lagged regression equation properties

Question

consider this equation 1: \begin{equation} \label{eq:1} C_{t} = \beta_{1} + \lambda C_{t-1} + \epsilon_{t} \end{equation} If the error term is independently and identically distributed (iid) with mean 0 and constant variance and \begin{equation} \label{eq:2} E(\epsilon_{t}\mid C_{t-1}) = 0 \end{equation} i.e., the error is not correlated with \begin{equation}C_{t-1})\end{equation} Is the OLS estimator of the coefficients in equation 1 unbiased and consistent under this new error specification? Why?

My answer: I would assume unbiased and consistent because the expectation of the errors and the lagged variable = 0 ? Thoughts?

Answer 1

OLS is consistent but biased. For consistency, we need $Cov(\epsilon_t, C_{t-1})=0$. The condition, $E[\epsilon_t|C_{t-1}]=0$ is sufficient for the covariance to be 0.

For unbiasedness, we need $E[\epsilon_t|C]=0$ where $C$ is the vector of $C_t$ at all time periods.