Cumulative sum test for parameter stability

Let's see it work

We have a simple model for unemployment. The model is that unemployment is a function of its lagged value. After we fit our model, we test for a structural break.

We have four series of unemployment, one for each Census region of the U.S. Those regions are West, South, New England, and Midwest. Variables west, south, ne, and midwest contain the corresponding unemployment. Our data are monthly and run from January 1990 to December 2008.

Here is a graph of the four series. Each of the series appears to have four turning points. They also appear to move together, but we will ignore that for the purpose of this example.

Perhaps that will be tracked by our model. Let's begin with the Western region. We will fit our model by typing

. regress west L.west
(output omitted)

We can now perform the test for a structural break:

. estat sbcusum, name(west, replace)

Cumulative sum test for parameter stability

Sample: 1990m2 thru 2008m12                Number of obs = 227
H0: No structural break


                   Test               Critical value          

      Type    statistic           1%           5%          10%

 Recursive       2.0281       1.1430       0.9479       0.8499

These results reject the null hypothesis of no structural break. Our simple model does not track the data. estat sbcusum also produced a graph:

The red line is the cumulative sum. If the red line is the shaded area, the test would have not rejected the null hypothesis at the 5 percent level. The graph for the West is seldom in the shaded area!

We repeated this for the other four regions. Here are the four graphs put together:

It appears that our model tracks the Midwestern region just fine, but it does not track the other three regions.

We chose to show you an example using four series and used sbcusum after fitting a simple model for each. When you use the command, you will probably be using it with one series and a more complicated, richer model that, you hope, will not reject the null hypothesis.