How do we test conditions, ie B_{1} = B_{2}

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

How do we test restictions

Using a t-test or f-test

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

Say we test B_{1}+B_{2}=1, rearrange to B_{1}+B_{2}-1=0 t-test = (B_{1}^+B_{2}^-1)/SE(B_{1}^+B_{2}^-1)

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

Using restricted least squares, eliminate as many estimates as restrictions. For B_{1}+B_{2} = 1, you get Yi = B_{1} + Xi(1-B_{1}). 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

What is meant by a structural change in the data?

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

How is a chow test performed?

1. Obtain full RSS, then RSS_{1} & RSS_{2} for the two seperate groups.

Now RSS restricted = RSS_{1}+RSS_{2}.

2. Make Fstat

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

-Constant variance in groups

-> can be tested by σ^^{2} / σ^{2} _{~ }F(n_{1}-k,n_{2}-k)

-identical & independent distributions

What is the Chow predictive failure test?

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

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

- 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