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.

Here’s an example to illustrate how the Chow test works. Let’s say we have a time series of data for the quarterly sales of a company over a period of 10 years. We want to determine whether there has been a change in the underlying relationship between the sales and one of its determinants, such as advertising expenditure.

To do this, we can first fit a linear regression model to the entire 10-year period, with sales as the dependent variable and advertising expenditure as the independent variable. This will give us the overall relationship between the two variables.

Next, we can divide the 10-year period into two sub-periods, such as the first 5 years and the last 5 years. We can then fit two separate regression models to each sub-period, with sales as the dependent variable and advertising expenditure as the independent variable in both cases.

Now, we can compare the two sub-periods to see if there has been a change in the relationship between sales and advertising expenditure. If there has been a change, we can use the Chow test to determine the point at which the change occurred.

To conduct the Chow test, we first need to calculate the sum of squared residuals (SSR) for each of the two regression models, which is the sum of the squared differences between the observed values of the dependent variable and the predicted values of the dependent variable from the regression model.

For example, if the first regression model has an SSR of 100, and the second regression model has an SSR of 120, we can then calculate the F-statistic using the formula:

F-statistic = (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.

If the calculated F-statistic is greater than the critical value for the F-distribution at a given significance level, we can reject the null hypothesis that there is no structural change in the regression model. This means that there is a statistically significant difference between the two regression models, and that a change in the underlying relationship has occurred.

To determine the point at which the change took place, we can conduct the Chow test for multiple sub-periods and compare the F-statistics for each sub-period. The sub-period with the highest F-statistic is likely the point at which the change occurred.

In conclusion, the Chow test is a useful tool for identifying structural changes in regression models, particularly in time series data. It allows us to determine whether a change in the underlying relationship has occurred, and if so, at what point the change took place.