16-11. Testing for a structural break, or testing for the equality of two regression equations, can be accomplished by performing the so-called Chow test.2 For instance, suppose we have a sample of n1 observations on the variables X and Y for one time period and a second sample of n2 data points on the same variables for another time period (e.g., we can model some measurable characteristic pre 9/11/01 and post 9/11/01). The relevant question is then, “Is there a change (either in intercepts, or slopes, or both) in the response of Y to X between the two periods?” That is, “Can each regression equation be viewed as coming from the same population?” Looked at a third way, “Is the regression relationship between X and Y structurally stable over the two time periods?”
To perform the Chow test for equality of two regression equations Yi1 = β01 + β11Xi1 + εi1, i = 1, . . . , n1; [Period 1] (16.E.2)
Yj2 = β02 + β12Xj2 + εj2, j = 1, . . . , n2; [Period 2] (16.E.3)
(here the second subscript refers to the subsample or period):
STEP1. Pool all n1+n2 = n observations and estimate a single regression equation Y = β0 + β1X + ε over the combined sample period. From this regression obtain the residual sum of squares SSEC with n1+n2−2 degrees of freedom.
STEP 2. Estimate the two subperiod regressions (16.E.2) and (16.E.3)
separately and obtain the residual sum of squares for each, denoted SSE1 with degrees of freedom n1 − 2 and SSE2 with degrees of freedom n2 − 2 respectively. Form SSE+ = SSE1 + SSE2 with n1 + n2 − 4 degrees of freedom.
STEP 3. To test H0: β01 = β02 and β11 = β12, against H1: H0 is not true, we use the test statistic F = (SSEC − SSE+)/2 SSE+/(n1 + n2 − 4)
∼ F2,n1+n2−4, (16.E.4)
where f is the sample realization of (16.E.4) and R = 9f |f > f1−α,2,n1+n2−4: .
For the data set given here, employ the Chow test to determine if, for α = 0.05, the two regression structures are the same. That is, at this level of significance, can we conclude that the two subsamples have been drawn from the same population regression structure?
Period 1 Period 2 X Y X Y 88 36 166 90 94 21 176 95 100 18 186 82 105 20 196 104 109 10 211 153 118 12 227 194 127 41 239 175 134 50 250 199 141 43 155 59