Fitting a linear model with non gaussian noise I am trying to evaluate the elasticity of prices of some goods. I am concerned about the gaussianity of the noise in the prices. With non gaussianity I am referring to the non existence of the firt/second moment fo the distribution of the error. Is there a way to fit linear model without the assumption of gaussianity of the error? 
 A: Yes, it is possible and it is used quite a bit nowadays. A possible alternative to the Gaussian distribution consists of using skew distributions. For example, skew versions of the Student t distributions. 
Let $f$ and $F$ be the Student $t$ density and distribution functions with $\nu>0$ degrees of freedom, respectively. Then, you can use the following asymmetric distributions for the errors:


*

*Two-piece Student t density.
$$s_1(x;\mu,\sigma,\gamma,\nu) = \dfrac{1}{\sigma}\left[f\left(\dfrac{x-\mu}{\sigma(1+\gamma)};\nu\right)I(x<\mu) + f\left(\dfrac{x-\mu}{\sigma(1-\gamma)};\nu\right)I(x\geq\mu) \right],$$
$\gamma\in(-1,1)$.


*

*Skew-symmetric Student t density.
$$s_2(x;\mu,\sigma,\gamma,\nu) = \dfrac{2}{\sigma}f\left(\dfrac{x-\mu}{\sigma};\nu\right)F\left(\lambda\dfrac{x-\mu}{\sigma};\nu\right),$$
$\lambda\in{\mathbb R}$.
These distributional assumptions can be used to produce a regression model which is more robust to departures from symmetry of the errors and the presence of outliers.
So, in the context of linear regression, you have the model
$$ y_j = x_j^{\top}\beta + \epsilon_j,$$
where $\epsilon_j\stackrel{ind.}{\sim} s_1\,\text{or}\,s_2$. So, your likelihood, for a sample with $n$ observations, becomes
$$L(\beta,\mu,\sigma,\gamma,\nu)=\prod_{j=1}^n s_i(y_j;x_j^{\top}\beta,\sigma,\gamma,\nu),\,\,\,i=1\,\,\,or\,\,\,2,$$
which can be maximised using your favourite optimisation method.
