# 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 $$\beta_1 = 0$$?

$$\beta_0, \beta_1, \beta_2$$ 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.