Standard Errors of Risk and Performance Estimators with Serially Correlated Returns

53 Pages Posted: 12 Dec 2017 Last revised: 31 Jan 2018

See all articles by Xin Chen

Xin Chen

University of Washington - Department of Applied Mathematics

R. Douglas Martin

University of Washington Applied Mathematics and Statistics

Date Written: January 30, 2018

Abstract

It is well known that small values of unsuspected returns serial correlation result in substantially inflated standard errors of sample mean estimates of mean returns, and that use of standard error es- timates based on assuming independent and identically distributed (i.i.d.) returns can result in serious under-estimation of the true standard errors. It turns out that nonparametric sample estimators of risk and performance measures may be represented as sums of influence function based nonlinear trans- formations of returns, thereby suffers from the same problem as sample means in the presence of serial correlation. We solve this problem by developing a new general method for computing the standard errors of risk and performance measure estimators for serially correlated returns. The method uses a frequency domain periodogram transformation of the time series of risk or performance measure influence functions as the observations for a polynomial based generalized linear model (GLM) fitting method for exponential distributions. The GLM fitting is a maximum likelihood method with elastic net regularization (glmExpEN). The latter alleviates collinearity problems associated with polynomial re- gression and encourages parsimonious model selection. We provide an example of applying the method to the returns of a collection of hedge funds, and compare the results to four alternative standard error computation methods: (a) an influence function based formula for standard errors based on i.i.d. returns, (b) a bootstrap method based on the i.i.d. assumptions, (c) a bootstrap method for serially correlated data, and (d) a Newey-West method. The results show that use of our new method often improves on all three of these methods when there is serial correlation. We also evaluate the performance of our method via Monte Carlo for a number of well-known risk and performance measures for AR(1) and MA(1) returns models, and in the process compare the performance of our method with the performance to the well-known Newey-West method. The results show that our method performs consistently well for mean, standard deviation, Sharpe ratio, Value-at-Risk and Expected Shortfall estimators for AR(1) and MA(1) returns with parameter values up to 0.5. Furthermore, the new method dominates the Newey- West method for those ranges of parameter values, often by substantial amounts. Our overall method, which we refer to as the seCorIF method, is implemented in the R package EstimatorStandardErrors.

Keywords: Risk/Performance Measure, Maximum-Likelihood Estimator (MLE), Influence Functions, Estimator Variance, Estimator Standard Error, Generalized Linear Model (GLM), Elastic Net Regularization, Serial Correlation

JEL Classification: C3, C14, G10

Suggested Citation

Chen, Xin and Martin, R. Douglas, Standard Errors of Risk and Performance Estimators with Serially Correlated Returns (January 30, 2018). Available at SSRN: https://ssrn.com/abstract=3085672 or http://dx.doi.org/10.2139/ssrn.3085672

Xin Chen

University of Washington - Department of Applied Mathematics ( email )

Box 352420
Seattle, WA 98195-2420
United States

R. Douglas Martin (Contact Author)

University of Washington Applied Mathematics and Statistics ( email )

Seattle, WA 98195
United States

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