A generalized method of moments comparison of the cox-ingersoll-ross and heath-jarrow-morton models

NORI~I- ~

A Generalized Method of Moments Comparison of the Cox-Ingersoll-Ross and Heath-Jarrow-Morton Models Mahendra Raj, Ah Boon Sim and David C. Thurston

In this paper, we compare two competing term structure models: the general equilibrium-based model of Cox, Ingersoll and Ross and the no-arbitrage-based model of Heath, Jarrow and Morton. We compare term structure fit using a set of US Treasury bills split in half around the structural break of October 1979, following the Federal Reserve Board's change in monetary policy. We use Hansen's generalized method of moments for parameter estimation. Non-nested comparison tests of model fit are made using Davidson-MacKinnon's J test. © 1997 Temple University

Keywords: Term structure; Generalized method of moments; Bond pricing; Cox, Ingersoll, and Ross; Heath, Jarrow, and Morton; Davidson-MacKinnon J test.

JEL classification: C51; C52; E43; G12

I. Introduction Studies of the term structure, the relationship between the rates of return of pure discount bonds and their maturities, have been undertaken for nearly a century. Currently, there are two major approaches used for pricing interest rate contingent claims. In the equilibrium-based approach, bond and contingent claim prices are determined endogenously. Information on investor preferences is required. A model inversion is performed with model parameters and the currently-observed term structure to ensure no-arbitrage opportunities. The second and most recent

Abtrust Chair of International Finance, Aberdeen Business School, Robert Gordon University, Aberdeen, Scotland (MR); School of Banking and Finance, University of New South Wales, Sydney, Australia (ABS); School of Business, Henderson State University, Arkadelphia, AR (DCT). Address correspondence to: D. C. Thurston, School of Business, Box 7512, Henderson State University, Arkadelphia, AR 71999; E-mail: [email protected]

Journal of Economics and Business 1997; 49:169-192 © 1997 Temple University

0148-6195/ 97 / $17.00 PII S0148-6195(97)00076-8

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M. Raj et al. approach is the no-arbitrage-based approach. This approach takes the currentlyobserved term structure as given. Contingent claims are determined exogenously in a risk-neutral world. In this paper, we compare the general equilibrium-based square root model of Cox, Ingersoll and Ross (CIR) (1985) with the no-arbitrage-based exponential and exponential square root models of Heath, Jarrow and Morton (HJM) (1992). We have developed testable empirical specifications to compare the fit of the CIR and HJM models, using thirty-year time series of US Treasury bills. We broke our data in half around the October 1979, structural break, following the Federal Reserve Board's change in monetary policy. Each data set half consists of 198 observations taken every four weeks on consecutive bond maturities ranging from one to twenty-six weeks. To test our models, we examined the difference between realized future rates and the models' implied future rates. We used Hansen's (1982) generalized method of moments to conduct our statistical estimation. Each bond maturity serves as a moment restriction. To compare model fit, we used Davidson and MacKinnon's (1981) non-nested comparison test. The rest of the paper is organized as follows. Section II discusses the HJM model and Section III discusses the CIR model. Section IV discusses the Treasury bill data sets and the statistical methodology. Section V discusses our empirical results and non-nested tests are described in Section VI, while Section VII concludes and discusses directions for future research.

II. The Heath-Jarrow-Morton Model The no-arbitrage-based approach originates with Ho and Lee (1986), who modeled the stochastic movement of pure discount bond prices, using a recombining binomial lattice. This approach makes fewer assumptions on investor behavior than the equilibrium-based approach. Additionally, the pricing of interest rate derivatives is consistent with the current term structure by construction. This is analogous to the way the current stock price is an input in the Black-Scholes (1973) stock option pricing formula. Building on Ho and Lee's work, Heath, Jarrow, and Morton (HJM) developed a no-arbitrage-based model based on the stochastic evolution of forward rates in discrete time (1990) and in continuous time (1992). Under no-arbitrage conditions, HJM completely characterized the dynamics of the term structure by specifying a valid forward rate volatility function(s), along with the currently-observed term structure. In contrast to Ho and Lee's bond-price-based dynamics, HJM's forwardrate-based dynamics provide a stationary series, stable parameters, and multiple factor extensions. The HJM paradigm is increasingly being used by investment houses to price and hedge interest-rate-sensitive securities. Interest-rate-derivative pricing models can usually be calibrated to approximately price-specific securities. However, what traders really want is a model which can be used to price their entire book of securities in a consistent manner. The HJM paradigm allows traders to accurately hedge their total exposure to interest rate and other risks. Flesaker (1994) tested the performance of the constant volatility HJM model, using daily data on Eurodollar futures options from the mid 1980s. He used GMM to conduct tests on a set of at-the-money options. Flesaker found that the constant

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171

model overvalues short-term options, and undervalues all options when interest rates are low. Amin and Morton (1994) used daily data on Eurodollar futures options to test a number of HJM models. They compared models based on their parameter stability and found the number of parameters has a significant effect on model behavior. They also found the one-factor models to be more stable than the two-factor models. Chan et al. (1992) tested eight single-factor models, using a nested specification of the spot-rate process. They employed GMM estimation on monthly data from the Chicago Research in Security Prices (CRSP) government bond files from 1964 to 1989. Contrary to almost all existing studies, Chan et ai. (1992) did no! find evidence for a structural break in October 1979. Using similar data to Chan et al. (1992) and GMM estimation, Abken (1993) tested a variety of HJM models. He found poor model fit, and suggested trying path-dependent models. Contrary to Chan et al., he found evidence for a structural break in October 1979. A drawback which both of these studies share is the assumption of a constant market price of risk. Several recently-published studies have tested discrete time HJM models, using GMM estimation without placing assumptions on the market price of risk. Thurston (1994) tested discrete-time constant and exponential decay models using Treasury bills. Using a simulated residuals approach based on Longstaff's (1989) method of comparing fitted parameter values with forecast rates of return, he found the constant model shows the best fit. Sim and Thurston (1996) tested the fit of discrete time constant and exponential decay HJM models, using a simultaneous set of bond price restrictions. They examined long time series of Treasury bills, CRSP monthly, and CRSP yearly data. They found the best model fit for the monthly data. They found that the exponential decay model performs marginally better. Brenner (1989) tested continuous time constant and exponential HJM models, using maximum-likelihood methods on US Treasury bills in a preference-free environment. He found evidence supporting the constant model.

Notation and Assumptions We now provide a brief discussion of our interest rate and related stochastic process notation, and assumptions placed on bond prices, used throughout the remainder of the paper. The trading interval is denoted by [0, r ]. Time is denoted by t~[0,~-], and bond maturity is denoted by T ~ [ 0 , r], where T > t . The differencing interval is denoted by d. Following Heath, Jarrow and Morton (1990), units of time (expressed in weeks) are denoted with a raised bar (e.g., for our four-week differencing interval, time is d = 4/52, while the number of time units is d = 4). We assume default-free, pure discount bonds. The price of a bond at time, t, with maturity date, T, is denoted by P(t, T). The (instantaneous) forward rate at time, t, for date, T > t, is denoted by f(t, T) while the (instantaneous) spot rate at time, t, is denoted by r(t). Spot rates from time t to T are denoted by r(t, T). The market price of risk at time, t, is denoted by ~b(t). The real-world and risk-neutral Brownian motions at time, t, are denoted by W,, and ~ , respectively. The forward rate drift function is denoted by or(.). The forward rate volatility function is denoted by or(t, T). The HJM exponential decay volatility function is given by or e x p { - A ( T - t)}, and the HJM exponential decay square volatility function is given by or e x p { - A ( T - t)}~/r-~. In the HJM exponential volatility

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M. Raj et al. functions, tr is the volatility coefficient, and A is the rate of decay. Finally, in the construction of our empirical specifications, we differentiate between realized and predicted future bond prices by denoting the model-predicted prices with a hat,

P(t, T). Calculation of the HJM Empirical Specifications HJM's (1992) expression for the evolution of forward rates in integrated form is:

f(t,T)

= f ( 0 , T) +

£a(s,T)ds + £o'(s,T)dW~,

(1)

where,

a(s,T)---~(s,T)(qb(s)-fro'(t,v)dv). HJM's seminal result is that in an arbitrage-free environment, given a valid forward rate volatility function, the drift term is uniquely specified in terms of the volatility function. To transform the HJM forward rate dynamics into an equivalent expression in terms of bond prices, we first integrated equation (1) over maturity (the second argument) to get:

ftTf(t,v)dv = ftTf(0, V ) d r + ftT£tot(s,v)dsdv+ ftTfotOr(s,u)dW s du.

(2)

Next, we used the following identity relationships between (instantaneous) forward rates and bond prices: P(t, T) = exp{ - fff(t, v) dr}, P(0, T) = exp{ - forf(O,v) dr}, and P(0, t) = exp{ - fdf(O, v) dr}, to express equation (2) in terms of bond prices. Explicitly we have: log/~(t, T) = log P(0, T) - log

P(O, t) + ftrfota(S, V) dsdv

+ ftr£to'(s,v) dW, dv.

(3)

Under conditions of certainty, the forward rate volatility is zero. In this case, the model-forecast price is implied from the currently-observed bond prices. Under conditions of uncertainty, the model-forecast future bond price will differ from the implied price. Subtracting both sides of equation (3) from the realized future bond price provides a good measure of model fit. The error between the realized future price and the model-forecast price is given by P(t, T) - 1;(t, T). Omitting the error term for clarity, we used a stochastic extension of Fubini's theorem [see HJM (1992)], to change the order of integration of the right-hand side of equation (3) and move all the bond prices to the left-hand side. Defining the

G M M Tests of CIR and HJM Models

forward bond price deviation F(t, T) -

173 as

log P ( 0 , t ) - log P ( 0 , t ) - log

P(t,

T),

(4)

we can e x p r e s s the r e s u l t i n g d y n a m i c s as:

F(t,T) =

fofr

(s,.) dvds + fot ftT~r(s,v) dvdW~.

(5)

E q u a t i o n s (1), (2), (3) a n d (5) all d e s c r i b e r e a l - w o r l d dynamics. H J M a v o i d e d specifying i n v e s t o r p r e f e r e n c e s in t h e i r c o n t i n g e n t claims pricing m o d e l by w o r k i n g in the r i s k - n e u t r a l world. O n e o f t h e m o s t a t t r a c t i v e f e a t u r e s o f t h e H J M p a r a d i g m is that p r i c e s o f i n t e r e s t r a t e d e r i v a t i v e securities, d e t e r m i n e d in t h e r i s k - n e u t r a l world, r e m a i n valid in t h e r e a l world, (like stock prices in t h e B l a c k - S c h o l e s model). W e u s e d a t w o - s t e p p r o c e d u r e to a v o i d specifying t h e m a r k e t price o f risk in the b o n d p r i c i n g e q u a t i o n (5). In o u r first step, we a p p l i e d G i r s a n o v ' s T h e o r e m 1 to express t h e b o n d p r i c e d y n a m i c s in t h e r i s k - n e u t r a l world:

F ( t , T ) = JoftT~r(s,u)(fs"O'(s,y)dy)dvds+

fotftTcr(s,v)dvdI~'s.

(6)

In o u r s e c o n d step, we s u b s t i t u t e d o u t for t h e stochastic t e r m with a n o t h e r b o n d o f m a t u r i t y , K, ( w h e r e K ~ T a n d K > t). 2 T o explicitly c a l c u l a t e the f o r w a r d b o n d price d e v i a t i o n s for t h e e x p o n e n t i a l m o d e l s , we s u b s t i t u t e d into e q u a t i o n (6) for t h e H J M e x p o n e n t i a l d e c a y a n d e x p o n e n t i a l s q u a r e r o o t volatility functions. 3 T h e f o r w a r d p r i c e d e v i a t i o n for the H J M e x p o n e n t i a l d e c a y m o d e l with a f o u r - w e e k d i f f e r e n c i n g interval, d, is t h e n given by: 4

( e ~ r - e Ad ) F ( d , T ) = e AK e Ad F(d, K) o. 2 +(e-~r _ e-aK)(e--ar _ e -ad)__~3 (e 2ad _ =Yg(II,r),

w h e r e 11 - { a , ¢ } ,

/3 = 0 for t h e e x p o n e n t i a l m o d e l , a n d model.

1)r(d) ~ (7)

/3 = .5 for the e x p o n e n t i a l s q u a r e r o o t

IThis removes the market price of risk from the drift term, giving a(s, v) = ~r(s, v)/','0"(t, y)dy. is now a Brownian motion under the equivalent martingale measure. The Girsanov transformation replaces the real-world measure, less the market price of risk, with the equivalent martingale measure: dWt = dWt = $(t). (The corresponding integrated form is: Wt = Wt - fd$(v)dL,.) 2In our empirical framework, when we substitute out for the risk-neutral world stochastic term with another forward bond price deviation, F(t, K), we are effectively removing the real-world stochastic term less the (implied) market price of risk. 3See the Appendix for details of the derivation. 4In the construction of a time series of forward bond price deviations, with this notation, every four weeks the current term structure is updated and time is reset to zero. In actual estimation, each forward bond price deviation is normalized for time by dividing both sides of equation (7) by the maturity of the future bond price.

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IIl. The Cox-Ingersoll-Ross Model In this paper, we examine CIR's nominal bond pricing model, often referred to as the "square root" model. CIR's square root model has been widely cited throughout the term structure literature and, as such, is an important benchmark for comparing the performance of new stochastic interest-rate models. The spot rate is used as a sufficient statistic for the currently-observed term structure and its evolution over time. One advantage of the square root model is its ability to price bonds and interest-rate contingent claims in terms of a single state variable, the (instantaneous) spot rate. A theoretical and practical drawback of the square root model is that contingent claim prices depend upon investor preferences, which are assumed to be logarithmic. Brown and Dybvig (1986) conducted one of the first major tests of CIR's model. They rewrote the original four-parameter model in terms of three observable parameters and the spot rate. They used monthly observations from the CRSP bond files from 1952 to 1983. Using maximum-likelihood methods, they obtained monthly cross-sectional estimates of the spot rate, its variance, and the mean reversion rate. They obtained their time series estimates by averaging their cross-sectional estimates. Criticisms of this study include: 1) model parameters are constrained to be constant across time, and 2) bond price residuals for all maturities are assumed to be identically distributed. Brown and Dybvig (1986) found that although the square root model consistently overestimates the spot rate, it provides a good forecast of the variance in changes in the spot-rate series. Finally, among bond price data, they found that CIR's model fits the T-bill data best. Chen and Scott (1992) tested the CIR square root model and multifactor extensions on two data sets using maximum-likelihood methods, and made nonnested comparisons following Vuong (1989). Their first data set was a monthly time series from McCulloch and Kwon's (1993) cubic spline-fitted database, while their second data set was a weekly time series of Treasury bills and bonds. Chen and Scott (1992) found a poor fit for the one-factor CIR model, and observed that adding a second factor improved the fit. Calculation o f the C I R Empirical Specification

CIR modeled the spot rate to evolve according to the following mean-reverting square root process: dr = K( tx - r ( t ) ) dt + o ' x / ~

d Wt

(8)

Attractive features of the square root model include: 1) interest rates are constrained to be non-negative; 2) bond pricing is based on a single factor; 3) volatility varies with the level of the spot rate, and 4) the model is Markovian in the spot rate. 5

5A function is a Markov process, if given past and present information, the future value of the security dependsonlyon present information.

GMM Tests of CIR and HJM Models

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One of the major drawbacks of using CIR's square root model to price intc,rest rate contingent claims is that the current pure discount bond prices are not matched by construction. Pricing derivatives is, thus, a two-step procedure. In the frst step, pure discount bonds are priced. In the second step, derivatives are priced and a complicated inversion is performed to match the currently-observed pure discount prices (and thus avoid arbitrage opportunities). Although the general equilibrium- and no-arbitrage-based approaches differ considerably, we can view both paradigms as obtaining future bond prices from the stochastic evolution of currently-observed bond prices. However, in the CIR model, as bond prices are determined endogenously, the market price of risk and other parameters are entangled with current bond prices, and a complicated nonlinear inversion of the model parameters and the current Term structure is required for full model identification.6 The identification problem with the CIR model has prevented the development of a completely satisfactory empirical estimation procedure. Gibbons and Ramaswamy (1993) used the steady state distribution of the spot rate in lieu of matching the current term structure. Other studies, such as Chen and Scott (1992), have used a nonlinear conditional density function and maximum-likelihood methods. Here, we have followed Brown and Dybvig (1986) to estimate a set of three observable parameters: {4)1,052,053}. We also used the one-week Treasu~ ~bill as a proxy to CIR's unobservable state variable, the insta~taneous spot rate. Pure discount bond prices in the square root model are given by: P(r(t); t, T) = A ( t , T)exp{ - B ( t , T ) r ( t ) } ,

(9)

where the functions A(-) and B(.) are given by:

A(t,T)B(t, T) -

(05, exp{05~(T- t)} )~ 052(ex~-05-~-_----~ 7_ ~ + 05, exp{051(T - t)} - 1 052(exp{051(T - t)} - 1) + 051

Brown and Dybvig's three observable parameters are related to the original four parameters as follows: 051 -= ( ( r

+ A) 2 +

20"2) I/2

(10a)

052 = (• + A + 051) / 2

(10b)

053 =- 2 t
t l0c)

The interpretation of CIR's original four parameters is as follows: /z is the steady state mean; K is the speed of adjustment; A is the market price of risk, 7 and 0- 2 is

6A difficult problem which any empirical study faces in trying to ensure a match of the CIR model to the current term structure is that CIR's model may not be consistent with the current term structure [see HJM (1992), pp. 96-97]. 7CIR's market price of risk parameter, A, is unrelated to HJM's exponential decay rate parameter, A.

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M. Raj et al. the variance of the instantaneous spot rate, given by: 0.2 = 2(~1~2 _ ~2).

(11)

The expectation of the infinitely maturing spot rate (long rate) is given by:

rL = (¢1 -- ~bz)~b3.

(12)

The spot rate process in C I R ' s model, calculated from equation (9), is:

r(t, T ) = ( r ( t ) B ( t , T ) - log A ( t , T ) ) / ( T - t).

(13)

To develop our testable empirical specification for C I R ' s square root model, we priced bonds using equation (9). We substituted from equation (9) into equation (4) for the T- and t-maturity bonds at time zero and the T - t-maturity bond at time t. For our empirical tests, we used a four-week differencing interval, denoted by d. Our forward bond price deviation for C I R ' s square root model is then given by: 8

F ( d , T ) = log A(0, T ) - log A(0, d) - log A ( d , T ) + B ( d , T ) r ( d ) + [B(0, d) - B(0, T ) l r ( 0 ) = $'(¢P, r),

where ¢ - {&l, ¢2, ¢b3}.

(14)

A positive instantaneous spot rate variance, or 2, requires ~b1 > ~b2 > 0. We imposed these constraints on the estimated p a r a m e t e r values by taking absolute values.

IV. Data and Methodology The Treasury Bill Data The data consist of the midpoints of Thursday bid and ask discounts for T-bills with maturities from one to twenty-six weeks, taken from The Wall Street Journal. 9 The Federal Reserve Board (FED) significantly changed its policy and, consequently, the dynamics of interest rate movements when it shifted its emphasis from the interest rate level (federal funds) to the m o n e y supply (bank reserves). This change occurred over a weekend and is reported in The Wall Street Journal on Monday, October 8, 1979. The data were split into two equal subsets, each set initially

8In the construction of a time series of forward bond price deviations, with this notation, every four weeks the current tetra structure is updated and time is reset to zero. In actual estimation, each forward bond price deviation is normalized for time by dividing both sides of equation (14) by the maturity of the future bond price. 9portions of the database originated from a Cornell University database. The next available quote was used in the event of a holiday, and suspicious quotes were cross-checked with the New York Times. Thursdays' quotes were used, as that is the maturity date for thirteen and twenty-sixweek T-bills. The T-bill quotes, recorded as banker's discounts based on a 360-dayyear, were converted to pure discount bond prices based on a 365-day year.

GMM Tests of CIR and HJM Models T--Bills:

177 1964

to

1979

Figure 1. Term structure from Data Set l.

containing 795 weekly observations. Data Set 1 runs from July 16, 1964 to October 10, 1979, while Data Set 2 runs from October 11, 1979 to D e c e m b e r 30, 1994. For comparability with other studies, and because of the large amount of data available, we analyzed term structure changes over nonoverlapping four-week periods. 1° Consequently, only a quarter of the available weekly data was used, but the resulting n u m b e r of observations in the cross-section is still large. Using this partitioning, the 795 weekly observations in each T-bill data set were reduced to subsets of 198 observations each, with four-week differencing. 1~ The four-week differencing reduced the n u m b e r of bond maturities available in the cross-section from twenty-six to twenty-two. D a t a Set 1 is illustrated in Figure 1. From an examination of Figure 1, we observe that for the first data set, short term rates are in general low, and the term structure is upward sloping. At the start of Data Set 1, in the mid 1960s, short-term rates were extremely fiat. We can observe two peaks in interest rates, in the late 1960s and the mid-1970s, when rates peaked and the term structure slopes downward. In the late 1970s, just prior to the Fed's monetary policy change, interest rates rose to a plateau of 10%. Data Set 2 is illustrated in Figure 2. From an examination of Figure 2, we first observe the extraordinary high levels of interest rates and downward sloping term structures which characterized the period immediately after the Fed's October 1979 change in monetary policy. In N o v e m b e r 1982, interest rates dropped significantly and remained at much lower levels. Even after the late 1982 drop in interest rate levels, short rates can be seen to be markedly more volatile than in the

l°G. O. Bierwag has suggested taking advantage of the weekly data and carrying out estimation over weekly differencing intervals to compare model performance for short-term hedging. UWe checked different quarterly subsets and our estimation results did not change substantially.

178

M. Raj et al. ~b eq"

T--Bills:

197g

to

1994

tO

Figure 2. Term structure from Data Set 2.

preceding period. In the late 1980s, interest rates rose again, although levels remained well below the levels we observed in the early 1980s. In the late 1980s, we also see a downward sloping term structure. Into the mid-1990s, interest rates remained at historically low levels.

The Generalized Method of Moments Estimation We used Hansen's (1982) generalized method of moments (GMM) technique to estimate model parameters, using first order moment restrictions. The goodness of fit of each model to the data is given by Hansen's J statistic, which measures the degree to which the moment conditions are satisfied. The vector of random disturbances, denoted by to(t, t + T), is added to equations (7) and (14) for the H J M and CIR models, respectively. The test of a model's fit is: H0:

E[Zito(t,t + T)] = 0,

i = 1,2 . . . . . I,

where, Zi~ ~ i

= 1, 2 . . . . . I are the instruments used, and I is the number of instruments. G M M estimation tests our hypothesis by measuring how close the sample moments of the model residuals are to the theoretical moments for each bond maturity. A

GMM Tests of CIR and HJM Models

179

measure of this closeness is: 1

N

kN(O) = "~t~=1 [ ( Z 4 t o ( t , t + T ) ) - E [ ( Z i to(t,t +

T))]],

i = 1,2,...,I, (15)

where, t = 1. . . . . N

where N is the n u m b e r of observations in the time series;

T = T~ . . . . . T B where B is the n u m b e r of bond maturities in the cross-section. The system of m o m e n t equations of the residuals, ~o(t, t + T ) is overidentified if the n u m b e r of p a r a m e t e r s in O, denoted by p, is less than the number of rows in the v e c t o r k N ( O ) , i.e.,

p
A k N ( O ) = 0.

(16)

Hansen and Singleton (1982) showed that solving equation (16) is equivalent to performing a two-step minimization. Hansen's (1982) J statistic is given by: JN((~)3) = k t N ( ~ 3 ) l Y H l ( ~ 2 ) k N ( ~ 3 ) . Asymptotically, the J statistic is distributed as X ~ with B × I - p freedom:

degrees of

NJ N ( ~) 3 ) ~ X2× , p . For our G M M estimation, we followed Sire and Thurston (1996) and used the Barlett kernel in our preliminary tests, and the Parzen kernel to obtain our final results.

M o m e n t Restrictions For G M M estimation, the n u m b e r of m o m e n t restrictions must equal or exceed the n u m b e r of estimated parameters, a requirement which is easily satisfied given the relatively large n u m b e r of bond maturities in our T-bill data sets. We used four-week differencing on our data sets, which contain Treasury bills with maturities ranging from one to twenty-six weeks. Additionally, following Thurston (1994), we used one bond to substitute out for the stochastic term in our empirical specification for the H J M models. 12 Consequently, prior to using instruments, we had twenty-one m o m e n t restrictions available.

~2The resulting moment restrictions are a function of the bond chosen to substitute out for the stochastic term. We checked our results using different bond substitutions, and found our results did not change substantially.

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M . R a j et al.

Table 1. E s t i m a t e d P a r a m e t e r V a l u e s : D a t a S e t 1a Panel A: CIR Square Root Model ~ = 1.3191, (0.1232), [10.70]; q~2 = 0.2965, (0.07584), [3.91]; ~3 = 0.2526, (0.07662), [3.30]; 6- = 0.7787, (0.1083), [7.19]; PL = 0.0749, (0.4100E - 02), [18.26]; J = 193 X 0.3817 = 73.66, [dr = 84 - 3], p value = .71 Instruments: C, S, S ( - 1 ) , S ( - 2); N M A = 2, Kernel = Parzen .~2 = .29, , ~ = .31, K2 = .36, ,~82 = .39, R92 = .41, ' ~ 0 = .42, "~1 = .42, ~'22 = .42, •~23 = .41, .~24 = .38, -~5 = .39, ~'~6 = .38, ,~27 = .37, R28 = A0, .~29 = .39, ~'220 = .39, R221 ~ .38, R~2 :

.37, R~3 = .36, R24 = .36, ~'2~5 = .35

Panel B: H J M Exponential Decay Model ([3 = 0) ~t = 2.4382, (0.2967), [8.21]; 6- = 0.2528, (0.04194), [6.03];/~ = 0.5275; J = 193 × 0.5097 = 98.38, [df = 84 - 2], p value = .10, Substitution bond: K = 26 Instruments: C, S, S ( - 1), S( - 2 ) ; N M A = 1, Kernel = Parzen •~52 = .39, R6z = .53, ~'27 = .59,/~2 : .66, ,~9z = .67, -~0 = .70, -~21 : .73, R~2 = .74, R~3 = .79, /~24 = .79, R25 = .80, R~Z6 = .81, R~7 = .83, ,~28 = .87, R~9 = .90, R220 = .91, •~ 1 = .91, R222 = .93, R23 = .94, / ~ 4 = .95, ~225 = .97

Panel C: HJM Exponential Decay Square Root Model = 2.6028, (0.2977), [8.74]; 6- = 0.9320, (0.2163), [4.31];/~ = 0.5686; J = 193 × 0.5111 = 98.65, [df = 84 - 2], p value = .10, Substitution bond: K = 26 Instruments: C, S, S ( - 1), S ( - 2); N M A = 1, Kernel = Parzen R~ = .29, R62 = .53, R~ = .58, ~,2 = .66, R92 = .67, ~'20 = .70, ~'~1 = .73, R~z = .74, Kz13 = .79, R24 : .79, ~,215 = .80, .~26 = .81, R~7 = .93, R2 s = .87, R29 = .89, "~o = .91, R2zl = .91, R2 z = .93, .R223 = .94, R24 = .95, R~5 = .97 aThe standard errors for the estimated parameter values are in parentheses,^while the corresponding t statistics are in brackets. The exponential model's estimated half-life is denoted by h. For the HJM models, the 2 i = 5,6 . . . . . 25 26-week maturity bond was used for the stochastic substitution. The adjusted ~ 2 values: Hi, refer to the 21 individual equation restrictions for the forward bond price deviations obtained from the corresponding bond maturities, with four-week differencing. For the CIR model, two lags were used to account for autocorrelation, while for the HJM models, one lag was used to account for autocorrelation. The degrees of freedom (df) are equal to 21 × p where p = 3 for panel A, and p = 2 for panels B and C. We have used the following four instruments: C, a vector of ones; S, the current (one-week proxy to the) instantaneous spot rate, and lagged values of the spot rate for four and eight weeks, (denoted by ( S ( - 1) and S ( - 2 ) , respectively).

V. E m p i r i c a l R e s u l t s The results from the GMM estimation for the CIR and HJM models, including the estimated parameter values; their standard errors and t statistics; individual equation ~2 values; J statistics, and p values are presented in Tables 1 and 2. We estimated each model, using a set of four instrumental variables. I3 The following

13We estimated the empirical specifications of the forward b o n d price deviations from o u r models with b o t h four-week differencing and four-week observation intervals. Consequently, lagged values of the forward b o n d price deviations, (which contain two current spot rates, and a four-week leading spot rate), do not contain information f r o m future data.

GMM Tests of CIR and HJM Models

181

Table 2. Estimated Parameter Values: Data Set 2" Panel A: CIR Square Root Model ~1 = 0.8484, (0.1064), [7.97]; ~2 = 0.1406, (0.06587), [2.14]; ~3 = 0.6637, (0.36512), [1.82]; 6- = 0.4462, (0.1135), [3.93]; fL = 0.0933, (0.8365E - 02), [11.61]; J = 193 x 0.2572 = 49.64, [df = 84 - 3], p v a l u e s = .99 I n s t r u m e n t s : C, S, S( - 1), S( - 2); N M A = 4, K e r n e l = P a r z e n R~ = .30, R~ = .12, ~'~ = .13,/~2 = .17, R92 = .18, -~0 = .17, ~ 2

= .18, R~z = .18,

R23 = .17, R24 = .18, R125 = .19, R126 = .20, R27 = .21, R2 s = .19, ,~(9 = .19, R~(I = .19, R~, = .19, ~'22 = .19, - ~ 3 = .19, R24 = .18, ~'~5 = .18

Panel B: H J M Exponential Decay Model (15 = 0) = 2.3836, (0.08946), [30.00]; 6- = 0.6953, (0.01951), [35.79]; h = 0.5816; J = 193 x 0.2554 = 49.28, [dr = 84 - 2], p v a l u e = .99, S u b s t i t u t i o n bond: R = 26 I n s t r u m e n t s : C, S, S( - 1), S( - 2); N M A = 4, K e r n e l = P a r z e n R~ = .23, R~ = .43, . ~ = .60, R82 = .60, .~92 = .71, -~0 = .81, R(l = .85, R~2 = .86,

R23 = .89, /~24 =

.92, R125=

.94, R~6 = .94, R~7 = .95, R~8 = .96, R~9 = .97, . ~ ( , = .97,

R~I = .98, R22 = .98, R23 = .99, R24 = .99, R~5 = .97

Panel C: HJM Exponential Decay Square Root Model (15 = 0.5) )t = 2.6223, (0.07646), [34.29]; 6" = 2.6229, (0.06981), [35.57];/~ = 0.5287; J = 193 x 0.2520 = 48.64, [dr = 84 - 2], p v a l u e = .99, S u b s t i t u t i o n bond: K = 26 I n s t r u m e n t s : C, S, S( - 1), S ( - 2); N M A = 4, K e r n e l = P a r z e n R5~ = .19, R62 = .41, ,~72 = .60, ~ 2 = .61, R92 = .71, R~0 = .81, R~, = .86, R~z = .87, - 2 = .90, R~4 = .92, R~5 = .94, R~6 = .94, n~7 = .95, R28 = .96, R~9 = .97, ~'~0 = .97, R,3 R21 = .98, R22 = .98,/~23 = .98, "~4 = .99, ~,225 = .99 "The standard errors for the estimated p a r a m e t e r values are in parentheses,Awhile the corresponding t statistics are in brackets. The exponential model's estimated half-life is denoted by h. For the HJM models, the 26-week maturity bond was used for the stochastic substitution. The adjusted R 2 values: ~/z, i = 5 , 6 , . . . , 25, refer to the 21 individual equation restrictions for the forward bond price deviations obtained from the corresponding bond maturities, with four-week differencing. For the C I R model, four lags were used to account for autocorrelation, for both models. The degrees of freedom (dr) are equal to 21 × p, where p = 3 for panel A and p = 2 for panels B and C. We have used the following four instruments, C, a vector of ones; S, the current (one-week proxy to the) instantaneous spot rate, and lagged values of the spot rate for four and eight weeks, (denoted by S ( - 1) and S ( - 2 ) , respectively).

four instruments were used: a vector of ones; the current spot rate; and lagged values of the spot rate for four and eight weeks. The one-week Treasury bill rate was used as a proxy for the spot rate. For the Data Set 1, two lags were used to account for autocorrelation for the CIR model, while one lag was used to account for autocorrelation in the HJM models. For Data Set 2, four lags were used to account for autocorrelation in all models. The values accompanying the Hansen-Singleton J statistics indicate the validity of the imposed moment restrictions. For the results from Data Set 1 in Table 1, the p value for the CIR model in panel A is .71, while the HJM models in panels B

182

M. Raj et al. and C both have p values of .10. For the results from Data Set 2 in Table 2, the p values for all models are .99. For estimation of the HJM models, we found it more difficult to estimate models for the first data set. It may be that the term structure for Data Set 1, composed of T-bill quotes over the time period prior to the October 1979 structural break, is too fiat to fit the exponential HJM model. Estimating the discrete HJM exponential, using a similar data set of T-bills, Thurston (1994) found the exponential model collapsed to the constant model, o-(t, T) = o-. The individual equation/~2 values for each estimated model indicate the degree of model fit to the data sets, given the imposed moment restrictions. Examining first the results from our estimated models with Data Set 1, reported in Table 1, we observe that for CIR's square root model, the Kz values are uniformly low across all 21 equations, ranging from .29 to .42. The individual ~2 values from the HJM models are similar to each other and, in sharp contrast to the CIR model, are much higher, ranging from .29 to .97. Turning next to the results from our estimated models with Data Set 2, reported in Table 2, we observe that the K2 values for the CIR model in panel A are much smaller than is the case for Data Set 1, ranging from .13 to .30. For the HJM models, the ~z values are again similar and much higher than the CIR model, ranging from .19 to .97. The evidence we obtained from the ~2 values for the individual equations indicates a better fit to both data sets for the HJM models. The better fit of the HJM models is particularly pronounced in Data Set 2, the period following the October 1979, structural break. However, strong conclusions cannot be drawn at this stage; results from a formal non-nested comparison will be examined in the next section. We next analyzed the estimated models more closely. We examined the estimated half-lives for the HJM models to see if they were feasible, as there have been few tests of the HJM term structure model to date. However, there have been many tests of the CIR model using different methodologies and data sets. Subsequently, we compared parameter values from our estimated CIR models with prior studies and examined the CIR models' implied term structures.

HJM Model Half-Lives The estimated rate of decay of the HJM exponential models is governed by the parameter, ~. Larger values of ] imply faster rates of decay. To examine the estimated time it takes for volatility to decrease by one-half, let the estimated half-life decay time be denoted by h. For the exponential decay models, we have

~e-i(t+h)r(t)t~ = Oe-~tr(t)~/2, where /3 = 0 for the exponential model and /3 = .5 for the exponential square root model. Solving for h gives the exponential decay models' estimated half-lives as: /~ = log2/~. The estimated half-life of the exponential model is approximately thirty weeks for both data sets, while the estimated half-life of the exponential square root model is approximately twenty-eight weeks for both data sets.

GMM Tests of CIR and HJM Models

183

Comparison with Maximum-Likelihood Tests of CIR Estimated parameter values from selected other studies are reported in Table 3. Maximum-likelihood studies are presented in panel A, while GMM studies are presented in panel B. Brown and Dybvig (1986) did not report their results for t~l , t~2 , and ~b3; consequently, in Table 3, we can only report their estimated variances and long rates. Their most similar time periods, corresponding to our Data Sets 1 and 2 are the periods from January 1967 to December 1976, and January 1977 to December 1983, respectively. Their values for the spot rate volatility are high and their corresponding standard errors for their parameter estimates are very large, encompassing the origin. Chen and Scott (1992) tested CIR, using two data sets. Their first data set is a monthly time series from 1960 to 1987, constructed from McCulloch and Kwon's (1993) cubic-spline-fitted data set, containing spot rates for three months, six months, five years, and the longest available maturity (10 to 25 years). Chen and Scott's second data set used weekly Thursday quotes, from The Wall Street Journal, from January 1980 to December 1992, on bond prices on 13- and 26-week Treasury bills, five-year Treasury bonds, and the longest maturity noncallable bonds. In their first data set, they obtained similar values for ~bI and th2, near their restrictions. They also obtained reasonable-looking results for their second data set. Brenner (1989) developed and used forward bond price deviations with maximum-likelihood techniques, using a single-bond substitution on several data sets of Table 3. Summary of CIR Results from Other Empirical Studies ~ Panel A: Studies Using Maximum Likelihood Brown-DybvigA

q~,=*

q~2= *

~3=

*

6.e = 2 . 0 4 1 E - 01

~t, = 0.0579

Brown-DybvigB

q~l= *

t~2=*

q~3= *

6.2 = 3 . 7 9 0 E - 01

PL = 0.0999

Chen-Scott A

~1 = 0.4400

~2 = 0.4321

q~3 =

8.5365

6-2 = 6.803E - 03

~t, = 0.06719

Chen-Scott B

~1 = 0.5529

q~2 = 0.5427

~3 = 10.4655

6.2 = 1.111E - 02

~L = 0.1071 ~t_ = 9 . 8 E -

Brenner A

q~l = 0 . 0 3 2 8 7

t~2 = 0.03287

~3 =

1.0000

6.2 = 3 . 1 3 6 E _

Brenner B

~ l = 0.02240

t~ 2 = 0.02240

~3 =

1.0001

6.2 = 5 . 0 1 8 E _ 0 8

13

~L = 2 . 2 E - 0 6

Godin

t~l = 0 . 4 1 4 4

~2 = 0.4074

q~3 = 11.998

6.2 = 5.703E - 03

rt~ = 0.08399

12

Panel B: Studies Using G M M This Study-A

~1 = 1.3191

~2 = 0.2965

~3 =

0.2526

6. 2 = 0.6064

Pt, = 0.0749

T h i s Study-B

~1 = 0.8484

~2 = 0.1406

~3 =

0.6637

6.2 = 0.1991

?t~ = 0.0933

Lougstaff

~1 = 0.8735

~2 = 0.8733

t~ 3 = 411.71

6.2 = 4 . 4 0 0 E - 04

?t~ = 0.1029

Thurston A

~1 = 0.5879

q~2 = 0.4765

~3 =

6.2 = 1.062E - 01

~t, = 0.1114

Thurston B

~ l = 0.2741

q~2 = 0.2667

~3 =

1.0000

6.2 = 3.971E - 03

fL = 0.007447

~2 = *

t~3 =

5.1780

~.2 = 7 . 3 0 0 E _ 0 2

rL = *

C h a n et al.

~l = *

1.0000

aThis Study-A and B denotes the results for our D a t a Sets 1 and 2, respectively. Standard errors for p a r a m e t e r estimates have been omitted, due to wide differences in reporting format and subsequent p a r a m e t e r transformations. Some studies, such as Brown and Dybvig (1986), and LongstafI (1992) reported such large standard errors that their p a r a m e t e r estimates were not significantly different from zero. Thurston (1992), Brenner (1989), Brown and Dybvig (1986), and Godin (1990) used the Brown and I)ybvig methodology. Brown and Dybvig did not report their estimates for 4~1, 4'2, and 4'3. Longstaff (1989) and Chen and Scott (1993) directly estimated C I R ' s original four parameters, while Chan et al. (1992) estimated parameters from the instantaneous spot rate dynamics. For Chan et al. it was not possible to infer values for 4'1, 4~2, and r L from their estimated parameters.

184

M. Raj et al.

short-term Treasury bill prices to test the CIR model. He used four-week differencing with a single-bond price substitution. His first data set runs from January 1973 to December 1979, while his second data set runs from January 1980 to December 1987. For the CIR model, he obtained low values for 4)] and ~b2, while his values for ~b3 were close to one. For both data sets, his values for 4)1, ~b2, and ~b3 were close to their restrictions; consequently, his values for the spot rate volatility and the long rate were close to zero. Godin (1990) used daily observations on over one hundred traded US Treasury strips, with maturities extending out to thirty years. His estimates were made for August 1986. His values for ~b1 and ~b2 were similar and close to their restrictions. His spot rate volatility was small, while his long rate looks reasonable.

Comparison with Other GMM Tests of CIR Longstaff (1989) used G M M to examine how well implied yields from CIR's square root model matched the current term structure using a set of two-, three-, fourand five-month Treasury bill returns from the CRSP bond files from June 1964 to December 1986. Four moment conditions were imposed, using expected bond price returns from equation (13). All four parameters of the square root model were estimated from the resulting exactly-identified system. Longstaff's method of moments is inappropriate for our comparative study, as a similar test of how well the HJM models fit the initial term structure would give a perfect fit! Longstaff's estimated volatility parameters are significantly smaller than our values for Data Sets 1 and 2. His values for 4'1 and 4)2 were similar and almost identical to their restrictions. Longstaff's estimated value for (])3 was large, counteracting the small difference between thl and (])2.14 Chan et al. (1992) carried out tests of CIR's model by estimating parameters from a general mean reverting specification of the instantaneous spot rate. 15

CIR's Implications for the Term Structure From panels A in Tables 1 and 2, we find the spot rate volatility to be approximately 78% for Data Set 1 and 45% for Data Set 2. This represents over a 42% decrease in the volatility of the instantaneous spot rate since the Fed's 1979 change

]4We attempted to replicate Longstaff's (1989) study, using similar data from the 1993 CRSP bond files. (We found three dates in 1973 with missing observations which we omitted, following Longstaff. This number is nine less than Longstaff; these nine additional values have been filled in over the years since this original 1986 data was used in Longstaff's study.) We obtained similar point estimates; however, we obtained standard errors which were over an order of magnitude larger. ]5They estimated the parameters a and /3 in a discretized version of the equation: dr = ( a + / 3 r ) r d t + ~

d Wt.

We obtained parameter values in Table 3 by making a comparison with the corresponding mean reverting process of the square root model in equation (8). We made a correspondence between K~ and K with a and /3, respectively. We obtained 4~3 by substituting into equation (10c).

GMM Tests of CIR and HJM Models

185 T--Bills:

1964

to

197g

c~

Figure3. Term structure implied by CIR for Data Set 1.

in monetary policy. The expected infinitely-maturing spot rates from equation (12) are: 7.49% for Data Set 1 and 9.33% for Data Set 2. These values for the long rate appear reasonable for our two data sets. To examine the reasonableness of these results, we compared CIR's implied term structures for Data Sets 1 and 2, illustrated in Figures 3 and 4, respectively, with the corresponding actual term structures, illustrated in Figures 1 and 2. Both sets of CIR's implied term structures based upon our estimated parameter values, appear to be reasonably similar to the general shape of the actual term structures. For Data Set 1, CIR's implied term structures, as illustrated in Figure 3, appear to be more steeply upward sloping than the actual term structures, as illustrated in Figure 1. Additionally, CIR's implied returns for the longer-term T-bills appear to exceed actual returns. For Data Set 2, CIR's implied term structures, as illustrated in Figure 4, although slightly upward sloping, appear to be flatter than the actual term structures, as illustrated in Figure 2. One of the implications of CIR's model is that the expected infinitely-maturing spot rate will be constant throughout the estimation period. This is probably true for real rates, approximately true for Data Set 1, as illustrated in Figure 1, but definitely not true for Data Set 2, as illustrated in Figure 2.

VI. Non-Nested Model Comparison Tests As formulated, our empirical specifications of CIR and HJM are non-nested models. To carry out our model comparisons and make a statistical statement of which model fits the data best, we used the Davidson-MacKinnon non-nested J test, which provides an artificial nesting of these two nonlinear models.

186

M. Raj et al. ¢N~

T--Sills:

1979

to

1994

:"-~ ~--..v-~ "

Figure 4. Term structure implied by CIR for Data Set 2.

The Davidson-MacKinnon Non-Nested J Test Let ~'((I), r) and ,~(II, r) be as defined in equations (14) and (7) for the CIR and H J M models, respectively. A simple artificial nesting of the two models can be created, using a linear combination such as: 16

F(d,T) = (1- 6)~'(~,r) + 6~(H,r),

0<6<1.

(17)

For most models, including the current ones, direct estimation of equation (17) is not practical. Davidson and MacKinnon (1981) suggested replacing either if(@, r) o r X ( I I , r) by its fitted value. The parameter of interest in equation (17) is 6. After replacing ,,~¢'(1-I,r) by X(I~I, r), we can estimate the parameter set, II, and the parameter, 6, in

F(d, T) = (1 - 6)~'(qb, r) + 6,g¢'(1~I, r),

0 < 6 < 1.

(18)

In doing so, f f ( ~ , r) is assumed to be the true model, and the null hypothesis is

H 0 : 8 = 0. When 6 4= 0, f f ( ~ , r) cannot be the true model, as X ( I I , r) has additional explanatory power over f f ( ~ , r). To complete this test, the roles of ff(d~, r) and ,,Ye'(FI, r) are reversed, ge'(Fl, r) is assumed to be the true model and ~'(@, r) is replaced by f f ( ~ , r). The null hypothesis is: H 0 : 6 = 1 in equation (17), or alternatively H0: y = 0 in the following equation:

F ( d , T ) = (1 - y ) ~ ( F l , r )

+ yff(~,r),

0 < y < 1.

(19)

16Other feasible artificial nestings can also be created using nonlinear combinations of g'(~, r) and ~(lI, r).

GMM Tests of CIR and HJM Models

187

Table 4. Non-Nested J Test Results a Panel A: Data Set 1

~c

P a n e l B: Data Set 2

t3 = 0

/3 = 0.5

/3 = 0

/3 = 0.5

0.895 (0.021) [41.70] 0.0018 (0.011) [1.69]

0.894 (0.021) [41.67] 0.020 (0.011) [1.84]

0.995 (0.004) [268.7] 0.00275 (0.00183) [1.503]

0.901 (0.011) [85.48] 0.00374 (00181) [2.069]

aThe fitted models are: Y -- ( l

yH)YcI R + yHyHj M

Y = (1

TC)YttjM q- "yCYcIR

The null hypotheses for these two equations are Hg: ytt = 0 (CIR is the true model), and Hh: y c = 0 (HJM is the true model), respectively. The upper half of each panel gives results under the null hypothesis that CIR is the true model, while the lower half of each panel gives results under the null hypothesis that HJM is the true model. Within each panel half, results for each H J M model are presented in columns. The first value is the estimate for 7 c or y " ; the standard error follows below in parentheses, and the t statistic follows below that in square brackets.

There are four possible outcomes from these tests, as either model can be accepted or rejected independently. We present the results of our non-nested J tests in Table 4. Panel A presents results for Data Set 1, while panel B presents results for Data Set 2. The upper half of each panel gives results under the null hypothesis that CIR is the true model: small values, here, for the test statistic pr~wide evidence of support for CIR. The lower half of each panel gives results under the null hypothesis that HJM is the true model: small values, here, for the test statistic provide support for HJM. Our results strongly reject the CIR model as the true model in favor of the HJM exponential decay models. Examining the results for Data Set 1, we first look at the upper half of panel A in Table 4, where we observe that given the CIR model, the HJM models have approximately 89% explanatory power. Looking next at the lower half of panel A, we observe that given the HJM model, the CIR model has 0.18% explanatory power for the exponential model and 2% explanatory power for the exponential square root model. Examining the results for Data Set 2, we first look at the upper half of panel B in Table 4. Here, we observe that given the CIR model, the HJM exponential model has approximately 99% explanatory power, while the exponential square root model has 90% explanatory power. Looking next at the lower half of panel B, we observe that given the HJM model, the CIR model has approximately 0.28% explanatory power for the exponential decay model and 0.37% explanatory power for the exponential square root model. Our results from the Davidson-MacKinnon non-nested comparison tests indicate that the HJM exponential models fit the data significantly better than the CIR square root model. VII. C o n c l u s i o n In this paper, we examined the performance of two of the leading dynamic term structure models--Cox, Ingersoll and Ross's (1985) equilibrium-based model, and

188

M. Raj et al. Heath, Jarrow and Morton's (1992) no-arbitrage-based model--on a data set of short-term Treasury bills split in half around the October 1979, structural break. GMM estimation was used with first order moment restrictions and four instruments. We compared models, using the Davidson-MacKinnon non-nested J test. From examination of the results from the GMM model estimations, we found that the ~2 values for the HJM models were larger than for the CIR model across both data sets. The comparison of the ~2 values provides informal evidence that the HJM models fit the data better, particularly for Data Set 2, following the Fed's change in monetary policy. Turning to the results from the Davidson-MacKinnon non-nested J tests, we found evidence that the HJM models fit our Treasury bill data sets significantly better than the CIR model, as the HJM models provide substantially more explanatory power. In conclusion, we obtained evidence in favor of the HJM exponential models over the CIR square root model for better term structure fit on two monthly time series of T-bills, each fifteen years in length, with four-week differencing, i

Directions for Future Research In this study, we compared the fit of two competing one-factor term structure models to the short end of the term structure. Future research is necessary to compare the relative performance of these competing models with longer-term data. Although one factor may be sufficient to explain the short end of the term structure, a different factor may drive long-term rates, as it is well known that short-term rates are more volatile than long-term rates. It is possible to extend our framework to test multifactor term structure models. Another avenue for future research, using our methodology, is to compare model performance on foreign term structures. Other term structure models which can be examined in our framework include the Vasicek (1977) model and Hull and White's (1990) extended Vasicek model. Comparison of the Vasicek model is of particular interest, as the Vasicek model can be shown to be equivalent to the HJM exponential decay model with a specified constant market price of risk. Direct comparison of these two models in our framework would provide a direct test of the effect of assuming a constant market price of risk. Other interesting topics for future research, using our methodology, include: comparisons between discrete and continuous models, comparisons between maximum likelihood and GMM estimators, further examination of different non-nested GMM model tests, and tests of duration measures.

Appendix We have derived the empirical specification for the HJM exponential models. From equation (6), the general form of forward bond price deviation for the continuous time HJM models is:

GMM Tests of CIR and HJM Models

189

The forward bond price deviation for the continuous time HJM exponential decay beta root model is found by specifying the forward rate volatility function, o-(t, T) = o-e- a(r- Or(t) p. Making the substitution gives: 17

F(t,T) = f' fro-e ~"-')r(s)o f"o-e-~'Y-"r(s) ~ dydvds ~O~t

Js

+ ]~fro'e-~(Y-"r(s)'8 dvdl~. Calculating the innermost integral gives:

F ( t , T ) =o-2for(S)2OfTe-a("

-1

")--A-

)

[e-

h(y

s)] t

dvds

)=s

i=t After the substitution of the integration limits from the first integral calculation, we have: _O. 2

F(t,T) = --

f/[fT-at.

A

S)(e-X(. s) _ 1) dv]r(s)2P ds

0 -2

ft(e-a(r-~) - e -a(t ~))r(s) ~ dl~. A Jo Simplification of the volatility component gives: _ o -2

F(t, T)

_ _ f t [ fr(e-2A(.-s) _ e-X(,'-S))dv]r(s)2P ds A ~0 I.~t A(t,T)

Or(e-AT A

e-At)fleXOr(s) ~ dl~s.

To compute the deterministic component, A(t, T), we first group like terms in the integrands:

A ( t , T ) = _ _ f t A" 0 1 . ' /

-2aV dv - exs't

-X" dv e2XSr(s) 2~ ds.

17Here, we m a d e use of the fact that this forward rate volatility function is time-maturity separable, to calculate our forward bond price deviation.

190

M. Raj et al. Next, after calculating the inner integrals, we have: __0-2 [ __-2As eaS ] A- J0/'t[[-e2A (e -zar - e -2At) + ---A-(e -at - e -at)] r(s) 2~ as.

A(t, T)

Rearranging the integrals in terms of the time variable, s, gives:

A(t, T) = ~

(e -2at -- e -2m)

e2a'r(s) 2~ ds

- ( e - A r - e-a') fotea~r(s)2~ ds]. Factoring out (e -at - e-At) yields: 0-2 -at - e -a') [1~ ( e -At + e_at)foteZasr(s)EtJd s - foteaSr(s)Z~ds] . A ( t , T ) =--~-i-(e Next, we substitute the simplified deterministic component, A(t, T) back into the forward price deviation. We then are able to divide out the common term, (e -At - e-At), from the deterministic and stochastic components: 0-2

F(t, T) e-AT _ e-At

Aa[l(e-ar+e-at)foteEasr(s)2'ds-foteASr(s)aads] -~foeA~r(s) ' diZZy.

To carry out our estimation procedure, we need to get rid of the Brownian motion. To do this, we solve for the stochastic term in terms of another forward bond price deviation, expressed in terms of a bond with maturity, K, where K ~ T and K ~ t:

"--A°~ fjeaSr(s) ~ dff'~ F(t, K) e-aK _ e-At

A2 ~ ( e - a t +

-

.

A f t e r substituting out for the stochastic term in the forward price deviation, F(t, T), with another bond with maturity, K, we can express the stochastic term in terms of the forward bond price deviation, F(t, K), and the modified deterministic component, expressed in terms of the K maturity bond.

[ e - A T - - e -At ) F(t, T) = ~ e_-:-f f e_at F(t, K) 2

+(e -at - e-ar)(e -At - e- At) ~-~-7A2£te2A'r(s)2O ds.

G M M Tests of CIR and HJM Models

191

Finally, to conduct empirical tests on our continuous time HJM model, we need to approximate the integrated instantaneous spot rates. For testing purposes, we assume constant spot rates over our differencing interval of four weeks. The approximated integral is then:

f~

e2A'r(s) 2/3 ds = r ( t )

2 /3 1

~-~(e

2At

- 1).

This p r o v i d e s o u r a p p r o x i m a t e d f o r w a r d b o n d price d e v i a t i o n for t h e H J M c o n t i n u o u s t i m e e x p o n e n t i a l b e t a r o o t m o d e l . 18

F(t,T) =

e-AT _ e - A t e_Al~ e ~ ) F ( t , K ) 0-2 + ( e - a t - e-AIC)(e -At - e-at) --4A 3 (e2A~ _ 1)r(t)2/3 .

To obtain the forward bond price deviations for the exponential decay and the e x p o n e n t i a l d e c a y s q u a r e r o o t m o d e l s , we set b e t a e q u a l to z e r o a n d o n e - h a l f , respectively.

We thank participants at the 1996 Financial Management Association and the 1996 Southern Finance Association meetings.

References 1. Abken, P. June 1993. Generalized method of moments tests of forward rate processes. Federal Reserve Bank of Atlanta Working Paper Series 93-7. 2. Amin, K. I., and Morton, A. 1994. Implied volatility functions in arbitrage-free term structure models. Journal of Finance and Economics 35(2): 141-180. 3. Black, F., and Scholes, M. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81(3):637-654. 4. Brenner, R. J. 1989. Three essays on the term structure. Ph.D. Thesis, Cornell University. 5. Brown, S. J., and Dybvig, P. H. 1986. The empirical implications of the Cox, Ingersoll, Ross theory of the term structure of interest rates. Journal of Finance 41(3):617-632. 6. Chen, R. R., and Scott, L. 1993. Maximum likelihood estimation for a multifactor equilibrium model of the term structure of interest rates. Journal of Fixed hwome 3(3):14-31. 7. Chan, K. C., Karolyi, G. A., Longstaff, F., and Sanders, A. B. 1994. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47(3):1209-1227.

18Earlier in the text, in equation (7), the forward bond price deviation is written with ? = d = 4, denoting four-week differencing.

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M. Raj et al. 8. Cox, J., Ingersoll, J. E., and Ross, S. A. 1985. A theory of the term structure of interest rates. Econometrica 53(2):385-407. 9. Davidson, R., and MacKinnon, J. 1993. Estimation and Interference in Econometrics. New York: Oxford University Press. 10. Davidson, R., and MacKinnon, J. 1981. Several tests for model specifications in the presence of alternative hypotheses. Econometrica 49(3):781-793. 11. Flesaker, B. 1994. Testing the Heath-Jarrow-Morton/Ho-Lee model of interest rate contingent claims pricing. Journal of Financial and Quantitative Analysis 28(4):483-495. 12. Gibbons, M. R., and Ramaswamy, K. 1993. A test of the Cox, Ingersoll, and Ross model of the term structure. Review of Financial Studies 6(3):619-658. 13. Godin, M. A. 1988. Equilibrium models of the term structure of interest rates: Applications to options in financial and insurance markets. Ph.D. Thesis, University of Pennsylvania, Wharton School of Business. 14. Godin, M. A. 1990. Interest rate volatility and equilibrium models of the term structure: Empirical evidence. Actuarial Research Clearing House (1):199-221. 15. Hansen, L. P. 1982. Large sample properties of generalized method of moments estimators. Econometrica 50(4):1029-1054. 16. Hansen, L. P., and Singleton, K. J. 1982. Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica 50(5):1269-1286. 17. Heath, D., Jarrow, R., and Morton, A. 1990. Bond pricing and the term structure of interest rates: A discrete time approximation. Journal of Financial and Quantitative Analysis 25(4):419-440. 18. Heath, D., Jarrow, R., and Morton, A. 1992. Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica 60(1):77-105. 19. Ho, T. S. Y., and Lee S.-B. 1986. Term structure movements and pricing interest rate contingent claims. Journal of Finance 41(5): 1011-1030. 20. Hull, J., and White, A. 1990. Pricing interest-rate-derivative securities. Review of Financial Studies 3(4):573-592. 21. Longstaff, F. A. 1989. A nonlinear general equilibrium model of the term structure of interest rates. Journal ofFinancialEconomics 23(2):195-224. 22. McCulloch, H. J., and Kwon, H.-C. March 1993. U.S. term structure data, 1946-1991. Working Paper, Ohio State University. 23. Sim, A. B., and Thurston, D. C. 1996. An empirical study of a new class of no-arbitragebased discrete models of the term structure. Journal of Financial Research, to appear. 24. Thurston, D. C. 1994. A generalized method of moments comparison of discrete Heath-Jarrow-Morton interest rate models. Asia Pacific Journal of Management 11(1):1-19. 25. Thurston, D. C. 1992. A generalized method of moments comparison of several discrete time stochastic models of the term structure in the Heath-Jarrow-Morton no-arbitragebased framework. Ph.D. Dissertation, University of Arizona. 26. Vasicek, O. 1977. An equilibrium characterization of the term structure. Journal of Financial Economics 5(2): 177-188. 27. Vuong, Q. A. 1989. Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57(2):307-333.