Abstract
This paper is concerned with optimizing the weights of the global minimum-variance portfolio (GMVP) in high-dimensional settings where both observation and population dimensions grow at a bounded ratio. Optimizing the GMVP weights is highly influenced by the data covariance matrix estimation. In a high-dimensional setting, it is well known that the sample covariance matrix is not a proper estimator of the true covariance matrix since it is not invertible when we have fewer observations than the data dimension. Even with more observations, the sample covariance matrix may not be well-conditioned. This paper determines the GMVP weights based on a regularized covariance matrix estimator to overcome the abovementioned difficulties. Unlike other methods, the proper selection of the regularization parameter is achieved by minimizing the mean squared error of an estimate of the noise vector that accounts for the uncertainty in the data mean estimation. Using random-matrix-theory tools, we derive a consistent estimator of the achievable mean squared error that allows us to find the optimal regularization parameter using a simple line search. Simulation results demonstrate the effectiveness of the proposed method when the data dimension is larger than, or of the same order of, the number of data samples.