How to create the exact Gaussian likelihood of an AR(1) process?

Question

I have read some book, they show me that the exact Gaussian likelihood of an AR(1) process is: the exact Gaussian likelihood of an AR(1) process

Please show me how to have the result like that. Thank you so much.

Answer 1

This is essentially a formula for a pdf of zero mean multivariate normal distribution. Assume you have an AR(1) process

$$x_t=\kappa x_{t-1}+\epsilon_t,$$

where $\epsilon_t \sim N(0,\sigma^2)$, i.i.d and $|\kappa|<1$. Now the unconditional mean of each observation is zero. One can see that the observations follow a multivariate normal distribution. Moreover,the "true" covariance matrix $\Gamma(\theta)$ depends on model parameters $\kappa$ and $\sigma^2$. E.g. assume you sample two observations for the process $(x_0,x_1)$, where the first comes from the stationary distribution. Now the variance of each observation is $\frac{\sigma^2}{1-\kappa^2}$ and the covariance is $\kappa^2\times \frac{\sigma^2}{1-\kappa^2}$. The likelihood is obtained by treating data as given, you will probably find this from the book.