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TESTING for STRUCTURAL CHANGE - many have asked questions about this issue. Suppose you have data on periods 1,2,...,T. For a linear model with one known single break in mean, the Chow test is often used. The Chow Test tests for a break at some fixed known breakpoint K. It tests to see if the regression coefficients are the same in periods 1,2,...,K and in K+1,...,T. The F-statistic for this test can be computed from the SSR (sum of squared residuals) of the two separate regressions in the first and second period, and the SSR of the regression on the combined data set. Let us call this statistic F(K) -- this F statistic tests for the existence of known breakpoint at K. There are two weaknesses in the Chow Test:

FIRST WEAKNESS in this test is that it ASSUMES that the VARIANCES remain the same before and after the break at K. It seems more likely that if a structural change occurs than both the regression coefficients and the variances will change.

SECOND WEAKNESS is that you have to KNOW what the breakpoint K is in order to use the test.

One way to remove the second weakness is to do a ROLLING CHOW test. That is, do a CHOW test at EACH breakpoint and use the BIGGEST value of the test statistic. This is called the sup F test, since it is of the form MAX (F(K1),F(K2),...,F(Ks)) where K1, K2, ... , Ks are the possible breakpoints. If we let K1,...,Ks take all feasible values, then this gives us a test for an UNKNOWN breakpoint, since all possible breakpoints are tested. There are many other tests for UNKNOWN breakpoints, including CUSUM, CUSUM of SQUARES, and others. It is also possible to use the AVERAGE F instead of the sup F. A problem with the rolling Chow Test is that it's critical values cannot be computed exactly. In a landmark paper, Andrews (1993) solved this problem by computing the asymptotic critical values, and he also showed that this test is superior to CUSUM and CUSUM of squares. Since this test takes the MAXIMUM value of many different F tests, it is called the SUP F test by Andrews. There is one important issue discovered by Andrews, which is that when testing for all possible breakpoints, we should start at a respectable distance AWAY from the two endpoints. That is, we should not test for breaks too close to the beginning or the end; otherwise there is significant loss of power. This is because with very small amount of data available in one of the two parts, the estimates are very imprecise, and the test cannot easily detect whether or not change occurred.

That was the state of the art in the 90's. Later research took the direction of making the assumptions about the error term weaker. Instead of assuming IID Normal errors as in standard regression, they created tests when errors form ARMA process and even more complex stochastic structures. This literature, and its applications, are reviewed in Hansen, B. (2001), "The New Econometrics of Structural Change: Dating Breaks in U.S. Labor Productivity,". Journal of Economic Perspectives,vol. 15, no. 4, pp. 117-128.

HOWEVER, all of this literature makes the same assumption that structural change occurs in the regression coefficients but NOT in the error process. Even the very complex error processes, with heteroskedasticity and autocorrelation (HAC), do not change in time. Recently, Maasoumi, Zaman, and Ahmad. (2010) published a paper (“Tests for Structural Change, Homogeneity, and Aggregation” Economic Modelling Vol 27 no.6 (2010): 1382-1391) which relaxes this assumption and allows variances to change at the breakpoint, while also allowing for multiple known breakpoints. This test is called the MZ (Maasoumi-Zaman) test by the authors. This test requires the breakpoints to be specified in advance. On the pattern of the rolling Chow, this test can also be used to detect unknown breakpoints by using it at all possible breakpoints, and then taking the maximum of all the statistics, leading to the sup MZ test. The performance of this test was evaluated recently by Ahmed, Mumtaz, Gulfam Haider, and Asad Zaman. "Detecting structural change with heteroskedasticity." Communications in Statistics-Theory and Methods (2017): 1-10. Theoretical evaluations shows that small changes in variance across the breakpoint cause significant deterioration in performance of the sup F test. Of course the sup MZ test is designed for this situation and hence works much better. Empirical evaluation was done by taking several macroeconomic series and testing them for structural change using both sup F and sup MZ. In these GNP series, tests indicated that structural change frequently involved both regression coefficients and the variance, leading to substantially superior performance of the sup MZ

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