Parameter Estimation from Overlapping Observations

Abstract

This paper examines parameter estimation (mean, volatility and correlation) for correlated Brownian processes making use of overlapping return observations. In doing so, we derive the minimum variance unbiased estimators within the space of linear (for the mean) and quadratic (for the variance and covariance) combinations of the observations. These estimators weight the observations using the inverse of the (known) correlation structure, for example, the variance estimator is given by: \[\sum_{i,j=1}^N\rho^{-1}_{ij}(x_i-\mu)(x_j-\mu)/(N-1)\], where \[x_i\] are the \[n\]-day overlapping return observations and $\mu$ is the estimated mean of the overlapping observations. These estimators (which are shown to be bias corrected versions of the maximum likelihood estimators) are shown to have standard errors that are not materially different from the standard error of the estimators which use non-overlapping, single-day observations. On the other hand, it is demonstrated that na\"{i}vely using standard estimators that equally-weight the observations (for example, for the variance estimate: \[\sum_{i=1}^N(x_i-\mu)^2/(N-1)\] as would be standard for non-overlapping observations) results in: \begin{enumerate} \item biased estimates, requiring the replacement of \[N-1\] with a factor that is very close to \[N-n\] to remove the bias, and \item estimates that are roughly \[\sqrt{2n/3}\] times noisier that estimates coming from the derived minimum variance estimators. \end{enumerate} These observations are demonstrated through Monte-Carlo experiments as well as using historical equity index data.