Joint Hypothesis Tests: F-Test for Multiple Coefficients, Lecture notes of Statistics

Download Joint Hypothesis Tests: F-Test for Multiple Coefficients and more Lecture notes Statistics in PDF only on Docsity!

7 – Joint Hypothesis Tests

Now that we have multiple “ X ” variables, and multiple β s, our

hypotheses might also involve more than one β.

• We shouldn’t use t - tests
• We should use the F - test

Example: CPS data again

summary(lm(wage ~ education + gender + age + experience))

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) - 1.9574 6.8350 - 0.286 0.

education 1.3073 1.1201 1.167 0.

genderfemale - 2.3442 0.3889 - 6.028 3.12e- 09 ***

age - 0.3675 1.1195 - 0.328 0.

experience 0.4811 1.1205 0.429 0.

---

Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.458 on 529 degrees of freedom

Multiple R-squared: 0.2533, Adjusted R-squared: 0.

F-statistic: 44.86 on 4 and 529 DF, p-value: < 2.2e- 16

The results of the above regression make me want to drop age and

experience.

This corresponds to the hypothesis:

H

0

: β

3

= 0 and β

4

H

A

: either β

3

≠ 0 or β

4

≠ 0 or both

Why would we want to drop variables?

A bigger problem: t

3

and t

4

are likely not independent

In the model:

0

1

1

2

2

3

3

4

4

• suppose that 𝑋

3

and 𝑋

4

are not independent (e.g. they are

correlated)

• then the OLS estimators b

3

and b

4

will be correlated - the

formula for b

3

(etc.) involves all of the “X” variables

(remember OVB)

• then t

3

and t

4

will be correlated!

Example

Suppose that 𝑋

3

and 𝑋

4

are positively correlated. Consider the

null:

H

0

: β

3

= 0 and β

4

• if b

3

and b

4

are both positive (or negative), it’s not that big of a

deal

• if one is positive and the other negative, that’s a big deal

Let’s try the F - test

I’m going to estimate two models:

• One model under the alternative hypothesis – we’ll call the

unrestricted model (the β s are allowed to be anything)

• One model under the null hypothesis – called the restricted

model. I get this model by taking the null hypothesis to heart.

That is, substitute in the values β

3

= 0 and β

4

= 0 into the full

model

Unrestricted model (under H A

unrestricted <- lm(wage ~ education + gender

• age + experience)

Restricted model (under H 0

restricted <- lm(wage ~ education + gender)

A formula for the F-test statistic

• The F - test takes into account the correlation between the

estimators that are involved in the test

• Note that if the unrestricted model “fits” significantly better

than the restricted model, we should reject the null.

• The difference in “fit” between the model under the null and

the model under the alternative leads to a formulation of the F -

test statistic, for testing joint hypotheses.

The RSS is a measure of fit:

𝑖

2

𝑛

𝑖= 1

where

e

𝑖

𝑖

𝑖

The F-test statistic may be written as:

𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑

𝑢𝑛𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑

𝑢𝑛𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑

𝑢𝑛𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑

where 𝑞 = # of restrictions, k = # of “ X ”s

F =

2 2

2

unrestricted restricted

unrestricted unrestricted

R R q

R n k

where:

2

restricted

R = the R

2

for the restricted regression

2

unrestricted

R = the R

2

for the unrestricted regression

q = the number of restrictions under the null

k unrestricted

= the number of regressors in the unrestricted regression.

The bigger the difference between the restricted and unrestricted

R

2

’s – the greater the improvement in fit by adding the variables in

question – the larger is the F statistic.

Testing you on the exam

• The F - test statistic can be obtained by comparing the R

2

in the

restricted model ( H

0

model) and the unrestricted model ( H

A

model).

• The decision to reject or not depends on whether the F - stat

exceeds the (5%) critical value:

q 5% critical value

• These values are only accurate if n is large (we’ll always

assume this)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) - 1.9574 6.8350 - 0.286 0.

education 1.3073 1.12 01 1.167 0.

genderfemale - 2.3442 0.3889 - 6.028 3.12e- 09 ***

age - 0.3675 1.1195 - 0.328 0.

experience 0.4811 1.1205 0.429 0.

---

Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.458 on 529 degrees of freedom

Multiple R-squared: 0.2533, Adjusted R-squared: 0.

F-statistic: 44.86 on 4 and 529 DF, p-value: < 2.2e- 16

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.21783 1.03632 0.210 0.

education 0.75128 0.07682 9.779 < 2e- 16 ***

genderfemale - 2.12406 0.40283 - 5.273 1.96e- 07 ***

---

Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.639 on 531 degrees of freedom

Multiple R-squared: 0.1884, Adjusted R-squared: 0.

F-statistic: 61.62 on 2 and 531 DF, p-value: < 2.2e- 16