Derive bias when AR(1) is approximated by MA(1)

Question

Consider the MA(1) process:

$$ y_t = \varepsilon_t + \theta_1 \varepsilon_{t-1} $$

where $\varepsilon$ is a white noise process with $\mathbb{E}(\varepsilon_t) = 0$ and $\operatorname{Var}(\varepsilon_t) = \sigma^2$, and $t=1,2,\dotsc,T$.   Assume now that given a realization of the MA(1) process you estimate instead an AR(1) model:

$$ y_t = \varphi_1 y_{t-1} + \varepsilon_t $$

where $0<\varphi_1<1$ is an unknown parameter, $\varepsilon$ is a white noise process, and $t=1,2,\dotsc,T$.

Derive the (large sample) bias in the OLS estimator $\varphi_1$.

Answer 1

Lets write the MA as an AR process, $$ \frac{1}{1-L\theta}y_t=\epsilon_t $$ that is an infinite order AR process, $$ \epsilon_t=\sum_{i=0}^{\infty}\theta^iy_{t-i} $$ that is $y_t= \sum_{i=1}^{\infty}\theta^iy_{t-i} +\epsilon_t$. Then you are are ignoring infinite number of terms as you just consider the first order of AR. Thus the bias is comming from misspecifing model. I show an example as you case is a more general case.

Let the true model be $y=x_1\beta_1+x_2\beta_2+e$ and the analyst considers just one variable, $x_1$ then OLS gives, $$ \hat\beta_1=(x'_1 x_1)^{-1}x'_1y $$ and $$ E(\hat\beta_1)=(x'_1 x_1)^{-1}x'_1(x_1\beta_1+x_2\beta_2+e)=\beta_1+E\{(x'_1 x_1)^{-1}x'_1x_2\}\beta_2 $$ where the second term in the right hand side is bias. Your case is more general to this example by infinite terms. On the other hand it is imposible to use OLS for an $AR(\infty)$ process as the covariance matrix is not invertible except $T\rightarrow \infty$.