Gaussian Estimation of a Two‐factor Continuous Time Model of the Short‐term Interest Rate

Abstract

This paper considers the econometric estimation of a two-factor model of the short-term interest rate. We develop a procedure for the time series estimation of its parameters, based on recently developed Gaussian estimation methods which are extended to handle unobservable state variables. The main methodological contribution is the derivation of an exact discrete model and the exact Gaussian likelihood function in terms of the discrete observations and structural parameters of the two-factor model. The model is estimated on one month euro-currency interest rate data for seven currencies ± the Belgium franc, Canadian dollar, Dutch guilder, French franc, German mark, Italian lira, and the US dollar over the period February 1981 to December 1995 ± and our results indicate that the method works well in practice. The empirical estimates reported in this study can be used to compare estimates from the calibration of an arbitrage-free analogue of the model to market prices of interest rate caps and swaptions for use in the ®nancial markets.(J.E.L.: C13, E43).

Rex Bergstrom’s Contributions to Continuous Time Macroeconometric Modeling

“…Babbs and Nowman (1999) developed a general multifactor model of interest rates and estimated the model using the Kalman filter. This model was also estimated using Bergstrom's Gaussian estimation methods in Bergstrom and Nowman (1999), where they derived the exact discrete model and exact Gaussian likelihood function. Wandasiewicz's (1993) thesis investigated monetary and fiscal policy feedbacks in the BW and BNW models using advanced optimal control theory developed in Bergstrom (1987;see also, Bergstrom, Nowman and Wandasiewicz, 1994).…”

Cyclical Trends in Continuous Time Models

“…This is a common feature of models of this kind and is imposed for identification purposes, but it can be quite restrictive. Bergstrom and Nowman (1999), for example, consider a model in which interest rates are a function of two unobservable variables for short and long-term economic news, which are modeled as a bivariate system of differential equations with correlated innovations. The discrete time model for interest rates is an ARMA(2,1), and there are too few autocorrelations of this moving average error to identify the continuoustime innovation variances unless the correlation parameter is fixed.…”

“…Thus, correlation between the innovations in the trend and cyclical components, i.e., σ ηκ and σ ζ κ , causes an identification problem for the variance parameters unless σ ηκ and σ ζ κ are fixed. We consequently follow Bergstrom and Nowman (1999) and set these parameters to zero. This is not too restrictive, since there is always dependence between the trend and cycle (given (1)), but in a trend-plus-cycle model, such restrictions imply independent components, which may be inappropriate for some processes.…”