# consistency and unbiasedness of ols estimator

Please explain why the OLS estimator is consistent and unbiased withregard to the equation below

$$$$\label{eq:1} C_{t} = \beta_{1} + \lambda C_{t-1} + \epsilon_{t}$$$$

We can write $$\varepsilon_t=\sum_{s=0}^\infty \rho^su_{t-s}$$.

Thus, $$Cov(\varepsilon_t, C_{t-1}) = Cov(\sum_{s=0}^\infty \rho^su_{t-s}, C_{t-1})$$.

We are given that $$E[u_t|C_{t-1},\varepsilon_{t-1}]=0$$. Thus, $$Cov(u_t, C_{t-1})=0$$. To conclude there is consistency also requires that $$Cov(u_{t-s},C_{t-1})=0$$ for all $$s>0$$.

OLS is definitely biased. For unbiasedness, we need $$E[u_t|C]=0$$ where $$C$$ is a vector of $$C_t$$ at all time periods. This is impossible because $$u_t$$ is definitely correlated with $$C_t$$ (at the same time period).

• and one last follow up: Apr 25, 2022 at 10:16
• Consider the following equation * $$\label{eq:1} C_{t} = \beta_{1} + \lambda Y_{t} + \epsilon_{t}$$ where, $$\label{eq:2} E(\epsilon_{t}\mid Y_{t}) = 0$$ $$\label{eq:3} \epsilon_{t} = \rho\epsilon_{t-1} + u_{t}$$ and the error component Ut, is iid with mean 0, constant variance, and$$\label{eq:4} E(u_{t}\mid Y_{t},\epsilon_{t-1}) = 0$$ questions: (i) Is the OLS estimator of the coefficients in (*) unbiased and consistent? Explain. Apr 25, 2022 at 10:17
• I would assume the ols estimator is unbiased and consistent because Yt covers all time periods unlike c(t-1). Apr 25, 2022 at 10:18
• Also how would you conduct hypothesis tests in this case? Why? Apr 25, 2022 at 10:50
• It's still possible that $Cov(u_t, Y_s)\ne 0$ for some $t\ne s$, which would cause bias. For inference, you can still do a standard t-test. Apr 25, 2022 at 16:14