# Why is a 'linear' regression model, with AR(1) error terms, a non-linear model?

Our model: $y_t=X_t\beta+u_t$

Our error terms: $u_t=\rho u_{t-1}+\epsilon_t$ with $\epsilon_t\sim IID(0,\sigma^2)$, and $|\rho|<1$.

This results in $y_t=\rho y_{t-1}+X_t\beta-\rho X_{t-1}\beta+\epsilon_t$. Why is this last model not linear in $\beta$ and $\rho$?

Is it because the parameters are multiplied with each other in the 3rd term? or is there something else?

Any help would be appreciated.

• I believe you're right. As described on the Wikipedia page (en.wikipedia.org/wiki/Linear_model), you need to be able to write it as a linear model. However, the coefficient on $X_{t-1}$ has a restriction on it that will always depend on $\rho$ or $\beta$. – jmbejara Apr 6 '16 at 3:39
• @jmbejara want to post an answer? I would like to put this question in the answered section. ;) – An old man in the sea. Apr 10 '16 at 14:22

In your example, as your intuition tells you, parameters $\rho$ and $\beta$ are multiplied themselves, making it a non-linear model. However, your model is a rather special kind of non-linear model because its parameters can still be recovered using linear methods. In fact, you do not need non-linear methods at all. Your model is over-identified as you can test whether the estimated combined term is consistent with the individual estimation of each parameter or not.