Introduction to Econometrics 7-1Technical Notes Introduction to Econometrics Lecture 7: Multiple Regression - Hypotheses about Coefficients and Sets of Coefficients An Outline of the Specification Problem With multiple regression we need to decide which explanatory variables to include in an equation. This is the problem of choosing a specification. Although economic theory can suggest variables, it is not informative enough to solve the problem, so we need statistical techniques. In a simple case these techniques can be regarded as a form of hypothesis testing. We need first to distinguish between nestedand non-nestedhypotheses. Consider three linear models (alternative regressions) explaining Y 1 Y = α+ βX + θZ + U12 Y = α+ βX + U23 Y = α+ θZ + U3Model 2 is nestedin Model 1: it is the special case in which θ= 0. Similarly Model 3 is nestedin Model 1. But Model 2 and Model 3 are not nested: neither is a special case of the other. Choosing between Model 1 and Model 2 can be based on a test of whether the null hypothesis H0: θ= 0 is rejected by the data at an appropriate significance level when Model 1 is estimated. Similarly the choice between Model 1 and Model 3 can be based on a test of H0: β= 0. Choosing between non-nested models (for example, choosing between Models 2 and 3) is much more difficult, and is not covered in this course. Significance Testing for Individual Coefficients If the specification problem can be reduced to a significance test on an individual coefficient then we can use a t-test. To test H0: β= 0 H1: β≠0 use the t-ratio βe/se[βe], which has the t-distribution with N-K degrees of freedom (K is the number of regressors includingthe intercept). To test H0: β= 1 H1: β≠1 the relevant statistic is t*= (βe- 1)/se[βe] which also has a t-distribution if H0is true. A version of this test is applicable whenever the null hypothesis is that the single coefficient βetakes a knownconstant value.
has intentionally blurred sections.
Sign up to view the full version.