Alastair R. Hall

Testing for a Unit Root in Time Series With Pretest Data-Based Model Selection

In this article we examine the impact of data-based lag-length estimation on the behavior of the augmented Dickey–Fuller (ADF) test for a unit root. We derive conditions under which the ADF test converges to the distribution tabulated by Dickey and Fuller and verify that these conditions are satisfied by several commonly employed lag-selection strategies. Simulation evidence indicates that the performance of the ADF test is considerably improved when the lag length is selected from the data. An application to inventory series illustrates that inference about a unit root can be very sensitive to the method of lag-length selection.

Generalized Method of Moments

1. Introduction 2. The Instrumental Variable Estimator in the Linear Regression Model 3. GMM Estimation in Correctly Specified Models 4. GMM Estimation in Misspecified Models 5. Hypothesis Testing 6. Asymptotic Theory and Finite Sample Behaviour 7. Moment Selection in Theory and in Practice 8. Alternative Approximations in Finite Sample Behaviour 9. Empirical Examples 10. Related Methods of Estimation Appendix: Mixing processes and Nonstationarity

Judging Instrument Relevance in Instrumental Variables Estimation

Recent research has emphasized the poor finite-sample performance of the instrumental variables estimator when the instruments are weakly correlated with the regressors. The authors show how the canonical correlations between regressors and instruments can provide a measure of instrument relevance in the general multiple-instrument-multiple-regressor case. However, their simulation results indicate that any such relevance measure probably has little practical merit, as its use may actually exacerbate the poor finite-sample properties of the instrumental variables estimator. Copyright 1996 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

Testing for unit roots in autoregressive moving average models: An instrumental variable approach

Abstract In this paper we propose an approach, based on an instrumental variable estimator, for testing the null hypothesis that a process Y t is an ARIMA( p , 1, q ) against the alternative that it is a stationary ARIMA( p + 1, 0, q ) process. Our approach is an extension of the procedure suggested by Hall (1989a) for the case p = 0. We derive the limiting distributions of the instrumental variable estimator when the estimated model is either (i) the true model, (ii) the true model with a shift in mean included, or (iii) the true model with a shift in mean and a linear time trend included. The performance of the test statistics is investigated using a Monte Carlo study. Generally speaking, the criteria based on the instrumental variable approach seem to perform as good as or better than the existing methods when the model is specified correctly. However, if the model is overspecified, then the empirical levels are higher than the nominal level in moderate-sized samples, whereas if the model is underspecified the instrumental variable estimators are inconsistent.

Predictive tests for structural change with unknown breakpoint

This paper considers predictive tests for structural change in models estimated via Generalized Method of Moments. Our analysis extends earlier work by Ghysels and Hall (1990a) by allowing for the instability to occur at an unknown point in the sample. We analyze various statistics based on continuous mappings of the sequence of predictive tests calculated for a set of possible breakpoints in the sample. The limiting distribution of these statistics is derived under both the null hypothesis and local alternatives. Percentiles are reported for the distribution under the null. A side product of our analysis is that we can illuminate the power properties of the predictive test and also compare its properties to those of the Wald, LR and LM tests for parameter variation. We study those power properties both via local asymptotic analysis and Monte Carlo. Cette etude generalise la procedure proposee par Ghysels et Hall (1990a) pour tester le changement structurel pour des modeles estimes par la methode de moments generalisee. Nous ne supposons plus le point de rupture comme etant connu et proposons plusieurs statistiques predictives avec changement structurel inconnu. Comme les distributions asymptotiques sont non standard, nous fournissons les valeurs critiques. Finalement, nous etudions la puissance des tests et faisons des comparaisons avec des tests du type Wald, LM et LR.

Are consumption-based intertemporal capital asset pricing models structural?

Abstract Hansen and Singleton (1982) and Dunn and Singleton (1986) have found supporting evidence for the overidentifying restrictions of two empirical consumption-based asset pricing models, when estimated with a particular set of single asset returns. In this paper, we submit these models to further scrutiny by testing whether they exhibit (structural) stability. A series of tests, recently developed by Ghysels and Hall (1990), are applied and a test for structuralinvariance is introduced based on the likelihood ratio type test procedure of Eichenbaum, Hansen, and Singleton (1988). There are a number of reasons why structural stability tests are particularly appropriate for diagnostic testing of Euler equation models, namely: (1) the Lucas econometric policy evaluation critique; (2) Euler equations are only a partial description of the data-generating process and parameter stability is one of the few assumptions imposed in their estimation via the generalized method of moments; and finally (3) it is demonstrated that Hansens overidentifying restrictions test has no power against a class of local alternatives characterized by a parameter drift.

The Information Matrix Test for the Linear Model

We derive the information matrix test, suggested by White, for the normal fixed regressor linear model, and show that the statistic decomposes asymptotically into the sum of three independent quadratic forms. One of these is Whites general test for heteroscedasticity and the remaining two components are quadratic forms in the third and fourth powers of the residuals respectively. Our results show that the test will fail to detect serial correlation and never be asymptotically optimal against heteroscedasticity, skewness and non-normal kurtosis.

On Periodic Structures and Testing for Seasonal Unit Roots

The standard testing procedures for seasonal unit roots developed so far have been based0501nly on time invariant ARMA processes with AR polynomials involving seasonal differencing. One attractive alternative is to employ periodic ARMA models in which the coefficients are allowed to vary with the season. In this paper, we present convenient procedures for testing for the presence of unit roots at the zero and seasonal frequencies in periodic time series. The limiting distributions of these statistics are derived and tabulated. Simulation evidence illustrates the advantages of allowing for periodicity in this context when it is present. The tests are illustrated via applications to macroeconomic and ozone level data. Les procedures standards pour tester la presence de racines unitaires aux frequences saisonnieres sont basees sur une representation invariante ARIMA. Une classe alternative de processus est celle des modeles a variations periodiques des parametres. Dans cette etude nous presentons des tests de racines unitaires qui prennent explicitement en compte une structure periodique. Les distributions asymptotiques sont derivees. Une etude Monte Carlo demontre les avantages de nos tests par rapport aux procedures standards.

Garp, Separability, And The Representative Agent

We examine whether annual, quarterly, and monthly U.S. aggregate consumption data could have been generated by a utility-maximizing representative agent with intertemporally separable utility. The model appears inapplicable over the full time periods covered by the NIPA data, which are the sample periods often used in the literature. The model does appear applicable, however, over long subsamples. The data also are inconsistent with separability assumptions routinely made in the literature. In particular, the main categories of consumption (nondurables, services, and durables) are not mutually separable. We consider the implications of our results for inference about consumption based on the representative-agent model.

Testing nonnested Euler conditions with quadrature-based methods of approximation

Abstract Singleton (1985) proposed a test of the validity of an Euler equation specification which incorporates information on a nonnested set of Euler equations from an alternative economic model. The information about the alternative model is introduced into the testing framework by constructing a sequence of local alternatives to the Euler equations of the maintained model which are in the direction of the Euler equations of the alternative model. Singletons test is easy to apply, but is designed to have power against an alternative that in many cases does not have a straightforward economic interpretation. In this paper, we reexamine the construction of local alternatives to Euler equations. We compare Singletons approach with a method based on perturbing the data-generation process (DGP) of the maintained model in the direction of the DGP of the alternative model. The second approach leads to tests which are always of interest, but require the complete specification of the DGP. In this paper, we derive a test for discriminating between two nonnested sets of Euler conditions which have been estimated us ing Generalized Method of Moments (GMM). The test is based on the Encompassing Principle of Cox (1961) and Mizon and Richard (1986), and uses Tauchens (1986) quadrature-based nethods for approximating the expectation of nonlinear functions of stationary random variables. Singletons empirical examples are also reconsidered.