
 Start

1824
 Prefix

Owners and managers of fixed
income portfolios will find accurate forecasts essential. Despite the sizable
body of research focusing on the term structure of interest rates, models based
on the analytics of this relationship–whether arbitragebased (e.g.
 Exact

Merton (1973), Heath et al. (1992))
 Suffix

or of a general equilibrium nature (e.g. Cox et al.
(1985a,b), Longstaff and Schwartz (1992))–have not proven to be reliable in the
prediction of shortterm interest rate movements. We are much better able to
identify arbitrage opportunities at a point in time than we are able to forecast
interest rate movements over a nearterm horizon.
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 Start

1912
 Prefix

Despite the sizable
body of research focusing on the term structure of interest rates, models based
on the analytics of this relationship–whether arbitragebased (e.g. Merton (1973), Heath et al. (1992)) or of a general equilibrium nature (e.g.
 Exact

Cox et al. (1985a,b), Longstaff and Schwartz (1992))
 Suffix

–have not proven to be reliable in the
prediction of shortterm interest rate movements. We are much better able to
identify arbitrage opportunities at a point in time than we are able to forecast
interest rate movements over a nearterm horizon.
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 Start

2285
 Prefix

We are much better able to
identify arbitrage opportunities at a point in time than we are able to forecast
interest rate movements over a nearterm horizon. Models of these
dynamics, such as
 Exact

Cox et al. (1985b),
 Suffix

generally rely upon restrictive
assumptions of linearity on the dynamics of the underlying processes and
stability of their conditional moments.
These assumptions would seem to be at odds with the data, as
nonlinearities in many highfrequency asset returns have been well
documented in the literature.
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 Start

4246
 Prefix

From an empirical
perspective, the presence of nonlinearities would form the basis for improved
predictability of interest rates.
Recent empirical research documents nonlinear dynamics both in the
mean and in the variance of interest rates.
 Exact

Hamilton (1988)
 Suffix

applies a Markov
switching model to U.S. shortterm interest rate data and finds that this
model fits the data better than a linear autoregressive model. Granger (1993)
shows that the U.
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 Start

4450
 Prefix

Hamilton (1988) applies a Markov
switching model to U.S. shortterm interest rate data and finds that this
model fits the data better than a linear autoregressive model.
 Exact

Granger (1993)
 Suffix

shows that the U.S. shortterm interest rate depends in a nonlinear manner
on the spread between long and short interest rates. Anderson (1994) provides
additional evidence for the types of nonlinear effects reported in Granger.
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 Start

4605
 Prefix

Hamilton (1988) applies a Markov
switching model to U.S. shortterm interest rate data and finds that this
model fits the data better than a linear autoregressive model. Granger (1993) shows that the U.S. shortterm interest rate depends in a nonlinear manner
on the spread between long and short interest rates.
 Exact

Anderson (1994)
 Suffix

provides
additional evidence for the types of nonlinear effects reported in Granger.
Kozicki (1994) finds asymmetry in the form of differing responses to positive
and negative shocks.
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 Start

4721
 Prefix

Granger (1993) shows that the U.S. shortterm interest rate depends in a nonlinear manner
on the spread between long and short interest rates. Anderson (1994) provides
additional evidence for the types of nonlinear effects reported in Granger.
 Exact

Kozicki (1994)
 Suffix

finds asymmetry in the form of differing responses to positive
and negative shocks. Naik and Lee (1993) and Das (1993) link the
nonlinearities to changes in economic regimes and stochastic jumps,
respectively.
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4834
 Prefix

Anderson (1994) provides
additional evidence for the types of nonlinear effects reported in Granger.
Kozicki (1994) finds asymmetry in the form of differing responses to positive
and negative shocks.
 Exact

Naik and Lee (1993) and Das (1993)
 Suffix

link the
nonlinearities to changes in economic regimes and stochastic jumps,
respectively. Finally, Pfann, Schotman, and Tscherning (1996) explore the
scope of nonlinear dynamics in shortterm interest rates and its implications
for the term structure.
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6513
 Prefix

approaches, our nonparametric
approach does not impose any specific type of nonlinearity in the estimation
process but, instead, lets the data determine a suitable regression function.
Therefore, the nonparametric approach avoids the parametricmodel
selection problem and allows for a wider array of nonlinear behavior.
Following
 Exact

Diebold and Nason (1990),
 Suffix

we use the locally weighted regression
method (henceforth LWR), a nonparametric estimation method, to model
nonlinearities in mean returns of the 90day U.S. Tbill rate. We measure the
forecasting accurancy of our LWR model using both root mean square error
(RMSE) and mean absolute deviation (MAD) criteria.
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8058
 Prefix

The Locally Weighted Regression (LWR) Method
We attempt to uncover nonlinear relationships in the 90day Tbill rate
using the nonparametric locally weighted regression (LWR) method. LWR is
a nearestneighbor (NN) estimation technique, first introduced by
 Exact

Cleveland (1979) and
 Suffix

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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8103
 Prefix

The Locally Weighted Regression (LWR) Method
We attempt to uncover nonlinear relationships in the 90day Tbill rate
using the nonparametric locally weighted regression (LWR) method. LWR is
a nearestneighbor (NN) 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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10001
 Prefix

To
set the observation weights, we use the tricube weighting function
wit=1−u
3
()
3
, where
u≡
xit−t
x∗
xq−t
x∗
(3)
The value of the regression surface at ∗x is then computed as
y ˆ ∗=ˆ g ∗x
()=∗x
′ˆ
,(4)
where
n
2
.(5)
ˆ =argmin
wt
t=1
∑ty−tx′()
 Exact

Stone (1977)
 Suffix

formulated the problem 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
6q
n→0.
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11475
 Prefix

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).
 Suffix

3. Empirical Estimates
A. Data and Preliminary Diagnostic Tests
Our data are quarterly observations for the 90day U.S. Tbill rate,
referred to as the Tbill rate hereafter. The sample period is 1957:1 to 1988:4
(training set) and observations from 1989:1 to 1993:4 (test set) are used for one7step ahead forecasts.
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12579
 Prefix

Tbill rate changes
are symmetric but leptokurtic.
We first investigate the lowfrequency properties of the TBill rate
series. To do so, we apply the PhillipsPerron tests (PP)
 Exact

(Phillips (1987),
 Suffix

Phillips and Perron (1988)) to both levels and first differences of the Tbill rate.
Table 2 presents the PP test results. Inference is robust to the order of serial
correlation allowed in the data.
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19461
 Prefix

The insample and outofsample superior performance of the LWR
methodology appears to be robust to autoregression order, window size, and
forecasting measure. This evidence is much more encouraging than that
found for exchange rates
 Exact

(Diebold and Nason (1990), Meese and Rose (1990)) and
 Suffix

stock returns (LeBaron (1988), Hsieh (1991)).
Our results could be extended to multiplestepahead forecasting
horizons. The empirical validity of the LWR methodology for other U.
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19535
 Prefix

The insample and outofsample superior performance of the LWR
methodology appears to be robust to autoregression order, window size, and
forecasting measure. This evidence is much more encouraging than that
found for exchange rates (Diebold and Nason (1990), Meese and Rose (1990)) and stock returns
 Exact

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

Our results could be extended to multiplestepahead forecasting
horizons. The empirical validity of the LWR methodology for other U.S.
interest rate series as well as for international interest rate series should also
be investigated.
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21761
 Prefix

,,T().
4
Dividing
the statistic by the estimate of the standard deviation gives
13W
m
(,T)=
TmC,T()−1C,T()m()
Vm,T()
12(A3)
which converges to a normal distribution with unit variance, i.e., N0,1().
Simulations presented by BDS show that this test has good power
against simple nonlinear deterministic systems as well as nonlinear stochastic
processes. Brock, Hsieh, and
 Exact

LeBaron (1991) and
 Suffix

Hsieh and LeBaron (1988)
also report Monte Carlo simulations showing that the asymptotic distribution
is a good approximation to the finite sample distribution when there are
more than 500 observations.
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21797
 Prefix

Simulations presented by BDS show that this test has good power
against simple nonlinear deterministic systems as well as nonlinear stochastic
processes. Brock, Hsieh, and LeBaron (1991) and Hsieh and
 Exact

LeBaron (1988)
 Suffix

also report Monte Carlo simulations showing that the asymptotic distribution
is a good approximation to the finite sample distribution when there are
more than 500 observations.
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22293
 Prefix

They recommend using between onehalf to
two times the standard deviation of the raw data. The accuracy of the
asymptotic distribution deteriorates for high embedding dimensions,
particularly when m is 10 and above.
14Endnotes
1
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Sims (1984), Abel (1988), Hodrick (1987), Baldwin and Lyons (1988), and Nason (1988)
 Suffix

have
shown that economic theory does not rule out the possibility of nonlinear dependence in
conditional means and higherorder conditional moments of asset returns.
2 The only exception was nonlinear autoregression of order one for which the performance of the
LWR fit was inferior to linear fits.
3 Note that i.i.d. implies that
Cm,T()=1C,T()
m but the converse is not tru
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22787
 Prefix

shown that economic theory does not rule out the possibility of nonlinear dependence in
conditional means and higherorder conditional moments of asset returns.
2 The only exception was nonlinear autoregression of order one for which the performance of the
LWR fit was inferior to linear fits.
3 Note that i.i.d. implies that
Cm,T()=1C,T()
m but the converse is not true.
 Exact

Dechert (1988)
 Suffix

offers several counter examples.
4 See Brock, Dechert, and Scheinkman (1987) for definition of the variance V.
15
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