Thomas J. Rothenberg

Efficient Tests for an Autoregressive Unit Root

This paper derives the asymptotic power envelope for tests of a unit autoregressive root for various trend specifications and stationary Gaussian autoregressive disturbances. A family of tests is proposed, members of which are asymptotically similar under a general 1(1) null (allowing nonnormality and general dependence) and which achieve the Gaussian power envelope. One of these tests, which is asymptotically point optimal at a power of 50%, is found (numerically) to be approximately uniformly most powerful (UMP) in the case of a constant deterministic term, and approximately uniformly most powerful invariant (UMPI) in the case of a linear trend, although strictly no UMP or UMPI test exists. We also examine a modification, suggested by the expression for the power envelope, of the Dickey-Fuller (1979) t-statistic; this test is also found to be approximately UMP (constant deterministic term case) and UMPI (time trend case). The power improvement of both new tests is large: in the demeaned case, the Pitman efficiency of the proposed tests relative to the standard Dickey-Fuller t-test is 1.9 at a power of 50%. A Monte Carlo experiment indicates that both proposed tests, particularly the modified Dickey-Fuller t-test, exhibit good power and small size distortions in finite samples with dependent errors.

Approximating the distributions of econometric estimators and test statistics

Publisher Summary Approximate distribution theory derives results from assumptions on the stochastic process generating the data. The quality of the approximation is not better than the quality of the specifications on which it is based. The models used by econometricians are, at best, crude and rather arbitrary. As most of the approximation methods employ information on the first four moments of the data whereas the usual asymptotic theory typically requires information only on the first two moments, some loss in robustness must be expected. However, if a rough idea about the degree of skewness and kurtosis is available, that information can be often exploited to obtain considerably improved approximations to sample statistics. The chapter discusses that sophisticated approximation theory is most appropriate in situations where the econometrician is able to make correct and detailed assumptions about the process being studied. In current practice, applied econometricians occasionally draw incorrect conclusions on the basis of alleged asymptotic properties of their procedures. In recent years, an extraordinary fondness for asymptotic theory has developed among econometricians. Considerable effort is devoted to showing that some new estimator or test is asymptotically normal and efficient. The assertion that a given estimator is approximately normal suggests that the speaker believes that it would be sensible to treat the estimator as though it were really normal. Accurate and convenient approximations for the distributions of econometric estimators and test statistics are of great value.

Inference in a nearly integrated autoregressive model with nonnormal innovations

Abstract Robust tests and estimators based on nonnormal quasi-likelihood functions are developed for autoregressive models with near unit root. Asymptotic power functions and power envelopes are derived for point-optimal tests of a unit root when the likelihood is correctly specified. The shapes of these power functions are found to be sensitive to the extent of nonnormality in the innovations. Power loss resulting from using least-squares unit-root tests in the presence of thick-tailed innovations appears to be greater than in stationary models.

Approximate power functions for some robust tests of regression coefficients

Edgeworth approximations are developed for the distribution functio ns of some statistics for testing a linear hypothesis on the coefficient s in a regression model with an unknown error covariance matrix. Adjust ments to the asymptotic critical values are found to insure that the tests have correct size to second order of approximation. The power loss due to the estimation of the error covariance matrix is calculated. Some examples involving heteroskedasticity and autocorrelation suggest that the null rejection probabilities of common robust regression tests are often considerably greater than their nominal level. Moreover, the cost of not knowing the error covariance matrix can be substantial. Copyright 1988 by The Econometric Society.

On Energy Policy Models

Many energy models cannot be relied upon in forecasting or policy analysis. The quality of the data is often poor, and the theoretical underpinnings tend to be inadequate. These points are illustrated by example.

The Effect of Uncertainty on Resource Allocation in a General Equilibrium Model

I. Introduction, 440. — II. Some partial equilibrium results, 442. — III. Model One: Bandom labor supply, 444.—IV. Model Two: Random production parameter, 453.—V. Summary and conclusions, 458.—Appendix, 458.

ASYMPTOTIC PROPERTIES OF SOME ESTIMATORS IN STRUCTURAL MODELS

Publisher Summary This chapter discusses asymptotic properties of some estimators in structural models. Under normality, the k-class estimators are asymptotically efficient and the members with a = 1 are equivalent to the class of bias-adjusted maximum likelihood estimators. The bias-adjusted maximum likelihood estimators constitute an essentially complete class of second-order optimal estimators. Although the k-class estimators are not asymptotically efficient in the absence of normality, they have identical skewness and kurtosis coefficients to order T-1. Under symmetry, the bias-adjusted least variance ratio estimators, although no longer maximum-likelihood based, dominate the other k-class estimators. The effect of skewness and kurtosis of the error distribution on the sampling properties of the alternative estimators is counterintuitive.

Simultaneous Equations Models

Models that attempt to explain the workings of the economy typically are written as interdependent systems of equations describing some hypothesized technological and behavioural relationships among economic variables. Supply and demand models, Walrasian general equilibrium models, and Keynesian macromodels are common examples. A large part of econometrics is concerned with specifying, testing, and estimating the parameters of such systems. Despite their common use, simultaneous equations models still generate controversy. In practice there is often considerable disagreement over their proper use and interpretation.

Simultaneous equations with covariance restrictions

Abstract Full-information maximum-likelihood and minimum-distance estimators are derived for the simultaneous-equations model with covariance restrictions. Linearization yields estimators that are easy to compute and have an instrumental-variables interpretation. The efficient three-stage least-squares estimator in the presence of covariance restrictions is shown to be a special case. Modified estimators which take into account possible nonnormality in the errors are also discussed.

Some elementary distribution theory for an autoregression fitted to a random walk

A permutation argument is used to derive some properties of regression coefficients in an autoregressive time series model with a polynomial trend and a unit root.