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I have a linear AR(2) regression model of an time series that looks like this:

$y_{t} = \beta_0 + \beta_1 y_{t-1} + \beta_2 y_{t-2} + \beta_3 x_t + \epsilon_t$

I am modelling the contribution of the time series $\{ y_{t-1}, y_{t-2}, x_t\}$ on tomorrow's $y$, $y_{t}$.

I want to test whether these contributions change over time. I do a rolling regression over a period of ~10 years giving me in total a set of about 2000 beta parameters $ \{\hat{\beta_1}, \hat{\beta_2}, \hat{\beta_3} \}$ over time. Indeed inspecting the change qualitatively reveals that there is a trend in the value the parameters over time.

My question: How can I test formally that there is indeed a structural change in the parameters over time?

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Chow test is what you want. Also, Google "parameter constancy".

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