It is postulated that the observed yield, denotedīy, is a noisy version of this unobserved (true) yield. Finally, note that the coefficient of in the third term is a unimodal function of that decays monotonically to 0 as and as therefore the third term is associated with the medium term bond yields. Next, note that as the coefficient of in the second term converges to 1 while that of in the third term converges to 0 therefore the second term can be thought of as a correction to the overall yield that isĪssociated with the short term bonds. It can also be thought of as the long term yield because as the other two terms vanish the coefficients of both and converge to 0 as (recall that is positive). The first term can be thought of as the overall yield level because it does not depend on, the bond maturity. For fixed time period t, the three terms in this model have relatively simple interpretation. This model is a dynamic version of a static model discussed in Nelson and Siegel ( 1987), where and are time invariant. One of these models depends on a positive, time-varying, scalar parameter and a time-varying three-dimensional vector parameter. Koopman, Mallee, and van der Wel ( 2010) discuss a variety of models for, which is called the yield surface. Even if time is not measured continuously and the bonds of only certain maturities are traded, is treated as a smooth function of two continuous variables, time t and maturity. Suppose that denotes the (idealized) yield at time t that is associated with a bond of maturity (in months). Values for this new bond, which is assigned the index of 18-that is, the value of mtype is 18. The following DATA step creates the necessary missing Has a maturity of 42 months and is not traded on the general exchange. In addition, suppose you are interested in extrapolating the fitted model to predict the yield of a hypothetical bond that The data have been extendedįor two more years by adding missing yields for the years 20, which causes the SSM procedure to produce model forecasts The variable date contains the observation date, yield contains the bond yield, maturity contains the associated bond maturity, and mtype contains an index (ranging from 1 to 17) that sequentially labels bonds of increasing maturity. The data are monthly bond yields that were recorded between the start of 1970 to the end ofĢ000 for 17 bonds of different maturities the maturities range from three months to 10 years (120 months). The following DATA step creates the yield-curve data set, Dns, that is used in this article. This example shows how you can fit the dynamic Nelson-Siegel (DNS) factor model discussed in Koopman, Mallee, and van der
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