The Chow test is a statistical and econometric test of whether the coefficients in two linear regressions on different data sets are equal. The Chow test was invented by economist Gregory Chow in 1960. In econometrics, the Chow test is most commonly used in time series analysis to test for the presence of a structural break. In program evaluation, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population.
structural break

program evaluation



At x=1.7 there is a structural break, regression on the subintervals [0,1.7] and [1.7,4] delivers a better modelling than the combined regression(dashed) over the whole interval.

Comparison of 2 different programs (red, green) existing in a common data set, separate regressions for both programs deliver a better modelling than a combined regression (black).

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_1x_{1t} + c_1x_{2t} + \varepsilon. \,
and

y_t=a_2+b_2x_{1t} + c_2x_{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_22k)}.
The test statistic follows the F distribution with k and N_1+N_22k degrees of freedom.
Remarks
 The global sum of squares (SSE) if often called 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.
References

Chow, Gregory C. (1960). "Tests of Equality Between Sets of Coefficients in Two Linear Regressions". Econometrica 28 (3): 591–605.

Doran, Howard E. (1989). Applied Regression Analysis in Econometrics. CRC Press. p. 146.

Dougherty, Christopher (2007). Introduction to Econometrics. Oxford University Press. p. 194.


External links

Computing the Chow statistic, Chow and Wald tests, Chow tests: Series of FAQ explanations from the Stata Corporation at https://www.stata.com/support/faqs/

[2]: Series of FAQ explanations from the SAS Corporation
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