GEL statistics under weak identification

Abstract

The central concern of this paper is the provision in a time series moment condition framework of practical recommendations of confidence regions for parameters whose coverage probabilities are robust to the strength or weakness of identification. To this end we develop Pearson-type test statistics based on GEL implied probabilities formed from general kernel smoothed versions of the moment indicators. We also modify the statistics suggested in Guggenberger and Smith (2008) for a general kernel smoothing function. Importantly for our conclusions, we provide GEL time series counterparts to GMM and GEL conditional likelihood ratio statistics given in Kleibergen (2005) and Smith (2007). Our analysis not only demonstrates that these statistics are asymptotically (conditionally) pivotal under both classical asymptotic theory and weak instrument asymptotics of Stock and Wright (2000) but also provides asymptotic power results in the weakly identified time series context. Consequently, the empirical null rejection probabilities of the associated tests and, thereby, the coverage probabilities of the corresponding confidence regions, should not be affected greatly by the strength or otherwise of identification. A comprehensive Monte Carlo study indicates that a number of the tests proposed here represent very competitive choices in comparison with those suggested elsewhere in the literature.

Introduction

In many situations empirical researchers are confronted with instrumental variables only weakly correlated with the endogenous variables with the consequence that structural parameters are only weakly identified. This paper contributes to what is now a voluminous literature on the topic of partial and weak identification initiated in Sargan (1983), Phillips (1989), and Staiger and Stock (1997). Those papers demonstrate that the correspondence between the limiting and finite-sample behaviours of estimators and test statistics may be poor in such circumstances.¹ The particular aim of this paper then is the practical recommendation of test statistics appropriate for nonlinear moment condition models with time series data that possess finite sample size properties relatively unaffected by the weakness or strength of identification and that also display good power. Consequently their acceptance regions may be inverted to provide reliable and robust confidence regions for parameters of interest.

The contributions of the paper are as follows. We propose new test statistics for a time series setting based on the implied probabilities generated by generalised empirical likelihood (GEL) estimation, see, e.g., Smith, 1997, Smith, 2011 and Newey and Smith (2004), appropriate for circumstances in which identification may be weak as discussed in Stock and Wright (2000), henceforth SW. These statistics generalise to the time series context versions of the Pearson-type GEL statistics suggested in Ramalho and Smith (2004) for independently and identically distributed (i.i.d.) observations and are defined in terms of smoothed moment indicators obtained using kernel function based weights which incorporate a bandwidth parameter; cf. Kitamura and Stutzer (1997) and Smith, 1997, Smith, 2000, Smith, 2011. We also reconsider the statistics developed in Guggenberger and Smith (2005), hereafter GSa, for i.i.d. observations and those proposed for time series data in Guggenberger and Smith (2008), henceforth GSb, that use a truncated kernel function for the smoothed moment indicators. In particular, since their finite sample behaviour may be affected by the choice of kernel function, we reformulate the GSb statistics for a general form of kernel function. Likewise, our Pearson-type GEL statistics also incorporate a general kernel function. The tests discussed here deal with both simple hypotheses on the full parameter vector and composite hypotheses on a subvector of parameters, the latter case, apart from one particular statistic, requiring that the parameters not under test be strongly identified, an assumption adopted in other papers, see, e.g., Kleibergen (2005), GSa, Otsu (2006) and GSb. In addition, and importantly for our conclusions, we propose GEL time series counterparts to the GMM conditional likelihood ratio (CLR) statistic in Kleibergen (2005) and the GEL CLR statistics developed in Smith (2007) for the i.i.d context; cf. Moreira (2003).

We investigate the large sample behaviour of the statistics proposed here at fixed alternative values for the weakly identified parameters, a scenario that permits a comparison of their relative asymptotic power properties in the weakly identified context. Importantly, under the null hypothesis, the statistics proposed here, in GSb and elsewhere, e.g., in Kleibergen (2005) and Otsu (2006), are asymptotically (conditionally) pivotal, unlike the corresponding Wald and likelihood ratio statistics. Indeed, with the notable exception of the GEL CLR statistics, whose large sample conditional null distribution may be straightforwardly simulated, our statistics are asymptotically chi-square under the null hypothesis. Thus, the empirical rejection probabilities under the null hypothesis (NRPs) of tests formed from these statistics may be expected to be relatively invariant to the strength or weakness of identification. Our elucidation of the asymptotic properties of the statistics also demonstrates certain asymptotic equivalences among classes of statistics. Hence, without recourse to higher order analysis, reliance must be placed on simulation experiments for statistical choice. Consequently, in order to inform our recommendations for econometric practice, we undertake a comprehensive Monte Carlo study to assess the finite sample size and power characteristics of the statistics proposed here based on a similar design to that in GSb. For comparative purposes we also include in this study a number of additional tests favourably reported in the recent weak instrument literature; see Kleibergen (2005), Otsu (2006) and GSb. To anticipate our conclusions, particular implementations of the GEL CLR and Pearson-type tests have satisfactory finite sample size properties and power characteristics that are very competitive with existing tests.

The paper is organised as follows. Section 2 reviews GEL estimation and associated constructs. Statistics appropriate for tests on the full vector of parameters are introduced in Section 3 and their limiting behaviour stated whereas those appropriate for the subvector case are discussed in Section 4. Section 5 describes the results of the Monte Carlo study. Section 6 concludes. The appendices contain the technical assumptions and proofs of results contained in the text.

The following notation is used in the paper. The symbols “$\stackrel{p}{\to }$”, “$\stackrel{d}{\to }$” and “$\Rightarrow$” denote convergence in distribution, convergence in probability and weak convergence of empirical processes, respectively, with “$\stackrel{a}{⪯}$” indicating that the limiting distribution of the left hand side expression is bounded above by that of the right hand side. Convergence “with probability approaching 1” is written as “w.p.a.1”. The space $C^i\left(S\right)$ contains all functions that are $i$ times continuously differentiable on the set $S$. The notation $\mathrm{vec}\left(M\right)$ stands for the column vectorisation of the $k\times p$ matrix $M$, i.e., if $M=(m_1,\dots ,m_p)$ then $\mathrm{vec}\left(M\right)≔{(m_1^{\prime },\dots ,m_p^{\prime })}^{\prime }$, “$M^{\prime }$” denotes the transpose matrix of $M$ and ${\left(M\right)}_{i,j}$ the element in the $i$-th row and $j$-th column. Furthermore, for a symmetric matrix $M$, “$M>0$” means that $M$ is positive definite, ${\lambda }_{\min }\left(M\right)$ and ${\lambda }_{\max }\left(M\right)$ are the smallest and largest eigenvalues of $M$, respectively, and $\Vert M\Vert ≔\sqrt{{\lambda }_{\max }\left(M^{\prime }M\right)}$. The vector inequality $y\le z$ is to be interpreted as the inequality $\le$ holding element-wise. For a full column rank matrix $A\in R^{k\times p}$ and $0<K\in R^{k\times k}$, we denote by $P_A\left(K\right)$ the oblique projection matrix $A{\left(A^{\prime }{K}^{-1}A\right)}^{-1}{A}^{\prime }{K}^{-1}$ and define $M_A\left(K\right)≔I_k-P_A\left(K\right)$, where $I_k$ is the $k$-dimensional identity matrix. We abbreviate this notation to $P_A$ and $M_A$ if $K=I_k$ and, if $p=0$, set $M_A=I_k$. Finally ${\chi }^2\left(k\right)$ and ${\chi }^2(k;\delta )$ respectively denote central chi-square and noncentral chi-square random variables with $k$ degrees of freedom and noncentrality parameter $\delta$.