Publications

Published and Forthcoming Papers (please print for personal use only)

Valid t-ratio Inference for IV, with Dave Lee, Justin McCrary, and Jack Porter. Supplement. Earlier version available at NBER. American Economic Review, 112(10): 3260-3290.

In the  single-IV model, researchers commonly rely on t  -ratio-based inference, even though the literature has quantified its potentially severe  large-sample distortions. Building on Stock and Yogo (2005), we introduce the  tF  critical value function, leading to a stan-dard error adjustment that is a smooth function of the  first-stage  F -statistic. For  one-quarter of specifications in 61 AER papers, cor-rected standard errors are at least 49 and 136 percent larger than conventional 2SLS standard errors at the  5 percent and  1 percent significance levels, respectively.  tF  confidence intervals have shorter expected length than those of  Anderson and Rubin (1949), whenever both are bounded.

Likelihood Inference and The Role of Initial Conditions for the Dynamic Panel Data Model, with Jose Diogo Barbosa. Also available at arXiv and cemap. Journal of Econometrics, 221(1), 160-179.

Lancaster (2002) proposes an estimator for the dynamic panel data model with homoskedastic errors and zero initial conditions. In this paper, we show this estimator is invariant to orthogonal transformations, but is inefficient because it ignores additional information available in the data. The zero initial condition is trivially satisfied by subtracting initial observations from the data. We show that differencing out the data further erodes efficiency compared to drawing inference conditional on the first observations. Finally, we compare the conditional method with standard random effects approaches for unobserved data. Standard approaches implicitly rely on normal approximations, which may not be reliable when unobserved data is very skewed with some mass at zero values. For example, panel data on firms naturally depend on the first period in which the firm enters on a new state. It seems unreasonable then to assume that the process determining unobserved data is known or stationary. We can instead make inference on structural parameters by conditioning on the initial observations.

Impossible Inference in Econometrics: Theory and Applications, with Marinho Bertanha. Journal of Econometrics, 218(2), 247-270.

This paper studies models in which hypothesis tests have trivial power, that is, power smaller than size. This testing impossibility, or impossibility type A, arises when any alternative is not distinguishable from the null. We also study settings where it is impossible to have almost surely bounded confidence sets for a parameter of interest. This second type of impossibility (type B) occurs under a condition weaker than the condition for type A impossibility: the parameter of interest must be nearly unidentified. Our theoretical framework connects many existing publications on impossible inference that rely on different notions of topologies to show models are not distinguishable or nearly unidentified. We also derive both types of impossibility using the weak topology induced by convergence in distribution. Impossibility in the weak topology is often easier to prove, it is applicable for many widely-used tests, and it is useful for robust hypothesis testing. We conclude by demonstrating impossible inference in multiple economic applications of models with discontinuity and time-series models.

Optimal Two-Sided Tests for Instrumental Variables Regression with Heteroskedastic and Autocorrelated Error, with Moreira. Supplement. Journal of Econometrics, 213(2), 398-433.

This paper considers two-sided tests for the parameter of an endogenous variable in an instrumental variable (IV) model with heteroskedastic and autocorrelated errors. We develop the finite-sample theory of weighted-average power (WAP) tests with normal errors and a known long-run variance. A typical weight choice yields tests with power near zero for parts of the parameter space. To illustrate this problem, we introduce two specific weights. The MM1 and MM2 weights depend on tuning parameters. As these parameters diverge to infinity, their associated WAP statistics simplify considerably. We denote these limiting statistics IL1 and IL2. While WAP tests using the MM1 (or IL1) weight can be severely biased, optimal tests based on the MM2 (or IL2) weight are naturally two-sided when errors are homoskedastic. Like the MM1 weight for homoskedastic errors, the MM2 typically yields biased tests with heteroskedastic and autocorrelated (HAC) errors.

The researcher has two natural choices. One possibility is to find a specific weight that yields naturally two-sided tests. The other alternative is to keep the chosen weight and impose additional boundary constraints that yield two-sided tests with HAC errors. In this paper, we propose two boundary conditions. The locally unbiased (LU) condition is related to the power around the null hypothesis and ...

Tests Based on t-Statistics for IV Regression with Weak Instruments, with Benjamin Mills and Lucas Vilela. Journal of Econometrics, 182(2), 351-363. Supplement.

This paper considers tests of the parameter of an endogenous variable in an instrumental variables regression model. The focus is on one-sided conditional t-tests. Theoretical and numerical work shows that the conditional 2SLS and Fuller t-tests perform well even when instruments are weakly correlated with the endogenous variable. When the population F-statistic is as small as two, their power is reasonably close to the power envelopes for similar and non-similar tests which are invariant to rotation transformations of the instruments. This finding is surprising considering the bad performance of two-sided conditional t-tests found in Andrews, Moreira, and Stock (2007). We show these tests have bad power because the conditional null distributions of t-statistics are asymmetric when instruments are weak. Taking this asymmetry into account, we propose two-sided tests based on t-statistics. These novel tests are approximately unbiased and can perform as well as the conditional likelihood ratio (CLR) test.

Signal Detection in High Dimension: The Multispiked Case, with Marc Hallin and Alexei Onatski. The Annals of Statistics, 42(1), 225-254. Supplement

This paper applies Le Cam's asymptotic theory of statistical experiments to the signal detection problem in high-dimension. We consider the problem of testing the null hypothesis of sphericity of a high-dimensional covariance matrix against an alternative of (unspecified) multiple symmetry-breaking directions (multispiked alternatives). Simple analytical expressions for the Gaussian asymptotic power envelope and the asymptotic powers of previously proposed tests are derived. Those asymptotic powers remain valid for non-Gaussian data satisfying mild moment restrictions. They appear to lie very substantially below the Gaussian power envelope, at least for small values of the number of symmetry-breaking directions. In contrast, the asymptotic power of Gaussian likelihood ratio tests based on the eigenvalues of the sample covariance matrix are shown to be very close to the envelope. Although based on Gaussian likelihoods, those tests remain valid under non-Gaussian densities satisfying mild moment conditions. The results of this paper extend to the case of multispiked alternatives and possibly non-Gaussian densities the findings of an earlier study (Onatski, Moreira and Hallin 2013a) of the single-spiked case. The methods we are using here, however, are entirely new, as the Laplace approximation methods considered in the single-spiked context do not extend to the multispiked case.

Asymptotic Power of Sphericity Tests for High-Dimensional Data, with Marc Hallin and Alexei Onatski. The Annals of Statistics, 41(3), 1204-1231. Supplement

This paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity. We establish the convergence, under the null hypothesis and contiguous alternatives, of the log ratio of the joint densities of the sample covariance eigenvalues to a Gaussian process indexed by the norm of the perturbation. When the perturbation norm is larger than the phase transition threshold studied in Baik et al. (2005), the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain. When the norm is below the threshold, the limiting process is non-degenerate, so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous. Using the asymptotic theory of statistical experiments, we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region. In particular, we show that the asymptotic power of the Tracy-Widom-type tests is trivial (that is, equals the asymptotic size), whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.

Tests with Correct Size when Instruments Can Be Arbitrarily Weak. Journal of Econometrics, 152(2), 131-140. Also available at Center for Labor Economics Working Paper Series 37, 2001, UC Berkeley.

This paper applies classical exponential-family statistical theory to develop a unifying framework for testing structural parameters in the simultaneous equations model under the assumption of normal errors with known reduced-form variance matrix. The results can be divided into the limited-information and full-information categories. In the limited-information model, it is possible to characterize the entire class of similar tests in a model with only one endogenous explanatory variable. In the full-information framework, this paper proposes a family of similar tests for subsets of endogenous variables' coefficients. For both limited- and full-information models, there exist power upper bounds for unbiased tests. When the model is just-identified, the Anderson_Rubin, score, and (pseudo) conditional likelihood ratio tests are optimal. When the model is over-identified, the (pseudo) conditional likelihood ratio test has power close to the power envelope when identification is strong.

Bootstrap Validity for the Score Test When Instruments May Be Weak, with Jack R. Porter and Gustavo A. Suarez. Journal of Econometrics, 149(1), 52-64.

It is well-known that size adjustments based on bootstrapping the t-statistic perform poorly when instruments are weakly correlated with the endogenous explanatory variable. In this paper, we provide a theoretical proof that guarantees the validity of the bootstrap for the score statistic. This theory does not follow from standard results, since the score statistic is not a smooth function of sample means and some parameters are not consistently estimable when the instruments are uncorrelated with the explanatory variable.

Decision Theory Applied to a Linear Panel Data Model, with Gary Chamberlain. Econometrica, 77(1), 107-133.

This paper applies some general concepts in decision theory to a linear panel data model. A simple version of the model is an autoregression with a separate intercept for each unit in the cross section, with errors that are independent and identically distributed with a normal distribution. There is a parameter of interest γ and a nuisance parameter τ, a N×K matrix, where N is the cross-section sample size. The focus is on dealing with the incidental parameters problem created by a potentially high-dimension nuisance parameter. We adopt a “fixed-effects” approach that seeks to protect against any sequence of incidental parameters. We transform τ to (δ, ρ, ω), where δ is a J×K matrix of coefficients from the least-squares projection of τ on a N×J matrix x of strictly exogenous variables, ρ is a K×K symmetric, positive semidefinite matrix obtained from the residual sums of squares and cross-products in the projection of τ on x, and ω is a (N−J) ×K matrix whose columns are orthogonal and have unit length. The model is invariant under the actions of a group on the sample space and the parameter space, and we find a maximal invariant statistic. The distribution of the maximal invariant statistic does not depend upon ω. There is a unique invariant distribution for ω. We use this invariant distribution as a prior distribution to obtain an integrated likelihood function. It depends upon the observation only through the maximal invariant statistic. We use the maximal invariant …

A Maximum Likelihood Method for the Incidental Parameter Problem. The Annals of Statistics, 37(6A), 3660-3696.

This paper uses the invariance principle to solve the incidental parameter problem of Neyman and Scott (1948). We seek group actions that preserve the structural parameter and yield a maximal invariant in the parameter space with fixed dimension. M-estimation from the likelihood of the maximal invariant statistic yields the maximum invariant likelihood estimator (MILE). Consistency of MILE for cases in which the likelihood of the maximal invariant is the product of marginal likelihoods is straightforward. We illustrate this result with a stationary autoregressive model with fixed effects and an agent-specific monotonic transformation model. Asymptotic properties of MILE, when the likelihood of the maximal invariant does not factorize, remain an open question. We are able to provide consistent, asymptotically normal and efficient results of MILE when invariance yields Wishart distributions. Two examples are an instrumental variable (IV) model and a dynamic panel data model with fixed effects.

Efficient Two-Sided Nonsimilar Invariant Tests in IV Regression with Weak Instruments, with Donald W.K. Andrews and James H. Stock. Journal of Econometrics, 146(2), 241-254.

As Nelson and Startz (1990a,b) dramatically demonstrated, standard hypothesis tests and confidence intervals in instrumental variables regression are invalid when instruments are weak. Recent work on hypothesis tests for the coefficient on a single included endogenous regressor when instruments may be weak has focused on similar tests. This paper extends that work to nonsimilar tests, of which similar tests are a subset. The power envelope for two-sided invariant (to rotations of the instruments) nonsimilar tests is characterized theoretically, then evaluated numerically for five IVs. The power envelopes for similar and nonsimilar tests differ theoretically, but are found to be very close numerically. The nonsimilar test power envelope is effectively achieved by the Moreira (2003) conditional likelihood ratio test, so that test is effectively uniformly most powerful invariant (UMPI). We also provide a new nonsimilar test, P*, which has chi-square-one critical values, is asymptotically efficient under strong instruments, involves only elementary functions, and is very nearly UMPI.

Performance of Conditional Wald Tests in IV Regression, with Weak Instruments, with Donald W.K. Andrews and James H. Stock. Journal of Econometrics, 139(1), 116-132.

We compare the powers of five tests of the coefficient on a single endogenous regressor in instrumental variables regression. Following Moreira (2003), all tests are implemented using critical values that depend on a statistic which is sufficient under the null hypothesis for the (unknown) concentration parameter, so these conditional tests are asymptotically valid under weak instrument asymptotics. Four of the tests are based on k-class Wald statistics (two-stage least squares, LIML, Fuller, and bias-adjusted TSLS); the fifth is Moreira’s (2003) conditional likelihood ratio (CLR) test. The heretofore unstudied conditional Wald (CW) tests are found to perform poorly, compared to the CLR test: in many cases, the CW tests have almost no power against a wide range of alternatives. Our analysis is facilitated by a new algorithm, presented here, for the computation of the asymptotic conditional p-value of the CLR test.

Optimal Inference in Regression Models with Nearly Integrated Regressors, with Michael Jansson. Econometrica, 74(3), 681-715.

This paper considers the problem of conducting inference on the regression coefficient in a bivariate regression model with a highly persistent regressor. Gaussian asymptotic power envelopes are obtained for a class of testing procedures that satisfy a conditionality restriction. In addition, the paper proposes testing procedures that attain these power envelopes whether or not the innovations of the regression model are normally distributed.

Optimal Two-Sided Invariant Similar Tests for Instrumental Variables Regression, with Donald W.K. Andrews and James H. Stock. Econometrica, 74(3), 715-752. Supplement

This paper considers tests of the parameter on an endogenous variable in an instrumental variables regression model. The focus is on determining tests that have some optimal power properties. We start by considering a model with normally distributed errors and known error covariance matrix. We consider tests that are similar and satisfy a natural rotational invariance condition. We determine a two-sided power envelope for invariant similar tests. This allows us to assess and compare the power properties of tests such as the conditional likelihood ratio (CLR), the Lagrange multiplier, and the Anderson–Rubin tests.We find that the CLR test is quite close to being uniformly most powerful invariant among a class of two-sided tests. The finite-sample results of the paper are extended to the case of unknown error covariance matrix and possibly nonnormal errors via weak instrument asymptotics. Strong instrument asymptotic results also are provided because we seek tests that perform well under both weak and strong instruments.

On the Validity of Econometric Techniques With Weak Instruments: Inference on Returns to Education Using Compulsory School Attendance Laws, with Luiz M. Cruz. Journal of Human Resources, 40(2), 393-410.

We evaluate Angrist and Krueger (1991) and Bound, Jaeger, and Baker (1995) by constructing reliable confidence regions around the 2SLS and LIML estimators for returns-to-schooling regardless of the quality of the instruments. The results indicate that the returns-to-schooling were between 8 and 25 percent in 1970 and between 4 and 14 percent in 1980. Although the estimates are less accurate than previously thought, most specifications by Angrist and Krueger (1991) are informative for returns-to-schooling. In particular, concern about the reliability of the model with 178 instruments is unfounded despite the low first-stage F-statistic. Finally, we briefly discuss bias-adjustment of estimators and pretesting procedures as solutions to the weak-instrument problem.

A Conditional Likelihood Ratio Test for Structural Models. Econometrica, 71 (4), 1027-1048.

This paper develops a general method for constructing exactly similar tests based on the conditional distribution of nonpivotal statistics in a simultaneous equations model with normal errors and known reduced-form covariance matrix. These tests are shown to be similar under weak-instrument asymptotics when the reduced-form covariance matrix is estimated and the errors are non-normal. The conditional test based on the likelihood ratio statistic is particularly simple and has good power properties. Like the score test, it is optimal under the usual local-to-null asymptotics, but it has better power when identification is weak.

Implementing Tests with Correct Size in the Simultaneous Equations Model, with Brian Poi. Stata Journal, 3 (1), 57-70.

This paper fixes size distortions of tests for structural parameters in the simultaneous equations model by computing critical value functions based on the conditional distribution of test statistics. The conditional tests can then be used to construct informative confidence regions for the structural parameter with correct coverage probability. Commands to implement these tests in Stata are also introduced.