Consider the AR(1) model (y_{t}=delta+theta y_{t-1}+e_{t}) where (|theta|)=0) and (operatorname{var}left(e_{t} mid I_{t-1} ight)=sigma^{2}). Let (bar{y}_{-1}=sum_{t=2}^{T} y_{t} /(T-1))

Consider the AR(1) model \(y_{t}=\delta+\theta y_{t-1}+e_{t}\) where \(|\theta|)=0\) and \(\operatorname{var}\left(e_{t} \mid I_{t-1}\right)=\sigma^{2}\). Let \(\bar{y}_{-1}=\sum_{t=2}^{T} y_{t} /(T-1)\) (the average of the observations on \(y\) with the first one missing) and \(\bar{y}_{-T}=\) \(\sum_{t=2}^{T} y_{t-1} /(T-1)\) (the average of the observations on \(y\) with the last one missing).

a. Show that the least squares estimator for \(\theta\) can be written as

b. Explain why \(\hat{\theta}\) is a biased estimator for \(\theta\).

c. Explain why \(\hat{\theta}\) is a consistent estimator for \(\theta\).

Transcribed Image Text:

0 = 0 + (1-1 -3-1) -2 (1-1-1-7)