Why is OLS estimator of AR(1) coefficient biased? I am trying to understand why OLS gives a biased estimator of an AR(1) process. Consider 
$$
\begin{aligned}
y_{t} &= \alpha + \beta y_{t-1} + \epsilon_{t}, \\
\epsilon_{t} &\stackrel{iid}{\sim} N(0,1).
\end{aligned}
$$
In this model, strict exogeneity is violated, i.e. $y_t$ and $\epsilon_t$ are correlated but $y_{t-1}$ and $\epsilon_t$ are uncorrelated. But if this is true, then why does the following simple derivation not hold?
$$
\begin{aligned}
\text{plim} \ \hat{\beta} &= \frac{\text{Cov}(y_{t},y_{t-1})}{\text{Var}(y_{t-1})} \\
&=\frac{\text{Cov}(\alpha + \beta y_{t-1}+\epsilon_{t}, y_{t-1})}{\text{Var}(y_{t-1})} \\
&= \beta+ \frac{\text{Cov}(\epsilon_{t}, y_{t-1})}{\text{Var}(y_{t-1})} \\
&=\beta. 
\end{aligned}
$$
 A: @Alecos nicely explains why a correct plim and unbiasedbess are not the same. As for the underlying reason why the estimator is not unbiased, recall that unbiasedness of an estimator requires that all error terms are mean independent of all regressor values, $E(\epsilon|X)=0$. 
In the present case, the regressor matrix consists of the values $y_1,\ldots,y_{T-1}$, so that - see mpiktas' comment - the condition translates into $E(\epsilon_s|y_1,\ldots,y_{T-1})=0$ for all $s=2,\ldots,T$.
Here, we have
\begin{equation*}
y_{t}=\beta y_{t-1}+\epsilon _{t},
\end{equation*}
Even under the assumption $E(\epsilon_{t}y_{t-1})=0$ we have that
\begin{equation*}
E(\epsilon_ty_{t})=E(\epsilon_t(\beta y_{t-1}+\epsilon _{t}))=E(\epsilon _{t}^{2})\neq 0.
\end{equation*}
But, $y_t$ is also a regressor for future values in ain AR model, as $y_{t+1}=\beta y_{t}+\epsilon_{t+1}$.
A: Expanding on two good answers. Write down the OLS estimator:
$$\hat\beta =\beta + \frac{\sum_{t=2}^Ty_{t-1}\varepsilon_t}{\sum_{t=2}^Ty_{t-1}^2}$$
For unbiasedness we need
$$E\left[\frac{\sum_{t=2}^Ty_{t-1}\varepsilon_t}{\sum_{t=2}^Ty_{t-1}^2}\right]=0.$$
But for that we need that $E(\varepsilon_t|y_{1},...,y_{T-1})=0,$ for each $t$. For AR(1) model this clearly fails, since $\varepsilon_t$ is related to the future values $y_{t},y_{t+1},...,y_{T}$.
A: As essentially discussed in the comments, unbiasedness is a finite sample property, and if it held it would be expressed as 
$$E (\hat \beta ) = \beta$$
(where the expected value is the first moment of the finite-sample distribution)
while consistency is an asymptotic property expressed as
$$\text{plim} \hat \beta = \beta$$
The OP shows that even though OLS in this context is biased, it is still consistent.
$$E (\hat \beta ) \neq \beta\;\;\; \text{but}\;\;\; \text{plim} \hat \beta = \beta$$
No contradiction here.
