1

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

2

Hypothesis Tests and Confidence Intervals for a Single Coefficient

3

Standard errors in multiple regression in STATA

4

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 …

5

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%.)

6

Size of a Test

7

The F-statistic testing 1 and 2

8

Large-sample distribution of the F-statistic

9

10

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

11

Recall the “subset” F-statistic from Stats class

12

The “restricted” and “unrestricted” regressions

13

Example

14

The homoskedasticity & normality-only F-statistic – Summary

15

The Fq,n–k–1 distribution

16

A history of statistics lesson …

17

Testing Single Restrictions on Multiple Coefficients

18

Testing Single Restrictions on Multiple Coefficients

19

Method 1: Rearrange (“transform”) the regression

20

Method 2: Perform the test directly

21

Confidence Sets for Multiple Coefficients

22

23

Confidence set based on inverting the F-statistic

24

Model Specification

25

Model Specification

26

California Test Score Application

27

More California data…

28

29

30

31

32

33