  In this paper, we perform a comprehensive study of different covariance and
precision matrix estimation methods in the context of minimum variance
portfolio allocation. The set of models studied by us can be broadly
categorized as: Gaussian Graphical Model (GGM) based methods, Shrinkage
Methods, Thresholding and Random Matrix Theory (RMT) based methods. Among
these, GGM methods estimate the precision matrix directly while the other
approaches estimate the covariance matrix. We perform a synthetic experiment to
study the network learning and sample complexity performance of GGM methods.
Thereafter, we compare all the covariance and precision matrix estimation
methods in terms of their predictive ability for daily, weekly and monthly
horizons. We consider portfolio risk as an indicator of estimation error and
employ it as a loss function for comparison of the methods under consideration.
We find that GGM methods outperform shrinkage and other approaches. Our
observations for the performance of GGM methods are consistent with the
synthetic experiment. We also propose a new criterion for the hyperparameter
tuning of GGM methods. Our tuning approach outperforms the existing methodology
in the synthetic setup. We further perform an empirical experiment where we
study the properties of the estimated precision matrix. The properties of the
estimated precision matrices calculated using our tuning approach are in
agreement with the algorithm performances observed in the synthetic experiment
and the empirical experiment for predictive ability performance comparison.
Apart from this, we perform another synthetic experiment which demonstrates the
direct relation between estimation error of the precision matrix and portfolio
risk.
