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∣i t.h /1 = c^+r∣i t.
We include the VAR forecasts for completeness, although one might expect them to be inferior to the AR
forecasts for at least two reasons. First, as is well-known from the macroeconomics literature,
unrestricted VARs tend to produce poor forecasts of economic variables even when there is important
cross-variable interaction, due to the large number of included parameters and the resulting potential for
in-sample overfitting.13 Second, our factors indeed display little cross-factor interaction and are not
highly correlated, so that an appropriate multivariate model is likely close to a stacked set of univariate
models.
In Figure 8 (right column) we provide some evidence on the goodness of fit of the AR(1) models
fit to the estimated level, slope and curvature factors, showing residual autocorrelation functions. The
autocorrelations are very small, indicating that the models accurately describe the conditional means of
level, slope and curvature .
Out-of-Sample Forecasting Performance of the Three-Factor Model
A good approximation to yield-curve dynamics should not only fit well in-sample, but also
forecast well out-of-sample. Because the yield curve depends only on {∣f1t, β2t, β3t}, forecasting the yield
curve is equivalent to forecasting {β1t, β2t, ∣33t}. In this section we undertake just such a forecasting
exercise. We estimate and forecast recursively, using data from 1985:1 to the time that the forecast is
made, beginning in 1994:1 and extending through 2000:12.
In Tables 4-6 we compare h -month-ahead out-of sample forecasting results from Nelson-Siegel
models to those of several natural competitors, for maturities of 3, 12, 36, 60 and 120 months, and
forecast horizons of h = 1, 6 and 12 months. Let us now describe the competitors in terms of how their
forecasts are generated.
(1) Random walk:
yt+h /1 (τ) = y( (τ).
The forecast is always “no change.”’
(2) Slope regression:
13 That, of course, is the reason for the ubiquitous use of Bayesian analysis, featuring strong
priors on the VAR coefficients, for VAR forecasting, as pioneered by Doan, Litterman and Sims (1984).
11