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I have this little problem and I would appreciate some help.

As part of my master thesis, I have to identify a trend in a univariate (GDP) time series for different countries. I have to separate the trend and the stochastic element in it for each country.

I have managed to do so by doing:

variable c @trend // for each country.

And then running a AR(1) on the residuals // for each country.

However, now I need to identify structural breaks in one of these countries. I've been reading and searching all over the internet and books and I've found that the test most people use to identify these structural changes is the Chow Test.

I know how to run the test, but I have't been able to figure out how to interpret the results, and decide whether there is a structural break or not.

Here there is an example of the results:

enter image description here

What puzzles me the most is the fact that, regardless the point I choose to break the series, I always get

Prob. F(2,47) 0.0016 //or any very significant value, with the same degrees of freedom.

Can someone please help me understand how I should interpret these results in order to identify where the breaks lie?

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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:

  1. 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).
  2. 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.
  3. 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.
  4. 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:

  1. Construct a dummy variable $D_t$ that is $=0$ before the break and $=1$ after the break.
  2. Run a regression: $$ y_t = \beta_0 + \beta_1 t + \gamma_0 D_t + \gamma_1 t D_t + \nu_t \qquad (2) $$
  3. 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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Chow test tests whether the two different models have the same coefficients. It follows $F$ distribution with $k$ and $N_1+N_2-2k$ degrees of freedom, where $k$ is the number of parameters and $N_1$ and $N_2$ are sample sizes of the data the two models are estimated on.

In your case two models are the same regression model estimated with the data before the potential break and after, hence $N_1+N_2=N$, where $N$ is the size of the full sample. This holds for any break. So the same degrees of freedom is perfectly normal. Now the fact that the test is always significant may indicate that the rejection occurs not because of the structural break but because of violations of the Chow test assumptions. Since you are testing GDP series this is possible, since GDP is usually a unit root process and this usually changes the distribution of the usual statistics.

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here the null hypothesis is H0: no structural break just look at p-value of f-statistics it is 0.0016 which is below 5% So reject H0 and there is structural break in your data. thx

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    $\begingroup$ What is your explanation of the constancy of the F-statistic? $\endgroup$ – whuber Dec 30 '12 at 14:44
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Ho: Parameter is structurally stable when your probability is less than 5% as in your case it is 0.0016 then do reject Ho (i.e., the null hypothesis). It means there is a structural break in your data. The best thing you can do is check each variable one by one by using the chow test.

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    $\begingroup$ Welcome to the site, @abida yousaf. Our goal is to build up a permanent repository of high-quality statistical information in the form of questions and answers, so please don't use texting-style abbreviations. $\endgroup$ – gung - Reinstate Monica Oct 24 '13 at 14:09
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I think that Ho is proved and H1 disproved. In addition the GDP has no unit root.

is meaning the trend remains. There is no break point.

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    $\begingroup$ Welcome to the site, @Luis Gonzaga Sarmiento. I tried to edit this to make the English smoother, but I was not sure what you meant in each case. Please ensure it still says what you want it to. In addition, can you clarify the phrase "is meaning the trend remains"? This seems like a fragment to me. Lastly, notice that your username, your identicon, & a link to your userpage is automatically added to your posts, so there is no need to sign your posts, & in fact, we prefer if you do not. $\endgroup$ – gung - Reinstate Monica Nov 6 '13 at 22:23
  • $\begingroup$ It's hard to prove/accept the null in the frequentist framework of NHST. Can you clarify? $\endgroup$ – chl Nov 6 '13 at 23:40
  • $\begingroup$ While I believe what you are saying is that the test suggests that the trend does not contain a breakpoint, an edit would help clarify. As a possible alternative to what @gung suggests, you may also edit your answer to include an explanation in your native language. Someone here should quickly be able to help translate it. Cheers. $\endgroup$ – cardinal Nov 7 '13 at 3:17
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Dear Stability test is used to determine the stability of coefficients of independent variable after OLS is peroformed. Stability test is performed by CUSUM and CUSUMQ. but if results of both contradict each other , then Chow breakpoint is used as alternative. for stability test Ho : Coefficients are stable or Coefficients are not different. here in this case p value is 0.0016 < 0.1 , that means our Ho got rejected. and we can say that our coeffiecients are different i.e. not stable. its all about from my side.

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An alternative for finding the exact year of structural break is to preform the regression: $$yt=β0+β_1t+γ_0Dt+γ_1tDt+νt(2)\;.$$ Move on the number of cases one by one below. From the point of structural break the regression coefficients will become significant continuously.

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  • $\begingroup$ "Move on," I could not fix. It is incomprehensible. $\endgroup$ – Carl Aug 15 '17 at 20:18

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