Lecture 7

# Lecture 7 - 1 Model Specification We do not know from...

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Unformatted text preview: 1 Model Specification We do not know from economic theory which regressors should be included in a multiple regression. This choice is a form of hypothesis testing We distinguish between nested and non-nested models Model 1 Y = α + β X + θ Z + U 1 Model 2 Y = α + β X + U 2 Model 3 Y = α + θ Z + U 3 Model 3 is special case of Model 1 (with β = 0), so it is nested in Model 1 - so is Model 2 But Models 2 and 3 are not nested - methods for choice between such non-nested models beyond scope of this course Individual significance tests in multiple regression models To test the significance of a single coefficient H : β = 0 H 1 : β ≠ use t[ β e ] = β e /SE( β e ) as the test statistic To test whether coefficient has known value β * H : β = β * H 1 : β ≠ β * use t * = ( β e- β * )/SE( β e ), so to test H : β = 1 use ( β e- 1)/SE( β e ) All statistics have t N-K distributions Joint significance tests in multiple regression Model 1 Y = α + β X + θ Z + φ W + U 1 H : β = 0 and θ = 0 H 1 : β ≠ 0 or θ ≠ 0 (or both) The regression corresponding to H is given by Model 2 Y = α + φ W + U 2 Comparing the ANOVA tables for Model 1 and Model 2 SSE Model 1 / σ 2 ~ χ 2 (3) SSE Model 2 / σ 2 ~ χ 2 (1) 2 Hence we can show that (SSE M1- SSE M2 )/ σ 2 ~ χ 2 (2) The ANOVA table also implies SSR M1 / σ 2 ~ χ 2 (N - 4) These results justify a test based on the F-statistic F = {(SSE M1- SSE M2 )/2}/{SSR M1 /(N - 4)} which has an F(2,N - 4) distribution Because both models have the same SST, this is the same statistic as F = {(SSR M2- SSR M1 )/2}/{SSR M1 /(N - 4)}...
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Lecture 7 - 1 Model Specification We do not know from...

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