Linear statistical inference for global and local minimum variance portfolios

Traditional portfolio optimization has been often criticized since it does not account for estimation risk. Theoretical considerations indicate that estimation risk is mainly driven by the parameter uncertainty regarding the expected asset returns rather than their variances and covariances. This is also demonstrated by several numerical studies. The global minimum variance portfolio has been advocated by many authors as an appropriate alternative to the traditional Markowitz approach since there are no expected asset returns which have to be estimated and thus the impact of estimation errors can be substantially reduced. But in many practical situations an investor is not willing to choose the global minimum variance portfolio, especially in the context of top down portfolio optimization. In that case the investor has to minimize the variance of the portfolio return by satisfying some specific constraints for the portfolio weights. Such a portfolio will be called ‘local minimum variance portfolio’. Some finite sample hypothesis tests for global and local minimum variance portfolios are presented as well as the unconditional finite sample distribution of the estimated portfolio weights and the first two moments of the estimated expected portfolio returns.