psi _{p}^{-1/2}M_{d}(\Psi _{p}^{-1})\Psi _{p}^{-1/2}G)^{-1/2}
\sqrt{2/{\Greekmath 0119} }.$ If ${\Greekmath 011E} _{1}$ and ${\Greekmath 011E} _{p}$ are estimated jointly, then
the weight matrix that minimizes the limiting MSE of $\widehat{{\Greekmath 011E} }_{1}$,
viz. $W_{opt},$ is the solution of $W_{opt}=(\Psi _{1}(W_{opt}))^{-1}$.
However, as $W_{opt}\neq c\Psi _{p}^{-1}$ for any $c\in
\mathbb{R}
_{+}$, $W_{opt}$ is not an optimal weight matrix for $\widehat{{\Greekmath 011E} }_{p}.$
Moreover, $W_{opt}=(\Psi _{1}(W_{opt}))^{-1}$ has no closed form solution.
Therefore, it is better to estimate ${\Greekmath 011E} _{1}$ and ${\Greekmath 011E} _{p}$ separately.
In that case $W_{1,opt}=(\Psi _{1}(\Psi _{p}^{-1}))^{-1}$ is an optimal
weight matrix for $\widehat{{\Greekmath 011E} }_{1}$. Let $\widehat{W}_{1,opt}$ denote a
consistent estimate of $W_{1,opt}=(\Psi _{1}(\Psi _{p}^{-1}))^{-1}$. Then we
obtain $\widehat{{\Greekmath 011E} }_{1,opt}$ by minimising $m_{T}^{\prime }({\Greekmath 011E} _{1};
\widehat{{\Greekmath 011E} }_{p,opt})\widehat{W}_{1,opt}m_{T}({\Greekmath 011E} _{1};\widehat{{\Greekmath 011E} }
_{p,opt}).$ $\widehat{W}_{1,opt}$ can be obtained by making use of the
equality $E((\mathcal{Z}_{0}+GV(W)/2)(\mathcal{Z}_{0}+GV(W)/2)^{\prime })=
\frac{1}{2}E((\mathcal{Z}_{0}-G\mathcal{Z}/{\Greekmath 011B} _{G})(\mathcal{Z}_{0}-G
\mathcal{Z}/{\Greekmath 011B} _{G})^{\prime })+\frac{1}{2}E(\mathcal{Z}_{0}\mathcal{Z}
_{0}^{\prime })$ with $W=\Psi _{p}^{-1}$ and using the inverse of a
consistent estimate of its right-hand side as $\widehat{W}_{1,opt}$.
Alternatively, $(\widehat{W}_{1,opt})^{-1}$ can be obtained by simulating
the second moment of $\mathcal{Z}_{0}+GV(\hat{\Psi}_{p}^{-1})/2,$\ see the
first paragraph of section 4.4 for details.\vspace{-0.12in}
\subsection{Examples with exact identification}
Kruiniger (2018a) derived the limiting distributions of two Modified MLEs
for the panel ARX(1) model with homoskedastic errors when the autoregressive
parameter equals one by viewing them as GMM\ estimators. In the unit root
case the autoregressive parameter is only second-order locally identified by
the objective functions of both Modified MLEs due to the nonlinear terms in
the modified score vector. Furthermore, the parameter vector is obviously
exactly identified and the condition $\Pr (\mathcal{R}_{2}=0)=0$ holds
because the condition $F+3!G_{1pp}HG+\frac{4!}{2}\tilde{{\Greekmath 0115}}_{3}\neq 0$
is satisfied. It is therefore unsurprising that the limiting distributions
obtained by Kruiniger (2018a) for both Modified MLEs of the autoregressive
parameter are in agreement with Theorem 1(b) above.
We now return to the example given in section 3. Recall that when ${\Greekmath 011A} =1$
and ${\Greekmath 011B} _{1}^{2}={\Greekmath 011B} _{2}^{2}$, then ${\Greekmath 011A} $ is second-order globally
identified by the four moment conditions mentioned in that example. However,
in this example $q=p=1$ and $F=0$ because there is only one nonlinear moment
condition that identifies ${\Greekmath 011A} $, namely $m_{AS,3}({\Greekmath 011A} )=0$, which is
quadratic in ${\Greekmath 011A} $. Thus in this case the limiting distribution of $
T^{1/4}(\widehat{{\Greekmath 011A} }-1)$ cannot be obtained from Theorem 1(b). This is
not due to a shortcoming of the theory; there are at least two explanations
for this result. Firstly, note that when ${\Greekmath 011A} =1,$ $m_{AS,3}(r)=E[(
{\Greekmath 0122} _{i,3}+(1-r)y_{i,2})(\Delta {\Greekmath 0122} _{i,2}+(1-r){\Greekmath 0122}
_{i,1})].$ Hence, when $N$ is large, the GMM estimator for ${\Greekmath 011A} $ is
approximately equal to the solution of $N^{-1/2}\mathop{\textstyle \sum }_{i=1}^{N}[(\widehat{
{\Greekmath 011A} }-1)^{2}y_{i,2}{\Greekmath 0122} _{i,1}-(\widehat{{\Greekmath 011A} }-1)({\Greekmath 0122}
_{i,1}{\Greekmath 0122} _{i,3}+y_{i,2}\Delta {\Greekmath 0122} _{i,2})+{\Greekmath 0122}
_{i,3}\Delta {\Greekmath 0122} _{i,2}]=0.$ However, the linear term $-(\widehat{
{\Greekmath 011A} }-1)N^{-1/2}\mathop{\textstyle \sum }_{i=1}^{N}({\Greekmath 0122} _{i,1}{\Greekmath 0122}
_{i,3}+y_{i,2}\Delta {\Greekmath 0122} _{i,2})=o_{p}(1)$ because $N^{1/4}(\widehat{
{\Greekmath 011A} }-1)=O_{p}(1)$ and $N^{-1/2}\mathop{\textstyle \sum }_{i=1}^{N}({\Greekmath 0122}
_{i,1}{\Greekmath 0122} _{i,3}+y_{i,2}\Delta {\Greekmath 0122} _{i,2})=O_{p}(1).$ Thus
when $N$ tends to infinity, the sample counterpart of $m_{AS,3}({\Greekmath 011A} )=0$
determines the distribution of $N^{1/2}(\widehat{{\Greekmath 011A} }-1)^{2},$ i.e., $
N^{1/2}(\widehat{{\Greekmath 011A} }-1)^{2}\overset{d}{\rightarrow }-\widetilde{\mathcal{Z
}}_{0}\mathbf{1}(\widetilde{\mathcal{Z}}_{0}<0)/{\Greekmath 011B} ^{2}$ with $
N^{-1/2}\mathop{\textstyle \sum }_{i=1}^{N}({\Greekmath 0122} _{i,3}\Delta {\Greekmath 0122} _{i,2})\overset{
d}{\rightarrow }\widetilde{\mathcal{Z}}_{0},$ but cannot determine the
distribution of the sign of $N^{1/4}(\widehat{{\Greekmath 011A} }-1)$. Secondly, if ${\Greekmath 011A}
=1$ and ${\Greekmath 011B} _{1}^{2}={\Greekmath 011B} _{2}^{2}$, then both roots of $m_{AS,3}(r)=0$
are equal to $1$, and hence \emph{both} roots of the sample counterpart of $
m_{AS,3}(r)=0$, viz. $\widehat{{\Greekmath 011A} }_{1}$ and $\widehat{{\Greekmath 011A} }_{2}$, are
consistent estimators of ${\Greekmath 011A} $. The limiting distributions of $N^{1/2}(
\widehat{{\Greekmath 011A} }_{1}-1)^{2}$ and $N^{1/2}(\widehat{{\Greekmath 011A} }_{2}-1)^{2}$ are the
same and given by Lemma 1(b). However, asymp-\linebreak totically $N^{1/4}(
\widehat{{\Greekmath 011A} }_{1}-1)$ and $N^{1/4}(\widehat{{\Greekmath 011A} }_{2}-1)$ have opposite
signs. Hence a theory that can determine the limiting distribution of the
sign of $N^{1/4}(\widehat{{\Greekmath 011A} }-1)$ cannot exist in this case.\vspace{
-0.12in}
\subsection{GMM-based inference under second-order identification}
The limiting distributions in Theorem 1 and part (b) of Lemma 1 are
non-standard but easy to simulate. Approximations to these limiting
distributions can be obtained by drawing randomly copies of $(\mathcal{Z}
_{0}^{\prime },$ $\mathcal{Z}_{1}^{\prime })^{\prime }$ from $N(0,\widehat{v}
)$, where $\widehat{v}$ is a consistent estimator of $v$, and using
consistent estimators of $W,$ $D,$ $G,$ $L,$ $G_{1p},$ $G_{1pp},$ $G_{1ppp},$
$F$ and $K_{k}$ for $k=1,2,\ldots ,q$ as required. One can use the quantiles
of the simulated distributions to construct confidence sets for the elements
of ${\Greekmath 011E} _{0}$.
Dovonon, Hall and Kleibergen (2020) studied and compared the local power
properties of various test-statistics for conducting inference in moment
conditions models that locally identify the parameters only to second order.
The tests considered include tests for $H_{0}:{\Greekmath 011E} _{0}=a,$ where $a$ is a
known vector, such as the conventional Wald and LM tests, the Generalized
Anderson-Rubin (GAR) test (Anderson and Rubin, 1949; Staiger and Stock,
1997; Stock and Wright, 2000), the KLM test (Kleibergen 2002; 2005) and the
GMM extension of Moreira's (2003) Conditional LR test, also known as the
GMM-M test (Kleibergen, 2005), and tests for $H_{0}:m({\Greekmath 011E} _{0})=0,$ such as
the identification-robust\linebreak J test of Kleibergen (2005) and the GAR
test. Under the null hypothesis the conventional LM and Wald test-statistics
have non-standard limiting distributions, although the LM test-statistic
converges to a ${\Greekmath 011F} ^{2}$ r.v. in a special case; the distribution of the
Wald test-statistic depends on $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$
only through $T^{1/2}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})^{2}$, which has a
limiting distribution that is a mixture of a half-normal distribution and $0$
. All the other test-statistics are robust to weak and second-order local
identification and have the same limiting distribution under the null
hypothesis as they would have under first-order local identification. Apart
from the Wald test, all the tests can also be used when the Jacobian is rank
deficient by more than one. Dovonon, Hall and Kleibergen (2020) found that
in a particular panel AR(1) model, the Wald test of the unit root hypothesis
has better power than the GAR, KLM, LM and GMM-M tests.
Kruiniger (2018b) discusses a Quasi LM\ test for $H_{0}:{\Greekmath 011E} _{0}=a,$ when $
{\Greekmath 011E} _{0,p}$ is possibly only second-order locally identified. Specifically,
Kruiniger's (2018b) Quasi LM\ test-statistic generalizes the LM
test-statistic $W_{n}^{(2)}({\Greekmath 0112} )$ in Bottai (2003), who studied the
asymptotic behaviour of several tests and confidence regions in identifiable
one-dimensional parametric models with a smooth likelihood function and
Fisher information equal to zero at some point in the parameter space, in
two ways, namely by allowing for several parameters in the model and by
relaxing Bottai's ML setup to a Quasi ML setup. Under $H_{0}$ both LM\
test-statistics have a ${\Greekmath 011F} ^{2}$-distribution, also when one of the
parameters that appears in the null hypothesis is only second-order locally
identified. In the latter case, the score that corresponds to that parameter
and appears in the LM\ test-statistic under first-order local identification
will be replaced by its first-derivative. Kruiniger (2018b) shows that his
Quasi LM\ test and the confidence region that is based on inverting his
test-statistic have correct asymptotic size in a uniform sense.
Finally, Lee and Liao (2018) pointed out that when (a part of) ${\Greekmath 011E} _{0}$
is only second-order locally identified by the original set of moment
conditions, then Jacobian-based moment conditions can be used to obtain GMM
estimators and overidentification-test-statistics with standard asymptotic
properties. They then noted that the asymptotic normal distributions of such
GMM estimators can be used to conduct standard inference on ${\Greekmath 011E} _{0}$.
However, their tests and confidence intervals are only valid when (a part
of) ${\Greekmath 011E} _{0}$ is only second-order locally identified by the original set
of moment conditions and hence they obviously do not have correct asymptotic
size in a uniform sense.$\vspace{-0.14in}$
\subsection{Monte Carlo results}
Using simulations, Kruiniger (2018a) and DH studied the finite sample
properties of specific GMM estimators under second-order identification in
the cases of exact and overidentification, respectively. Both papers found
that the GMM\ estimator of ${\Greekmath 011E} _{0,p}$\thinspace is biased. These findings
are related to the asymmetry of the limiting distributions of $T^{1/4}(
\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ in these cases, which are given in Theorem
1 above.\footnote{
Recall that in the case of exact identification Theorem 1 of DH does not
provide a limiting distribution of $T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$
because $\Pr (
\mathbb{R}
_{1}=0)\neq 0$ and hence its sign is not well characterized in the limit in
this case.} DH also studied the coverage rates of two kinds of confidence
intervals for ${\Greekmath 011E} _{0,p}$ in a model with $p=1$ that are based on a GMM\
estimator that exploits $q>p$ moment conditions and use analytic quantiles
of its limiting distribution given in Lemma 1(b) above and simulated
quantiles of its limiting distribution given in Theorem 1(1a) above,
respectively, when ${\Greekmath 011E} _{0,p}$ is locally identified at the second-order
and they concluded that these limiting distributions give a reasonable
approximation to the behaviour of the GMM\ estimator for ${\Greekmath 011E} _{0,p}$.$
\vspace{-0.14in}$
\subsection{Indirect Inference}
Dovonon and Hall (2018) also considered the limiting distribution of an
Indirect Inference (II) estimator under second-order local identification.
Specifically, DH focused on an II estimator for the parameter vector ${\Greekmath 0112}
_{0}\in \Omega \subset \mathcal{
\mathbb{R}
}^{p}$ which is defined by the following set-up: the auxiliary model
consists of a set of $q$ population moment conditions indexed by a vector of
auxiliary parameters $h\in \mathcal{H}\subset \mathcal{
\mathbb{R}
}^{l}$ and the target function for the II estimation is a GMM estimator of
the auxiliary parameter vector. Within this framework, there are two types
of identification conditions: one set involving the binding function, and
the other involving the auxiliary parameters. The standard first-order
asymptotic theory is premised on the assumption that the binding function
satisfies global and first-order local identification conditions and the
auxiliary parameters are globally and first-order locally identified within
the auxiliary model. DH presents the limiting distribution of the II
estimator under the following two scenarios: (i) the binding function
satisfies the global and first-order local identification conditions and the
auxiliary parameters are globally identified but only locally identified at
second order; (ii) the binding function satisfies the global identification
condition but only satisfies the local identification condition at second
order, and the auxiliary parameters are globally and first-order locally
identified.
Unsurprisingly, the limiting distribution theories of DH for II estimation
under these two different scenarios with second-order identification, i.e.,
their Theorems 2 and 3(b) are incomplete for similar reasons as their
limiting distribution theory for GMM\ estimation is: their limiting
distributions for scenario (i) and scenario (ii) are only valid in the case
of overidentification, that is, when $q>l$ and when $l>p,$ respectively, or
in terms of their conditions, when $\Pr (\mathcal{R}_{1}^{(a)}=0)=0$ and $
\Pr (\mathcal{R}_{1}^{(b)}(s)=0)=0$, respectively, where $\mathcal{R}
_{1}^{(a)}$ and $\mathcal{R}_{1}^{(b)}(s)$ are defined similarly as $
\mathcal{R}_{1}$, see DH. Our results and derivations for GMM estimation
under second-order local identification can be used to complete both
theories and in particular can be used to straightforwardly derive the
limiting distributions of the II estimators in the case of exact
identification under either scenario. The representations of these
distributions are obtained by replacing $\mathcal{R}_{1}^{(a)}$ and $
\mathcal{R}_{1}^{(b)}(s)$ (implicit) in Theorems 2 and 3(b) in DH by $
\mathcal{R}_{2}^{(a)}$ and $\mathcal{R}_{2}^{(b)}(s)$, respectively, which
are defined similarly as $\mathcal{R}_{2}$ above just like $\mathcal{R}
_{1}^{(a)}$ and $\mathcal{R}_{1}^{(b)}(s)$ are defined similarly as $
\mathcal{R}_{1}$.\vspace{-0.12in}
\section{Concluding remarks}
The limiting distribution theory of DH (2018) for GMM\ estimators under
second-order\ local identification depends on a non-transparent condition,
namely that $\Pr (\mathcal{R}_{1}=0)=0$ where $\mathcal{R}_{1}$ is defined
in (\ref{f7}). We have shown that this condition is only satisfied when $
{\Greekmath 011E} $ is overidentified and derived the limiting distribution of $T^{1/4}(
\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ for the case where ${\Greekmath 011E} $ is exactly
identified. This distribution is different from that of $T^{1/4}(\widehat{
{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ given in DH\ for the case where ${\Greekmath 011E} $ is
overidentified. In particular, the limiting distributions of the sign of $
T^{1/4}(\widehat{{\Greekmath 011E} }_{p}-{\Greekmath 011E} _{0,p})$ for the cases of exact and
overidentification, respectively, are different and are obtained by using
expansions of the GMM objective function of different orders. We have also
pointed out that the limiting distribution theories of DH for Indirect
Inference (II) estimation under two different scenarios with second-order
identification where the target function is a GMM\ estimator of the
auxiliary parameter vector, are incomplete for similar reasons and we have
discussed how they can be completed.
The asymptotic theory for GMM\ estimators that has been discussed in this
paper can be generalized in two directions: (i) one can consider cases where
the (expected) Jacobian matrix has a rank deficiency that is higher than
one, and/or (ii) local identification of an order that is higher than two.\
In the case of second-order identification where the Jacobian has a rank
deficiency of $rd$ (with $rd\in
\mathbb{N}
\backslash \{0,1\}$), one can reparametrize the model in such a way that the
last $rd$ columns of the Jacobian are zero, and we expect that the
asymptotic theory is similar to the theory for the case of a rank deficiency
of one apart from the fact that in the current case there are now $rd$ GMM\
estimators that converge at rate $T^{-1/4}$. In the case of local
identification of order $s$ (with $s>1$), we expect that the GMM\
estimator(s) of the higher-order identified parameter(s) converge(s) at rate
$T^{-1/(2s)}.$ Like Rotnitzky et al. (2000), one also needs to distinguish
between cases where $s$ is even and cases where $s$ is odd: when $s$ is
even, we expect that the limiting distribution(s) of the GMM\ estimator(s)
of the higher-order identified parameter(s) is/are a mixture of a spike at
the true value and a non-standard distribution, while when $s$ is odd, we
expect that their limiting distribution(s) is/are equal to the distribution
of the $s-th$ root of a normal random variable. Furthermore, when $s$ is
even, the GMM estimators of the remaining parameters converge at the usual
rate $T^{-1/2}$ and have a limiting distribution that is a mixture of two
normal distributions, whereas when $s$ is odd, they converge at the usual
rate $T^{-1/2}$ and have a normal limiting distribution.\vspace{-0.12in}
\pagebreak