<p>We can write <span class="math-container">$\varepsilon_t=\sum_{s=0}^\infty \rho^su_{t-s}$</span>.</p>
<p>Thus, <span class="math-container">$Cov(\varepsilon_t, C_{t-1}) = Cov(\sum_{s=0}^\infty \rho^su_{t-s}, C_{t-1})$</span>.</p>
<p>We are given that <span class="math-container">$E[u_t|C_{t-1},\varepsilon_{t-1}]=0$</span>. Thus, <span class="math-container">$Cov(u_t, C_{t-1})=0$</span>. To conclude there is consistency also requires that <span class="math-container">$Cov(u_{t-s},C_{t-1})=0$</span> for all <span class="math-container">$s&gt;0$</span>.</p>
<p>OLS is definitely biased. For unbiasedness, we need <span class="math-container">$E[u_t|C]=0$</span> where <span class="math-container">$C$</span> is a vector of <span class="math-container">$C_t$</span> at all time periods. This is impossible because <span class="math-container">$u_t$</span> is definitely correlated with <span class="math-container">$C_t$</span> (at the same time period).</p>

CC BY-SA 4.0
2022-04-25T10:16:24.567Z
and one last follow up:
John Williams
CC BY-SA 4.0
2022-04-25T10:17:07.403Z
Consider the following equation * \begin{equation} \label{eq:1}
C_{t} = \beta_{1} + \lambda   Y_{t} + \epsilon_{t}
\end{equation}
where,
\begin{equation} \label{eq:2}
E(\epsilon_{t}\mid Y_{t}) = 0
\end{equation}
\begin{equation} \label{eq:3}
\epsilon_{t} = \rho\epsilon_{t-1} + u_{t}
\end{equation}
and the error component Ut, is iid with mean 0, constant variance, and\begin{equation} \label{eq:4}
E(u_{t}\mid   Y_{t},\epsilon_{t-1}) = 0
\end{equation}

questions:
(i) Is the OLS estimator of the coefficients in (*) unbiased and consistent? Explain.
John Williams
CC BY-SA 4.0
2022-04-25T10:18:17.273Z
I would assume the ols estimator is  unbiased and consistent because Yt covers all time periods unlike c(t-1).
John Williams
CC BY-SA 4.0
2022-04-25T10:50:50.133Z
Also how would you conduct hypothesis tests in this case? Why?
John Williams
CC BY-SA 4.0
2022-04-25T16:14:44.483Z
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.
Michael Gmeiner
CC BY-SA 4.0
2022-04-25T08:42:45.957Z
2022-04-25T08:42:45.957Z
Michael Gmeiner
<p><a href="https://i.sstatic.net/vDCxa.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/vDCxa.png" alt="enter image description here" /></a></p>
<p>Please explain why the OLS estimator is consistent and unbiased withregard to the equation below</p>
<p><span class="math-container">\begin{equation} \label{eq:1}
C_{t} = \beta_{1} + \lambda   C_{t-1} + \epsilon_{t}
\end{equation}</span></p>

<econometrics><regression><statistics>
consistency and unbiasedness of ols estimator
John Williams
# consistency and unbiasedness of ols estimator

(Asked by: username_0 on 2022-04-25)

[![enter image description here](https://i.sstatic.net/vDCxa.png)](https://i.sstatic.net/vDCxa.png)

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

\begin{equation} \label{eq:1}
C\_{t} = \beta\_{1} + \lambda C\_{t-1} + \epsilon\_{t}
\end{equation}

**Tags:** econometrics, regression, statistics

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## 1 answers

### Answer by username_1 on 2022-04-25. Score: 2

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).

> **Comments:**
> **username_0**: and one last follow up:  
> **username_0**: Consider the following equation * \begin{equation} \label{eq:1} C_{t} = \beta_{1} + \lambda   Y_{t} + \epsilon_{t} \end{equation} where, \begin{equation} \label{eq:2} E(\epsilon_{t}\mid Y_{t}) = 0 \end{equation} \begin{equation} \label{eq:3} \epsilon_{t} = \rho\epsilon_{t-1} + u_{t} \end{equation} and the error component Ut, is iid with mean 0, constant variance, and\begin{equation} \label{eq:4} E(u_{t}\mid   Y_{t},\epsilon_{t-1}) = 0 \end{equation}  questions: (i) Is the OLS estimator of the coefficients in (*) unbiased and consistent? Explain.  
> **username_0**: I would assume the ols estimator is  unbiased and consistent because Yt covers all time periods unlike c(t-1).  
> **username_0**: Also how would you conduct hypothesis tests in this case? Why?  
> **username_1**: 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.  

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