In ordinary least squares, for example in an experimental design case, I obtain the regression coefficents by:
$ \hat B = {({X^t}{X})}^{-1}X^ty$
Then, my null hypothesis for each coefficent is:
$H_0 : \hat B_i = 0$
$H_1 : \hat B_i \neq 0$
Next, by assuming the coefficents follows $t$ distribution:
$t_{calculated} = \dfrac {\hat B_i}{se({\hat B_i})}$ where $se({\hat B_i})$ is the standard error of coefficent $\hat B_i$
Again, for OLS, I keep reading that $se({\hat B_i})$ is square root of the corresponding diagonal element ($i^{th}$ diagonal element) of ${\hat \sigma^2}{({X^t}{X})}^{-1}$ where ${\hat \sigma^2}$ is mean square error and is defined as $\sum(y-\hat y)^2$
My questions are:
What if I obtain my regression coefficients by other regression methods such as Ridge, LASSO, ElasticNet, Partial Least Squares etc..? Is this test and/or the way of calculating $t$ value is still valid? I am suspecting it is not, because there are regression techniques which do not involve the calculation of ${({X^t}{X})}^{-1}$ at all.
Why and how do we assume ${\hat \sigma^2}{({X^t}{X})}^{-1}$ is standard error of regression coefficients $B$ in the first place? I think I don't understand the concept of standard error of a regression coefficient in first place.
Small note which may be relevant: I am dealing with a matrix having multicollinearity and OLS fails badly. I think the multicollinearity issue adds another dimension to significance tests and makes things even more complicated.