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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

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.

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  • $\begingroup$ Hi: Check out Andrew Harvey's "Econometric Analysis of Time Series". This book has beautiful, step by step explanations for how to construct MLE's for that type of problem. $\endgroup$
    – mlofton
    Sep 18 at 5:23
  • $\begingroup$ I have no chance to get this book. How can I find this book? Or can you share the related part of the book as an answer? I really need to learn how to solve for this question. Can you help me to solve this question? Thank you for your helps @mlofton $\endgroup$
    – 1190
    Sep 18 at 14:55
  • $\begingroup$ I can send the relevant section to you through email but let me first see if I can find a pdf of the book on the internet. I bought it on amazon but Harvey's books are generally expensive. The title is "The Econometric Analysis of Time Series" by Andrew Harvey. Let me see if I can find it somewhere and get back to you. $\endgroup$
    – mlofton
    Sep 19 at 15:33
  • $\begingroup$ Hi: You can buy it for 3 dollars from Abe books. It says that condition is poor but who cares really. abebooks.com/Econometric-Analysis-Time-Series-Harvey-A/…-comus_shopp_textbook--naa-_-naa&msclkid=3d436d9c8eda1f4da91a0eb402d17799 Still let me see if I can find a pdf of the book on the net. $\endgroup$
    – mlofton
    Sep 19 at 15:36
  • $\begingroup$ This is not Harvey's but Pollock pretty much explains the same thing. le.ac.uk/users/dsgp1/COURSES/ELOMET/LECTURE5.PDF. $\endgroup$
    – mlofton
    Sep 19 at 15:44

2 Answers 2

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Note that $\hat{u}_t = \phi \hat{u}_{t-1} + v_t,\, v_t \sim \mathcal N\left(0, \sigma^2_{v}\right)$ implies $\hat{u}_t | \hat{u}_{t-1}\sim \mathcal N\left(\phi \hat{u}_{t-1}, \sigma^2_{v}\right)$ for $t = 2, \ldots, n$.
With the additional assumption $\hat{u}_1 \sim \mathcal N\left(0, \sigma^2_{v}/\left(1 - \phi^2\right)\right)$ you get the full log-likelihood kernel
$$ l\left(\phi,\sigma^2_v;\hat{u}\right) = -\frac{n}{2} \cdot \ln\left(\sigma^2_v\right) + \frac{1}{2} \cdot \ln\left(1-\phi^2\right) - \frac{1}{2\sigma^2_v} \left[\sum_{t=2}^n\left(\hat{u}_t-\phi\hat{u}_{t-1}\right)^2 + \left(1 - \phi^2\right)\hat{u}_1^2 \right]. $$ There is no closed-form solution for the maximum likelihood estimates $\hat{\phi}$ and $\hat{\sigma}^2_v$ in this case, but they can be determined iteratively.

Alternatively, you could treat $\hat{u}_1$ as deterministic/known and just maximize the conditional likelihood by maximizing the conditional log-likelihood kernel $$ l_c\left(\phi,\sigma^2_v;\hat{u}\right) = -\frac{n-1}{2} \cdot \ln\left(\sigma^2_v\right) - \frac{1}{2\sigma^2_v} \sum_{t=2}^n\left(\hat{u}_t-\phi\hat{u}_{t-1}\right)^2. $$ The corresponding score equations $$ \begin{align} \frac{\partial l_c}{\partial \phi}\left(\phi,\sigma^2_v;\hat{u}\right) = \frac{1}{\sigma^2_v} \sum_{t=2}^n\left(\hat{u}_t-\phi\hat{u}_{t-1}\right) \cdot \hat{u}_{t-1} \overset{!}{=} 0 \\ \iff \sum_{t=2}^n\left(\hat{u}_t-\phi\hat{u}_{t-1}\right) \cdot \hat{u}_{t-1} = 0 \end{align} $$
and $$ \begin{align} \frac{\partial l_c}{\partial \sigma^2_v}\left(\phi,\sigma^2_v;\hat{u}\right) = -\frac{n-1}{2\sigma^2_v} + \frac{1}{2\left(\sigma^2_v\right)^2} \sum_{t=2}^n\left(\hat{u}_t-\phi\hat{u}_{t-1}\right)^2 \overset{!}{=} 0 \\ \iff -\left(n - 1\right) + \frac{1}{\sigma^2_v} \sum_{t=2}^n\left(\hat{u}_t-\phi\hat{u}_{t-1}\right)^2 = 0 \end{align} $$
then yield the maximum conditional likelihood estimates $$ \begin{align} &\hat{\phi} = \frac{\sum_{t=2}^n\hat{u}_t\hat{u}_{t-1}}{\sum_{t=2}^n\hat{u}_{t-1}^2}, \\ &\hat{\sigma}^2_v = \frac{1}{n-1} \sum_{t=2}^n\left(\hat{u}_t-\hat{\phi}\hat{u}_{t-1}\right)^2, \end{align} $$ from which you only need $\hat{\phi}$ to calculate $\hat{\beta}_\text{FGLS}$ since the matrix $V$ does not depend on $\sigma^2_v$.


Instead of the two-step FGLS procedure you could also use a (full) maximum likelihood approach with the original data that, under the assumption of a multivariate normal $u$, computes $\left(\hat{\beta}, \hat{\sigma}^2, \hat{\phi}\right)$ as maximizer of the likelihood $$ \mathcal{L}\left(\beta, \sigma^2, \phi; y, X\right) = \left(2\pi\sigma^2\right)^{-n/2}|V|^{-1/2} \exp\left(-\frac{1}{2\sigma^2}\left(y-X\beta\right)^\top V^{-1} \left(y-X\beta\right)\right), $$ where $$ \begin{align} &V = \frac{1}{1-\phi^2} \begin{pmatrix} 1 & \phi & \phi^2 & \cdots & \phi^{n-1}\\ \phi & 1 & \phi & \cdots & \phi^{n-2}\\ \phi^2 & \phi & 1 & \cdots & \phi^{n-3}\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ \phi^{n-1} & \phi^{n-2} & \phi^{n-3}& \cdots & 1 \end{pmatrix},|V|=\left(1-\phi^2\right)^{-1},\\ &V^{-1} = \begin{pmatrix} 1 & -\phi & 0 & \cdots & 0 & 0\\ -\phi & 1+\phi^2 & -\phi & \cdots & 0 & 0\\ 0 & -\phi & 1+\phi^2 & \cdots & 0 & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \cdots & 1+\phi^2 & -\phi\\ 0 & 0 & 0 & \cdots & -\phi & 1 \end{pmatrix}. \end{align} $$ Usually, this would be done by iteratively computing the maximizer of the log-likelihood kernel $$ l\left(\beta, \sigma^2, \phi; y, X\right) = -\frac{n}{2} \cdot \ln\left(\sigma^2\right) + \frac{1}{2} \cdot \ln\left(1-\phi^2\right) - \frac{1}{2\sigma^2}\left(y-X\beta\right)^\top V^{-1} \left(y-X\beta\right). $$

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  • $\begingroup$ Thank you very much! $\endgroup$
    – 1190
    Sep 21 at 22:40
  • $\begingroup$ @1190 You're welcome, I'm glad I could help. $\endgroup$
    – statmerkur
    Sep 22 at 11:52
  • $\begingroup$ @statmerkur: very nice answer but, in your second approach, where do the $\hat{u_t}$ come from ? also, in your third approach, would you be using some non-linear optimization method like BFGS or something like that ? Thanks. Oh, basically, what I'm saying is that that I don't think there is a closed form MLE for the OP's problem ? Thanks. $\endgroup$
    – mlofton
    Sep 22 at 18:17
  • $\begingroup$ @mlofton The $\hat{u}_t$'s are the OLS residuals resulting from regressing $y$ on $X$. Yes, there is no closed-form MLE in the third approach$-$that's why I wrote "iteratively". I would go for a Newton-Raphson or quasi-Newton (e.g. BFGS) type of optimization algorithm. $\endgroup$
    – statmerkur
    Sep 22 at 18:49
  • $\begingroup$ @statmerkur In the closed-form MLE in the third approach, only a 1D search over $\phi$ needs to be performed, as I describe in my answer, because to each specific value of $\phi$ corresponds a specific matrix $V_\phi$ and the intermediate solution for $\beta_{\phi}$ has an optimal closed-form GLS solution, i.e., for each specific value of $\phi$ we are maximizing the likelihood. $\endgroup$
    – Number
    Sep 22 at 21:42
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It would appear that Feasible Generalized Least Square (FGLS) should be used when the covariance matrix of the errors $cov(u) = \sigma^2V$ has a completely unknown form. However, in this problem, you have very specific prior knowledge of its form: it's the autocorrelation matrix of a 1st order autoregressive process, AR(1), which is a Toeplitz matrix. You even wrote this matrix explicitly.

Therefore, I feel the most appropriate and exact solution is to "directly" maximize the conditional log likelihood (conditioned on $\phi$), equivalently minimize the negative log likelihood conditioned on $\phi$, and then maximize over $\phi$:

For each specific value of $\phi$, the log likelihood of $u$ conditioned on $\phi$ is a multivariate Gaussian of $u$ ($u_{\phi} = y - X\beta_{\phi}$), but it's necessary to keep the $det(V_{\phi})$ term, as explained below. For a specific value of $\phi$, i.e., conditioned on $\phi$, this is maximized by the Generalized Least Squares (GLS): $\beta_{\phi} = (X'V^{-1}_{\phi}X)^{-1}X'V^{-1}_{\phi}y$.

But in order to maximize the overall unconditional (log) likelihood, it is necessary to compare likelihood values for different values of $\phi$, which requires to keep the $det(V_{\phi})$ term in the expression of the log likelihood.

Since the value of the unknown $\phi$ is in the bounded 1D range $-1 < \phi < 1$, it is a simple matter to maximize the unconditional likelihood over this range by a numerical 1D search (for example by a Fibonacci search for the maximum, etc.), or analytically by computing the $det(V_{\phi})$ as a function of $\phi$ and differentiating by $\phi$. (The analytical expression for $det(V_{\phi})$ as a function of $\phi$ can be obtained "easily", since $V$ has a very specific "easy" Toeplitz form.)

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  • $\begingroup$ Thank you a lot. $\endgroup$
    – 1190
    Sep 21 at 22:41
  • $\begingroup$ You have to incorporate $\sigma^2$ (without which the (log-)likelihood can't be evaluated) as well. Maximizing the profile (log-)likelihood of $\phi$ (treating both $\beta$ and $\sigma^2$, whose MLEs have closed-form expressions for a fixed $\phi$, as nuisance parameters) in a one dimensional optimization would work. $\endgroup$
    – statmerkur
    Sep 24 at 11:26

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