<p>I have the following regression</p>
<p><span class="math-container">$$y = X\beta +u$$</span></p>
<p>where <span class="math-container">$y$</span> and <span class="math-container">$u$</span> are <span class="math-container">$(n\times 1)$</span> and <span class="math-container">$X$</span> is a fixed <span class="math-container">$(n \times k)$</span> matrix with full column rank and <span class="math-container">$\beta$</span> is an unknown <span class="math-container">$(k\times 1)$</span> vector of parameters.</p>
<p><span class="math-container">$E(u)=0$</span> and <span class="math-container">$var(u)=\sigma^2 V$</span> with <span class="math-container">$(n\times n) $</span> symmetric and positive definite non-diagonal matrix <span class="math-container">$V$</span>.</p>
<p>Let <span class="math-container">$\hat {\beta_{OLS}}= (X'X)^{-1} X'y$</span></p>
<p>Suppose <span class="math-container">$$u_t = \phi u_{t-1} +v_t $$</span> where <span class="math-container">$v_t$</span> is a white noise process with variance <span class="math-container">$\sigma_v^2$</span> and <span class="math-container">$\phi$</span> is an unknown parameter with <span class="math-container">$|\phi|&lt;1$</span>.</p>
<p>Describe how to get the feasible generalized least square (FGLS) estimator for <span class="math-container">$\beta$</span>.</p>
<p>In the FGLS estimation procedure, we should use the maximized likelihood estimator <span class="math-container">$\hat {\phi_{MLE}}$</span>, which should be obtained through the conditional log-likelihood  function set conditional on the first observation.</p>
<hr />
<p><strong>What I did</strong></p>
<p>I obtain the variance-covariance matrix as follows:</p>
<p><a href="https://i.sstatic.net/Ap1Pl.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Ap1Pl.jpg" alt="enter image description here" /></a></p>
<p>Since <span class="math-container">$\phi$</span> is unknown, the matrix <span class="math-container">$V$</span> is also unknown. Therefore, we need to use FGLS as follows:</p>
<p><em>Step 1:</em> Estimate the generalized regression <span class="math-container">$y=X\beta + u$</span> by OLS and obtain <span class="math-container">$\hat {u}$</span>.</p>
<p><strong>Step 2:</strong> Construct the auxiliary regression <span class="math-container">$\hat {u_t}= \phi \hat {u_{t-1}}+ v_t$</span> in order to obtain <span class="math-container">$\hat {\phi_{MLE}}$</span> and <span class="math-container">$\hat {v_{MLE}}$</span>.</p>
<p>For that, I write down a likelihood function</p>
<p><span class="math-container">$$v_t \sim N(0, \sigma_v^2)$$</span></p>
<p><span class="math-container">$\hat {u_t} \sim N(0, \sigma_v^2)$</span></p>
<p><span class="math-container">$I(\theta, u)=- \frac{n}{2} log(2\pi) - \frac{n}{2} log(\sigma_v^2) - \frac{n}{2\sigma_v^2} (\hat {u_t}- \phi \hat {u_{t-1}})'(\hat {u_t} - \phi \hat {u_{t-1}}) $</span></p>
<p>where <span class="math-container">$\theta = [\phi \sigma_v^2]$</span></p>
<p>FOC:</p>
<p><span class="math-container">$$\frac{\partial I(\theta, u)}{\partial \phi} = 0$$</span></p>
<p><span class="math-container">$$\hat {\phi_{MLE}} = (\hat {u_{t-1}}' \hat {u_{t-1}})\hat {u_{t-1}}' \hat {u_{t}}$$</span></p>
<p><em>Step 3:</em></p>
<p>By using <span class="math-container">$\hat {\phi_{MLE}}$</span> and <span class="math-container">$\hat {v_{MLE}}$</span>, I will obtain the matrix <span class="math-container">$V$</span> and calculate <span class="math-container">$\hat {\beta_{FGLS}} = (X'V^{-1}X)^{-1}X'V^{-1}y$</span></p>
<hr />
<p>I couldn't do the step 2. The question states the conditional log-likelihood function, however I cannot write down this log-likelihood function as conditional function to solve for <span class="math-container">$\hat {\phi_{MLE}}$</span>  and  <span class="math-container">$\hat {v_{MLE}}$</span>.</p>
<p>Please help me how to calculate the step 2. Thank you for your all kinds of helps and suggestions.</p>
<p>This is duplicate post.</p>

CC BY-SA 4.0
2022-09-19T03:13:51.333Z
See the "correcting for AR(1)" section in this. https://web.stanford.edu/class/stats191/notebooks/Correlated_errors.html. The two stage regression procedure described immediately below that should give you what you need.
mark leeds
CC BY-SA 4.0
2022-09-19T09:25:10.593Z
Thank you @markleeds I took a look. However, on this page MLE is not applied. My actual questing is that.
studentp
<econometrics><regression><self-study><time-series><mle>
Application of Maximum Likelihood estimation (MLE) to the Generalized Least Square (GLS) model
studentp
# Application of Maximum Likelihood estimation (MLE) to the Generalized Least Square (GLS) model

(Asked by: username_0 on 2022-09-18)

I have the following regression

$$y = X\beta +u$$

where $y$ and $u$ are $(n\times 1)$ and $X$ is a fixed $(n \times k)$ matrix with full column rank and $\beta$ is an unknown $(k\times 1)$ vector of parameters.

$E(u)=0$ and $var(u)=\sigma^2 V$ with $(n\times n) $ symmetric and positive definite non-diagonal matrix $V$.

Let $\hat {\beta\_{OLS}}= (X'X)^{-1} X'y$

Suppose $$u\_t = \phi u\_{t-1} +v\_t $$ where $v\_t$ is a white noise process with variance $\sigma\_v^2$ and $\phi$ is an unknown parameter with $|\phi|<1$.

Describe how to get the feasible generalized least square (FGLS) estimator for $\beta$.

In the FGLS estimation procedure, we should use the maximized likelihood estimator $\hat {\phi\_{MLE}}$, which should be obtained through the conditional log-likelihood function set conditional on the first observation.

---

**What I did**

I obtain the variance-covariance matrix as follows:

[![enter image description here](https://i.sstatic.net/Ap1Pl.jpg)](https://i.sstatic.net/Ap1Pl.jpg)

Since $\phi$ is unknown, the matrix $V$ is also unknown. Therefore, we need to use FGLS as follows:

*Step 1:* Estimate the generalized regression $y=X\beta + u$ by OLS and obtain $\hat {u}$.

**Step 2:** Construct the auxiliary regression $\hat {u\_t}= \phi \hat {u\_{t-1}}+ v\_t$ in order to obtain $\hat {\phi\_{MLE}}$ and $\hat {v\_{MLE}}$.

For that, I write down a likelihood function

$$v\_t \sim N(0, \sigma\_v^2)$$

$\hat {u\_t} \sim N(0, \sigma\_v^2)$

$I(\theta, u)=- \frac{n}{2} log(2\pi) - \frac{n}{2} log(\sigma\_v^2) - \frac{n}{2\sigma\_v^2} (\hat {u\_t}- \phi \hat {u\_{t-1}})'(\hat {u\_t} - \phi \hat {u\_{t-1}}) $

where $\theta = [\phi \sigma\_v^2]$

FOC:

$$\frac{\partial I(\theta, u)}{\partial \phi} = 0$$

$$\hat {\phi\_{MLE}} = (\hat {u\_{t-1}}' \hat {u\_{t-1}})\hat {u\_{t-1}}' \hat {u\_{t}}$$

*Step 3:*

By using $\hat {\phi\_{MLE}}$ and $\hat {v\_{MLE}}$, I will obtain the matrix $V$ and calculate $\hat {\beta\_{FGLS}} = (X'V^{-1}X)^{-1}X'V^{-1}y$

---

I couldn't do the step 2. The question states the conditional log-likelihood function, however I cannot write down this log-likelihood function as conditional function to solve for $\hat {\phi\_{MLE}}$ and $\hat {v\_{MLE}}$.

Please help me how to calculate the step 2. Thank you for your all kinds of helps and suggestions.

This is duplicate post.

**Tags:** econometrics, regression, self-study, time-series, mle

> **Comments:**
> **username_1**: See the "correcting for AR(1)" section in this. https://web.stanford.edu/class/stats191/notebooks/Correlated_errors.html. The two stage regression procedure described immediately below that should give you what you need.  
> **username_0**: Thank you @markleeds I took a look. However, on this page MLE is not applied. My actual questing is that.  

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## 0 answers

