Consistency of an OLS estimator in time series

Question

In the ARMA(2,1) time series

$y_t = \beta_0+\beta_1y_{t-1}+\beta_2y_{t-2}+ u_t + \phi_1u_{t-2}$

$u_t$ and $u_{t-2}$ are white noise shocks. The time series are stationary and ergodic.

In the solution it says that the estimators of $\beta_0$, $\beta_1$ and $\beta_2$ are consistent if $\beta_1 = 0$ becasue "$y_{t-1}$ is still included in the model, which is correlated with part of the error term $u_{t-1}$"

(The question was "Would the OLS estimator of $\beta_0$, $\beta_1$ and $\beta_2$ when regressing $y_t$ on a constant, $y_{t-1}$ and $y_{t-2}$ consistent if $\beta_1 = 0$?")

Is it because it is a time series model and (different from usual simultaneous equation system) $y_{t-1}$ can still be included as the dependent variable of another equation and is correlated with $u_{t-1}$, which makes the whole equation system unidentified?

Or does "$y_{t-1}$ is still included in the model" mean that it is still considered when regression was done as one does not know before regressing it that $\beta_1 = 0$?

Answer 1

$\beta_0, \beta_1, \beta_2$ are all well identified.

When $\beta_1 = 0$, we have $y_t = (1 - \beta_2 L^2)^- (\beta_0 + u_t + \phi_1 u_{t-2})$. So $y_t$ is a weighted average of $u_t, u_{t-2}, u_{t-4}$, ... plus a constant (assuming $|\beta_2| < 1$ and $\beta_2 \ne -\phi_1$). (When $\beta_2 = -\phi_1$, then $y_t$ is merely $u_t$ + a constant, but the conclusion below still stands.)

So in your original OLS, the regressor $y_{t-2}$ is correlated with the residual term (assuming $\phi_1 \ne 0$) and this makes $\hat{\beta_2}$ inconsistent.