# Consistency of an OLS estimator in time series

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 $$\beta_1 = 0$$?