The Nobel Memorial Prize for Robert F. Engle

We could estimate the parameters 0. = {β_1t, β_2t, β_3t, λ_t} by nonlinear least squares, for each month t.
Following standard practice tracing to Nelson and Siegel (1987), however, we instead fix λ_t at a
prespecified value, which lets us compute the values of the two regressors (factor loadings) and use
ordinary least squares to estimate the betas (factors), for each month t. Doing so enhances not only
simplicity and convenience, but also numerical trustworthiness, by enabling us to replace hundreds of
potentially challenging numerical optimizations with trivial least-squares regressions. The question
arises, of course, as to an appropriate value for λ_t. Recall that λ_t determines the maturity at which the
loading on the medium-term, or curvature, factor achieves it maximum. Two- or three-year maturities are
commonly used in that regard, so we simply picked the average, 30 months. The λ_t value that
maximizes the loading on the medium-term factor at exactly 30 months is λ_t=0.0609.

Applying ordinary least squares to the yield data for each month gives us a time series of
estimates of {fi₁₁, β₂₁, β₃₁} and a corresponding panel of residuals, or pricing errors. Note that, because
the maturities are not equally spaced, we implicitly weight the most “active” region of the yield curve
most heavily when fitting the model.⁹ There are many aspects to a full assessment of the “fit” of our
model. In Figure 4 we plot the implied average fitted yield curve against the average actual yield curve.
The two agree quite closely. In Figure 5 we dig deeper by plotting the raw yield curve and the three-
factor fitted yield curve for some selected dates. Clearly the three-factor model is capable of replicating
a variety of yield curve shapes: upward sloping, downward sloping, humped, and inverted humped. It
does, however, have difficulties at some dates, especially when yields are dispersed, with multiple
interior minima and maxima. Overall, however, the residual plot in Figure 6 indicates a good fit.

In Table 2 we present statistics that describe the in-sample fit. The residual sample
autocorrelations indicate that pricing errors are persistent. As noted in Bliss (1997b), regardless of the
term structure estimation method used, there is a persistent discrepancy between actual bond prices and
prices estimated from term structure models. Presumably these discrepancies arise from persistent tax
and/or liquidity effects.¹⁰ However, because they persist, they should vanish from fitted yield changes.

In Figure 7 we plot {β₁₁ ,β₂₁ ,∣3₃₁} along with the empirical level, slope and curvature defined
earlier. The figure confirms our assertion that the three factors in our model correspond to level, slope

⁹ Other weightings and loss functions have been explored by Bliss (1997b), Soderlind and
Svensson (1997), and Bates (1999).

¹⁰ Although, as discussed earlier, we attempted to remove illiquid bonds, complete elimination is
not possible.

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