Facing Up to Conditioned Diffusions

Abstract

Most data used in finance are generated naturally rather than experimentally. While researchers are typically interested in estimates of model parameters that are not conditional on the particular sample, actual estimates are necessarily conditional on the data. Recent research on survivorship bias in equity returns and the estimation of term structure models from time-series of interest rate data suggests that failing to account for the implicit conditioning can seriously bias the results of empirical research. This paper develops theoretical and numerical tools that make it possible to account for the implicit conditioning when the underlying data are generated by a time-homogeneous univariate diffusion, and carries out a detailed analysis for three specific conditioning events that are of interest in finance. The techniques are illustrated by obtaining estimates of the drift and diffusion coefficients of a term-structure model from a standard time-series of interest rate data both with and without conditioning on these three events. The estimates indicate that the conditioning events have an important impact on the estimated drift coefficient but little effect on the estimated diffusion coefficient. A test statistic fails to reject linearity of the drift coefficient of the short rate process regardless of which of the conditioning events is assumed.