Current Research

Online Paper Repository

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Working Papers

What to Do When You Can't Use '1.96' Confidence Intervals for IV.

To address the well-established large-sample invalidity of the +/-1.96 critical values for the t-ratio in the single variable just-identified IV model, applied research typically qualifies the inference based on the first-stage-F (Staiger and Stock (1997) and Stock and Yogo (2005)). We fully extend this F-based approach to its logical conclusion by presenting new critical values for the t-ratio to additionally accommodate values of F that do not meet existing thresholds needed for validity. These new t-ratio critical values simultaneously fix the main problem of over-rejection (invalidity) and the under-appreciated possibility of under-rejection (conservativeness) that can occur when relying solely on the usual 1.96 critical value. We show that the corresponding new confidence intervals are generally expected to be substantially shorter than competing “robust to weak instrument” intervals, including those from the recommended benchmark of Anderson and Rubin (1949) (AR). In a sample of 89 specifications from 10 recent empirical studies drawn from five general interest journals, the new “VtF” intervals are shorter than AR intervals 100 percent of the time, and even more likely to produce statistically significant results than the usual +/-1.96 procedure.

Robust Conditional Wald Inference for Over-Identified IV.

For the over-identified linear instrumental variables model, researchers commonly report the 2SLS estimate along with the robust standard error and seek to conduct inference with these quantities. If errors are homoskedastic, one can control the degree of inferential distortion using the first-stage F critical values from Stock and Yogo (2005), or use the robust-to-weak instruments Conditional Wald critical values of Moreira (2003). If errors are non-homoskedastic, these methods do not apply. We derive the generalization of Conditional Wald critical values that is robust to non-homoskedastic errors (e.g., heteroskedasticity or clustered variance structures), which can also be applied to nonlinear weakly-identified models (e.g. weakly-identified GMM).

Efficiency Loss of Asymptotically Efficient Tests in An Instrumental Variables Regression. Supplement. Earlier version available at cemap.

In an instrumental variable model, the score statistic can be bounded for any alternative in parts of the parameter space. These regions involve a constraint on the .first-stage regression coe.fficients and the reduced-form covariance matrix. Consequently, the Lagrange Multiplier test can have power close to size, despite being effi.cient under standard asymptotics. This information loss limits the power of conditional tests which use only the Anderson-Rubin and the score statistic. The conditional quasi-likelihood ratio test also su.ffers severe losses because it can be bounded for any alternative. A necessary condition for drastic power loss to occur is that the Hermitian of the reduced-form covariance matrix has eigenvalues of opposite signs. These cases are denoted impossibility designs (ID). We show this happens in practice, by applying our theory to the problem of inference on the intertemporal elasticity of substitution (IES). Of eleven countries studied by Yogo (2004) and Andrews (2016), nine are consistent with ID at the 95% level.

Optimal Invariant Tests in an Instrumental Variables Regression With Heteroskedastic and Autocorrelated Errors. Also available at arXiv.

This paper uses model symmetries in the instrumental variable (IV) regression to derive an invariant test for the causal structural parameter. Contrary to popular belief, we show there exist model symmetries when equation errors are heteroskedastic and autocorrelated (HAC). Our theory is consistent with existing results for the homoskedastic model (Andrews, Moreira, and Stock (2006) and Chamberlain (2007)), but in general uses information on the structural parameter beyond the Anderson-Rubin, score, and rank statistics. This suggests that tests based only the Anderson-Rubin and score statistics discard information on the causal parameter of interest. We apply our theory to construct designs in which these tests indeed have power arbitrarily close to size. Other tests, including other adaptations to the CLR test, do not suffer the same deficiencies. Finally, we use the model symmetries to propose novel weighted-average power tests for the HAC-IV model.

A Critical Value Function Approach, with an Application to Persistent Time-Series. Supplement. Also available at arXiv, cemap, and Ensaios Economicos.

Researchers often rely on the t-statistic to make inference on parameters in statistical models. It is common practice to obtain critical values by simulation techniques. This paper proposes a novel numerical method to obtain an approximately similar test. This test rejects the null hypothesis when the test statistic is larger than a critical value function (CVF) of the data. We illustrate this procedure when regressors are highly persistent, a case in which commonly used simulation methods encounter difficulties in controlling size uniformly. Our approach works satisfactorily, controls size, and yields a test which outperforms the two other known similar tests.

Contributions to the Theory of Optimal Tests. Supplement. A previous version (2010) was presented at Columbia, Michigan, Wisconsin (Dec-2010); Berkeley (ARE), Harvard-MIT, Princeton, UCSD, Yale (Jan-2011); and Duke-NCS-UNC Triangle, Georgetown (Jan-2013). Also available at Ensaios Economicos.

This paper considers tests which maximize the weighted average power (WAP). The focus is on determining WAP tests subject to an uncountable number of equalities and/or inequalities. The unifying theory allows us to obtain tests with correct size, similar tests, and unbiased tests, among others. A WAP test may be randomized and its characterization is not always possible. We show how to approximate the power of the optimal test by sequences of nonrandomized tests. Two alternative approximations are considered. The first approach considers a sequence of similar tests for an increasing number of boundary conditions. This discretization allows us to implement the WAP tests in practice. The second method finds a sequence of tests which approximate the WAP test uniformly. This approximation allows us to show that WAP similar tests are admissible. The theoretical framework is readily applicable to several econometric models, including the important class of the curved-exponential family. In this paper, we consider the instrumental variable model with heteroskedastic and autocorrelated errors (HAC-IV) and the nearly integrated regressor model. In both models, we find WAP similar and (locally) unbiased tests which dominate other available tests.

Tests with Correct Size in the Simultaneous Equations Model. 2002, UC Berkeley.

Classical statistical theory is employed to find tests for structural parameters with correct size even when identification is weak. The family of exactly similar tests is characterized in the limited-information model where the reduced-form errors are normal with known covariance matrix. A version of the score test and a particular two-step procedure based on a preliminary test for identification are shown to be members of this family of similar tests. In addition, a power bound is derived for the family. The Anderson-Rubin test is shown to be optimal when the model is just identified; no test is uniformly most powerful when the model is overidentified. Again assuming normal reduced form errors with known covariance matrix, a general method is proposed for finding a similar test based on the conditional distribution of an arbitrary test statistic. The method is applied to the two-stage least-squares Wald statistic and to the likelihood ratio test statistic. Dropping the assumption of known error distribution, it is found that slightly modified versions of the these conditional tests are similar under weak-instrument asymptotics. Monte Carlo simulations suggest that the conditional likelihood ratio test is essentially optimal when identification is strong and also dominates other similar tests ...