Chow Test

TL;DR

• Tests whether a regression relationship changes between sub-periods or groups.
• Compares sum of squared residuals from separate regressions and uses an F-statistic to decide significance.
• Can be applied repeatedly across candidate break points; the sub-period with the largest F-statistic is the most likely change point.

Definition

The Chow test is a statistical test used to determine whether there is a structural change in a regression model. It is commonly used in econometrics and time series analysis to determine whether a change in the underlying model has occurred, and if so, at what point the change took place.

Explanation

• Fit a regression model to the full sample to obtain an overall relationship.
• Split the sample into two sub-periods (or groups) and fit separate regression models to each sub-period.
• Calculate the sum of squared residuals (SSR) for each sub-period model.
• Compute an F-statistic from the SSRs to test the null hypothesis that there is no structural change (i.e., the coefficients are the same across sub-periods).
• If the calculated F-statistic exceeds the critical value from the F-distribution at a chosen significance level, reject the null hypothesis and conclude a statistically significant change in the regression relationship.
• To locate the timing of a change, repeat the test across multiple candidate break points; the sub-period (break) with the highest F-statistic indicates the most likely point of change.

Examples

Quarterly sales and advertising expenditure (10-year series)

• Data: quarterly sales of a company over a period of 10 years and advertising expenditure as a determinant.
• Procedure:

  1. Fit a linear regression over the entire 10-year period with sales as the dependent variable and advertising expenditure as the independent variable.

  2. Divide the 10-year period into two sub-periods: the first 5 years and the last 5 years.

  3. Fit two separate regressions (one to each sub-period) using the same dependent and independent variables.

  4. Compute the SSR for each sub-period model. For example, the first regression model has an SSR of 100, and the second regression model has an SSR of 120.

  5. Calculate the F-statistic using the given formula:

     $$F\text{-statistic} = \frac{(SSR1 - SSR2)/(k1 - k2)}{(SSR2/n2)}$$

     where SSR1 is the sum of squared residuals for the first regression model, SSR2 is the sum of squared residuals for the second regression model, k1 is the number of independent variables in the first regression model, k2 is the number of independent variables in the second regression model, and n2 is the sample size for the second regression model.

  6. Compare the calculated F-statistic to the critical value from the F-distribution. If it is greater, reject the null hypothesis of no structural change.

  7. To find the point of change, perform the Chow test for multiple sub-period splits and identify the split with the highest F-statistic.

Use cases

• Econometrics
• Time series analysis
• Identifying whether and when the relationship between dependent and independent variables changed in a regression model

Related terms

• Regression model
• Sum of squared residuals (SSR)
• F-statistic
• Structural change