Chow test

The Chow test is an econometric test of whether the coefficients in two linear regressions on different data are equal. The Chow test is most commonly used in time series analysis to test for the presence of a structural break.

Suppose that we model our data as



y=a+bx_1 + cx_2 + \varepsilon.\, $$

If we split our data into two groups, then we have



y=a_1+b_1x_1 + c_1x_2 + \varepsilon. \,$$

and



y=a_2+b_2x_1 + c_2x_2 + \varepsilon. \, $$

The Chow test is a test that asserts that $$a_1=a_2$$, $$b_1=b_2$$, and $$c_1=c_2$$.

Let $$S_C$$ be the sum of squared residuals from the combined data, $$S_1$$ be the sum of squares from the first group, and $$S_2$$ be the sum of squares from the second group. $$N_1$$ and $$N_2$$ are the number of observations in each group and $$k$$ is the total number of parameters (in this case, 3). Then the Chow test statistic is



\frac{(S_C -(S_1+S_2))/k}{(S_1+S_2)/(N_1+N_2-2k)}. $$

The test statistic follows the F distribution with $$k$$ and $$N_1+N_2-2k$$ degrees of freedom.