Topic 8: Testing & Structural change

Question 1

How do we test conditions, ie B₁ = B₂

Answer 1

We rearrange to get B^₁ - B^₂ = 0 then consider that any addition of normal variables are normally distributed

Question 2

How do we test restictions

Answer 2

Using a t-test or f-test

Question 3

How do we test a restriction, t-test style?

Answer 3

Say we test B₁+B₂=1, rearrange to B₁+B₂-1=0 t-test = (B₁^+B₂^-1)/SE(B₁^+B₂^-1)

Question 4

How do we test a restriction, F-test style?

Answer 4

Using restricted least squares, eliminate as many estimates as restrictions. For B₁+B₂ = 1, you get Yi = B₁ + Xi(1-B₁). Then we get u^_r, the residuals for our restricted regression. Our F-test is then shown below; where m = number of restrictions, n = observations k = parameters in unrestricted regression

note, missing divide by m in diagram

Question 5

What is meant by a structural change in the data?

Answer 5

A change in the underlying PRF that occurs in the observations, present usually in time series

Question 6

How is a chow test performed?

Answer 6

1. Obtain full RSS, then RSS₁ & RSS₂ for the two seperate groups.

Now RSS restricted = RSS₁+RSS₂.

2. Make Fstat

Question 7

What does the chow test assume? Are there problems/ How do we diagnose?

Answer 7

• Constant variance in groups
• > can be tested by σ^² / σ² _~ F(n₁-k,n₂-k)
• identical & independent distributions

Question 8

What is the Chow predictive failure test?

Answer 8

A test which considers the whole sample and one group within the sample, used when the other group has a small number of observations. B1^ & B2^ are used to predict observations for the second group. RSS for the first group is RSSur, RSS for the entire sample is RSSr. Test is

Question 9

Name some tests for non-linear models. How do thse tests work?

Answer 9

• Likelyhood ratio, -Wald - Lagrange multiplier

With large sample sizes they are identical

1. Find log-likelyhood ratio without restrictions (ULLF)

\2. Find log-likelyhood ratio with restrictions (RLLF)

3. calculate LR = 2(ULLR-RLLR)
4. use R _~ chi(m) to run test