We study large sample properties of likelihood ratio tests of the unit-root hypothesis in an autoregressive model of arbitrary order. Earlier research on this testing problem has developed likelihood ratio tests in the autoregressive model of order 1, but resorted to a plug-in approach when dealing with higher-order models. In contrast, we consider the full model and derive the relevant large sample properties of likelihood ratio tests under a local-to-unity asymptotic framework. As in the simpler model, we show that the full likelihood ratio tests are nearly efficient, in the sense that their asymptotic local power functions are virtually indistinguishable from the Gaussian power envelopes. Extensions to sieve-type approximations and different classes of alternatives are also considered.

This paper highlights a tension between semiparametric efficiency and bootstrap consistency in the context of a canonical semiparametric estimation problem, namely the problem of estimating the average density. It is shown that although simple plug-in estimators suffer from bias problems preventing them from achieving semiparametric efficiency under minimal smoothness conditions, the nonparametric bootstrap automatically corrects for this bias and that, as a result, these seemingly inferior estimators achieve bootstrap consistency under minimal smoothness conditions. In contrast, several “debiased” estimators that achieve semiparametric efficiency under minimal smoothness conditions do not achieve bootstrap consistency under those same conditions.

Many empirical studies estimate the structural effect of some variable on an outcome of interest while allowing for many covariates. We present inference methods that account for many covariates. The methods are based on asymptotics where the number of covariates grows as fast as the sample size. We find a limiting normal distribution with variance that is larger than the standard one. We also find that with homoskedasticity this larger variance can be accounted for by using degrees-of-freedom-adjusted standard errors. We link this asymptotic theory to previous results for many instruments and for small bandwidth(s) distributional approximations.

We investigate the properties of several bootstrap-based inference procedures for semiparametric density-weighted average derivatives. The key innovation in this paper is to employ an alternative asymptotic framework to assess the properties of these inference procedures. This theoretical approach is conceptually distinct from the traditional approach (based on asymptotic linearity of the estimator and Edgeworth expansions), and leads to different theoretical prescriptions for bootstrap-based semiparametric inference. First, we show that the conventional bootstrap-based approximations to the distribution of the estimator and its classical studentized version are both invalid in general. This result shows a fundamental lack of “robustness” of the associated, classical bootstrap-based inference procedures with respect to the bandwidth choice. Second, we present a new bootstrap-based inference procedure for density-weighted average derivatives that is more “robust” to perturbations of the bandwidth choice, and hence exhibits demonstrable superior theoretical statistical properties over the traditional bootstrap-based inference procedures. Finally, we also examine the validity and invalidity of related bootstrap-based inference procedures and discuss additional results that may be of independent interest. Some simulation evidence is also presented.

This paper proposes (apparently) novel standard error formulas for the density-weighted average derivative estimator of Powell, Stock, and Stoker (Econometrica 57, 1989). Asymptotic validity of the standard errors developed in this paper does not require the use of higher-order kernels, and the standard errors are “robust” in the sense that they accommodate (but do not require) bandwidths that are smaller than those for which conventional standard errors are valid. Moreover, the results of a Monte Carlo experiment suggest that the finite sample coverage rates of confidence intervals constructed using the standard errors developed in this papercoincide (approximately) with the nominal coverage rates across a nontrivial range of bandwidths.

This paper studies a model widely used in the weakinstruments literature and establishes admissibilityof the weighted average power likelihood ratio testsrecently derived by Andrews, Moreira, and Stock(2004, NBER Technical Working Paper 199). The classof tests covered by this admissibility resultcontains the Anderson and Rubin (1949,Annals of Mathematical Statistics20, 46–63) test. Thus, there is no conventionalstatistical sense in which the Anderson and Rubin(1949) test “wastes degrees of freedom.” Inaddition, it is shown that the test proposed byMoreira (2003, Econometrica 71,1027–1048) belongs to the closure of (i.e., can beinterpreted as a limiting case of) the class oftests covered by our admissibility result.

This paper studies the asymptotic behavior of aGaussian linear instrumental variables model inwhich the number of instruments diverges with thesample size. Asymptotic efficiency bounds areobtained for rotation invariant inference proceduresand are shown to be attainable by procedures basedon the limited information maximum likelihoodestimator. The bounds are obtained by characterizingthe limiting experiment associated with the modelinduced by the rotation invariance restriction.

Unbiasedness of the OLS estimator with random regressors—solution.

Two new stationarity tests are proposed. Both tests can be viewed as generalizations of existing stationarity tests and dominate these in terms of local asymptotic power. Improvements are achieved by accommodating stationary covariates. A Monte Carlo investigation of the small sample properties of the tests is conducted, and an empirical illustration from international finance is provided.This paper has benefited from the comments of Pentti Saikkonen (the co-editor), two anonymous referees, and seminar participants at University of Aarhus, Indiana University, Purdue University, Stanford University, UC Riverside, the 2001 Nordic Econometric Meeting, and the 2001 NBER Summer Institute. A MATLAB program that implements the tests proposed in this paper is available at http://elsa.Berkeley.EDU/users/mjansson.

Unbiasedness of the OLS estimator with random regressors.

Consistency of kernel estimators of the long-run covariance matrix of a linear process is established under weak moment and memory conditions. In addition, it is pointed out that some existing consistency proofs are in error as they stand.

This paper proposes a notion of near cointegration and generalizes several existing results from the cointegration literature to the case of near cointegration. In particular, the properties of conventional cointegration methods under near cointegration are characterized, thereby investigating the robustness of cointegration methods. In addition, we obtain local asymptotic power functions of five cointegration tests that take cointegration as the null hypothesis.