I am wondering about the Markowitz theory of portfolio construction in practice. Hence, if one wants to know the efficient frontier, what variances can one use. The only method that I can think is the MLE estimation from past data, but using past data is not effective in my opinion? What do you think?
In literature you'll find many approaches to compute the variance. As mentioned already, the standard ideas are to use MLE, Shrinkage on the Covariance Matrix (Ledoit, Wolf), Shrinkage on the inverse of the Covariance Matrix (Kourtis,Dotsis) which makes sense as in fact the inverse of the Covariance Matrix determines the shape of the efficient frontier. Incorporating dynamics into the estimation as for example with the complete battery of GARCH models is another approach. As you mentioned, relaying on past data is not always effective, as the possibility of structural breaks in the Covariance Matrix exist which would not be observable in past data. In order to overcome this problem there are also approach to use forward-looking data in order to estimate the Covariance Matrix. The core idea of this paper (Kempf) is to rely solely on current option prices when estimating the covariance matrix instead of using historical return information. They argue 'since option prices reflect the expectations of market participants about risk, this approach - unlike the backward-looking approaches used so far - is inherently forward looking'
This is why Markowitz says that the diversification of the portfolio is always preferable. You have a lot of certain past data and some fallible speculations to evaluate the variance and expected return of a title. Inherently the best possible evaluation method, and there are several main ones, is not a foolproof inference. But, if you diversify your portfolio, you minimize the impact of bad assessment, including spurious correlations and of wrong speculation.