# Identification robust inference for moments based analysis of linear dynamic
panel data models

Maurice J.G. Bun Economics and Research Division (De Nederlandsche Bank) and
Amsterdam School of Economics (University of Amsterdam). Email:
<EMAIL_ADDRESS>The research of the author has been funded by the NWO
Vernieuwingsimpuls research grant ‘Causal Inference with Panel Data’. Frank
Kleibergen Amsterdam School of Economics, University of Amsterdam,
Roetersstraat 11, 1018 WB Amsterdam, The Netherlands. Email:
<EMAIL_ADDRESS>
Acknowledgements: We thank Manuel Arellano, Richard Blundell, Steve Bond,
Peter Boswijk, Geert Dhaene, Frank Windmeijer and participants of seminars at
Bristol, CEMFI, CORE, the Cowles Summer Conference at Yale, the EC2 meeting in
Maastricht, Groningen, Leuven, Monash, the 19-th International conference on
panel data in London, Oxford, the Tinbergen Institute in Amsterdam, Toulouse
and UCL for helpful comments and discussion.

###### Abstract

We use identification robust tests to show that difference, level and non-
linear moment conditions, as proposed by Arellano and Bond (1991), Arellano
and Bover (1995), Blundell and Bond (1998) and Ahn and Schmidt (1995) for the
linear dynamic panel data model, do not separately identify the autoregressive
parameter when its true value is close to one and the variance of the initial
observations is large. We prove that combinations of these moment conditions,
however, do so when there are more than three time series observations. This
identification then solely results from a set of, so-called, robust moment
conditions. These robust moments are spanned by the combined difference, level
and non-linear moment conditions and only depend on differenced data. We show
that, when only the robust moments contain identifying information on the
autoregressive parameter, the discriminatory power of the Kleibergen (2005) LM
test using the combined moments is identical to the largest rejection
frequencies that can be obtained from solely using the robust moments. This
shows that the KLM test implicitly uses the robust moments when only they
contain information on the autoregressive parameter.

JEL codes: C12, C23, C26

##  1 Introduction

It is common to estimate the parameters of linear dynamic panel data models
using the Generalized Method of Moments (GMM, Hansen (1982)). The moment
conditions for the linear dynamic panel data model either analyze it in first
differences using lagged levels of the series as instruments, in levels using
lagged first differences as instruments or using a product of levels and first
differences. We refer to the first set of moment conditions as Dif(ference)
moment conditions, see Arellano and Bond (1991), the second set as Lev(el)
moment conditions, see Arellano and Bover (1995), Blundell and Bond (1998) and
the third set as N(on-)L(inear) moment conditions, see Ahn and Schmidt (1995).

The Dif, Lev and NL moment conditions can be used separately to identify the
parameters of dynamic panel data models. To exhaust all information, however,
two particular combinations of Dif, Lev and NL moment conditions have been
proposed. We refer to the combined Dif and Lev moment conditions as the
Sys(tem) moment conditions and the combination of the Dif and NL moment
conditions as the A(hn-)S(chmidt) moment conditions.111Note that in a
combination of all three sets of moments conditions, the NL moment conditions
are redundant. The Sys moment conditions exhaust all information on the
autoregressive parameter that is present under mean stationarity, see Arellano
and Bover (1995) and Blundell and Bond (1998). The AS moment conditions
exhaust all information whilst not assuming mean stationarity, see Ahn and
Schmidt (1995).

We analyze the identification of the autoregressive parameter by the various
sets of moment conditions for a range of true values including the case of
highly persistent panel data. All moment conditions involve first differences
of the series to remove individual specific effects. The first difference
operator removes information in the time series at the unit root value of the
autoregressive parameter. It is well known that the Dif moment conditions
therefore do not identify the autoregressive parameter when its true value is
(close to) one, since lagged levels are then weak predictors of first
differences. This has led to the development of the NL and Lev, and hence AS
and Sys, moment conditions which were originally considered to identify the
autoregressive parameter when the panel data are highly persistent.

To show the identification issues at specific values of the autoregressive
parameter, we use identification robust tests, i.e. the GMM-A(nderson-)R(ubin)
statistic of Anderson and Rubin (1949) and Stock and Wright (2000), and the
K(leibergen) L(agrange) M(ultiplier) statistic of Kleibergen (2005). At values
of the parameters where identification issues occur, the rejection frequency
of these tests provenly coincides with the significance level so the
identification issues are relatively easy to detect by inspecting the power
curves. Using power curves of the KLM test, we show that Dif, Lev and NL
moment conditions separately do not identify the autoregressive parameter for
persistent values of it when paired with a large variance of the initial
observations. The same holds for the Sys moment conditions with three times
series observations. The power curves further show that Sys and AS moment
conditions generally identify the autoregressive parameter when the number of
time series observations exceeds three.

We formally prove these identification results using an asymptotic sampling
scheme in which we jointly let the variance of the initial observations and
the number of cross section observations go to infinity. For a range of
relative convergences rates of the variance of the initial observations
compared to the cross section sample size, the Dif, Lev and NL sample moments
and their derivatives diverge. Both the population moment and the Jacobian
identification condition are then ill defined which implies that the
autoregressive parameter is not separately identified by the Dif, NL or Lev
moment conditions. These results confirm and extend earlier findings in Madsen
(2003), Bond $et$ $al.$ (2005), Hahn $et$ $al.$ (2007), Kruiniger (2009) and
Phillips (2018).

Using our asymptotic sampling scheme, we also prove that AS and Sys moment
conditions identify the autoregressive parameter irrespective of the variance
of the initial observation when the number of time series observations exceeds
three. When the variance of the initial observations is large, the
identification results from a set of, so-called, robust sample moments that
are a combination of the Dif, Lev and NL sample moments (other than AS and
Sys) and only depend on differenced data. These robust sample moments are
spanned by the Sys sample moments and also by the AS sample moments. They
identify the autoregressive parameter irrespective of the variance of the
initial observation and including the case of highly persistent data. They are
a subset of the moment conditions in Kruiniger (2002), which are derived under
the additional assumption of time series homoskedasticity.

Despite these positive identification results for the Sys and AS moments, the
large sample distributions of corresponding one step and two step GMM
estimators are known to be non-standard when the variance of the initial
observation is large and the autoregressive parameter is close to one. This
makes it hard to infer if and how standard GMM inference using the original AS
or Sys sample moments exploits the information contained in the robust sample
moments that they encompass. The non-standard limiting behavior results since
the identification of the autoregressive parameter is then of, so-called,
second order since the Jacobian of the robust sample moments is rank deficient
but the Hessian is not, see $e.g.$ Dovonon and Renault (2013), Dovonon and
Hall (2018) and Dovonon $et$ $al.$ (2020). It explains the large biases of the
one step and two step GMM estimators and the size distortions of their
corresponding t-statistics when the series are persistent, see $e.g.$ Madsen
(2003), Bond and Windmeijer (2005), Bond $et$ $al.$ (2005), Dhaene and
Jochmans (2016), Hahn $et$ $al.$ (2007), Kruiniger (2009) and Bun and
Windmeijer (2010). Because of the second order identification, GMM estimators
based on the robust sample moments also have non-standard asymptotic
distributions when the data are persistent, see Dovonon $et$ $al.$ (2020).

We therefore analyze how identification robust test statistics exploit the
identifying information in the robust sample moments. We prove that the
identification robust KLM test procedure based on either AS or Sys sample
moments exploits all the identifying information contained in the robust
sample moments. We do so by first determining the (infeasible) optimal
weighted average of the robust sample moments that maximizes the
discriminatory power of a GMM-AR test of the autoregressive parameter in
settings where only the robust sample moments contain identifying information.
Next we determine the discriminatory power of KLM tests, based on AS or Sys
moment conditions, under such settings and prove that it equals that of the
GMM-AR test using the optimal weighted average of the robust sample moments.
KLM tests using AS or Sys moment conditions thus resort to just using the
robust sample moments when only the latter contain information on the
autoregressive parameter. It is therefore not necessary to explicitly use the
robust sample moments, which provide identification under mild conditions,
since they are implicitly used in the KLM test based on either AS or Sys
sample moments.

The paper is organized as follows. Section 2 introduces the linear dynamic
panel data model and the different moment conditions we use to identify its
parameters. It also discusses identification robust statistics, specifically
the KLM test, that we use to illustrate the identification issues that occur
at persistent values of the autoregressive parameter. In Section 3, we use a
representation theorem, akin to the cointegration representation theorem, see
Engle and Granger (1987) and Johansen (1991), to pin down the identification
properties of the different moment conditions. This theorem also allows us to
obtain the robust sample moments. In Section 4, we define the GMM-AR test that
uses the (infeasible) optimal weighted average of the robust sample moments
and derive the large sample distribution of the KLM test using AS or Sys
moment conditions under settings where only the robust sample moments contain
information on the autoregressive parameter. The fifth (final) section
concludes. Proofs of theorems and definitions of sample moments are provided
in the Appendix. We use the following notation throughout the paper: vec($A)$
stands for the (column) vectorization of the $k\times n$ matrix $A,$
vec($A)=(a_{1}^{\prime}\ldots a_{n}^{\prime})^{\prime}$ for $A=(a_{1}\ldots
a_{n}),$ $P_{A}=A(A^{\prime}A)^{-1}A^{\prime}$ is a projection on the columns
of the full rank matrix $A$ and $M_{A}=I_{N}-P_{A}$ is a projection on the
space orthogonal to $A.$ Convergence in probability is denoted by
“$\underset{p}{\rightarrow}$”, convergence in distribution by
“$\underset{d}{\rightarrow}$” and “$\underset{a}{=}$” means asymptotically
equivalent.

##  2 Identification robust GMM inference for dynamic panel data models

In this section, we briefly describe the dynamic panel data model and the
different sets of moment conditions. Thereafter we discuss identification
robust GMM inference including the construction of confidence intervals.
Finally, we illustrate the identification issues that occur when using the
different moment conditions for dynamic panel data models, by computing power
curves based on the identification robust KLM statistic.

###  2.1 Model and moment conditions

We analyze the first-order autoregressive linear dynamic panel data model

$\begin{array}[c]{cll}y_{it}=&c_{i}+\theta
y_{it-1}+u_{it},&i=1,\ldots,N,\text{ }t=2,\ldots,T,\end{array}$ (1)

with $T$ the number of time periods and $N$ the number of cross section
observations. We assume that the initial observation $y_{i1}$ is observed and
that the vector of observations $(y_{i1},...,y_{iT})$ for individual $i$ is
independently distributed across the $N$ individuals. We will later on make
further assumptions on the initial observations to properly define the process
in (1). For expository purposes, we analyze the simple dynamic panel data
model in (1) which can be extended with additional lags of $y_{it}$ and
explanatory variables.222The extension to other explanatory variables would
depend on the nature of these. For some settings such an extension would be
trivial but for others not so. Estimation of the parameter $\theta$ by means
of least squares leads to an inconsistent estimator in samples with a finite
value of $T$ and large $N,$ see $e.g.$ Nickell (1981). We therefore estimate
it using GMM. We obtain the GMM moment conditions from the unconditional
moment assumptions:

$\begin{array}[c]{rll}E[u_{it}]=&0,&t=2,\ldots,T,\text{ }\\\
E[u_{it}u_{is}]=&0,&s\neq t;\text{ }s,\text{ }t=2,\ldots,T,\text{ }\\\
E[u_{it}c_{i}]=&0,&t=2,\ldots,T,\\\
E[u_{it}y_{i1}]=&0,&t=2,\ldots,T.\end{array}$ (2)

Under these assumptions, the moments of the $T(T-1)$ interactions of $\Delta
y_{it}$ and $y_{it}:$

$\begin{array}[c]{c}E[\Delta y_{it}y_{ij}],\qquad j=1,\ldots,T,\text{
}t=2,\ldots,T\end{array}$ (3)

can be used to construct functions which identify the parameter of interest
$\theta.$ We do not use products of $\Delta y_{it}$ to identify $\theta$ since
we would need further assumptions, $i.e.$ homoskedasticity or initial
condition assumptions, see $e.g.$ Han and Phillips (2010).

Two different sets of moment conditions, which are functions of the moments in
(3), are commonly used to identify $\theta:$

  1. 1.

Difference (Dif) moment conditions:

$\begin{array}[c]{c}E[y_{ij}(\Delta y_{it}-\theta\Delta y_{it-1})]=0,\qquad
j=1,\ldots,t-2;\text{ }t=3,\ldots,T,\end{array}$ (4)

as proposed by $e.g.$ Anderson and Hsiao (1981) and Arellano and Bond (1991).
The Dif moment conditions solely result from the conditions in (2).

  2. 2.

Level (Lev) moment conditions:

$\begin{array}[c]{c}E[\Delta y_{it-1}(y_{it}-\theta y_{it-1})]=0,\qquad
t=3,\ldots,T,\text{ }\end{array}$ (5)

as proposed by Arellano and Bover (1995), see also Blundell and Bond (1998).
Besides the conditions in (2), the Lev moment conditions use

$E\left[\Delta y_{it}c_{i}\right]=0,$ (6)

which implies that the original data in levels have constant correlation over
time with the individual-specific effects. The Lev moment conditions (5) hold
under the following conditions regarding the initial observations $y_{i1}$
$(i=1,...,N):$

$y_{i1}=\mu_{i}+u_{i1},$ (7) $\mu_{i}=c_{i}/(1-\theta),$ (8)
$\begin{array}[c]{rl}E[u_{i1}]=&0,\\\ E[u_{i1}c_{i}]=&0,\\\
E[u_{i1}u_{it}]=&0,\qquad t>1.\end{array}$ (9)

The specification of the initial observations in (7)-(9) is often referred to
as mean stationarity. In our analysis we maintain the assumption of mean
stationarity.

The Dif and Lev moments can be used separately or jointly to identify
$\theta.$ When we use the moment conditions in (4) and (5) jointly, we refer
to them as system (Sys) moment conditions,333We could extend the Lev moment
conditions to $\frac{1}{2}(T-1)(T-2)$ sample moments by including additional
interactions of $\Delta y_{it-j}$ and $y_{it}-\theta y_{it-1},$ for
$j=2,\ldots,t-2.$ It can be shown, however, that all conditions on top of
those in (5) can be constructed as linear combinations of the Dif conditions
in (4) and the Lev conditions in (5). see Arellano and Bover (1995) and
Blundell and Bond (1998). Another set of nonlinear (NL) moment conditions,
which just like the Dif moments only use the conditions in (2), results from
Ahn and Schmidt (1995):

$\begin{array}[c]{c}E[(y_{it}-\theta y_{it-1})(\Delta y_{it-1}-\theta\Delta
y_{it-2})]=0\qquad t=4,\ldots,T.\end{array}$ (10)

The NL moments can be used separately or jointly with the Dif moments to
identify $\theta.$ When we use the moment conditions in (4) and (10) jointly,
we refer to them as Ahn-Schmidt (AS) moment conditions.

Ahn and Schmidt (1995) show that their AS moment conditions exhaust the
information on $\theta$ in the moment conditions (2) and are therefore
complete. Mean stationarity adds one moment condition (6) to the moment
conditions in (2). Hence, the complete set of moment conditions under (2) and
(6) equals the AS moment conditions and (6). Upon rewriting we can show that
these combined moment conditions are identical to the Sys moment conditions so
they are complete under (2) and (6).

###  2.2 Identification robust GMM tests

In GMM, we consider a $k$-dimensional vector of moment conditions, see Hansen
(1982):

$E[f_{i}(\theta_{0})]=0,\qquad i=1,\ldots,N,$ (11)

where $f_{i}(\theta)$ is a $k$-dimensional (continuous and continuously
differentiable) function of the observed data for individual $i$ and the
unknown parameter vector $\theta$ whose functional expression is identical for
all individuals. There is a unique true value of the $p$-dimensional vector
$\theta$ where the moment conditions are satisfied, which we denote by
$\theta_{0},$ and $k$ is at least as large as $p.$ We only analyze the first-
order autoregressive panel data model so $p=1$ for our setting. The population
moments in (11) are estimated using the sample moments,

$\begin{array}[c]{c}f_{N}(\theta)=\frac{1}{N}\sum_{i=1}^{N}f_{i}(\theta).\end{array}$
(12)

The $k\times p$ dimensional matrix $q_{N}(\theta)$ contains the derivative of
$f_{N}(\theta)$ with respect to $\theta:$

$\begin{array}[c]{c}q_{N}(\theta)=\frac{\partial}{\partial\theta^{\prime}}f_{N}(\theta)=\frac{1}{N}\sum_{i=1}^{N}q_{i}(\theta),\end{array}$
(13)

with $q_{i}(\theta)=\frac{\partial}{\partial\theta^{\prime}}f_{i}(\theta).$
Specifications of the sample moment functions $f_{N}(\theta)$ and
$q_{N}(\theta)$ for the Dif, Lev, Sys, NL and AS moment conditions are
provided in the Appendix.

Statistical inference based on the two step GMM estimator is known to be of
poor quality in the case of weak identification, which leads to an
inconsistent estimator with non-standard behavior of its corresponding
t-statistic, see e.g. Phillips (1989), Staiger and Stock (1997) and Stock and
Wright (2000). The non-standard limiting behavior of one and two step GMM
estimators for dynamic panel data models in the case of weak identification
has been documented in e.g. Madsen (2003), Kruiniger (2009) and Phillips
(2018).

In this study we therefore use identification robust GMM statistics to
overcome the aforementioned problems. The main advantage of identification
robust statistics is that, unlike conventional two step GMM statistics, their
limiting distributions are unaffected by the identification strength. Define
$\theta^{\ast}$ as the hypothesized value under the null hypothesis. A
particularly simple to compute identification robust GMM statistic to test
H${}_{0}:\theta=\theta^{\ast}$ is the GMM extension of the Anderson-Rubin
statistic, see Anderson and Rubin (1949) and Stock and Wright (2000):

$\begin{array}[c]{c}GMM\text{-}AR(\theta^{\ast})=Nf_{N}(\theta^{\ast})^{\prime}\hat{V}_{ff}(\theta^{\ast})^{-1}f_{N}(\theta^{\ast}),\end{array}$
(14)

with $\hat{V}_{ff}(\theta)$ the Eicker-White covariance matrix estimator:

$\begin{array}[c]{cc}\hat{V}_{ff}(\theta)=&\frac{1}{N}\sum_{i=1}^{N}(f_{i}(\theta)-f_{N}(\theta))(f_{i}(\theta)-f_{N}(\theta))^{\prime}.\end{array}$
(15)

The GMM-AR statistic equals the continuous updating objective function (Hansen
et al., 1996) evaluated in $\theta^{\ast}.$ A possible drawback of the GMM-AR
statistic is its lower power in the case of overidentified models. The KLM
statistic of Kleibergen (2005) partly overcomes this. The KLM statistic is a
quadratic form of the score of the GMM-AR statistic with respect to $\theta$:

$\begin{array}[c]{cl}KLM(\theta^{\ast})=&Nf_{N}(\theta^{\ast})^{\prime}\hat{V}_{ff}(\theta^{\ast})^{-1}\hat{D}_{N}(\theta^{\ast})\left[\hat{D}_{N}(\theta^{\ast})^{\prime}\hat{V}_{ff}(\theta^{\ast})^{-1}\hat{D}_{N}(\theta^{\ast})\right]^{-1}\\\
&\hat{D}_{N}(\theta^{\ast})^{\prime}\hat{V}_{ff}(\theta^{\ast})^{-1}f_{N}(\theta^{\ast}),\end{array}$
(16)

with $\hat{D}_{N}(\theta)$ a $k\times p$ dimensional matrix,

$\begin{array}[c]{c}\text{vec}(\hat{D}_{N}(\theta))=\text{vec}(q_{N}(\theta))-\hat{V}_{qf}(\theta)\hat{V}_{ff}(\theta)^{-1}f_{N}(\theta),\end{array}$
(17)

and

$\begin{array}[c]{cc}\hat{V}_{qf}(\theta)=&\frac{1}{N}\sum_{i=1}^{N}\left(\text{vec}[q_{i}(\theta)-q_{N}(\theta)]\right)(f_{i}(\theta)-f_{N}(\theta))^{\prime}.\end{array}$
(18)

The limiting distributions of the identification robust GMM-AR and KLM
statistics apply under less restrictive assumptions than those of the
traditional test statistics based on two step GMM. The GMM-KLM and GMM-AR
statistics converge under H0 to $\chi^{2}(p)$ and $\chi^{2}(k)$ distributed
random variables even when the Jacobian, $J(\theta_{0})=E(q_{i}(\theta_{0})),$
does not have a full rank value, see Stock and Wright (2000), Kleibergen
(2005) and Newey and Windmeijer (2009). Other identification robust statistics
for GMM are proposed in Kleibergen (2005), Andrews (2016) and Andrews and
Mikusheva (2016) which all provide extensions of the conditional likelihood
ratio statistic of Moreira (2003) to GMM. The conditional likelihood ratio
statistic is optimal for the homoskedastic linear instrumental variables
regression model with one included endogenous variable, see Andrews $et$ $al.$
(2006). None of its extensions to GMM has, however, shown to be optimal for
our setting of the dynamic linear panel autoregression so we just use the
easier to implement GMM-AR and KLM statistics.444Andrews et al. (2006)
establish the optimality of the likelihood ratio test for the iid linear
instrumental variables regression model using the Neymann-Pearson lemma. We
cannot do so here since the identification of $\theta$ depends on other
nuisance parameters besides the Jacobian, like the initial observations, so it
is not obvious how optimality can be established.

The identification robust tests can be inverted to obtain corresponding
identification robust confidence sets. The $100\times(1-\alpha)\%$ confidence
set for $\theta$ (denoted by $\text{CS}_{\theta}\text{(}\alpha)$ below)
consists of all values of $\theta^{\ast}$ for which the respective
identification robust test does not reject using its $100\times\alpha\%$
asymptotic critical value:

$\begin{array}[c]{c}\text{CS}_{\theta}\text{(}\alpha)=\left\\{\theta^{\ast}:IRT\text{(}\theta^{\ast})\leq
CDF_{IRT}(\alpha)\right\\},\end{array}$ (19)

with $IRT(\theta^{\ast})$ the identification robust statistic evaluated at
$\theta^{\ast}$ and $CDF_{IRT}(\alpha)$ the $(1-\alpha)\times 100$-th
percentile of the limiting distribution of $IRT$($\theta_{0}).$

The identification robust tests are not quadratic functions of $\theta^{\ast}$
so they cannot directly be inverted to obtain the confidence set.555An
exception is the GMM-AR statistic in the homoskedastic linear instrumental
variables regression model, see Dufour and Taamouti (2003). The confidence
sets resulting from them do therefore not have the usual expression of an
estimator plus or minus a multiple of the standard error. Instead, we have to
specify a $p$-dimensional grid of values of $\theta^{\ast}$ and compute the
identification robust statistic for every value of $\theta^{\ast}$ on the grid
to determine if it is less than the appropriate critical value so
$\theta^{\ast}$ is part of the confidence set.

Specifically, the confidence set in (19) can have three distinct shapes:

  1. 1.

Bounded and convex: there is a closed compact set of values of $\theta^{\ast}$
for which the identification robust test statistic does not exceed the
critical value.

  2. 2.

Unbounded: this occurs either when there are no values of $\theta^{\ast}$ for
which the identification robust test statistic exceeds the critical value
(unbounded and convex), or when there are bounded sets of values of
$\theta^{\ast}$ for which the identification robust test statistic exceeds the
critical value (unbounded and disjoint).

  3. 3.

Empty: the identification robust test statistic exceeds the critical value for
all values of $\theta^{\ast}.$

Bounded and convex confidence sets occur when the parameters of interest are
well identified. Unbounded confidence sets are indicative of weak
identification so if we then test H${}_{0}:\theta=\theta^{\ast}$ at a very
large, possibly infinite, value of $\theta^{\ast}$ using an identification
robust test at, say, the 5% significance level, it does not necessarily
reject. For such instances, we thus often do not reject the hypothesis of an
infinite value of $\theta$ so we obtain an unbounded 95% confidence set. In
Dufour (1997, Theorems 3.3 and 3.6), it is shown that any size correct
procedure used to test parameters which can be non-identified must have a
positive probability of producing an unbounded 95% confidence set. Conversely,
also any test procedure, like, for example, the Wald $t$-test, which can not
generate an unbounded 95% confidence set, can not be a size correct test
procedure when the tested parameter can be non-identified. Empty confidence
sets occur when the model is misspecified so there is no value of $\theta$ for
which the moment condition holds. Since the GMM-AR statistic tests whether all
moment conditions hold, it also tests misspecification. It can therefore
result in empty confidence sets but the KLM test cannot since it is equal to
zero at the continuous updating estimator of Hansen $et$ $al.$ (1996), which
is the minimizer of the GMM-AR statistic.

The identification robust statistics conduct tests on the full parameter
vector $\theta.$ Valid $(1-\alpha)\times 100\%$ confidence sets for the
individual elements of $\theta$ then result by projecting the joint
$p$-dimensional $(1-\alpha)\times 100\%$ confidence set for $\theta$ on the
$p$ different axes. These projection based confidence sets are size correct so
they contain the true value of $\theta$ with a probability which is at least
$(1-\alpha)\times 100\%$ irrespective of the strength of identification.
Projection based confidence sets can face computational issues when $p$ is
rather large given the large number of points on the $p$-dimensional grid for
which the statistic then has to be computed.

Confidence sets for the individual elements of $\theta$ can also be obtained
by plugging in an estimator for the remaining elements of $\theta$ after which
the (conditional) limiting distribution can be sharpened using the usual
degrees of freedom correction of the $\chi^{2}$ limiting distributions. The
resulting confidence sets only have correct coverage when these remaining
parameters are well identified, see Kleibergen (2005). Just in some isolated
cases, such, as for example, when using the GMM-AR statistic in the
homoskedastic linear instrumental variables regression model or in the linear
factor model for determining risk premia in finance, can we prove that these
confidence sets are valid without requiring the partialled out parameters to
be well identified, see Guggenberger $et$ $al.$ (2012, 2019), Kleibergen and
Zhan (2020), Kleibergen (2020) and Kleibergen $et$ $al.$ (2020).

###  2.3 Using identification robust tests to highlight identification issues

Identification robust GMM tests are size correct irrespective of the
identification strength. Therefore, their rejection frequencies can be used in
a straightforward manner to illustrate the identification issues at particular
values of the autoregressive parameter in the dynamic panel data model. The
conventional $t$-test based on the two step GMM estimator is not suitable for
this purpose as it is size distorted in the case of weak identification and,
hence, rejection frequencies would not equal the significance level.

To illustrate the identification issues for the different moment conditions,
we compute the rejection frequencies of 5% significance KLM tests of
H${}_{0}:\theta=0.5$ for a range of (true data generating) values
$\theta_{0}.$ We do so by simulating data from the panel autoregressive model
in (1) with three or four time series observations, so $T=3$ or $4,$ and two
hundred and fifty individuals, so $N=250.$ The individual specific effects
$c_{i}$ and idiosyncratic errors $u_{it}$ are independently generated from
$N(0,\sigma_{c}^{2})$ and $N(0,1)$ distributions respectively. We vary the
value of $\sigma_{c}^{2}$ to show the sensitivity of the identification of
$\theta$ using the panel moment conditions to the variance of the initial
observations. We assume mean stationarity so (7)-(9) hold.

We consider four KLM tests based on Dif, Lev, Sys and AS moment conditions,
which have been calculated according to equation (16) using
$\theta^{\ast}=0.5$. The figures in Panels 1 and 2 show the rejection
frequencies of KLM tests of H${}_{0}:\theta=0.5$ with 5% significance for four
values of $\sigma_{c}^{2}$ and a range of true values $\theta_{0}$. Panel 1
does so for three times series observations while Panel 2 covers four time
series observations. The simulation experiment is designed such that the
variance of the initial observations becomes very large when $\theta_{0}$ gets
close to one and $\sigma_{c}^{2}$ exceeds zero.

$\begin{array}[c]{c}\text{Panel 1. Rejection frequencies of KLM test of
H}_{0}:\theta=0.5\text{ with 5\% significance }\\\ \text{using different
moment conditions for }T=3,\text{ }N=250\text{ and }\sigma_{c}^{2}=0\text{
(dashed), }\\\ \text{0.5 (solid), one (dashed-dot) and two (dotted)}\\\
\begin{array}[c]{ccc}\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/rejt3ab.eps}}&&\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/rejt3bb.eps}}\\\
\text{Figure 1.1: Dif moment conditions}&&\text{Figure 1.2: Lev moment
conditions}\end{array}\\\
\begin{array}[c]{c}\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/rejt3sys.eps}}\\\
\text{Figure 1.3: Sys moment conditions}\end{array}\end{array}$

Figures 1.1 and 2.1 show that the rejection frequencies of the KLM test with
Dif moment conditions for $\theta_{0}$ close to one converges to the
significance level of 5%. It is well known that the Jacobian of the Dif moment
conditions is zero when $\theta_{0}$ equals one so they then do not identify
$\theta.$ The KLM test is identification robust which explains why the
rejection frequency equals the significance level both at the hypothesized
value of $\theta^{\ast}=0.5$ and when $\theta_{0}$ is close to one for all
values of $\sigma_{c}^{2}$. The latter results since the Dif moment conditions
do then not identify $\theta$, hence the KLM test has no discriminating power
so the power of the KLM test equals the significance level.

$\begin{array}[c]{c}\text{Panel 2. Rejection frequencies of KLM test of
H}_{0}:\theta=0.5\text{ with 5\% significance }\\\ \text{using different
moment conditions for }T=4,\text{ }N=250\text{ and }\sigma_{c}^{2}=0\text{
(dashed), }\\\ \text{0.5 (solid), one (dashed-dot) and two (dotted)}\\\
\begin{array}[c]{ccc}\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/rejt4ab.eps}}&&\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/rejt4bb.eps}}\\\
\text{Figure 2.1: Dif moment conditions}&&\text{Figure 2.2: Lev moment
conditions}\end{array}\\\
\begin{array}[c]{ccc}\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/rejt4sys.eps}}&&\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/rejt4as.eps}}\\\
\text{Figure 2.3: Sys moment conditions}&&\text{Figure 2.4: AS moment
conditions}\end{array}\end{array}$

Figures 1.2 and 2.2 show the rejection frequencies of 5% significance tests of
H${}_{0}:\theta=0.5$ using the KLM test with Lev moment conditions.
Interestingly, these figures show that the Lev moment conditions only identify
$\theta$ when the true value $\theta_{0}$ is close to one when
$\sigma_{c}^{2}=0.$ Non-zero values of $\sigma_{c}^{2}$ correspond with a
large variance of the initial observations when $\theta_{0}$ is close to one
and Figures 1.2 and 2.2 show that the Lev moment conditions do not identify
$\theta$ in this case. This contradicts the common perception that the Lev
moment conditions generally identify $\theta$ irrespective of the setting of
nuisance parameters, like, the variance of the initial observations.

Figures 1.3 and 2.3 show the rejection frequencies of 5% significance tests of
H${}_{0}:\theta=0.5$ using the KLM test with Sys moment conditions.
Surprisingly, these figures show that the Sys moment conditions do not
identify $\theta$ when $\theta_{0}$ is close to one and $\sigma_{c}^{2}>0$
when $T=3$ but do so when $T=4.$

Figure 2.4 shows the rejection frequencies of 5% significance tests of
H${}_{0}:\theta=0.5$ using the KLM test with AS moment conditions. These
rejection frequencies show that the AS moment conditions, which are not
defined for $T=3,$ identify $\theta$ when its true value is close to one and
the variance of the initial observations is very large. Interestingly, the
rejection frequencies of KLM tests of H0 using the Sys and AS moment
conditions are very close when $\theta_{0}$ is near one when paired with large
variances of the initial observations.

Summarizing, Panels 1 and 2 illustrate a few stylized facts that concern the
identification of $\theta$ for the DGP used in the simulation experiment:

  1. 1.

Dif moment conditions do not identify $\theta$ when $\theta_{0}$ is close to
one for general $T.$

  2. 2.

Lev moment conditions do not identify $\theta$ when $\theta_{0}$ is close to
one for large variances of the initial observations for general $T.$

  3. 3.

Sys moment conditions do not identify $\theta$ when $\theta_{0}$ is close to
one for large variances of the initial observations when $T=3.$

  4. 4.

Sys and AS moment conditions identify $\theta$ when $\theta_{0}$ is close to
one for large variances of the initial observations when $T$ exceeds $3.$

  5. 5.

The rejection frequencies of KLM tests of H0 using AS and Sys moment
conditions when $\theta_{0}$ is close to one and the variance of the initial
observations is large are almost identical.

Except for the first stylized fact, a theory backing them up is lacking so we
aim to provide one in the sections ahead. In doing so, we show that all
information regarding $\theta,$ when its true value is close to one and the
variance of the initial observations is large, is contained in a set of, so-
called, robust moment conditions which are a combination of either the AS or
Sys moment conditions. We furthermore show that the KLM test based on the
original AS or Sys moment conditions, as reported in Panels 1 and 2, makes
optimal use of these robust sample moments when only they contain information
on $\theta$.

Alongside the identification issues we can infer from the rejection
frequencies in Panels 1 and 2, they are also indicative of the different kind
of confidence sets that can result from the identification robust tests as
discussed previously. For example, the low rejection frequencies occurring for
$\theta_{0}$ around one, that result from the identification issues, show that
the 95% confidence sets for $\theta$ are then typically very wide, possibly
unbounded, when $\theta_{0}$ has such a value paired with a large variance of
the initial observations. To visualize this further, Panel 3 contains the (one
minus the) $p$-value plots of KLM tests using AS, Dif, Lev and Sys moment
conditions for four data sets using the same DGPs as in Panels 1-2 with
$N=250$ and $\theta_{0}=0.95.$666We note that the figures in Panel 3 show (one
minus) the p-value for one realized data set and do not show the simulated
empirical distribution function of the test under the null hypothesis which is
sometimes also referred to as a $p$-value plot, see Davidson and MacKinnon
(2002).  The DGPs used for the four figures differ over the values of $T$ and
$\sigma_{c}^{2}.$ The intersections of the depicted $p$-value plots with the
line at 0.95 indicate the 95% confidence sets of KLM tests with the respective
moment condition.

In Figures 3.1 and 3.3, $\sigma_{c}^{2}=0$ so identification issues only occur
at $\theta_{0}$ close to one when using the Dif moment conditions. Since
$\theta_{0}$ is 0.95, this explains why the $p$-value plots of the KLM test
with the Dif moments conditions do not cross the line at 0.95 in Figures 3.1
and 3.3 so the resulting 95% confidence sets are very wide. The $p$-value
plots in Figures 3.1 and 3.3 of KLM tests with Sys and Lev moment conditions
show that they lead to bounded 95% confidence sets since these moment
conditions have no identification issues when $T=3$ and $\sigma_{c}^{2}=0.$

In Figure 3.2, where $T=3$ and $\sigma_{c}^{2}=0.5,$ none of the $p$-value
plots crosses the line at 0.95 so 95% confidence sets that result from KLM
tests with Dif, Lev and Sys moment conditions are all very wide and possibly
unbounded. This is indicative of the identification issues when $T=3$ and
$\sigma_{c}^{2}=0.5$ for true values of $\theta$ close to one.

In Figure 3.4, where $T=4$ and $\sigma_{c}^{2}=0.5,$ KLM tests with Sys and AS
moment conditions both result in finite 95% confidence sets while the KLM test
with Dif and Lev moment conditions leads to very wide possibly unbounded
confidence sets. Hence, Sys and AS moment conditions have no identification
issues while Dif and Lev moment conditions do. The AS moment conditions are
quadratic functions of $\theta$ which explains the somewhat unusual shape of
their $p$-value plots in Figures 3.3 and 3.4.

$\begin{array}[c]{c}\text{Panel 3. One minus }p\text{-value plots of KLM tests
using different moments conditions: Sys }\\\ \text{(solid), AS (dotted), Lev
(dashed), Dif (dash-dot) for }\theta_{0}=0.95\text{ and }N=250.\text{ }\\\ \\\
\begin{array}[c]{ccc}\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/klmt3th095f0interval01.eps}}&&\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/klmt3th095f05interval01.eps}}\\\
\text{Figure 3.1: }T=3,\text{ }\sigma_{c}^{2}=0&&\text{Figure 3.2: }T=3,\text{
}\sigma_{c}^{2}=0.5\end{array}\\\
\begin{array}[c]{ccc}\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/klmt4th095f0interval01.eps}}&&\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/klmt4th095f05interval01.eps}}\\\
\text{Figure 3.3: }T=4,\text{ }\sigma_{c}^{2}=0&&\text{Figure 3.4: }T=4,\text{
}\sigma_{c}^{2}=0.5\end{array}\end{array}$

##  3 Identification from different moment conditions

Stylized facts 1-4 illustrated by the figures in Panels 1-3 show the
identification issues that occur for the autoregressive parameter $\theta$
when the variance of the initial observations is large and $\theta_{0}$, i.e.
the true value in the DGP, is close to one. To pin these identification issues
down precisely, we use an asymptotic sampling scheme which consists of joint
drifting sequences for the autoregressive parameter and the variance of the
initial observation. We indicate this dependence on the sample size $N$ by
$\theta_{0,N}$ and $h_{N}(\theta_{0,N})=\frac{1}{\sqrt{var(y_{i1})}}$
respectively. The true value of $\theta,$ previously denoted by $\theta_{0},$
is from now on therefore denoted by $\theta_{0,N}$. Assumptions 1 and 2 group
the different requirements needed to obtain our results.

#### Assumption 1.

a. The drifting sequences of the autoregressive parameter and variance of the
initial observations are such that:

$\begin{array}[c]{rl}\lim_{N\rightarrow\infty}\theta_{0,N}=&1\\\
\lx@intercol\hfil\lim_{N\rightarrow\infty}h_{N}(\theta_{0,N})=\hfil\lx@intercol&d_{1},\end{array}$
(20)

with $d_{1}$ a finite, possibly zero constant.

b. The initial observations satisfy the mean stationarity conditions in
(7)-(9).

c. The joint limit behavior of the variance of $u_{i1}$ and $(1-\theta_{0,N})$
is such that

$\begin{array}[c]{c}\lim_{N\rightarrow\infty}(1-\theta_{0,N})\sigma_{1,N}^{2}=d_{2},\end{array}$
(21)

with $\sigma_{1,N}^{2}=$var($u_{i1})$, $d_{2}$ a finite, possibly zero
constant and $(1-\theta_{0,N})^{\frac{1}{2}}u_{i1}$ is a random variable with
finite fourth order moments.

d. The variance of the product of the initial observation $y_{i1}$ and the
disturbances $u_{it}$ is such that

$\begin{array}[c]{c}var(u_{it}y_{i1})=\sigma_{t}^{2}var(y_{i1}),\text{
}t=2,\ldots,T,\end{array}$ (22)

with $\sigma_{t}^{2}=var(u_{it}),$ $t=2,\ldots,T.$

e. The errors $u_{i1}/\sigma_{1,N},$ $u_{i2}\ldots,u_{iT}$ and $c_{i},$
$i=1,\ldots,N,$ are independently distributed within individuals and over the
different individuals and have mean zero, finite variance and finite fourth
order moments and satisfy the conditions in (2).

Assumption 1a concerns the joint limit behavior of the variance of the initial
observations and $\theta_{0,N}$. By the definition of $\mu_{i}$ in (8) and
Assumption 1a, $\mu_{i}$ is also drifting with the sample size since it is a
function of $\theta_{0,N}$ and so are $y_{i1}$ and $\sigma_{1,N}^{2}.$
Assumption 1b specifies that the initial observations follow the mean
stationarity assumption, which is necessary for the Lev and Sys moment
conditions to hold. Assumptions 1c-e are mainly technical assumptions, which
are needed to obtain our theoretical results. Assumption 1c sets an upper
bound on the rate at which the variance of $u_{i1}$ can diverge. It implies
that the variance of $u_{i1}$ is at most proportional to
$(1-\theta_{0,N})^{-1}$ (so covariance stationarity is allowed for).
Assumption 1d holds under independence of $u_{it}$ and $y_{i1}$ but it can
also hold under less stringent conditions. In the sequel, we analyze the
identification of $\theta$ when the variance of the initial observations gets
large compared to that of the subsequent disturbances. Assumption 1d enables
such settings. Assumption 1e is a technical assumption which is needed to use
a central limit theorem.

Assumption 1a allows the variance of the initial observations to be large
jointly with a large value for the autoregressive parameter. When $d_{1}$ in
(20) equals zero, the rate at which $h_{N}(\theta_{0,N})$ goes to zero, or the
variance of the initial observation goes to infinity, is key to the
identification of $\theta$ from the sample moment conditions. We therefore put
down two alternative assumptions regarding the joint convergence of the sample
size and the variance of the initial observations under which there is
identification or identification is problematic for specific moment
conditions.

#### Assumption 2.

a. $d_{1}=0$ and the drifting sequence of the variance of the initial
observation is such that:

$\begin{array}[c]{c}h_{N}(\theta_{0,N})\sqrt{N}\underset{N\rightarrow\infty}{\rightarrow}0.\end{array}$
(23)

b. $d_{1}\neq 0$ or the drifting sequence of the variance of the initial
observation is such that:

$\begin{array}[c]{c}h_{N}(\theta_{0,N})\sqrt{N}\underset{N\rightarrow\infty}{\rightarrow}\infty.\end{array}$
(24)

Identification generically holds under Assumption 2b but can become
problematic under Assumption 2a and then depends on the particular moment
condition and number of time series observations as we show later on. In the
intermediate case where $h_{N}(\theta_{0,N})\sqrt{N}$ converges to a finite,
but non-zero constant, we are in a case similar to that discussed in the weak
instrument literature where the sample Jacobian converges to a random variable
which leads to inconsistent estimators with non-standard behavior of their
corresponding t-statistics. Because of the practical similarities with
Assumption 2a, however, we do not separately discuss it.

Since any assumption about the convergence rates of the sample size and the
variance of the initial observations is to a large extent arbitrary, also the
identification of $\theta$ by these conditions is arbitrary for DGPs for which
the true value of $\theta$ is close to one and the variance of the initial
observations is infinite when the true value of $\theta$ equals one. Some
plausible DGPs, all of which accord with mean stationarity (7)-(9), for the
initial observations belong to this category:

DGP 1.

$\sigma_{c}^{2}=$var($c_{i}),$ $\sigma_{1,N}^{2}=\sigma_{1}^{2},$
$h(\theta_{0,N})^{-2}=\sigma_{c}^{2}/(1-\theta_{0,N})^{2}+\sigma_{1}^{2},$ so
when $\theta_{0,N}\underset{N\rightarrow\infty}{\rightarrow}1,$
$(1-\theta_{0,N})^{-1}h(\theta_{0,N})\underset{N\rightarrow\infty}{\rightarrow}\sigma_{c}^{-1}.$

DGP 2.

$\sigma_{c}^{2}=$var($c_{i}),$
$\sigma_{1,N}^{2}=\frac{\sigma^{2}}{1-\theta_{0,N}^{2}},$
$\sigma^{2}=var(u_{it}),$ $t=2,\ldots,T,$
$h(\theta_{0,N})^{-2}=\sigma_{c}^{2}/(1-\theta_{0,N})^{2}+\sigma^{2}/(1-\theta_{0,N}^{2}),$
so when $\theta_{0,N}\underset{N\rightarrow\infty}{\rightarrow}1,$
$(1-\theta_{0,N})^{-1}h(\theta_{0,N})\underset{N\rightarrow\infty}{\rightarrow}\sigma_{c}^{-1}.$

DGP 3.

$\sigma_{\mu}^{2}=$var($\mu_{i}),$
$\sigma_{1,N}^{2}=\frac{\sigma^{2}}{1-\theta_{0,N}^{2}},$
$\sigma^{2}=var(u_{it}),$ $t=2,\ldots,T,$
$h(\theta_{0,N})^{-2}=\sigma_{\mu}^{2}+\sigma^{2}/(1-\theta_{0,N}^{2}),$ so
when $\theta_{0,N}\underset{N\rightarrow\infty}{\rightarrow}1,$
$(1-\theta_{0,N}^{2})^{-\frac{1}{2}}h(\theta_{0,N})\underset{N\rightarrow\infty}{\rightarrow}\sigma^{-1}.$

DGP 4.

$\sigma_{\mu}^{2}=$var($\mu_{i}),$
$\sigma_{1,N}^{2}=\sigma^{2}\frac{1-\theta_{0,N}^{2(g+1)}}{1-\theta_{0,N}^{2}},$
$\sigma^{2}=var(u_{it}),$ $t=2,\ldots,T,$
$h(\theta_{0,N})^{-2}=\sigma_{\mu}^{2}+\sigma^{2}\frac{1-\theta_{0,N}^{2(g+1)}}{1-\theta_{0,N}^{2}},$
so when $\theta_{0,N}\underset{N\rightarrow\infty}{\rightarrow}1,$
$\left(\frac{1-\theta_{0,N}^{2}}{1-\theta_{0,N}^{2(g+1)}}\right)^{-\frac{1}{2}}h(\theta_{0,N})\underset{N\rightarrow\infty}{\rightarrow}\sigma^{-1}.$

DGP 5.

$\sigma_{c}^{2}=$var($c_{i}),$
$\sigma_{1,N}^{2}=\sigma^{2}\frac{1-\theta_{0,N}^{2(g+1)}}{1-\theta_{0,N}^{2}},$
$\sigma^{2}=var(u_{it}),$ $t=2,\ldots,T,$
$h(\theta_{0,N})^{-2}=\sigma_{c}^{2}/(1-\theta_{0,N})^{2}+\sigma^{2}\frac{1-\theta_{0,N}^{2(g+1)}}{1-\theta_{0,N}^{2}},$
so when $\theta_{0,N}\underset{N\rightarrow\infty}{\rightarrow}1,$
$\left(1-\theta_{0,N}\right)^{-1}h(\theta_{0,N})\underset{N\rightarrow\infty}{\rightarrow}\sigma_{c}^{-1}.$

DGPs 4 and 5 characterize an autoregressive process of order one that has
started $g$ periods in the past while the initial observations that result
from DGP 2 and 3 result from an autoregressive process that has started an
infinite number of periods in the past. DGPs 2 and 3 are also used by Blundell
and Bond (1998) and Arellano and Bover (1995) use DGP 2, but these studies
keep the variance of the initial observations fixed.

For DGPs 1-5 to imply Assumption 2a, the limiting sequence $\theta_{0,N}$ has
to be such that:

$\begin{array}[c]{rlcl}\text{DGP }1,\text{ }2,\text{
}5:&(1-\theta_{0,N})\sqrt{N}\underset{N\rightarrow\infty}{\rightarrow}0\text{
}&\text{for which it is sufficient
that}&\theta_{0,N}=1-\frac{e}{N^{\frac{1}{2}(1+\epsilon)}}\\\ \text{DGP
}3:&(1-\theta_{0,N}^{2})N\underset{N\rightarrow\infty}{\rightarrow}0&\text{for
which it is sufficient that}&\theta_{0,N}=1-\frac{e}{N^{1+\epsilon}}\\\
\text{DGP }4:&\frac{N}{g}\underset{N\rightarrow\infty,\text{
}g\rightarrow\infty}{\rightarrow}0,&&\end{array}$ (25)

with $e$ a constant and $\epsilon$ some real number larger than zero. In the
case of DGP 4, (25) implies that the process has been running longer than the
sample size $N.$ Kruiniger (2009) uses the above specification of DGP 3 with
$\epsilon=0$ and DGP 4 with $N/g$ converging to a constant to construct local
to unity asymptotic approximations of the distributions of two step GMM
estimators that use the Dif, Lev or Sys moment conditions.

We do not confine ourselves to a specific DGP for the initial observations so
we obtain results that apply more generally. While the (non-) identification
conditions for identifying $\theta$ that result from the above data generating
processes might be (in)plausible, it is the arbitrariness of them which is
problematic. Additionally, the identification condition might hold but it can
still lead to large size distortions of Wald test statistics, like, the
t-test.

To analyze the identification of $\theta$ by the different moment conditions
for a general number of time periods $T$, we start out with a representation
theorem. For the different moment conditions, it states the behavior of the
sample moments and their derivatives under Assumptions 1 and 2a.

#### Theorem 1 (Representation Theorem).

Under Assumptions 1 and 2a, we can characterize the large sample behavior of
the Dif, Lev, NL, AS and Sys sample moments for $T$ time series observations
and their derivatives by:

$\begin{array}[c]{ll}\left(\begin{array}[c]{c}f_{N}^{j}(\theta)\\\
q_{N}^{j}(\theta)\end{array}\right)=&\left(\begin{array}[c]{c}A_{f}^{j}(\theta)\\\
A_{q}^{j}(\theta)\end{array}\right)\left[\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left(\psi-
h_{N}(\theta_{0,N})\sigma_{1,N}\iota_{T-1}\psi_{c}\right)-\iota_{T-1}d_{2}\right]+\\\
&\left(\begin{array}[c]{c}\mu_{f}^{j}(\theta,\bar{\sigma}^{2})\\\
\mu_{q}^{j}(\theta,\bar{\sigma}^{2})\end{array}\right)+o_{p}(1),\end{array}$
(26)

with $j=Dif,$ $Lev,$ $NL,$ $AS,$ $Sys$. The specifications of the
$k_{j}$-dimensional sample moments $f_{N}^{j}(\theta)$ and derivatives
$q_{N}^{j}(\theta)$ are given in the Appendix. Furthermore,
$A_{f}^{j}(\theta),$ $A_{q}^{j}(\theta),$
$\mu_{f}^{j}(\theta,\bar{\sigma}^{2})$ and
$\mu_{q}^{j}(\theta,\bar{\sigma}^{2})$ are constant $k_{j}\times(T-1),$
$k_{j}\times(T-1),$ $k_{j}\times 1$ and $k_{j}\times 1$ dimensional matrices,
$\bar{\sigma}^{2}=(\sigma_{2}^{2}\ldots\sigma_{T}^{2}),$

$\begin{array}[c]{rl}\frac{h_{N}(\theta_{0,N})}{\sqrt{N}}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i1}u_{i2}\\\
\vdots\\\ y_{i1}u_{iT}\end{array}\right)\underset{d}{\rightarrow}&\psi\\\
\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\frac{u_{i1}}{\sigma_{1,N}}c_{i}\underset{d}{\rightarrow}&\psi_{c},\end{array}$
(27)

so $\psi$ is a $(T-1)$-dimensional normal random vector, $\psi\sim
N(0,$diag($\sigma_{2}^{2}\ldots\sigma_{T}^{2}))$, $\psi_{c}\sim
N(0,$var($c_{i}))$ and independent from $\psi,$ and $\iota_{T-1}$ is a
$(T-1)$-dimensional vector of ones.

The specifications of $A_{f}^{j}(\theta),$ $A_{q}^{j}(\theta),\mathit{\
}\mu_{f}^{j}(\theta,\bar{\sigma}^{2}),\mathit{\
}\mu_{q}^{j}(\theta,\bar{\sigma}^{2})$ for values of $T$ equal to 3-5 are all
stated in the Appendix.

Proof. see the Appendix.

The representation theorem in Theorem 1 is reminiscent of the cointegration
representation theorem, see $e.g.$ Engle and Granger (1987) and Johansen
(1991). Identical to that representation theorem, Theorem 1 shows that the
behavior of the moment series changes over different directions.

Theorem 1 implies that the sample moment and its derivative diverge in the
direction of $\binom{A_{f}^{j}(\theta)}{A_{q}^{j}(\theta)}$ since the latter
components get multiplied by $\frac{1}{h(\theta_{0,N})\sqrt{N}},$ which under
Assumption 2a goes off to infinity when the sample size increases. The only
identifying information for $\theta$ then results from that part of the sample
moment which does not depend on $\psi$. Since $\psi$ only affects the part of
the sample moments spanned by $A_{f}^{j}(\theta),$ the sample moments are
independent of $\psi$ in the direction of the maximal non-degenerate space
spanned by vectors orthogonal to $A_{f}^{j}(\theta)$ to which we refer as the
orthogonal complement of $A_{f}^{j}(\theta)$. We construct the orthogonal
complement, which we denote by $A_{f}^{j}(\theta)_{\perp}$, as the full rank
matrix projecting on the orthogonal complement of the range space of
$A_{f}^{j}(\theta)$. It consists of the minimal set of vectors spanning the
null space of the columns of $A_{f}^{j}(\theta)$. In the case the null space
has dimension zero, a full rank specification of $A_{f}^{j}(\theta)_{\perp}$
can not be constructed.

When we pre-multiply the sample moments by the orthogonal complement of
$A_{f}^{j}(\theta),$ we obtain

$\begin{array}[c]{cl}A_{f}^{j}(\theta)_{\perp}^{\prime}f_{N}^{j}(\theta)=&A_{f}^{j}(\theta)_{\perp}^{\prime}\mu_{f}^{j}(\theta,\bar{\sigma}^{2})+o_{p}(1).\end{array}$
(28)

Compared with expression (26) in Theorem 1, the elements multiplied by
$A_{f}^{j}(\theta)$ have dropped out since
$A_{f}^{j}(\theta)_{\perp}^{\prime}A_{f}^{j}(\theta)\equiv 0.$ The right hand
side of (28) now contains all remaining identifying elements of the original
moment conditions. From expression (28), it is seen that identification
results only when (1) $A_{f}^{j}(\theta)_{\perp}$ is a full rank matrix; (2)
$A_{f}^{j}(\theta)_{\perp}^{\prime}\mu_{f}^{j}(\theta,\bar{\sigma}^{2})\neq 0$
for all $\theta\neq\theta_{0,N}$.

For an illustrative example of Theorem 1, consider the large sample behavior
for $T=3$ of the Lev sample moment, $\frac{1}{N}\sum_{i=1}^{N}\Delta
y_{i2}(y_{i3}-\theta y_{i2}),$ and its derivative,
$-\frac{1}{N}\sum_{i=1}^{N}y_{i2}\Delta y_{i2},$ when $\theta_{0,N}$ converges
to one according to (20) and mean stationarity (8)-(9) applies. The Lev moment
condition has been proposed by Arellano and Bover (1995) and Blundell and Bond
(1998) to overcome the identification problems of the Dif moment condition
near the unit root. Under Assumption 1, the relevant elements for the large
sample behavior are:

$\begin{array}[c]{rl}f_{N}^{Lev}(\theta)=&\frac{1}{N}\sum_{i=1}^{N}\Delta
y_{i2}(y_{i3}-\theta y_{i2})\\\
=&(1-\theta)\left\\{\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2}+\frac{1}{N}\sum_{i=1}^{N}u_{i2}y_{i1}+\right.\\\
&\left.\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)u_{i1}y_{i1}\right\\}+o_{p}(1),\\\
q_{N}^{Lev}(\theta)=&-\frac{1}{N}\sum_{i=1}^{N}y_{i2}\Delta y_{i2}\\\
=&-\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2}-\frac{1}{N}\sum_{i=1}^{N}u_{i2}y_{i1}\\\
&-\frac{1}{N}\sum_{i=1}^{N}(1-\theta_{0,N})u_{i1}y_{i1}+o_{p}(1),\end{array}$
(29)

see the proof of Theorem 1 in the Appendix for a derivation. The $o_{p}(1)$
remainder terms contain all elements in (29) that can not dominate the large
sample behavior when $\theta_{0,N}$ goes to one according to the drifting
parameter sequences defined in Assumption 1. The components explicitly
specified in (29) either have a non-zero mean or depend on the initial
observations $y_{i1}$. Under Assumption 1, we have that

$\begin{array}[c]{c}h_{N}(\theta_{0,N})\frac{1}{\sqrt{N}}\sum_{i=1}^{N}u_{i2}y_{i1}\underset{d}{\rightarrow}\psi_{2},\text{
}\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\frac{u_{i1}}{\sigma_{1,N}}c_{i}\underset{d}{\rightarrow}\psi_{c},\end{array}$
(30)

which is proven in Lemma 1 in the Appendix and where $\psi_{2}$ and $\psi_{c}$
are independent normal random variables with mean zero and variance
$\sigma_{2}^{2}$ and $\sigma_{c}^{2},$ $\sigma_{c}^{2}=$var($c_{i})$. It
explains why $\frac{1}{N}\sum_{i=1}^{N}u_{i2}y_{i1}$ and
$\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)u_{i1}y_{i1}=\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)u_{i1}^{2}+\frac{1}{N}\sum_{i=1}^{N}u_{i1}c_{i}$
explicitly appear in (29). When $d_{1}$ in (20) equals zero, the rate at which
$h_{N}(\theta_{0,N})$ goes to zero, or the variance of the initial observation
goes to infinity, determines the behavior of the sample moments in (29). For
example, when $d_{1}=0$ and these sequences are as in Assumption 2b, it holds
that

$\begin{array}[c]{rcl}\frac{1}{N}\sum_{i=1}^{N}y_{i2}\Delta
y_{i2}&\underset{p}{\rightarrow}&\sigma_{2}^{2}-d_{2}.\end{array}$ (31)

Although Assumption 1 does not fully pin down $d_{2}$, which value depends on
the particular DGP for the initial observations, it is clear that the
probability limit of the sample Jacobian typically differs from zero. Hence,
the Lev moment condition seems to identify $\theta$ irrespective of its true
value, see Arellano and Bover (1995) and Blundell and Bond (1998). There is a
caveat though since, under Assumption 2a, Theorem 1 shows that:

$\begin{array}[c]{rl}f_{N}^{Lev}(\theta)=&\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\frac{h_{N}(\theta_{0,N})}{\sqrt{N}}\sum_{i=1}^{N}\Delta
y_{i2}(y_{i3}-\theta y_{i2})\\\
=&(1-\theta)\left\\{\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}(\psi_{2}-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c})+(\sigma_{2}^{2}-d_{2})\right\\}+o_{p}(1),\\\
q_{N}^{Lev}(\theta)=&-\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\frac{h_{N}(\theta_{0,N})}{\sqrt{N}}\sum_{i=1}^{N}y_{i2}\Delta
y_{i2},\\\
=&-\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}(\psi_{2}-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c})-(\sigma_{2}^{2}-d_{2})+o_{p}(1),\end{array}$
(32)

which implies that the sample moments of the Lev population moment and
Jacobian diverge when the sample size increases. The Lev sample moment then no
longer identifies $\theta$ since the components that would identify $\theta$
in the Jacobian identification condition, i.e.
$\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2},$ gets dominated by the component
$\frac{1}{N}\sum_{i=1}^{N}u_{i2}y_{i1}$ and possibly
$\frac{1}{N}\sum_{i=1}^{N}(1-\theta_{0,N})u_{i1}y_{i1}.$

We next discuss what Theorem 1 implies for the different sets of moment
conditions discussed previously and their respective orthogonal complements of
$A_{f}(\theta).$

#### Dif and Lev conditions

When $T=3$ or $4,$ the specifications of
$\mu_{f}^{j}(\theta,\bar{\sigma}^{2})$, $A_{f}^{j}(\theta)$ and
$A_{f}^{j}(\theta)_{\perp}$ for the Dif and Lev moment conditions, which are
stated in the proof of Theorem 1 in the Appendix, are:

$\begin{array}[c]{cll}\text{{Dif:}}&\mathbf{T=3}&\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})=0,\text{
}A_{f}^{Dif}(\theta)=(-\theta\text{ }1),\text{
}A_{f}^{Dif}(\theta)_{\perp}=(1\text{ }\theta)\text{ }\\\
&\mathbf{T=4}&\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})=\left(\begin{array}[c]{c}0\\\
0\\\ 0\end{array}\right),\text{
}A_{f}^{Dif}(\theta)=\left(\begin{array}[c]{ccc}-\theta&1&0\\\ 0&-\theta&1\\\
0&-\theta&1\end{array}\right),\text{
}A_{f}^{Dif}(\theta)_{\perp}=\left(\begin{array}[c]{c}0\\\ -1\\\
1\end{array}\right).\\\
\text{{Lev:}}&\mathbf{T=3}&\mu_{f}^{Lev}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}\sigma_{2}^{2}\\\
0\end{array}\right),\text{ }A_{f}^{Lev}(\theta)=(1-\theta\text{ }0),\text{
}A_{f}^{Lev}(\theta)_{\perp}\text{ does not exist}\\\
&\mathbf{T=4}&\mu_{f}^{Lev}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}\sigma_{2}^{2}\\\
\sigma_{3}^{2}\\\ 0\end{array}\right),\text{
}A_{f}^{Lev}(\theta)=\left(\begin{array}[c]{ccc}1-\theta&0&0\\\
0&1-\theta&0\end{array}\right),\\\ &&A_{f}^{Lev}(\theta)_{\perp}\text{ does
not exist.}\end{array}$ (33)

The expressions of $A_{f}^{Lev}(\theta)$ are all such that we cannot specify a
non-zero matrix $A_{f}^{Lev}(\theta)_{\perp}$ such that
$A_{f}^{Lev}(\theta)_{\perp}^{\prime}A_{f}^{Lev}(\theta)=0.$ This remains so
when $T$ exceeds four, see the Appendix. Hence, $A_{f}^{Lev}(\theta)_{\perp}$
does not exist (as a non-zero matrix). Regarding the Dif moments, when $T>3$
the rank of the orthogonal complement of $A_{f}^{Dif}(\theta),$
$A_{f}^{Dif}(\theta)_{\perp},$ is larger than zero. However, since
$\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})$ equals zero for any value of $T$,
$A_{f}^{Dif}(\theta)_{\perp}^{\prime}\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})=0$
so the Dif moment conditions do not identify $\theta.$ Summarizing, we have:

$\begin{array}[c]{rll}\text{{Dif}:}&\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})\text{
is vector of all zeros.}&\text{No identification when }T\geq 3.\\\
\text{{Lev:}}&A_{f}^{Lev}(\theta)_{\perp}\text{ does not exist.}&\text{No
identification when }T\geq 3.\end{array}$ (34)

#### NL condition

The NL moment condition is not defined for $T=3.$ When $T=4,$ the expressions
of $\mu_{f}^{j}(\theta,\bar{\sigma}^{2})$, $A_{f}^{j}(\theta)$ and
$A_{f}^{j}(\theta)_{\perp}$ read

$\begin{array}[c]{cl}\text{{NL:}}&\mu_{f}^{NL}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\sigma_{3}^{2}-\theta\sigma_{2}^{2}\right),\text{
}A_{f}^{NL}(\theta)=\left(\begin{array}[c]{ccc}\theta(\theta-1)&1-\theta&0\end{array}\right),\text{
}\\\ &A_{f}^{NL}(\theta)_{\perp}\text{ does not exist.}\end{array}$ (35)

Since the orthogonal complement does not exist, the NL moment condition does
not identify $\theta.$ The expression of $A_{f}^{NL}(\theta)$ for a larger
number of time series observations (see the Appendix) is also such that the
orthogonal complement $A_{f}^{NL}(\theta)_{\perp}$ also does not exist. Hence
for larger values of $T,$ the NL moment conditions also do not identify
$\theta.$

#### AS and Sys conditions

The expressions of $\mu_{f}^{j}(\theta,\bar{\sigma}^{2})$, $A_{f}^{j}(\theta)$
and $A_{f}^{j}(\theta)_{\perp}$ when $T=3,$ $4$ for the AS and Sys moment
conditions result from stacking those of the Dif and NL and Dif and Lev moment
conditions respectively:

$\begin{array}[c]{cll}\text{{AS:}}&\mathbf{T=4}&\mu_{f}^{AS}(\theta,\bar{\sigma}^{2})=\left(\begin{array}[c]{c}0\\\
0\\\ 0\\\
\left(1-\theta\right)\left(\sigma_{3}^{2}-\theta\sigma_{2}^{2}\right)\end{array}\right),\text{
}A_{f}^{AS}(\theta)=\left(\begin{array}[c]{ccc}-\theta&1&0\\\ 0&-\theta&1\\\
0&-\theta&1\\\ \theta(\theta-1)&1-\theta&0\end{array}\right),\text{ }\\\
&&A_{f}^{AS}(\theta)_{\perp}=\left(\begin{array}[c]{cc}\theta-1&0\\\ 0&-1\\\
0&1\\\ 1&0\end{array}\right).\\\
\text{{Sys:}}&\mathbf{T=3}&\mu_{f}^{Sys}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}0\\\
\sigma_{2}^{2}\end{array}\right),\text{
}A_{f}^{Sys}(\theta)=\left(\begin{array}[c]{cc}-\theta&1\\\
1-\theta&0\end{array}\right),\text{ }\\\ &&A_{f}^{Sys}(\theta)_{\perp}\text{
does not exist.}\end{array}$
$\begin{array}[c]{cll}\text{{Sys:}}&\mathbf{T=4}&\mu_{f}^{Sys}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}0\\\
0\\\ 0\\\ \sigma_{2}^{2}\\\ \sigma_{3}^{2}\end{array}\right),\text{
}A_{f}^{Sys}(\theta)=\left(\begin{array}[c]{ccc}-\theta&1&0\\\ 0&-\theta&1\\\
0&-\theta&1\\\ 1-\theta&0&0\\\ 0&1-\theta&0\end{array}\right),\text{ }\\\
&&A_{f}^{Sys}(\theta)_{\perp}=\left(\begin{array}[c]{cc}\theta-1&0\\\ 0&-1\\\
0&1\\\ -\theta&0\\\ 1&0\end{array}\right).\end{array}$ (36)

When $T=3,$ $A_{f}^{Sys}(\theta)$ is a full rank square matrix so its
orthogonal complement does not exist. It implies that the Sys moment
conditions do not identify $\theta$ when $T=3.$ When $T=4$, the orthogonal
complement of $A_{f}^{j}(\theta),$ $A_{f}^{j}(\theta)_{\perp},$ has rank
larger than zero for both AS and Sys moments. Furthermore, the specification
of $\mu_{f}^{j}(\theta,\bar{\sigma}^{2})$ for the AS and Sys moment conditions
in (36) is such that
$A_{f}^{j}(\theta)_{\perp}^{\prime}\mu_{f}^{j}(\theta,\bar{\sigma}^{2})\neq 0$
for all $\theta\neq\theta_{0,N}$, while it is not difficult to see that
$\lim_{N\rightarrow\infty}A_{f}^{j}(\theta_{0,N})_{\perp}^{\prime}\mu_{f}^{j}(\theta_{0,N},\bar{\sigma}^{2})=0$
which just reflects that the moment conditions hold at the true value. This
implies that although the AS and Sys sample moments diverge in the direction
of $A_{f}^{j}(\theta),$ so that part cannot be used to identify $\theta,$ the
AS and Sys sample moments identify $\theta$ by their part which is spanned by
the orthogonal complement of $A_{f}^{j}(\theta).$ The expressions of
$\mu_{f}^{j}(\theta,\bar{\sigma}^{2})$ and $A_{f}^{j}(\theta)$ in the proof of
Theorem 1 in the Appendix show that this argument extends to all values of $T$
larger than three.

Our preceding analysis is summarized by Corollary 1:

#### Corollary 1 (Identification of $\theta$).

Under Assumptions 1 and 2a, $\theta$ is identified by the AS and Sys moment
conditions when $T$ exceeds three. Furthermore, $\theta$ is not identified by
the Dif, Lev and NL moment conditions separately for any value of T and the
Sys moment conditions when T equals three.

Corollary 1 proves stylized facts 1-4 from Section 3, which are illustrated by
Panels 1-2. It also shows that the identification from the Lev moment
condition remains problematic for larger values of $T$ but the Sys and AS
moment conditions generally identify $\theta$ for values of $T$ larger than
three.

Regarding the NL moments we find that they are not robust to all settings of
nuisance parameters like the variance of the initial observations. Alvarez and
Arellano (2004) and Kruiniger (2013) have shown that, when the data, including
the initial observation, have finite second moments and the autoregressive
parameter equals one, $\theta$ is identified by the NL and, hence, the AS
moment conditions if and only if $T\geq 4.$ Furthermore, if $T\geq 4$,
$\theta$ is only locally identified when the unconditional variances of the
errors change at a constant rate of growth between $t=2$ and $t=T-1$ and only
second-order but globally identified when the unconditional variances between
$t=2$ and $t=T-1$ are equal. Unlike Alvarez and Arellano (2004) and Kruiniger
(2013), our limiting sequence for the variance of the initial observations
allows for unbounded values. Theorem 1 then shows that identification by the
NL moment conditions is lost when its convergence rate accords with (23). The
intuition is that the NL moment conditions are a product of levels and first
differences so they are unlikely to identify the parameters in limit sequences
where the variance of the initial observations increases faster than the
sample size.

Theorem 1 can be used to construct the non-standard limiting behavior of one
and two step GMM estimators that result from the different moment conditions.
These are similar to the non-standard results in e.g. Madsen (2003) and
Kruiniger (2009) so we, for reasons of brevity, refrain from stating them.

#### Robust sample moments

Theorem 1 shows that the identification of $\theta$ when the variance of the
initial observations is large results from the part of the (AS or Sys) moment
conditions that lies in the direction of $A_{f}^{j}(\theta)_{\perp}$.
Expressions of the orthogonal complements of $A_{f}^{j}(\theta)$ for $T=4$ and
5 for the AS and Sys moment conditions are stated in (36). They can be
specified (see the Appendix) as

$A_{f}^{j}(\theta)_{\perp}=(G_{f,T}^{j}(\theta)\text{ }\vdots\text{
}G_{2,T}^{j}),$ (37)

where $T$ indicates the number of time periods and $G_{2,T}^{j}$ is such that
$G_{2,T}^{j\prime}\mu_{f}^{j}(\theta,\bar{\sigma}^{2})=0$ for all $\theta.$
Furthermore, $G_{f,T}^{j}(\theta)$ is the only part of
$A_{f}^{j}(\theta)_{\perp}$ that depends on $\theta$. The orthogonal
complements are then such that the resulting, what we refer to as, robust
moment conditions are quadratic in $\theta:$

$g_{f,T}^{j}(\theta)=A_{f}(\theta)_{\perp}^{j\prime}f_{N}^{j}(\theta)=a\theta^{2}+b\theta+d,$
(38)

where the expressions for $a,$ $b$ and $d$ are constructed in the Appendix:

T=4:

Sys

$a=\frac{1}{N}\sum_{i=1}^{N}\binom{(\Delta y_{i2})^{2}}{0},$
$b=-\frac{1}{N}\sum_{i=1}^{N}\binom{(y_{i3}-y_{i1})^{2}}{\Delta y_{i2}\Delta
y_{i3}},$ $d=\frac{1}{N}\sum_{i=1}^{N}\binom{(y_{i4}-y_{i1})\Delta
y_{i3}}{\Delta y_{i2}\Delta y_{i4}}.$

AS

$a=\frac{1}{N}\sum_{i=1}^{N}\binom{(y_{i3}-y_{i1})\Delta y_{i2}}{0},$
$b=-\frac{1}{N}\sum_{i=1}^{N}\binom{(y_{i3}-y_{i1})\Delta
y_{i3}+(y_{i4}-y_{i1})\Delta y_{i2}}{\Delta y_{i2}\Delta y_{i3}},\allowbreak$

$d=\frac{1}{N}\sum_{i=1}^{N}\binom{(y_{i4}-y_{i1})\Delta y_{i3}}{\Delta
y_{i2}\Delta y_{i4}}.$

T=5:

Sys

$a=\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}(\Delta y_{i2})^{2}\\\
(y_{i3}-y_{i1})\Delta y_{i3}\\\ (\Delta y_{i3})^{2}\\\ 0\\\
0\end{array}\right),$
$b=-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}(y_{i3}-y_{i1})^{2}\\\
(y_{i4}-y_{i1})(y_{i4}-y_{i2})\\\ (y_{i4}-y_{i2})^{2}\\\ \Delta y_{i2}\Delta
y_{i4}\\\ \Delta y_{i3}\Delta y_{i4}\end{array}\right),$

$d=\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}(y_{i4}-y_{i1})\Delta
y_{i3}\\\ (y_{i5}-y_{i1})\Delta y_{i4}\\\ (y_{i5}-y_{i2})\Delta y_{i4}\\\
\Delta y_{i2}\Delta y_{i5}\\\ \Delta y_{i3}\Delta y_{i5}\end{array}\right).$

AS

$a=\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}(y_{i3}-y_{i1})\Delta
y_{i2}\\\ (y_{i4}-y_{i1})\Delta y_{i3}\\\ (y_{i4}-y_{i2})\Delta y_{i3}\\\ 0\\\
0\end{array}\right),$
$b=-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}(y_{i4}-y_{i1})\Delta
y_{i2}+(y_{i3}-y_{i1})\Delta y_{i3}\\\ (y_{i4}-y_{i1})\Delta
y_{i4}+(y_{i5}-y_{i1})\Delta y_{i3}\\\ (y_{i4}-y_{i2})\Delta
y_{i4}+(y_{i5}-y_{i2})\Delta y_{i3}\\\ \Delta y_{i2}\Delta y_{i4}\\\ \Delta
y_{i3}\Delta y_{i4}\end{array}\right),$

$d=\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}(y_{i4}-y_{i1})\Delta
y_{i3}\\\ (y_{i5}-y_{i1})\Delta y_{i4}\\\ (y_{i5}-y_{i2})\Delta y_{i4}\\\
\Delta y_{i2}\Delta y_{i5}\\\ \Delta y_{i3}\Delta y_{i5}\end{array}\right),$

and similar specifications of $a,$ $b$ and $d$ result for larger values of
$T.$

It is interesting to see that these robust moments only depend on differences
of the data so the initial observations get differenced out. This explains why
these moments are robust to the variance of the initial observations. When the
autoregressive parameter equals one and in the case of iid normal errors and
time series homoskedasticity, Ahn and Thomas (2006) and Kruiniger (2013) show
that the maximum likelihood estimator of Hsiao et al. (2002) and the random
effects estimator of Anderson and Hsiao (1982) have the same limiting
distributions. These results show that, similar to our findings, moment
conditions involving levels of the data are redundant in this setting and only
moment conditions using differences of the data, like our robust moment
conditions, are informative.

#### Large individual effect variance

So far we have focused on highly persistent panel data resulting from a large
autoregressive parameter. However, the representation theorem for the moment
conditions and their derivatives in Theorem 1 applies to any setting where the
variance of the initial observations gets large. The expression of the initial
observation in (7) shows that its variance becomes large when either the
variance of the initial disturbance term, $u_{i1},$ or the individual specific
effect, $\mu_{i},$ becomes large. Theorem 1 focusses on a large variance that
results from the autoregressive parameter converging to one. Theorem 1 does,
however, extend to the case where jointly with the sample size, the individual
specific effect variance becomes large in such a manner that Assumption 2a
holds. This drifting sequence applies to any value of the autoregressive
parameter so the resulting identification issues are then no longer confined
to the unit root value. Hence, they also apply to the cases with only moderate
autoregressive dynamics, but a large variance of the unobserved heterogeneity.
The robust moments in (38) also apply to this case. Kruiniger (2002)
extensively analyzes the setting of a large variance of the individual
specific effects. He shows that only moment conditions based on differences of
the data yield a consistent estimator so moment conditions involving levels
are redundant. He also constructs the set of optimal moment conditions
assuming time series homoskedasticity. Our robust moments (38) extend his set
of optimal moment conditions since they remain valid under a large variance of
the individual specific effect and also allow for time series
heteroskedasticity.

##  4 KLM test and robust sample moments

Theorem 1 establishes identification results for the AS and Sys moment
conditions, which are based on the robust sample moments. It is not clear,
however, how an identification robust test procedure makes use of it. In this
section, we show that the KLM test based on the original AS or Sys moment
conditions just uses the robust sample moments when only the latter contain
identifying information on the autoregressive parameter. We show that, under
large variances of the initial observation and when the true value of $\theta$
is close to one, the KLM test based on either the AS or Sys moment conditions
exploits the identifying information from the robust moment conditions in an
optimal manner. For practical purposes, this implies that we do not have to
explicitly use the robust sample moments since they are implicitly used when
conducting a KLM test using AS or Sys moment conditions.

We obtain the above result in four steps. First, we characterize the limit
behavior of the robust sample moments. Second, we use it to determine
asymptotic sequences for the true and hypothesized values so the power
properties of the corresponding identification robust test statistics when
using the robust moments are not trivial and stay informative. Third, we
construct the largest (infeasible) discriminatory power that can be obtained
from combining the robust moments. Fourth, we show that it coincides with the
rejection frequency of KLM tests using either AS or Sys moment conditions.
Summarizing, the KLM test based on original AS or Sys moment conditions
implicitly resorts to using the robust sample moments in an optimal manner
when only these contain information on $\theta.$

###  4.1 Large sample behavior of robust sample moments

To construct the limiting behavior of the robust sample moments for settings
where only they contain information on $\theta,$ we first state the
probability limits of the quantities $a,$ $b$ and $d$ in (38) under Assumption
1. The components that comprise the robust sample moments do not depend on the
variance of the initial observations so they are not affected by Assumption 2.
Since we analyze the behavior when the true value $\theta_{0,N}$ is converging
to one, we specify this convergence behavior of $\theta_{0,N}$ so it is
dominated by the random components present in the limit behavior of $a,$ $b$
and $d$ which are of order $O_{p}(N^{-\frac{1}{2}}).$ This then implies that
$\theta_{0,N}$ converges rather rapidly to one with a convergence rate that is
faster than $N^{-\frac{1}{2}}.$ Hence, $\theta_{0,N}$ is considered to be in
the close neighborhood of one.

#### Theorem 2.

Under Assumption 1, the limit behavior of the different components of
$g_{f,T}^{j}(\theta)$, $j=AS,$ $Sys,$ for $\theta_{0,N}=1+\frac{l}{N^{\tau}}$
with l a fixed constant, $l<0,$ and $\tau>\frac{1}{2},$ is characterized by:

#### T=4:

$a=\binom{\sigma_{2}^{2}}{0}+O_{p}(N^{-\frac{1}{2}}),$
$b=-\binom{\sigma_{2}^{2}+\sigma_{3}^{2}}{0}+O_{p}(N^{-\frac{1}{2}}),$
$d=\binom{\sigma_{3}^{2}}{0}+O_{p}(N^{-\frac{1}{2}}).$

#### T=5:

$\begin{array}[c]{c}a=\left(\begin{array}[c]{c}\sigma_{2}^{2}\\\
\sigma_{3}^{2}\\\ \sigma_{3}^{2}\\\ 0\\\
0\end{array}\right)+O_{p}(N^{-\frac{1}{2}}),\text{
}b=\left(\begin{array}[c]{c}\sigma_{2}^{2}+\sigma_{3}^{2}\\\
\sigma_{3}^{2}+\sigma_{4}^{2}\\\ \sigma_{3}^{2}+\sigma_{4}^{2}\\\ 0\\\
0\end{array}\right)+O_{p}(N^{-\frac{1}{2}}),\text{
}d=\left(\begin{array}[c]{c}\sigma_{3}^{2}\\\ \sigma_{4}^{2}\\\
\sigma_{4}^{2}\\\ 0\\\
0\end{array}\right)+O_{p}(N^{-\frac{1}{2}}).\end{array}$

Proof. see the Appendix.

Although AS and Sys robust moments are different, Theorem 2 implies that under
Assumption 1 the probability limits of $a$, $b$ and $d$ are identical.
Furthermore, Theorem 2 implies that the Jacobian of the robust moment equation
(38) is of full column rank when $\sigma_{t}^{2}\neq\sigma^{2}$ for at least
one value of $t=2,...,T$. This fulfills one of the sufficient conditions for
standard asymptotic theory for GMM inference based on the robust sample
moments, which since the other sufficient conditions can be shown to hold as
well, applies for these settings.

###  4.2 Asymptotic sequence for the hypothesized value

We want to compare tests of H${}_{0}:\theta=\theta^{\ast}$ using the robust
sample moments to KLM tests of H0 using the original AS and Sys moments for
settings where the identification can be problematic, which occurred for true
values of $\theta$ close to one and large variances of the initial
observations. Because we want to analyze local asymptotic power while the true
value $\theta_{0,N}$ is converging to one according to
$\theta_{0,N}=1+\frac{l}{N^{\tau}}$, we also consider a local to unity
drifting sequence for the hypothesized value $\theta^{\ast},$ which we denote
by $\theta(e)$ with $e<0$ the localizing parameter. Although less common in
asymptotic power analysis, the advantage of a drifting hypothesized value is
that our results hold for a range of hypothesized values.

The asymptotic sequence $\theta(e)$ is such that the behavior of the
identification robust tests is not diverging and informative about $\theta,$
when the true value $\theta_{0,N}$ is converging to one. Theorem 3 establishes
the particular rate at which $\theta(e)$ converges to one which makes these
conditions hold. Note that there is a slight abuse of notation as from now on
we suppress the superscript $j$ in $g_{f,T}^{j}(\theta(e))$, $j=AS,$ $Sys,$
which is inconsequential for the results to follow.

#### Theorem 3.

Under Assumption 1, $\theta_{0,N}=1+\frac{l}{N^{\tau}}$ with l a fixed
constant, $l<0,$ and $\tau>\frac{1}{2},$ the robust moments
$\sqrt{N}g_{f,T}(\theta(e))$ are informative about $\theta$ and converge to a
bounded in probability, non-degenerate random variable under the following
local to unity drifting sequence $\theta(e)$:

1\. $\theta(e)=1+\frac{e}{\sqrt[4]{N}}$ in the case of
$\sigma_{t}^{2}=\sigma^{2},$ $t=2,\ldots T,$

2. $\theta(e)=1+\frac{e}{\sqrt{N}}$ when $\sigma_{t}^{2}\neq\sigma^{2},$ for at least one value of $t,$ $t=2,\ldots T-1,$

with $e<0$ a finite constant.

Proof. see the Appendix.

The quartic root convergence rate in Theorem 3.1 results since the Jacobian of
the robust moment equation (38) is then equal to zero but the Hessian is not.
It is thus a setting of so-called second order identification with first order
underidentification. Estimators then generally have quartic root convergence
rates, see $e.g.$ Dovonon and Renault (2013), Dovonon and Hall (2018) and
Dovonon $et$ $al.$ (2020). A quartic root convergence rate for estimators in
dynamic panel data models is also found by Ahn and Thomas (2006) and Kruiniger
(2013).

The quartic root convergence rate for the robust sample moments results from
specifying $\theta(e)=1+\frac{e}{N^{1/4}}$ and $\sigma_{t}^{2}=\sigma^{2},$
$t=2,\ldots T.$ All elements of the robust sample moments which are linear in
$e$ then cancel out in the limit. We are then left with a quadratic term in
$e$ and components that converge at the rate $\frac{1}{\sqrt{N}}$. A quartic
root convergence rate makes all these components of the same order of
magnitude. Theorem 3 shows that error variances which are constant over time,
$\sigma_{t}^{2}=\sigma^{2},$ $t=2,\ldots T,$ lead to this slow convergence
rate.

###  4.3 Largest rejection frequencies of robust sample moments

To show that the KLM test of H0 using AS and Sys moment conditions just uses
the robust sample moments when only these contain information on $\theta,$ we
use the largest rejection frequencies that result in such instances from the
robust sample moments. To obtain these largest rejection frequencies, we first
consider the GMM-AR test of H${}_{p}:\theta(e)=1+\frac{e}{\sqrt[4]{N}}$ using
the robust sample moments, which is specified as:

$\begin{array}[c]{c}\text{GMM-
AR(}\theta(e))=Ng_{f,T}(\theta(e))^{\prime}\hat{V}_{gg}(\theta(e))^{-1}g_{f,T}(\theta(e)),\end{array}$
(39)

with $g_{f,T}(\theta(e))$ the moments in (38) evaluated at
$\theta(e)=1+\frac{e}{\sqrt[4]{N}}$ and $\hat{V}_{gg}(\theta(e))$ the (Eicker-
White) covariance matrix estimator of the covariance matrix of
$g_{f,T}(\theta(e)).$ For $T=4$ and 5:777We thank an anonymous referee for
showing this.

$\begin{array}[c]{ccl}\mathbf{T=4:}&g_{f,T=4}^{AS}(\theta(e))=&\left(\begin{array}[c]{lc}1&-\theta(e)\\\
0&1\end{array}\right)g_{f,T=4}^{Sys}(\theta(e))\\\
\mathbf{T=5:}&g_{f,T=5}^{AS}(\theta(e))=&\left(\begin{array}[c]{ccccc}1&-\theta(e)/(1-\theta(e))&\theta(e)/(1-\theta(e))&0&0\\\
0&1&0&0&-\theta(e)\\\ 0&0&1&0&-\theta(e)\\\ 0&0&0&1&0\\\
0&0&0&0&1\end{array}\right)g_{f,T=5}^{Sys}(\theta(e))\end{array}$

so GMM-AR($\theta(e))$ is equivalent for the AS and Sys moment conditions
since the invertible matrix by which $g_{f,T}^{Sys}(\theta(e))$ has to be pre-
multiplied to obtain $g_{f,T}^{AS}(\theta(e))$ cancels out in GMM-
AR($\theta(e)).$ This result can be extended to larger values of $T.$

#### Theorem 4.

Under Assumption 1, $\theta_{0,N}=1+\frac{l}{N^{\tau}}$ with l a fixed
constant, $l<0,$ and $\tau>\frac{1}{2},$ $\sigma_{t}^{2}=\sigma^{2},$
$t=2,\ldots T,$ the large sample distribution of the GMM-AR statistic (39) for
testing H${}_{p}:\theta(e)=1+\frac{e}{\sqrt[4]{N}},$ in a sample of size $N$
is characterized by

$\begin{array}[c]{c}\chi^{2}(\delta(N),p_{\max}),\end{array}$ (40)

with
$\delta(N)=(e\sigma)^{4}\binom{\iota_{p}}{0}^{\prime}(B(N)^{\prime}V_{abd}B(N))^{-1}\binom{\iota_{p}}{0},$
$p$ the number of columns G${}_{f,T}(\theta),$ so when $T=4,$ $p=1$ and when
$T=5,$ $p=3,$ and $p_{\max}$ the number of elements of $g_{f,T}(\theta(e))$,
so, when $T=4,$ $p_{\max}=2,$ while $p_{\max}=5$ for $T=5,$

$\begin{array}[c]{c}B(N)=(\iota_{3}\otimes
I_{p_{\max}})+\frac{e}{\sqrt[4]{N}}\left[(2+\frac{e}{\sqrt[4]{N}})(e_{1,3}\otimes
I_{p_{\max}})+(e_{2,3}\otimes I_{p_{\max}})\right],\end{array}$ (41)

$V_{abd}$ the covariance matrix of $a,$ $b$ and $d$, $I_{p_{\max}}$ the
$p_{\max}\times p_{\max}$ dimensional identity matrix, $e_{1,3}$ and $e_{2,3}$
the first and second $3\times 1$ dimensional unity vectors and
$\chi^{2}(\delta,p_{\max})$ a non-central $\chi^{2}$ distribution with non-
centrality parameter $\delta$ and $p_{\max}$ degrees of freedom.

Proof. see the Appendix.

The expression of the large sample distribution in Theorem 4 depends on the
sample size. Given the quartic root convergence rate, convergence to the
limiting distribution is very slow so it is important for the accuracy of the
approximation of the finite sample distribution to incorporate higher order
components. The proof of Theorem 4 in the Appendix therefore from the outset
considers all higher order components of $g_{f,T}(\theta(e))$ in order to
construct a large sample approximation of the distribution of GMM-
AR($\theta(e)).$

To obtain the maximal rejection frequencies using the robust sample moments,
we use a (infeasible) weighted average of the moment equations in
$g_{f,T}(\theta(e))$ where the weights are chosen such that the non-centrality
parameter equals the one of the non-central $\chi^{2}$ limiting distribution
of the GMM-AR statistic while the degrees of freedom is equal to one ($i.e.$
the number of elements of $\theta$). This value of the non-centrality
parameter is also the maximal one that can be obtained using a weighted
average of the robust sample moments.

#### Theorem 5.

Under Assumption 1, $\theta_{0,N}=1+\frac{l}{N^{\tau}}$ with l a fixed
constant, $l<0,$ and $\tau>\frac{1}{2},$ $\sigma_{t}^{2}=\sigma^{2},$
$t=2,\ldots T,$ an optimal (infeasible) GMM-AR test of
H${}_{p}:\theta(e)=1+\frac{e}{\sqrt[4]{N}}$ that uses a weighted average of
the robust sample moments can be constructed that has approximately a

$\begin{array}[c]{c}\chi^{2}(\delta(N),1),\end{array}$ (42)

distribution in large samples of size $N$.

Proof. see the Appendix.

The GMM-AR statistics in Theorems 4 and 5 both have non-central $\chi^{2}$
distributions with the same non-centrality parameter so the one with the
smallest number of degrees of freedom, i.e. the statistic in Theorem 5, has
the largest power.

Figure 4 illustrates Theorem 5 and shows the maximal rejection frequencies
based on combining the robust sample moments based on either AS or Sys moment
condition in a GMM-AR test888We use the covariance matrix estimator for each
simulated data set to compute the GMM-AR statistics. for $T=4$ and $5.$ It
uses DGP 1 from Section 3 with a true value of $\theta$ which is very close to
one (0.99) and a large value of $\sigma_{c}^{2}$ (ten) compared to
$\sigma^{2}$ (one), which amplifies the variance of the initial conditions.
The DGP thus satisfies mean stationarity (7)-(9) and also time series
homoskedasticity, $i.e.$ $\sigma_{t}^{2}=\sigma^{2}$ for $t=2,\ldots,T$. We
use $N=2000$, a relatively large value and test for a wide range of values for
$\theta$, which together with $N$ provides a mapping to the constant $e$
$(=\sqrt[4]{N}(\theta-1))$ in Figure 4 (horizontal axis). The usual power
curve, as shown earlier in the Figures in Panels 1 and 2, reports the
rejection frequencies of tests of the hypothesized parameter value as a
function of the parameter value used in the DGP where the data is simulated
from. Figure 4, however, reports for a fixed parameter value equal to one in
the DGP used to simulate the data, the rejection frequencies as a function of
a varying localizing parameter $e$ and, hence, autoregressive parameter
$\theta(e),$ under the tested null hypothesis. The rejection frequencies in
Figure 4 thus report those observed at one for a range of the usual power
curves where the tested parameter values correspond with those on the
horizontal axes in Figure 4.

Because of the equivalence of the GMM-AR test for the AS and Sys robust
moments, the rejection frequencies are identical for the AS and Sys based
robust sample moments and only differ over $T.$ Any remaining differences in
Figure 4 are due to sampling noise.

$\begin{array}[c]{c}\text{Figure 4. Rejection frequencies of GMM-AR tests of
H}_{p}:\theta(e)=1+\frac{e}{\sqrt[4]{N}}\\\ \text{using weighted robust sample
moments}\\\
\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/powerenvelopesysast45.eps}}\\\
\lx@intercol\text{Note: 5\% significance level, true value of }\theta\text{\
is 0.99, }N=2000\text{, Sys \& }T=4\text{ (dashed), AS \&}\hfil\lx@intercol\\\
\lx@intercol\text{ }T=4\text{ (dotted), Sys \& }T=5\text{ (solid), AS \&
}T=5\text{ (dash-dotted).}\hfil\lx@intercol\end{array}$

###  4.4 Large sample behavior of the KLM test

Finally, we construct the large sample distribution of KLM tests of
H${}_{p}:\theta(e)=1+\frac{e}{\sqrt[4]{N}}$ using AS and Sys moment conditions
when $\theta_{0,N}$ accords with the drifting sequences in Assumptions 1 and
2a so only the robust sample moments contain information on $\theta.$

#### Theorem 6.

Under Assumptions 1 and 2a, $\theta_{0,N}=1+\frac{l}{N^{\tau}}$ with l a fixed
constant, $l<0,$ and $\tau>\frac{1}{2},$ $\sigma_{t}^{2}=\sigma^{2},\mathit{\
}t=2,\ldots T,$ the large sample distribution of the KLM statistic using the
AS or Sys moments for testing the hypothesis
H${}_{p}:\theta(e)=1+\frac{e}{\sqrt[4]{N}}$ is characterized by

$\begin{array}[c]{rl}\text{KLM(}\theta(e))\sim&\chi^{2}(\delta(N),1)\end{array}$
(43)

with $\delta(N)$ defined in Theorem 4.

Proof. see the Appendix.

Under Assumptions 1 and 2a, Theorem 1 implies that the GMM sample moments
diverge in one direction and converge in another one. Identical to tests for
cointegration, Theorem 6 shows that the diverging parts of the GMM sample
moments cancel out in the large sample distribution of the KLM test so it only
contains elements from the converging part of the GMM sample moments. The
proof of the large sample distribution of the KLM test is therefore rather
elaborate since this has to be shown for each of the different components of
the KLM test.

Theorem 6 shows that the large sample distribution of the KLM test using AS or
Sys moment conditions when only the robust sample moments contain information
on $\theta$ is identical to the limiting distribution of the GMM-AR test that
optimally combines the robust sample moments for these settings. It proves
that KLM tests using the AS and Sys moment conditions then only use the robust
sample moments. It is similar to what happens in cointegration where since the
cointegrating vector and stochastic trends operate orthogonally, a likelihood
ratio test on the cointegration vector also does not depend on the stochastic
trends, see e.g. Johansen (1991).

Theorem 6 is illustrated by the Figures in Panel 5, which show the rejection
frequencies of 5% significance tests using a KLM test of
H${}_{p}:\theta(e)=1+\frac{e}{\sqrt[4]{N}}$ with AS and Sys moment conditions
when $T$ equals four, Figure 5.1, and five, Figure 5.2, respectively. It uses
the same DGP as for Figure 4. Also identical to Figure 4, the rejection
frequencies in Panel 5 report the rejection frequencies when using a fixed
parameter value in the DGP where we simulate the data from, as a function of a
varying parameter value under the tested hypothesis.

Panel 5 shows, for both $T=4$ and $T=5$, that the rejection frequencies that
result from using the KLM test with either AS or Sys moment conditions are
equal to the largest rejection frequencies, that can be obtained with the
robust moments when only they contain information on $\theta$. It illustrates
that the robust sample moments are (implicitly) used when you conduct KLM
tests with AS or Sys moment conditions. Hence, in practice one can just use AS
or Sys moment conditions in the construction of the KLM test, i.e. there is no
need to switch to the robust sample moments.

Panel 5 also provides a visual proof of stylized fact 5 from Section 3, i.e.
rejection frequencies for the KLM test using AS or Sys moment conditions are
almost identical when the true value of $\theta$ is close to one and for large
variances of the initial observations, and that it is not specific for the
tested values used there but holds generally for different tested values of
$\theta.$

$\begin{array}[c]{c}\text{Panel 5: Rejection frequencies of KLM tests of
H}_{p}:\theta(e)=1+\frac{e}{\sqrt[4]{N}}\text{ using AS (dashed) }\\\
\text{and Sys (dash-dotted) and GMM-AR tests using (infeasible) optimal
weighted average }\\\ \text{of robust sample moments (solid line)}\\\ \\\
\begin{array}[c]{ccc}\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/powert4klm.eps}}&&\raisebox{0.0pt}{\includegraphics[height=159.99838pt,width=212.37251pt]{graphics/powert5klm.eps}}\\\
\text{Figure 5.1: }T=4&&\text{Figure 5.2: }T=5\end{array}\\\ \\\
\lx@intercol\text{Note: 5\% significance level, true value of }\theta\text{\
is 0.99, N=2000.}\hfil\lx@intercol\end{array}$

##  5 Conclusions

We have analyzed GMM inference for dynamic panel data models involving highly
persistent panel data. We show that the Dif, Lev and NL moment conditions
separately do not identify the parameters in dynamic panel data models for a
general number of time periods. This results from the divergence of the
initial observations for some plausible data generating processes involving
highly persistent panel data. When there are more than three time periods, the
AS and Sys moment conditions, however, do lead to identification. The
identification based on the AS and Sys moment conditions for the problematic
cases of divergent initial observations results from so-called robust sample
moments. They are combinations of either the AS or Sys sample moments and do
not depend on the initial observations.

Despite the positive identification results for AS and Sys moment conditions,
conventional inference based on two step GMM estimators is not valid since
these estimators have non-standard limiting distributions near the unit root.
Similar results hold for two step GMM estimators based on our robust sample
moments. We have therefore analyzed the large sample properties of
identification robust GMM test procedures. These test statistics are size
correct, easy to implement and have been used in a variety of models analyzed
using GMM. We show that the identification robust KLM statistic based on the
AS and Sys sample moments implicitly resorts to using the robust sample
moments when only the latter contain identifying information.

Based on the theoretical analysis and numerical results a number of remarks
can be made regarding the implementation of GMM inference for applied linear
dynamic panel data analysis. First, statistical inference, i.e. hypothesis
testing and confidence intervals, should be based on identification robust
tests, like, for example, the KLM or GMM-AR test. The non-standard limiting
behavior of the two step GMM coefficient estimator makes the use of
conventional GMM inference hazardous in applied research when there are
identification issues. Second, one should always use either AS or Sys moment
conditions since these deliver identification under more general conditions
when $T>3.$ An advantage of the AS moments is that they are valid under less
restrictive assumptions than the Sys moments. Third, when mean stationarity
applies, the Sys moments are preferred. Although AS and Sys moments contain
the same amount of identifying information when $\theta$ is close to one and
the variance of the initial observations is large, in practice the opposite
may well be the case if one is not close to the unit root (or if time series
heteroskedasticity is present). This is shown, for example, by our simulated
KLM power curves in Section 2. Fourth, the original AS or Sys moments should
be used in an identification robust GMM test statistic and not the implied
robust sample moments. Although only the latter preserve identification when
the variance of the initial observations is large, we have shown that the
identification robust KLM test based on the AS or Sys moments implicitly uses
the robust sample moments.

Finally, for expository purposes we have only analyzed the first-order
autoregressive panel data model. The extension to panel data models with
multiple endogenous regressors, e.g. dynamic models with additional endogenous
regressors, is an important area for future research.

Appendix. Specification of GMM sample moments and proofs

#### Specification of sample moment functions

For the Dif moment conditions in (4), $k_{Dif}$ equals $\frac{1}{2}(T-2)(T-1)$
while $f_{i}^{Dif}(\theta)$ and $q_{i}^{Dif}(\theta)$ read

$\begin{array}[c]{rl}f_{i}^{Dif}(\theta)=&Z_{i}^{Dif}\varphi_{i}^{Dif}(\theta)\\\
q_{i}^{Dif}(\theta)=&-Z_{i}^{Dif}\Delta y_{-1,i},\end{array}$

with $\varphi_{i}^{Dif}(\theta)=(\Delta y_{i3}-\theta\Delta y_{i2}\ldots\Delta
y_{iT}-\theta\Delta y_{iT-1})^{\prime},$ $\Delta y_{-1,i}=(\Delta
y_{i2}\ldots\Delta y_{iT-1})^{\prime}$ and

$\begin{array}[c]{c}Z_{i}^{Dif}=\left(\begin{array}[c]{ccc}y_{i1}&0\ldots
0&0\\\ 0&\ddots&0\\\ 0&0\ldots 0&\left(\begin{array}[c]{c}y_{i1}\\\ \vdots\\\
y_{iT-2}\end{array}\right)\end{array}\right):\frac{1}{2}(T-1)(T-2)\times(T-2).\end{array}$

For the Lev moment conditions in (5), $k_{Lev}$ equals $T-2$ while the sample
moment functions are

$\begin{array}[c]{rl}f_{i}^{Lev}(\theta)=&Z_{i}^{Lev}\varphi_{i}^{Lev}(\theta)\\\
q_{i}^{Lev}(\theta)=&-Z_{i}^{Lev}y_{-1,i},\end{array}$

with $\varphi_{i}^{Lev}(\theta)=(y_{i3}-\theta y_{i2}\ldots y_{iT}-\theta
y_{iT-1})^{\prime},$ $y_{-1,i}=(y_{i2}\ldots y_{iT-1})^{\prime},$ and

$\begin{array}[c]{c}Z_{i}^{Lev}=\left(\begin{array}[c]{ccc}\Delta
y_{i2}&0\ldots 0&0\\\ 0&\ddots&0\\\ 0&0\ldots 0&\Delta
y_{iT-1}\end{array}\right):(T-2)\times(T-2).\end{array}$

For the NL moment conditions in (10), $k_{NL}$ equals $T-3$ while the sample
moment functions can be specified as

$\begin{array}[c]{rl}f_{i}^{NL}(\theta)=&Z_{i}^{NL}(\theta)\varphi_{i}^{NL}(\theta)\\\
q_{i}^{NL}(\theta)=&\left(\frac{\partial}{\partial\theta}Z_{i}^{NL}(\theta)\right)\varphi_{i}^{NL}(\theta)+Z_{i}^{NL}(\theta)\left(\frac{\partial}{\partial\theta}\varphi_{i}^{NL}(\theta)\right),\end{array}$

with $\varphi_{i}^{NL}(\theta)=((y_{i4}-\theta y_{i3})\ldots(y_{iT}-\theta
y_{iT-1}))^{\prime}$ and

$\begin{array}[c]{c}Z_{i}^{NL}(\theta)=\left(\begin{array}[c]{ccc}(\Delta
y_{i3}-\theta\Delta y_{i2})&0\ldots 0&0\\\ 0&\ddots&0\\\ 0&0\ldots 0&(\Delta
y_{iT-1}-\theta\Delta
y_{iT-2})\end{array}\right):(T-3)\times(T-3).\end{array}$

The sample moments for the AS moment conditions result by just stacking the
appropriate sample moments stated above so $k_{AS}$ equals
$\frac{1}{2}(T-1)(T-2)+T-3$. In a similar manner, the Sys sample moments
result so $k_{Sys}$ equals $\frac{1}{2}(T+1)(T-2)$.

#### Lemma 1.

We state some intermediate results, which involve the different terms in the
sample moments and their derivatives. Assumption 1 implies the following:

$\begin{array}[c]{rrll}\mathbf{i.}&\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)y_{i1}u_{i1}=&-d_{2}-\frac{\sigma_{1,N}}{\sqrt{N}}\psi_{c}+o_{p}(1),&\\\
\mathbf{ii.}&\frac{1}{N}\sum_{i=1}^{N}(1-\theta_{0,N})u_{i1}u_{it}\underset{p}{\rightarrow}&0,&t>1,\\\
\mathbf{iii.}&\frac{1}{N}\sum_{i=1}^{N}u_{it}^{2}\underset{p}{\rightarrow}&\sigma_{t}^{2},&t>1,\\\
\mathbf{iv.}&\frac{1}{N}\sum_{i=1}^{N}\Delta y_{it}\Delta
y_{it}\underset{p}{\rightarrow}&\sigma_{t}^{2},&t>1,\\\
\mathbf{v.}&\frac{1}{N}\sum_{i=1}^{N}\Delta y_{it}\Delta
y_{is}\underset{p}{\rightarrow}&0,&t,s>1,t\neq s.\\\
\mathbf{vi.}&\frac{h_{N}(\theta_{0,N})}{\sqrt{N}}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i1}u_{i2}\\\
\vdots\\\
y_{i1}u_{iT}\end{array}\right)\underset{d}{\rightarrow}&\psi,&\end{array}$

with $\psi=(\psi_{2}\ldots\psi_{T})^{\prime}\sim
N(0,$diag($\sigma_{2}^{2},\ldots,\sigma_{T}^{2}))$ independent from
$\psi_{c}\sim N(0,\sigma_{c}^{2}),$ $\sigma_{c}^{2}=$var($c_{i}$).

#### Proof of Lemma 1.

i. Under mean stationarity, we have:

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)y_{i1}u_{i1}=\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)u_{i1}^{2}+\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)u_{i1}\mu_{i}.\end{array}$

Assumption 1c implies that $(1-\theta_{0,N})^{\frac{1}{2}}u_{i1}$ is a random
variable with finite fourth moments so a law of large numbers applies:

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)u_{i1}^{2}\underset{p}{\rightarrow}-d_{2}.\end{array}$

Since $c_{i}=(1-\theta_{0,N})\mu_{i},$ we can specify:

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)u_{i1}\mu_{i}=-\frac{1}{N}\sum_{i=1}^{N}u_{i1}c_{i}=-\frac{\sigma_{1,N}}{\sqrt{N}}\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\frac{u_{i1}}{\sigma_{1,N}}c_{i},\end{array}$

because

$\begin{array}[c]{c}\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\frac{u_{i1}}{\sigma_{1,N}}c_{i}\underset{d}{\rightarrow}\psi_{c},\end{array}$

with $\psi_{c}$ independent of $\psi_{j},$ $j=2,\ldots,T,$ as $c_{i}$ is
independent from $u_{ij},$ $j=2,,\ldots,T.$ Upon combining, we obtain:

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)y_{i1}\mu_{i}=-d_{2}-\frac{\sigma_{1,N}}{\sqrt{N}}\psi_{c}+o_{p}(1).\end{array}$

ii. Since $u_{it}$ are independently distributed, $t=1,\ldots,T,$ and
$(1-\theta_{0,N})^{\frac{1}{2}}u_{i1}$ is a random variable with finite fourth
moments, a law of large numbers applies:

$\begin{array}[c]{rll}\frac{1}{N}\sum_{i=1}^{N}(1-\theta_{0,N})u_{i1}u_{it}\underset{p}{\rightarrow}&0,&t>1.\end{array}$

iii. Finite fourth moments of $u_{it}$ implies that a law of large numbers
applies:

$\begin{array}[c]{rll}\frac{1}{N}\sum_{i=1}^{N}u_{it}^{2}\underset{p}{\rightarrow}&\sigma_{t}^{2},&t>1.\end{array}$

iv. Mean stationarity implies $\Delta
y_{i2}=u_{i2}+\left(\theta_{0,N}-1\right)u_{i1},$ so

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}\Delta y_{i2}\Delta
y_{i2}=\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2}+\left(\theta_{0,N}-1\right)\frac{1}{N}\sum_{i=1}^{N}\left(\theta_{0,N}-1\right)u_{i1}^{2}+\frac{2}{N}\sum_{i=1}^{N}\left(\theta_{0,N}-1\right)u_{i1}u_{i2}.\end{array}$

Because
$\frac{1}{N}\sum_{i=1}^{N}(1-\theta_{0,N})u_{i1}^{2}\underset{p}{\rightarrow}d_{2}$
and $(1-\theta_{0,N})\underset{N\rightarrow\infty}{\rightarrow}0$, we have

$\begin{array}[c]{c}\left(\theta_{0,N}-1\right)\frac{1}{N}\sum_{i=1}^{N}\left(\theta_{0,N}-1\right)u_{i1}^{2}\underset{p}{\rightarrow}0,\end{array}$

which shows that
$\left(\theta_{0,N}-1\right)\frac{1}{N}\sum_{i=1}^{N}\left(\theta_{0,N}-1\right)u_{i1}^{2}=o_{p}(1)$.
Furthermore, since both $\left(\theta_{0,N}-1\right)^{\frac{1}{2}}u_{i1}$ and
$u_{i2}$ have finite fourth moments and are independent,
$\frac{2}{N}\sum_{i=1}^{N}\left(\theta_{0,N}-1\right)u_{i1}u_{i2}=o_{p}(1)$,
which implies that

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}\Delta y_{i2}\Delta
y_{i2}=\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2}+o_{p}(1).\end{array}$

Finally, we have $E\left(u_{i2}^{2}\right)=\sigma_{2}^{2}$ and finite fourth
moment, hence

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}\Delta
y_{i2}^{2}\underset{p}{\rightarrow}\sigma_{2}^{2}.\end{array}$

Along the same lines as the above this can be shown to hold for other values
of $t$ as well.

v. Similar to the above, when substituting for $\Delta y_{i2}$ and $\Delta
y_{i3}$ we have

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}\Delta y_{i2}\Delta
y_{i3}=\frac{1}{N}\sum_{i=1}^{N}u_{i2}u_{i3}+\left(\theta_{0,N}-1\right)\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2}+\theta_{0,N}\left(\theta_{0,N}-1\right)\frac{1}{N}\sum_{i=1}^{N}u_{i1}u_{i2}+\\\
+(\theta_{0,N}-1)\frac{1}{N}\sum_{i=1}^{N}u_{i1}u_{i3}+(\theta_{0,N}-1)^{2}\frac{1}{N}\sum_{i=1}^{N}u_{i1}u_{i2}+\theta_{0,N}(\theta_{0,N}-1)^{2}\frac{1}{N}\sum_{i=1}^{N}u_{i1}^{2}.\end{array}$

Similar derivations as before show that
$\frac{1}{N}\sum_{i=1}^{N}\theta_{0,N}\left(\theta_{0,N}-1\right)^{2}u_{i1}^{2}\underset{p}{\rightarrow}0,$
$\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)^{2}u_{i1}u_{i2}\underset{p}{\rightarrow}0,$
$\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)u_{i1}u_{i3}\underset{p}{\rightarrow}0,$
$\frac{1}{N}\sum_{i=1}^{N}\theta_{0,N}\left(\theta_{0,N}-1\right)u_{i1}u_{i2}\underset{p}{\rightarrow}0,$
$\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)u_{i2}^{2}\underset{p}{\rightarrow}0$,
$\frac{1}{N}\sum_{i=1}^{N}u_{i2}u_{i3}\underset{p}{\rightarrow}0$, so all
these terms are $o_{p}(1)$ and have probability limit 0, implying that

$\begin{array}[c]{rl}\frac{1}{N}\sum_{i=1}^{N}\Delta y_{i2}\Delta
y_{i3}\underset{p}{\rightarrow}&0.\end{array}$

Along similar lines this can be proven to extend to the first differences at
other time periods.

vi. Since $h_{N}(\theta_{0,N})^{-2}=var(y_{i1}),$ the random variable
$h_{N}(\theta_{0,N})y_{i1}$ has variance equal to one. Since $y_{i1}$ and
$u_{it},$ $t>1,$ are independent, because of Assumption 1e,
$E(h_{N}(\theta_{0,N})y_{i1}u_{it})=0$. Furthermore, Assumption 1d implies
that $Var\left(h_{N}(\theta_{0,N})y_{i1}u_{it}\right)=\sigma_{t}^{2}$, which
is finite. A central limit theorem therefore applies:

$\begin{array}[c]{rl}\frac{h_{N}(\theta_{0,N})}{\sqrt{N}}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i1}u_{i2}\\\
\vdots\\\
y_{i1}u_{iT}\end{array}\right)=\frac{1}{\sqrt{N}}\sum_{i=1}^{N}\left(\begin{array}[c]{c}h_{N}(\theta_{0,N})y_{i1}u_{i2}\\\
\vdots\\\
h_{N}(\theta_{0,N})y_{i1}u_{iT}\end{array}\right)\underset{d}{\rightarrow}&\psi,\end{array}$

with $\psi=(\psi_{y_{1i}u_{i2}}\ldots\psi_{y_{1i}u_{iT}})^{\prime}$ a $T-1$
dimensional, mean zero normal random vector. Assumption 1e states that
$u_{i1}/\sigma_{1,N},$ $u_{i2}\ldots,u_{iT}$ and $c_{i}$ are independently
distributed within individuals and over the different individuals. It implies
that $u_{i1}c_{i}$ and $y_{i1}u_{it}$ are uncorrelated. Since $\psi$ and
$\psi_{c}$ are the limits of the scaled sums of $y_{i1}u_{it}$ and
$u_{i1}c_{i}$, they are uncorrelated normal random variables and therefore
independent. As a result of this, the $T\times T$ covariance matrix of $\psi$
and $\psi_{c}$ is diagonal:

$\begin{array}[c]{rl}V_{\binom{\psi}{\psi_{c}}\binom{\psi}{\psi_{c}}}=&var(\psi,\psi_{c})=\left(\begin{array}[c]{cc}V_{\psi\psi}&V_{\psi\psi_{c}}\\\
V_{\psi_{c}\psi}&V_{\psi_{c}\psi_{c}}\end{array}\right)\\\
=&E\left[\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}h_{N}(\theta_{0,N})y_{i1}u_{i2}\\\
\vdots\\\ h_{N}(\theta_{0,N})y_{i1}u_{iT}\\\
\frac{u_{i1}}{\sigma_{1,N}}c_{i}\end{array}\right)\left(\begin{array}[c]{c}h_{N}(\theta_{0,N})y_{i1}u_{i2}\\\
\vdots\\\ h_{N}(\theta_{0,N})y_{i1}u_{iT}\\\
\frac{u_{i1}}{\sigma_{1,N}}c_{i}\end{array}\right)^{\prime}\right]\\\
=&E\left[\lim_{N\rightarrow\infty}\left(\begin{array}[c]{c}h_{N}(\theta_{0,N})y_{i1}u_{i2}\\\
\vdots\\\ h_{N}(\theta_{0,N})y_{i1}u_{iT}\\\
\frac{u_{i1}}{\sigma_{1,N}}c_{i}\end{array}\right)\left(\begin{array}[c]{c}h_{N}(\theta_{0,N})y_{i1}u_{i2}\\\
\vdots\\\ h_{N}(\theta_{0,N})y_{i1}u_{iT}\\\
\frac{u_{i1}}{\sigma_{1,N}}c_{i}\end{array}\right)^{\prime}\right]\\\
=&\text{diag(}\sigma_{2}^{2}\ldots\sigma_{T}^{2}\text{
}\sigma_{c}^{2}).\end{array}$

#### Proof of Theorem 1.

T=3. Under mean stationarity we have

$\begin{array}[c]{l}\Delta y_{i2}=u_{i2}+\left(\theta_{0,N}-1\right)u_{i1}\\\
\Delta
y_{i3}=u_{i3}+\left(\theta_{0,N}-1\right)u_{i2}+\theta_{0,N}\left(\theta_{0,N}-1\right)u_{i1}.\end{array}$

Substituting these expressions, we can specify the Dif sample moment and its
derivative as

$\begin{array}[c]{rl}f_{N}^{Dif}(\theta)=&\frac{1}{N}\sum_{i=1}^{N}\left(y_{i1}\Delta
y_{i3}-\theta y_{i1}\Delta y_{i2}\right)\\\
=&\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i3}+\left(\theta_{0,N}-1-\theta\right)\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i2}+(\theta_{0,N}-\theta)\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)y_{i1}u_{i1},\\\
q_{N}^{Dif}(\theta)=&-\frac{1}{N}\sum_{i=1}^{N}y_{i1}\Delta y_{i2}\\\
=&-\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i2}-\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)y_{i1}u_{i1}.\end{array}$

Combining convergence results stated in Lemma 1, the large sample behavior of
the Dif sample moment and derivative can thus be characterized by

$\begin{array}[c]{l}f_{N}^{Dif}(\theta)=\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left[\left(\psi_{3}-\theta\psi_{2}\right)-(1-\theta)h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\right]-\left(1-\theta\right)d_{2}+o_{p}\left(1\right),\\\
q_{N}^{Dif}(\theta)=-\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left[\psi_{2}-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\right]+d_{2}+o_{p}\left(1\right),\end{array}$

where we note that $h_{N}(\theta_{0,N})\sigma_{1,N}\leq 1,$ since
var($y_{i1})\geq$var($u_{i1}),$ from which it is readily seen that

$\begin{array}[c]{l}A_{f}^{Dif}(\theta)=\left(\begin{array}[c]{cc}-\theta&1\end{array}\right),\text{
}\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})=0,\\\
A_{q}^{Dif}(\theta)=\left(\begin{array}[c]{cc}-1&0\end{array}\right),\text{
}\mu_{q}^{Dif}(\theta,\bar{\sigma}^{2})=0.\end{array}$

Regarding the Lev moment, using

$\begin{array}[c]{l}y_{i2}=\Delta y_{i2}+y_{i1}\\\ y_{i3}=\Delta y_{i3}+\Delta
y_{i2}+y_{i1},\end{array}$

we have

$\begin{array}[c]{rl}f_{N}^{Lev}(\theta)=&\frac{1}{N}\sum_{i=1}^{N}(y_{i3}-\theta
y_{i2})\Delta y_{i2}\\\ =&\frac{1}{N}\sum_{i=1}^{N}\left(\Delta
y_{i3}+(1-\theta)\Delta y_{i2}\right)\Delta
y_{i2}+(1-\theta)\frac{1}{N}\sum_{i=1}^{N}y_{i1}\Delta y_{i2}.\end{array}$

Exploiting mean stationarity and substituting for $\Delta y_{i2}$ and $\Delta
y_{i3},$ we write

$\begin{array}[c]{c}(1-\theta)\frac{1}{N}\sum_{i=1}^{N}y_{i1}\Delta
y_{i2}=(1-\theta)\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i2}+(1-\theta)\left(\theta_{0,N}-1\right)\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i1},\end{array}$

and using Lemma 1, we have

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}\left(\Delta
y_{i3}+(1-\theta)\Delta y_{i2}\right)\Delta
y_{i2}=\left(1-\theta\right)\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2}+o_{p}\left(1\right).\end{array}$

Regarding the Lev derivative, we have

$\begin{array}[c]{rl}q_{N}^{Lev}(\theta)=&-\frac{1}{N}\sum_{i=1}^{N}y_{i2}\Delta
y_{i2}\\\ =&-\frac{1}{N}\sum_{i=1}^{N}\Delta y_{i2}\Delta
y_{i2}-\frac{1}{N}\sum_{i=1}^{N}y_{i1}\Delta y_{i2},\end{array}$

where

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}y_{i1}\Delta
y_{i2}=\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i2}+\left(\theta_{0,N}-1\right)\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i1},\end{array}$

and

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}\Delta y_{i2}\Delta
y_{i2}=\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2}+o_{p}\left(1\right).\end{array}$

Therefore, we can write the Lev moment condition and derivative as

$\begin{array}[c]{rl}f_{N}^{Lev}(\theta)=&(1-\theta)\left\\{\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2}+\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i2}+\right.\\\
&\left.\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)y_{i1}u_{i1}\right\\}+o_{p}(1).\\\
q_{N}^{Lev}(\theta)=&-\frac{1}{N}\sum_{i=1}^{N}u_{i2}^{2}-\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i2}\\\
&-\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)y_{i1}u_{i1}+o_{p}(1).\end{array}$

Combining this and other convergence results from Lemma 1, the large sample
behavior of the Lev sample moment and derivative can thus be characterized by

$\begin{array}[c]{rl}f_{N}^{Lev}(\theta)=&(1-\theta)\left\\{\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left[\psi_{2}-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\right]+\left(\sigma_{2}^{2}-d_{2}\right)\right\\}+o_{p}\left(1\right)\\\
q_{N}^{Lev}(\theta)=&-\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left[\psi_{2}-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\right]-\left(\sigma_{2}^{2}-d_{2}\right)+o_{p}\left(1\right),\end{array}$

so this implies that

$\begin{array}[c]{l}A_{f}^{Lev}(\theta)=(1-\theta\text{ }0),\text{
}\mu_{f}^{Lev}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\sigma_{2}^{2},\\\
A_{q}^{Lev}(\theta)=(-1\text{ }0),\text{
}\mu_{q}^{Lev}(\theta,\bar{\sigma}^{2})=-\sigma_{2}^{2}.\end{array}$

From this last result, it is not difficult to see that, under Assumption 2b,
we have

$\begin{array}[c]{c}\frac{1}{N}\sum_{i=1}^{N}y_{i2}\Delta
y_{i2}\underset{p}{\rightarrow}\sigma_{2}^{2}-d_{2}.\end{array}$

The reason for this is that Assumption 2b amounts to
$h_{N}(\theta_{0,N})\sqrt{N}=\frac{\sqrt{N}}{\sqrt{var(y_{i1})}}\underset{N\rightarrow\infty}{\rightarrow}\infty$
and, since var($y_{i1})\geq$var($u_{i1})$, it implies that
$\sigma_{1,N}^{2}/N\underset{N\rightarrow\infty}{\rightarrow}0.$ Finally, the
Sys sample moment and derivative simply result from stacking the Dif and Lev
sample moments and derivatives:

$\begin{array}[c]{rl}f_{N}^{Sys}(\theta)=&\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i1}\Delta
y_{i3}-\theta y_{i1}\Delta y_{i2}\\\ y_{i3}\Delta y_{i2}-\theta y_{i2}\Delta
y_{i2}\end{array}\right),\\\
q_{N}^{Sys}(\theta)=&-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i1}\Delta
y_{i2}\\\ y_{i2}\Delta y_{i2}\end{array}\right).\end{array}$

Combining earlier convergence results, the large sample behavior of the Sys
sample moment and derivative can thus be characterized by

$\begin{array}[c]{rl}f_{N}^{Sys}(\theta)=&\left(\begin{array}[c]{cc}-\theta&1\\\
1-\theta&0\end{array}\right)\left[\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left\\{\left(\begin{array}[c]{c}\psi_{2}\\\
\psi_{3}\end{array}\right)-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\iota_{2}\right\\}-\iota_{2}d_{2}\right]+\\\
&\left(1-\theta\right)\left(\begin{array}[c]{c}0\\\
\sigma_{2}^{2}\end{array}\right)+o_{p}(1),\\\
q_{N}^{Sys}(\theta)=&-\left(\begin{array}[c]{cc}1&0\\\
1&0\end{array}\right)\left[\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left\\{\left(\begin{array}[c]{c}\psi_{2}\\\
\psi_{3}\end{array}\right)-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\iota_{2}\right\\}-\iota_{2}d_{2}\right]-\\\
&\left(\begin{array}[c]{c}0\\\
\sigma_{2}^{2}\end{array}\right)+o_{p}(1),\end{array}$

from which it is readily seen that

$\begin{array}[c]{l}A_{f}^{Sys}(\theta)=\left(\begin{array}[c]{cc}-\theta&1\\\
1-\theta&0\end{array}\right),\text{
}\mu_{f}^{Sys}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}0\\\
\sigma_{2}^{2}\end{array}\right),\\\
A_{q}^{Sys}(\theta)=\left(\begin{array}[c]{cc}-1&0\\\
-1&0\end{array}\right),\text{
}\mu_{q}^{Sys}(\theta,\bar{\sigma}^{2})=\left(\begin{array}[c]{c}0\\\
-\sigma_{2}^{2}\end{array}\right).\end{array}$

T=4. Under mean stationarity, we have

$\begin{array}[c]{l}\Delta y_{i2}=u_{i2}+\left(\theta_{0,N}-1\right)u_{i1}\\\
\Delta
y_{i3}=u_{i3}+\left(\theta_{0,N}-1\right)u_{i2}+\theta_{0,N}\left(\theta_{0,N}-1\right)u_{i1}\\\
\Delta
y_{i4}=u_{i4}+\left(\theta_{0,N}-1\right)u_{i3}+\theta_{0,N}\left(\theta_{0,N}-1\right)u_{i2}+\theta_{0,N}^{2}\left(\theta_{0,N}-1\right)u_{i1}.\end{array}$

Substituting these expressions and $y_{i2}=\Delta y_{i2}+y_{i1}$, we can
specify the Dif sample moments and their derivatives as

$\begin{array}[c]{l}f_{N}^{Dif}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i1}\Delta
y_{i3}-\theta y_{i1}\Delta y_{i2}\\\ y_{i1}\Delta y_{i4}-\theta y_{i1}\Delta
y_{i3}\\\ y_{i2}\Delta y_{i4}-\theta y_{i2}\Delta y_{i3}\end{array}\right)\\\
=\left(\begin{array}[c]{ccc}\theta_{0,N}-1-\theta&1&0\\\
\left(\theta_{0,N}-\theta\right)(\theta_{0,N}-1)&\theta_{0,N}-1-\theta&1\\\
\left(\theta_{0,N}-\theta\right)(\theta_{0,N}-1)&\theta_{0,N}-1-\theta&1\end{array}\right)\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i1}u_{i2}\\\
y_{i1}u_{i3}\\\ y_{i1}u_{i4}\end{array}\right)+\\\
(\theta_{0,N}-\theta)(\theta_{0,N}-1)\left(\begin{array}[c]{c}1\\\
\theta_{0,N}\\\
\theta_{0,N}\end{array}\right)\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i1}+\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}0\\\
0\\\ \Delta y_{i2}(\Delta y_{i4}-\theta\Delta
y_{i3})\end{array}\right),\end{array}$

$\begin{array}[c]{l}q_{N}^{Dif}(\theta)=-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i1}\Delta
y_{i2}\\\ y_{i1}\Delta y_{i3}\\\ y_{i2}\Delta y_{i3}\end{array}\right)\\\
=-\left(\begin{array}[c]{ccc}1&0&0\\\ \theta_{0,N}-1&1&0\\\
\theta_{0,N}-1&1&0\end{array}\right)\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i1}u_{i2}\\\
y_{i1}u_{i3}\\\
y_{i1}u_{i4}\end{array}\right)-(\theta_{0,N}-1)\left(\begin{array}[c]{c}1\\\
\theta_{0,N}\\\
\theta_{0,N}\end{array}\right)\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i1}\\\
-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}0\\\ 0\\\ \Delta
y_{i2}\Delta y_{i3}\end{array}\right).\end{array}$

The limit behavior of the first two terms in each expression has been
established before. Furthermore, Lemma 1 shows that the last term in each
expression is $o_{p}(1)$. Therefore, the large Dif sample moment and
derivative can be expressed as:

$\displaystyle f_{N}^{Dif}(\theta)$
$\displaystyle=\left(\begin{array}[c]{ccc}-\theta&1&0\\\ 0&-\theta&1\\\
0&-\theta&1\end{array}\right)\left[\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left\\{\left(\begin{array}[c]{c}\psi_{2}\\\
\psi_{3}\\\
\psi_{4}\end{array}\right)-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\iota_{3}\right\\}-\iota_{3}d_{2}\right]+o_{p}(1),$
$\displaystyle q_{N}^{Dif}(\theta)$
$\displaystyle=-\left(\begin{array}[c]{ccc}1&0&0\\\ 0&1&0\\\
0&1&0\end{array}\right)\left[\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left\\{\left(\begin{array}[c]{c}\psi_{2}\\\
\psi_{3}\\\
\psi_{4}\end{array}\right)-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\iota_{3}\right\\}-\iota_{3}d_{2}\right]+o_{p}(1),$

from which it is readily seen that

$\begin{array}[c]{l}A_{f}^{Dif}(\theta)=\left(\begin{array}[c]{ccc}-\theta&1&0\\\
0&-\theta&1\\\ 0&-\theta&1\end{array}\right),\text{
}\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})=\left(\begin{array}[c]{c}0\\\ 0\\\
0\end{array}\right),\\\
A_{q}^{Dif}(\theta)=-\left(\begin{array}[c]{ccc}1&0&0\\\ 0&1&0\\\
0&1&0\end{array}\right),\text{
}\mu_{q}^{Dif}(\theta,\bar{\sigma}^{2})=\left(\begin{array}[c]{c}0\\\ 0\\\
0\end{array}\right).\end{array}$

After some algebra, we can specify the Lev sample moments and their
derivatives as

$\begin{array}[c]{l}f_{N}^{Lev}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i3}\Delta
y_{i2}-\theta y_{i2}\Delta y_{i2}\\\ y_{i4}\Delta y_{i3}-\theta y_{i3}\Delta
y_{i3}\end{array}\right)\\\
=\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{ccc}1-\theta&0&0\\\
(1-\theta)(\theta_{0,N}-1)&1-\theta&0\end{array}\right)\left(\begin{array}[c]{c}y_{i1}u_{i2}\\\
y_{i1}u_{i3}\\\
y_{i1}u_{i4}\end{array}\right)+(1-\theta)(\theta_{0,N}-1)\left(\begin{array}[c]{c}1\\\
\theta_{0,N}\end{array}\right)\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i1}\\\
+(1-\theta)\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}\Delta
y_{i2}\Delta y_{i2}\\\ \Delta y_{i3}\Delta
y_{i3}\end{array}\right)+\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}\Delta
y_{i3}\Delta y_{i2}\\\ \left(\Delta y_{i4}+(1-\theta)\Delta
y_{i2}\right)\Delta y_{i3}\end{array}\right),\end{array}$
$\begin{array}[c]{l}q_{N}^{Lev}(\theta)=-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}y_{i2}\Delta
y_{i2}\\\ y_{i3}\Delta y_{i3}\end{array}\right)\\\
=-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{ccc}1&0&0\\\
\theta_{0,N}-1&1&0\end{array}\right)\left(\begin{array}[c]{c}y_{i1}u_{i2}\\\
y_{i1}u_{i3}\\\
y_{i1}u_{i4}\end{array}\right)-(\theta_{0,N}-1)\left(\begin{array}[c]{c}1\\\
\theta_{0,N}\end{array}\right)\frac{1}{N}\sum_{i=1}^{N}y_{i1}u_{i1}\\\
-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}\Delta y_{i2}\Delta
y_{i2}\\\ \Delta y_{i3}\Delta
y_{i3}\end{array}\right)-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{c}0\\\
\Delta y_{i2}\Delta y_{i3}\end{array}\right).\end{array}$

Using Lemma 1, the large sample behavior of these expressions is equal to:

$\displaystyle f_{N}^{Lev}(\theta)$
$\displaystyle=\left(\begin{array}[c]{ccc}1-\theta&0&0\\\
0&1-\theta&0\end{array}\right)\left[\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left\\{\left(\begin{array}[c]{c}\psi_{2}\\\
\psi_{3}\\\
\psi_{4}\end{array}\right)-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\iota_{2}\right\\}-\iota_{2}d_{2}\right]+$
$\displaystyle\left(1-\theta\right)\left(\begin{array}[c]{c}\sigma_{2}^{2}\\\
\sigma_{3}^{2}\end{array}\right)+o_{p}(1),$

$\displaystyle q_{N}^{Lev}(\theta)$
$\displaystyle=-\left(\begin{array}[c]{ccc}1&0&0\\\
0&1&0\end{array}\right)\left[\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left\\{\left(\begin{array}[c]{c}\psi_{2}\\\
\psi_{3}\\\
\psi_{4}\end{array}\right)-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\iota_{2}\right\\}-\iota_{2}d_{2}\right]-$
$\displaystyle\left(\begin{array}[c]{c}\sigma_{2}^{2}\\\
\sigma_{3}^{2}\end{array}\right)+o_{p}(1),$

so this implies that

$\begin{array}[c]{l}A_{f}^{Lev}(\theta)=\left(\begin{array}[c]{ccc}1-\theta&0&0\\\
0&1-\theta&0\end{array}\right),\text{
}\mu_{f}^{Lev}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}\sigma_{2}^{2}\\\
\sigma_{3}^{2}\end{array}\right)\\\
A_{q}^{Lev}(\theta)=-\left(\begin{array}[c]{ccc}1&0&0\\\
0&1&0\end{array}\right),\text{
}\mu_{q}^{Lev}(\theta,\bar{\sigma}^{2})=-\left(\begin{array}[c]{c}\sigma_{2}^{2}\\\
\sigma_{3}^{2}\end{array}\right).\end{array}$

We can specify the NL sample moment and its derivative as

$\displaystyle\begin{array}[c]{l}f_{N}^{NL}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\left(y_{i4}-\theta
y_{i3}\right)\left(\Delta y_{i3}-\theta\Delta y_{i2}\right)\\\
=\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{ccc}(1-\theta)(\theta_{0,N}-\theta-1)&(1-\theta)&0\end{array}\right)\left(\begin{array}[c]{c}y_{i1}u_{i2}\\\
y_{i1}u_{i3}\\\ y_{i1}u_{i4}\end{array}\right)+\\\
\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)(\theta_{0,N}-\theta)(1-\theta)y_{i1}u_{i1}+\left(1-\theta\right)\frac{1}{N}\sum_{i=1}^{N}\left(\Delta
y_{i3}\Delta y_{i3}-\theta\Delta y_{i2}\Delta y_{i2}\right)+\\\
\frac{1}{N}\sum_{i=1}^{N}\left((\Delta y_{i4}+\left(1-\theta\right)\Delta
y_{i2})\Delta y_{i3}-(\Delta y_{i4}+\left(1-\theta\right)\Delta
y_{i3})\theta\Delta y_{i2}\right),\end{array}$
$\displaystyle\begin{array}[c]{l}q_{N}^{NL}(\theta)=-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{ccc}\theta_{0,N}-2\theta&-1&0\end{array}\right)\left(\begin{array}[c]{c}y_{i1}u_{i2}\\\
y_{i1}u_{i3}\\\
y_{i1}u_{i4}\end{array}\right)+\frac{1}{N}\sum_{i=1}^{N}(\theta_{0,N}-1)(1+\theta_{0,N}-2\theta)y_{i1}u_{i1}-\\\
\frac{1}{N}\sum_{i=1}^{N}\left(\Delta y_{i3}\Delta
y_{i3}+\left(1-2\theta\right)\Delta y_{i2}\Delta
y_{i2}\right)-\frac{1}{N}\sum_{i=1}^{N}\left(\Delta y_{i2}\Delta
y_{i3}+\left(\Delta y_{i4}+\left(1-2\theta\right)\Delta y_{i3}\right)\Delta
y_{i2}\right).\end{array}$

Using Lemma 1, the large sample behavior of these expressions is equal to:

$\begin{array}[c]{l}f_{N}^{NL}(\theta)=\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{ccc}\theta(\theta-1)&1-\theta&0\end{array}\right)\left[\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left\\{\left(\begin{array}[c]{c}\psi_{2}\\\
\psi_{3}\\\
\psi_{4}\end{array}\right)-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\iota_{2}\right\\}-\iota_{2}d_{2}\right]+\\\
\left(1-\theta\right)\left(\sigma_{3}^{2}-\theta\sigma_{2}^{2}\right)+o_{p}(1),\end{array}$
$\begin{array}[c]{l}q_{N}^{NL}(\theta)=-\frac{1}{N}\sum_{i=1}^{N}\left(\begin{array}[c]{ccc}1-2\theta&1&0\end{array}\right)\left[\frac{1}{h_{N}(\theta_{0,N})\sqrt{N}}\left\\{\left(\begin{array}[c]{c}\psi_{2}\\\
\psi_{3}\\\
\psi_{4}\end{array}\right)-h_{N}(\theta_{0,N})\sigma_{1,N}\psi_{c}\iota_{2}\right\\}-\iota_{2}d_{2}\right]-\\\
\sigma_{3}^{2}-(1-2\theta)\sigma_{2}^{2}+o_{p}(1),\end{array}$

so this implies that:

$\begin{array}[c]{l}A_{f}^{NL}(\theta)=\left(\begin{array}[c]{ccc}\theta(\theta-1)&1-\theta&0\end{array}\right),\text{
}\mu_{f}^{NL}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\sigma_{3}^{2}-\theta\sigma_{2}^{2}\right)\\\
A_{q}^{NL}(\theta)=\left(\begin{array}[c]{ccc}2\theta-1&-1&0\end{array}\right),\text{
}\mu_{q}^{NL}(\theta,\bar{\sigma}^{2})=\left(2\theta-1\right)\sigma_{2}^{2}-\sigma_{3}^{2}.\end{array}$

Finally, regarding AS and Sys moment conditions, we simply have

$\begin{array}[c]{l}A_{f}^{Sys}(\theta)=\left(\begin{array}[c]{c}A_{f}^{Dif}(\theta)\\\
A_{f}^{Lev}(\theta)\text{ }\vdots\text{ }0\end{array}\right),\text{
}\mu_{f}^{Sys}(\theta,\bar{\sigma}^{2})=\left(\begin{tabular}[c]{c}$\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})$\\\
$\mu_{f}^{Lev}(\theta,\bar{\sigma}^{2})$\end{tabular}\right),\\\
A_{q}^{Sys}(\theta)=\left(\begin{array}[c]{c}A_{q}^{Dif}(\theta)\\\
A_{q}^{Lev}(\theta)\text{ }\vdots\text{ }0\end{array}\right),\text{
}\mu_{q}^{Sys}(\theta,\bar{\sigma}^{2})=\left(\begin{tabular}[c]{c}$\mu_{q}^{Dif}(\theta,\bar{\sigma}^{2})$\\\
$\mu_{q}^{Lev}(\theta,\bar{\sigma}^{2})$\end{tabular}\right).\end{array}$
$\begin{array}[c]{l}A_{f}^{AS}(\theta)=\left(\begin{array}[c]{c}A_{f}^{Dif}(\theta)\\\
A_{f}^{NL}(\theta)\text{ }\vdots\text{ }0\end{array}\right),\text{
}\mu_{f}^{AS}(\theta,\bar{\sigma}^{2})=\left(\begin{tabular}[c]{c}$\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})$\\\
$\mu_{f}^{NL}(\theta,\bar{\sigma}^{2})$\end{tabular}\right),\\\
A_{q}^{AS}(\theta)=\left(\begin{array}[c]{c}A_{q}^{Dif}(\theta)\\\
A_{q}^{NL}(\theta)\text{ }\vdots\text{ }0\end{array}\right),\text{
}\mu_{q}^{AS}(\theta,\bar{\sigma}^{2})=\left(\begin{tabular}[c]{c}$\mu_{q}^{Dif}(\theta,\bar{\sigma}^{2})$\\\
$\mu_{q}^{NL}(\theta,\bar{\sigma}^{2})$\end{tabular}\right).\end{array}$

T=5. Using similar calculations, we obtain:

$A_{f}^{Dif}(\theta)=\left(\begin{array}[c]{cccc}-\theta&1&0&0\\\
0&-\theta&1&0\\\ 0&-\theta&1&0\\\ 0&0&-\theta&1\\\ 0&0&-\theta&1\\\
0&0&-\theta&1\end{array}\right),\text{
}\mu_{f}^{Dif}(\theta,\bar{\sigma}^{2})=\left(\begin{array}[c]{c}0\\\ 0\\\
0\\\ 0\\\ 0\\\ 0\end{array}\right),$
$A_{f}^{Lev}(\theta)=\left(\begin{array}[c]{cccc}1-\theta&0&0&0\\\
0&1-\theta&0&0\\\ 0&0&1-\theta&0\end{array}\right),\text{
}\mu_{f}^{Lev}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}\sigma_{2}^{2}\\\
\sigma_{3}^{2}\\\ \sigma_{4}^{2}\end{array}\right),$
$A_{f}^{NL}(\theta)=\left(\begin{array}[c]{cccc}\theta(\theta-1)&1-\theta&0&0\\\
0&\theta(\theta-1)&1-\theta&0\end{array}\right),\text{
}\mu_{f}^{NL}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}\sigma_{3}^{2}-\theta\sigma_{2}^{2}\\\
\sigma_{4}^{2}-\theta\sigma_{3}^{2}\end{array}\right).$

General T. Along the lines of the above, it is also possible to construct the
expressions of $A_{f}^{j}(\theta),A_{q}^{j}(\theta),$
$\mu_{f}^{j}(\theta,\bar{\sigma}^{2})$ and
$\mu_{q}^{j}(\theta,\bar{\sigma}^{2})$ for larger values of $T$ which we, for
reasons of brevity, refrain from.

#### Orthogonal complements of $A_{f}^{AS}(\theta)$ and $A_{f}^{Sys}(\theta)$
for $T=4$ and $5$ and the specification of the robust sample moments

We specify the orthogonal complements as in (37), which we repeat here for
convenience:

$A_{f}^{j}(\theta)_{\perp}=(G_{f,T}^{j}(\theta)\text{ }\vdots\text{
}G_{2,T}^{j}),$

where $T$ indicates the number of time periods and $G_{2,T}^{j}$ is such that
$G_{2,T}^{j\prime}\mu_{f}^{j}(\theta,\bar{\sigma}^{2})=0.$ This notation is
used in the proofs of subsequent theorems.

T=4. From the expressions of $A_{f}^{j}(\theta)$ and
$\mu_{f}^{j}(\theta,\bar{\sigma}^{2})$ in (36), $G_{f,T=4}^{j}(\theta)$ and
$G_{2,T=4}^{j}$ for $j=AS,$ $Sys$ result as:

$\displaystyle G_{f,T=4}^{AS}(\theta)$
$\displaystyle=\left(\begin{array}[c]{c}-(1-\theta)\\\ 0\\\ 0\\\
1\end{array}\right),\text{ }G_{2,T=4}^{AS}=\left(\begin{array}[c]{c}0\\\ -1\\\
1\\\ 0\end{array}\right),$ $\displaystyle G_{f,T=4}^{Sys}(\theta)$
$\displaystyle=\left(\begin{array}[c]{c}-\left(1-\theta\right)\\\ 0\\\ 0\\\
-\theta\\\ 1\end{array}\right),\text{
}G_{2,T=4}^{Sys}=\left(\begin{array}[c]{c}0\\\ -1\\\ 1\\\ 0\\\
0\end{array}\right).$

From these expressions and (36), it is easily seen that

$\displaystyle
A_{f}^{AS}(\theta)_{\perp}^{\prime}\mu_{f}^{AS}(\theta,\bar{\sigma}^{2})$
$\displaystyle=\left(\begin{array}[c]{c}\left(1-\theta\right)\left(\sigma_{3}^{2}-\theta\sigma_{2}^{2}\right)\\\
0\end{array}\right),$ $\displaystyle
A_{f}^{Sys}(\theta)_{\perp}^{\prime}\mu_{f}^{Sys}(\theta,\bar{\sigma}^{2})$
$\displaystyle=\left(\begin{array}[c]{c}\sigma_{3}^{2}-\theta\sigma_{2}^{2}\\\
0\end{array}\right),$

from which follows that
$A_{f}^{j}(\theta)_{\perp}^{\prime}\mu_{f}^{j}(\theta,\bar{\sigma}^{2})\neq 0$
for all $\theta\neq\theta_{0,N},$ $j=AS,$ $Sys.$

T=5. The expressions for $A_{f}^{j}(\theta),$
$\mu_{f}^{j}(\theta,\bar{\sigma}^{2}),$ $G_{f,T=5}^{j}(\theta)$ and
$G_{2,T=5}^{j}$ for $j=AS,$ $Sys$ are:

$\displaystyle A_{f}^{AS}(\theta)$
$\displaystyle=\left(\begin{array}[c]{cccc}-\theta&1&0&0\\\ 0&-\theta&1&0\\\
0&-\theta&1&0\\\ 0&0&-\theta&1\\\ 0&0&-\theta&1\\\ 0&0&-\theta&1\\\
\theta(\theta-1)&1-\theta&0&0\\\
0&\theta(\theta-1)&1-\theta&0\end{array}\right)\text{,
}\mu_{f}^{AS}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}0\\\
0\\\ 0\\\ 0\\\ 0\\\ 0\\\ \sigma_{3}^{2}-\theta\sigma_{2}^{2}\\\
\sigma_{4}^{2}-\theta\sigma_{3}^{2}\end{array}\right),$ $\displaystyle
G_{f,T=5}^{AS}(\theta)$
$\displaystyle=\left(\begin{array}[c]{ccc}-(1-\theta)&0&0\\\
0&-(1-\theta)&0\\\ 0&0&-(1-\theta)\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\ 1&0&0\\\
0&1&1\end{array}\right),\text{
}G_{2,T=5}^{AS}=\left(\begin{array}[c]{cc}0&0\\\ 0&0\\\ 0&0\\\ -1&0\\\ 1&-1\\\
0&1\\\ 0&0\\\ 0&0\end{array}\right),$ $\displaystyle A_{f}^{Sys}(\theta)$
$\displaystyle=\left(\begin{array}[c]{cccc}-\theta&1&0&0\\\ 0&-\theta&1&0\\\
0&-\theta&1&0\\\ 0&0&-\theta&1\\\ 0&0&-\theta&1\\\ 0&0&-\theta&1\\\
1-\theta&0&0&0\\\ 0&1-\theta&0&0\\\ 0&0&1-\theta&0\end{array}\right)\text{,
}\mu_{f}^{Sys}(\theta,\bar{\sigma}^{2})=\left(1-\theta\right)\left(\begin{array}[c]{c}0\\\
0\\\ 0\\\ 0\\\ 0\\\ 0\\\ \sigma_{2}^{2}\\\ \sigma_{3}^{2}\\\
\sigma_{4}^{2}\end{array}\right),$ $\displaystyle G_{f,T=5}^{Sys}(\theta)$
$\displaystyle=\left(\begin{array}[c]{ccc}-(1-\theta)&0&0\\\
0&-(1-\theta)&0\\\ 0&0&-(1-\theta)\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\
-\theta&0&0\\\ 1&-\theta&-\theta\\\ 0&1&1\end{array}\right),\text{
}G_{2,T=5}^{Sys}=\left(\begin{array}[c]{cc}0&0\\\ 0&0\\\ 0&0\\\ -1&0\\\
1&-1\\\ 0&1\\\ 0&0\\\ 0&0\\\ 0&0\end{array}\right).$

Straightforward algebra shows that
$A_{f}^{j}(\theta)_{\perp}^{\prime}\mu_{f}^{j}(\theta,\bar{\sigma}^{2})\neq 0$
for all $\theta\neq\theta_{0,N},$ $j=AS,$ $Sys.$

The robust sample moments are defined as

$g_{f,T}^{j}(\theta)=A_{f}(\theta)_{\perp}^{j\prime}f_{N}^{j}(\theta),$

with $A_{f}(\theta)_{\perp}^{j}=(G_{f,T}^{j}(\theta)$ $\vdots$ $G_{2,T}^{j}).$
For the Sys moment conditions, $G_{f,T}^{j}(\theta)$ is a linear function of
$\theta$ and $G_{2,T}^{j}$ does not depend on $\theta.$ Since
$f_{N}^{j}(\theta)$ is linear in $\theta$ as well for the Sys sample moments,
the part of $g_{f,T}^{j}(\theta)$ resulting from
$G_{f,T}^{j}(\theta)^{\prime}f_{N}^{j}(\theta)$ is quadratic in $\theta$ while
the part that results from $G_{2,T}^{j\prime}f_{N}^{j}(\theta)$ is linear in
$\theta.$ Given the specification of $G_{f,T}^{j}(\theta),$ $G_{2,T}^{j}$ and
$f_{N}^{j}(\theta),$ it is then straightforward to compute the specification
of $a,$ $b$ and $d.$

For the AS moment conditions, $G_{f,T}^{j}(\theta)$ is a linear function of
$\theta$ and $G_{2,T}^{j}$ does not depend on $\theta.$ For the AS sample
moments, $f_{N}^{j}(\theta)$ is quadratic in $\theta$ but the part of