Marcelo J. Moreira

A Conditional Likelihood Ratio Test for Structural Models

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.

Asymptotic power of sphericity tests for high-dimensional data

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 the alternative, 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 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, whereas that of the eigenvalue-based likelihood ratio test is strictly larger than the size, and close to the power envelope.

Signal detection in high dimension: The multispiked case

This paper deals with the local asymptotic structure, in the sense ofLe Cam’s asymptotic theory of statistical experiments, of the signal detectionproblem in high dimension. More precisely, we consider the problemof testing the null hypothesis of sphericity of a high-dimensional covariancematrix against an alternative of (unspecified) multiple symmetry-breakingdirections (multispiked alternatives). Simple analytical expressions for theasymptotic power envelope and the asymptotic powers of previously proposedtests are derived. These asymptotic powers are shown to lie verysubstantially below the envelope, at least for relatively small values of thenumber of symmetry-breaking directions under the alternative. In contrast,the asymptotic power of the likelihood ratio test based on the eigenvalues ofthe sample covariance matrix is shown to be close to that envelope. Theseresults extend to the case of multispiked alternatives the findings of an earlierstudy (Onatski, Moreira and Hallin, 2011) of the single-spiked case. The methods we are using here, however, are entirely new, as the Laplace approximationsconsidered in the single-spiked context do not extend to themultispiked case.

A Maximum Likelihood Method for the Incidental Parameter Problem

This paper uses the invariance principle to solve the incidental parameter problem. 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). We apply our method to (i) a stationary autoregressive model with fixed effects; (ii) an agent-specific monotonic transformation model; (iii) an instrumental variable (IV) model; and (iv) a dynamic panel data model with fixed effects. In the first two examples, there exist group actions that completely discard the incidental parameters. In a stationary autoregressive model with fixed effects, MILE coincides with existing conditional and integrated likelihood methods. The invariance principle also gives a new perspective to the marginal likelihood approach. In an agent-specific monotonic transformation model, our approach yields an estimator that is consistent and asymptotically normal when errors are Gaussian. In an instrumental variable (IV) model, this paper unifies asymptotic results under strong instruments (SIV) and many weak instruments (MWIV) frameworks. We obtain consistency, asymptotic normality, and optimality results for the limited information maximum likelihood estimator directly from the invariant likelihood. Our approach is parallel to M-estimation in problems in which the number of parameters does not change with the sample size. In a dynamic panel data model with N individuals and T time periods, MILE is consistent as long as NT goes to infinity. We obtain a large N, fixed T bound; this bound coincides with Hahn and Kuersteiners (2002) bound when T goes to infinity. MILE reaches (i) our bound when N is large and T is fixed; and (ii) Hahn and Kuersteiners (2002) bound when both N and T are large.

Group invariance, likelihood ratio tests, and the incidental parameter problem in a high-dimensional linear model

This paper considers a linear panel data model with reduced rank regressors and interactive fixed effects. The leading example is a factor model where some of the factors are observed, some others not. Invariance considerations yield a maximal invariant statistic whose density does not depend on incidental parameters. It is natural to consider a likelihood ratio test based on the maximal invariant statistic. Its density can be found by using as a prior the unique invariant distribution for the incidental parameters. That invariant distribution is least favorable and leads to minimax optimality properties. Combining the invariant distribution with a prior for the remaining parameters gives a class of admissible tests. A particular choice of distribution yields the spiked covariance model of Johnstone (2001). Numerical simulations suggest that the maximal invariant likelihood ratio test outperforms the standard likelihood ratio test. Tests which are not invariant to data transformations (i) are uniquely represented as randomized tests of the maximal invariant statistic and (ii) do not solve the incidental parameter problem.