# Beta hat marginal distribution in Strong OLS setting

If in OLS setting I assume: Epsilon|X is normally distributed (mean= 0; variance= sigma^2 * Identity matrix)

We have that the estimated coefficients (each "beta hat j") is conditional on X normally distributed.

But if we consider "beta hat j" not conditional on X in this setting, what is its distribution? Does it keep Gaussian distribution with stochastic parameters? If it is not like that, why?

• To answer this question, we would need to know (or make assumptions about) the distribution of $X,$ so it looks like this won't be answerable unless you can supply that additional information. – whuber Dec 15 '18 at 20:32