APPROXIMATE POWER FUNCTIONS FOR SOME ROBUST TESTS OF REGRESSION COEFFICIENTS Thomas J . Rothenberg Department o f Economics U n i v e r s i t y o f C a l i f o r n i a , Berkeley Research Papers i n Economics No. 84-1 Summary Edgeworth expansions are developed f o r the d i s t r i b u t i o n functions o f some t e s t s t a t i s t i c s i n the normal l i n e a r regression model where the e r r o r covariance matrix i s unknown. Tests based on generalized l e a s t squares estimates and also on ordinary l e a s t squares estimates are considered. In both cases, adjustments t o the asymptotic c r i t i c a l The approxi values are found and approximate l o c a l power c a l c u l a t e d . and a u t o c o r r e l a t i o n . mations are applied to a number o f examples, i n c l u d i n g h e t e r o s c e d a s t i c i t y

[When the error covariance matrix in a linear model depends on a few unknown parameters, the regression coefficients can be estimated by a two-step procedure. Consistent estimates of the covariance parameters are first obtained and then used in a generalized least squares regression. Under the assumption that the errors are normal and the covariance parameter estimates are well behaved, an asymptotic expansion is developed for the distribution function of the two-step GLS estimate. the error in treating the estimate as normal is found to be of order n extasciicircum- extasciicircum2 as the sample size n tends to infinity.]

A WELL DEVELOPED EXACT STATISTICAL THEORY exists for hypothesis testing in the normal linear regression model when the errors are independent and homoscedastic. In the more general case where the error covariance matrix is nonscalar and depends on a set of unknown parameters, exact analysis is difficult and reliance is usually placed on asymptotic approximations for large sample size n. In this paper, higher-order asymptotic expansions are developed for comparing the size and power of some common procedures for testing linear hypotheses on the regression coefficients in a class of generalized normal linear models. The class investigated is essentially the same as in Breusch [4] and Magnus [6] and includes many of the examples of heteroscedasticity and autocorrelation discussed in the literature. We assume simply that the regressors are nonrandom and that the error covariance matrix is a smooth function of a few parameters that can be efficiently estimated by maximum likelihood. Tests based on the Wald, likelihood ratio, and Lagrange multiplier principles are considered. These principles lead to three tests which, though distinct in finite samples, are locally asymptotically equivalent and share certain asymptotic optimality properties. Of course, there are infinitely many other tests that are asymptotically equivalent to the ones examined here. Although the techniques of this paper can be applied to any of them, our results concern only the tests arising from the three traditional principles. We show that, to a second order of approximation under local alternatives, the likelihood ratio test statistic is a simple average of the Wald statistic and the Lagrange multiplier statistic. When the null hypothesis is one dimensional, the three tests are, to second order, equally powerful; that is, after the critical regions are adjusted so that the tests have (to order n - i) the same size, the local power functions differ by terms of smaller order than n -. When the null hypothesis contains more than one

A theory of identification is developed for general stochastic model whose probability law is determined by finite number of parameters. It is shown under weak regularity conditions that local identifiability of the unknown parameter vector is equivalent to nonsingularity of the information matrix. The use of reduced-form parameters to establish identifiability is also analyzed. The general results are applied to the familiar problem of determining whether the coefficients of system of linear simultaneous equations are identifiable. THE IDENTIFICATION PROBLEM concerns the possibility of drawing inferences from observed samples to an underlying theoretical An important part of econometric theory involves the derivation of conditions under which given structure will be identifiable. The basic results for linear simultaneous equation systems under linear parameter constraints were given by Koopmans and Rubin [10] in 1950. Extensions to nonlinear systems and nonlinear constraints were made by Wald [15], Fisher [4, 5, 6], and others. A summary of these results can be found in Fisher's comprehensive study [7]. The identification problem has also been thoroughly analyzed in the context of the classical single-equation errors-in-variables model. The basic papers here are by Neyman [12] and Reiers0l [13]. Most of this previous work on the identification problem has emphasized the special features of the particular model being examined. This has tended to obscure the fact that the problem of structural identification is very general one. It is not restricted to simultaneous-equation or errors-in-variables models. As Koopmans and Reiers0l [9] emphasize, the identification problem is a general and fundamental problem arising, in many fields of inquiry, as concomitant of the scientific procedure that postulates the existence of structure. In their important paper Koopmans and Reiers0l define the basic characteristics of the general identification problem. In the present paper we shall, in the case of general parametric model, derive some identifiability criteria. These criteria include the standard rank conditions for linear models as special cases. Our approach is based in part on the information matrix of classical mathematical statistics. Since this matrix is measure of the amount of information about the unknown parameters available in the sample, it is not surprising that it should be related to identification. For lack of identification is simply the lack of sufficient information to distinguish between alternative structures. The following results make this relationship more precise.2

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.

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.