Lecture 7

Lecture 7 - Introduction to Econometrics Lecture 7:...

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Introduction to Econometrics 7-1 Technical 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 nested and non-nested hypotheses. Consider three linear models (alternative regressions) explaining Y 1 Y = α + β X + θ Z + U 1 2 Y = α + β X + U 2 3 Y = α + θ Z + U 3 Model 2 is nested in Model 1: it is the special case in which θ = 0. Similarly Model 3 is nested in 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 H 0 : 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 H 0 : 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 H 0 : β = 0 H 1 : β 0 use the t-ratio β e /se[ β e ], which has the t-distribution with N-K degrees of freedom (K is the number of regressors including the intercept). To test H 0 : β = 1 H 1 : β 1 the relevant statistic is t * = ( β e - 1)/se[ β e ] which also has a t-distribution if H 0 is true. A version of this test is applicable whenever the null hypothesis is that the single coefficient β e takes a known constant value.
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Introduction to Econometrics 7-2 Technical Notes Significance Testing for Groups of Coefficients Consider the slightly more complex model 1 Y = α + β X + θ Z + φ W + U and the joint null hypothesis about the pair of coefficients β and θ . H 0 : β = 0 and θ = 0 H 1: not ( β = 0 and θ = 0) which is equivalent to β 0 or θ 0 (or both) This implies that the model which corresponds to the null is 2 Y = α + φ W + U From the AN alysis O f VA riance Tables for the two regressions we know that SSE Model 1 / σ 2 = { β 2 Var(X) + θ 2 Var(Z) + φ 2 Var(W) + 2 βθ Cov(X, Z) + 2 βφ Cov(X,W) + 2 θφ Cov(Z,W)} which is distributed as χ 2 (3) SSE Model 2 / σ 2 = φ 2 Var(W)/ σ 2 which is distributed as χ 2 (1) Hence under the null H 0 , which implies β = θ = 0, the difference between the explained sums of squares, {SSE Model 1 - SSE Model 2 }/ σ 2 , has zero mean and is distributed as χ 2 (2). Unfortunately we can’t use this result directly, because we don’t know
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Lecture 7 - Introduction to Econometrics Lecture 7:...

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