Pitfalls in the Estimation of Continuous Time Interest Rate Models : The Case of the CIR Model

The difficulties in estimating continuous time interest rate models stem in large parts from measurement problems. The observations are in discrete time and the infinitesimal rate is unobservable and replaced by an observed short term rate with non infinitesimal time-to-maturity. These two measurement features create discretization bias and time-to-maturity bias, respectively. To discuss this question we present an analysis of the different estimation methods used to estimate the historical mean-reverting parameter in the CIR model. Moreover the nonlinear dynamics and the parameter dependent domain of the observed rates can explain the lack of accuracy and even the nonconsistency of several standard estimation approaches.