Tests for Over-identifying Restrictions in Partially Identified Linear Structural Equations

Cragg and Donald (1996) have pointed out that the asymptotic size of tests for overidentifying restrictions can be much smaller than the asymptotic nominal size when the structural equation is partially identified. This may lead to misleading inference if the critical values are obtained from a chi-square distribution. To overcome this problem we derive the exact asymptotic distribution of the Byron test statistic. This allows the calculation of asymptotic critical values and p-values corrected for possible failure of identification.Invariant Tests, Over-identifying restrictions, Partially identified structural equation

The Exact Cumulative Distribution Function of a Ratio of Quadratic Forms in Normal Variables with Application to the AR(1) Model.

Often neither the exact density nor the exact cumulative distribution function (CDF) of a statistic of interest are available in the statistics and econometrics literature (for example the maximum likelihood estimator of the autocorrelation coefficient in a simple Gaussian AR(1) model with zero start-up value). In other cases the exact CDF of a statistic of interest is very complicated despite the statistic being “simple” (for example the circular serial correlation coefficient, or a quadratic form of a vector uniformly distributed over the unit n-sphere). The first part of the paper tries to explain why this is the case by studying the analytic properties of the CDF of a statistic under very general assumptions. Differential geometric considerations show that there can be points where the CDF of a given statistic is not analytic, and such points do not depend on the parameters of the model but only on the properties of the statistic itself. The second part of the paper derives the exact CDF of a ratio of quadratic forms in normal variables, and for the first time a closed form solution is found. These results are then specialised to the maximum likelihood estimator of the autoregressive parameter in a Gaussian AR(1) model with zero start-up value, which is shown to have precisely those properties highlighted in the first part of the paper.

On the Bimodality of the Exact Distribution of the TSLS Estimator

Nelson and Startz (Econometrica, 58, 1990), Maddala and Jong (Econometrica, 60, 1992) and Wolgrom (Econometrica, 69, 2001) have shown that the density of the two-stage least squares estimator may be bimodal in a just identified structural equation. This paper further investigates the conditions under which bimodality may arise in a just over-identified model.Bimodality, Identification, Structural equation, Two Stage Least Squares.

The Asymptotic distribution of the LIML Estimator in a Partially Identified Structural Equation

We derive general formulae for the asymptotic distribution of the LIML estimator for the coefficients of both endogenous and exogenous variables in a partially identified linear structural equation. We extend previous results of Phillips (1989) and Choi and Phillips (1992) where the focus was on IV estimators. We show that partial failure of identification affects the LIML in that its moments do not exist even asymptotically.LIML estimator, Partial Identification, Linear structural equation, Asymptotic distribution

Weighted Average Power Similar Tests for Structural Change for the Gaussian Linear Regression Model

The average exponential tests for structural change of Andrews and Ploberger (Econometrica, 62, 1994) and Andrews, Lee and Ploberger (Journal of Econometrics 70, 1996) and modifications thereof maximize a weighted average power which incorporates specific weighting functions in order to make the resulting test statistics simple. Generalizations of these tests involve the numerical evaluation of (potentially) complicated integrals. In this paper we suggest a uniform Laplace approximation to evaluate weighted average power test statistics for which a simple closed form does not exist. We also show that a modification of the avg-F test is optimal under a very large class of weighting functions and can be written as a ratio of quadratic forms. Finally, we discuss how the computational burden of averaging over all possible change-points can be addressed.Linear Regression Model, Similar Tests, Invariant Tests, Structural Change, Weighted Average Power Tests, Laplace Approximation, Uniform Laplace Approximation.

The Density of the Sufficient Statistics for a Gaussian AR(1) Model in Terms of Generalized Functions

This paper derives the exact joint distribution of the minimal sufficient statistics in the first-order AR(1) model with Gaussian errors and zero start-up value. The results are fundamental to an exact distribution theory for the statistics that are typically of interest in this model.Gaussian AR(1); Generalized Functions.

Conditional inference for possibly unidentified structural equations

The possibility that a structural equation may not be identified casts doubt on the measures of estimator precision that are normally used. We argue that the observed identifiability test statistic is directly relevant to the precision with which the structural parameters can be estimated, and hence argue that inference in such models should be conditioned on the observed value of that statistic (or statistics). We examine in detail the effects of conditioning on the properties of the ordinary least squares (OLS) and two-stage least squares (TSLS) estimators for the coefficients of the endogenous variables in a single structural equation. We show that: (a) conditioning has very little impact on the properties of the OLS estimator, but a substantial impact on those of the TSLS estimator; (b) the conditional variance of the TSLS estimator can be very much larger than its unconditional variance (when the identifiability statistic is small), or very much smaller (when the identifiability statistic is large); and (c) conditional mean-square-error comparisons of the two estimators favour the OLS estimator when the sample evidence only weakly supports the identifiablity hypothesis, can favour TSLS slightly when that evidence is moderately favourable, but there is nothing to choose between the two estimators when the data strongly supports the identification hypothesis

Ill-posed Problems and Instruments' Weakness

Potscher (Econometrica, 2002) has pointed out that several estimation problems in econometrics are ill-posed. This paper further studies the nature of ill-posed problems in parametric models. Our starting point is that both parameters and estimators may be seen as maps from the manifold of density functions to an m-dimensional Euclidean space, and we investigate the properties that these maps have to transmit perturbations. In the special case of structural equations models, we argue that this framework provides coherent measures of instruments' weaknessIll-posed Problems, Weak Instruments, Parametric Models