Hypothesiss Testing For Multiple Regression Models

Question 1

Single linear restrictions tests we can do (3)

Answer 1

1. Whether individual parameters take on particular values e.g common ones are
   H₀: β₁=1 (test for unity e.g additional £1=additional £1 consumption!)
   Or β₁=0 to test significance (if not 0 = significant!)
2. If regressors have same impact on dependent variable. E.g if degree and vocational equal impact income so we would set β₁=B₂ in our hypothesis!
3. Testing sum of slope parameters (β) = 1

Question 2

Tests of multiple linear restrictions (joint tests)

Answer 2

1. Whether multiple parameters take on specific values.
2. Whether slope parameters all =0
3. Testing several restrictions involving multiple parameters e.g 𝐻0: 𝛽2 + 𝛽5 =
   −1 and 𝛽₃ = 𝛽₄ and β₇=0

Question 3

1st possible test:
Whether multiple parameters take on specific values example

Answer 3

Two variables with a one-for-one impact on the dependent variable.

E.g 𝐻0: 𝛽1 = 𝛽3 = 1

This is 2 restrictions at the same time (as saying parameters are equal to each other, but also equal to 1)

Question 4

2nd possible test:
Test of overall significance
Whether slope parameters jointly =0

Answer 4

𝐻0: 𝛽1 = 𝛽2 = ⋯ = 𝛽𝑘 = 0

Different from the one in single restrictions, where we can test the SUM of slope parameters = 1 (so only 1 equal sign/restriction in the previous!!!)

If we accept null it means the regression line is not significant at all.

Question 5

So now we know what can be tested, how do we test this?

A) for single linear restrictions
B) Multiple linear restrictions

Answer 5

A) T and F tests

B) f test only.

Question 6

Relationship between t statistic and f statistic

Answer 6

T statistic ( result)squared is same as f statistic result

T²=F

Question 7

F test

Answer 7

(RSSr - RSSu)/q
/
RSSu / [n-(k+1)]

~Fq, n - (k+1)

𝑅𝑆𝑆𝑟 is the residual sums of squares from the restricted model

𝑅𝑆𝑆𝑢 is the residual sums of squares from the unrestricted model

q is no. of restrictions imposed (1 in a single restriction)

Question 8

1st possible case in single linear restrictions: Testing single linear restrictions involving multiple parameters e.g β₂=β₃

Which method is better to use in this situation?

(vocational=degree example)

Answer 8

F is easier

For F test
In the restricted model (assume β₂=β₃ just replace β₃ for b₂.

then factorise, and rename the factored bracket e.g Wi (remember we made W for groups of X)

Question 9

Testing multiple restrictions (so can only do F test)

First example :

𝐻₀:𝛽₁ =𝛽₄ =1

Question 10

Second example

Question 11

Example 3: test of overall significant (All parameters jointly =0

What would the null hypothesis be?

Answer 11

𝐻₀:𝛽₁ =𝛽₂ =⋯=𝛽𝑘 =0
Same as saying 𝐻₀: 𝑅² = 0

Question 12

Then, for overall significance tests we use an adjusted F test for R²

Answer 12

F=
R²/k
/
(1-R²) / [n-(k+1)]

~
Fk,n-(k+1)

In this instance Q=k since we are testing all parameters (k) are =0

Question 13

We saw the best way for single linear with multiple parameters eg B2=B3 is using F test.

What is the formula for t test for single linear restrictions involving multiple parameters. E.g B2=B3

Problem set4q3d does this

Answer 13

B2-B3
/
SE(B2+B3) - 2σ₂β₂β₃