The 12 reference contexts in paper Christopher F. Baum, Olin Liu (1994) “An Alternative Strategy for Estimation of a Nonlinear Model of the Term Structure of Interest Rates” / RePEc:boc:bocoec:275

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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 time-varying parameters via a “moving window” nonlinear system strategy which places no constraints on the motion of parameters nor on their variance-covariance matrix.
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    The model’s fit varies meaningfully over the postwar era, while the in-sample 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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    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 one-factor 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=qm-r()dt+sdz(1) Arbitrage arguments are used to derive a partial differential equation which all default-free discount bond prices must satisfy in equilibrium.
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    follows a meanreverting process of the form dr=qm-r()dt+sdz(1) Arbitrage arguments are used to derive a partial differential equation which all default-free 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 Black-Scholes type of arbitrage argument that assumes that a riskless portfolio can be formed using three default-free 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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    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.
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    Richard (1978)
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    uses a Black-Scholes type of arbitrage argument that assumes that a riskless portfolio can be formed using three default-free 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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    Richard (1978) uses a Black-Scholes type of arbitrage argument that assumes that a riskless portfolio can be formed using three default-free 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 two-factor 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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    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=qm-r()dt+srdz(t)(3) where dz
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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 one-factor 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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    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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    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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    To implement equation (33) in this manner, we require prices of zero-coupon bonds of various tenors, as well as a proxy for the instantaneous interest rate. We utilize the zero-coupon 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 18-months, as well as for 2, 3, 4, 5, 7, and 10 years. CFI provide monthly quotations on price and yield of zero-coupon bonds for 1955 through 1992.
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