How do we test conditions, ie B1 = B2
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 B1+B2=1, rearrange to B1+B2-1=0 t-test = (B1^+B2^-1)/SE(B1^+B2^-1)
How do we test a restriction, F-test style?
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
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 RSS1 & RSS2 for the two seperate groups.
Now RSS restricted = RSS1+RSS2.
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 / σ2 ~ F(n1-k,n2-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