TheChow test (Chinese:鄒檢定), proposed byeconometricianGregory Chow in 1960, is astatistical test of whether the true coefficients in twolinear regressions on different data sets are equal. In econometrics, it is most commonly used intime series analysis to test for the presence of astructural break at a period which can be assumed to be knowna priori (for instance, a major historical event such as a war). Inprogram evaluation, the Chow test is often used to determine whether the independent variables have different impacts on different subgroups of the population.

Applications of the Chow test

Structural break (slopes differ)	Program evaluation (intercepts differ)
At${\displaystyle x=1.7}$ there is a structural break; separate regressions on the subintervals${\displaystyle [0,1.7]}$ and${\displaystyle [1.7,4]}$ 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).

Suppose that we model our data as

${\displaystyle y_t=a+b{x}_{1t}+c{x}_{2t}+\epsilon .}$

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

${\displaystyle y_t=a_1+b_1{x}_{1t}+c_1{x}_{2t}+\epsilon }$

and

${\displaystyle y_t=a_2+b_2{x}_{1t}+c_2{x}_{2t}+\epsilon .}$

Thenull hypothesis of the Chow test asserts that${\displaystyle a_1=a_2}$,${\displaystyle b_1=b_2}$, and${\displaystyle c_1=c_2}$, and there is the assumption that themodel errors${\displaystyle \epsilon }$ areindependent and identically distributed from anormal distribution with unknownvariance.

Let${\displaystyle S_C}$ be the sum of squaredresiduals from the combined data,${\displaystyle S_1}$ be the sum of squared residuals from the first group, and${\displaystyle S_2}$ be the sum of squared residuals from the second group.${\displaystyle N_1}$ and${\displaystyle N_2}$ are the number of observations in each group and${\displaystyle k}$ is the total number of parameters (in this case 3, i.e. 2 independent variables coefficients + intercept). Then the Chow test statistic is

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

The test statistic follows theF-distribution with${\displaystyle k}$ and${\displaystyle N_1+N_2-2k}$degrees of freedom.

The same result can be achieved via dummy variables.

Consider the two data sets which are being compared. Firstly there is the 'primary' data set i={1,...,${\displaystyle n_1}$} and the 'secondary' data set i={${\displaystyle n_1}$+1,...,n}. Then there is the union of these two sets: i={1,...,n}. If there is no structural change between the primary and secondary data sets a regression can be run over the union without the issue of biased estimators arising.

Consider the regression:

${\displaystyle y_t={\beta }_0+{\beta }_1{x}_{1t}+{\beta }_2{x}_{2t}+...+{\beta }_k{x}_{kt}+{\gamma }_0{D}_t+\sum _{i=1}^k{\gamma }_i{x}_{it}{D}_t+{\epsilon }_t.}$

Which is run over i={1,...,n}.

D is a dummy variable taking a value of 1 for i={${\displaystyle n_1}$+1,...,n} and 0 otherwise.

If both data sets can be explained fully by${\displaystyle ({\beta }_0,{\beta }_1,...,{\beta }_k)}$ then there is no use in the dummy variable as the data set is explained fully by the restricted equation. That is, under the assumption of no structural change we have a null and alternative hypothesis of:

${\displaystyle H_0:{\gamma }_0=0,{\gamma }_1=0,...,{\gamma }_k=0}$

${\displaystyle H_1:\text{otherwise}}$

The null hypothesis of joint insignificance of D can be run as an F-test with${\displaystyle n-2(k+1)}$ degrees of freedom (DoF). That is:${\displaystyle F=\frac{(RS{S}^R-RS{S}^U)/(k+1)}{RS{S}^U/DoF}}$.

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${\displaystyle 2k}$ assumptions (with${\displaystyle k}$ 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.

• Chow, Gregory C. (1960)."Tests of Equality Between Sets of Coefficients in Two Linear Regressions"(PDF).Econometrica.28 (3):591–605.doi:10.2307/1910133.JSTOR 1910133.S2CID 116311724. Archived fromthe original(PDF) on 2019-12-28.
• Doran, Howard E. (1989).Applied Regression Analysis in Econometrics. CRC Press. p. 146.ISBN 978-0-8247-8049-4.
• Dougherty, Christopher (2007).Introduction to Econometrics. Oxford University Press. p. 194.ISBN 978-0-19-928096-4.
• Kmenta, Jan (1986).Elements of Econometrics (Second ed.). New York: Macmillan. pp. 412–423.ISBN 978-0-472-10886-2.
• Wooldridge, Jeffrey M. (2009).Introductory Econometrics: A Modern Approach (Fourth ed.). Mason: South-Western. pp. 243–246.ISBN 978-0-324-66054-8.

Wikimedia Commons has media related toChow test.

• Computing the Chow statistic,Chow and Wald tests,Chow tests: Series of FAQ explanations from theStata Corporation athttps://www.stata.com/support/faqs/
• [1]: Series of FAQ explanations from theSAS Corporation

• Time series statistical tests
• Regression diagnostics

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