<p>OLS is consistent but biased. For consistency, we need <span class="math-container">$Cov(\epsilon_t, C_{t-1})=0$</span>. The condition, <span class="math-container">$E[\epsilon_t|C_{t-1}]=0$</span> is sufficient for the covariance to be 0.</p>
<p>For unbiasedness, we need <span class="math-container">$E[\epsilon_t|C]=0$</span> where <span class="math-container">$C$</span> is the vector of <span class="math-container">$C_t$</span> at all time periods.</p>

CC BY-SA 4.0
2022-04-23T14:24:09.977Z
Just a follow up: Consider the following regression equation (* )    \begin{equation} \label{eq:1}
C_{t} = \beta_{1} + \lambda   C_{t-1} + \epsilon_{t}
\end{equation}
and let \begin{equation} \label{eq:3}
\epsilon_{t} = \rho\epsilon_{t-1} + u_{t}
\end{equation} where the error component \begin{equation} u_{t}\end{equation} is iid with mean 0 and constant
variance, and \begin{equation} \label{eq:4}
E(u_{t}\mid   C_{t-1},\epsilon_{t-1}) = 0
\end{equation}
Is the OLS estimator of the coefficients in (*)
unbiased and consistent under this error specification? Why?
John Williams
CC BY-SA 4.0
2022-04-25T08:44:06.013Z
OLS will be biased for the same reason, although it may be consistent. I responded on your other question.
Michael Gmeiner
CC BY-SA 4.0
2022-04-21T14:21:38.980Z
2022-04-21T14:21:38.980Z
Michael Gmeiner
<p>consider this equation 1: <span class="math-container">\begin{equation} \label{eq:1}
C_{t} = \beta_{1} + \lambda   C_{t-1} + \epsilon_{t}
\end{equation}</span>
If the error term is independently and identically distributed (iid) with mean 0 and
constant variance and <span class="math-container">\begin{equation} \label{eq:2}
E(\epsilon_{t}\mid C_{t-1}) = 0
\end{equation}</span>
i.e., the error is not correlated with  <span class="math-container">\begin{equation}C_{t-1})\end{equation}</span>
Is the OLS estimator of the coefficients in equation 1
unbiased and consistent under this new error specification? Why?</p>
<p>My answer: I would assume unbiased and consistent because the expectation of the errors and the lagged variable = 0 ?
Thoughts?</p>

<econometrics><regression><statistics>
lagged regression equation properties
John Williams
# lagged regression equation properties

(Asked by: username_0 on 2022-04-21)

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?

**Tags:** econometrics, regression, statistics

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## 1 answers

### Answer by username_1 on 2022-04-21. Score: 2

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.

> **Comments:**
> **username_0**: Just a follow up: Consider the following regression equation (* )    \begin{equation} \label{eq:1} C_{t} = \beta_{1} + \lambda   C_{t-1} + \epsilon_{t} \end{equation} and let \begin{equation} \label{eq:3} \epsilon_{t} = \rho\epsilon_{t-1} + u_{t} \end{equation} where the error component \begin{equation} u_{t}\end{equation} is iid with mean 0 and constant variance, and \begin{equation} \label{eq:4} E(u_{t}\mid   C_{t-1},\epsilon_{t-1}) = 0 \end{equation} Is the OLS estimator of the coefficients in (*) unbiased and consistent under this error specification? Why?  
> **username_1**: OLS will be biased for the same reason, although it may be consistent. I responded on your other question.  

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