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.

  • $\begingroup$ 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 $\beta$. $\endgroup$ – jmbejara Apr 6 '16 at 3:39
  • $\begingroup$ @jmbejara want to post an answer? I would like to put this question in the answered section. ;) $\endgroup$ – An old man in the sea. Apr 10 '16 at 14:22

To be precise, a model is said to be a linear regression model when it is linear in its parameters. For extension, a model is said to be non-linear when it is non-linear in its parameter (Wooldridge, 2010, p.397).

As such, a linear model can have non-linear variables. A standard example is the Mincer equation, where wage is a linear function of education, experience and experience squared.

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.

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  • $\begingroup$ Lucho, sorry for the delay of 3 years... =D Could you tell me how you can recover the parameters using a linear methods? $\endgroup$ – An old man in the sea. Jun 15 '19 at 11:15

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