Take $y=\alpha + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_3 x_5 + \epsilon$

The Bai-Perron sequential approach allows to identify multiple structural break dates for the regression coefficients in the regression above. However, the structural breaks are common across coefficients. What if I have good reason to believe the structural breaks are different for $x_1$ and $x_2$ and perhaps nonexistent for $x_3$ but still find it relevant to include all variables in the regression for the sake of controlling for omitted variable biases?

I could in principle test for all possible combinations of arbitrary breaks in the regression coefficients using the chow tests. But not only is this computationally unfeasible in my case due to the large sample size, it is also the case that the chow tests are for known break dates. The Andrews sup-F test for unknown break dates focus on single breaks and do not detect multiple break dates.

Which methods would you recommed to do this?

up vote 2 down vote accepted

If you estimate the breakpoints for all coefficients simultaneously, then you will get a break when at least one of the coefficients changes (provided that the power to detect these changes is large enough). Then you can do post hoc tests to check which coefficients actually change at which breaks.

For obtaining coefficient-wise breaks I wouldn't know of a solution that is guaranteed to find globally optimal breaks. However, you could recursively adjust the dependent variable to $y - \beta_1 x_1 + \dots \beta_{j - 1} x_{j - 1} + \beta_{j + 1} x_{j + 1} + \dots + \beta_k x_k = \alpha + \beta_j x_j$ and then only estimate the breaks wrt $x_j$. In each step you could select the variable $x_j$ with the most significant structural break test. But, of course, this is not guaranteed to yield the best-possible solution.

As for Andrews' sup$F$ test: Although this is designed for a single-break alternative, it has good power against many other structural-change patterns as well, including certain smooth transitions or certain multiple breaks.

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