# Conditional maximum likelihood of AR(1) UNIFORM PROCESS

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.

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:

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

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