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

Question

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.

Answer 1

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.

Answer 2

In response to a comment-request by @Anoldmaninthesea, a linear method to estimate a linear regression with 1st-order auto correlated residuals,

$$y_t = \alpha + x_t\beta + u_t, \qquad u_t = \rho u_{t-1} + e_t,\;\;\; e_t \sim {\rm WN}$$

is the Cochrane-Orcutt one.

Note that the above can be re-written as

$$y^*_t = \delta + \beta x_t^* + e_t,\qquad y^*_t=y_t - \rho y_{t-1},\;\;\;x^*_t=y_t - \rho x_{t-1},\;\;\; \delta = \alpha(1-\rho).$$

The method is iterative. Summarily:

Step 1: Estimate by OLS $y_t = \alpha + x_t\beta$, and obtain the residuals.

Step 2: Regress the residuals on their lag, and obtain an estimate of $\rho$.

Step 3: Form the series $\hat y^*_t = y_t - \hat \rho y_{t-1}$ and $\hat x^*_t = x_t - \hat \rho x_{t-1}$.

Step 4: Run the OLS regression $\hat y^*_t = \delta + \beta \hat x_t^*$ an dobtain the residuals

Continue with Step 2 etc, until the estimate for $\hat \rho$ is stabilized.

Many econometric software include this algorithm as a ready-made command.