Abstract
This article develops unbiased weighted variance and skewness estimators for overlapping return distributions. These estimators extend the variance estimation methods constructed in Bod et. al. (Applied Financial Economics 12:155-158, 2002) and Lo and MacKinlay (Review of Financial Studies 1:41-66, 1988). In addition, they may be used in overlapping return variance or skewness ratio tests as in Charles and Darné (Journal of Economic Surveys 3:503-527, 2009) and Wong (Cardiff Economics Working Papers, 2016). An example using synthetic overlapping returns from a model fit to data from the SPY S&P 500 exchange traded fund is given in order to demonstrate under which circumstances the unbiased correction becomes significant in skewness estimation. Finally, we compare the effect of the HAC weighting schemes of Andrews (Econometrica 53:817-858, 1991) as a function of sample size and overlapping return window length.
Highlights
Overlapping returns are used in many contexts in the finance and econometrics literature
Standard statistical inference and estimation techniques applied to overlapping return financial time series are typically biased
This motivates the development of unbiased analogues of such estimators which we explore in the cases of the variance and skewness statistics
Summary
Overlapping returns are used in many contexts in the finance and econometrics literature. Standard statistical inference and estimation techniques applied to overlapping return financial time series are typically biased. For such series, recent data is regularly viewed as more relevant than past information, which has resulted in the creation of weighted generalizations of estimation methodologies. This motivates the development of unbiased analogues of such estimators which we explore in the cases of the variance and skewness statistics. Our main contribution is to extend these results by developing weighted unbiased variance and skewness estimators for overlapping return time series. We compare the estimation of the weighted volatility and skewness of the overlapping return distribution of the S&P 500 index for various weighting schemes, sample sizes, and overlapping lengths and conclude with potential additional questions to explore
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