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

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

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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.

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Example F c = 4.038 Reject the null hypothesis

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Testing the significance of a model Restricted model

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Example F c = 3.187

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An extended model

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The significance of advertising F c =3.120

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

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Is this significantly different from $40000? T-test t c = 1.993

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

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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 F c =3.120

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

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A restricted model

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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.

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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.

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

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

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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.

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The consequences of collinearity Exact collinearity renders OLS inoperable. Near exact leads to increased standard errors. R 2 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.

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Identifying and mitigating collinearity Identifying: –Large standard errors with high R 2. –Pairwise correlation coefficients in excess of 0.8 –Auxiliary regressions. Mitigating –Additional data. –Parameter restrictions

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