Documento 4 - Hypothesis test and confidence intervals with multiple regressors

STR

hypothesis at the 5% significance level

Standard errors in multiple regression in

STATA

reg testscr str pctel, robust;

Regression with robust standard errors Number of obs = 420

F( 2, 417) = 223.82

Prob > F = 0.0000

R-squared = 0.4264

Root MSE = 14.464

------------------------------------------------------------------------------

| Robust

testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

str | -1.101296 .4328472 -2.54 0.011 -1.95213 -.2504616

pctel | -.6497768 .0310318 -20.94 0.000 -.710775 -.5887786

_cons | 686.0322 8.728224 78.60 0.000 668.8754 703.189

------------------------------------------------------------------------------

   

TestScore 686.0 1.10 STR 0.650 PctEL

(8.7

) (

0.

43

) (

0.031

) –

We use heteroskedasticity-robust standard errors for exactly the same reason as

in the case of a single regressor.

Tests of Joint Hypotheses (SW Section 7.2)

(1 of 2)

Let Expn = expenditures per pupil and consider the population

regression model: β β β β

TestScore = + STR + Expn + PctEL + u

i 0 1 i 2 i 3 i i

The null hypothesis that “school resources don’t matter,” and the

alternative that they do, corresponds to:

β β

H : = 0 and = 0

0 1 2

β ≠ β ≠

vs. H : either 0 or 0 or both

1 1 2

β β β β

TestScore = + STR + Expn + PctEL + u

i 0 1 i 2 i 3 i i

Tests of Joint Hypotheses (SW Section 7.2)

(2 of 2)

• β β

H : = 0 and = 0

0 1 2

• β ≠ β ≠

vs. H : either 0 or 0 or both

1 1 2

• 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

β β

example above, q = 2, and the two restrictions are = 0 and = 0.

1 2

• 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%)!

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

“common

probability of incorrectly rejecting the null using the sense”

test based on the two individual t-statistics. To simplify the calculation,

suppose that and are independently distributed (this isn’t true in

–

general just in this example). Let t and t be the t-statistics:

1 2

ˆ ˆ

 

 

0 0

 

1 2

t and t

ˆ ˆ

 

1 2

SE ( ) SE ( )

1 2

“one at time” test is:

The β β

reject H : = = 0 if |t | > 1.96 and/or |t | > 1.96

0 1 2 1 2

“one at a time” test rejects

What is the probability that this H , when H

0 0

is actually true? (It should be 5%.)

Suppose t and t are independent

1 2

(for this example).

The probability of incorrectly rejecting the null hypothesis using

the “one at a time” test

  

Pr [| t | 1.96 and/or | t | 1.96]

H 1 2

0

   

1 Pr [| t | 1.96 and | t | 1.96]

H 1 2

0

    

1 Pr [| t | 1.96] Pr [| t | 1.96]

H 1 H 2

0 0

(because t and t are independent by assumption)

1 2

– 2

= 1 (.95) –

= .0975 = 9.75% which is not the desired 5%!!

The size of a test is the actual rejection

rate under the null hypothesis.

• The size of the “common sense” test isn’t 5%!

• In fact, its size depends on the correlation between t and t

1 2

ˆ ˆ

 

(and thus on the correlation between and ).

1 2

Two Solutions:

• –

Use a different critical value in this procedure not 1.96 (this is

the “Bonferroni –

method see SW App. 7.1) (this method is

rarely used in practice however)

• β β

Use a different test statistic designed to test both and at

1 2

once: the F-statistic (this is common practice)

The F-statistic

The F-statistic tests all parts of a joint hypothesis at once.

β β

Formula for the special case of the joint hypothesis = and

1 1,0

β β

= in a regression with two regressors:

2 2,0  



  ˆ

2 2

t t 2 t t

1

 1 2 t , t 1 2

 

F 1 2

 



 ˆ 2

2 1

 

t , t

1 2



ˆ

where estimates the correlation between t and t .

t , t 1 2

1 2

Reject when F is large (how large?)

β β

The F-statistic testing and :

1 2

 



  ˆ

2 2

t t 2 t t

1

 1 2 t , t 1 2

 

F 1 2

 



 ˆ 2

2 1

 

t , t

1 2

• The F-statistic is large when t and/or t is large

1 2

• The F-statistic corrects (in just the right way) for the correlation

between t and t .

1 2

• β’s

The formula for more than two is nasty unless you use matrix

algebra.

• This gives the F-statistic a nice large-sample approximate

distribution, which is…

Large-sample distribution of the F-statistic

p

 

ˆ

Consider the special case that t and t are independent, so 0;

1 2 t , t

1 2

in large samples the formula becomes

 



  ˆ

2 2

t t 2 t t

1 1

  

1 2 t , t 1 2

  2 2

F (

t t )

1 2

 



 ˆ 1 2

2

2 1 2

 

t ,

t

1 2

• Under the null, t and t have standard normal distributions that,

1 2

in this special case, are independent

• The large-sample distribution of the F-statistic is the distribution

of the average of two independently distributed squared standard

normal random variables.

The chi-squared distribution  2

The chi - squared distribution with q degrees of freedom ( ) is

q

defined to be the distribution of the sum of q independent squared

standard normal random variables.  2

In large samples, F is distributed as / q .

q  2

Selected large-sample critical values of / q

q

q 5% critical value

1 3.84 (why?)

2 3.00 (the case q = 2 above)

3 2.60

4 2.37

5 2.21

Computing the p-value using the F-statistic:



 2

p -value tail probability of the / q distribution beyond

q

the F -statistic actually computed.

Implementation in STATA

Use the “test” command after the regression

Example: Test the joint hypothesis that the population coefficients

on STR and expenditures per pupil (expn_stu) are both zero, against

the alternative that at least one of the population coefficients is

nonzero.

F-test example, California class size data:

reg testscr str expn_stu pctel, r;

Regression with robust standard errors Number of obs = 420

F( 3, 416) = 147.20

Prob > F = 0.0000

R-squared = 0.4366

Root MSE = 14.353

------------------------------------------------------------------------------

| Robust

testscr | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

str | -.2863992 .4820728 -0.59 0.553 -1.234001 .661203

expn_stu | .0038679 .0015807 2.45 0.015 .0007607 .0069751

pctel | -.6560227 .0317844 -20.64 0.000 -.7185008 -.5935446

_cons | 649.5779 15.45834 42.02 0.000 619.1917 679.9641

------------------------------------------------------------------------------

NOTE

test str expn_stu; The test command follows the regression

( 1) str = 0.0 There are q=2 restrictions being tested

( 2) expn_stu = 0.0

F( 2, 416) = 5.43 The 5% critical value for q=2 is 3.00

Prob > F = 0.0047 Stata computes the p-value for you

More on F-statistics.

There is a simple formula for the F-statistic that holds only under

homoskedasticity (so it isn’t very useful) but which nevertheless

might help you understand what the F-statistic is doing.

The homoskedasticity-only F-statistic

When the errors are homoskedastic, there is a simple formula for

computing the “homoskedasticity-only” F-statistic:

• Run two regressions, one under the null hypothesis (the

“restricted” regression) and one under the alternative hypothesis

“unrestricted”

(the regression).

• – –

2

Compare the fits of the regressions the R s if the

“unrestricted” model fits sufficiently better, reject the null

“restricted” “unrestricted”

The and regressions

Example: are the coefficients on STR and Expn zero?

Unrestricted population regression (under H ):

1

β β β β

TestScore = + STR + Expn + PctEL + u

i 0 1 i 2 i 3 i i

Restricted population regression (that is, under H ):

0

β β

TestScore = + PctEL + u (why?)

i 0 3 i i

• The number of restrictions under H is q = 2 (why?).

0

• 2

The fit will be better (R will be higher) in the unrestricted

regression (why?) 2

By how much must the R increase for the coefficients on Expn

and PctEL to be judged statistically significant?

Simple formula for the homoskedasticity-only

F-statistic: 

2 2

( R R )/ q

 unrestricted restricted

F   

2

(1 R )/( n k 1)

unrestricted unrestricted

where: 

2 2

R the R for the restricted regression

restricted 

2 2

R the R for the unrestricted regression

unrestricted

q = the number of restrictions under the null

k = the number of regressors in the unrestricted regression.

unrestricted

• The bigger the difference between the restricted and unrestricted

–

2

R s the greater the improvement in fit by adding the variables in

–

question the larger is the homoskedasticity-only F.

Example:

Restricted regression:

  

2

TestScore 644.7 0.671

PctEL

, R 0.4149

restricted

(1.0) (0.032)

Unrestricted regression:

   

TestScore 649.6 0.29 STR 3.87 Expn 0.656 PctEL

(15.5) (0.48) (1.59) (0.032)

  

2

R 0.4366, k 3, q 2

unrestricted unrestricted 

2 2

( R R )/ q

 unrestricted restricted

So F   

2

(1 R )/( n k 1)

unrestricted unres