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 The MODEL Procedure

## Chow Tests

The Chow test is used to test for break points or structural changes in a model. The problem is posed as a partitioning of the data into two parts of size n1 and n2. The null hypothesis to be tested is

where is estimated using the first part of the data and is estimated using the second part.

The test is performed as follows (refer to Davidson and MacKinnon 1993, p. 380).

1. The p parameters of the model are estimated.
2. A second linear regression is performed on the residuals, , from the nonlinear estimation in step one.
where is Jacobian columns that are evaluated at the parameter estimates. If the estimation is an instrumental variables estimation with matrix of instruments W, then the following regression is performed:
where is the projection matrix.
3. The restricted SSE (RSSE) from this regression is obtained. An SSE for each subsample is then obtained using the same linear regression.
4. The F statistic is then
This test has p and n-2p degrees of freedom.

Chow's test is not applicable if min(n1,n2) < p, since one of the two subsamples does not contain enough data to estimate .In this instance, the predictive Chow test can be used. The predictive Chow test is defined as

where n1 > p . This test can be derived from the Chow test by noting that the SSE2 = 0 when n2 <= p and by adjusting the degrees of freedom appropriately.

You can select the Chow test and the predictive Chow test by specifying the CHOW=arg and the PCHOW=arg options in the FIT statement, where arg is either the number of observations in the first sample or a parenthesized list of first sample sizes. If the sizes for the second or the first group are less than the number of parameters, then a PCHOW test is automatically used. These tests statistics are not produced for GMM and FIML estimations.

The following is an example of the use of the Chow test.

      data exp;
x=0;
do time=1 to 100;
if time=50 then x=1;
y = 35 * exp( 0.01 * time ) + rannor( 123 ) + x * 5;
output;
end;
run;

proc model data=exp;
parm zo 35 b;
dert.z = b * z;
y=z;
fit y init=(z=zo) / chow =(40 50 60) pchow=90;
run;

The data set introduced an artificial structural change into the model (the structural change effects the intercept parameter). The output from the requested Chow tests are shown in Figure 14.46.

 The MODEL Procedure

 Structural Change Test Test Break Point Num DF Den DF F Value Pr > F Chow 40 2 96 12.95 <.0001 Chow 50 2 96 101.37 <.0001 Chow 60 2 96 26.43 <.0001 Predictive Chow 90 11 87 1.86 0.0566

Figure 14.46: Chow's Test Results

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