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?

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#### Best Answer

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:

$$s_1(x;mu,sigma,gamma,nu) = dfrac{1}{sigma}left[fleft(dfrac{x-mu}{sigma(1+gamma)};nuright)I(x<mu) + fleft(dfrac{x-mu}{sigma(1-gamma)};nuright)I(xgeqmu) right],$$

$gammain(-1,1)$.

$$s_2(x;mu,sigma,gamma,nu) = dfrac{2}{sigma}fleft(dfrac{x-mu}{sigma};nuright)Fleft(lambdadfrac{x-mu}{sigma};nuright),$$

$lambdain{mathbb R}$.

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 $epsilon_jstackrel{ind.}{sim} s_1,text{or},s_2$. 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.

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