I am running a regression to predict a variable Y as follows:
I am trying to get a distribution of the expected value of Y given standard errors in the model estimate. To do this, I am planning to make random draws of alpha and beta, normally distributed with mean equal to their estimates and standard deviation equal to their standard error, and feed an X value into this model. The output will generate the distribution of Y. However, I don't know how to get the correlation between $\alpha$ and $\beta$, which I can feed into a Cholesky decomposition to create correlated random draws. How do I find this correlation? Also, does this method make sense?