Outline

1
Outline
2
Hypothesis Tests and Confidence Intervals for a
Single Coefficient in Multiple Regression (SW
Section 7.1)
3
Standard errors in multiple regression in STATA
4
Tests of Joint Hypotheses(SW Section 7.2)

• A joint hypothesis specifies a value for 2 or
  more coefficients, also called imposing
  restrictions on q coefficients
• Consider the population regression model
• Expn expenditures per pupil
• Test the null hypothesis that school resources
  dont matter vs. the alternative that they do
  

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Why cant we just test the coefficients one at a
time?

• The one at time test is
• Reject H0 ?1 ?2 0 if t1 gt 1.96
  and/or t2 gt 1.96
• What is the probability that this one at a time
  test rejects H0, when H0 is actually true? (It
  should be 5.)

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Size of a Test
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The F-statistic testing ?1 and ?2
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Large-sample distribution of the F-statistic
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Computing the p-value using the F-statistic
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F-test example, California class size data
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The restricted and unrestricted regressions
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Simple formula for the homoskedasticity-only
F-statistic
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Example
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The homoskedasticity-only F-statistic summary
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Digression The F distribution
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The Fq,nk1 distribution
19
Another digression A little history of
statistics
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A little history of statistics, ctd
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Summary the homoskedasticity-only F-statistic
and the F distribution
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Summary testing joint hypotheses
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Testing Single Restrictions on Multiple
Coefficients (SW Section 7.3)
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Testing single restrictions on multiple
coefficients, ctd.
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Method 1 Rearrange (transform) the regression
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Rearrange the regression, ctd.
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Method 2 Perform the test directly
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Confidence Sets for Multiple Coefficients (SW
Section 7.4)
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Joint confidence sets ctd.
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Confidence set based on inverting the F-statistic
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An example of a multiple regression analysis
and how to decide which variables to include in a
regression
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A general approach to variable selection and
model specification
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Digression about measures of fit
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Back to the test score application
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More California data
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Digression on presentation of regression results
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Summary Multiple Regression