Just to make it simple, let us assume we have data $(x_1,y_1), (x_2, y_2),...,(x_n,y_n)$ and perform a linear regression:
$y = \alpha + \beta x + \epsilon$
I know how to compute the exact confidence interval for $\beta$, but how should we compute the asymptotic confidence interval for $\beta$ without assuming the Gaussian distribution of the error term $\epsilon$?
Thank you in advance!!
Short answer: The asymptotic confidence interval for $\beta$ without assuming a Gaussian error is the same as the exact confidence interval when you do assume a Gaussian error. The reason for this is that the asymptotic interval comes from using the central limit theorem, which tells us that the distribution of the OLS estimator converges to a normal distribution as $n \rightarrow \infty$.
Longer answer: The OLS estimator in linear regression can be writted as:
$$\hat{\boldsymbol{\beta}} = \boldsymbol{\beta} + (\mathbf{x}^\text{T} \mathbf{x})^{-1} \mathbf{x}^\text{T} \boldsymbol{\varepsilon}.$$
Assuming linearity and homoskedasticity of the error term (but not assuming it is Gaussian) we have:
$$\hat{\boldsymbol{\beta}} \sim \text{Dist}(\text{Mean} = \boldsymbol{\beta}, \text{Var} = \sigma^2 (\mathbf{x}^\text{T} \mathbf{x})^{-1}).$$
As $n \rightarrow \infty$ we can employ the central limit theorem to obtain the asymptotic distribution $\hat{\boldsymbol{\beta}} \sim \text{N}$. This requires an assumption on the asymptotic matrix $(\mathbf{x}^\text{T} \mathbf{x})^{-1}$ to ensure that its elements do not balloon out too fast (i.e., to ensure finite asymptotic variance), but so long as we are willing to make standard asymptotic assumptions here, then for large $n$ we will get:
$$\hat{\boldsymbol{\beta}} \overset{\text{Approx}}{\sim} \text{N}(\text{Mean} = \boldsymbol{\beta}, \text{Var} = \sigma^2 (\mathbf{x}^\text{T} \mathbf{x})^{-1}).$$
This approximation is the exact result in the case of Gaussian error terms, so at this point we are using the approximation given by that exact case. The confidence intervals, etc., that follow from this are therefore the same ones that occur as exact intervals in the Gaussian case.