Solved – What to do with negative Chow test

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I have just done a Chow test on a regression in order to see whether there is a structural break. I am a bit stumped however as my Chow test returns a negative number. Now what do I do?

More specifically, this expression (from the Wikipedia entry for the Chow test):

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

turns out negative for me.

EDIT: It turns out it was a simple programming error.

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You've made a mistake somewhere in your calculations. It's not possible for the sum of the squared residuals from a single regression using the combined data to be less than the sum of the sums of squared residuals from the regressions using the two separate sets of data.

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Solved – Chow test or not

Your question is most interesting to me and it's solution has been my primary research for a number of years.

There are a number of ways that "a structural break" may occur.

If there is a change in the Intercept or a change in Trend in "the latter portion of the time series" then one would be better suited to perform Intervention Detection (N.B. this is the empirical identification of the significant impact of an unspecified Deterministic Variable such as a Level Shift or a Change in Trend or the onset of a Seasonal Pulse ). Intervention Detection then is a pre-cursor to Intervention Modelling where a suggested variable is included in the model. You can find information on the web by googling "AUTOMATIC INTERVENTION DETECTION" . Some authors use the term "OUTLIER DETECTION" but like a lot of statistical language this can be confusing/imprecise . Detected Interventions can be any of the following (detecting a significant change in the mean of the residuals );

a 1 period change in Level ( i.e. a Pulse )
a multi-period contiguous change in Level ( i.e. a change in Intercept )
a systematic Pulse ( i.e. a Seasonal Pulse )
a trend change (i.e. 1,2,3,4,5,7,9,11,13,15 ..... )

These procedures are easily programmed IN R/SAS/Matlab and routinely available in a number of commercially available time series packages however there are many pitfalls that you need to be wary of such as whether to detect the stochastic structure first or do Intervention detection on the original series. This is like the chicken and egg problem. Early work in this area was limited to type 1's and as such will probably be insufficient for your needs .

If no such phenomenon is detected then one might consider the CHOW TEST which normally requires the user to pre-specify the point of hypothesized change. I have been researching and implementing procedures to DETECT the point of change by evaluating alternative hypothetical points in time to determine the most likely break point.

In closing one might also be sensitive to the possibility that there might have been a structural change in the error variance thus that might mask the CHOW TEST leading to a false acceptance of the null hypothesis of no significant break points in parameters.

Solved – How to identify structural change using a Chow test on Eviews

I am assuming that you are treating each country separately, and are attempting to determine if there is a break-point in the level of a series. Here are three (EDIT: four) main points that I hope will help:

The Chow test assumes that there is a known break-point in the series. If this point is not know, the Chow test is not appropriate (there are alternatives, although inference will be difficult in such a small sample).
The degrees of freedom in the F-test will be the same for each test of break-point. That is, it will always be F(2,47). The F-statistic calculated (7.438332 in your example) should be different at each tested point. However, given that you have a relatively small sample, such a test may suggest that there is a structural break at every point in the series.
Have you considered alternatives to the full structural break? For example, including a dummy variable for 1991 that could pick up an exogenous shock (such as a policy implementation that impacted GDP growth only in that period, but the economy returned to trend after). Alternatively, you could consider a broken trend model, if you think that the trend growth in GDP has shifted but not the intercept.
EDIT: Following from another user's point (mpiktas) that GDP may have a unit root. You should probably be looking at GDP as a natural logarithm (as we often see GDP moving with an exponential trend, due to the nature of population growth, etc.). Inference from a trend model on the log of GDP should be fine (log-GDP is probably trend-stationary - although you should do some testing - which implies that once accounting for the trend the residual series is stationary).

From your example: $$ y_t = \beta_0 + \beta_1 t + \epsilon_t \qquad (1)$$ The basic form of the Chow test is:

Construct a dummy variable $D_t$ that is $=0$ before the break and $=1$ after the break.
Run a regression: $$ y_t = \beta_0 + \beta_1 t + \gamma_0 D_t + \gamma_1 t D_t + \nu_t \qquad (2) $$
Test the sum of squared residuals from (1) against (2) where: $$ H_0 : \gamma_0 = \gamma_1 = 0 $$ $$ H_1 \text{: At least one coefficient not equal to zero} $$ And, $ F = \frac{SSR_{(1)} - SSR_{(2)}}{SSR_{(1)}} \frac{N-k}{q} $ Where $q$ is the number of restrictions (the number of equals signs in the null hypothesis $H_0$ above, and $k$ is the number of parameters in the restricted model (after applying the null hypothesis, so just $\beta_0$ and $\beta_1$).

Hope this helps.

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Solved – Chow test or not

Solved – How to identify structural change using a Chow test on Eviews

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