<p>Let $Z_t = \phi Z_{t-1} + u_t$ where $u_t \sim uniform[-1,1]$ and $|\phi|&lt;1$ I 
I am facing problems coming up with conditional maximum likelihood estimate of an AR(1) process with <strong>uniform errors</strong>. I want to compute the conditional maximum likelihood estimate of $\phi$. Specifically, I want to see how I can come up with: </p>

<ul>
<li><p>the joint distribution of the uniform random variables conditional on $Z_1$ (e.g. if initial values for $Z$ where $Z_1 = 0.3$, $Z_2 = 0.8$ and $Z_3 = 1.2$, $Z_4 = 1.6$) how can I get the joint distribution</p>

<ul>
<li>What will be the conditional maximum likelihood estimate of $\phi$</li>
</ul></li>
</ul>

<p>I have tried to approach the problem as follows but end up with no solution:
Conditional on $Z_1$, $Z_2$ is distributed as: </p>

<p>$Z_2 \sim uniform[-1 + 0.3 \phi, 1 + 0.3 \phi]$ marginal distribution $ 1/2$</p>

<p>$Z_3 \sim uniform[-1 + 0.8\phi, 1 + 0.8 \phi$ marginal distribution $ 1/2$</p>

<p>$Z_4 \sim uniform[-1 + 1.2\phi , 1 + 1.2\phi$ marginal distribution $ 1/2$</p>

<p>I got the marginal distribution to be (1/2) for all of them got confused on how I can maximize the likelihood if the marginal distribution of the variables is not dependent on $\phi$, I am sure I missed something, any suggestion:</p>

<p>Assuming the processes are independent, I took the joint distribution to be a multiple of the individual marginals (1/2), but I think this is somehow wrong and it should depend on the interval they can exist together.</p>

<p>$ f_{z_2,z_3,z_4} = \prod 1 I(x_1 &lt; \phi &lt; x_n)$ but I couldn't get how I should approach this indicator function and also identify what should be the common interval where the marginal distributions overlap.</p>

CC BY-SA 3.0
2017-12-10T22:42:49.397Z
While this is a good question, I feel like you would get more responses in cross validated.
user15543
CC BY-SA 3.0
2017-12-11T13:18:37.700Z
thanks I will also post it on cross validated.
Daniel
CC BY-SA 3.0
2017-12-12T18:37:20.727Z
You should note, however, that this is a perfectly acceptable question for economics.SE. It's up to you to post it here or there. See here: https://economics.stackexchange.com/help/on-topic
jmbejara
CC BY-SA 3.0
2018-01-17T18:53:39.147Z
@jmbejara, you might know better than me what is on topic and what is not, but this looks like a statistical question without an econometric flavour. There is a difference between econometrics and statistics, and this one looks purely statistical. (To briefly illustrate the difference, an econometric question would be dealing with identification, instrumental variables, endogeneity, etc.) I would be cautious about the overlap between Economics SE and Cross Validated and would be inclined to keep the statistical questions on one site, i.e. Cross Validated.
Richard Hardy
CC BY-SA 3.0
2018-01-18T09:58:00.320Z
I'm voting to close this question as off-topic because it is an exact duplicate of https://stats.stackexchange.com/questions/318118/conditional-maximum-likelihood-of-ar1-uniform-process
Ubiquitous
<econometrics><statistics><time-series>
Conditional maximum likelihood of AR(1) UNIFORM PROCESS
Daniel
# Conditional maximum likelihood of AR(1) UNIFORM PROCESS

(Asked by: username_0 on 2017-12-10)

Let $Z\_t = \phi Z\_{t-1} + u\_t$ where $u\_t \sim uniform[-1,1]$ and $|\phi|<1$ I
I am facing problems coming up with conditional maximum likelihood estimate of an AR(1) process with **uniform errors**. I want to compute the conditional maximum likelihood estimate of $\phi$. Specifically, I want to see how I can come up with:

* the joint distribution of the uniform random variables conditional on $Z\_1$ (e.g. if initial values for $Z$ where $Z\_1 = 0.3$, $Z\_2 = 0.8$ and $Z\_3 = 1.2$, $Z\_4 = 1.6$) how can I get the joint distribution

  + What will be the conditional maximum likelihood estimate of $\phi$

I have tried to approach the problem as follows but end up with no solution:
Conditional on $Z\_1$, $Z\_2$ is distributed as:

$Z\_2 \sim uniform[-1 + 0.3 \phi, 1 + 0.3 \phi]$ marginal distribution $ 1/2$

$Z\_3 \sim uniform[-1 + 0.8\phi, 1 + 0.8 \phi$ marginal distribution $ 1/2$

$Z\_4 \sim uniform[-1 + 1.2\phi , 1 + 1.2\phi$ marginal distribution $ 1/2$

I got the marginal distribution to be (1/2) for all of them got confused on how I can maximize the likelihood if the marginal distribution of the variables is not dependent on $\phi$, I am sure I missed something, any suggestion:

Assuming the processes are independent, I took the joint distribution to be a multiple of the individual marginals (1/2), but I think this is somehow wrong and it should depend on the interval they can exist together.

$ f\_{z\_2,z\_3,z\_4} = \prod 1 I(x\_1 < \phi < x\_n)$ but I couldn't get how I should approach this indicator function and also identify what should be the common interval where the marginal distributions overlap.

**Tags:** econometrics, statistics, time-series

> **Comments:**
> **username_1**: While this is a good question, I feel like you would get more responses in cross validated.  
> **username_0**: thanks I will also post it on cross validated.  
> **username_2**: You should note, however, that this is a perfectly acceptable question for economics.SE. It's up to you to post it here or there. See here: https://economics.stackexchange.com/help/on-topic  
> **username_3**: @username_2, you might know better than me what is on topic and what is not, but this looks like a statistical question without an econometric flavour. There is a difference between econometrics and statistics, and this one looks purely statistical. (To briefly illustrate the difference, an econometric question would be dealing with identification, instrumental variables, endogeneity, etc.) I would be cautious about the overlap between Economics SE and Cross Validated and would be inclined to keep the statistical questions on one site, i.e. Cross Validated.  
> **username_4**: I'm voting to close this question as off-topic because it is an exact duplicate of https://stats.stackexchange.com/questions/318118/conditional-maximum-likelihood-of-ar1-uniform-process  

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