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5159
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Second, we
recognise that models of the term structure of interest rates are theories of
predicting the term structures for given parameters and stochastic processes, and
the resulting theoretical term structures usually fail to match the actual term
structures. A plausible explanation for this failure is time variation in the model’s
parameters, as
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Hull and White (1990)
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suggested. Thus, the primary focus of our
empirical application is the estimation of timevarying parameters via a “moving
window” nonlinear system strategy which places no constraints on the motion of
parameters nor on their variancecovariance matrix.
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5975
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The model’s fit varies meaningfully over the postwar era,
while the insample error variance appears to be correlated with common
macroeconomic factors.
This paper is organized as follows. In Section II, we highlight the key assumptions
of the CIR (1985b) and
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Longstaff (1989)
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models, and discuss some of the relevant
results of their papers. Our version of the single state variable nonlinear model is
derived in Section III, and a closed form solution is obtained for constant
parameters.
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6576
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Section V presents our empirical application to interwar and
postwar U.S. term structure data, while Section VI summarizes our results.
II. Review of the Literature
Economists have tried a number of different techniques to model the term structure
of interest rates on discount bonds.
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Vasicek (1977)
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derived a general form of a
partial equilibrium onefactor model of the term structure of interest rates, in which
the instantaneous interest rate, r, is the only state variable, and follows a meanreverting process of the form
dr=qmr()dt+sdz(1)
Arbitrage arguments are used to derive a partial differential equation which all
defaultfree discount bond prices must satisfy in equilibrium.
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7164
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follows a meanreverting process of the form
dr=qmr()dt+sdz(1)
Arbitrage arguments are used to derive a partial differential equation which all
defaultfree discount bond prices must satisfy in equilibrium. In economic terms,
the excess expected return on the default free discount bond with maturity T must
equal the risk premium of the same security. Extensions alone the same line were
made by
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Dothan (1978), Richard (1978), Brennan and Schwartz (1979) and
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many
others.
Richard (1978) uses a BlackScholes type of arbitrage argument that assumes that a
riskless portfolio can be formed using three defaultfree discount bonds with distinct
maturities. That portfolio can be treated as a perfect substitute for any default free
discount bond of other maturities.
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7239
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In economic terms,
the excess expected return on the default free discount bond with maturity T must
equal the risk premium of the same security. Extensions alone the same line were
made by Dothan (1978), Richard (1978), Brennan and Schwartz (1979) and many
others.
 Exact

Richard (1978)
 Suffix

uses a BlackScholes type of arbitrage argument that assumes that a
riskless portfolio can be formed using three defaultfree discount bonds with distinct
maturities. That portfolio can be treated as a perfect substitute for any default free
discount bond of other maturities.
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7529
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Richard (1978) uses a BlackScholes type of arbitrage argument that assumes that a
riskless portfolio can be formed using three defaultfree discount bonds with distinct
maturities. That portfolio can be treated as a perfect substitute for any default free
discount bond of other maturities.
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Brennan and Schwartz (1979)
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developed a twofactor arbitrage model of the term
structure of interest rates. They assume that at any point in time, the term structure
can be written as a function of time and the yields on the default free discount bonds
with the shortest and longest maturities.
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10603
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The solution of this partial differential equation automatically
guarantees that the equilibrium discount bond pricing model will eliminate arbitrage
opportunities and will also be consistent with the underlying economy.
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Longstaff (1989)
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extended the CIR model through a nonlinear version of the term
structure of interest rates. He introduced two forms of nonlinearity: first, he
assumes that technological change affects production returns nonlinearly through a
form of increasing return to scale; and second, he derives the instantaneous interest
rate which can be expressed by the stochastic process
dr=qmr()dt+srdz(t)(3)
where dz
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13666
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The result shows that the nonlinear
general equilibrium model of the term structure of interest rates from this paper has
several analytical improvements over other onefactor models in providing realistic
shapes of the yield curve and term premium.
III.The Model
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Cox, Ingersoll and Ross (1985b)
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develop an intertemporal general equilibrium
model for the term structure of interest rates. It can be summarized as follows:
C1: There is a single good produced in the economy, which may be allocated to
either consumption and investment.
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16035
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In
equilibrium, CIR show that a discount bond price P(r,t) must satisfy the partial
differential equation:
Pt+Prqm(q+l)r[]+
1
2
Prrs
2
(r)rP=0(8)
where l is the negative coefficient of the linear factor risk premium lrPr, or the
price of risk.
To derive the nonlinear model of the term structure of interest rates and obtain a
closed form solution,
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Longstaff (1989)
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works within the CIR framework but
assumes that the single state variable follows a process with random walk behavior
with constant drift and constant variance:
dY=adt+bdz(t)(9)
Longstaff introduces nonlinearity through two different channels.
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16743
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More specifically, the
mean and variance of the production rate of return are proportional to Y2, or
dX(t)=aY
2
dt+bYdz(t)(10)
The instantaneous interest rate is then r(Y,t) = cY2, a nonlinear relationship. This
extension describes a different set of technologies than those in the linear case. It
also induces mean reversion in the equilibrium interest rate, as discussed by
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Sundaresan (1984).
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But Longstaff does not derive the factor risk premium internally
from the CIR framework; he merely assumes that the factor risk premium is linear
in r, and equal to lrPr, the linear form chosen by CIR.
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It has a variance s2r, which depends on r, and s>0, so
the variance increases as the instantaneous interest rate increases.
(2) At r=0, the variance is zero and drift is qμ >0, so that negative interest rates are
precluded.
(3) The boundary study of the stochastic process of the interest rate based on
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Feller (1951), and Karlin and Taylor (1981)
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by Longstaff shows that if the initial interest
rate is nonnegative, then subsequent interest rates from the process will be
nonnegative.
(4) Even if the interest rate reaches zero, since μ>0, the interest rate will
subsequently become positive.
(5) The interest rate dynamics represented above are mean reverting towards μ2 at
the speed of adjustment q.
(6) The interest rate movement can be de
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42010
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To implement equation (33) in this manner, we require prices of zerocoupon bonds
of various tenors, as well as a proxy for the instantaneous interest rate. We utilize
the zerocoupon bond prices constructed by
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Coleman, Fisher and Ibbotson (1989, 1993)
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for the postwar era. We make use of their monthly estimates of zero coupon
bond prices for 12 tenors: 1, 3, 6, 9, 12 and 18months, as well as for 2, 3, 4, 5, 7,
and 10 years. CFI provide monthly quotations on price and yield of zerocoupon
bonds for 1955 through 1992.
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