Conditional maximum likelihood of AR(1) UNIFORM PROCESS

Question

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.

Answer 1

Applying the range restriction $-1 \leqslant u_t \leqslant 1$ to your recursive equation you can obtain:

$$\begin{matrix} - R_t \leqslant \phi \leqslant R_t & & R_t \equiv | \frac{1 - Z_t}{Z_{t-1}} |. \end{matrix}$$

This gives a restriction for one consecutive pair of observed values. Taking this restriction over all the values in the data gives the intersection of these ranges, which is:

$$\begin{matrix} - M_n \leqslant \phi \leqslant M_n & & M_n \equiv \min R_t. \end{matrix}$$

So your likelihood function is:

$$L_{\boldsymbol{z}} (\phi) = \mathbb{I}(- M_n \leqslant \phi \leqslant M_n).$$

Any parameter value within the allowable range gives the same value of the likelihood function (which is not surprising since you have uniform errors). This means that the MLE is undefined.

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One way you could deal with this problem is to look at the MLE conditional only on the first and last values. This would give you a likelihood function that is not flat, which would allow you to obtain a solution that would probably be unique.