Isaiah Andrews

Maximum Likelihood Inference in Weakly Identified DSGE Models

This paper examines the problem of weak identification in maximum likelihood, motivated by problems with estimation and inference a multi-dimensional, non-linear DSGE model. We suggest a test for a simple hypothesis concerning the full parameter vector which is robust to weak identification. We also suggest a test for a composite hypothesis regarding a sub-vector of parameters. The suggested test is shown to be asymptotically exact when the nuisance parameter is strongly identified, and in some cases when the nuisance parameter is weakly identified. We pay particular attention to the question of how to estimate Fisher’s information, and make extensive use of martingale theory.

Conditional Linear Combination Tests for Weakly Identified Models

We introduce the class of conditional linear combination tests, which reject null hypotheses concerning model parameters when a data-dependent convex combination of two identification-robust statistics is large. These tests control size under weak identification and have a number of optimality properties in a conditional problem. We show that the conditional likelihood ratio test of Moreira, 2003 is a conditional linear combination test in models with one endogenous regressor, and that the class of conditional linear combination tests is equivalent to a class of quasi-conditional likelihood ratio tests. We suggest using minimax regret conditional linear combination tests and propose a computationally tractable class of tests that plug in an estimator for a nuisance parameter. These plug-in tests perform well in simulation and have optimal power in many strongly identified models, thus allowing powerful identification-robust inference in a wide range of linear and nonlinear models without sacrificing efficiency if identification is strong.

A Geometric Approach to Nonlinear Econometric Models

Conventional tests for composite hypotheses in minimum distance models can be unreliable when the relationship between the structural and reduced‐form parameters is highly nonlinear. Such nonlinearity may arise for a variety of reasons, including weak identification. In this note, we begin by studying the problem of testing a “curved null” in a finite‐sample Gaussian model. Using the curvature of the model, we develop new finite‐sample bounds on the distribution of minimum‐distance statistics. These bounds allow us to construct tests for composite hypotheses which are uniformly asymptotically valid over a large class of data generating processes and structural models.

Conditional Inference With a Functional Nuisance Parameter

This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite-dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi-likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.

Valid Two-Step Identification-Robust Confidence Sets for GMM

In models with potentially weak identification, researchers often decide whether to report a robust confidence set based on an initial assessment of model identification. Two-step procedures of this sort can generate large coverage distortions for reported confidence sets, and existing procedures for controlling these distortions are quite limited. This paper introduces a generally applicable approach to detecting weak identification and constructing two-step confidence sets in GMM. This approach controls coverage distortions under weak identification and indicates strong identification, with probability tending to 1 when the model is well identified.

Unbiased Instrumental Variables Estimation under Known First-Stage Sign

We derive mean-unbiased estimators for the structural parameter in instrumental variables models with a single endogenous regressor where the sign of one or more first stage coecients is known. In the case with a single instrument, the unbiased estimator is unique. For cases with multiple instruments we propose a class of unbiased estimators and show that an estimator within this class is ecient when the instruments are strong. We show numerically that unbiasedness does not come at a cost of increased dispersion in models with a single instrument: in this case the unbiased estimator is less dispersed than the 2SLS estimator. Our finite-sample results apply to normal models with known variance for the reducedform errors, and imply analogous results under weak instrument asymptotics with an unknown error distribution.

Robust Two-Step Confidence Sets, and the Trouble with the First Stage F-Statistic

When weak identification is a concern researchers frequently calculate confidence sets in two steps, first assessing the strength of identification and then, on the basis of this initial assessment, deciding whether to use an identification-robust confidence set. Unfortunately, two-step procedures of this sort can generate highly misleading confidence sets, and we demonstrate that two-step confidence sets based on the first stage F-statistic can have extremely poor coverage in linear instrumental variables models with heteroskedastic errors. To remedy this issue, we introduce a simple approach to detecting weak identification and constructing two-step confidence sets which we show controls coverage distortions under weak identification in general nonlinear GMM models, while also indicating strong identification with probability tending to one if the model is well-identified. Applying our approach to linear IV we show that it is competitive with approaches based on the first-stage F-statistic under homoscedasticity but performs far better under heteroskedasticity.

Unbiased Instrumental Variables Estimation Under Known First-Stage Sign

We derive mean-unbiased estimators for the structural parameter in instrumental variables models with a single endogenous regressor where the sign of one or more first stage coefficients is known. In the case with a single instrument, there is a unique non-randomized unbiased estimator based on the reduced-form and first-stage regression estimates. For cases with multiple instruments we propose a class of unbiased estimators and show that an estimator within this class is efficient when the instruments are strong. We show numerically that unbiasedness does not come at a cost of increased dispersion in models with a single instrument: in this case the unbiased estimator is less dispersed than the 2SLS estimator. Our finite-sample results apply to normal models with known variance for the reduced-form errors, and imply analogous results under weak instrument asymptotics with an unknown error distribution.