Busy immersing myself in some cointegration papers and my understanding is that you check for a cointegrating relationship because running regression on a nonstationary series will lead to "spurious regression". I'm just not sure what part of OLS Beta estimation breaks down with a loss of normality. I think it has something to do with the fact that a nonstationary series has autocorrelation which means that returns are not independent and so you can't use the Strong Law of Large Numbers / Central Limit Theorem and the t-test stuff for normal regression.Sorry that it's such a basic question... appreciate your time! thank you

A pure random walk (Brownian motion) is a non-stationary series (variance blows up with time) and the increments are uncorrelated! On the other hand, AR(1), for e.g. is a stationary process with finite autocorrelation!! On beta estimation, I suppose ordinary least squares assumes that the residuals are normally (Gaussian) distributed; the problem can therefore also be looked in the sense of maximum likelihood comprising of Gaussian distributions. If you assume there is deviation from normality, you can replace the Gaussian function with some other pdf (maybe univariate t-distribution for e.g. or perhaps some kind of Gram-Charlier expansion..) and your beta is simply one of the fitting parameters