Chow test

The Chow test, proposed by econometrician Gregory Chow in 1960, is a test of whether the true coefficients in two linear regressions on different data sets are equal. In econometrics, it is most commonly used in time series analysis to test for the presence of a structural break at a period which can be assumed to be known a priori (for instance, a major historical event such as a war). In program evaluation, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population.

Illustrations

Applications of the Chow test
Structural break (slopes differ) Program evaluation (intercepts differ)

At $x=1.7$  there is a structural break; separate regressions on the subintervals $[0,1.7]$  and $[1.7,4]$  delivers a better model than the combined regression (dashed) over the whole interval. Comparison of two different programs (red, green) in a common data set: separate regressions for both programs deliver a better model than a combined regression (black).

Mathematical details

Suppose that we model our data as

$y_{t}=a+bx_{1t}+cx_{2t}+\varepsilon .\,$

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

$y_{t}=a_{1}+b_{1}x_{1t}+c_{1}x_{2t}+\varepsilon \,$

and

$y_{t}=a_{2}+b_{2}x_{1t}+c_{2}x_{2t}+\varepsilon .\,$

The null hypothesis of the Chow test asserts that $a_{1}=a_{2}$ , $b_{1}=b_{2}$ , and $c_{1}=c_{2}$ , and there is the assumption that the model errors $\varepsilon$  are independent and identically distributed from a normal distribution with unknown variance.

Let $S_{C}$  be the sum of squared residuals from the combined data, $S_{1}$  be the sum of squared residuals from the first group, and $S_{2}$  be the sum of squared residuals 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.

Remarks

• The global sum of squares (SSE) is often called the Restricted Sum of Squares (RSSM) as we basically test a constrained model where we have $2k$  assumptions (with $k$  the number of regressors).
• Some software like SAS will use a predictive Chow test when the size of a subsample is less than the number of regressors.