Breakpoint for bivariate data

Question

The breakpoint(s) estimation approach implemented in the strucchange package (Zeilei & al) seems to work very well (based on my little experience with this package on real case studies).

Is there an existing similar approach for multivariate regression models ? Otherwise, would it be easy to generalize the strucchange method ? (Unfortunately I do not have time to study this question myself).

Update

Here is an animated example (thanks to Yihui Xie's animation package). I have a bivariate dataset, each pair $(x,y)$ has been recorded at some date:

         x        y       date
1 3.690131 6.797999 25/01/2012
2 3.552278 7.055363 07/02/2012
3 3.623821 6.745984 10/02/2012
4 3.600735 6.847450 10/02/2012
5 3.726609 6.894321 14/02/2012
6 3.578204 6.823344 17/02/2012
...

Sarting $n_1=40$, I estimate the shape of the isodensity ellipses of the first $n_1$ observations as well as the shape of the isodensity ellipses of the remaining $n_2=n-n_1$ observations. These are simulated data. For some $n_0>40$ I have simulated Gaussian bivariate i.i.d pairs $(x_i,y_i)$, $i=1, \ldots, n_0$, and then I have simulated Gaussian bivariate i.i.d pairs $(x_i,y_i)$, $i=n_0+1, \ldots, n$ with different parameters. The question is: find $n_0$, which is here a breakpoint for the model lm(cbind(x,y)~1) in R syntax.

Answer 1

Stephane, I answer this thread because you linked it in another post.
Sadly I have not come across calculations for a QLR test with a set of dependent variables instead of one. Seeing as it (at least it used to be) considerable hard to calculate a good table of critical values for the QLR test, I don't know if such a thing exists.

Now you say that your variables are i.i.d. If they are jointly normal but independent, you can go ahead and do a split regression on both regressands and QLR test this. I don't know if you'd theoretically needed different critical values for this as well, but I suggest you err on the side of caution when it comes to significance. Other than that this should at least give a good indication where the structural break might be.

If the variables are not independent I'd probably also do this, and maybe build a normal multiple regression model out of the thing, for example x=y+t or x-y=t or something and run a few simulations on how this tests QLR wise, but this is of course trial and error if you don't want to create a whole new table on your own.

I am sorry I couldn't be of further help, I am not versed enough to give you a definite solution on the problem as I doubt one would get around having to calculate the fitting statistic yourself. Hope this gives you some ideas.

When if comes to implementing the QLR test, just loop F-Tests over the model with break at t/vs. model without, while cutting off ~20% of the dataset at both sides (depends on the table of critical values you use, it will be stated in the table). Display the F-Values and pick the one which seems most likely, pretty straightforward.