Further Inference in the Multiple Regression Model

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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
Fc 4.038
Reject the null hypothesis
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Testing the significance of a model
Restricted model
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Example
Fc 3.187
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An extended model
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The significance of advertising
Fc3.120
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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
tc 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
Fc3.970
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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
Fc3.120
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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.
• 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.

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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