Statistical Properties of the Roll Serial Covariance Bid/Ask Spread Estimator

Abstract

ABSTRACT Exact small sample population moments of the standard serial covariance and variance estimators are derived under the assumptions of the Roll bid/ask spread model. Noise explains why serial covariance estimates are often positive in annual samples of daily and weekly returns. Small sample estimator bias partially explains why weekly estimates are more negative than daily estimates. Noise causes the Roll spread estimator to be severely biased by Jensen's inequality. The French‐Roll adjusted variance estimator is unbiased but noisy. Empirical tests confirm the major implications.