Patrik Guggenberger

A BIAS-REDUCED LOG-PERIODOGRAM REGRESSION ESTIMATOR FOR THE LONG-MEMORY PARAMETER

The widely used log-periodogram regression estimator of the long-memory parameter d proposed by Geweke and Porter-Hudak (1983) (GPH) has been criticized because of its finite-sample bias, see Agiakloglou, Newbold, and Wohar (1993). In this paper, we propose a simple bias-reduced log-periodogram regression estimator, {d hat}r, that eliminates the first- and higher-order biases of the GPH estimator. The bias-reduced estimator is the same as the GPH estimator except that one includes frequencies to the power 2k for k = 1,...,r, for some positive integer r, as additional regressors in the pseudo-regression model that yields the GPH estimator. The reduction in bias is obtained using assumptions on the spectrum only in a neighborhood of the zero frequency, which is consistent with the semiparametric nature of the long-memory model under consideration. Following the work of Robinson (1995b) and Hurvich, Deo, and Brodsky (1998), we establish the asymptotic bias, variance, and mean-squared error (MSE) of {d hat}r, determine the MSE optimal choice of the number of frequencies, m to include in the regression, and establish the asymptotic normality of {d hat}r. These results show that the bias of {d hat}r goes to zero at a faster rate than that of the GPH estimator when the normalized spectrum at zero is sufficiently smooth, but that its variance only is increased by a multiplicative constant. In consequence, the optimal rate of convergence to zero of the MSE of {d hat}dr is faster than that of the GPH estimator. We establish the optimal rate of convergence of a minimax risk criterion for estimators of d when the normalized spectral density is in a class that includes those that are smooth of order s > 1 at zero. We show that the bias-reduced estimator {d hat}r attains this rate when r > (s-2)/2 and m is chosen appropriately. For s > 2, the GPH estimator does not attain this rate. The proof of these results uses results of Giraitis, Robinson, and Samarov (1997). Some Monte Carlo simulation results for stationary Gaussian ARFIMA(1,d,1) models show that the bias-reduced estimators perform well relative to the standard log-periodogram estimator.

Hybrid and Size‐Corrected Subsampling Methods

This paper considers inference in a broad class of nonregular models. The models considered are nonregular in the sense that standard test statistics have asymptotic distributions that are discontinuous in some parameters. It is shown in Andrews and Guggenberger (2009a) that standard fixed critical value, subsampling, and m out of n bootstrap methods often have incorrect asymptotic size in such models. This paper introduces general methods of constructing tests and confidence intervals that have correct asymptotic size. In particular, we consider a hybrid subsampling/fixed-critical-value method and size-correction methods. The paper discusses two examples in detail. They are (i) confidence intervals in an autoregressive model with a root that may be close to unity and conditional heteroskedasticity of unknown form and (ii) tests and confidence intervals based on a post-conservative model selection estimator. Copyright 2009 The Econometric Society.

ASYMPTOTIC SIZE AND A PROBLEM WITH SUBSAMPLING AND WITH THE m OUT OF n BOOTSTRAP

This paper considers inference based on a test statistic that has a limit distribution that is discontinuous in a parameter. The paper shows that subsampling and m out of n bootstrap tests based on such a test statistic often have asymptotic size—defined as the limit of exact size—that is greater than the nominal level of the tests. This is due to a lack of uniformity in the pointwise asymptotics. We determine precisely the asymptotic size of such tests under a general set of high-level conditions that are relatively easy to verify. The results show that the asymptotic size of subsampling and m out of n bootstrap tests is distorted in some examples but not in others.

On the Asymptotic Size Distortion of Tests When Instruments Locally Violate the Exogeneity Assumption

In the linear instrumental variables model with possibly weak instruments we derive the asymptotic size of testing procedures when instruments locally violate the exogeneity assumption. We study the tests by Anderson and Rubin (1949, The Annals of Mathematical Statistics 20, 46–63), Moreira (2003, Econometrica 71, 1027–1048), and Kleibergen (2005, Econometrica 73, 1103–1123) and their generalized empirical likelihood versions. These tests have asymptotic size equal to nominal size when the instruments are exogenous but are size distorted otherwise. While in just-identified models all the tests that we consider are equally size-distorted asymptotically, the Anderson-Rubin type tests are less size-distorted than the tests of Moreira (2003) and Kleibergen in over-identified situations. On the other hand, we also show that there are parameter sequences under which the former test asymptotically overrejects more frequently. Given that strict exogeneity of instruments is often a questionable assumption, our findings should be important to applied researchers who are concerned about the degree of size distortion of their inference procedure. We suggest robustness of asymptotic size under local model violations as a new alternative measure to choose among competing testing procedures. We also investigate the subsampling and hybrid tests introduced in Andrews and Guggenberger (2010a, Journal of Econometrics 158, 285–305) and show that they do not offer any improvement in terms of size-distortion reduction over the Anderson-Rubin type tests.

The Impact Of A Hausman Pretest On The Asymptotic Size Of A Hypothesis Test

This paper investigates the asymptotic size properties of a two-stage test in the linear instrumental variables model when in the first stage a Hausman (1978) specification test is used as a pretest of exogeneity of a regressor. In the second stage, a simple hypothesis about a component of the structural parameter vector is tested, using a t -statistic that is based on either the ordinary least squares (OLS) or the two-stage least squares estimator (2SLS), depending on the outcome of the Hausman pretest. The asymptotic size of the two-stage test is derived in a model where weak instruments are ruled out by imposing a positive lower bound on the strength of the instruments. The asymptotic size equals 1 for empirically relevant choices of the parameter space. The size distortion is caused by a discontinuity of the asymptotic distribution of the test statistic in the correlation parameter between the structural and reduced form error terms. The Hausman pretest does not have sufficient power against correlations that are local to zero while the OLS-based t -statistic takes on large values for such nonzero correlations. Instead of using the two-stage procedure, the recommendation then is to use a t -statistic based on the 2SLS estimator or, if weak instruments are a concern, the conditional likelihood ratio test by Moreira (2003).

GENERALIZED EMPIRICAL LIKELIHOOD ESTIMATORS AND TESTS UNDER PARTIAL, WEAK, AND STRONG IDENTIFICATION

The principal purpose of this paper is to describe the performance of generalized empirical likelihood (GEL) methods for time series instrumental variable models specified by nonlinear moment restrictions when identification may be weak. The paper makes two main contributions. Firstly, we show that all GEL estimators are first-order equivalent under weak identification. The GEL estimator under weak identification is inconsistent and has a nonstandard asymptotic distribution. Secondly, the paper proposes new GEL test statistics, which have chi-square asymptotic null distributions independent of the strength or weakness of identification. Consequently, unlike those for Wald and likelihood ratio statistics, the size of tests formed from these statistics is not distorted by the strength or weakness of iden- tification. Modified versions of the statistics are presented for tests of hypotheses on parameter subvectors when the parameters not under test are strongly identified. Monte Carlo results for the linear instrumental variable regression model suggest that tests based on these statistics have very good size properties even in the presence of conditional heteroskedasticity. The tests have competitive power properties, especially for thick tailed or asymmetric error distributions.

Finite Sample Evidence Suggesting a Heavy Tail Problem of the Generalized Empirical Likelihood Estimator

Comprehensive Monte Carlo evidence is provided that compares the finite sample properties of generalized empirical likelihood (GEL) estimators to the ones of k-class estimators in the linear instrumental variables (IV) model. We focus on sample median, mean, mean squared error, and on the coverage probability and length of confidence intervals obtained from inverting a t-statistic based on the various estimators. The results indicate that in terms of the above criteria, all the GEL estimators and the limited information maximum likelihood (LIML) estimator behave very similarly. This suggests that GEL estimators might also share the “no-moment” problem of LIML. At sample sizes as in our Monte Carlo study, there is no systematic bias advantage of GEL estimators over k-class estimators. On the other hand, the standard deviation of GEL estimators is pronouncedly higher than for some of the k-class estimators. Therefore, if mean squared error is used as the underlying loss function, our study suggests the use of computationally simple estimators, such as two-stage least squares, in the linear IV model rather than GEL. Based on the properties of confidence intervals, we cannot recommend the use of GEL estimators either in the linear IV model.

On the asymptotic sizes of subset Anderson-Rubin and Lagrange multiplier tests in linear instrumental variables regression

We consider tests of a simple null hypothesis on a subset of the coefficients of the exogenous and endogenous regressors in a single-equation linear instrumental variables regression model with potentially weak identification. Existing methods of subset inference (i) rely on the assumption that the parameters not under test are strongly identified, or (ii) are based on projection-type arguments. We show that, under homoskedasticity, the subset Anderson and Rubin (1949) test that replaces unknown parameters by limited information maximum likelihood estimates has correct asymptotic size without imposing additional identification assumptions, but that the corresponding subset Lagrange multiplier test is size distorted asymptotically.

Asymptotics for Stationary Very Nearly Unit Root Processes

This paper considers a mean zero stationary first-order autoregressive (AR) model. It is shown that the least squares estimator and t statistic have Cauchy and standard normal asymptotic distributions, respectively, when the AR parameter rho_n is very near to one in the sense that 1 - rho_n = (n^{-1}).

Generic Results for Establishing the Asymptotic Size of Confidence Sets and Tests

This paper provides a set of results that can be used to establish the asymptotic size and/or similarity in a uniform sense of confidence sets and tests. The results are generic in that they can be applied to a broad range of problems. They are most useful in scenarios where the pointwise asymptotic distribution of a test statistic has a discontinuity in its limit distribution. The results are illustrated in three examples. These are: (i) the conditional likelihood ratio test of Moreira (2003) for linear instrumental variables models with instruments that may be weak, extended to the case of heteroskedastic errors; (ii) the grid bootstrap confidence interval of Hansen (1999) for the sum of the AR coefficients in a k-th order autoregressive model with unknown innovation distribution, and (iii) the standard quasi-likelihood ratio test in a nonlinear regression model where identification is lost when the coefficient on the nonlinear regressor is zero.