# General Hypothesis. Testing

Anil K. Bera and Gamini Premaratne*

1 Introduction

The history of statistical hypothesis testing is, indeed, very long. Neyman and Pearson (1933) traced its origin to Bayes (1763). However, systematic applications of hypothesis testing began only after the publication of Karl Pearson’s (1900) goodness-of-fit test, which is regarded as one of the 20 most important scientific breakthroughs in this century. In terms of the development of statistical methods, Ronald Fisher took up where Pearson left off. Fisher (1922) can be regarded as the analytical beginning of statistical methods. In his paper Fisher advocated the use of maximum likelihood estimation and provided the general theory of parametric statistical inference. In order to develop various statistical techniques, Fisher (1922) also introduced such basic concepts as consistency, efficiency, and sufficiency that are now part of our day-to-day vocabulary. Fisher, however, was not particularly interested in testing per se, and he occupied himself mostly in solving problems of estimation and sampling distributions. Neyman and Pearson (1928) suggested the likelihood ratio (LR) test, but that was mostly based on intuitive arguments. The foundation of the theory of hypothesis testing was laid by Neyman and Pearson (1933), and for the first time the concept of "optimal test" was introduced through the analysis of "power function." The result was the celebrated Neyman-Pearson (N-P) lemma. This lemma provides a way to find the most powerful (MP) and uniformly most powerful (UMP) tests. Neyman and Pearson (1936) generalized the basic N-P lemma to restrict optimal tests to suitable subclasses since the UMP test rarely exists. On the basis of Neyman – Pearson’s foundation in testing, several general test principles were gradually developed, such as Neyman’s (1937) smooth test, Wald (1943), Rao’s (1948) score test and Neyman’s (1959) C(a) test. During the last four decades no new funda­mental test principle has emerged. However, econometricians have produced a

large number of general test procedures such as those in Hausman (1978), Newey (1985), Tauchen (1985), and White (1982). Also, simultaneously, econometricians applied the basic test principles, most notably Rao’s score test, and developed model diagnostic and evaluation techniques for the basic assumptions such as serial independence, homoskedasticity, and normality for the regression models, and these procedures are now routinely used in applied econometrics.

The aim of this chapter is very modest. Our main purpose is to explain the basic test principles with examples in a simple way and discuss how they have been used by econometricians to develop procedures to suit their needs. In the next section, we review the general test procedures suggested in the statistics literature. In Section 3, we discuss some standard tests in econometrics and demonstrate how they are linked to some basic principles. The last section offers some concluding remarks. At the outset we should state that there is nothing original about the material covered in this chapter. Students of econometrics are sometimes not aware of the origin of many of the tests they use. Here, we try to provide intuitive descriptions of some test principles with simple examples and to show that various econometric model evaluations and diagnostic pro­cedures have their origins in some of the basic statistical test principles. Much of what we cover can be found, though in a scattered fashion, in Lehmann (1986, 1999), Godfrey (1988), Bera and Ullah (1991), Davidson and MacKinnon (1993), Gourieroux and Monfort (1995), Bera (2000), Bera and Billias (2000) and Rao (2000).