In a linear regression I have two variables that are correlated with rho = 0.8. Given two multiple linear models where the two variables go in mutually exclusive, the estimare for each one is highly statistical significant. If I plug in both variables, the significance almost vanishes for both, likely due to the high correlation among the two variables
Unfortunately, I cannot collect anymore data to tackle the problem. So I thought maybe ridge and/or lasso regression can help: Both methods can be used in case of high collinearity among the regressors and can be used for model selection. As standard errors for ridge and lasso regressions are mostly meaningess, my reasoning is now: look at a ridge or lasso model at "optimal" lambda value and check if one or both of the variables are driven out of the model (estimate (close to) zero) by the shrinkage or close to being driven out (for larger lambda values).
Does that sound like a sane suggestion to support an effect of both variables? I am not really interested in the estimates themselves (maybe their sign) but rather want to know if both of them stand for an independent effect (on their own) in the model when including both.