Further Inference in the Multiple Regression Model

1. Further Inference in the Multiple Regression Model Hill et al Chapter 8

2. The F-Test Used to test hypotheses on one or more parameters Unrestricted model: Restricted model

3. The F-statistic Are the differences in SSE significant? If the null hypothesis is true, then the statistic F has an F-distribution with J numerator degrees of freedom and T-K denominator degrees of freedom.

4. Example Fc= 4.038 Reject the null hypothesis

5. Testing the significance of a model Restricted model

6. Example Fc = 3.187

7. An extended model

8. The significance of advertising Fc=3.120

9. The optimal level of advertising Marginal benefit from advertising: Marginal benefit equals marginal cost:

10. Is this significantly different from $40000? T-test tc = 1.993

11. Is this significantly different from $40000? F-test Restricted model obtained by writing the equation under the assumption that the null is true: Fc=3.970

12. Testing two conjectures • Optimal advertising is $40000 • If advertising is $40000 and price is $2, revenue will be 175000 Two hypotheses to substitute in to get restricted model Fc=3.120

13. Incorporating non-sample information Multiplying each price and income in a demand equation by a constant  has no effect on demand

14. A restricted model

15. Omitted and irrelevant variables • An omitted variable which is correlated with other variables in the regression will lead to bias. • The omission of ‘insignificant’ variables may lead to bias (remember all you have done is failed to reject a null) • Including irrelevant variables will inflate the variances of the estimated parameters.

16. The RESET test: principle • If we omit variables and they are correlated with existing variables, including a function of these variables may allow us to pick up some of the effect of the omitted variables. • If we can artificially improve the model by including powers of the predictions of the model, then a better functional form may exist. • Overall: if we can improve a model by including powers of the predictions the model is inadequate.

17. The RESET test: practice In both cases the null is of no mis-specification

18. The RESET test: example The linear model is mis-specified.

19. Collinear Economic Variables • Explanatory variables move together in systematic ways. • Attribute the increase in TR that is the consequence of two types of advertising. • Identify the effects of increasing input quantities when technology is of the fixed proportions type.

20. The consequences of collinearity • Exact collinearity renders OLS inoperable. • Near exact leads to increased standard errors. • R2 may be high but individual coefficients are likely to be insignificant. • Estimates will be sensitive to the addition of a few observations. • Accurate prediction may still be possible.

21. Identifying and mitigating collinearity • Identifying: • Large standard errors with high R2. • Pairwise correlation coefficients in excess of 0.8 • Auxiliary regressions. • Mitigating • Additional data. • Parameter restrictions