I am trying to estimate the regression model, say standard linear model, with the error term having a Pareto distribution instead of normal. Although it is fairly straightforward to construct the maximum likelihood function, it is practically not easy because we do know how to obtain the estimate for the minimum of the error term as it is an unobserved variable. What I can think of to do it is to estimate the coefficient and Pareto parameters with the grid of predetermined the minimum threshold Pareto parameters and choose the one with the largest likelihood value. Is that how people practically do that?
The model I have in mind is:
$y_i=\beta_0 + \beta_1 x_i + \epsilon_i$ where $\epsilon_i \sim Pareto(k,\alpha)$
where $\epsilon_i>0$, and $\epsilon_i >k>0$. So, you would estimate $\beta_0$, $\beta_1$, $k$ and $\alpha$.