A Note on the Relation Between Principal Components and Dynamic Factors in Affine Term Structure Models

Abstract

In econometric applications of the term structure, affine models are among the most used ones. Nevertheless, even presenting a closed form characteristic function, its estimation procedure still presents many points to be understood and difficulties to be removed. In this note, we address one of these points. Suppose we estimate an affine dynamic term structure model, and also apply principal component analysis to the interest rate database available. A very plausible question would inquire about the relation (if any) between the principal components obtained assuming no dynamic restrictions, and the dynamic factors estimated using the proposed term structure model. We answer this question when estimating a standard affine model using zero coupon data. We show that each principal component can be approximated by a linear transformation of the dynamic factors. Although simple, this is an important step to the understanding of the mechanics of dynamic affine term structure models. A numerical example using U.S. zero data illustrates the result.

Are multi-factor Gaussian term structure models still useful? An empirical analysis on Italian BTPs

“…The three factors can be given the usual interpretation of level, steepness, and curvature. Almeida (2005) explains the relation between principal components obtained assuming no dynamic restrictions and the dynamic factors estimated using multi-factor Gaussian term structure models. The author finds that the linear structure embedded in dynamic affine term structure models directly translates into an approximation of the nonnegligible principal components by linear transformations of the state vector.…”