A COMPANION TO Theoretical Econometrics

Parametric and. Nonparametric Tests. of Limited Domain and. Ordered Hypotheses. in Economics

Esfandiar Maasoumi*

1 Introduction

In this survey, technical and conceptual advances in testing multivariate linear and nonlinear inequality hypotheses in econometrics are summarized. This is discussed for economic applications in which either the null, or the alternative, or both hypotheses define more limited domains than the two-sided alternatives typically tested. The desired goal is increased power which is laudable given the endemic power problems of most of the classical asymptotic tests. The impedi­ments are a lack of familiarity with implementation procedures, and characteri­zation problems of distributions under some composite hypotheses.

Several empirically important cases are identified in which practical "one-sided" tests can be conducted by either the %2-distribution, or the union intersection mechanisms based on the Gaussian variate, or the increasingly feasible and popular resampling/simulation techniques. Point optimal testing and its derivatives find a natural medium here whenever unique characterization of the null distribu­tions for the "least favorable" cases is not possible.

Most of the recent econometric literature in this area is parametric deriv­ing from the multivariate extensions of the classical Gaussian means test with

ordered alternatives. Tests for variance components, random coefficients, over­dispersion, heteroskedasticity, regime change, ARCH effects, curvature regula­rity conditions on flexible supply, demand, and other economic functions, are examples. But nonparametric tests for ordered relations between distributions, or their quantiles, or curvature regularity conditions on nonparametric economic relations, have witnessed rapid development and applications in economics and finance. We detail tests for Stochastic Dominance which indicate a major departure in the practice of empirical decision making in, so far, the areas of welfare and optimal financial strategy.

The additional information available when hypotheses can restrict attention to subspaces of the usual two-sided (unrestricted) hypotheses, can enhance the power of tests. Since good power is a rare commodity the interest in inequality restricted hypothesis tests has increased dramatically. In addition, the two-sided formulation is occasionally too vague to be of help when more sharply ordered alternatives are of interest. An example is the test of order relations (e. g. stochastic dominance) amongst investment strategies, or among income/welfare distribu­tions. The two-sided formulation fails to distinguish between "equivalent" and "unrankable" cases.

In statistics, D. J. Bartholomew, H. Chernoff, V. J. Chacko, A. Kudo, and P. E. Nuesch are among the first to refine and extend the Neyman-Pearson testing procedure for one-sided alternatives, first in the one and then in the multivariate settings. For example, see Bartholomew (1959a, 1959b) and Kudo (1963). Later refinements and advances were obtained by Nuesch, Feder, Perlman, and others. At least in low dimensional cases, the power gains over the two-sided counter­parts have been shown to be substantial, see Bartholomew (1959b), and Barlow et al. (1972). While Chernoff and Feder clarified the local nature of tests and gave some solutions when the true parameter value is "near" the boundaries of the hypotheses regions (see Wolak, 1989), Kudo, Nuesch, Perlman, and Shorack were among the first to develop the elements of the x2-distribution theory for the like­lihood ratio and other classical tests. See Barlow et al. (1972) for references.

In econometrics, Gourieroux, Holly, and Monfort (1980, 1982), heretofore GHM, are seminal contributions which introduced and extended this literature to linear and nonlinear econometric/regression models. The focus in GHM (1982) is on the following testing situation:

y = xp + u

RP > r, R : q x K, q < K, the dimension of p.

We wish to test

H0 : Rp = r,... vs... H1 : Rp > r.

u is assumed to be a Gaussian variate with zero-mean and finite variance Q. Gourieroux et al. (1982) derive the Lagrange multiplier (LM)/Kuhn-Tucker (KT) test, as well as the likelihood ratio (LR) and the Wald (W) tests with known and
unknown covariance matrix, Q, of the regression errors. With known covariance the three tests are equivalent and distributed exactly as a %2-distribution. They note that the problem considered here is essentially equivalent to the following in the earlier statistical literature: Let there be T independent observations from a p-dimensional N(p, X). Test

H0 : p = 0, against the alternative

H1 : p > 0, all i, with at least one strict inequality (25.3)

The LR test of this hypothesis has the %2-distribution which is a mixture of chi-squared distributions given by:

p

X w( p, j )X(j), (25.4)

j=0

with x(20) = 1 at the origin. The weights w() are probabilities to be computed in a multivariate setting over the space of alternatives. This is one of the practical impediments in this area, inviting a variety of solutions which we shall touch upon. These include obtaining bounds, exact tests for low dimension cases, and resampling/Monte Carlo techniques.

When Q is unknown but depends on a finite set of parameters, GHM (1982) and others have shown that the same distribution theory applies asymptotically. GHM show that these tests are asymptotically equivalent and satisfy the usual inequality: W > LR > LM(KT). We'll give the detailed form of these test statistics. In particular the LM version may be desirable as it can avoid the quadratic programming (QP) routine needed to obtain estimators under the inequality restrictions. We also point to routines that are readily available in Fortran and GAUSS (but alas not yet in the standard econometric software packages).

It should be noted that this simultaneous procedure competes with another approach based on the union intersection technique. In the latter, each univariate hypothesis is tested, with the decision being a rejection of the joint null if the least significant statistic is greater than the а-critical level of a standard Gaussian variate. Consistency of such tests has been established. We will discuss examples of these alternatives. Also, the nonexistence generally of an optimal test in the multivariable case has led to consideration of point optimal testing, and tests that attempt to maximize power in the least favorable case, or on suitable "averages". This is similar to recent attempts to deal with power computation when alter­natives depend on nuisance parameters. For example see King and Wu (1997) and their references.

In the case of nonlinear models and/or nonlinear inequality restrictions, GHM (1980) and Wolak (1989, 1991) discuss the distribution of the same %2 tests, while Dufour considers modified classical tests. In this setting, however, there is another problem, as pointed out by Wolak (1989, 1991). When q < K there is generally no unique solution for the "true P" from Rp = r (or its nonlinear counterpart). But
convention dictates that in this-type case of composite hypotheses, power be computed for the "least favorable" case which arises at the boundary Rfi = r. It then follows that the asymptotic distribution (when Q is consistently estimated in customary ways) cannot, in general, be uniquely characterized for the least favorable case. Sufficient conditions for a unique distribution are given in Wolak (1991) and will be discussed here. In the absence of these conditions, a "localized" version of the hypothesis is testable with the same x 2-distribution.

All of the above developments are parametric. There is at least an old tradi­tion for the nonparametric "two sample" testing of homogeneity between two distributions, often assumed to belong to the same family. Pearson type and Kolmogorov-Smirnov (KS) tests are prominent, as well as the Wilcoxan rank test. In the case of inequality or ordered hypotheses regarding relations between two unknown distributions, Anderson (1996) is an example of the modified Pearson tests based on relative cell frequencies, and Xu, Fisher, and Wilson (1995) is an example of quantile-based tests which incorporate the inequality information in the hypotheses and, hence lead to the use of x2-distribution theory. The multivariate versions of the KS test have been studied by McFadden (1989), Klecan, McFadden, and McFadden (1991), Kaur, Rao, and Singh (1994), and Maasoumi, Mills, and Zandvakili (1997). The union intersection alternative is also fully discussed in Davidson and Duclos (1998), representing a culmination of this line of develop­ment. The union intersection techniques do not exploit the inequality informa­tion and are expected to be less powerful. We discuss the main features of these alternatives.

In Section 2 we introduce the classical multivariate means problem and a general variant of it that makes it amenable to immediate application to very general econometric models in which an asymptotically normal estimator can be obtained. At this level of generality, one can treat very wide classes of processes, as well as linear and non-linear models, as described in Potscher and Prucha (1991a, 1991b). The linear model is given as an example, and the asymptotic dis­tribution of the classical tests is described.

The next section describes the nonlinear models and the local nature of the hypothesis than can be tested. Section 4 is devoted to the nonparametric setting. Examples from economics and finance are cited throughout the chapter. Section 5 concludes.