1 hypothesis tests & confidence intervals (sw ch. 7) 1.for a single coefficient 2.for multiple...
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Hypothesis Tests & Confidence Intervals (SW Ch. 7)
1. For a single coefficient2. For multiple coefficients3. Other types of hypotheses involving multiple coefficients4. Model specification
• how to decide what variables to include in a regression model
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Hypothesis Tests and Confidence Intervals for a Single Coefficient
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Standard errors in multiple regression in STATA
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Tests of Joint Hypotheses• 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 don’t matter” vs. the alternative that they do …
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Why can’t we just test the coefficients “one at a time”?
• The “one at time” test is: Reject H0: 1 = 2 = 0 if |t1| > 1.96 and/or |t2| > 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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p-value for the F-statistic
. display Ftail(2, 416, 5.43) = .00469934 . display 1 – chi2(2, 2*5.43) = .0043831
. display invchi2(2, .95) / 2 = 2.9957
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Recall the “subset” F-statistic from Stats class
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The “restricted” and “unrestricted” regressions
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Example
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The homoskedasticity & normality-only F-statistic – Summary
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The Fq,n–k–1 distribution
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A history of statistics lesson …
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Testing Single Restrictions on Multiple Coefficients
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Testing Single Restrictions on Multiple Coefficients
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Method 1: Rearrange (“transform”) the regression
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Method 2: Perform the test directly
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Confidence Sets for Multiple Coefficients
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Confidence set based on inverting the F-statistic
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Model Specification
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Model Specification
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California Test Score Application
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More California data…
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