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Uniform Quasi ML based inference for the panel AR(1) model.
JEL\ classification: C12, C13, C23.
Keywords: dynamic panel data model, identification robust inference, quasi Lagrange multiplier test, score test, second-order identification, singular information matrix, uniform inference, Wald test.
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Maximum Likelihood (ML) offers attractive alternatives to Generalized Method of Moments (GMM)\ estimators for dynamic panel data models. However, to date no identification robust inference methods exist that can be used in conjunction with the ML estimators for these models. In this paper we propose ML based inference methods for panel AR(1) models with arbitrary initial conditions and heteroskedasticity that are robust to the strength of identification. We show that (Quasi) Lagrange Multiplier (LM) tests and confidence sets (CSs) that use the expected Hessian rather than the observed Hessian of the log-likelihood function have correct asymptotic size and coverage probability in a uniform sense, respectively. Such Quasi LM\ tests and CSs are also robust to misspecification of the distribution of the data and to heterogeneity, including heteroskedasticity. We derive the power envelope of a Fixed Effects version of such an LM test for hypotheses involving the autoregressive parameter when the average information matrix is estimated by a centered OPG\ estimator and the model is only second-order identified, and show that it coincides with the maximal attainable power curve in the worst case setting. We also study the empirical size and power properties of these (Quasi) LM tests and find that the hypothesis that the (Quasi) LM test has correct size cannot be rejected.
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This paper proposes new inference methods for panel AR models with arbitrary initial conditions and heteroskedasticity and possibly additional regressors that are robust to the strength of identification. Specifically, we consider several Maximum Likelihood based methods of constructing tests and confidence sets (CSs) and show that (Quasi) LM tests and CSs that use the expected Hessian rather than the observed Hessian of the log-likelihood have correct asymptotic size in a uniform sense.
There exists a vast literature devoted to estimation of versions of the panel AR(1) model or, more generally, dynamic panel data models. The estimators in this literature can roughly be classified into two groups: Instrumental Variables/Generalized Method of Moments (IV/GMM) type estimators and Maximum Likelihood (ML) type estimators. Seminal contributions to the first group include Anderson and Hsiao (1981, 1982), Holtz-Eakin et al. (1988), Arellano and Bond (AB, 1991), Arellano and Bover (ABov, 1995) and Ahn and Schmidt (AS, 1995), while seminal contributions to the second group include Chamberlain (1980), Anderson and Hsiao (1981, 1982), MaCurdy (1982) and Hsiao et al. (2002). The last two papers considered Fixed Effects (FE) ML\ estimators, that is, estimators that are based on data in first differences and are consistent under minimal assumptions, whereas the other two papers in the second group considered (correlated) Random Effects (RE) ML estimators.\footnote{ The RE approach assumes finite second moments of the data in levels while the FE approach only assumes finite second moments of the data in differences, cf. Kruiniger (2001, 2022).} \footnote{ The Transformed MLE of Hsiao et al. (2002) and the FEMLE in Kruiniger (2001, 2013, 2022) are the same estimator and are the FE counterpart of the REMLE.}
Kruiniger (2001) has shown that when the data are i.i.d. and normal, then the RE and FE MLEs of the autoregressive parameter ${\Greekmath 011A} $ in a panel AR(1) model with homoskedastic errors and $\left\vert {\Greekmath 011A} \right\vert <1$ are asymptotically equivalent to optimal AS-type GMM\ estimators, which in this case also exploit moment conditions that rely on time-series homoskedasticity, and that when ${\Greekmath 011A} =1,$ the information matrix associated with the FEMLE for the panel AR(1) model with homoskedastic errors is singular. Kruiniger (2013) and Alvarez and Arellano (2022) have developed RE\ and FE MLEs for the panel AR(1) model that allows for time-series heteroskedasticity. These estimators of ${\Greekmath 011A} $ remain large $N$, fixed $T$ consistent when the data are non-normal or i.h.d., where $N$ and $T$ are the dimensions of the panel. Kruiniger (2013) and Ahn and Thomas (2023)\ have shown that when ${\Greekmath 011A} =1$ and the errors bar possibly the error of the last period are homoskedastic over time, then RE and FE MLEs of ${\Greekmath 011A} $ that allow for time-series heteroskedasticity are $N^{1/4}$-consistent and have non-normal\ limiting distributions. The slower rate of convergence is related to the fact that in this case the information matrix is singular so that ${\Greekmath 011A} $ is only second-order locally identified, cf. Sargan (1983) and Rotnitzky et al. (2000).
Monte Carlo studies in Hsiao et al. (2002), Kruiniger (2013) and Hayakawa and Pesaran (2015) have found that the (Quasi) MLEs for the panel AR(1) and ARX(1) models have very good finite sample properties but that when ${\Greekmath 011A} $ is close to one, Wald tests for hypotheses about ${\Greekmath 011A} $ are oversized and asymptotic confidence intervals based on the QMLEs of ${\Greekmath 011A} $ are too narrow because ${\Greekmath 011A} $ is weakly identified. However, to date no iden- tification robust inference methods have been proposed that are related to these QMLEs.
Weak identification including second- rather than first-order local identification affect the rate of convergence and the limiting distributions of the estimators and pose a challenge to conducting inference both in the case of GMM and the ML method. For reliable inference it is important that tests and/or confidence sets (CSs) have correct asymptotic size in a uniform sense. As Andrews et al. (2020) explain, pointwise asymptotics often provide very poor approximations to the finite-sample size when the test statistic of interest has a discontinuous pointwise asymptotic distribution. For this reason Wald-type tests and CSs for (hypotheses/parameter vectors that include) ${\Greekmath 011A} $ will generally not have correct asymptotic size in a uniform sense. Also likelihood based (Quasi) LR-type tests and CSs for (hypotheses/parameter vectors that include) ${\Greekmath 011A} $ will in many cases not have correct (locally) uniform asymptotic size, cf. Rotnitzky et al. (2000) and Bottai (2003). In this paper we will instead propose tests and CSs for (hypotheses/parameter vectors that include) ${\Greekmath 011A} $ that are based on (Quasi) LM test statistics that use the expected Hessian of a RE or FE log-likelihood and show that they have correct uniform asymptotic size.
Our approach generalizes that of Bottai (2003), who considered CSs based on similar LM test statistics in the context of identifiable one-dimensional parametric models with a smooth likelihood function and information equal to zero at a critical point, in at least two ways. Firstly, we consider multi-parameter models and hypotheses. To show that our (Quasi) LM tests and CSs have correct uniform asymptotic size, we make use of the fact that in suitably reparametrized versions of the panel AR(1) model, the parameters other than ${\Greekmath 011A} $ are still strongly identified when ${\Greekmath 011A} =1.$ Secondly, by using Quasi LM test statistics, we also allow for misspecification of the distribution of the data and heterogeneity.
Bottai (2003) explains for the one-parameter case why an LM\ test that is based on the observed Hessian rather than the expected Hessian will not have correct uniform asymptotic size when the parameter, say ${\Greekmath 0112} $, is second-order identified at a critical point ${\Greekmath 0112} ^{\ast }$. In this case the LM\ test statistic does not converge in distribution to a ${\Greekmath 011F} ^{2}(1)$ random variable under the sequence ${\Greekmath 0112} _{n}={\Greekmath 0112} ^{\ast }-cn^{-1/4}$ as the sample size $n$ tends to infinity although it does converge to a $ {\Greekmath 011F} ^{2}(1)$ random variable under sequences ${\Greekmath 0112} _{n}={\Greekmath 0112} ^{\ast }-cn^{-b}$ with $b>1/4.$ The same situation arises in the multi-parameter case.
We will now discuss alternative, GMM-based methods. When the autoregressive parameter ${\Greekmath 011A} $ is local to unity, then the AB GMM\ estimator has poor finite sample properties and is inconsistent due to weak instruments, cf. ABov, Blundell and Bond (BB, 1998) and Kruiniger (K, 2009). If in addition the individual time series are covariance stationary or the number of pre-sample realizations is large, then the System estimator of Abov and BB may have a non-normal limiting distribution and its rate of convergence may depend on the choice of the weight matrix, cf. K2009. Furthermore, when mean-stationarity holds but the ratio of the variance of the individual effects to the variance of the idiosyncratic errors is large, then the AB GMM and the System estimator can also suffer from a weak instruments problem and be biased, cf. Hayakawa (2007), Bun and Windmeijer (2010) and Kruiniger (2001, 2022). Incidentally, Bun and Kleibergen (2022) and Kruiniger (2022) have shown that provided that $T$ is not too small, ${\Greekmath 011A} $ is identified, albeit possibly only second-order identified when ${\Greekmath 011A} =1$, by a set of linear and quadratic AS-type moment conditions that only depend on a lack of serial correlation assumption for the errors and on differenced data and hence do not require mean-stationarity to hold.
GMM-based tests (and CSs) for ${\Greekmath 011A} $ that have correct uniform asymptotic size when ${\Greekmath 011A} =1$ include (CSs based on) the Newey and West (1987) type GMM\ LM\ test(-statistic)s that exploit ABov and SYS moment conditions, respectively, see Madsen (2003) and K2009, and identification-robust test(-statistics)s such as the GMM AR test of Stock and Wright (2000) and the KLM and GMM-CLR tests of Kleibergen (2005) that exploit AB, ABov, SYS and AS moments conditions, cf. Bun and Kleibergen (2014).\footnote{ These GMM LM\ tests have correct uniform asymptotic size when ${\Greekmath 011A} =1$ because the scaled sample moment conditions they depend on (evaluated at $ {\Greekmath 011A} =1$) and their first derivates are asymptotically independent. As a result these GMM\ LM test statistics have the same limiting distribution as they would have under strong identification, i.e., $N(0,1)$ (or ${\Greekmath 011F} ^{2}(1) $ when squared), see Kruiniger (2009). However, in general GMM LM tests are not robust to lack of identification. For instance, if ${\Greekmath 011A} =1,$ the GMM LM test that only exploits the moment conditions of AB does not have correct uniform asymptotic size and shouldn't be used,\thinspace cf.\thinspace Bond and Windmeijer (2005).} Results in Newey and Windmeijer (NW, 2009) imply that most of these identification robust test statistics also have correct asymptotic size under many weak moment conditions asymptotics with a restriction on the relative rate at which the number of moment conditions and $N$ tend to infinity. The only possible exceptions are the tests that exploit the non-linear AS moment conditions when NW's second assumption, in particular global identification, does not hold. However, this only happens when $T\geq 3$ and the variances of the errors change over time at a constant rate, cf. Alvarez and Arellano (2022). Andrews and Guggenberger (2017) have shown that under rather general conditions the KLM and GMM-CLR tests for nonlinear moment condition models have correct asymptotic size as $N\rightarrow \infty $ although they note that for GMM-CLR tests this result depends in the multi-parameter case on how the conditioning statistic, upon which the GMM-CLR test depends, is weighted.
Bun and Kleibergen (2014) have shown for $T\geq 4$ that in a worst case scenario, where the variance of the initial observations goes to infinity, the true value of ${\Greekmath 011A} $ is equal to one and the errors are homoskedastic so that ${\Greekmath 011A} $ is only second-order locally identified, the power envelope of the KLM test based on the AS\ moment conditions and a centered optimal weight matrix, viz. the AS\ KLM\ test, coincides with the maximal attainable power curve for testing $H_{0}:{\Greekmath 011A} =1-cN^{-1/4},$ and that in such a scenario the AS GMM\ AR\ and the AS\ GMM LM tests have less power than the AS KLM test when $T\geq 5$.\footnote{ In the worst case scenario effectively only moment conditions based on second moments of differences of the data are exploited, which do not require mean-stationarity when $\left\vert {\Greekmath 011A} \right\vert <1$.}\ The analysis of local power of these tests that is provided in Bun and Kleibergen (2014) is different from that in Dovonon et al. (2020). The latter assumes that the true value of ${\Greekmath 011A} $ equals $1-cN^{-1/4}$ and tests $H_{0}:{\Greekmath 011A} =1$, and finds in a simulation study that the AS\ GMM\ AR test has better power properties than the AS\ KLM\ and AS\ GMM-CLR tests.
Andrews et al. (2019) list some limitations and weaknesses of inference procedures that are based on identification robust test statistics. Firstly, the KLM and GMM-CLR tests for subvectors of ${\Greekmath 0112} $ generally no longer have correct asymptotic size when the nuisance parameters are not strongly identified, although the GMM AR\ test for subvectors of ${\Greekmath 0112} $ still has correct asymptotic size in this case when the errors are homoskedastic, cf. Guggenberger et al. (2012, 2019) and Kleibergen (2021). Secondly, a large number of inference procedures have been proposed for overidentified models with nonhomoskedastic errors, some of which are based on analogs or generalizations of the GMM-CLR test, but there is no consensus on what procedures to use in practice, beyond the recommendation to use methods that are efficient when the instruments are strong. Recently, Andrews (2018) introduced a widely applicable approach to detecting weak identification and constructing two-step confidence sets that controls coverage distortions under weak identification. In cases where the model is well identified, his method indicates this and reports nonrobust confidence sets with probability tending to 1. Finally, it has been found that increases in the number of moment conditions can lead to size distortions of the identification robust tests in finite samples, cf. Kleibergen and Mavroeidis (2009), Kleibergen (2019) and Bun et al. (2020). The GMM LM test is also susceptible to this problem. In their Monte Carlo studies based on various panel AR(1) and panel ARX(1) models with parameter values that correspond to different degrees of identification strength both Hayakawa and Pesaran (2015) and Bun and Poldermans (2015) find evidence for size distortions for the GMM\ LM, GMM AR, KLM and GMM-CLR tests that exploit different sets of moment conditions (e.g. AB, AS, SYS and subsets and collapsed versions of these sets) and that the distortions get worse as $T$ or the number of moment conditions increases. The size distortions become smaller when $N$ gets larger. Bun et al. (2020) show that when the number of moment conditions is moderately large, then the GMM AR\ test that is based on a weighting matrix that uses centered moment conditions is oversized, whereas the uncentered version of the GMM AR test is conservative. Bun et al. (2020) also propose a degrees-of-freedom corrected centered GMM\ AR test that has good size properties.
The paper proceeds as follows. In section 2 we present the panel AR(1) model, the assumptions, the (Q)ML estimators and their asymptotic properties including their local-to-unit root limiting distributions. In section 3 we discuss the asymptotic size properties of various ML\ based tests including Quasi LM tests that use the expected Hessian of the RE\ or the FE log-likelihood function. We also derive the power envelope of the FE version of such an QLM test for $H_{0}:{\Greekmath 011A} =1-cN^{-1/4}$ when the average information matrix is estimated by a centered OPG\ estimator and the model is only second-order identified, and the maximal attainable power curve for testing $H_{0}:{\Greekmath 011A} =1-cN^{-1/4}$ in the worst case scenario. In section 4 we conduct a Monte Carlo study into the empirical size and power properties of the QLM tests. Section 5 concludes and the appendix contains the proofs.$ \vspace*{-0.22in}$
Consider the panel AR(1) model with individual effects:$\vspace*{-0.12in}$
for $i=1,...,N$ and $t=2,...,T.$ When deriving the asymptotics results, we let $N$ become large while the number of observations per `individual', $T,$ remains fixed. The analysis below can be extended straightforwardly for models that also include strictly exogenous covariates; we have omitted them from the model to keep the presentation simple.
We assume that $-1<{\Greekmath 011A} \leq 1.$ Note that the model can be rewritten as $ y_{i,t}-{\Greekmath 0116} _{i}={\Greekmath 011A} (y_{i,t-1}-{\Greekmath 0116} _{i})+{\Greekmath 0122} _{i,t}$ and that the $ {\Greekmath 0111} _{i}$ disappear (the ${\Greekmath 0116} _{i}$ drop out) when ${\Greekmath 011A} =1$. The parame- trization ${\Greekmath 0111} _{i}=(1-{\Greekmath 011A} ){\Greekmath 0116} _{i}$ prevents the individual effects from turning into individual trends at ${\Greekmath 011A} =1$ and thereby avoids a discontinuity in the data generating process at ${\Greekmath 011A} =1$.
The vectors of idiosyncratic errors ${\Greekmath 0122} _{i}=({\Greekmath 0122} _{i,2}$ $ ...$ ${\Greekmath 0122} _{i,T})^{\prime }$ are independently distributed across individuals and satisfy the following Standard Assumptions, SA$k$, for $k=2$ or $k=4$:$\vspace{-0.12in}$
where ${\Greekmath 0126} \geq 0$ is an arbitrarily small constant. In the sequel SA2 is denoted by SA.
The individual effects can often be treated as random effects. In this case we make the following Random Effects Assumptions, REA$k$, for $k=2$ or $k=4$: $\vspace{-0.12in}$
In addition, we let ${\Greekmath 011B} _{{\Greekmath 0111} }^{2}=E({\Greekmath 0111} _{i}^{2})$ and ${\Greekmath 011B} _{{\Greekmath 0111} y}=E({\Greekmath 0111} _{i}y_{i,1}).$ The i.i.d. assumption in ((ref)) is only made for presentational convenience. It can easily be relaxed. The assumption $E\left\vert {\Greekmath 0116} _{i}\right\vert ^{k+{\Greekmath 0126} }<\infty $ (for $ k=2$ or $k=4$) ensures that under covariance stationarity the means of the data, i.e., ${\Greekmath 0111} _{i}/(1-{\Greekmath 011A} )={\Greekmath 0116} _{i},$ $i=1,...,N$, are drawn from a distribution with a finite variance rather than a variance that tends to infinity when ${\Greekmath 011A} $ approaches one.
Unlike the RE ML\ estimators, the FE ML\ estimators only exploit data in first differences. This reflects the fact that the FE approach entails making minimal assumptions about the ${\Greekmath 0116} _{i}$ and the $y_{i,1}.$ In the FE case we assume that $v_{i,1}\equiv y_{i,1}-{\Greekmath 0116} _{i},$ $i=1,...,N$, satisfy a Fixed Effects Assumption, FEA$k$, for $k=2$ or $k=4$, cf. Kruiniger (2013):$ \vspace{-0.12in}$
The i.i.d. assumption in ((ref)) is made for presentational convenience. It can be relaxed.
Suppose that $y_{i,1}$ depends on ${\Greekmath 0116} _{i}$ in a linear fashion, i.e. $ y_{i,1}=E(y_{i,1})+{\Greekmath 010B} _{1}{\Greekmath 0116} _{i}+{\Greekmath 0122} _{i,1}$ with $ E({\Greekmath 0122} _{i,1})=0,$ and ${\Greekmath 0116} _{i}\perp {\Greekmath 0122} _{i,1}$. In the important case that ${\Greekmath 010B} _{1}=1,$ we have ${\Greekmath 0116} _{i}\perp v_{i,1}$ and FEA$ k$ does not impose any restrictions on ${\Greekmath 0116} _{i}$ and $y_{i,1}$ other than those on $y_{i,1}-{\Greekmath 0116} _{i}$. However when ${\Greekmath 010B} _{1}\neq 1,$ FEA$k$ implies restrictions on the ${\Greekmath 0116} _{i}$ themselves$.$
In the sequel REA2 and FEA2 are denoted by REA and FEA, respectively.
We add assumption B (Basic), which in the RE\ case amounts to
and
and in the FE case amounts to ((ref)) and
We sometimes use an augmented version of assumption B, i.e., assumption B$ ^{\prime }$\ that in addi- tion to ((ref)) and ((ref)) assumes that ${\Greekmath 0122} _{i,l}{\Greekmath 0122} _{i,s}{\Greekmath 0122} _{i,t}\perp y_{i,1}$ for all $l,s,t\in \{2,...,T\}$ and $i=1,...,N$.
In the paper we will also make reference to the assumption of mean-stationarity:
We can allow for heteroskedasticity of the ${\Greekmath 0122} _{i,t}$ in the time-series dimension:
In some cases we make the stronger assumption of Time Series Homoskedasticity (TSH):
Let $\overline{{\Greekmath 011B} }_{s}^{2}=N^{-1}\sum_{i=1}^{N}E({\Greekmath 0122} _{i,s}^{2}) $, then we say that assumption TSH$_{t}^{\ast }$ holds if and only if $\overline{{\Greekmath 011B} }_{s}^{2}=\overline{{\Greekmath 011B} }_{2}^{2},$ for $ s=3,...,t.$ We will often make a `homoskedasticity' assumption that is weaker than TSH, namely TSH$^{\ast }$, which is short for TSH$_{T}^{\ast }.$ Similarly we can use a weaker version of time-series heteroskedasticity, namely $\overline{{\Greekmath 011B} }_{t}^{2}={\Greekmath 0115} _{t}^{2}<\infty ,$ for $ t=2,...,T. $
Direct application of the Maximum Likelihood method to the nonstationary panel AR(1) model with RE will generally yield an inconsistent estimator for ${\Greekmath 011A} $ due to correlation between the individual effects (${\Greekmath 0111} _{i}$) and the regressors ($y_{i,t-1},$ $t=2,...,T$). However, a consistent ML estimator for ${\Greekmath 011A} $ can be obtained after reformulating the model. Following Chamberlain (1980) we can decompose the ${\Greekmath 0111} _{i}$ into a term that depends on the initial observation, $y_{i,1},$ and a term that does not: \footnote{ For the sake of a simple exposition we assume that $E(y_{i,1})=0$. A situation where $E(y_{i,1})\neq 0$ can be handled by including an intercept term in ((ref)) and in ((ref)). }
where $v_{i}$ is a new individual effect with $E(v_{i})=0$, ${\Greekmath 0119} (1-{\Greekmath 011A} )=$ plim$_{N\rightarrow \infty }\sum_{i=1}^{N}({\Greekmath 0111} _{i}y_{i,1})/$ $ \sum_{i=1}^{N}y_{i,1}^{2}$ and plim$_{N\rightarrow \infty }N^{-1}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}(y_{i,1}v_{i})=0.$
Let $y_{i}=(y_{i,2}$ $...$ $y_{i,T})^{\prime }$ and $y_{i,-1}=(y_{i,1}$ $...$ $y_{i,T-1})^{\prime }$ and let ${\Greekmath 0113} $ denote a vector of ones. Then using the decomposition of the `correlated effects' ${\Greekmath 0111} _{i}$ given in ((ref) ), we can rewrite the panel AR(1) model with RE\ as
Let plim$_{N\rightarrow \infty }N^{-1}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}v_{i}^{2}={\Greekmath 011B} _{v}^{2}$ and $\Phi =(1-{\Greekmath 011A} )^{2}{\Greekmath 011B} _{v}^{2}{\Greekmath 0113} {\Greekmath 0113} ^{\prime }+\Psi ({\Greekmath 0110} )$. Then it is easily verified that plim$_{N\rightarrow \infty }N^{-1}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}(y_{i,0}{\Greekmath 0113} ^{\prime }\Phi ^{-1}u_{i})=0$ and plim$_{N\rightarrow \infty }N^{-1}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}(y_{i,-1}^{\prime }\Phi ^{-1}u_{i})=0,$ cf. similar results under homoskedasticity in Blundell and Bond (1998). After imposing the assumption that the error components are i.i.d. and normal, i.e., $u_{i}\sim i.i.d.$ $N(0,\Phi )$, application of the ML method to ((ref)) yields the RE ML\ estimator of ${\Greekmath 011A} $, ${\Greekmath 0119} $, ${\Greekmath 011B} _{v}^{2}$ and ${\Greekmath 0110} =({\Greekmath 0115} _{2}^{2}$ ${\Greekmath 0115} _{3}^{2}$ $...$ ${\Greekmath 0115} _{T}^{2})^{\prime }$. This estimator will still be consistent when the $ {\Greekmath 0122} _{i,t}$ are heteroskedastic across both dimensions of the panel.$ \,$
When calculating the REMLE it is convenient to use the reparameterization $ \tilde{{\Greekmath 0119}}={\Greekmath 0119} (1-{\Greekmath 011A} )$ and $\widetilde{{\Greekmath 011B} }_{v}^{2}=(1-{\Greekmath 011A} )^{2}{\Greekmath 011B} _{v}^{2}.$ Let ${\Greekmath 0112} _{0}=({\Greekmath 011A} ,\tilde{{\Greekmath 0119}},\widetilde{{\Greekmath 011B} }_{v}^{2},{\Greekmath 0110} ^{\prime })^{\prime }.$ Then the log-likelihood function for the above model will be denoted by $l_{RE}({\Greekmath 0112} )=\mathop{\textstyle \sum }\nolimits_{i=1}^{N}l_{RE,i}({\Greekmath 0112} )$ where ${\Greekmath 0112} =(r,\widetilde{p}, \widetilde{s}_{v}^{2},z^{\prime })^{\prime }$ and $l_{RE,i}({\Greekmath 0112} )$ is the contribution to $l_{RE}({\Greekmath 0112} )$ from `individual' $i$. The REMLE of $ {\Greekmath 0112} _{0}$ will be denoted by $\widehat{{\Greekmath 0112} }_{RE}$ or simply by $ \widehat{{\Greekmath 0112} }.$
Hsiao et al. (2002) has proposed the Transformed MLE for the panel AR(1) model, which can be viewed as the FE counterpart of the REMLE, cf. Kruiniger (2001). One can obtain the FEMLE for the model that allows for heteroskedasticity by replacing ${\Greekmath 0116} _{i}$ in ((ref)) by $y_{i,1}+v_{i},$ and imposing that the $(v_{i}$ ${\Greekmath 0122} _{i}^{\prime })$ are i.i.d. and normal with $v_{i}\perp {\Greekmath 0122} _{i}$ when $\left\vert {\Greekmath 011A} \right\vert <1$.\footnote{ For the sake of a simple exposition we assume that $E(v_{i})=0$. A situation where $E(v_{i})\neq 0$ can be handled by including an intercept term in ((ref)). } This amounts to imposing the restriction ${\Greekmath 0119} =1 $ on the model in ((ref)) and leads to the following formulation of the nonstationary panel AR(1) model with FE:
and where $v_{i}=-v_{i,1}$ satisfy assumption ((ref)). After imposing $ u_{i}\sim i.i.d.$ $N(0,\Phi )$ with $\Phi =\widetilde{{\Greekmath 011B} }_{v}^{2}{\Greekmath 0113} {\Greekmath 0113} ^{\prime }+\Psi ({\Greekmath 0110} )$, application of the ML method to ((ref)) yields the FE MLE of ${\Greekmath 011A} $, $\widetilde{{\Greekmath 011B} }_{v}^{2}$, ${\Greekmath 0110} =({\Greekmath 0115} _{2}^{2}$ ${\Greekmath 0115} _{3}^{2}$ $...$ ${\Greekmath 0115} _{T}^{2})^{\prime }$. This estimator will also be consistent when the ${\Greekmath 0122} _{i,t}$ are heteroskedastic across both dimensions of the panel.$\,$\ The log-likelihood function for the above model will be denoted by $l_{FE}({\Greekmath 0112} )$ where in this case ${\Greekmath 0112} =(r,\widetilde{s}_{v}^{2},z^{\prime })^{\prime }$.
In the previous subsection we allowed for cross-sectional heteroskedasticity but otherwise assumed homogeneity of the distributions of the (standardized) idiosyncratic errors. However, such strong distributional assumptions with respect to the errors are almost never satisfied by panel data. K2013 has shown that when the data exhibit heterogeneity the ML method still yields consistent RE and FE Quasi ML\ estimators for ${\Greekmath 011A} $ if SA, REA (or FEA) and B hold, and $T\geq 3$ when $\left\vert {\Greekmath 011A} \right\vert <1$ and $T\geq 4$ when ${\Greekmath 011A} =1$ (cf. Theorem 2 in K2013). Under stronger conditions on the $ {\Greekmath 0122} _{i},$ $y_{i,1}$ and ${\Greekmath 0116} _{i}$ (or $v_{i,1}$), namely under SA4, REA4 (or FEA4), B and an appropriate Lindeberg condition, K2013 has also shown that the first-order fixed parameter asymptotic distributions of the RE and FE QMLEs of ${\Greekmath 0112} _{0}$ are normal when either ${\Greekmath 011A} =1$ and the average information matrix and the Expected Hessian of the log-likelihood function are nonsingular or when $\left\vert {\Greekmath 011A} \right\vert <1$, that is, $\sqrt{N}(\widehat{{\Greekmath 0112} }-{\Greekmath 0112} _{0})\overset{d}{ \rightarrow }N(0,H({\Greekmath 0112} _{0})^{-1}G({\Greekmath 0112} _{0})H({\Greekmath 0112} _{0})^{-1})$ in these cases, where $H({\Greekmath 0112} _{0})$ is the asymptotic Hessian of the log-likelihood function and $G({\Greekmath 0112} _{0})$ is the asymptotic information matrix (cf. Theorem 3 in K2013).
K2001 has shown that if ${\Greekmath 011A} =1,$ the ${\Greekmath 0122} _{i}$ are i.i.d. and normal, and assumption TSH holds and has been imposed on the likelihood function, then the Expected Hessian of $N^{-1}l_{FE}({\Greekmath 0112} )$ (and thus also the information matrix) is singular. Ahn and Thomas (2023) and Bond et al. (2005) have established an analogous result for the Expected Hessian of $ N^{-1}l_{RE}({\Greekmath 0112} ).$ In these cases standard ML theory cannot be used to derive the limiting distribution of the MLE. Applying the asymptotic theory recently developed by Rotnitzky, Cox, Bottai and Robins (2000, henceforth RCBR) for cases with an information matrix of rank one less than full, Ahn and Thomas (2023) have derived the non-normal limiting distribution of the REMLE. The theory of RCBR is based on using suitable Taylor expansions of the possibly reparametrized log-likelihood and score functions around the value of the parameter vector ${\Greekmath 0112} $\ at which the information matrix is singular, viz. ${\Greekmath 0112} _{\ast }.$
Unsurprisingly, when ${\Greekmath 011A} =1$ and TSH$^{\ast }$ holds but TSH\ has not been imposed, the Expected Hessians of $N^{-1}l_{RE}({\Greekmath 0112} )$ and $ N^{-1}l_{FE}({\Greekmath 0112} )$ are singular as well. By extending the theory of RCBR to allow for i.h.d. data, K2013 has generalized the result of Ahn and Thomas (2023) as follows.
Application of the asymptotic results of RCBR or their extension for i.h.d. data to the (Q)MLEs requires that one reparametrizes the RE and FE models described in section 2.1 so that three conditions, viz. (B1)-(B3), which are stated just above Theorem 1, are satisfied. They have been verified in K2013 and are similar to conditions given in RCBR. The reparametrizations are such that the scores at ${\Greekmath 0112} _{\ast }$ for all the new parameters but one are linearly independent, the QMLEs of these parameters converge at rate $ O_{p}(N^{-1/2})$ and the score at ${\Greekmath 0112} _{\ast }$ for the remaining new parameter, which is ${\Greekmath 011A} $, is equal to zero w.p.1.
In the case of the RE(Q)MLE we need to use the following new parametrization (indicated by the subscript $n$): ${\Greekmath 0112} _{0,n}=({\Greekmath 011A} _{n},\widetilde{ {\Greekmath 011B} }_{v,n}^{2},{\Greekmath 0110} _{n}^{\prime },\tilde{{\Greekmath 0119}}_{n})^{\prime }$ where $ {\Greekmath 011A} _{n}={\Greekmath 011A} ,$ $\widetilde{{\Greekmath 011B} }_{v,n}^{2}=\widetilde{{\Greekmath 011B} } _{v}^{2}/{\Greekmath 011B} _{2}^{2}-(1-{\Greekmath 011A} ),$ ${\Greekmath 0110} _{n}=({\Greekmath 011B} _{2,n}^{2},{\Greekmath 011B} _{3,n}^{2},...,{\Greekmath 011B} _{T,n}^{2})^{\prime }$ with ${\Greekmath 011B} _{t,n}^{2}={\Greekmath 011B} _{t}^{2}/{\Greekmath 011A} ,$ $t=2,...,T,$ so that ${\Greekmath 0110} _{n}={\Greekmath 0110} /{\Greekmath 011A} $, and $\tilde{ {\Greekmath 0119}}_{n}=\tilde{{\Greekmath 0119}}-(1-{\Greekmath 011A} ).$ Under this reparametrization the regression equation becomes $y_{i}-y_{i,1}{\Greekmath 0113} ={\Greekmath 011A} _{n}(y_{i,-1}-y_{i,1}{\Greekmath 0113} )+ \tilde{{\Greekmath 0119}}_{n}y_{i,1}{\Greekmath 0113} +u_{i}.$ Noting that we can express the elements of ${\Greekmath 0112} $ as functions of the elements of ${\Greekmath 0112} _{n}=(r_{n},\widetilde{s }_{v,n}^{2},z_{n}^{\prime },\widetilde{p}_{n})^{\prime },$ viz. ${\Greekmath 0112} ({\Greekmath 0112} _{n})=(r_{n},r_{n}s_{2,n}^{2}(\widetilde{s} _{v,n}^{2}+(1-r_{n})),r_{n}z_{n}^{\prime },\widetilde{p}_{n}+(1-r_{n}))^{ \prime },$ the reparameterized log-likelihood function is given by $ l_{n,RE}({\Greekmath 0112} _{n})=\sum_{i=1}^{N}l_{n,RE,i}({\Greekmath 0112} _{n})$ where $ l_{n,RE,i}({\Greekmath 0112} _{n})=l_{RE,i}({\Greekmath 0112} ({\Greekmath 0112} _{n})).\,$To obtain results for the FE(Q)MLE one leaves out the last elements of ${\Greekmath 0112} _{0},$ ${\Greekmath 0112} , $ ${\Greekmath 0112} _{0,n}$ and ${\Greekmath 0112} _{n},$ i.e., the elements that correpond to $ \widetilde{{\Greekmath 0119} },$ $\widetilde{p},$ $\widetilde{{\Greekmath 0119} }_{n}$\ and $\widetilde{p }_{n}$, respectively. Hence it suffices to focus on the RE(Q)MLE.
Note that if ${\Greekmath 011A} =1$ and TSH$^{\ast }$ holds, then ${\Greekmath 0112} _{0}={\Greekmath 0112} _{0,n}=(1,0,{\Greekmath 011B} ^{2}{\Greekmath 0113} ^{\prime },0)^{\prime }$ for some ${\Greekmath 011B} ^{2}$ . Thus ${\Greekmath 0112} _{\ast }=(1,0,{\Greekmath 011B} ^{2}{\Greekmath 0113} ^{\prime },0)^{\prime }$ for some ${\Greekmath 011B} ^{2}$. We may partition ${\Greekmath 0112} _{0}$ as ${\Greekmath 0112} _{0}=({\Greekmath 011A} ,{\Greekmath 010E} ^{\prime })^{\prime }$, partition ${\Greekmath 0112} _{\ast }$ as ${\Greekmath 0112} _{\ast }=({\Greekmath 011A} _{\ast },{\Greekmath 010E} _{\ast }^{\prime })^{\prime }=(1,{\Greekmath 010E} _{\ast }^{\prime })^{\prime }$, partition ${\Greekmath 0112} _{0,n}$ as ${\Greekmath 0112} _{0,n}=({\Greekmath 011A} _{n},{\Greekmath 010E} _{n}^{\prime })^{\prime }$ and partition ${\Greekmath 0112} _{n}$ as ${\Greekmath 0112} _{n}=(r_{n},d_{n}^{\prime })^{\prime }.$ We will use $ l_{i}, $ $l_{n}$ and $l_{n,i}$ as short for $l_{RE,i}({\Greekmath 0112} ),$ $ l_{n,RE}({\Greekmath 0112} _{n})$ and $l_{n,RE,i}({\Greekmath 0112} _{n}),$ respectively. We also define:
Note that $U_{2,N}=(N^{-1/2}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}S_{i,1},$ $ N^{-1/2}\mathop{\textstyle \sum }\nolimits_{i=1}^{N}S_{i,2}^{\prime })^{\prime }$ and that condition (B2), which requires that for all $K\in \mathbb{R} ^{T+1}$ $\frac{\partial ^{2}l_{n}}{\partial {\Greekmath 011A} _{n}^{2}}|_{{\Greekmath 0112} _{\ast }}\neq K^{\prime }\frac{\partial l_{n}}{\partial {\Greekmath 010E} _{n}}|_{{\Greekmath 0112} _{\ast }}$ with positive probability, implies that $\mathcal{I}$ is nonsingular.
Condition (B3), which requires that for all $\overline{K}=(k$ $K^{\prime })^{\prime }\in \mathbb{R} ^{T+2}$\ $\frac{\partial ^{3}l_{n}}{\partial {\Greekmath 011A} _{n}^{3}}|_{{\Greekmath 0112} _{\ast }}\neq k\frac{\partial ^{2}l_{n}}{\partial {\Greekmath 011A} _{n}^{2}}|_{{\Greekmath 0112} _{\ast }}+K^{\prime }\frac{\partial l_{n}}{\partial {\Greekmath 010E} _{n}}|_{{\Greekmath 0112} _{\ast }}$ with positive probability, ensures that $U_{N}$ is not identically equal to zero.
Condition (B1) requires that $\frac{\partial l_{n}}{\partial {\Greekmath 011A} _{n}} |_{{\Greekmath 0112} _{\ast }}=0$ w.p.1.
In order to obtain the parametrization for which conditions (B1)-(B3) are met K2013 has used the formula ${\Greekmath 0112} _{n}={\Greekmath 0112} +[0,\widetilde{K }^{\prime }]^{\prime }(r-1),$ where $\widetilde{K}=\mathop{\textstyle \sum }_{i}E((\frac{ \partial l_{i}}{\partial {\Greekmath 011A} }|_{{\Greekmath 0112} _{\ast }})(\frac{\partial l_{i}}{ \partial {\Greekmath 010E} }|_{{\Greekmath 0112} _{\ast }}))\times $\ $[\mathop{\textstyle \sum }_{i}E(( \frac{\partial l_{i}}{\partial {\Greekmath 010E} }|_{{\Greekmath 0112} _{\ast }})(\frac{\partial l_{i}}{\partial {\Greekmath 010E} }|_{{\Greekmath 0112} _{\ast }})^{\prime })]^{-1}$, cf. RCBR p. 264.
Let $\mathbf{1}\{\breve{Z}_{1}>0\}=1$ if $\breve{Z}_{1}>0,$ let $\mathbf{1}\{ \breve{Z}_{1}>0\}=0$ if $\breve{Z}_{1}\leq 0,$ let $\mathbf{1}\{\breve{Z} _{1}\leq 0\}=1-\mathbf{1}\{\breve{Z}_{1}>0\},$ and let ${\Greekmath 0112} _{a,n}=({\Greekmath 011A} _{n},\widetilde{{\Greekmath 011B} }_{v,n}^{2},{\Greekmath 0110} _{n}^{\prime })^{\prime }.$ Then K2013 obtained the following result when the ${\Greekmath 0122} _{i}$ (or, equivalently, the data) are i.h.d.:
The QMLEs of ${\Greekmath 0112} $ are given by $\widehat{{\Greekmath 0112} }={\Greekmath 0112} (\widehat{ {\Greekmath 0112} }_{n})$. The parameter ${\Greekmath 011A} $ is second-order identified (cf. Sargan, 1983) because $\frac{\partial l_{n}}{\partial r_{n}}|_{{\Greekmath 0112} _{\ast }}=0,$ while $\frac{\partial ^{2}l_{n,i}}{\partial r_{n}^{2}}|_{{\Greekmath 0112} _{\ast }}\neq 0$, and because for all $K\in \mathbb{R} ^{T+1}$ $\frac{\partial ^{2}l_{n}}{\partial r_{n}^{2}}|_{{\Greekmath 0112} _{\ast }}\neq K^{\prime }\frac{\partial l_{n}}{\partial d_{n}}|_{{\Greekmath 0112} _{\ast }}$ with positive probability, that is, (B.2) holds. As a result the expansion of the log-likelihood involves $N^{1/4}(\widetilde{{\Greekmath 011A} }-1)$ rather than $ N^{1/2}(\widetilde{{\Greekmath 011A} }-1)$ and the QMLEs of $\left\vert {\Greekmath 011A} -1\right\vert $ converge at rate $O_{p}(N^{-1/4}).$ Furthermore, when $ {\Greekmath 0112} _{0,n}={\Greekmath 0112} _{\ast }$ and $\breve{Z}_{1,N}>0,$ $l_{n}({\Greekmath 0112} _{n})$ is bimodal. In this case the QMLE of ${\Greekmath 011A} $ is determined by higher-order terms in the expansion of $l_{n}({\Greekmath 0112} _{n})$ around ${\Greekmath 0112} _{n}={\Greekmath 0112} _{\ast }$, i.e., $U_{N}^{\prime }\breve{Z}_{N}$, cf. K2013. Although $\frac{ \partial l_{n}}{\partial d_{n}}|_{{\Greekmath 0112} _{\ast }}\neq 0$ a.s.\ and $ \widehat{{\Greekmath 010E} }_{n}-{\Greekmath 010E} _{n}=O_{p}(N^{-1/2}),$ whereas $\frac{\partial l_{n}}{\partial r_{n}}|_{{\Greekmath 0112} _{\ast }}=0$ a.s.\ and $\widehat{{\Greekmath 011A} } _{n}-1=O_{p}(N^{-1/4}),$ all elements of $\widehat{{\Greekmath 0112} }_{a,n}$ have a non-normal limiting distribution. All these findings would still hold if the model also included exogenous regressors.
Theorem 1 holds for i.h.d. data. In the special case where the ${\Greekmath 0122} _{i}$ are i.i.d. and normal, $\mathcal{H}=-\mathcal{I}$, $\breve{Z}\sim N(0, \mathcal{I}^{-1}),$ $U\perp \breve{Z}$ and hence $\Pr (\breve{B}=0|\breve{Z} _{1})=1/2,$ $\widehat{{\Greekmath 011A} }$ has a symmetric limiting distribution, and $ \widehat{{\Greekmath 011A} }_{RE}$ is asymptotically equivalent to $\widehat{{\Greekmath 011A} }_{FE}$ . However, in general (i.e., except for some special cases) if the $ {\Greekmath 0122} _{i}$ are i.i.d and non-normal or i.h.d., then $\mathcal{H}\neq - \mathcal{I}$, $E(U\breve{Z}^{\prime })\neq 0$ and hence $\Pr (\breve{B}=0| \breve{Z}_{1})\neq \Pr (\breve{B}=0)$, and $\widehat{{\Greekmath 011A} }$ has an asymmetric limiting distribution. Moreover, $\widehat{{\Greekmath 011A} }_{RE}$ is not asymptotically equivalent to $\widehat{{\Greekmath 011A} }_{FE}$.
It is well-known that (Quasi) ML estimators can be reinterpreted as GMM estimators, cf. e.g. Newey and McFadden (NMcF, 1994). The underlying moment conditions can be obtained by setting the expected score vector equal to zero. It follows that the Expected Hessian of the (quasi) log-likelihood function equals the first derivative of the vector of moment conditions exploited by the (Q)MLE with respect to the parameters. Therefore when the Expected Hessian of the (quasi) log-likelihood function is almost singular, the (Q)MLE suffers from a `weak moment conditions problem.' K2013 gives necessary and sufficient conditions for this situation to arise when $ Var(y_{i,1}-{\Greekmath 0116} _{i})\propto (1-{\Greekmath 011A} )^{0}=1$ $\forall i\in \{1,2,...,N\}$:
If the Expected Hessian of the (quasi) log-likelihood function is almost singular, one can obtain a better approximation to the finite sample distribution of the (Q)MLE than the usual approximation by using local asymptotics. K2013 obtained the following result:
Note that since $\widetilde{{\Greekmath 011B} }_{v,n}^{2}=\widetilde{{\Greekmath 011B} } _{v}^{2}/{\Greekmath 011B} _{2}^{2}-(1-{\Greekmath 011A} )=(1-{\Greekmath 011A} )((1-{\Greekmath 011A} ){\Greekmath 011B} _{v}^{2}/{\Greekmath 011B} _{2}^{2}-1)$ and $\tilde{{\Greekmath 0119}}_{n}=({\Greekmath 0119} -1)(1-{\Greekmath 011A} )$, both condition $ N^{1/2}(\widetilde{{\Greekmath 011B} }_{v,n,N}^{2}-0)=o(1)$ and condition $N^{1/2}( \tilde{{\Greekmath 0119}}_{n,N}-0)=o(1)$ normally require that $N^{1/2}({\Greekmath 011A} _{n,N}-1)=o(1) $ rather than $N^{1/4}({\Greekmath 011A} _{n,N}-1)=o(1).$ However, when for instance ${\Greekmath 0119} =1$ and ${\Greekmath 011B} _{v}^{2}=2{\Greekmath 011B} _{2}^{2}/(1-{\Greekmath 011A} ^{2}),$ then the results of theorem 3 hold under any $\{{\Greekmath 011A} _{n,N}\}$ such that $ N^{1/4}({\Greekmath 011A} _{n,N}-1)=o(1).$ In any case, when both $N^{1/4}({\Greekmath 011A} _{n,N}-1)=o(1)$ and $N^{1/2}({\Greekmath 010E} _{n,N}-{\Greekmath 010E} _{\ast })=o(1)$ for some $ {\Greekmath 011B} _{2}^{2},$ the RE-\ and the FEQMLE of ${\Greekmath 011A} $ are $N^{1/4}-$ consistent.
Results in RCBR suggest that in principle under $({\Greekmath 011A} _{n,N}-1)=-{\Greekmath 0115} N^{-b}$ different local asymptotic distributions and rates of convergence are obtained for the QMLE of ${\Greekmath 011A} $ when $b=1/4,$ $1/6<b<1/4$, $b=1/6$ or $ 0<b<1/6,$ respectively.
Wald test statistics, some versions of (Quasi) LM test statistics, and (Quasi) LR test statistics that are used for testing an hypothesis about $ {\Greekmath 011A} $ and are based on the reparametrized RE or FE\ likelihood do not uniformly converge to their fixed parameter first-order limiting distributions near the singularity point, cf. Bottai (2003). As a consequence these tests do not have correct asymptotic size in a uniform sense. (Q)LM test statistics that are standardised by (using a sandwich formula involving) the expected rather than the observed average Hessian are exceptions to this rule. Bottai (2003) explains why this is the case in the context of models with a single parameter. As we will show in the proof of theorem 4, when testing a hypothesis that includes a restriction on the parameter that has zero score at the singularity point, these test statistics still converge uniformly to their fixed parameter first-order limiting distribution, which is a central ${\Greekmath 011F} ^{2}$ -distribution, near the singularity point and these tests have correct asymptotic size in a uniform sense. Crucially, the expected average Hessian of $l_{n}({\Greekmath 0112} _{n})$, i.e., $\overline{H}(\mathcal{\breve{{\Greekmath 0112}}} _{n})\equiv E_{\mathcal{\breve{{\Greekmath 0112}}}_{n}}(\frac{1}{N}\frac{\partial ^{2}l_{n}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}\partial {\Greekmath 0112} _{n}^{\prime }} |_{\mathcal{\breve{{\Greekmath 0112}}}_{n}})$, is negative definite and hence nonsingular for any value of $\mathcal{\breve{{\Greekmath 0112}}}_{n}$ that differs from the singu- larity point ${\Greekmath 0112} _{\ast }$. Note that the values of the elements of $\overline{H}(\mathcal{\breve{{\Greekmath 0112}}}_{n})$ do not depend on the true distribution of the data. Alternative testing approaches that are based on a QLR test statistic but use a so-called type 2 robust critical value, which is determined by a statistic that indicates the closeness to the singularity point, or a least favourable critical value (cf. Andrews and Cheng, 2013) either do not work due to non-monotonicity of the limiting distributions of the QLR test statistic near the singularity point or are conservative and have low power.
We will now introduce the QLM test statistic $QLM(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ for testing $H_{0}:$ $A\mathcal{{\Greekmath 0112} }_{0,n}=a$, which includes a restriction on ${\Greekmath 011A} $, where $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}$ is a restricted QML\ estimate of $\mathcal{{\Greekmath 0112} }_{0,n}$ such that $A\widetilde{ \mathcal{{\Greekmath 0112} }}_{n}=a,$ $A$ is a $J\times \dim (\mathcal{{\Greekmath 0112} })$ constant matrix of rank $J,$ and $a\ $is a constant vector. Define the average information matrix $\overline{\mathcal{J}}\mathcal{(\mathcal{\breve{ {\Greekmath 0112}}}}_{n}\mathcal{)}=N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}\mathcal{J}_{i}\mathcal{( \mathcal{\breve{{\Greekmath 0112}}}}_{n}\mathcal{)}$ with $\mathcal{J}_{i}\mathcal{( \breve{{\Greekmath 0112}}}_{n}\mathcal{)}=\left( \frac{\partial l_{n,i}({\Greekmath 0112} _{n})}{ \partial {\Greekmath 0112} _{n}}|_{\mathcal{\breve{{\Greekmath 0112}}}_{n}}\right) \left( \frac{ \partial l_{n,i}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}^{\prime }}|_{\mathcal{ \breve{{\Greekmath 0112}}}_{n}}\right) ,$ where $l_{n,i}({\Greekmath 0112} _{n})$ is the contribution to the reparametrized log-likelihood function, $l_{n}({\Greekmath 0112} _{n})$, by individual $i$. If $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112} _{\ast }$ for all ${\Greekmath 011B} ^{2}>0$, then $QLM(\widetilde{\mathcal{{\Greekmath 0112} }} _{n})$ is given by, cf. White (1994, p. 173):
If $H_{0}$ is true and $\mathcal{{\Greekmath 0112} }_{0,n}\neq {\Greekmath 0112} _{\ast }$ (for all ${\Greekmath 011B} ^{2}>0$), then $QLM(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})\overset{d} {\rightarrow }{\Greekmath 011F} ^{2}(J).$ To test $H_{0}:$ ${\Greekmath 011A} =a$ when $\widetilde{ \mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112} _{\ast }$ (for all ${\Greekmath 011B} ^{2}>0$,) for some known value of $a\in (-1,1]$, one can use $QLM(\widetilde{\mathcal{ {\Greekmath 0112} }}_{n})$ in ((ref)) with $A=(1$ $\mathbf{0}^{\prime })$ and $ \frac{\partial l_{n}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})}{\partial {\Greekmath 0112} _{n} }=A^{\prime }\frac{\partial l_{n}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})}{ \partial {\Greekmath 011A} }$. Note also that, unlike the values of the (Quasi) Wald and Hausman test-statistics that use $\overline{H}^{-1}(\widehat{ \mathcal{{\Greekmath 0112} }}_{n})$ (cf. White, 1994, p. 173), the value of $QLM( \widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ remains the same when $\overline{H}^{-1}( \mathcal{\breve{{\Greekmath 0112}}}_{n})$ for some $\mathcal{\breve{{\Greekmath 0112}}}_{n}$ is replaced by $adj(\overline{H}(\mathcal{\breve{{\Greekmath 0112}}}_{n}))$. Furthermore, if $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112} _{\ast }$, then $\overline{ \mathcal{J}}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ and p$\lim_{N\rightarrow \infty }\overline{\mathcal{J}}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ are positive definite.
If $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$ for some ${\Greekmath 011B} ^{2}>0,$ $rk(\overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}))=\dim ( \overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}))-1$ and hence $rk(adj( \overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})))=1.$ Furthermore, $ \overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})_{i,j}=0$ iff $i=1$ and/or $ j=1,$ $adj(\overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}))_{i,j}\neq 0$ iff $i=j=1,$ and $adj(\overline{H}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}))(1$ $ \mathbf{0}^{\prime })^{\prime }\propto (1$ $\mathbf{0}^{\prime })^{\prime }.$ Hence if $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$ (for some $ {\Greekmath 011B} ^{2}>0$), the QLM test statistic in ((ref)) for $H_{0}:$ ${\Greekmath 011A} =a$ would be equal to $(\frac{\partial l_{n}(\mathcal{{\Greekmath 0112} }_{n})}{\partial r} |_{{\Greekmath 0112} _{\ast }})^{2}/(\mathop{\textstyle \sum }_{i=1}^{N}(\frac{\partial l_{n,i}(\mathcal{ {\Greekmath 0112} }_{n})}{\partial r}|_{{\Greekmath 0112} _{\ast }})^{2})$. However, this test statistic cannot be used, because $\frac{\partial l_{n,i}(\mathcal{{\Greekmath 0112} } _{n})}{\partial r}|_{{\Greekmath 0112} _{\ast }}=0$ a.s.\thinspace and p$ \lim_{N\rightarrow \infty }N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}(\frac{\partial l_{n,i}( \mathcal{{\Greekmath 0112} }_{n})}{\partial r}|_{{\Greekmath 0112} _{\ast }})^{2}=0$. More generally, $N^{-1/2}A\,adj(\overline{H}({\Greekmath 0112} _{\ast }))\frac{\partial l_{n}(\mathcal{{\Greekmath 0112} }_{n})}{\partial \mathcal{{\Greekmath 0112} }_{n}}|_{{\Greekmath 0112} _{\ast }}=0$ a.s. and p$\lim_{N\rightarrow \infty }[A\,adj(\overline{H} ({\Greekmath 0112} _{\ast }))\times $ $\overline{\mathcal{J}}({\Greekmath 0112} _{\ast })adj( \overline{H}({\Greekmath 0112} _{\ast }))A^{\prime }]=\mathbf{0.}$ To test $ H_{0}:\nolinebreak $ $\nolinebreak A\mathcal{{\Greekmath 0112} }_{0,n}=a$ when $ \widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$ (for some ${\Greekmath 011B} ^{2}>0$ ), one can use the following QLM test statistic, cf. Bottai (2003):
with
$\vspace*{-0.25in}$and
where we have used $l_{n}$ and $l_{n,i}$ as short for $l_{n}({\Greekmath 0112} _{n})$ and $l_{n,i}({\Greekmath 0112} _{n})$. In contrast, the Wald and Hausman test-statistics that use $\overline{H}^{-1}(\widehat{\mathcal{{\Greekmath 0112} }}_{n})$ when $\widehat{\mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112} _{\ast }$ cannot be defined when $\widehat{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }.$ If $H_{0}$ is true and $\mathcal{{\Greekmath 0112} }_{0,n}={\Greekmath 0112} _{\ast }$ for some ${\Greekmath 011B} ^{2}>0$, then $QLM({\Greekmath 0112} _{\ast })\overset{d}{\rightarrow }{\Greekmath 011F} ^{2}(J).$ To test $H_{0}:$ ${\Greekmath 011A} =a$ when $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$, one can use $QLM(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ in ((ref)) with $A=(1$ $ \mathbf{0}^{\prime })$ and $\widetilde{S}(\widetilde{\mathcal{{\Greekmath 0112} }} _{n})=A^{\prime }(\sum\nolimits_{i=1}^{N}S_{i,1})$. It can easily be shown that the test statistic $QLM(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ for testing $H_{0}:$ ${\Greekmath 011A} =a$ given by ((ref)) and ((ref)) is continuous at $ \widetilde{\mathcal{{\Greekmath 0112} }}_{n}={\Greekmath 0112} _{\ast }$ (for any ${\Greekmath 011B} ^{2}>0$) by using de l'H\^{o}pital's rule twice.
Confidence Sets (CSs) that have correct uniform asymptotic size can be obtained by "inverting" the QLM test, i.e., $QLM(\widetilde{\mathcal{{\Greekmath 0112} } }_{n})$ given by ((ref)) and ((ref)), or any of the other aforementioned tests that have correct uniform asymptotic size. For instance, a CS for ${\Greekmath 011A} $ of level $(1-{\Greekmath 010B} )\ast 100\%$ constructed in this way is the set of points $a\in {\mathbb{R}}$ for which the test fails to reject $H_{0}:{\Greekmath 011A} =a$ at significance level ${\Greekmath 010B} .$
When the data are i.i.d. and normal and ${\Greekmath 011A} $ is not local or equal to one, the (Q)LM test that uses the expected Hessian has optimal local power properties. Next we derive the power envelope of a centered version of $QLM( \widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ for testing $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} / \sqrt[4]{N}$, where ${\Greekmath 0114} $ is a constant, viz. $QLM^{c}(\widetilde{ \mathcal{{\Greekmath 0112} }}_{n}),$ when the data are i.i.d. and normal and ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast }.$ Let $\overline{\mathcal{J}}^{c}\mathcal{({\Greekmath 0112} }_{n} \mathcal{)}=N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}\mathcal{J}_{i}^{c}\mathcal{({\Greekmath 0112} }_{n} \mathcal{)}$ with $\mathcal{J}_{i}^{c}\mathcal{(\mathcal{\mathcal{\breve{ {\Greekmath 0112}}}}}_{n}\mathcal{)}=(\frac{\partial l_{n,i}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}}|_{\mathcal{\breve{{\Greekmath 0112}}}_{n}}-N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}(\frac{ \partial l_{n,i}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}}|_{\mathcal{\breve{{\Greekmath 0112} }}_{n}}))(\frac{\partial l_{n,i}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}^{\prime } }|_{\mathcal{\breve{{\Greekmath 0112}}}_{n}}-N^{-1}\mathop{\textstyle \sum }_{i=1}^{N}(\frac{\partial l_{n,i}({\Greekmath 0112} _{n})}{\partial {\Greekmath 0112} _{n}^{\prime }}|_{\mathcal{\breve{ {\Greekmath 0112}}}_{n}}))$. If $\widetilde{\mathcal{{\Greekmath 0112} }}_{n}\neq {\Greekmath 0112} _{\ast },$ we have
where $A=(1$ $\mathbf{0}^{\prime }).$ If $\widetilde{\mathcal{{\Greekmath 0112} }} _{n}={\Greekmath 0112} _{\ast },$ $QLM^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ is similar to ((ref)) with $\widetilde{S}(\widetilde{\mathcal{{\Greekmath 0112} }} _{n})=A^{\prime }(\sum\nolimits_{i=1}^{N}S_{i,1})$ and $\widetilde{\mathcal{J }}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ replaced by $\widetilde{\mathcal{J}} ^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})=N^{-1}\sum\nolimits_{i=1}^{N}(S_{i}- \overline{S})(S_{i}-\overline{S})^{\prime },$ where $\overline{S} =N^{-1}\sum\nolimits_{i=1}^{N}S_{i}$.
If mean stationarity ((ref)) holds when $\left\vert {\Greekmath 011A} \right\vert <1 $ and $Var(\Delta y_{i,t})/Var(y_{i,1})=o(N^{-1}),$ $i=1,2,...,N,$ then only the full set of moment conditions based on differenced data matters asymptotically for identification of ${\Greekmath 011A} $ and there is no loss in efficiency when exploiting only the latter for estimation or testing purposes; additional moment conditions involving levels of the data are redundant (cf. Kruiniger, 2022, and Bun and Kleibergen, 2022). Note that $Var(\Delta y_{i,t})=O(1)$ for any ${\Greekmath 011A} \in (-1,1].$ We will investigate the power of the $QLM^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ tests under a worst case scenario where ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast }$ and $ 1/Var(y_{i,1})=o(N^{-1})$ $i=1,2,...,N$ and consider a sequence of null hypotheses that is local-to-unity: $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ (cf. Bun and Kleibergen, 2022). In this scenario it is sufficient to focus on the FE version of $QLM^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}),$ viz. $ QLM_{FE}^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n}).$
The quartic root rate in the sequence of hypotheses is related to the fact that ${\Greekmath 011A} $ is only second-order identified when ${\Greekmath 0112} _{0}={\Greekmath 0112} _{\ast }$. When we impose TSH on $l_{n}({\Greekmath 0112} _{n}),$ we obtain the following result:
We will obtain the maximal attainable power (MAP) curve for testing $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ by first considering the asymptotic distribution of an GMM-Anderson-Rubin statistic which tests $H_{0}$ by using all the moment conditions based on first-differences of the data whilst (the true value of) ${\Greekmath 011A} =1.$ This GMM-AR statistic is given by
with$\vspace{-0.12in}$
where $D_{r}$ is a $(T-1)\times (T-1)$ band matrix with $(D_{r})_{i,i}=1$ and $(D_{r})_{i+1,i}=-r$ for $i=1,2,...,T-2,($and $T-1)$ and $ (D_{r})_{i,j}=0 $ elsewhere; $P=(0$ $g$ $I_{p})$ is a $p\times \frac{1}{2} (T-1)T$ matrix where $p=\frac{1}{2}T(T-1)-2$ and $g$ is a $p$-vector such that $(1,-1,$ $g^{\prime })^{\prime }=vech(D_{r}D_{r}^{\prime })$; and $ \widehat{V}_{mm}({\Greekmath 011A} )$ is the Eicker-White estimator of the covariance matrix of $m({\Greekmath 011A} ).$
We obtain the MAP curve for testing $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ by deriving the asymptotic distribution of an GMM-Anderson-Rubin statistic which tests $H_{0}$ by using the (infeasible) weighted average of the moment conditions in $E(m({\Greekmath 011A} ))=0_{p}$ (with $m(r)$ given in ((ref))) that leads to the largest value of the non-centrality parameter of this distribution whilst (the true value of) ${\Greekmath 011A} =1$.
This result implies that, like the KLM-statistic, the centered LM-statistic $ QLM^{c}(\widetilde{\mathcal{{\Greekmath 0112} }}_{n})$ is efficient both when ${\Greekmath 011A} $ is less than one and when ${\Greekmath 011A} $ is equal to one.
To test $H_{0}:$ ${\Greekmath 011A} =1$ one could also use a Wald test based on $\sqrt{N}( \widehat{{\Greekmath 011A} }-1)^{2}$ where $\widehat{{\Greekmath 011A} }$ is the REQMLE or FEQMLE of $ {\Greekmath 011A} .$ Under $H_{0}\ $we have $\sqrt{N}(\widehat{{\Greekmath 011A} }-1)^{2}\overset{d}{ \rightarrow }\breve{Z}_{1}\mathbf{1}\{\breve{Z}_{1}>0\},$ cf. Theorem 1. Recall that $\breve{Z}_{N}=(\breve{Z}_{1,N},\breve{Z}_{2,N}^{\prime })^{\prime }=-\mathcal{H}^{-1}U_{2,N}\overset{d}{\rightarrow }\breve{Z}$ with $U_{2,N}=(\frac{1}{2!}N^{-1/2}\frac{\partial ^{2}l_{n}}{\partial r_{n}^{2}}|_{{\Greekmath 0112} _{\ast }},N^{-1/2}\frac{\partial l_{n}}{\partial d_{n}^{\prime }}|_{{\Greekmath 0112} _{\ast }})^{\prime }$ and $\breve{Z}=(\breve{Z} _{1},\breve{Z}_{2}^{\prime })^{\prime }\sim N(0,\mathcal{H}^{-1}\mathcal{IH} ^{-1}).$ When the data are i.i.d. non-normal or i.h.d., one can bootstrap the distribution of $U_{2,N}$. To do this, one can make use of the fact that under $H_{0}$, ${\Greekmath 0122} _{i}=y_{i}-y_{i,-1}$ for $i=1,...,N.$
In this section we investigate through Monte Carlo simulations the empirical size and power properties of QLM-tests for testing a simple hypothesis of the type $H_{0}:$ ${\Greekmath 011A} =a$, namely $QLM(\mathcal{{\Greekmath 011A} }),$ that are based on the RE and FE\ likelihood functions for various panel AR(1) models without covariates. The data were generated using the panel AR(1) model given in ((ref)). We study how the properties of these tests are affected if we change (1) the value of ${\Greekmath 011A} ,$ (2) the distributions of the initial conditions $v_{i,1}=y_{i,1}-{\Greekmath 0116} _{i}$, (3) the distributions of the idiosyncratic errors (the ${\Greekmath 0122} _{i,t}$) and/or (4) the ratio of the variances of the error components. We conducted the simulation experiments for $(T,N)=(4,100),$ $(9,100),$ $(4,250)$ or $(9,250).$ The nominal size of the tests was $0.05$ and for each scenario the number of replications was 2500.
We calculated the empirical size of the tests for ${\Greekmath 011A} =0.2,$ $0.5,$ $0.8,$ $0.9,$ $0.95,$ $0.98$ or $0.99$ and we calculated the empirical power of the tests for $H_{0}:{\Greekmath 011A} =0.8$ when ${\Greekmath 011A} =0.5,$ $0.6,$ $0.7,$ $0.9,$ $0.95$ or $0.99.$
In all simulation experiments the individual effects, the ${\Greekmath 0116} _{i}$, were i.i.d. $N(0,{\Greekmath 011B} _{{\Greekmath 0116} }^{2})$ with ${\Greekmath 011B} _{{\Greekmath 0116} }^{2}=1$ or $25.$ The errors, the ${\Greekmath 0122} _{i,t}$, were either i.i.d. $N(0,1)$ or i.i.d. $ ({\Greekmath 011F} ^{2}(1)-1)/\sqrt{2}.$ Note that in both cases $Var({\Greekmath 0122} _{i,t})=1$ for $i=1,...,N$ and $t=2,...,T$.
In order to assess how the assumptions with respect to $y_{i,1}-{\Greekmath 0116} _{i}$, $ i=1,...,N,$ affect the finite sample properties of the tests, we conducted three different kinds of experiments: in one set, labeled NS-Normal, the initial observations are non-stationary, i.e., $y_{i,1}-{\Greekmath 0116} _{i}=0$, $ i=1,...,N,$ and the ${\Greekmath 0122} _{i,t}\sim N(0,1)$, whereas in the other two sets, labeled S-Normal and S-ChiSq., respectively, the initial observations are drawn from stationary distributions, i.e., either $ (y_{i,1}-{\Greekmath 0116} _{i})\sim N(0,1/(1-{\Greekmath 011A} ^{2}))$ when the ${\Greekmath 0122} _{i,t}\sim N(0,1)$ or $(y_{i,1}-{\Greekmath 0116} _{i})\sim ({\Greekmath 011F} ^{2}(1)-1)/\sqrt{ 2(1-{\Greekmath 011A} ^{2})}$ when the ${\Greekmath 0122} _{i,t}\sim ({\Greekmath 011F} ^{2}(1)-1)/\sqrt{2}$ . Note that in design NS-Normal the data are still mean stationary, i.e., $ E(y_{i,1}-{\Greekmath 0116} _{i})=0$ and $E({\Greekmath 0116} _{i}(y_{i,1}-{\Greekmath 0116} _{i}))=0$, and that in all designs $E(y_{i,t}-y_{i,t-1})=0$.
In the cases of the RE and FE models, $(1-{\Greekmath 011A} ){\Greekmath 0116} _{i}+{\Greekmath 0122} _{i}$ is decomposed as $(1-{\Greekmath 011A} ){\Greekmath 0119} y_{i,1}+(1-{\Greekmath 011A} )v_{i}+{\Greekmath 0122} _{i}=(1-{\Greekmath 011A} ){\Greekmath 0119} y_{i,1}+u_{i}$ with ${\Greekmath 0119} =1$ for the FE case. In the experiments we imposed homoskedasticity on the likelihood functions and added the restrictions ${\Greekmath 011B} ^{2}>0$ and $\widetilde{{\Greekmath 011B} }_{v}^{2}\geq 0 $ or the restrictions ${\Greekmath 011B} ^{2}>0$ and $(T-1)\widetilde{{\Greekmath 011B} } _{v}^{2}+{\Greekmath 011B} ^{2}>0$ in case the restriction $\widetilde{{\Greekmath 011B} } _{v}^{2}\geq 0$ was binding to ensure that the estimates of $ E(u_{i}u_{i}^{\prime })$ were positive definite.
We allowed for time effects by subtracting cross-sectional averages from the data.
Note that the QML estimators suffer from a weak moment conditions problem when ${\Greekmath 011A} $ is close to one.
Tables 1-12 report the simulation results in terms of the relative rejection frequenties of the tests. Tables 1-4, 9, 10 report results concerning the empirical size of the tests whereas tables 5-8, 11, 12 report results concerning the empirical power of the tests. The tables also differ with respect to the the value of ${\Greekmath 011B} _{{\Greekmath 0116} }^{2}$: tables 1-8 correspond to $ {\Greekmath 011B} _{{\Greekmath 0116} }^{2}=1$ and report results for both RE and FE versions of the QLM tests whereas tables 9-12 correspond to ${\Greekmath 011B} _{{\Greekmath 0116} }^{2}=25$ and report results for the RE version of the QLM\ test only because changes in the value of ${\Greekmath 011B} _{{\Greekmath 0116} }^{2}$ do not affect the FE affects version of the QLM test by construction. If the tests have correct size (i.e., 0.05), then the standard errors of the estimates of the empirical size are $\sqrt{ 0.05\ast (1-0.05)}/\sqrt{2500}\approx 0.0044.$
Inspection of the results in tables 1-4, 9 and 10 leads to the following conclusions regarding the empirical size of the QLM tests:
Inspection of the results in tables 5-8, 11 and 12 leads to the following conclusions regarding the empirical power of the QLM tests:
In this paper we proposed new ML based inference methods for panel AR models with arbitrary initial conditions and heteroskedasticity and possibly additional regressors that are robust to the strength of identification. Specifically, we showed that (Quasi) LM tests and CSs that use the expected Hessian rather than the observed Hessian of the RE or the FE\ log-likelihood function have correct asymptotic size in a uniform sense. We also derived the power envelope of a FE version of such an LM test for testing $H_{0}:$ $ {\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ when the average information matrix is estimated by a centered OPG\ estimator and the model is only second-order identified, and showed that it coincides with the maximal attainable power curve for testing $H_{0}:$ ${\Greekmath 011A} =a=1-{\Greekmath 0114} /\sqrt[4]{N}$ in the worst case setting. In a Monte Carlo study that included a variety of experiments we found that these (Quasi) LM tests have correct empirical size and good empirical power properties.
None of the existing ML based inference methods for dynamic panel data models have correct uniform asymptotic size close to the point in the parameter space at which such a model is only second-order identified and therefore the methods proposed in this paper will result in more reliable inference. The proposed methods can be adapted to generalizations of the panel AR(1) model that was considered in this paper including dynamic panel models with a factor structure and panel VAR models. Similarly, Quasi LM\ test that have correct uniform asymptotic size can also be developed for other multi-parameter models that are only second-order identified at some point in the parameter space, such as the examples mentioned in e.g. Rotnitzky et. al. (2000) and Dovonon and Hall (2018).
An issue that requires additional study is the effect of the choice of the estimator for the average information matrix on the power of the QLM tests.