Chapter 7 Hypothesis Tests and Confidence Intervals in

Chapter 7 Hypothesis Tests and Confidence Intervals in Multiple Regression

Outline 2

Hypothesis Tests and Confidence Intervals for a Single Coefficient in Multiple Regression (SW Section 7. 1) 3

Example: The California class size data 4

Standard errors in multiple regression in STATA 5

Tests of Joint Hypotheses (SW Section 7. 2) 6

Tests of joint hypotheses, ctd. 7

Why can’t we just test the coefficients one at a time? 8

Suppose t 1 and t 2 are independent (for this calculation). 9

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The F-statistic 11

The F-statistic testing 1 and 2: 12

Large-sample distribution of the F-statistic 13

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Computing the p-value using the F-statistic: 15

F-test example, California class size data: 16

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The “restricted” and “unrestricted” regressions 18

Simple formula for the homoskedasticity-only F-statistic: 19

Example: 20

The homoskedasticity-only F-statistic – summary 21

Digression: The F distribution 22

The Fq, n–k– 1 distribution: 23

Another digression: A little history of statistics… 24

A little history of statistics, ctd… 25

Summary: the homoskedasticity-only Fstatistic and the F distribution 26

Summary: testing joint hypotheses 27

Testing Single Restrictions on Multiple Coefficients (SW Section 7. 3) 28

Testing single restrictions on multiple coefficients, ctd. 29

Method 1: Rearrange (“transform”) the regression 30

Rearrange the regression, ctd. 31

Method 2: Perform the test directly 32

Confidence Sets for Multiple Coefficients (SW Section 7. 4) 33

Joint confidence sets ctd. 34

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Confidence set based on inverting the F-statistic 36

An example of a multiple regression analysis – and how to decide which variables to include in a regression… 37

A general approach to variable selection and “model specification” 38

Digression about measures of fit… 39

Back to the test score application: 40

More California data… 41

Digression on presentation of regression results 42

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Summary: Multiple Regression 44