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Abstract:
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Decision making in finance often requires an accurate estimate of the coskewness matrix to optimize the allocation to random variables with asymmetric distributions. The classical sample estimator of the coskewness matrix performs poorly in terms of mean squared error (MSE) when the sample size is small. A solution is to use shrinkage estimators, defined as the convex combination between the sample coskewness matrix and a target matrix, with the aim of minimizing the MSE. In this paper, we propose unbiased consistent estimators for the MSE loss function and include the possibility of having multiple target matrices. Simulations show that these improvements lead to a substantial reduction in the MSE when estimating the third order comoment matrix of asymmetric distributions, as well as for the estimation of the skewness of a linear combination of random variables. In a financial portfolio application, we find that the proposed shrinkage coskewness estimators are effective in determining the linear combination with a lower out-of-sample variance and higher skewness.
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