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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 $\beta_1 = 0$?

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