# lagged regression equation properties

consider this equation 1: $$$$\label{eq:1} C_{t} = \beta_{1} + \lambda C_{t-1} + \epsilon_{t}$$$$ If the error term is independently and identically distributed (iid) with mean 0 and constant variance and $$$$\label{eq:2} E(\epsilon_{t}\mid C_{t-1}) = 0$$$$ i.e., the error is not correlated with $$$$C_{t-1})$$$$ 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?

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.
• Just a follow up: Consider the following regression equation (* ) $$\label{eq:1} C_{t} = \beta_{1} + \lambda C_{t-1} + \epsilon_{t}$$ and let $$\label{eq:3} \epsilon_{t} = \rho\epsilon_{t-1} +$$u_{t}$$$$ where the error component $$u_{t}$$ is iid with mean 0 and constant variance, and $$\label{eq:4} E(u_{t}\mid C_{t-1},\epsilon_{t-1}) = 0$$ Is the OLS estimator of the coefficients in (*) unbiased and consistent under this error specification? Why? Commented Apr 23, 2022 at 14:24