2У((24)-yt(3)-yt(120) = .00053β2t+.37 β3t. Alternatively, note that an increase in β3t will have little
effect on very short or very long yields, which load minimally on it, but will increase medium-term
yields, which load more heavily on it, thereby increasing yield curve curvature.
Now that we have interpreted Nelson-Siegel as a three-factor of level, slope and curvature, it is
appropriate to contrast it to Litzenberger, Squassi and Weir (1995), which is highly related yet distinct.
First, although Litzenberger et al. model the discount curve Pt (τ) using exponential components and we
model the yield curve yt (τ) using exponential components, the yield curve is a log transformation of the
discount curve because yt (τ) = -log Pt (τ)∕τ, so the two approaches are equivalent in the one-factor case.
In the multi-factor case, however, a sum of factors in the yield curve will not be a sum in the discount
curve, so there is generally no simple mapping between the approaches. Second, both we and
Litzenberger et al. provide novel interpretations of the parameters of fitted curves. Litzenberger et al.,
however, do not interpret parameters directly as factors.
In closing this sub-section, it is worth noting that what we have called the “Nelson-Siegel curve”’
is actually a different factorization than the one originally advocated by Nelson and Siegel (1987), who
used
1 -e ^λtτ
У( (t) = b 1t + b 2 t-y---- - b 3 te'''.
Obviously the Nelson-Siegel factorization matches ours with b 11 = β11, b 21 = β21+β31, and b 31 = β31. Ours is
preferable, however, for reasons that we are now in a position to appreciate. First, (1 -e λtτ) / λtτ and
e λtτ have similar monotonically decreasing shape, so if we were to interpret b2 and b3 as factors, then
their loadings would be forced to be very similar, which creates at least two problems. First,
conceptually, it would be hard to provide intuitive interpretations of the factors in the original Nelson-
Siegel framework. Second, operationally, it would be difficult to estimate the factors precisely, because
the high coherence in the factors produces multicolinearity.
Stylized Facts of the Yield Curve and the Model’s Potential Ability to Replicate Them
A good model of yield curve dynamics should be able to reproduce the historical stylized facts
concerning the average shape of the yield curve, the variety of shapes assumed at different times, the
strong persistence of yields and weak persistence of spreads, and so on. It is not easy for a parsimonious
model to accord with all such facts. Duffee (2002), for example, shows that multi-factor affine models
are inconsistent with many of the facts, perhaps because term premia may not be adequately captured by
affine models.