Chapter 7 - linear regression with multiple regressors

Chapter 7 - linear regression with multiple regressors -...

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7-1 Hypothesis Tests and Confidence Intervals in Multiple Regression (SW Chapter 7) Outline 1. Hypothesis tests and confidence intervals for a single coefficient 2. Joint hypothesis tests on multiple coefficients 3. Other types of hypotheses involving multiple coefficients 4. How to decide what variables to include in a regression model?
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Hypothesis Tests and Confidence Intervals for a Single Coefficient in Multiple Regression (SW Section 7.1) 11 1 ˆˆ () ˆ var( ) E ββ β is approximately distributed N (0,1) (CLT). Thus hypotheses on 1 can be tested using the usual t - statistic, and confidence intervals are constructed as { 1 ˆ ± 1.96 × SE ( 1 ˆ )}. So too for 2 ,…, k . 1 ˆ and 2 ˆ are generally not independently distributed – so neither are their t -statistics (more on this later). 7-2
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Example : The California class size data (1) = 698.9 – 2.28 ¥ STR n TestScore (10.4) (0.52) (2) = 686.0 – 1.10 ¥ STR – 0.650 PctEL n TestScore (8.7) (0.43) (0.031) The coefficient on STR in (2) is the effect on TestScore of a unit change in STR , holding constant the percentage of English Learners in the district The coefficient on STR falls by one-half 7-3
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7-4 The 95% confidence interval for coefficient on STR in (2) is {–1.10 ± 1.96 × 0.43} = (–1.95, –0.26) The t -statistic testing β STR = 0 is t = –1.10/0.43 = –2.54, so we reject the hypothesis at the 5% significance level We use heteroskedasticity-robust standard errors – for exactly the same reason as in the case of a single regressor.
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7-5 Tests of Joint Hypotheses (SW Section 7.2) Let Expn = expenditures per pupil and consider the population regression model: TestScore i = β 0 + 1 STR i + 2 Expn i + 3 PctEL i + u i The null hypothesis that “school resources don’t matter,” and the alternative that they do, corresponds to: H 0 : 1 = 0 and 2 = 0 vs. H 1 : either 1 0 or 2 0 or both TestScore i = 0 + 1 STR i + 2 Expn i + 3 PctEL i + u i
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7-6 Tests of joint hypotheses, ctd. H 0 : β 1 = 0 and 2 = 0 vs. H 1 : either 1 0 or 2 0 or both A joint hypothesis specifies a value for two or more coefficients, that is, it imposes a restriction on two or more coefficients. In general, a joint hypothesis will involve q restrictions. In the example above, q = 2, and the two restrictions are 1 = 0 and 2 = 0. A “common sense” idea is to reject if either of the individual t -statistics exceeds 1.96 in absolute value. But this “one at a time” test isn’t valid: the resulting test rejects too often under the null hypothesis (more than 5%)!
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Why can’t we just test the coefficients one at a time? Because the rejection rate under the null isn’t 5%. We’ll calculate the probability of incorrectly rejecting the null using the “common sense” test based on the two individual t - statistics. To simplify the calculation, suppose that 1 ˆ β and 2 ˆ are independently distributed. Let t 1 and t 2 be the t -statistics: t 1 = 1 1 ˆ 0 ˆ () SE and t 2 = 2 2 ˆ 0 ˆ SE The “one at time” test is: reject H 0 : 1 = 2 = 0 if | t 1 | > 1.96 and/or | t 2 | > 1.96 What is the probability that this “one at a time” test rejects H 0 , when H 0 is actually true? (It should be 5%.) 7-7
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Suppose t 1
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Chapter 7 - linear regression with multiple regressors -...

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