
 Start

1977
 Prefix

Models of the term structure of interest rates fall into two categories:
those based upon arbitrage arguments and those based on a general equilibrium
formulation. In the former category, singlefactor models of the term structure of interest
rates have been proposed by
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Merton (1973), Vasicek (1977), Dothan (1978), Schaefer and Schwartz (1978), and
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many others. Multifactor term structure models have been proposed by
Richard (1987), Brennan and Schwartz (1979), Langetieg (1980), Schaefer and Schwartz
(1984), and Heath et al. (1992).
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2132
 Prefix

In the former category, singlefactor models of the term structure of interest
rates have been proposed by Merton (1973), Vasicek (1977), Dothan (1978), Schaefer and Schwartz (1978), and many others. Multifactor term structure models have been proposed by
 Exact

Richard (1987), Brennan and Schwartz (1979), Langetieg (1980), Schaefer and Schwartz (1984), and Heath et al. (1992).
 Suffix

Models representing a complete general equilibrium
specification of the term structure have been put forth by Cox, Ingersoll and Ross (1985a,b),
Longstaff and Schwartz (1992) and many others. Chan, Karolyi, Longstaff and Sanders (1992)
and Broze, Scaillet, and Zakoian (1995) provided empirical comparisons of the adequacy of
the models’ explanation of the data.
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2378
 Prefix

Multifactor term structure models have been proposed by
Richard (1987), Brennan and Schwartz (1979), Langetieg (1980), Schaefer and Schwartz (1984), and Heath et al. (1992). Models representing a complete general equilibrium
specification of the term structure have been put forth by
 Exact

Cox, Ingersoll and Ross (1985a,b), Longstaff and Schwartz (1992) and
 Suffix

many others. Chan, Karolyi, Longstaff and Sanders (1992)
and Broze, Scaillet, and Zakoian (1995) provided empirical comparisons of the adequacy of
the models’ explanation of the data. Both theoretical and empirical results from these two
branches of research suggest that term structure models that allow yield nonlinearity can
provide additional insight and explanatory powe
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2459
 Prefix

Multifactor term structure models have been proposed by
Richard (1987), Brennan and Schwartz (1979), Langetieg (1980), Schaefer and Schwartz (1984), and Heath et al. (1992). Models representing a complete general equilibrium
specification of the term structure have been put forth by Cox, Ingersoll and Ross (1985a,b), Longstaff and Schwartz (1992) and many others.
 Exact

Chan, Karolyi, Longstaff and Sanders (1992) and
 Suffix

Broze, Scaillet, and Zakoian (1995) provided empirical comparisons of the adequacy of
the models’ explanation of the data. Both theoretical and empirical results from these two
branches of research suggest that term structure models that allow yield nonlinearity can
provide additional insight and explanatory power for the modelling of equilibrium interest
rates.
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5031
 Prefix

Econometric Methodology
We attempt to uncover nonlinear relationships in U.S. Tbill and bond yields using the
nonparametric locally weighted regression (LWR) method. LWR is a nearestneighbor
estimation technique, first introduced by
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Cleveland (1979) and
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further developed by
Cleveland and Devlin (1988), and Cleveland, Devlin, and Grosse (1988). It is a way of
estimating a regression surface through a multivariate smoothing procedure, fitting a
function of independent variables locally and in a movingaverage manner.
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5078
 Prefix

Tbill and bond yields using the
nonparametric locally weighted regression (LWR) method. LWR is a nearestneighbor
estimation technique, first introduced by Cleveland (1979) and further developed by
 Exact

Cleveland and Devlin (1988), and
 Suffix

Cleveland, Devlin, and Grosse (1988). It is a way of
estimating a regression surface through a multivariate smoothing procedure, fitting a
function of independent variables locally and in a movingaverage manner.
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6968
 Prefix

We also tried the tricube function,
wit=1−u
3
()
3
, suggested by Cleveland, as well as locally unweighted regression but (3)
proved slightly superior empirically.
The value of the regression surface at
x∗
is then computed as
yˆ∗=ˆg∗x
()=∗x
′ˆ
β,
(4)
where
n
2
.
(5)
βˆ=arg min
wt
t=1
∑ty−tx′β()
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Stone (1977)
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addressed the issue of consistent estimation through regularity
conditions on weights of the neighbors. Consistency of NN estimators (and therefore LWR)
requires that the number of NNs used go to infinity with sample size, but at a slower rate,
that is as n→∞,q→∞,but
q
n→0.
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8331
 Prefix

Although the exact distribution of the error sum of squares is not
χ2
(as the
eigenvalues of I−L() need not be all ones or zeros), it can be approximated by a constant
multiplied by a
χ2
variable. The constant and degrees of freedom are chosen so that the first
two moments of the approximating distribution match those of the distribution of the error
sum of squares
 Exact

(Kendall and Stuart, 1977).
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3. Data and Diagnostic Tests
Our data set consists of a variety of short and longterm U.S. Treasury interest rates:
monthly observations on the Federal Funds rate, 3month, 6month, and 12month U.
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10185
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As expected, yield changes for
short maturities exhibit greater variability than those for longer maturities.
We first investigate the lowfrequency properties of the yield series. To do so, we
apply the PhillipsPerron tests (PP)
 Exact

(Phillips (1987), Phillips and Perron (1988)) to
 Suffix

both levels
and first differences of our sample series, with results presented in Table 2. Inference is
robust to the order of serial correlation allowed in the data. All PP tests fail to reject the unit
root null hypothesis in the yield series but strongly reject the unit root null in yield changes.
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10742
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All PP tests fail to reject the unit
root null hypothesis in the yield series but strongly reject the unit root null in yield changes.
The PP test results therefore strongly support the hypothesis of a single unit root in the yield
series. Given the low power of standard unit root tests against fractional alternatives
 Exact

(Diebold and Rudebusch (1991))
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we apply the seminonparametric procedure suggested by
Geweke and PorterHudak (GPH, 1983) to the yield series. The GPH test avoids the knifeedged I(1) and I(0) distinction in the PP test by allowing the integration order to take on
any real value (fractional integration).
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12199
 Prefix

i.i.d. null hypothesis in the BDS test is consistent with some type of
dependence in the data, which could result from a linear stochastic system, a nonlinear
stochastic system, or a nonlinear deterministic system. Under the null hypothesis, the BDS
test statistic asymptotically converges to a standard normal variate. However, Monte Carlo
simulations by Brock, Hsieh, and
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LeBaron (1991) and Hsieh and LeBaron (1988)
 Suffix

suggest that
the asymptotic distribution is a poor approximation to the finite sample distribution when
2 We also applied the KPSS test (Kwiatkowski, Phillips, Schmidt, and Shin (1992)) in which the null hypothesis
is stationarity.
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13609
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19, 19, 6, and 22, respectively (the
maximum order allowed is 24), and the ARCH orders are 3, 4, 2, 2, 2, and 4, respectively.4
We applied the BDS test to these three sets of series for embedding dimensions of m=2, 3, 4
and 5. For each m,ε is set to 0.5 and 1.0 standard deviations (σ) of the data. We use the
quantiles from the small sample simulations reported by
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Brock et al. (1991)
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as
approximations to the finitesample critical values of our BDS statistics. The i.i.d. null
hypothesis is overwhelmingly rejected in all cases for yield changes. When the BDS test is
applied to the ARfiltered series we still obtain strong rejections of the i.i.d. null hypothesis
suggesting that linear dependence in the first moments does not fully account for rejection
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15220
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in linear models, extends to some nonlinear models but not to ARCH
models.
4 The orders for the conditional variance equation are chosen on the basis of superior performance of
diagnostic tests for serial correlation in the standardized and squared standardized residuals obtained from
estimating the corresponding ARARCH models.
5 Using the BDS test,
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Hsieh (1989)
 Suffix

found no evidence of inmean nonlinearities in daily foreign exchange rates
after properly specifying their conditional distributions and time variation in conditional volatility. However,
using neural networks, Kuan and Liu (1995) provided statistically significant out of sample forecasting
improvements over the random walk model for daily exchange rates.
nonlinear struct
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15458
 Prefix

correlation in the standardized and squared standardized residuals obtained from
estimating the corresponding ARARCH models.
5 Using the BDS test, Hsieh (1989) found no evidence of inmean nonlinearities in daily foreign exchange rates
after properly specifying their conditional distributions and time variation in conditional volatility. However,
using neural networks,
 Exact

Kuan and Liu (1995)
 Suffix

provided statistically significant out of sample forecasting
improvements over the random walk model for daily exchange rates.
nonlinear structure in the conditional mean of yieldchange series, we now turn to modeling
nonlinearities in their first moments by means of localfitting methodology.
4.
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20573
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Full results are available upon request from the
authors.
window size enhances the view that the estimated nonlinearities are not a statistical artifact,
but rather capture essential aspects of the data generating process.
In order to formally evaluate model performance, we apply the forecast comparison
test of
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Granger and Newbold (1986,
 Suffix

pp. 278280) to test the hypothesis that there is no
difference in the forecasting accuracy between the linear and nonlinear models. Given that
the nonparametric fit generally achieves a lower RMSE, we test the null hypothesis of no
difference in the forecasting performance between the AR and LWR models against the onesided alternative that the LWR model has superior forecas
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22943
 Prefix

Since it appears that the dynamical behavior of US interest
rates is very local in nature, letting the data determine the regression function pays
dividends in terms of improvements in the forecasting accuracy of interest rate models.
Although the LWR estimation method has failed to successfully predict stock returns
 Exact

(Hsieh (1991), LeBaron (1988)) and
 Suffix

exchange rates (Diebold and Nason (1990), Meese and Rose
(1990, 1991), Mizrach (1992)) it appears to be very useful in modeling conditional mean
changes in interest rate series.9 In addition to nonlinearities in variances and possibly higher
moments, significant nonlinearities in the mean clearly exist for U.
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22998
 Prefix

it appears that the dynamical behavior of US interest
rates is very local in nature, letting the data determine the regression function pays
dividends in terms of improvements in the forecasting accuracy of interest rate models.
Although the LWR estimation method has failed to successfully predict stock returns (Hsieh (1991), LeBaron (1988)) and exchange rates
 Exact

(Diebold and Nason (1990), Meese and Rose (1990, 1991), Mizrach (1992))
 Suffix

it appears to be very useful in modeling conditional mean
changes in interest rate series.9 In addition to nonlinearities in variances and possibly higher
moments, significant nonlinearities in the mean clearly exist for U.
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23947
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Second, our positive results invite the use of alternative
nonparametric methods to be employed as forecasting tools for U.S. interest rates. Finally, an
obvious avenue of future research is to apply the LWR methodology to interest rate series
from other industrial countries.
9
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LeBaron (1992)
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did provide some forecast improvements for stock returns and foreign exchange rates using
the locally unweighted regression method with the level of volatility as the crucial element of conditioning
information.
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