- Email: [email protected]
Some international evidence on the stochastic behavior of interest rates
rI l l
E
•Ti TN E EoR
w 0 R T I-I M A N N
Journal o f lnternational Money and Finance, Vol. 14, No. 5, pp. 721-738, 1995 Copyright ©1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0261-5606(95)00030-5 0261-5606/95 $10.00 + 0.00
Some international evidence on the stochastic behavior of interest rates Y K TSE
Department of Economics and Statistics, National University of Singapore, Kent Ridge, Singapore 0511 This paper examines the stochastic behavior of short-term interest rates in eleven countries. Following Chan et al. (1992a), we consider eight stochastic models of short-term interest rates, all of which are nested within a discretized approximation to a differential equation. We use the generalized method of moments to estimate the model. The empirical results show that no single model can satisfactorily describe the stochastic structure of interest rates for all countries. On balance, if we classify the sample countries according to the size of the elasticity of variance of short-term interest rates, France, Holland and the USA belong to the group of high elasticity. In comparison, Australia, Belgium, Germany and Japan belong to the group of medium elasticity, while Canada, Italy, Switzerland and the UK belong to the group of low elasticity. (JEL 315).
The interest rate is a fundamental determinant of asset prices in financial markets. Many stochastic models have been developed by academic researchers in the continuous-time setting (see, e.g. Vasicek, 1977; Brennan and Schwartz, 1979; Cox et al, 1985; and Constantinides, 1992). These models provide a rich framework for specifying the dynamic behavior of interest rates. Based on an assumed stochastic model interest rates can be generated to mimic possible future movements. The generated interest rates may be adjusted to provide arbitrage-free term structures--a concept studied by Harrison and Pliska (1981), Ho and Lee (1986) and Heath et al. (1992), among others. Alternatively, the interest rates may be adjusted to reflect an individual's subjective view of the economic world. Arbitrage-free interest rates are important in pricing interest rate contingent cash flows, some of which admit analytic closed form solutions while others may require numerical evaluations. On the other hand, simulating interest rates in the real-world setting may be useful when risk-reward analysis is conducted under different scenarios. Recently Chan et al. (1992a) (CKLS hereafter) examined empirically the stochastic behavior of short-term interest rates in the USA. They considered eight continuous-time interest rate models, all of which can be nested within a general stochastic differential equation. Applying discrete approximation they estimated the models using the generalized method of moments (GMM) and conducted tests of restrictions to evaluate each model's performance. In this
International evidence on interest rates: Y K Tse
paper we adopt this frame-work of analysis and consider the short-term interest rates in eleven countries, including the USA. The results show that no single model can satisfactorily describe the stochastic structure of interest rates for all countries. On balance, if we classify the sample countries according to the size of the elasticity of variance of the short-term interest rates, France, Holland and the USA belong to the group of high elasticity. In comparison, Australia, Belgium, Germany and Japan belong to the group of medium elasticity, while Canada, Italy, Switzerland and the UK belong to the group of low elasticity. The selection of a stochastic model for short-term interest rates is an issue that requires theoretical consideration as well as empirical evidence. On theoretical bases, models that permit closed form solutions for contingent claims may be desirable since they provide greater analytical insights than numerical solutions. 1 On the other hand, models that represent reality inaccurately, or rely on unrealistic assumptions, may incur model risk. A subtle balance between the two issues is required for a successful evaluation of interest rate sensitive assets. The empirical results reported in this paper may provide some information that is useful for the selection of models. The plan of this paper is as follows. In Section 2 we outline the framework of stochastic interest rate models as summarized by CKLS. We describe the various models that are nested within the framework and explain the methodology of the econometric estimation. The data and some preliminary statistical results are summarized in Section 3. In Section 4 we report the empirical results of the GMM estimation of the eleven countries considered. Finally, some concluding remarks are given in Section 5. I. The continuous-time framework and estimation method
Various stochastic models in the continuous-time framework have been proposed for the movements of short-term interest rates. These models have been applied to price interest rate contingent cash flows. Analytic solutions of the valuation models may or may not be available depending on the complexity of the interest rate process assumed. The issue of comparing these stochastic models empirically is important when the objective is to select an appropriate model for pricing interest rate contingent claims, such as callable bonds and bond options. CKLS pointed out that many of the models considered in the literature can be nested within the following differential equation (1)
dr = ( ~ ' + f l ' r ) d t + t r r ~ d Z ,
where r is interest rate, t is time and d Z is a Wiener process. Specifically, the following models are nested within equation (1) above 1. Merton (1973) 2. Vasicek (1977) 3. Cox-Ingersoll-Ross (1985) square root process (CIR SR) 4. Dothan (1978) 722
dr = a 'dt + t r d Z dr = ( a ' + f l ' r ) d t + ~ d Z dr = ( a ' + f l ' r ) d t + trrl/EdZ dr = ~rr d Z
Journal of Intemational Money and Finance 1995 Volume 14 Number 5
International evidence on interest rates: Y K Tse
5. Geometric Brownian Motion (GBM) 6. Brennan-Schwartz (1979) (as) 7. Cox-Ingersoll-Ross (1980) variable rate model (CIR VR) 8. Constant elasticity of variance (CEV)
dr -- [3 'r dt + ~ r d Z d r = ( a ' + [3'r)dt + trr d Z dr = t r r a / Z d Z dr = [3 'r dt + trrVdZ.
Model 1 is a simple Brownian motion (generalized Wiener process) with drift parameter a'. It was used by Merton (1973) to derive the values of pure discount bonds. Model 2 is the Ornstein-Uhlenbeck process studied by Vasicek (1977). It implies that interest rates follow an autoregressive process of order 1. Thus, both Models 1 and 2 permit interest rates to take negative values. In contrast, the CIR SR model does not suffer from this drawback. This model implies that interest rate changes are proportional to a non-central X2 variate. The Dothan and CIR VR models specify that interest rate changes are without drift. The GBM model has been used extensively in the valuation of stock options. It implies that interest rates have a log-normal distribution. The BS model is similar to the CIR SR model in that the interest rates are not permitted to take negative values. Model 8 is conveniently defined as the CEV model, following the terminology of CKLS. It has the property that the elasticity of the instantaneous variance with respect to interest rate is 2y, which is independent of the level of interest rates. Set in the above framework the exact maximum likelihood estimation (MLE) of the interest rate models requires knowledge of the exact likelihood function of the continuous-time process. Lo (1988) considered the problem of maximum likelihood estimation using discretized approximations to the exact continuous-time solution. He argued that the discretized MLE is in general inconsistent. The inconsistency does not vanish with increasing sample size, but depends on the size of the sampling interval. The errors in discretization, however, may not be serious if the sampling interval is short, as demonstrated by Tse (1992, 1994). Following the discretization approach, CKLS considered the following approximation to equation (1)2 (2)
Ft+ 1 -- rt ~ Ol ÷ [31 t ÷ ~'t+ 1'
where et+ 1 is assumed to satisfy (3)
E(et+l)=OandE(etz+l)=tr2r
2v.
An alternative framework that incorporates some of the models considered can be described by the following equation (4)
dr = K( Ix - r ) d t + trrVdZ,
where r is the speed-of-adjustment coefficient and /x is the long-run interest rate. Thus, the constraints /z > 0 and r >_0 are assumed. Equation (4) has the advantage that the parameters have straightforward economic interpretations. Journal of International Money and Finance 1995 Volume 14 Number 5
723
International evidence on interest rates: Y K Tse
In discretized form it can be written as <5)
rt+ 1 -- rt =
KIxAt - K r t A t + et+ 1.
Comparing equations (2) and (5), we observe that equation (5) (with constraints o n / x and K, and hence on a and 13) is a special case of equation (2), with (6)
a = K/zAt and /3 = - K A t ,
so that a = - / 3 / z . Although equations (2> and (5) are very similar the following differences should be noted. First, as /3 = 0 implies a = 0, the generalized Wiener process (Merton model) with /3 = 0 and a :~ 0 is not derivable from equation (5>. Second, if we accept equation (5> we would expect, from equation (6), a > 0 and /3 < 0. Such constraints, however, are not imposed on equation (2). Third, as /z > 0, equation (6) implies that if a = 0 then r = 0 and hence /3 = 0. Thus, the G B M and CEV models (with a = 0 and /3 4: 0) are not derivable from equation (5>. While equation (5> can be used as the basic framework for comparing alternative interest rate models it incorporates a less rich class of submodels. Thus, in this paper we follow the CKLS framework for the purpose of comparison and better generality. 3 To estimate the parameters of equation (2>, CKLS used the generalized method of moments (GMM), due to its robustness against misspecifications in the distribution of the residuals. Defining 0 as the parameter vector consisting of a, /3, tr and Y, CKLS considered a v e c t o r ft(O) given by
I Et+1 l et+
]
lrt
(7)
ft(O)=l,?+l_o.2r2V
,
[(e2+x--O-2r2"/)rt
where Et+ 1 = 1",+1 - rt - a - / 3 r t, for t = 0,..., n - 1. Under the null hypothesis specified in equations (2> and (3>, E[ft(0)] = 0. The G M M estimator of 0, 0, is the value that minimizes the quadratic form (8>
J(O) =g'(O)W(O)g(O),
where n
(9)
g ( O ) = n1 2 f t ( O )
t=l
'
and W (0) is the weighting matrix given by (10)
W(O) = S-1(0),
with S ( O ) = E[ft(O)f't(O)]. 4 All models considered can be estimated by this method. For the unrestricted model, J ( 0 ) is zero. In general we define 0 0 as the k-element (k < 4) subvector of 0 consisting of unrestricted parameters.
724
Journal o/International Money and Finance 1995 Volume 14 Number 5
International evidence on interest rates: Y K Tse
We also define D(O) as the 4 x k matrix of partial derivatives of g(O) with respect to 00, i.e.
D(O) = 3g(O) 30' o
(11)
Then the asymptotic covariance matrix of 00 can be consistently estimated by (12)
l [D'( Oo)S-X( Oo)D( Oo)] -1
To evaluate the various restrictions imposed on the unrestricted model we calculate the following test statistic 5
AJ=n[JOo-J(O)],
(13)
which is distributed approximately as a X 2 with 4 - k degrees of freedom. It should be pointed out that equation (7) is not the only possible vector to define the objective function. For example, the following vector ~?t+ 1
rt2v Et+ 1
rt2v - 1 (14)
f*(O)
=
,2+1
0 -2
(, r 2 ~,
rt
also satisfies the requirement that E[ f * (0)] = 0, and thus can be used to define the objective function. To facilitate comparison with CKLS for the results on US data, however, we use ft(0) to calculate the G M M estimates. 6
II The data used in this study were extracted from various issues of The Economist. Annualized three-month money market rates at approximately the end of the month were collected. The countries included in the study and the sampling period for each country are summarized in Table 1.7 Thus, the sample size ranges from 130 to 214. Table 2 summarizes the unconditional distribution of the three-month money market rates, which we denote generically as rt, for various countries. The first panel presents the first four sample moments of r r Under the normality assumption, the skewness and kurtosis have expected values of zero and three, Journal of International Money and Finance 1995 Volume 14 Number 5
725
International evidence on interest rates: Y K Tse
TABLE 1. Short-term interest rate data.
Country
Code
Data Period
Australia Belgium Canada France Germany Holland Italy Japan Switzerland UK USA
Au Be Ca Fr Ge Ho It Ja Sw UK US
83/5-94/5 76/8-94/5 83/8-94/5 76/8-94/5 77/1-94/5 77/4-94/5 77/2-94/5 83/4-94/5 76/8-94/5 76/8-94/5 76/8-94/5
Number of Observations 133 214 130 214 209 206 208 134 214 214 214
respectively. It is found that only two countries (France and the USA) have kurtosis higher than three, while Japan has the lowest kurtosis. Also, except for Australia, Canada, Japan and the UK, all countries have positive skewness. 8 Judging from the standard deviations and the ranges of the unconditional distributions, Australia, Italy and the USA have the largest variability. 9 On the other hand, Japan is the country where interest rates are the most stable. To examine the serial correlation of r t in the first and second moments, we compute two sets of autocorrelation functions. We calculate t31xi, which is the ith order sample autocorrelation coefficient of r r Similarly, P2,i, is the ith order sample autocorrelation coefficient of ( r t - ~ . ) 2 , where r is the sample mean of r r These autocorrelation coefficients, up to order 24, are used to calculate the Box-Pierce statistics, which we denote as Qj(24) for j = 1, 2. These statistics are approximately distributed as X2 if there is no serial dependence. Thus, the results show that interest rates are highly autocorrelated in both the first and second moments. The high value of Q1 and the fact that the sample autocorrelation function decays very slowly suggest that interest rates may be highly autocorrelated or even non-stationary. However, as interest rates may not be normal, we further perform tests of serial correlation using the non-parametric runs tests. These are denoted by R 1 and R2, respectively, for tests of the first and second moments. The R statistics are approximately standard normal if there is no serial correlation, with negative values indicative of positive serial correlation. The results strongly indicate that there is positive serial correlation in both the mean and the variance for all countries. In Table 3 we present the summary statistics of the differenced interest rates, A r t = r t - r t _ 1. Here the Q and R statistics for serial correlation often give conflicting results. Judging from the R statistic, which is more robust, four countries (Australia, Belgium, Canada and the USA) have significant serial correlation in squared interest rate changes. On the other hand, all countries 726
Journal o f lntemational Money and Finance 1995 Volume 14 Number 5
to --4
t.~
~,
~"
,~
~,
•~
9.82 2.80 5.19 17.50 0.800 2.912 0.959 0.909 0.855 0.811 1946.6 - 12.72 0.898 0.766 0.665 0.583 949.7 -11.92
Be 9.17 2.55 3.62 14.00 -0.251 2.384 0.970 0.939 0.895 0.851 1084.7 - 10.21 0.916 0.841 0.734 0.640 498.2 -9.40
Ca 10.26 2.59 5.50 19.00 0.894 3.761 0.948 0.904 0.869 0.817 1877.5 - 12.13 0.859 0.744 0.689 0.561 674.1 -10.47
Fr 6.82 2.41 3.35 13.20 0.529 2.484 0.984 0.968 0.947 0.920 2116.5 - 13.93 0.937 0.855 0.770 0.671 821.3 -11.56
Ge 7.43 2.25 2.88 15.50 0.562 2.958 0.942 0.902 0.846 0.783 1505.7 - 13.54 0.660 0.590 0.444 0.310 361.4 -9.51
Ho 14.26 3.40 7.81 21.75 0.494 2.259 0.971 0.939 0.906 0.869 2307.8 - 13.46 0.943 0.889 0.847 0.804 1321.8 -11.52
It 5.10 1.52 1.93 7.65 -0.126 2.004 0.978 0.949 0.916 0.875 1085.1 - 10.75 0.932 0.851 0.763 0.678 546.6 -10.52
Ja 5.02 2.50 0.12 10.75 0.161 2.352 0.943 0.911 0.884 0.841 1723.9 - 11.31 0.883 0.798 0.717 0.627 694.8 -11.64
Sw
11.26 3.01 4.94 18.13 -0.118 2.492 0.959 0.907 0.849 0.794 1306.9 - 12.88 0.916 0.842 0.749 0.658 747.5 -12.28
UK
8.09 3.23 3.12 17.75 0.718 3.346 0.962 0.921 0.884 0.839 2303.8 - 13.29 0.858 0.745 0.669 0.538 945.5 -13.22
US
(r t _
Notes:
Interest rates are given in percentages, t31,i is the ith sample autocorrelation coefficient of r t a n d P2.i is the ith sample autocorrelation coefficient of f)2. Qj (24), j = 1,2, are the Box-Pierce portmanteau statistics based on autocorrelation coefficients of order up to 24. Similarly, Rj, j = 1, 2, are the runs tests for serial correlation. Under the null hypothesis of white noise, Q is approximately a x 2 with 24 degrees of freedom ( X224,0.o5 = 36.4) and R is approximately a standard normal.
R 2
Q2(24)
P2,4
t32~ t32,3
P2,1
R 1
0.912 0.879 1367.5 - 9.98 0.940 0.878 0.807 0.742 731.9 -9.56
0.947
t31,3 Pl,4 Q~(24)
0.976
Pl,2
12.20 4.29 4.57 19.80 -0.413 2.027
Pl,1
Mean Standard deviation Minimum Maximum Skewness Kurtosis
Au
TABLE 2. Summary statistics of r t
~"
~. ~"
8"
"~
~.
~.
~ ~"
~'
- 0.062 0.862 - 2.500 2.750 0.569 2.669 4.610 3.776 0.102 0.118 - 0.063 - 0.148 40.0 - 3.45 0.257 0.082 0.101 0.043 104.1 - 1.52
- 0.031 0.772 - 3.750 3.350 0.577 3.439 9.177 18.403 0.116 0.047 - 0.102 - 0.081 19.1 - 4.59 0.239 0.010 0.031 - 0.019 43.8 - 4.20
Be - 0.022 0.660 - 1.700 2.720 0.672 3.116 5.468 5.721 - 0.024 0.143 - 0.070 - 0.060 43.0 - 3.07 0.314 0.300 0.009 - 0.035 35.3 - 4.22
Ca - 0.020 0.810 - 5.190 4.500 - 0.093 -0.553 17.670 43.704 - 0.088 -0.066 0.165 - 0.042 35.4 - 0.98 0.375 0.017 0.008 0.012 40.9 - 4.14
Fr 0.001 0.454 - 1.700 2.600 1.383 8.145 10.965 23.448 0.024 0.133 0.163 0.109 48.0 - 0.79 0.319 0.166 0.099 0.016 101.3 - 3.76
Ge 0.003 0.750 - 4.370 4.000 0.170 0.995 14.461 33.495 - 0.196 0.136 0.066 - 0.123 60.3 - 1.40 0.443 0.105 0.099 0.025 111.8 - 4.23
Ho
rt -
It - 0.040 0.745 - 3.320 3.040 0.085 0.497 6.866 11.353 0.074 0.041 0.063 - 0.054 21.0 - 1.94 0.152 0.084 0.267 0.044 45.7 - 2.15
r t _ 1.
- 0.026 0.255 - 0.910 1.090 0.171 0.805 8.752 13.540 0.300 0.060 0.240 0.055 47.5 - 2.01 0.067 0.232 0.119 0.071 26.5 - 2.17
Ja 0.015 0.836 - 3.940 5.940 1.152 6.864 16.414 39.960 - 0.220 -0.045 0.160 0.002 52.6 - 1.12 0.334 0.034 0.130 0.137 59.5 - 0.94
Sw - 0.029 0.805 - 2.680 2.870 0.671 3.996 4.887 5.620 0.159 0.079 - 0.058 - 0.051 35.0 - 1.10 0.041 0.036 -0.055 0.050 35.1 - 1.62
UK
- 0.004 0.897 - 4.750 4.120 - 0.857 -5.104 12.075 27.036 0.012 -0.060 0.087 - 0.300 93.8 - 2.45 0.170 0.267 0.068 0.177 198.9 - 7.84
US
freedom ( X224,0.05= 36.4) and R is approximately a standard normal.
Notes:
Interest rates are given in percentages, t31, i is the ith sample autocorrelation coefficient of A r t = r t - r t_ 1 and t32.i is the ith sample autocorrelation coefficient of ( A r t - - ~ i ) 2 where ~-7 is the sample m e a n of A r t. Q j (24), j = 1,2, are the Box-Pierce p o r t m a n t e a u statistics based on autocorrelation coefficients of order up to 24. Similarly, Rj, j = 1,2, are the runs tests for serial correlation. U n d e r the null hypothesis of white noise, Q is approximately a )(2 with 24 degrees of
R1 132,1 132,2 132,3 132,4 Q2 (24) R2
01 (24)
Mean Standard deviation Minimum Maximum Skewness Standardized skewness Kurtosis Standardized kurtosis 131,1 131,2 131,3 131,4
Au
TABLE 3. S u m m a r y s t a t i s t i c s o f
~..
_~.
~.
g~
~.
g~
International evidence on interest rates: Y K Tse
have significant serial correlation in squared interest rate changes except for Australia, Switzerland and the UK. To give some indication of the appropriateness of the normality assumption we calculate the standardized skewness and kurtosis. Skewness is standardized with respect to mean zero and variance 6/n, while kurtosis is standardized with respect to mean 3 and variance 24/n. These statistics are approximately standard normal if Ar t is normal and serially uncorrelated. The results show that all interest rates are significantly leptokurtic (i.e. kurtosis larger than three). France, Holland, Italy and Japan have symmetric interest rate changes. Surprisingly, the USA is the only country where interest rate changes are significantly negatively skewed. In summary, the differenced data for short-term interest rates demonstrate departure from normality. Serial correlation appears to be more serious in the variance as compared to the mean.
Ill. Empirical results The estimation results for the unrestricted model is presented in Table 4. Table 5 summarizes the results for the unrestricted models. Figures in parentheses are the estimated standard errors. From Table 4 we can see that ~ varies considerably across the countries. Four countries (Canada, Italy, Switzerland and the UK) have estimated values of 7 that are not statistically significant. TM The USA has the largest estimated value of 7, followed by France. Using data from June 1964 through December 1989 CKLS ( C h a n e t al., 1992a) reported the estimated value of 7 to be 1.50 for the USA. Compared to our estimated value of 1.71, the difference may be interpreted as an indication that interest rate volatility has become more sensitive to the interest rate level in later years, or alternatively, that the variance elasticity has increased. In their study on the Japanese market, CKLS (Chan et al., 1992b) reported ~ to be 2.44, much larger than the value of 0.62 given here. 11 CKLS argued that ~/ is the most important parameter in determining the term structure of interest rates as it directly affects the conditional volatility. Of the eleven countries considered, values of ~/ range from -0.36 to 1.73, indicating that the sensitivity of volatility covers a broad spectrum of values. For the parameters a and /3, with the exception of Switzerland, none of the estimated values is statistically significantly different from zero. Thus, the data do not provide precise information about these parameters. As discussed earlier, if we start with equation (5) we would expect a > 0 and /3 < 0. Both constraints are violated for Japan. 12 Otherwise, the estimated values of a and /3 for other countries have the expected signs. Furthermore, based on equation (5) a and /3 are related through the equation a = -/3/z. Thus, replacing /x by t h e sample mean of interest rates, r, we would expect 3 to be close to -/3~ if equation (5) holds. The last column of Table 4 summarizes the value of/3F. It can be seen that 3 and /37 are remarkably close. This is an indication that equation (5) may provide a useful basic framework for conducting model comparison. As imposing relevant restrictions may improve the power of inference this is an Journal of lntemational Money and Finance 1995 Volume 14 Number 5
729
International evidence on interest rates: Y K Tse
interesting topic for future research. Finally, using the relationship K = -~3/At, we can see that Switzerland has the highest s p e e d - o f - a d j u s t m e n t coefficient, followed by Holland. A t the o t h e r extreme, Australia has the lowest s p e e d - o f - a d j u s t m e n t coefficient. 13 O f all the m o d e l s considered, the Vasicek, C I R SR and BS models are p r o b a b l y the m o s t serious c o n t e n d e r s for contingent claims valuation. T h e s e m o d e l s i m p o s e no restrictions on a and /3, but m a k e different a s s u m p t i o n s a b o u t the value of y. F o r the Vasicek m o d e l ( y = 0), m o s t values of t~ and /3 TABLE 4. The estimated unrestricted models. Parameters Country
&
/3
8"
~,
/~
Au
0.0516 (0.1453)
-0.0092 (0.0141)
0.1553 (0.0817)
0.6763 (0.1987)
0.1122
Be
0.2939 (0.1937)
- 0.0330 (0.0211)
0.1331 (0.0981)
0.7579 (0.3168)
0.3241
Ca
0.2476 (0.2499)
- 0.0293 (0.0251)
1.4093 (0.9250)
- 0.3600 (0.2945)
0.2687
Fr
0.4073 (0.2891)
- 0.0415 (0.0311)
0.0161 (0.0144)
1.6289 (0.3656)
0.4258
Ge
0.1250 (0.0834)
-0.0181 (0.0137)
0.1122 (0.0641)
0.7141 (0.3070)
0.1234
Ho
0.4263 (0.2790)
- 0.0570 (0.0407)
0.0257 (0.0327)
1.5997 (0.5868)
0.4235
It
0.1893 (0.1915)
- 0.0160 (0.0134)
0.3806 (0.3434)
0.2519 (0.3394)
0.2282
Ja
- 0.0270 (0.0592)
0.0001 (0.0127)
0.0920 (0.0416)
0.6187 (0.2796)
- 0.0051
Sw
0.3206 (0.1119)
- 0.0609 (0.0200)
0.7714 (0.2128)
0.0424 (0.1351)
0.3057
UK
0.2697 (0.1863)
-0.0264 (0.0161)
0.6096 (0.2880)
0.1132 (0.1991)
0.2973
USA
0.3011 (0.1934)
- 0.0376 ( - 0.0297)
0.0179 (0.0082)
1.7283 (0.2044)
0.3042
Notes: The estimated model is rt+ 1 - rt = ot + fir t + Et+ 1, with E(•t+ 1) = 0 and E(e2t+ 1~ = a2r2L Figures in parentheses are standard errors.
730
Journal o f International Money and Finance 1995 Volume 14 N um be r 5
~,
C I R SR ( y = 0.5)
Vasicek (7 = 0)
Merton (7 = 0)
- 0.009 (0.014) 0.717 (0.073)
/~
- 0.011 (0.014) 0.243 (0.020)
/~
d-
0.078 (0.143)
t~
d-
0.046 (0.145)
- 0.036 (0.062) 0.710 (0.073)
- 0.005 (0.006) 0.150 (0.078) 0.689 (0.198)
Au
&
~
&
~,
~-
/~
t~
CEV
Model
,~
~
,~
- 0.038 (0.020) 0.237 (0.023)
0.349 (0.181)
- 0.045 (0.020) 0.676 (0.071)
0.408 (0.186)
0.025 (0.048) 0.649 (0.073)
- 0.002 (0.005) 0.085 (0.060) 0.936 (0.297)
Be
0.002 (0.021) 0.188 (0.016)
- 0.090 (0.193)
- 0.014 (0.021) 0.618 (0.051)
0.081 (0.200)
- 0.047 (0.051) 0.604 (0.049)
- 0.005 (0.006) 0.860 (0.611) - 0.154 (0.321)
Ca
- 0.008 (0.029) 0.195 (0.030)
0.099 (0.273)
0.000 (0.029) 0.566 (0.094)
0.022 (0.273)
0.023 (0.043) 0.566 (0.092)
0.002 (0.005) 0.025 (0.025) 1.431 (0.417)
Fr
- 0.023 (0.011) 0.164 (0.016)
0.156 (0.069)
- 0.033 (0.011) 0.372 (0.038)
0.203 (0.071)
0.020 (0.027) 0.357 (0.038)
0.002 (0.005) 0.069 (0.031) 0.967 (0.226)
Ge
- 0.027 (0.038) 0.239 (0.028)
0.219 (0.261)
- 0.015 (0.038) 0.614 (0.073)
0.136 (0.262)
0.036 (0.044) 0.624 (0.070)
0.005 (0.006) 0.090 (0.143) 1.000 (0.779)
Ho
- 0.016 (0.013) 0.195 (0.016)
0.168 (0.190)
- 0.015 (0.013) 0.733 (0.062)
0.193 (0.191)
- 0.019 (0.049) 0.705 (0.059)
- 0.003 (0.004) 0.332 (0.321) 0.290 (0.364)
It
TABLE 5. The estimated restricted models.
0.002 (0.012) 0.110 (0.013)
- 0.036 (0.056)
0.009 (0.012) 0.211 (0.027)
- 0.068 (0.057)
- 0.030 (0.020) 0.205 (0.027)
- 0.005 (0.004) 0.086 (0.037) 0.661 (0.261)
Ja
- 0.055 (0.020) 0.290 (0.042)
0.232 (0.102)
- 0.060 (0.020) 0.830 (0.105)
0.328 (0.110)
0.027 (0.048) 0.666 (0.121)
- 0.011 (0.010) 0.437 (0.280) 0.237 (0.318)
Sw
- 0.016 (0.015) 0.236 (0.016)
0.129 (0.173)
- 0.029 (0.015) 0.792 (0.053)
- 0.302 (0.177)
- 0.010 (0.055) 0.776 (0.054)
0.004 (0.005) 0.467 (0.229) 0.214 (0.206)
UK
0.030 (0.030) 0.182 (0.021)
- 0.145 (0.197)
0.039 (0.031) 0.459 (0.056)
- 0.207 (0.200)
0.043 (0.033) 0.429 (0.053)
0.008 (0.005) 0.016 (0.008) 1.749 (0.211)
US
~
5'
~'
~"
0.016 (0.001)
0.009 (0.141) - 0.008 (0.014) 0.064 (0.005)
Notes: T h e
Be
rt+ 1 - -
0.022 (0.002)
0.232 (0.176) - 0.028 (0.020) 0.075 (0.007)
-0.002 (0.005) 0.073 (0.007)
0.074 (0.007)
estimated models are parentheses are standard errors.
~"
6"
~ ~.
CIR V R (3' = 1.5)
/3
&
E
BS (3' = 1)
/~
GBM (3' = 1)
~
0.067 (0.005)
~"
Dothan (3' = 1)
-0.007 (0.006) 0.064 (0.005)
Au
(Continued).
Model
TABLE 5
rt
=
Fr
with
0 and
0.023 (0.002)
0.079 (0.068) - 0.011 (0.011) 0.065 (0.006)
0.002 (0.005) 0.065 (0.006)
0.064 (0.006)
Ge
E ( E t + 1) =
0.021 (0.003)
0.230 (0.276) - 0.022 (0.030) 0.066 (0.010)
0.003 (0.005) 0.066 (0.010)
0.064 (0.009)
ot + f i r t + et+ 1,
0.017 (0.002)
- 0.195 (0.191) 0.012 (0.020) 0.056 (0.005)
-0.008 (0.006) 0.057 (0.005)
0.056 (0.005)
Ca
= o ' 2 r 2~'.
0.031 (0.003)
0.318 (0.264) - 0.041 (0.039) 0.089 (0.010)
0.005 (0.006) 0.090 (0.010)
0.087 (0.010)
Ho
0.018 (0.003)
- 0.003 (0.056) - 0.005 (0.012) 0.048 (0.006)
-0.006 (0.004) 0.048 (0.006)
0.049 (0.006)
Ja
0.024 (0.007)
0.187 (0.099) - 0.049 (0.019) 0.096 (0.016)
-0.017 (0.009) 0.089 (0.017)
0.074 (0.019)
Sw
0.017 (0.001)
- 0.064 (0.173) - 0.003 (0.015) 0.063 (0.005)
-0.008 (0.005) 0.063 (0.005)
0.067 (0.005)
UK
0.027 (0.003)
- 0.006 (0.196) 0.009 (0.030) 0.070 (0.007)
0.008 (0.005) 0.070 (0.007)
0.069 (0.007)
US
Reported estimates are for the unrestricted parameters. Figures in
0.011 (0.001)
0.085 (0.187) - 0.011 (0.013) 0.048 (0.004)
-0.006 (0.003) 0.047 (0.004)
0.047 (0.004)
It
~"
~
International evidence on interest rates: Y K Tse
remain statistically insignificant. Exceptions to this are the Belgium, G e r m a n y and Switzerland equations, in which both t~ and fi are f o u n d to be significant. F o r the France, J a p a n and U S A equations, estimates of a and /3 have the wrong signs. As the estimates of the unrestricted model show that ~ for these countries are significantly different from zero, imposing zero restriction for y has caused misspecification. Similar anomalies exist in the Canada, Switzerland and U S A equations for the C I R SR model, and the C a n a d a and U S A equations for the BS model, where inappropriate restrictions on 3/ have b e e n imposed. T o examine the validity of the restrictions imposed by various models we p e r f o r m model specification tests based on AJ. T h e results are summarized in Table 6, from which the following conclusions seem to emerge. First, Holland is a country for which there is no model that is clearly favored. T h e Vasicek, C I R SR and BS models are not rejected against the unrestricted model. On the contrary, the above three models are rejected at the 5 percent level for the USA. As ~ is quite large for the U S A equation and the largest y assumed in these three models is one (for the BS model), the rejection of these three models is perhaps not surprising. Second, for the France equation, both the Vasicek and the C I R SR models, but not the BS model, are rejected at the 10 percent level. Third, for Australia, Belgium, G e r m a n y and Japan, the Vasicek model is rejected at the 10 percent level. However, both the C I R SR and the BS models cannot be rejected for these countries. Fourth, for Canada, Switzerland and the UK, the Vasicek model cannot be rejected at the 10 percent level. Both the C I R SR and the BS models, however, are rejected. Fifth, for Italy the TABLE6. Tests of model restrictions. Unrestricted Restricted alternative null Au Unrestricted Unrestricted Unrestricted Unrestricted Unrestricted Unrestricted Unrestricted Unrestricted Vasicek GBM BS BS CEV CEV CEV
Merton Vasicek CIR SR Dothan GBM BS CIR VR CEV Merton Dothan Dothan GBM Dothan GBM CIRVR
Be
7.51 8.22 6.74 5.37 1.83 2.43 0.97 3.81 0.39 3.65 0.24 0.04 3.59 3.44 0.18 3.20 0.77 2.85 0.58 0.16 0.73 3.77 0.15 3.61 0.79 0.61 0.21 0.45 3.41 0.24
Ca
Fr
1.19 1.17 1.58 5.36 4.70 1.60 5.86 0.46 0.02 0.66 3.76 3.10 4.90 4.24 5.40
4.77 4.70 3.98 6.03 3.96 2.77 5.03 1.68 0.07 2.07 3.26 1.19 4.35 2.28 3.35
Ge
Ho
10.19 2.92 3.45 2.70 0.91 0.86 2.21 5.20 1.61 5.08 0.66 0.86 4.56 7.93 1.61 2.88 6.74 0.18 0.60 0.12 1.55 4.34 0.95 4.22 0.60 2.32 0.00 2.20 2.95 5.05
It
Ja
3.04 0.17 0.28 3.83 3.26 1.33 4.05 2.74 2.87 0.57 2.50 1.93 1.09 0.52 1.31
4.08 4.00 2.52 1.60 0.71 0.69 0.72 0.02 0.08 0.89 0.91 0.02 1.58 0.69 0.70
Sw UK 8.48 1.92 4.61 5.15 4.71 4.46 4.47 4.46 6.56 0.44 0.69 0.25 0.69 0.25 0.01
2.90 0.28 4.25 7.12 6.59 4.91 6.76 2.36 2.62 0.53 2.21 1.68 4.76 4.23 2.55
US 11.46 10.93 10.58 8.93 8.89 8.28 2.57 0.62 0.53 0.04 0.65 0.61 8.31 8.27 1.95
d.f. 2 1 1 3 2 1 3 1 1 1 2 1 2 1 2
d.f. = degrees of freedom. All test statistics are distributed as X2. The (upper tail) initial values at the 5 percent level of significancefor X2 with 1, 2 and 3 degrees of freedom are, respectively,3.84, 5.99 and 7.81. The correspondingfigures at 10 percent level are, respectively,2.71, 4.61 and 6.25.
Notes:
Journal of lnternational Money and Finance 1995 Volume 14 Number 5
733
Merton
Vasicek
CIR SR
~" ~
~ ~"
~
t~
CEV
C1
Unrestricted
C1 C2 C3
C1 C2 C3
C1 C2 C3
C3
C2
C1
C2 C3
Stat
Model
~
.-d
0.003 0.032 0.228
0.002 0.000 0.000
0.000 0.000 0.000
0.001 0.053 0.224
0.002 0.052 0.221
Au
0.019 0.009 0.151
0.026 0.000 0.000
0.000 0.000 0.000
0.000 0.033 0.154
0.014 0.023 0.149
Be
0.000 0.009 - 0.123
0.003 0.000 0.000
0.000 0.000 0.000
0.000 0.002 0.134
0.013 0.016 0.132
Ca
0.001 0.001 0.230
0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.030 0.244
0.017 0.054 0.223
Fr
0.016 0.009 0.150
0.030 0.000 0.000
0.000 0.000 0.000
0.000 0.052 0.157
0.009 0.023 0.150
Ge
0.007 0.004 0.269
0.002 0.000 0.000
0.000 0.000 0.000
0.000 0.021 0.350
0.029 0.078 0.384
Ho
TABLE 7. Summary statistics of models,
0.005 0.010 0.049
0.005 0.000 0.000
0.000 0.000 0.000
0.000 0.003 0.053
0.005 0.002 0.055
It
0.000 0.010 0.126
0.003 0.000 0.000
0.000 0.000 0.000
0.001 0.018 0.121
0.000 0.016 0.123
Ja
0.027 0.006 0.014
0.033 0.000 0.000
0.000 0.000 0.000
0.001 0.002 0.022
0.033 0.000 0.034
Sw
0.004 0.017 0.033
0.012 0.000 0.000
0.000 0.000 0.000
0.000 0.003 0.043
0.010 0.001 0.044
UK
0.011 0.001 0.418
0.020 0.000 0.000
0.000 0.000 0.000
0.001 0.155 0.419
0.018 0.201 0.389
US
~
t~
~" ~"
0.000 0.135 0.201
0.002 0.081 0.205
0.001 0.080 0.206
0.000 0.099 0.222
Au
0.000 0.097 0.148
0.010 0.042 0.145
0.000 0.038 0.153
0.000 0.040 0.156
Be
0.000 0.040 - 0.086
0.002 0.022 -0.110
0.001 0.026 -0.106
0.000 0.025 - 0.100
Ca
0.000 0.032 0.241
0.005 0.010 0.232
0.000 0.009 0.242
0.000 0.008 0.239
Fr
0.000 0.133 0.137
0.003 0.055 0.147
0.000 0.057 0.156
0.000 0.053 0.153
Ge
0.000 0.052 0.414
0.015 0.022 0.324
0.000 0.021 0.350
0.000 0.018 0.343
Ho
0.000 0.050 0.024
0.003 0.032 0.034
0.001 0.030 0.033
0.000 0.029 0.038
It
0.000 0.052 0.099
0.001 0.038 0.111
0.001 0.038 0.111
0.000 0.039 0.114
Ja
0.000 0.003 0.000
0.022 0.008 -0.001
0.003 0.006 -0.003
0.000 0.003 0.057
Sw
0.000 0.087 0.015
0.000 0.041 0.014
0.001 0.042 0.015
0.000 0.056 0.027
UK
0.000 0.075 0.420
0.001 0.012 0.435
0.001 0.012 0.434
0.000 0.012 0.429
US
squared residuals and the fitted conditional variance,
Notes: C l and C 2 are, respectively, the coefficient of determination of the mean and variance equations of the model. C 3 is the correlation coefficient between the
C3
C1 C2
C3
C2
C1
CIR VR
GBM
,~
C3
C1 C2
C1 C2 C3
Dothan
~ ~"
Stat
(Continued).
BS
Model
~
~n
TABLE 7
~'
g"
~'
International evidence on interest rates: Y K Tse
BS model is rejected while there is little to choose between the Vasicek and the CIR SR models. In summary, there is no single model that describes satisfactorily the stochastic behavior of interest rates for all the countries considered. If we classify the countries according to the size of ~/, France, Holland and the USA belong to the group of high elasticity. The estimated elasticities for these countries are larger than 1.5. In comparison, Australia, Belgium, Germany and Japan belong to the group of medium elasticity, while Canada, Italy, Switzerland and the UK belong to the group of low elasticity. For the last group of countries, ~/ is not statistically different from zero. To further assess the relative performance of the alternative models we compute the coefficient of determination of the mean and the variance equations, denoted by C 1 and C2, respectively. 14 In addition, we also report the simple correlation coefficient between the squared estimated residuals and the fitted conditional variance. This statistic is denoted by C 3. The results are reported in Table 7. It can be seen that by the C 2 criterion, the CIR VR model, which has the largest specified value of 31 (1.5), ranks the highest for most of the countries. The notable exceptions are France, Holland and the USA, for which cases ~/ are larger than 1.5. In comparison, C 3 favors models for which 3, is not specified, namely, the unrestricted model and the CEV model. By the C 3 criterion, these two models rank the highest for five countries. Also, in the group of countries with moderate ~,, the CIR SR model ranks very high by the C 3 criterion. Overall, the correlation between the estimated volatility and the squared residual is closely aligned to the appropriate value of the elasticity parameter. IV. Conclusions
Our analysis on the short-term interest rates of eleven countries shows that interest rate changes are leptokurtic. The unconditional variance is highest for the USA and lowest for Japan. As non-normality is evident we avoid using MLE. Following CKLS, we consider eight stochastic models of interest rates, all of which are nested within a discretized approximation to a differential equation. The GMM approach is used in preference to the MLE. The empirical results show that no single model can satisfactorily describe the stochastic structures of interest rates for all countries. The conditional volatility of interest rates can be very sensitive to the level of interest rates, as in the cases of France, Holland and the USA. On the other hand, Canada, Italy, Switzerland and the UK are the countries where the elasticity of variance is low. The Vasicek model may be preferred for these countries. For Australia, Belgium, Germany and Japan, the elasticity of variance is moderate. For these countries, there is no clear-cut statistical evidence for the choice between the CIR SR model and the BS model. Notes 1. A partial listing of analytic solutions of interest rate contingent claims includes those by Vasicek (1977), Cox et al. (1985), Jamshidian (1989), Amin and Jarrow (1991), Chen 736
Journal of lntemational Money and Finance 1995 Volume 14 Number 5
International evidence on interest rates: Y K Tse 2. 3.
4. 5. 6.
7.
8. 9. 10. 11.
12. 13. 14.
and Scott (1992) and Chen (1992). Parameters in equations (1) and (2) are related as follows: a = a ' A t and /3 = /3'At, where At is the sampling interval. One could also argue that the models admitted by equation (5) (i.e. the Vasicek, CIR SR, BS, Dothan and CIR VR models) are serious contenders for the purpose of evaluating interest rate contingent claims, while those that are excluded (i.e. the CEV, GBM and Merton models) may be rejected on grounds of unrealistic implications. The methodology described in this paper can be modified to use equation (5) as the unrestricted model. Whether imposing more structure on the basic model may lead to different conclusions is a topic for future research. In the actual calibration, S (0) was replaced by its sample analogue. The application of this test statistic can be extended to pairwise nesting. The use of ft*(O) has the advantage that the corresponding function g(O) is the score function if the residuals are normal. Thus, under normality the G M M estimates and the ML estimates of the unrestricted model are the same. To the extent that the MLE is optimal when the residuals are correctly specified, the GMM estimates based on ft* (0) do not lose optimality when errors are normal. On the other hand, the variance estimates of 0 using equation (12) are consistent even if the residuals are non-normal. As shown in Table 3 below, however, et is significantly non-normal. When f * ( O ) was used for the estimation, results were found to be rather unstable, as demonstrated by the large estimated standard errors. Thus, in this paper we only report results based on ft(O). While data for Japan were available from 1974 we decided to use data starting from April 1983, when banks in Japan were allowed to participate in over-the-counter sales of newly issued bonds. Although CKLS (Chan et al. 1992b) reported that no structural change was detected when Japanese data from February 1980 to September 1989 were examined, casual data perusal showed that for the 25 months prior to April 1984 the interest rates in Japan remained unchanged. As the interest rate observations do not form a random sample, formal significance tests on skewness and kurtosis are not performed. Variability here refers to the variation of the unconditional distribution. This should be distinguished from the conditional volatility of the interest rate changes. The anomaly that ~ of the Canada equation is negative is not disturbing as ~ is not statistically significantly different from zero. The CKLS result, however, is based on MLE. Using the CKLS sampling period ( 8 0 / 2 - 8 9 / 9 ) , with data collected from the same source as the data used in this paper, the MLE of 3' is 2.38. In comparison, the GMM estimate is 1.57. Thus, the discrepancy in the results could be attributed to the differences in the method of estimation and the sampling period. On the other hand, the closeness of our MLE to that of CKLS shows that the difference in data source does not appear to cause the discrepancy. It should be noted, however, that the magnitudes of both & and /3 are quite small. Also, the t-ratios of these parameters are statistically insignificant. As noted above, the speed-of-adjustment coefficient for Japan is practically zero. These statistics were suggested by CKLS as additional measures of the model performance. These measures, however, may not be appropriate if the estimation is not based on least squares. While the G M M estimation of the mean equation is a (weighted) least squares estimation, the estimation of the variance equation is not. Indeed, C 2 is biased in favor of a model that gives rise to large variations in the estimated conditional variance, irrespective of its performance as a forecast for the conditional variance. Thus, C 2 would favor a model with large elasticity of variance, such as the CIR VR model.
References
AMIN, K.I. A N D R. A. JARROW,'Pricing foreign currency options under stochastic interest rates,' Journal o f International Money and Finance, September 1991, 10: 310-329. Journal of International Money and Finance 1995 Volume 14 Number 5
737
Internationalevidence on interestrates: Y K Tse BP,Er~AN, M.J.M. AND E. S. SCHWARTZ,'A continuous time approach to the pricing of bonds,' Journal of Banking and Finance, 1979, 3: 133-155. CHAN, K. C., G. A. KAROVLI, F. A. LONGSTAFF AND A. B. SANDERS, 'An empirical comparison of alternative models of the short-term-interest rate,' Journal of Finance, July 1992, 47:1209-1227 (1992a). CHAN, K. C., G. A. KAROVLt, F. A. LONGSTAFFAND A. B. SANDERS, 'The volatility of Japanese interest rates: A comparison of alternative term structure models,' Pacific-Basin Capital Markets Research, Vol III, 1992, 119-136 (1992b). CHEN, r. r., 'Exact solutions for futures and European futures options on pure discount bonds,' Journal of Financial and Quantitative Analysis, March 1992, 27: 97-107. CHEN, R. R. AND L. Scow, 'Pricing interest rate options in a two-factor Cox-Ingersoll-Ross model of term structure,' The Review of Financial Studies, 1992, 5: 613-636. CONSTANTINIDES, G. M., 'A theory of the nominal term structure of interest rates,' The Review of Financial Studies, 5: 531-552. Cox, J. C., J. E. INGERSOLLAND S. A. Ross, 'An analysis of variable rate loan contracts,' Journal of Finance, May 1980, 35: 389-403. Cox, J. C., J. E. INGERSOLLAND S. A. ROSS, 'A theory of the term structure of interest rates,' Econometrica, March 1985, 53: 385-407. DOTHAN, U. L., 'On the term structure of interest rates,' Journal of Financial Economics, March 1978, 6: 59-69. HARRISON, J. M. AND S. PLISKA, 'Martingales and stochastic integrals in the theory of continuous trading,' Stochastic Processes and Their Applications, 1981, 11: 215-260. HEATH, D., R. JARROWAND A. MORTON, 'Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation,'Econometrica, January 1992, 60: 77-105. Ho, T. S. AND S. LEE, 'Term structure movements and pricing interest rate contingent claims,' Journal of Finance, December 1986, 41: 1011-1028. JAMSHIDIAN, F., 'An exact bond option formula,' Journal of Finance, March 1989, 44: 205-209. Lo, A. W., Maximum likelihood estimation of generalized Ito processes with discretely sampled data,' Econometric Theory, 1988, 4: 231-247. MERTON, R. C., 'Theory of rational option pricing,' Bell Journal of Economics and Management Science, 1973, 4: 141-183. TSE, Y. K., 'MLE of some continuous time financial models: Some Monte Carlo results,' Mathematics and Computers in Simulation, 1992, 33: 575-580. TSE, Y. IC, 'Maximum likelihood estimation of Cox-Ingersoll-Ross and Brennan-Schwartz models of interest rate movement,' Hong Kong Economic Papers, forthcoming, 1994. VASlCEK, O., 'An equilibrium characterization of the term structure,' Journal of Financial Economics, November 1977, 5: 177-188.
738
Journal of lnternationaIMoneyand Finance1995 Volume 14 Number5
E
•Ti TN E EoR
w 0 R T I-I M A N N
Journal o f lnternational Money and Finance, Vol. 14, No. 5, pp. 721-738, 1995 Copyright ©1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0261-5606(95)00030-5 0261-5606/95 $10.00 + 0.00
Some international evidence on the stochastic behavior of interest rates Y K TSE
Department of Economics and Statistics, National University of Singapore, Kent Ridge, Singapore 0511 This paper examines the stochastic behavior of short-term interest rates in eleven countries. Following Chan et al. (1992a), we consider eight stochastic models of short-term interest rates, all of which are nested within a discretized approximation to a differential equation. We use the generalized method of moments to estimate the model. The empirical results show that no single model can satisfactorily describe the stochastic structure of interest rates for all countries. On balance, if we classify the sample countries according to the size of the elasticity of variance of short-term interest rates, France, Holland and the USA belong to the group of high elasticity. In comparison, Australia, Belgium, Germany and Japan belong to the group of medium elasticity, while Canada, Italy, Switzerland and the UK belong to the group of low elasticity. (JEL 315).
The interest rate is a fundamental determinant of asset prices in financial markets. Many stochastic models have been developed by academic researchers in the continuous-time setting (see, e.g. Vasicek, 1977; Brennan and Schwartz, 1979; Cox et al, 1985; and Constantinides, 1992). These models provide a rich framework for specifying the dynamic behavior of interest rates. Based on an assumed stochastic model interest rates can be generated to mimic possible future movements. The generated interest rates may be adjusted to provide arbitrage-free term structures--a concept studied by Harrison and Pliska (1981), Ho and Lee (1986) and Heath et al. (1992), among others. Alternatively, the interest rates may be adjusted to reflect an individual's subjective view of the economic world. Arbitrage-free interest rates are important in pricing interest rate contingent cash flows, some of which admit analytic closed form solutions while others may require numerical evaluations. On the other hand, simulating interest rates in the real-world setting may be useful when risk-reward analysis is conducted under different scenarios. Recently Chan et al. (1992a) (CKLS hereafter) examined empirically the stochastic behavior of short-term interest rates in the USA. They considered eight continuous-time interest rate models, all of which can be nested within a general stochastic differential equation. Applying discrete approximation they estimated the models using the generalized method of moments (GMM) and conducted tests of restrictions to evaluate each model's performance. In this
International evidence on interest rates: Y K Tse
paper we adopt this frame-work of analysis and consider the short-term interest rates in eleven countries, including the USA. The results show that no single model can satisfactorily describe the stochastic structure of interest rates for all countries. On balance, if we classify the sample countries according to the size of the elasticity of variance of the short-term interest rates, France, Holland and the USA belong to the group of high elasticity. In comparison, Australia, Belgium, Germany and Japan belong to the group of medium elasticity, while Canada, Italy, Switzerland and the UK belong to the group of low elasticity. The selection of a stochastic model for short-term interest rates is an issue that requires theoretical consideration as well as empirical evidence. On theoretical bases, models that permit closed form solutions for contingent claims may be desirable since they provide greater analytical insights than numerical solutions. 1 On the other hand, models that represent reality inaccurately, or rely on unrealistic assumptions, may incur model risk. A subtle balance between the two issues is required for a successful evaluation of interest rate sensitive assets. The empirical results reported in this paper may provide some information that is useful for the selection of models. The plan of this paper is as follows. In Section 2 we outline the framework of stochastic interest rate models as summarized by CKLS. We describe the various models that are nested within the framework and explain the methodology of the econometric estimation. The data and some preliminary statistical results are summarized in Section 3. In Section 4 we report the empirical results of the GMM estimation of the eleven countries considered. Finally, some concluding remarks are given in Section 5. I. The continuous-time framework and estimation method
Various stochastic models in the continuous-time framework have been proposed for the movements of short-term interest rates. These models have been applied to price interest rate contingent cash flows. Analytic solutions of the valuation models may or may not be available depending on the complexity of the interest rate process assumed. The issue of comparing these stochastic models empirically is important when the objective is to select an appropriate model for pricing interest rate contingent claims, such as callable bonds and bond options. CKLS pointed out that many of the models considered in the literature can be nested within the following differential equation (1)
dr = ( ~ ' + f l ' r ) d t + t r r ~ d Z ,
where r is interest rate, t is time and d Z is a Wiener process. Specifically, the following models are nested within equation (1) above 1. Merton (1973) 2. Vasicek (1977) 3. Cox-Ingersoll-Ross (1985) square root process (CIR SR) 4. Dothan (1978) 722
dr = a 'dt + t r d Z dr = ( a ' + f l ' r ) d t + ~ d Z dr = ( a ' + f l ' r ) d t + trrl/EdZ dr = ~rr d Z
Journal of Intemational Money and Finance 1995 Volume 14 Number 5
International evidence on interest rates: Y K Tse
5. Geometric Brownian Motion (GBM) 6. Brennan-Schwartz (1979) (as) 7. Cox-Ingersoll-Ross (1980) variable rate model (CIR VR) 8. Constant elasticity of variance (CEV)
dr -- [3 'r dt + ~ r d Z d r = ( a ' + [3'r)dt + trr d Z dr = t r r a / Z d Z dr = [3 'r dt + trrVdZ.
Model 1 is a simple Brownian motion (generalized Wiener process) with drift parameter a'. It was used by Merton (1973) to derive the values of pure discount bonds. Model 2 is the Ornstein-Uhlenbeck process studied by Vasicek (1977). It implies that interest rates follow an autoregressive process of order 1. Thus, both Models 1 and 2 permit interest rates to take negative values. In contrast, the CIR SR model does not suffer from this drawback. This model implies that interest rate changes are proportional to a non-central X2 variate. The Dothan and CIR VR models specify that interest rate changes are without drift. The GBM model has been used extensively in the valuation of stock options. It implies that interest rates have a log-normal distribution. The BS model is similar to the CIR SR model in that the interest rates are not permitted to take negative values. Model 8 is conveniently defined as the CEV model, following the terminology of CKLS. It has the property that the elasticity of the instantaneous variance with respect to interest rate is 2y, which is independent of the level of interest rates. Set in the above framework the exact maximum likelihood estimation (MLE) of the interest rate models requires knowledge of the exact likelihood function of the continuous-time process. Lo (1988) considered the problem of maximum likelihood estimation using discretized approximations to the exact continuous-time solution. He argued that the discretized MLE is in general inconsistent. The inconsistency does not vanish with increasing sample size, but depends on the size of the sampling interval. The errors in discretization, however, may not be serious if the sampling interval is short, as demonstrated by Tse (1992, 1994). Following the discretization approach, CKLS considered the following approximation to equation (1)2 (2)
Ft+ 1 -- rt ~ Ol ÷ [31 t ÷ ~'t+ 1'
where et+ 1 is assumed to satisfy (3)
E(et+l)=OandE(etz+l)=tr2r
2v.
An alternative framework that incorporates some of the models considered can be described by the following equation (4)
dr = K( Ix - r ) d t + trrVdZ,
where r is the speed-of-adjustment coefficient and /x is the long-run interest rate. Thus, the constraints /z > 0 and r >_0 are assumed. Equation (4) has the advantage that the parameters have straightforward economic interpretations. Journal of International Money and Finance 1995 Volume 14 Number 5
723
International evidence on interest rates: Y K Tse
In discretized form it can be written as <5)
rt+ 1 -- rt =
KIxAt - K r t A t + et+ 1.
Comparing equations (2) and (5), we observe that equation (5) (with constraints o n / x and K, and hence on a and 13) is a special case of equation (2), with (6)
a = K/zAt and /3 = - K A t ,
so that a = - / 3 / z . Although equations (2> and (5) are very similar the following differences should be noted. First, as /3 = 0 implies a = 0, the generalized Wiener process (Merton model) with /3 = 0 and a :~ 0 is not derivable from equation (5>. Second, if we accept equation (5> we would expect, from equation (6), a > 0 and /3 < 0. Such constraints, however, are not imposed on equation (2). Third, as /z > 0, equation (6) implies that if a = 0 then r = 0 and hence /3 = 0. Thus, the G B M and CEV models (with a = 0 and /3 4: 0) are not derivable from equation (5>. While equation (5> can be used as the basic framework for comparing alternative interest rate models it incorporates a less rich class of submodels. Thus, in this paper we follow the CKLS framework for the purpose of comparison and better generality. 3 To estimate the parameters of equation (2>, CKLS used the generalized method of moments (GMM), due to its robustness against misspecifications in the distribution of the residuals. Defining 0 as the parameter vector consisting of a, /3, tr and Y, CKLS considered a v e c t o r ft(O) given by
I Et+1 l et+
]
lrt
(7)
ft(O)=l,?+l_o.2r2V
,
[(e2+x--O-2r2"/)rt
where Et+ 1 = 1",+1 - rt - a - / 3 r t, for t = 0,..., n - 1. Under the null hypothesis specified in equations (2> and (3>, E[ft(0)] = 0. The G M M estimator of 0, 0, is the value that minimizes the quadratic form (8>
J(O) =g'(O)W(O)g(O),
where n
(9)
g ( O ) = n1 2 f t ( O )
t=l
'
and W (0) is the weighting matrix given by (10)
W(O) = S-1(0),
with S ( O ) = E[ft(O)f't(O)]. 4 All models considered can be estimated by this method. For the unrestricted model, J ( 0 ) is zero. In general we define 0 0 as the k-element (k < 4) subvector of 0 consisting of unrestricted parameters.
724
Journal o/International Money and Finance 1995 Volume 14 Number 5
International evidence on interest rates: Y K Tse
We also define D(O) as the 4 x k matrix of partial derivatives of g(O) with respect to 00, i.e.
D(O) = 3g(O) 30' o
(11)
Then the asymptotic covariance matrix of 00 can be consistently estimated by (12)
l [D'( Oo)S-X( Oo)D( Oo)] -1
To evaluate the various restrictions imposed on the unrestricted model we calculate the following test statistic 5
AJ=n[JOo-J(O)],
(13)
which is distributed approximately as a X 2 with 4 - k degrees of freedom. It should be pointed out that equation (7) is not the only possible vector to define the objective function. For example, the following vector ~?t+ 1
rt2v Et+ 1
rt2v - 1 (14)
f*(O)
=
,2+1
0 -2
(, r 2 ~,
rt
also satisfies the requirement that E[ f * (0)] = 0, and thus can be used to define the objective function. To facilitate comparison with CKLS for the results on US data, however, we use ft(0) to calculate the G M M estimates. 6
II The data used in this study were extracted from various issues of The Economist. Annualized three-month money market rates at approximately the end of the month were collected. The countries included in the study and the sampling period for each country are summarized in Table 1.7 Thus, the sample size ranges from 130 to 214. Table 2 summarizes the unconditional distribution of the three-month money market rates, which we denote generically as rt, for various countries. The first panel presents the first four sample moments of r r Under the normality assumption, the skewness and kurtosis have expected values of zero and three, Journal of International Money and Finance 1995 Volume 14 Number 5
725
International evidence on interest rates: Y K Tse
TABLE 1. Short-term interest rate data.
Country
Code
Data Period
Australia Belgium Canada France Germany Holland Italy Japan Switzerland UK USA
Au Be Ca Fr Ge Ho It Ja Sw UK US
83/5-94/5 76/8-94/5 83/8-94/5 76/8-94/5 77/1-94/5 77/4-94/5 77/2-94/5 83/4-94/5 76/8-94/5 76/8-94/5 76/8-94/5
Number of Observations 133 214 130 214 209 206 208 134 214 214 214
respectively. It is found that only two countries (France and the USA) have kurtosis higher than three, while Japan has the lowest kurtosis. Also, except for Australia, Canada, Japan and the UK, all countries have positive skewness. 8 Judging from the standard deviations and the ranges of the unconditional distributions, Australia, Italy and the USA have the largest variability. 9 On the other hand, Japan is the country where interest rates are the most stable. To examine the serial correlation of r t in the first and second moments, we compute two sets of autocorrelation functions. We calculate t31xi, which is the ith order sample autocorrelation coefficient of r r Similarly, P2,i, is the ith order sample autocorrelation coefficient of ( r t - ~ . ) 2 , where r is the sample mean of r r These autocorrelation coefficients, up to order 24, are used to calculate the Box-Pierce statistics, which we denote as Qj(24) for j = 1, 2. These statistics are approximately distributed as X2 if there is no serial dependence. Thus, the results show that interest rates are highly autocorrelated in both the first and second moments. The high value of Q1 and the fact that the sample autocorrelation function decays very slowly suggest that interest rates may be highly autocorrelated or even non-stationary. However, as interest rates may not be normal, we further perform tests of serial correlation using the non-parametric runs tests. These are denoted by R 1 and R2, respectively, for tests of the first and second moments. The R statistics are approximately standard normal if there is no serial correlation, with negative values indicative of positive serial correlation. The results strongly indicate that there is positive serial correlation in both the mean and the variance for all countries. In Table 3 we present the summary statistics of the differenced interest rates, A r t = r t - r t _ 1. Here the Q and R statistics for serial correlation often give conflicting results. Judging from the R statistic, which is more robust, four countries (Australia, Belgium, Canada and the USA) have significant serial correlation in squared interest rate changes. On the other hand, all countries 726
Journal o f lntemational Money and Finance 1995 Volume 14 Number 5
to --4
t.~
~,
~"
,~
~,
•~
9.82 2.80 5.19 17.50 0.800 2.912 0.959 0.909 0.855 0.811 1946.6 - 12.72 0.898 0.766 0.665 0.583 949.7 -11.92
Be 9.17 2.55 3.62 14.00 -0.251 2.384 0.970 0.939 0.895 0.851 1084.7 - 10.21 0.916 0.841 0.734 0.640 498.2 -9.40
Ca 10.26 2.59 5.50 19.00 0.894 3.761 0.948 0.904 0.869 0.817 1877.5 - 12.13 0.859 0.744 0.689 0.561 674.1 -10.47
Fr 6.82 2.41 3.35 13.20 0.529 2.484 0.984 0.968 0.947 0.920 2116.5 - 13.93 0.937 0.855 0.770 0.671 821.3 -11.56
Ge 7.43 2.25 2.88 15.50 0.562 2.958 0.942 0.902 0.846 0.783 1505.7 - 13.54 0.660 0.590 0.444 0.310 361.4 -9.51
Ho 14.26 3.40 7.81 21.75 0.494 2.259 0.971 0.939 0.906 0.869 2307.8 - 13.46 0.943 0.889 0.847 0.804 1321.8 -11.52
It 5.10 1.52 1.93 7.65 -0.126 2.004 0.978 0.949 0.916 0.875 1085.1 - 10.75 0.932 0.851 0.763 0.678 546.6 -10.52
Ja 5.02 2.50 0.12 10.75 0.161 2.352 0.943 0.911 0.884 0.841 1723.9 - 11.31 0.883 0.798 0.717 0.627 694.8 -11.64
Sw
11.26 3.01 4.94 18.13 -0.118 2.492 0.959 0.907 0.849 0.794 1306.9 - 12.88 0.916 0.842 0.749 0.658 747.5 -12.28
UK
8.09 3.23 3.12 17.75 0.718 3.346 0.962 0.921 0.884 0.839 2303.8 - 13.29 0.858 0.745 0.669 0.538 945.5 -13.22
US
(r t _
Notes:
Interest rates are given in percentages, t31,i is the ith sample autocorrelation coefficient of r t a n d P2.i is the ith sample autocorrelation coefficient of f)2. Qj (24), j = 1,2, are the Box-Pierce portmanteau statistics based on autocorrelation coefficients of order up to 24. Similarly, Rj, j = 1, 2, are the runs tests for serial correlation. Under the null hypothesis of white noise, Q is approximately a x 2 with 24 degrees of freedom ( X224,0.o5 = 36.4) and R is approximately a standard normal.
R 2
Q2(24)
P2,4
t32~ t32,3
P2,1
R 1
0.912 0.879 1367.5 - 9.98 0.940 0.878 0.807 0.742 731.9 -9.56
0.947
t31,3 Pl,4 Q~(24)
0.976
Pl,2
12.20 4.29 4.57 19.80 -0.413 2.027
Pl,1
Mean Standard deviation Minimum Maximum Skewness Kurtosis
Au
TABLE 2. Summary statistics of r t
~"
~. ~"
8"
"~
~.
~.
~ ~"
~'
- 0.062 0.862 - 2.500 2.750 0.569 2.669 4.610 3.776 0.102 0.118 - 0.063 - 0.148 40.0 - 3.45 0.257 0.082 0.101 0.043 104.1 - 1.52
- 0.031 0.772 - 3.750 3.350 0.577 3.439 9.177 18.403 0.116 0.047 - 0.102 - 0.081 19.1 - 4.59 0.239 0.010 0.031 - 0.019 43.8 - 4.20
Be - 0.022 0.660 - 1.700 2.720 0.672 3.116 5.468 5.721 - 0.024 0.143 - 0.070 - 0.060 43.0 - 3.07 0.314 0.300 0.009 - 0.035 35.3 - 4.22
Ca - 0.020 0.810 - 5.190 4.500 - 0.093 -0.553 17.670 43.704 - 0.088 -0.066 0.165 - 0.042 35.4 - 0.98 0.375 0.017 0.008 0.012 40.9 - 4.14
Fr 0.001 0.454 - 1.700 2.600 1.383 8.145 10.965 23.448 0.024 0.133 0.163 0.109 48.0 - 0.79 0.319 0.166 0.099 0.016 101.3 - 3.76
Ge 0.003 0.750 - 4.370 4.000 0.170 0.995 14.461 33.495 - 0.196 0.136 0.066 - 0.123 60.3 - 1.40 0.443 0.105 0.099 0.025 111.8 - 4.23
Ho
rt -
It - 0.040 0.745 - 3.320 3.040 0.085 0.497 6.866 11.353 0.074 0.041 0.063 - 0.054 21.0 - 1.94 0.152 0.084 0.267 0.044 45.7 - 2.15
r t _ 1.
- 0.026 0.255 - 0.910 1.090 0.171 0.805 8.752 13.540 0.300 0.060 0.240 0.055 47.5 - 2.01 0.067 0.232 0.119 0.071 26.5 - 2.17
Ja 0.015 0.836 - 3.940 5.940 1.152 6.864 16.414 39.960 - 0.220 -0.045 0.160 0.002 52.6 - 1.12 0.334 0.034 0.130 0.137 59.5 - 0.94
Sw - 0.029 0.805 - 2.680 2.870 0.671 3.996 4.887 5.620 0.159 0.079 - 0.058 - 0.051 35.0 - 1.10 0.041 0.036 -0.055 0.050 35.1 - 1.62
UK
- 0.004 0.897 - 4.750 4.120 - 0.857 -5.104 12.075 27.036 0.012 -0.060 0.087 - 0.300 93.8 - 2.45 0.170 0.267 0.068 0.177 198.9 - 7.84
US
freedom ( X224,0.05= 36.4) and R is approximately a standard normal.
Notes:
Interest rates are given in percentages, t31, i is the ith sample autocorrelation coefficient of A r t = r t - r t_ 1 and t32.i is the ith sample autocorrelation coefficient of ( A r t - - ~ i ) 2 where ~-7 is the sample m e a n of A r t. Q j (24), j = 1,2, are the Box-Pierce p o r t m a n t e a u statistics based on autocorrelation coefficients of order up to 24. Similarly, Rj, j = 1,2, are the runs tests for serial correlation. U n d e r the null hypothesis of white noise, Q is approximately a )(2 with 24 degrees of
R1 132,1 132,2 132,3 132,4 Q2 (24) R2
01 (24)
Mean Standard deviation Minimum Maximum Skewness Standardized skewness Kurtosis Standardized kurtosis 131,1 131,2 131,3 131,4
Au
TABLE 3. S u m m a r y s t a t i s t i c s o f
~..
_~.
~.
g~
~.
g~
International evidence on interest rates: Y K Tse
have significant serial correlation in squared interest rate changes except for Australia, Switzerland and the UK. To give some indication of the appropriateness of the normality assumption we calculate the standardized skewness and kurtosis. Skewness is standardized with respect to mean zero and variance 6/n, while kurtosis is standardized with respect to mean 3 and variance 24/n. These statistics are approximately standard normal if Ar t is normal and serially uncorrelated. The results show that all interest rates are significantly leptokurtic (i.e. kurtosis larger than three). France, Holland, Italy and Japan have symmetric interest rate changes. Surprisingly, the USA is the only country where interest rate changes are significantly negatively skewed. In summary, the differenced data for short-term interest rates demonstrate departure from normality. Serial correlation appears to be more serious in the variance as compared to the mean.
Ill. Empirical results The estimation results for the unrestricted model is presented in Table 4. Table 5 summarizes the results for the unrestricted models. Figures in parentheses are the estimated standard errors. From Table 4 we can see that ~ varies considerably across the countries. Four countries (Canada, Italy, Switzerland and the UK) have estimated values of 7 that are not statistically significant. TM The USA has the largest estimated value of 7, followed by France. Using data from June 1964 through December 1989 CKLS ( C h a n e t al., 1992a) reported the estimated value of 7 to be 1.50 for the USA. Compared to our estimated value of 1.71, the difference may be interpreted as an indication that interest rate volatility has become more sensitive to the interest rate level in later years, or alternatively, that the variance elasticity has increased. In their study on the Japanese market, CKLS (Chan et al., 1992b) reported ~ to be 2.44, much larger than the value of 0.62 given here. 11 CKLS argued that ~/ is the most important parameter in determining the term structure of interest rates as it directly affects the conditional volatility. Of the eleven countries considered, values of ~/ range from -0.36 to 1.73, indicating that the sensitivity of volatility covers a broad spectrum of values. For the parameters a and /3, with the exception of Switzerland, none of the estimated values is statistically significantly different from zero. Thus, the data do not provide precise information about these parameters. As discussed earlier, if we start with equation (5) we would expect a > 0 and /3 < 0. Both constraints are violated for Japan. 12 Otherwise, the estimated values of a and /3 for other countries have the expected signs. Furthermore, based on equation (5) a and /3 are related through the equation a = -/3/z. Thus, replacing /x by t h e sample mean of interest rates, r, we would expect 3 to be close to -/3~ if equation (5) holds. The last column of Table 4 summarizes the value of/3F. It can be seen that 3 and /37 are remarkably close. This is an indication that equation (5) may provide a useful basic framework for conducting model comparison. As imposing relevant restrictions may improve the power of inference this is an Journal of lntemational Money and Finance 1995 Volume 14 Number 5
729
International evidence on interest rates: Y K Tse
interesting topic for future research. Finally, using the relationship K = -~3/At, we can see that Switzerland has the highest s p e e d - o f - a d j u s t m e n t coefficient, followed by Holland. A t the o t h e r extreme, Australia has the lowest s p e e d - o f - a d j u s t m e n t coefficient. 13 O f all the m o d e l s considered, the Vasicek, C I R SR and BS models are p r o b a b l y the m o s t serious c o n t e n d e r s for contingent claims valuation. T h e s e m o d e l s i m p o s e no restrictions on a and /3, but m a k e different a s s u m p t i o n s a b o u t the value of y. F o r the Vasicek m o d e l ( y = 0), m o s t values of t~ and /3 TABLE 4. The estimated unrestricted models. Parameters Country
&
/3
8"
~,
/~
Au
0.0516 (0.1453)
-0.0092 (0.0141)
0.1553 (0.0817)
0.6763 (0.1987)
0.1122
Be
0.2939 (0.1937)
- 0.0330 (0.0211)
0.1331 (0.0981)
0.7579 (0.3168)
0.3241
Ca
0.2476 (0.2499)
- 0.0293 (0.0251)
1.4093 (0.9250)
- 0.3600 (0.2945)
0.2687
Fr
0.4073 (0.2891)
- 0.0415 (0.0311)
0.0161 (0.0144)
1.6289 (0.3656)
0.4258
Ge
0.1250 (0.0834)
-0.0181 (0.0137)
0.1122 (0.0641)
0.7141 (0.3070)
0.1234
Ho
0.4263 (0.2790)
- 0.0570 (0.0407)
0.0257 (0.0327)
1.5997 (0.5868)
0.4235
It
0.1893 (0.1915)
- 0.0160 (0.0134)
0.3806 (0.3434)
0.2519 (0.3394)
0.2282
Ja
- 0.0270 (0.0592)
0.0001 (0.0127)
0.0920 (0.0416)
0.6187 (0.2796)
- 0.0051
Sw
0.3206 (0.1119)
- 0.0609 (0.0200)
0.7714 (0.2128)
0.0424 (0.1351)
0.3057
UK
0.2697 (0.1863)
-0.0264 (0.0161)
0.6096 (0.2880)
0.1132 (0.1991)
0.2973
USA
0.3011 (0.1934)
- 0.0376 ( - 0.0297)
0.0179 (0.0082)
1.7283 (0.2044)
0.3042
Notes: The estimated model is rt+ 1 - rt = ot + fir t + Et+ 1, with E(•t+ 1) = 0 and E(e2t+ 1~ = a2r2L Figures in parentheses are standard errors.
730
Journal o f International Money and Finance 1995 Volume 14 N um be r 5
~,
C I R SR ( y = 0.5)
Vasicek (7 = 0)
Merton (7 = 0)
- 0.009 (0.014) 0.717 (0.073)
/~
- 0.011 (0.014) 0.243 (0.020)
/~
d-
0.078 (0.143)
t~
d-
0.046 (0.145)
- 0.036 (0.062) 0.710 (0.073)
- 0.005 (0.006) 0.150 (0.078) 0.689 (0.198)
Au
&
~
&
~,
~-
/~
t~
CEV
Model
,~
~
,~
- 0.038 (0.020) 0.237 (0.023)
0.349 (0.181)
- 0.045 (0.020) 0.676 (0.071)
0.408 (0.186)
0.025 (0.048) 0.649 (0.073)
- 0.002 (0.005) 0.085 (0.060) 0.936 (0.297)
Be
0.002 (0.021) 0.188 (0.016)
- 0.090 (0.193)
- 0.014 (0.021) 0.618 (0.051)
0.081 (0.200)
- 0.047 (0.051) 0.604 (0.049)
- 0.005 (0.006) 0.860 (0.611) - 0.154 (0.321)
Ca
- 0.008 (0.029) 0.195 (0.030)
0.099 (0.273)
0.000 (0.029) 0.566 (0.094)
0.022 (0.273)
0.023 (0.043) 0.566 (0.092)
0.002 (0.005) 0.025 (0.025) 1.431 (0.417)
Fr
- 0.023 (0.011) 0.164 (0.016)
0.156 (0.069)
- 0.033 (0.011) 0.372 (0.038)
0.203 (0.071)
0.020 (0.027) 0.357 (0.038)
0.002 (0.005) 0.069 (0.031) 0.967 (0.226)
Ge
- 0.027 (0.038) 0.239 (0.028)
0.219 (0.261)
- 0.015 (0.038) 0.614 (0.073)
0.136 (0.262)
0.036 (0.044) 0.624 (0.070)
0.005 (0.006) 0.090 (0.143) 1.000 (0.779)
Ho
- 0.016 (0.013) 0.195 (0.016)
0.168 (0.190)
- 0.015 (0.013) 0.733 (0.062)
0.193 (0.191)
- 0.019 (0.049) 0.705 (0.059)
- 0.003 (0.004) 0.332 (0.321) 0.290 (0.364)
It
TABLE 5. The estimated restricted models.
0.002 (0.012) 0.110 (0.013)
- 0.036 (0.056)
0.009 (0.012) 0.211 (0.027)
- 0.068 (0.057)
- 0.030 (0.020) 0.205 (0.027)
- 0.005 (0.004) 0.086 (0.037) 0.661 (0.261)
Ja
- 0.055 (0.020) 0.290 (0.042)
0.232 (0.102)
- 0.060 (0.020) 0.830 (0.105)
0.328 (0.110)
0.027 (0.048) 0.666 (0.121)
- 0.011 (0.010) 0.437 (0.280) 0.237 (0.318)
Sw
- 0.016 (0.015) 0.236 (0.016)
0.129 (0.173)
- 0.029 (0.015) 0.792 (0.053)
- 0.302 (0.177)
- 0.010 (0.055) 0.776 (0.054)
0.004 (0.005) 0.467 (0.229) 0.214 (0.206)
UK
0.030 (0.030) 0.182 (0.021)
- 0.145 (0.197)
0.039 (0.031) 0.459 (0.056)
- 0.207 (0.200)
0.043 (0.033) 0.429 (0.053)
0.008 (0.005) 0.016 (0.008) 1.749 (0.211)
US
~
5'
~'
~"
0.016 (0.001)
0.009 (0.141) - 0.008 (0.014) 0.064 (0.005)
Notes: T h e
Be
rt+ 1 - -
0.022 (0.002)
0.232 (0.176) - 0.028 (0.020) 0.075 (0.007)
-0.002 (0.005) 0.073 (0.007)
0.074 (0.007)
estimated models are parentheses are standard errors.
~"
6"
~ ~.
CIR V R (3' = 1.5)
/3
&
E
BS (3' = 1)
/~
GBM (3' = 1)
~
0.067 (0.005)
~"
Dothan (3' = 1)
-0.007 (0.006) 0.064 (0.005)
Au
(Continued).
Model
TABLE 5
rt
=
Fr
with
0 and
0.023 (0.002)
0.079 (0.068) - 0.011 (0.011) 0.065 (0.006)
0.002 (0.005) 0.065 (0.006)
0.064 (0.006)
Ge
E ( E t + 1) =
0.021 (0.003)
0.230 (0.276) - 0.022 (0.030) 0.066 (0.010)
0.003 (0.005) 0.066 (0.010)
0.064 (0.009)
ot + f i r t + et+ 1,
0.017 (0.002)
- 0.195 (0.191) 0.012 (0.020) 0.056 (0.005)
-0.008 (0.006) 0.057 (0.005)
0.056 (0.005)
Ca
= o ' 2 r 2~'.
0.031 (0.003)
0.318 (0.264) - 0.041 (0.039) 0.089 (0.010)
0.005 (0.006) 0.090 (0.010)
0.087 (0.010)
Ho
0.018 (0.003)
- 0.003 (0.056) - 0.005 (0.012) 0.048 (0.006)
-0.006 (0.004) 0.048 (0.006)
0.049 (0.006)
Ja
0.024 (0.007)
0.187 (0.099) - 0.049 (0.019) 0.096 (0.016)
-0.017 (0.009) 0.089 (0.017)
0.074 (0.019)
Sw
0.017 (0.001)
- 0.064 (0.173) - 0.003 (0.015) 0.063 (0.005)
-0.008 (0.005) 0.063 (0.005)
0.067 (0.005)
UK
0.027 (0.003)
- 0.006 (0.196) 0.009 (0.030) 0.070 (0.007)
0.008 (0.005) 0.070 (0.007)
0.069 (0.007)
US
Reported estimates are for the unrestricted parameters. Figures in
0.011 (0.001)
0.085 (0.187) - 0.011 (0.013) 0.048 (0.004)
-0.006 (0.003) 0.047 (0.004)
0.047 (0.004)
It
~"
~
International evidence on interest rates: Y K Tse
remain statistically insignificant. Exceptions to this are the Belgium, G e r m a n y and Switzerland equations, in which both t~ and fi are f o u n d to be significant. F o r the France, J a p a n and U S A equations, estimates of a and /3 have the wrong signs. As the estimates of the unrestricted model show that ~ for these countries are significantly different from zero, imposing zero restriction for y has caused misspecification. Similar anomalies exist in the Canada, Switzerland and U S A equations for the C I R SR model, and the C a n a d a and U S A equations for the BS model, where inappropriate restrictions on 3/ have b e e n imposed. T o examine the validity of the restrictions imposed by various models we p e r f o r m model specification tests based on AJ. T h e results are summarized in Table 6, from which the following conclusions seem to emerge. First, Holland is a country for which there is no model that is clearly favored. T h e Vasicek, C I R SR and BS models are not rejected against the unrestricted model. On the contrary, the above three models are rejected at the 5 percent level for the USA. As ~ is quite large for the U S A equation and the largest y assumed in these three models is one (for the BS model), the rejection of these three models is perhaps not surprising. Second, for the France equation, both the Vasicek and the C I R SR models, but not the BS model, are rejected at the 10 percent level. Third, for Australia, Belgium, G e r m a n y and Japan, the Vasicek model is rejected at the 10 percent level. However, both the C I R SR and the BS models cannot be rejected for these countries. Fourth, for Canada, Switzerland and the UK, the Vasicek model cannot be rejected at the 10 percent level. Both the C I R SR and the BS models, however, are rejected. Fifth, for Italy the TABLE6. Tests of model restrictions. Unrestricted Restricted alternative null Au Unrestricted Unrestricted Unrestricted Unrestricted Unrestricted Unrestricted Unrestricted Unrestricted Vasicek GBM BS BS CEV CEV CEV
Merton Vasicek CIR SR Dothan GBM BS CIR VR CEV Merton Dothan Dothan GBM Dothan GBM CIRVR
Be
7.51 8.22 6.74 5.37 1.83 2.43 0.97 3.81 0.39 3.65 0.24 0.04 3.59 3.44 0.18 3.20 0.77 2.85 0.58 0.16 0.73 3.77 0.15 3.61 0.79 0.61 0.21 0.45 3.41 0.24
Ca
Fr
1.19 1.17 1.58 5.36 4.70 1.60 5.86 0.46 0.02 0.66 3.76 3.10 4.90 4.24 5.40
4.77 4.70 3.98 6.03 3.96 2.77 5.03 1.68 0.07 2.07 3.26 1.19 4.35 2.28 3.35
Ge
Ho
10.19 2.92 3.45 2.70 0.91 0.86 2.21 5.20 1.61 5.08 0.66 0.86 4.56 7.93 1.61 2.88 6.74 0.18 0.60 0.12 1.55 4.34 0.95 4.22 0.60 2.32 0.00 2.20 2.95 5.05
It
Ja
3.04 0.17 0.28 3.83 3.26 1.33 4.05 2.74 2.87 0.57 2.50 1.93 1.09 0.52 1.31
4.08 4.00 2.52 1.60 0.71 0.69 0.72 0.02 0.08 0.89 0.91 0.02 1.58 0.69 0.70
Sw UK 8.48 1.92 4.61 5.15 4.71 4.46 4.47 4.46 6.56 0.44 0.69 0.25 0.69 0.25 0.01
2.90 0.28 4.25 7.12 6.59 4.91 6.76 2.36 2.62 0.53 2.21 1.68 4.76 4.23 2.55
US 11.46 10.93 10.58 8.93 8.89 8.28 2.57 0.62 0.53 0.04 0.65 0.61 8.31 8.27 1.95
d.f. 2 1 1 3 2 1 3 1 1 1 2 1 2 1 2
d.f. = degrees of freedom. All test statistics are distributed as X2. The (upper tail) initial values at the 5 percent level of significancefor X2 with 1, 2 and 3 degrees of freedom are, respectively,3.84, 5.99 and 7.81. The correspondingfigures at 10 percent level are, respectively,2.71, 4.61 and 6.25.
Notes:
Journal of lnternational Money and Finance 1995 Volume 14 Number 5
733
Merton
Vasicek
CIR SR
~" ~
~ ~"
~
t~
CEV
C1
Unrestricted
C1 C2 C3
C1 C2 C3
C1 C2 C3
C3
C2
C1
C2 C3
Stat
Model
~
.-d
0.003 0.032 0.228
0.002 0.000 0.000
0.000 0.000 0.000
0.001 0.053 0.224
0.002 0.052 0.221
Au
0.019 0.009 0.151
0.026 0.000 0.000
0.000 0.000 0.000
0.000 0.033 0.154
0.014 0.023 0.149
Be
0.000 0.009 - 0.123
0.003 0.000 0.000
0.000 0.000 0.000
0.000 0.002 0.134
0.013 0.016 0.132
Ca
0.001 0.001 0.230
0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.030 0.244
0.017 0.054 0.223
Fr
0.016 0.009 0.150
0.030 0.000 0.000
0.000 0.000 0.000
0.000 0.052 0.157
0.009 0.023 0.150
Ge
0.007 0.004 0.269
0.002 0.000 0.000
0.000 0.000 0.000
0.000 0.021 0.350
0.029 0.078 0.384
Ho
TABLE 7. Summary statistics of models,
0.005 0.010 0.049
0.005 0.000 0.000
0.000 0.000 0.000
0.000 0.003 0.053
0.005 0.002 0.055
It
0.000 0.010 0.126
0.003 0.000 0.000
0.000 0.000 0.000
0.001 0.018 0.121
0.000 0.016 0.123
Ja
0.027 0.006 0.014
0.033 0.000 0.000
0.000 0.000 0.000
0.001 0.002 0.022
0.033 0.000 0.034
Sw
0.004 0.017 0.033
0.012 0.000 0.000
0.000 0.000 0.000
0.000 0.003 0.043
0.010 0.001 0.044
UK
0.011 0.001 0.418
0.020 0.000 0.000
0.000 0.000 0.000
0.001 0.155 0.419
0.018 0.201 0.389
US
~
t~
~" ~"
0.000 0.135 0.201
0.002 0.081 0.205
0.001 0.080 0.206
0.000 0.099 0.222
Au
0.000 0.097 0.148
0.010 0.042 0.145
0.000 0.038 0.153
0.000 0.040 0.156
Be
0.000 0.040 - 0.086
0.002 0.022 -0.110
0.001 0.026 -0.106
0.000 0.025 - 0.100
Ca
0.000 0.032 0.241
0.005 0.010 0.232
0.000 0.009 0.242
0.000 0.008 0.239
Fr
0.000 0.133 0.137
0.003 0.055 0.147
0.000 0.057 0.156
0.000 0.053 0.153
Ge
0.000 0.052 0.414
0.015 0.022 0.324
0.000 0.021 0.350
0.000 0.018 0.343
Ho
0.000 0.050 0.024
0.003 0.032 0.034
0.001 0.030 0.033
0.000 0.029 0.038
It
0.000 0.052 0.099
0.001 0.038 0.111
0.001 0.038 0.111
0.000 0.039 0.114
Ja
0.000 0.003 0.000
0.022 0.008 -0.001
0.003 0.006 -0.003
0.000 0.003 0.057
Sw
0.000 0.087 0.015
0.000 0.041 0.014
0.001 0.042 0.015
0.000 0.056 0.027
UK
0.000 0.075 0.420
0.001 0.012 0.435
0.001 0.012 0.434
0.000 0.012 0.429
US
squared residuals and the fitted conditional variance,
Notes: C l and C 2 are, respectively, the coefficient of determination of the mean and variance equations of the model. C 3 is the correlation coefficient between the
C3
C1 C2
C3
C2
C1
CIR VR
GBM
,~
C3
C1 C2
C1 C2 C3
Dothan
~ ~"
Stat
(Continued).
BS
Model
~
~n
TABLE 7
~'
g"
~'
International evidence on interest rates: Y K Tse
BS model is rejected while there is little to choose between the Vasicek and the CIR SR models. In summary, there is no single model that describes satisfactorily the stochastic behavior of interest rates for all the countries considered. If we classify the countries according to the size of ~/, France, Holland and the USA belong to the group of high elasticity. The estimated elasticities for these countries are larger than 1.5. In comparison, Australia, Belgium, Germany and Japan belong to the group of medium elasticity, while Canada, Italy, Switzerland and the UK belong to the group of low elasticity. For the last group of countries, ~/ is not statistically different from zero. To further assess the relative performance of the alternative models we compute the coefficient of determination of the mean and the variance equations, denoted by C 1 and C2, respectively. 14 In addition, we also report the simple correlation coefficient between the squared estimated residuals and the fitted conditional variance. This statistic is denoted by C 3. The results are reported in Table 7. It can be seen that by the C 2 criterion, the CIR VR model, which has the largest specified value of 31 (1.5), ranks the highest for most of the countries. The notable exceptions are France, Holland and the USA, for which cases ~/ are larger than 1.5. In comparison, C 3 favors models for which 3, is not specified, namely, the unrestricted model and the CEV model. By the C 3 criterion, these two models rank the highest for five countries. Also, in the group of countries with moderate ~,, the CIR SR model ranks very high by the C 3 criterion. Overall, the correlation between the estimated volatility and the squared residual is closely aligned to the appropriate value of the elasticity parameter. IV. Conclusions
Our analysis on the short-term interest rates of eleven countries shows that interest rate changes are leptokurtic. The unconditional variance is highest for the USA and lowest for Japan. As non-normality is evident we avoid using MLE. Following CKLS, we consider eight stochastic models of interest rates, all of which are nested within a discretized approximation to a differential equation. The GMM approach is used in preference to the MLE. The empirical results show that no single model can satisfactorily describe the stochastic structures of interest rates for all countries. The conditional volatility of interest rates can be very sensitive to the level of interest rates, as in the cases of France, Holland and the USA. On the other hand, Canada, Italy, Switzerland and the UK are the countries where the elasticity of variance is low. The Vasicek model may be preferred for these countries. For Australia, Belgium, Germany and Japan, the elasticity of variance is moderate. For these countries, there is no clear-cut statistical evidence for the choice between the CIR SR model and the BS model. Notes 1. A partial listing of analytic solutions of interest rate contingent claims includes those by Vasicek (1977), Cox et al. (1985), Jamshidian (1989), Amin and Jarrow (1991), Chen 736
Journal of lntemational Money and Finance 1995 Volume 14 Number 5
International evidence on interest rates: Y K Tse 2. 3.
4. 5. 6.
7.
8. 9. 10. 11.
12. 13. 14.
and Scott (1992) and Chen (1992). Parameters in equations (1) and (2) are related as follows: a = a ' A t and /3 = /3'At, where At is the sampling interval. One could also argue that the models admitted by equation (5) (i.e. the Vasicek, CIR SR, BS, Dothan and CIR VR models) are serious contenders for the purpose of evaluating interest rate contingent claims, while those that are excluded (i.e. the CEV, GBM and Merton models) may be rejected on grounds of unrealistic implications. The methodology described in this paper can be modified to use equation (5) as the unrestricted model. Whether imposing more structure on the basic model may lead to different conclusions is a topic for future research. In the actual calibration, S (0) was replaced by its sample analogue. The application of this test statistic can be extended to pairwise nesting. The use of ft*(O) has the advantage that the corresponding function g(O) is the score function if the residuals are normal. Thus, under normality the G M M estimates and the ML estimates of the unrestricted model are the same. To the extent that the MLE is optimal when the residuals are correctly specified, the GMM estimates based on ft* (0) do not lose optimality when errors are normal. On the other hand, the variance estimates of 0 using equation (12) are consistent even if the residuals are non-normal. As shown in Table 3 below, however, et is significantly non-normal. When f * ( O ) was used for the estimation, results were found to be rather unstable, as demonstrated by the large estimated standard errors. Thus, in this paper we only report results based on ft(O). While data for Japan were available from 1974 we decided to use data starting from April 1983, when banks in Japan were allowed to participate in over-the-counter sales of newly issued bonds. Although CKLS (Chan et al. 1992b) reported that no structural change was detected when Japanese data from February 1980 to September 1989 were examined, casual data perusal showed that for the 25 months prior to April 1984 the interest rates in Japan remained unchanged. As the interest rate observations do not form a random sample, formal significance tests on skewness and kurtosis are not performed. Variability here refers to the variation of the unconditional distribution. This should be distinguished from the conditional volatility of the interest rate changes. The anomaly that ~ of the Canada equation is negative is not disturbing as ~ is not statistically significantly different from zero. The CKLS result, however, is based on MLE. Using the CKLS sampling period ( 8 0 / 2 - 8 9 / 9 ) , with data collected from the same source as the data used in this paper, the MLE of 3' is 2.38. In comparison, the GMM estimate is 1.57. Thus, the discrepancy in the results could be attributed to the differences in the method of estimation and the sampling period. On the other hand, the closeness of our MLE to that of CKLS shows that the difference in data source does not appear to cause the discrepancy. It should be noted, however, that the magnitudes of both & and /3 are quite small. Also, the t-ratios of these parameters are statistically insignificant. As noted above, the speed-of-adjustment coefficient for Japan is practically zero. These statistics were suggested by CKLS as additional measures of the model performance. These measures, however, may not be appropriate if the estimation is not based on least squares. While the G M M estimation of the mean equation is a (weighted) least squares estimation, the estimation of the variance equation is not. Indeed, C 2 is biased in favor of a model that gives rise to large variations in the estimated conditional variance, irrespective of its performance as a forecast for the conditional variance. Thus, C 2 would favor a model with large elasticity of variance, such as the CIR VR model.
References
AMIN, K.I. A N D R. A. JARROW,'Pricing foreign currency options under stochastic interest rates,' Journal o f International Money and Finance, September 1991, 10: 310-329. Journal of International Money and Finance 1995 Volume 14 Number 5
737
Internationalevidence on interestrates: Y K Tse BP,Er~AN, M.J.M. AND E. S. SCHWARTZ,'A continuous time approach to the pricing of bonds,' Journal of Banking and Finance, 1979, 3: 133-155. CHAN, K. C., G. A. KAROVLI, F. A. LONGSTAFF AND A. B. SANDERS, 'An empirical comparison of alternative models of the short-term-interest rate,' Journal of Finance, July 1992, 47:1209-1227 (1992a). CHAN, K. C., G. A. KAROVLt, F. A. LONGSTAFFAND A. B. SANDERS, 'The volatility of Japanese interest rates: A comparison of alternative term structure models,' Pacific-Basin Capital Markets Research, Vol III, 1992, 119-136 (1992b). CHEN, r. r., 'Exact solutions for futures and European futures options on pure discount bonds,' Journal of Financial and Quantitative Analysis, March 1992, 27: 97-107. CHEN, R. R. AND L. Scow, 'Pricing interest rate options in a two-factor Cox-Ingersoll-Ross model of term structure,' The Review of Financial Studies, 1992, 5: 613-636. CONSTANTINIDES, G. M., 'A theory of the nominal term structure of interest rates,' The Review of Financial Studies, 5: 531-552. Cox, J. C., J. E. INGERSOLLAND S. A. Ross, 'An analysis of variable rate loan contracts,' Journal of Finance, May 1980, 35: 389-403. Cox, J. C., J. E. INGERSOLLAND S. A. ROSS, 'A theory of the term structure of interest rates,' Econometrica, March 1985, 53: 385-407. DOTHAN, U. L., 'On the term structure of interest rates,' Journal of Financial Economics, March 1978, 6: 59-69. HARRISON, J. M. AND S. PLISKA, 'Martingales and stochastic integrals in the theory of continuous trading,' Stochastic Processes and Their Applications, 1981, 11: 215-260. HEATH, D., R. JARROWAND A. MORTON, 'Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation,'Econometrica, January 1992, 60: 77-105. Ho, T. S. AND S. LEE, 'Term structure movements and pricing interest rate contingent claims,' Journal of Finance, December 1986, 41: 1011-1028. JAMSHIDIAN, F., 'An exact bond option formula,' Journal of Finance, March 1989, 44: 205-209. Lo, A. W., Maximum likelihood estimation of generalized Ito processes with discretely sampled data,' Econometric Theory, 1988, 4: 231-247. MERTON, R. C., 'Theory of rational option pricing,' Bell Journal of Economics and Management Science, 1973, 4: 141-183. TSE, Y. K., 'MLE of some continuous time financial models: Some Monte Carlo results,' Mathematics and Computers in Simulation, 1992, 33: 575-580. TSE, Y. IC, 'Maximum likelihood estimation of Cox-Ingersoll-Ross and Brennan-Schwartz models of interest rate movement,' Hong Kong Economic Papers, forthcoming, 1994. VASlCEK, O., 'An equilibrium characterization of the term structure,' Journal of Financial Economics, November 1977, 5: 177-188.
738
Journal of lnternationaIMoneyand Finance1995 Volume 14 Number5