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Forecasting U.K. and U.S. interest rates using continuous time term structure models
Forecasting U.K. and U.S. Interest Rates Using Continuous Time Term Structure Models
S. L. BYERS AND K. B. NOWMAN
In this paper we compare the forecasting performance of different one factor interest rate models commonly used in the financial markets. In particular we estimate the general CKLS model and the special cases of Merton, Vasicek, CIRSR, Dothan, GBM, Brennan-Schwartz, CIRVR, and CEV models using the approach of Nowman (1997). The models are estimated using weekly Euro-currency data for the U.K. and U.S. over a range of maturities. We find the forecasting performance varies across models and markets.
JEL classification: E43; E47 Keywords: Gaussian Estimation; Forecasting; Continuous Time
I.
INTRODUCTION
In recent years extensive empirical testing of different single factor continuous time models of the term structure of interest rates has been carried out based on the seminal paper by Chan, Karolyi, Longstaff, and Sanders (1992a, hereafter CKLS). Recent examples include Chan, Karolyi, Longstaff, and Sanders (1992b), Tse (1995), Dahlquist (1996), Hiraki and Takezawa (1996), Shoji and Ozaki (1996), and N o w m a n (1997, 1998a). In comparison little work has been done on comparing the forecasting performance of the CKLS model and the special cases o f Merton (1973), Vasicek (1977), Cox, Ingersoll and Ross (1985, hereafter CIRSR), Dothan (1978), Geometric Brownian Motion (GBM) process of Black and Scholes (1973), Brennan and Schwartz (1980, hereafter BS), Cox, Ingersoll, and Ross (1980, hereafter CIRVR), and the Constant Elasticity of Variance (CEV) process used in Cox (1975) and Cox and Ross (1976). An important component of these term structure models is what relationship exists between the level of rates and the volatilty of rates. C K L S found using monthly U.S. T-bill data that term structure models with volatilities more highly sensitive to the level of interest rates have a better fit to the data. In contrast, N o w m a n (1997) found the opposite using monthly U.K. interbank data. Generally the relationship varies across currencies, see for example, Tse (1995), Dahlquist (1996), and N o w m a n (1998a). In this paper we estimate the continuous time term structure models considered by C K L S utilizing the discrete model used in N o w m a n (1997) and the Gaussian estimation methods of Bergstrom (1983, 1984, 1985, S. L. Byers and K. B. Nowman • Department of Investment, Risk Management and Insurance, City University Business School, Barbican Centre, Frobisher Cresent, London, U.K., EC2Y 8HB; Phone: 0171-477-8698; Fax: 0171-477-8885. International Review of Financial Analysis, Vol. 7, No. 3, 1998, pp. 191-206 Copyright © 1998 by JAI PRESS Inc., All rights of reproduction in any form reserved.
ISSN: 1057-5219
192
INTERNATIONAL REVIEW OF FINANCIAL ANALYSIS / Vol. 7(3)
1986). (See Brennan & Schwartz, 1979, who also used an exact discrete model to estimate a term structure model). More recently Ball and Torous (1998) have used monthly and weekly sampled one month rates within a augmented CKLS model with stochastic volatility. Ball and Torous (1998) find that weekly sampled data has the advantage of giving more reliable estimates of the relationship between the volatility of rates and the level of rates. We therefore use weekly sampled data for the U.K. and U.S. Eurocurrency rates. Compared to the previous studies (except Hiraki & Takezawa, 1996) we also use a wider range of maturities including the one week and one, three, six and twelve month rates. Comparisons of the forecasting performance of continuous time models that have been estimated using the Gaussian estimation methods and the forecastng algorithm of Bergstrom (1989) have been undertaken recently by Bergstrom and Chambers (1990), Bergstrom, Nowman, and Wymer (1992), Chambers (1991 a, 1993), Chambers and Nowman (1997) and Nowman (1996, 1998b). We find the forecasting performance varies across models and markets. The outline of the remainder of the paper is as follows. In Section II the different interest rate models used in this study are presented and the Gaussian estimation econometric methodology discussed. Section III describes the data used in the study and Section IV presents the empirical and forecasting results. Section V offers some conclusions.
II.
CONTINUOUS T I M E M O D E L S
The CKLS general specification is given by Equation 1 below. The equation allows the conditional mean and variance to depend on the level r. d r (t) = {o~ + ~ r ( t ) } d t + t~r r ( t ) d Z
(t _>0)
(1)
where {r(t), t > 0} is a real continuous time random process, tx, I], y, and o are unknown structural parameters and d Z is a Wiener process. From the CKLS model we can obtain the important special cases: Merton (~ = 0, y = 0), Vasicek (y = 0), CIR (y = 0.5), Dothan (c~ = 0, [~ = 0, y = 1), GBM (tx = 0, y = 1), BS (y= 1), CIRVR ((x = 0, ~ = 0, y = 3/2), and CEV (c~ = 0). To estimate the parameters of continuous time models from discrete data we have to use a discrete model derived from the solution of the continuous time model. In a series of papers Bergstrom (1983, 1985, 1986, 1990, 1997) (see also, Chambers, 1991b; Gandolfo, 1993; Nowman, 1991, 1993) has been concerned with obtaining asymptotically efficient estimates using an exact discrete model which takes account of the exact restrictions on the distribution of the discrete data implied by the continuous time model. Bergstrom (1983) provided a detailed analysis of using exact discrete models in obtaining the exact Gaussian estimates (estimates which would be exact maximum likelihood estimates if the innovations were Brownian motion) of the parameters of the continuous time model. In Bergstrom (1985, 1986, 1997) the methods where extended to include nonstationary systems, exogenous variables and stochastic trends. The new methodology has recently been surveyed by Bergstrom (1996) and applied in macroeconomics by Bergstrom, Nowman, and Wymer (1992) (see also, Nowman, 1996, 1998b) to a second order continuous time macroeconometric model of the United Kingdom. In Nowman (1997) a discrete model of Bergstrom (1984, Theorem 2) was modified for hetroscedasticity to estimate the CKLS model. The discrete model used by Nowman (1997) is given by Equation 2 below
Forecasting U.K. and U.S. Interest Rates
193
r(t) = e ~ r ( t - 1) + "~(e I] - 1 ) + tit (t = 1, 2 . . . . .
w h e r e tit (t = 1 , 2 . . . . . T) s a t i s f i e d t h e c o n d i t i o n s g i v e n in N o w m a n
T)
(2)
(1997). Following Nowman
(1997) we let L (0) be minus twice the logarithm of the Gaussian likelihood function where
Table 1 S u m m a r y Statistics: U K and U S E u r o e u r r e n c y Data 1986--1997 Variables
T
Mean
Standard Deviation
Pl
92
P3
P4
P5
P6
627 626
0.0917 -0.0001
0.0327 0.0105
0.95 -0.4 7
0.94 -0.0 1
0.94 0.01
0.94 -0.04
0.94 0.08
0.93 -0.0 1
627 626
0.0925 -0.0001
0.0319 0.0026
0.99 -0.1 7
0.99 0.01
0.99 0.11
0.98 -0.00
0.98 0.03
0.98 0.10
627 626
0.0930 -0.0001
0.0314 0.0020
0.99 0.12
0.99 0.07
0.99 0.09
0.98 0.06
0.98 0.06
0.98 0.05
627 626
0.0931 -0.0001
0.0306 0.0022
0.99 0.09
0.99 0.09
0.99 0.08
0.98 0.06
0.98 0.04
0.97 -0.0 0
627 626
0.0937 -0.0001
0.0290 0.0025
0.99 0.05
0.99 0.05
0.98 0.08
0.98 0.05
0.97 0.02
0.96 --0.0 0
627 626
0.0601 -0.0000
0.0185 0.0049
0.96 -0.3 9
0.95 ~).0 9
0.95 0.04
0.95 -0.02
0.94 0.03
0.94 -0.0 3
627 626
0.0607 -0.0000
0.0181 0.0018
0.99 -0.0 8
0.99 -0.04'
0.98 0.11
0.98 -0.11
0.97 -0.04
0.97 -0.02
627 626
0.0615 -0.0000
0.0179 0.0014
0.99 0.03
9.99 0.01
0.99 0.04
0.98 0.05
0.97 0.05
0.97 -0.0 4
627 626
0.0626 -0.0000
0.0178 0.0016
0.99 -0.0 2
0.99 0.03
0.98 0.04
0.98 0.07
0.97 0.06
0.97 -0.0 2
627 626
0,0649 -0,0000
0.0175 0.0017
0.99 0.02
0.99 0.07
0.98 0.05
0.97 0.08
0.97 0.07
0.96 -0.0 3
UK 7D r(t) dr(t)
UK 1M r(t) dr(t)
UK 3M r(t) dr(t)
UK 6M r(t) dr(t)
UK 12M r(t) dr(t) US 7D r(t) dr(t) US 1M r(t) dr(t) US 3M r(t) dr(t) US 6M r(t) dr(t) US 12M r(t) dr(t)
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INTERNATIONAL REVIEW OF FINANCIAL ANALYSIS / Vol. 7(3)
Table 2
Estimates of UK Short Term Interest Rate Models: 7-Day Model
O~
CKLS
0.0008 (0.0007)
Merton
-0.0001 (0.0004)
Vasicek
0.0049 (0.0013)
CIRSR
0.0020 (0.0010)
Dothan
0.0
GBM
0.0
BS
0.0010 (0.0008)
CIRVR
0,0
CEV
0,0
~
¢32
-0.0076 (0.0107) 0.0
Log Likelihood
"~
0.3713 (0.1828)
1.8090 (0.0999)
)~2 Test
df
2780.2089
0.0001 (<0.0001 )
0.0
2 5 3 9 . 3 4 3 7 481.730 4
2
~).0537 (0.01341
0.0001 (<0.000l)
0.0
2547.6922
465.033 4
1
-0.0224 (0.0118)
0.0009 (<0.0(X)1)
0.5
2677.1458
206.126 2
1
0.0078 (0.0004)
1.0
2745.0624
70.2930
3
0.0020 (0.0035)
0.0078 (0.0004)
1.0
2745.2317
69.9544
2
-0.0104 (0.01101
0.0079 (0.0004)
1.0
2745.9684
68.4810
1
0.0825 ({I.0046)
1.5
2774.5189
11.3800
3
2779.6383
1.1412
l
0.0
0.0 0.0041 (0.003 l)
0.3583 (0.1763)
1.8{}38 (0.0997)
the c o m p l e t e v e c t o r of parameters is 0 = [~, 13, 7, ~2]. T h e n the Gaussian estimates are obtained f r o m E q u a t i o n 3 (defined in N o w m a n , 1997) using s o m e numerical o p t i m i z a t i o n routine.
l
T L(0) =
r ( t ) - efJr(t - 1 ) - ~ ( e f3 - 1 )
21°gmtt +
2 mtt
t=l
III.
t21
(3)
DATA
T h e short-term interest rates used in this study are E u r o - c u r r e n c y rates on U.K. and U.S. currencies (middle rate) obtained from D a t a s t r e a m TM. W e use w e e k l y data on a W e d e n s d a y in the e m p i r i c a l analysis to a v o i d missing observations and w e e k - d a y effects. The data c o v e r the period f r o m January 1986 to D e c e m b e r 1997 g i v i n g a total of 627 observations. The maturity s p e c t r u m of the data c o v e r s the seven day and one, three, six and t w e l v e m o n t h rates. Table 1 reports the descriptive statistics. The table displays the means, standard deviations and first six autocorrelations o f the rates and changes in the rates. For the U.K. w e h a v e the f o l l o w i n g a v e r a g e levels and standard deviations o f the rates: seven day rate m e a n is 9.17% with standard deviation
195
Forecasting U.K. and U.S. Interest Rates
Table 3 E s t i m a t e s of U K Short T e r m Interest Rate Models: 1-Month Model
~
~
CKLS
0.0003 (0.0002)
Merton
-0.0001 (0.0001)
Vasicek
-0.0037 (0.0034) 0.0
132 0.0055 (0.0026)
"~
1.4316 (0.0954)
Log Likelihood
X2 Test
df
3506.5602
<0.0001 (<0.0001 )
0.0
3406.6791 199.7622
2
0.0003 (0.0003)
-0.0037 <0.0001 (0.0033) (<0.0001)
0.0
3407.3061 198.5082
1
CIRSR
0.0003 (0.0003)
-0.0036 0.0001 (0.0032) (<0.0001)
0.5
3462.0745 88.9714
1
Dothan
0.0
0.0007 (.a).OOOl)
1.0
3495.9327 21.2550
3
GBM
0.0
-0.0004 (0.0011)
0.0007 (<0.0001)
1.0
3 4 9 5 . 9 9 5 2 21.1300
2
BS
0.0003 (0.0002)
-0.0036 0.0007 (0.0033) (<0.0001)
1.0
3496.5595 20.0014
1
CIRVR
0.0
1.5
3505.6510
1.8184
3
CEV
0.0
3505.9332
1.2540
1
0.0
0.0 -0.0001 (0.0010)
0.0077 (0.00O4) 0.0055 (0.0004)
1.4298 (0.0947)
3.27%; one month rate mean is 9.25% with standard deviation 3.19%; three month rate mean is 9.30% with standard deviation 3.14%; six month rate mean is 9.31% with standard deviation 3.06%; twelve month rate mean is 9.37% with standard deviation 2.90%. For the U.S. we have the following average levels and standard deviations of the rates: seven day rate mean is 6.01% with standard deviation 1.85%; one month rate mean is 6.07% with standard deviation 1.81%; three month rate mean is 6.15% with standard deviation 1.79%; six month rate mean is 6.26% with standard deviation 1.78%; twelve month rate mean is 6.49% with standard deviation 1.75 %.
IV.
E M P I R I C A L AND F O R E C A S T I N G RESULTS
We now present the empirical results of the models for the U.K. and U.S. interest rates. In each of the Tables 2-11 the Gaussian estimates (standard errors in brackets) and maximized log likelihoods of the different continuous time models of the short-term interest rate are presented. Following Chan, Karolyi, Longstaff, and Sanders (1992b) and Nowman (1997) the explanatory power compared to the unrestricted model is compared using the maximized Gaussian log likelihood function values.
U.K. Results For the U.K. seven day rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV model performs the best followed by the CIRVR, BS, GBM,
196
I N T E R N A T I O N A L R E V I E W O F F I N A N C I A L A N A L Y S I S / Vol. 7(3)
Table 4 Estimates of UK Short Term Interest Rate Models: 3-Month Model
0¢
[3
o2
"[
CKLS
0.0002 (0.0002)
-0.0031 (0.0028)
0.0024 (0.0012)
1.3578 (0.1030)
Merton
-0.0OOl
0.0
(0.0001)
<0.0001
Log Likelihood
)~2 Test
df
3648.5757
0.0
3567.7554
161.6406
2
(<0.0001)
Vasicek
0.0002 (0.0002)
-0.0024 (0.0026)
<0.0001 (<0.0001 )
0.0
3568.1943
160.7628
1
CIRSR
0.0002 (0.0002)
-0.0027 (0.0025)
<0.0001 (<0.0001)
0.5
3615.5955
65.9604
1
0.0004
1.0
3642.2430
12.6654
3
Dothan
0.0
0.0
(<0.OO01 ) GBM BS
0.0 0.0002 (0.0002)
CIRVR
0.0
CEV
0.0
-0.0005 (0.0008)
0.0004 (
1.0
3642.4311
12.2892
2
-0.0030 (0.0026)
0.0004 (<0.OO01)
1.0
3642.9327
11.2860
l
0.0048 (0.0O03)
1.5
3647.1649
2.8216
3
3648.2308
0.6898
1
0.0 -0.0003 (0.OO08)
0.0023 (0.OO 12)
1.3531 (0.1030)
Table 5 Estimates of UK Short Term Interest Rate Models: 6-Month Log
Z2
Model
O~
~
02
~t
Likelihood
Test
CKLS
0.0003 (0.0002)
-0.0036 (0.0030)
0.0018 (0.0009)
1.2656 (0.0997)
3595.1866
Merton
-0.OO01 (0.0002)
Vasicek
0.0002 (0.0003)
CIRSR
0.0002 (0.0002)
Dothan
0.0
0.0
df
<0.OO01 (<0.0001 )
0.0
3520.1661
150.0410
2
-0.0029 (0.0029)
<0.0001 (<0.0001)
0.0
3520.6677
149.0378
1
-0.0031 (0.0029)
0.0001 (<0.0001 )
0.5
3567.0343
56.3046
1
0.0005
1.0
3591.2052
7.9628
3
0.0
(<0.0001 ) GBM BS
0.0 0.0003 (0.0002)
CIRVR
0.0
CEV
0.0
-0.0004 (0.0009)
0.0005 (<0.0001)
1.0
3591.3306
7.7120
2
-0.0034 (0.0029)
0.0005 (<0.0001)
1.0
3591.9262
6.5208
1
0.0/157 (0.00O3)
1.5
3591.8008
6.7716
3
3594.7164
0.9404
1
0.0 -0.0003 (0.0009)
0.0018 (0.0008)
1.2606 (0.0998)
197
Forecasting U.K. and U.S. Interest Rates
Table 6 Estimates of UK Short Term Interest Rate Models: 12-Month Model
OL
~
CKLS
0.0003 (0.0003)
Merton
-0.0001 (o.oool)
Vasicek
0.0003 (0.0003)
CIRSR
0.0003 (0.0003)
Dothan
0.0
GBM
0.0
-0.0004 (0.0009)
BS
0.0003 (0.0003)
CIRVR
0.0
CEV
0.0
-0.0042 (0.0034)
~2
Log
~2
Likelihood
Test
df
1.3500 (0.1001)
3533.2704
0.0
3450.7572 165.0264
2
-0.0039 <0.0001 (0.0034) (<0.0001)
0.0
3451.3842 163.7724
1
-0.0040 0.0001 (0.0033) (<0.0001)
0.5
3 4 9 9 . 2 8 7 0 67.9668
1
0.0006 (
1.0
3526.4988
13.5432
3
0.0006 (<0.0001)
1.0
3526.5928
13.3552
2
-0.0041 0.0006 (0.0033) (<0.0001)
1.0
3527.2825
11.9758
1
1.5
3531.2953
3.9502
3
3532.4553
1.6302
1
0.0
0.0
0.0 -0.0001 (0.0009)
0.0032 (0.0015)
'~
<0.0001 (<0.0001)
0.0066 (0.0OO4) 0.0031 (0.0015)
1.3472 (0.1009)
Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate of y is 1.8090 and highly significant. The unrestricted model also implies only weak evidence of mean reversion in the short rate; the parameter [3 is insignificant. Based on the Z 2 likelihood ratio test under the null hypothesis that the nested models restrictions are valid the results imply that we can reject all models except the CEV model at the five percent level. For the U.K. one month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model the CEV and C I R V R models perform the best followed by the BS, GBM, Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate of y is 1.4316 and significant. This compares to Dahlquist's (1996) estimate of 0.1562 using the monthly one month Eurocurrency rate; N o w m a n ' s (1997) estimate using the monthly one month interbank rate of 0.2898 and N o w m a n and Sorwar's (in press) estimate of 1.0461 using the monthly one month Eurocurrency rate. In Ball and Torous (1998) they report the stochastic volatility model estimates of 0.398 using weekly one month Eurocurrency rate over the period 1985-1995 and using monthly and weekly interbank data estimates of 0.261 and 0.486 respectively. Based on the )~2 likelihood ratio test implies that we can reject all models except the CIRVR and CEV models. For the U.K. three month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV and CIRVR models perform the best followed by the BS, GBM, Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model,the Gaussian estimate o f y i s 1.3578 and significant. This estimate compares to the estimate of Tse (1995) on monthly three month money market data of 0.1132 and N o w m a n and Sorwar' s (1998) estimate of 1.3564 using the monthly three month Eurocurrency rate. Based on the ~2 likelihood ratio test implies that we can reject all models except the CIRVR and CEV models.
198
I N T E R N A T I O N A L R E V I E W O F F I N A N C I A L A N A L Y S I S / Vol. 7(3)
Table 7 Estimates of US Short Term Interest Rate Models: 7-Day Log
X2
Model
~
~
~2
It
Likelihood
Test
df
CKLS
0.0001 (0.0003)
0.0003 (0.0078)
0.8746 (0.4730)
1.9458 (0.0940)
3219.6136
Merton
-0.0001 (0.0002)
Vasicek
0.0027 (0.0007)
CIRSR
0.0014 (0.0005)
Dothan
0.0
GBM
0.0
BS
0.0006 (0.0004)
CIRVR
0.0
CEV
0.0
0.0
<0.0001 (<0.0001 )
0.0
2993.3137
452.5998
2
-0.0467 (0.01131
<0.0001 (<0.000l)
0.0
3002.0321
435.1630
1
-0.0242 (0.0092)
0.00113 (<0.00011
0.5
3096.0695
247.0880
1
0.0047 (0.001)3)
1.0
3164.0928
111.0416
3
0.0009 (0.0027)
0.0047 (0.0/)/)3)
1.0
3164.1555
110.9162
2
-0.0111 (0.0070)
0.0047 (0.00113)
1.0
3165.3781
108.4710
1
0.0712 (0.O040)
1.5
3206.3526
26.5220
3
3219.4255
0.3762
1
0.0
0.0 0.0031 (0.0022)
0.8930 (0.4459)
1.9498 (0.0867)
Table 8 Estimates of US Short Term Interest Rate Models: 1-Month Log
X2
Model
O~
~
~2
It
Likelihood
Test
CKLS
0.0004 (0.0002)
-0.0082 (0.0038)
0.0008 (0.0003)
0.9544 (0.0614)
3615.0939
Merton
-0.0001 (0.0001)
Vasicek
0.0009 (0.0003)
CIRSR
0.0006 (0.0002)
Dothan
0.0
0.0
df
<0.0001 (<0.0001)
0.0
3490.1014
249.9850
2
-0.0160 (0.0050)
<0.0001 (<0.0001 )
0.0
3495.0234
240.1410
1
-0.0107 (0.0043)
0.0001 (<0.0001)
0.5
3583.4931
63.2016
1
0.0011
1.0
3610.6108
8.9662
3
0.0
(<0.0001 ) GBM BS
0.0 0.0004 (0.0002)
CIRVR
0.0
CEV
0.0
-0.0005 (0.0013)
0.0011 (<0.0001)
1.0
3610.7049
8.7780
2
-0.0081 (0.0039)
0.0011 (<0.0001 )
1.0
3612.7113
4.7652
1
0.0213 (0.0012)
1.5
3573.4297
83.3284
3
3610.7989
8.5900
1
0.0 -0.0006 (0.0013)
0.0009 (0.0003)
0.9714 (0.0606)
199
Forecasting U.K. and U.S. Interest Rates
Table 9 E s t i m a t e s o f US S h o r t T e r m I n t e r e s t R a t e M o d e l s : 3 - M o n t h Model
(X
CKLS
0.0003 (0.0001)
Merton
-0.0001 (0.0001)
Vasicek
~
Log
X2
Likelihood
Test
df
3761.3416
<0.0001 (<0.0001)
0.0
3552.2058 418.2716
2
0.0008 (0.0003)
-0.0146 <0.0001 (0.0048) (<0.0001)
0.0
3557.1277 408.4278
1
CIRSR
0.0005 (0.0002)
-0.0093 <0.0001 (0.0038) (<0.0001)
0.5
3674.7843 173.1146
1
Dothan
0.0
1.0
3742.7511
37.1810
3
GBM
0.0
-0.0008
1.0
3 7 4 3 . 0 3 3 2 36.6168
2
BS
0.0003 (0.0002)
(0.0010) (<0.0001) -0.0066 0.0007 (0.0032) (<0.0001)
1.0
3744.7261
33.2310
1
1.5
3755.4478
11.7876
3
6.7088
1
0.0
CEV
0.0
0.0
0.0
0.0 -0.0003 (0.0010)
0.0044 (0.0017)
~t 1.3367 (0.0654)
CIRVR
-0.0055 (0.0031)
(y2
0.0007 (<0.0001) 0.0007
0.0113 (0.0006) 0.0049 (0.0018)
1.3532 (0.0649)
3757.9872
For the U.K. six month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV model performs the best followed by the BS, CIRVR, G B M , Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate of ~' is 1.2656 and significant. Based on the ~2 likelihood ratio test implies that we can reject all models except the C I R V R and CEV models. For the U.K. twelve month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV model performs the best followed by the CIRVR, BS, GBM, Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate o f 7 is 1.3500 and significant. Based on the ~2 likelihood ratio test implies that we can reject all models except the CIRVR and CEV models.
U.S. Results For the U.S. seven day rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV model performs the best followed by the CIRVR, BS, GBM, Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate of 7 is 1.9458 and significant. The unrestricted model also implies no evidence of mean reversion. Based on the Z 2 likelihood ratio test implies that we can reject all models except the C E V model. For the U.S. one month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the BS, CEV, G B M , and Dothan models perform the best followed by the CIRSR, CIRVR, Vasicek and Merton models. Regarding the unre-
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I N T E R N A T I O N A L R E V I E W O F F I N A N C I A L A N A L Y S I S / Vol. 7(3)
Table 10 Estimates of US Short Term Interest Rate Models: 6-Month Model
(X
~
•2
~t
CKLS
0.0003 (0.0002)
-0.0062 (0.0035)
0.0044 (0.0017)
1.3096 (0.0702)
Merton
-0.0001 (0.0001)
Vasicek
0.0009 (0.0003)
C1RSR
0.0005 (0.0002)
Dothan
0.0
GBM
0.0
BS
0.0004 (0.0002)
CIRVR
0.0
CEV
0.0
0.0
Log Likelihood
X2 Test
df
3699.2686
<0.0001 (
0.0
3520.3228
357.8916
2
-0.0153 (0.0049)
<0.0001 (<0.0001)
0.0
3525.1194
348.2984
1
-0.0101 (0.0042)
0.0001 (<0.0001)
0.5
3628.1355
142.2662
1
0./)008 (<0.0001)
1.0
3684.7222
29.0928
3
-0.0007 (0.00 11 )
0.0008 (<0.1)001)
1.0
3684.9103
28.7166
2
-0.0072 (0.00 11 )
0.00(/8 (<0.0001 )
1.0
3686.6973
25.1426
1
0.0131 10.0/)/)7)
1.5
3692.9673
12.6026
3
3696.0082
6.5208
1
0.0
0.0 -0.0002 10.0011 )
0.0(149 (0.00 191
1.3269 (0.0698)
Table 11 Estimates of US Short Term Interest Rate Models: 12-Month Log
Model
O~
~
~2
y
CKLS
0.0004 (0.0002)
-0.0079 (0.0039)
0.0020 (0.0008)
1.1577 10.0714)
Merton
-41.0001 (0.0001 )
Vasicek
0.0009 (0.0003)
CIRSR
0.0006 (0.0003)
Dothan
0.0
GBM
0.0
BS
0.0005 (0.0002)
CIRVR
0.0
CEV
0.0
0.0
Likelihood
X2
Test
df
3643.0581
<0.0001 (<0.0001 )
0.0
3504.0208
278.0746
2
-0.0156 (0.0051 )
<0.0001 (<0.0001 )
0.0
3508.5979
268.9204
1
-0.0109 (0.0044)
0.1101/1 (<0.0001)
0.5
3595.5315
95.0532
1
0.00119 (<0.0001 )
1.0
3635.7535
14.6092
3
-0.0006 (0.0012)
0.0009 (<0.00011
1.0
3635.9103
14.2956
2
-0.0084 (0.0040)
0.0009 (<0.0001 )
1.0
3637.9480
10.2202
1
0.0142 (0.0008)
1.5
3628.6057
28.9048
3
3639.0453
8.0256
I
0.0
0.0 -0.0004 10.0012)
0.0023 (0.00119)
1.1760 (0.0703)
201
Forecasting U.K. and U.S. Interest Rates
T a b l e 12
UK Summary Statistics for the Forecasts Model
CKLS
Merton
Vasicek
CIRSR
Dothan
GBM
BS
C1RVR
CEV
7D RMSE MAE ME U
0.0010 0.0009 0.0006 0.0072
0.0018 0.0015 --0.001 5 0.0125
0.0048 0.0042 0.0042 0.0318
0.001 0.001 0.001 0.011
0.0012 0.0010 -0.000 9 0.0085
0.0008 0.0006 -<.000 1 0.0053
0.0010 0.0009 0.0006 0.0070
0.0012 0.0010 -0.000 9 0.0085
0.0013 0.0011 0.0009 0.0091
1M RMSE MAE ME U
0.0008 0.0007 0.0006 0.0053
0.0007 0.0006 ~).000 1 0.0048
0.0008 0.0007 0.0006 0.0053
0.000 8 0.000 7 0.000 7 0.005 5
0.0007 0.0005 0.0005 0.0047
0.0006 0.0005 0.0003 0.0042
0.0008 0.0007 0.0007 0.0055
0.0007 0.0005 0.0005 0.0047
0.0007 0.0005 0.0005 0.0045
3M RMSE MAE ME U
0.0005 0.0005 0.0004 0.0036
0.0004 0.0003 <.0000 1 0.0026
0.0008 0.0007 0.0007 0.0055
0.000 7 0.000 6 0.000 6 0.004 6
0.0007 0.0006 0.0006 0.0047
0.0005 0.0005 0.0004 0.0034
0.0006 0.0005 0.0005 0.0038
0.0007 0.0006 0.0006 0.0047
0.0006 0.0005 0.0005 0.0039
6M RMSE MAE ME U
0.0014 0.0014 0.0014 0.0095
0.0008 0.0007 0.0006 0.0055
0.0012 0.0011 0.0011 0.0078
0.001 0.001 0.001 0.007
1 0 0 3
0.0013 0.0012 0.0012 0.0086
0.0011 0.0010 0.0010 0.0075
0.0015 0.0015 0.0015 0.0101
0.0013 0.0012 0.0012 0.0086
0.0012 0.0011 0.0011 0.0077
12M RMSE MAE ME U
0.0014 0.0012 0.0012 0.0091
0.0010 0.0008 0.0007 0.0069
0.0015 0.0013 0.0013 0.0098
0.001 4 0.001 3 0.001 3 0.009 6
0.0014 0.0013 0.0013 0.0097
0.0013 0.0011 0.0011 0.0087
0.0014 0.0012 0.0012 0.0093
0.0014 0.0013 0.0013 0.0097
0.0014 0.0013 0.0013 0.0094
6 3 3 2
stricted m o d e l , t h e G a u s s i a n e s t i m a t e o f T is 0 . 9 5 4 4 . T h i s c o m p a r e s to S h o j i a n d O z a k i ' s ( 1 9 9 6 ) e s t i m a t e o f 1.1473 for m o n t h l y o n e m o n t h T.bill rate a n d C K L S ' s e s t i m a t e o f 1.4999 u s i n g m o n t h l y o n e m o n t h T . b i l l rate. In N o w m a n ( 1 9 9 8 a ) u s i n g m o n t h l y o n e m o n t h E u r o c u r r e n c y rate a n e s t i m a t e o f 1.0519 u p to M a r c h 1995 is r e p o r t e d . B a l l a n d T o r o u s ( 1 9 9 8 ) r e p o r t a n e s t i m a t e o f 0 . 7 5 4 b a s e d o n w e e k l y o n e m o n t h E u r o c u r r e n c y rate o v e r the p e r i o d 1 9 8 5 - 1 9 9 5 . B a s e d o n the ~2 l i k e l i h o o d ratio test i m p l i e s t h a t w e c a n r e j e c t all m o d e l s . F o r the U.S. t h r e e m o n t h rate b a s e d o n the m a x i m i z e d G a u s s i a n l o g l i k e l i h o o d v a l u e s c o m p a r e d to the u n r e s t r i c t e d m o d e l , the C E V a n d C I R V R m o d e l s p e r f o r m t h e b e s t f o l l o w e d b y t h e B S , G B M , D o t h a n , C I R S R , V a s i c e k a n d M e r t o n m o d e l s . R e g a r d i n g the u n r e s t r i c t e d m o d e l , the G a u s s i a n e s t i m a t e o f T is 1.3367. T h i s c o m p a r e s to T s e ' s ( 1 9 9 5 ) e s t i m a t e for m o n t h l y t h r e e m o n t h m o n e y m a r k e t d a t a o f 1.7283. B a s e d o n the ~2 l i k e l i h o o d ratio test i m p l i e s t h a t w e c a n r e j e c t all m o d e l s . F o r the U.S. six m o n t h rate b a s e d o n the m a x i m i z e d G a u s s i a n log l i k e l i h o o d v a l u e s c o m p a r e d to the u n r e s t r i c t e d m o d e l , the C E V a n d C I R V R m o d e l s p e r f o r m the b e s t f o l l o w e d b y the B S , G B M , D o t h a n , C I R S R , V a s i c e k a n d M e r t o n m o d e l s . R e g a r d i n g t h e u n r e s t r i c t e d m o d e l , the
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Table 13 US Summary
Statistics for the
Forecasts
Model
CKLS
Merton
Vasicek
CIRSR
Dothan
GBM
BS
CIRVR
CEV
7D RMSE MAE ME U
0.0011 0.0010 0.0008 0.0098
0.0007 0.0005 -0.0005 0.0063
0.0009 0.0008 0.0006 0.0080
0.0007 0.0006 0.0004 0.0063
0.0005 0.0004 0.0001 0.0047
0.0007 0.0006 0.0004 0.0064
0.0005 0.0004 <0.0001 0.0044
0.0005 0.0004 0.0001 0.0047
0.0014 0.0013 0.0012 0.0129
1M RMSE MAE ME U
0.0044 0.0044 0.0044 0.0382
0.0044 0.0043 0.0043 0.0379
0.0046 0.0046 0.0046 0.0395
0.0047 0.0047 0.0047 0.0404
0.0050 0.0049 0.0049 0.0427
0.0048 0.0048 0.0048 0.0412
0.0044 0.0044 0.0044 0.0384
0.0050 0.0049 0.0049 0.0427
0.0047 0.0047 0.0047 0.0409
3M RMSE MAE ME U
0.0016 0.0015 0.0015 0.0145
0.0012 0.0011 0.0010 0.0107
0.0015 0.0014 0.0014 0.0134
0.0015 0.0014 0.0014 0.0136
0.0017 0.0016 0.0016 0.0153
0.0015 0.0014 0.0013 0.0131
0.0013 0.0012 0.0012 0.0116
0.0017 0.0016 0.0016 0.0153
0.0016 0.0015 0.0015 0.0144
6M RMSE MAE ME U
0.0015 0.0014 0.0014 0.0134
0.0013 0.0012 0.0011 0.0117
0.0019 0.0018 0.0018 0.0168
0.0014 0.0013 0.0012 0.0124
0.0018 0.0017 0.0017 0.0161
0.0016 0.0015 0.0015 0.0142
0.0017 0.0016 0.0016 0.0153
0.0018 0.0017 0.0017 0.0161
0.0018 0.0016 0.0016 0.0155
12M RMSE MAE ME U
0.0021 0.0018 0.0016 0.0179
0.0019 0.0017 0.0014 0.0168
0.0023 0.0020 0.0019 0.0195
0.0022 0.0019 0.0018 0.0187
0.0024 0.0021 0.0020 0.0203
0.0022 0.0019 0.0018 0.0189
0.0024 0.0021 0.0021 0.0205
0.0024 0.0021 0.0020 0.0203
0.0022 0.0020 0.0019 0.0194
G a u s s i a n e s t i m a t e o f 7 i s 1.3096. B a s e d o n the ~2 l i k e l i h o o d ratio test i m p l i e s t h a t w e c a n reject all m o d e l s . F o r the U.S. t w e l v e m o n t h rate b a s e d o n the m a x i m i z e d G a u s s i a n log l i k e l i h o o d v a l u e s c o m p a r e d to the u n r e s t r i c t e d m o d e l , the C E V , B S , G B M , a n d D o t h a n m o d e l s p e r f o r m the b e s t f o l l o w e d b y the C I R V R , C I R S R , V a s i c e k a n d M e r t o n m o d e l s . R e g a r d i n g t h e u n r e s t r i c t e d m o d e l , t h e G a u s s i a n e s t i m a t e o f T is 1.1577. B a s e d o n the X 2 l i k e l i h o o d r a t i o test u n d e r the null h y p o t h e s i s t h a t the n e s t e d m o d e l s r e s t r i c t i o n s are v a l i d the results i m p l y that w e c a n r e j e c t all m o d e l s .
Forecast Analysis W e n o w c o m p a r e the f o r e c a s t i n g p e r f o r m a n c e o f the d i f f e r e n t s i n g l e f a c t o r c o n t i n u o u s t i m e t e r m s t r u c t u r e m o d e l s . In T a b l e s 12-13 w e r e p o r t a n u m b e r o f s u m m a r y f o r e c a s t statistics (see P i n d y c k & R u b i n f e l d , 1998). In p a r t i c u l a r w e c o m p u t e t h e r o o t m e a n s q u a r e error (RMSE), mean absolute error (MAE), mean error (ME) and Theil's inequality measure (U). I f w e let the f o r e c a s t v a l u e o f r(t) b e r * ( t ) the f o u r m e a s u r e s are t h e n g i v e n b y
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Forecasting U.K. and U.S. Interest Rates
~/[ T-1 ]~ (r* (t) - r(t))2l
RMSE
=
MAE
=
T-I~I
r*(t) -
r(t)l
ME
=
T-1]~(r*(t) -
r(t))
U
=
,J[ T-1 ]~(r* (t) - r(t)) 2] J [ T - l ~ ( r * ( t ) ) 2]
+
~[T-I~-" (r(t)) 2]
and the summations run from t = T + 1 to T + K. If the forecasts are perfect then each of the above measures is zero. If U = 1, the predictive performance of the model is as bad as it could possibly be. It should be noted that the forecasts are dynamic K step-ahead forecasts. We take the forecast period as K = 11 starting beginning of 1998. Turning to the U.K. forecasting results in Table 12. For the seven day rate based on the RMSE's the GBM model performs the best followed by CKLS, BS, Dothan, CIRVR, CEV, CIRSR, Merton and Vasicek models. For the one month rate based on the RMSE's the GBM model performs the best followed by Merton, Dothan, CIRVR, CEV, Vasicek, CIRSR, CKLS, and BS models. For the three month rate based on the RMSE's the Merton model performs the best followed by CKLS, GBM, BS, CEV, CIRSR, Dothan, CIRVR, and Vasicek models. For the six month rate the Merton model performs the best followed by the CIRSR, GBM, CEV, Vasicek, Dothan, CIRVR, CKLS and BS models. For the twelve month rate based on the RMSE's the Merton model performs the best followed by GBM, CKLS, CIRSR, Dothan, BS, CIRVR, CEV and Vasicek models. Turning to the U.S. forecasting results in Table 13. For the seven day rate based on the RMSE's the BS, Dothan and CIRVR models perform the best followed by Merton, CIRSR, GBM, Vasicek, CKLS, and CEV models. For the one month rate based on the RMSE's the CKLS, Merton, and BS models perform the best followed by Vasicek, CIRSR, CEV, GBM Dothan and CIRVR models. For the three month rate based on the RMSE's the Merton model performs the best followed by the BS, Vasicek, CIRSR, GBM, CKLS, CEV, Dothan and CIRVR models. For the six month rate based on the RMSE's the Merton model performs the best followed by the CIRSR, CKLS, GBM, BS, Dothan, CIRVR, CEV and Vasicek models. For the twelve month rate based on the RMSE's the Merton model performs the best followed by the CKLS, CIRSR, GBM, CEV, Vasicek, Dothan, BS and CIRVR models. The reader is refereed to Tables 12 - 13 for conclusions based on the other summary statistics. Generally the results imply that the models forecasting performance differs across models and currencies.
V.
CONCLUSIONS
In this paper we have compared the forecasting performance of different one factor interest rate models commonly used in the financial markets. In particular we have estimated the general CKLS model and the special cases of Merton, Vasicek, CIRSR, Dothan, GBM, Brennan-Schwartz, CIRVR, and CEV models using the approach of Nowman (1997). The models were estimated using weekly Euro-currency data for the U.K. and U.S. over a range of maturities. We have compared the forecasting performance of the models and our results imply that the forecasting performance varies across models and markets.
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ACKNOWLEDGMENT The second author is grateful to Professor Rex Bergstrom for advice and his continued support in his continuous time research.
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S. L. BYERS AND K. B. NOWMAN
In this paper we compare the forecasting performance of different one factor interest rate models commonly used in the financial markets. In particular we estimate the general CKLS model and the special cases of Merton, Vasicek, CIRSR, Dothan, GBM, Brennan-Schwartz, CIRVR, and CEV models using the approach of Nowman (1997). The models are estimated using weekly Euro-currency data for the U.K. and U.S. over a range of maturities. We find the forecasting performance varies across models and markets.
JEL classification: E43; E47 Keywords: Gaussian Estimation; Forecasting; Continuous Time
I.
INTRODUCTION
In recent years extensive empirical testing of different single factor continuous time models of the term structure of interest rates has been carried out based on the seminal paper by Chan, Karolyi, Longstaff, and Sanders (1992a, hereafter CKLS). Recent examples include Chan, Karolyi, Longstaff, and Sanders (1992b), Tse (1995), Dahlquist (1996), Hiraki and Takezawa (1996), Shoji and Ozaki (1996), and N o w m a n (1997, 1998a). In comparison little work has been done on comparing the forecasting performance of the CKLS model and the special cases o f Merton (1973), Vasicek (1977), Cox, Ingersoll and Ross (1985, hereafter CIRSR), Dothan (1978), Geometric Brownian Motion (GBM) process of Black and Scholes (1973), Brennan and Schwartz (1980, hereafter BS), Cox, Ingersoll, and Ross (1980, hereafter CIRVR), and the Constant Elasticity of Variance (CEV) process used in Cox (1975) and Cox and Ross (1976). An important component of these term structure models is what relationship exists between the level of rates and the volatilty of rates. C K L S found using monthly U.S. T-bill data that term structure models with volatilities more highly sensitive to the level of interest rates have a better fit to the data. In contrast, N o w m a n (1997) found the opposite using monthly U.K. interbank data. Generally the relationship varies across currencies, see for example, Tse (1995), Dahlquist (1996), and N o w m a n (1998a). In this paper we estimate the continuous time term structure models considered by C K L S utilizing the discrete model used in N o w m a n (1997) and the Gaussian estimation methods of Bergstrom (1983, 1984, 1985, S. L. Byers and K. B. Nowman • Department of Investment, Risk Management and Insurance, City University Business School, Barbican Centre, Frobisher Cresent, London, U.K., EC2Y 8HB; Phone: 0171-477-8698; Fax: 0171-477-8885. International Review of Financial Analysis, Vol. 7, No. 3, 1998, pp. 191-206 Copyright © 1998 by JAI PRESS Inc., All rights of reproduction in any form reserved.
ISSN: 1057-5219
192
INTERNATIONAL REVIEW OF FINANCIAL ANALYSIS / Vol. 7(3)
1986). (See Brennan & Schwartz, 1979, who also used an exact discrete model to estimate a term structure model). More recently Ball and Torous (1998) have used monthly and weekly sampled one month rates within a augmented CKLS model with stochastic volatility. Ball and Torous (1998) find that weekly sampled data has the advantage of giving more reliable estimates of the relationship between the volatility of rates and the level of rates. We therefore use weekly sampled data for the U.K. and U.S. Eurocurrency rates. Compared to the previous studies (except Hiraki & Takezawa, 1996) we also use a wider range of maturities including the one week and one, three, six and twelve month rates. Comparisons of the forecasting performance of continuous time models that have been estimated using the Gaussian estimation methods and the forecastng algorithm of Bergstrom (1989) have been undertaken recently by Bergstrom and Chambers (1990), Bergstrom, Nowman, and Wymer (1992), Chambers (1991 a, 1993), Chambers and Nowman (1997) and Nowman (1996, 1998b). We find the forecasting performance varies across models and markets. The outline of the remainder of the paper is as follows. In Section II the different interest rate models used in this study are presented and the Gaussian estimation econometric methodology discussed. Section III describes the data used in the study and Section IV presents the empirical and forecasting results. Section V offers some conclusions.
II.
CONTINUOUS T I M E M O D E L S
The CKLS general specification is given by Equation 1 below. The equation allows the conditional mean and variance to depend on the level r. d r (t) = {o~ + ~ r ( t ) } d t + t~r r ( t ) d Z
(t _>0)
(1)
where {r(t), t > 0} is a real continuous time random process, tx, I], y, and o are unknown structural parameters and d Z is a Wiener process. From the CKLS model we can obtain the important special cases: Merton (~ = 0, y = 0), Vasicek (y = 0), CIR (y = 0.5), Dothan (c~ = 0, [~ = 0, y = 1), GBM (tx = 0, y = 1), BS (y= 1), CIRVR ((x = 0, ~ = 0, y = 3/2), and CEV (c~ = 0). To estimate the parameters of continuous time models from discrete data we have to use a discrete model derived from the solution of the continuous time model. In a series of papers Bergstrom (1983, 1985, 1986, 1990, 1997) (see also, Chambers, 1991b; Gandolfo, 1993; Nowman, 1991, 1993) has been concerned with obtaining asymptotically efficient estimates using an exact discrete model which takes account of the exact restrictions on the distribution of the discrete data implied by the continuous time model. Bergstrom (1983) provided a detailed analysis of using exact discrete models in obtaining the exact Gaussian estimates (estimates which would be exact maximum likelihood estimates if the innovations were Brownian motion) of the parameters of the continuous time model. In Bergstrom (1985, 1986, 1997) the methods where extended to include nonstationary systems, exogenous variables and stochastic trends. The new methodology has recently been surveyed by Bergstrom (1996) and applied in macroeconomics by Bergstrom, Nowman, and Wymer (1992) (see also, Nowman, 1996, 1998b) to a second order continuous time macroeconometric model of the United Kingdom. In Nowman (1997) a discrete model of Bergstrom (1984, Theorem 2) was modified for hetroscedasticity to estimate the CKLS model. The discrete model used by Nowman (1997) is given by Equation 2 below
Forecasting U.K. and U.S. Interest Rates
193
r(t) = e ~ r ( t - 1) + "~(e I] - 1 ) + tit (t = 1, 2 . . . . .
w h e r e tit (t = 1 , 2 . . . . . T) s a t i s f i e d t h e c o n d i t i o n s g i v e n in N o w m a n
T)
(2)
(1997). Following Nowman
(1997) we let L (0) be minus twice the logarithm of the Gaussian likelihood function where
Table 1 S u m m a r y Statistics: U K and U S E u r o e u r r e n c y Data 1986--1997 Variables
T
Mean
Standard Deviation
Pl
92
P3
P4
P5
P6
627 626
0.0917 -0.0001
0.0327 0.0105
0.95 -0.4 7
0.94 -0.0 1
0.94 0.01
0.94 -0.04
0.94 0.08
0.93 -0.0 1
627 626
0.0925 -0.0001
0.0319 0.0026
0.99 -0.1 7
0.99 0.01
0.99 0.11
0.98 -0.00
0.98 0.03
0.98 0.10
627 626
0.0930 -0.0001
0.0314 0.0020
0.99 0.12
0.99 0.07
0.99 0.09
0.98 0.06
0.98 0.06
0.98 0.05
627 626
0.0931 -0.0001
0.0306 0.0022
0.99 0.09
0.99 0.09
0.99 0.08
0.98 0.06
0.98 0.04
0.97 -0.0 0
627 626
0.0937 -0.0001
0.0290 0.0025
0.99 0.05
0.99 0.05
0.98 0.08
0.98 0.05
0.97 0.02
0.96 --0.0 0
627 626
0.0601 -0.0000
0.0185 0.0049
0.96 -0.3 9
0.95 ~).0 9
0.95 0.04
0.95 -0.02
0.94 0.03
0.94 -0.0 3
627 626
0.0607 -0.0000
0.0181 0.0018
0.99 -0.0 8
0.99 -0.04'
0.98 0.11
0.98 -0.11
0.97 -0.04
0.97 -0.02
627 626
0.0615 -0.0000
0.0179 0.0014
0.99 0.03
9.99 0.01
0.99 0.04
0.98 0.05
0.97 0.05
0.97 -0.0 4
627 626
0.0626 -0.0000
0.0178 0.0016
0.99 -0.0 2
0.99 0.03
0.98 0.04
0.98 0.07
0.97 0.06
0.97 -0.0 2
627 626
0,0649 -0,0000
0.0175 0.0017
0.99 0.02
0.99 0.07
0.98 0.05
0.97 0.08
0.97 0.07
0.96 -0.0 3
UK 7D r(t) dr(t)
UK 1M r(t) dr(t)
UK 3M r(t) dr(t)
UK 6M r(t) dr(t)
UK 12M r(t) dr(t) US 7D r(t) dr(t) US 1M r(t) dr(t) US 3M r(t) dr(t) US 6M r(t) dr(t) US 12M r(t) dr(t)
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INTERNATIONAL REVIEW OF FINANCIAL ANALYSIS / Vol. 7(3)
Table 2
Estimates of UK Short Term Interest Rate Models: 7-Day Model
O~
CKLS
0.0008 (0.0007)
Merton
-0.0001 (0.0004)
Vasicek
0.0049 (0.0013)
CIRSR
0.0020 (0.0010)
Dothan
0.0
GBM
0.0
BS
0.0010 (0.0008)
CIRVR
0,0
CEV
0,0
~
¢32
-0.0076 (0.0107) 0.0
Log Likelihood
"~
0.3713 (0.1828)
1.8090 (0.0999)
)~2 Test
df
2780.2089
0.0001 (<0.0001 )
0.0
2 5 3 9 . 3 4 3 7 481.730 4
2
~).0537 (0.01341
0.0001 (<0.000l)
0.0
2547.6922
465.033 4
1
-0.0224 (0.0118)
0.0009 (<0.0(X)1)
0.5
2677.1458
206.126 2
1
0.0078 (0.0004)
1.0
2745.0624
70.2930
3
0.0020 (0.0035)
0.0078 (0.0004)
1.0
2745.2317
69.9544
2
-0.0104 (0.01101
0.0079 (0.0004)
1.0
2745.9684
68.4810
1
0.0825 ({I.0046)
1.5
2774.5189
11.3800
3
2779.6383
1.1412
l
0.0
0.0 0.0041 (0.003 l)
0.3583 (0.1763)
1.8{}38 (0.0997)
the c o m p l e t e v e c t o r of parameters is 0 = [~, 13, 7, ~2]. T h e n the Gaussian estimates are obtained f r o m E q u a t i o n 3 (defined in N o w m a n , 1997) using s o m e numerical o p t i m i z a t i o n routine.
l
T L(0) =
r ( t ) - efJr(t - 1 ) - ~ ( e f3 - 1 )
21°gmtt +
2 mtt
t=l
III.
t21
(3)
DATA
T h e short-term interest rates used in this study are E u r o - c u r r e n c y rates on U.K. and U.S. currencies (middle rate) obtained from D a t a s t r e a m TM. W e use w e e k l y data on a W e d e n s d a y in the e m p i r i c a l analysis to a v o i d missing observations and w e e k - d a y effects. The data c o v e r the period f r o m January 1986 to D e c e m b e r 1997 g i v i n g a total of 627 observations. The maturity s p e c t r u m of the data c o v e r s the seven day and one, three, six and t w e l v e m o n t h rates. Table 1 reports the descriptive statistics. The table displays the means, standard deviations and first six autocorrelations o f the rates and changes in the rates. For the U.K. w e h a v e the f o l l o w i n g a v e r a g e levels and standard deviations o f the rates: seven day rate m e a n is 9.17% with standard deviation
195
Forecasting U.K. and U.S. Interest Rates
Table 3 E s t i m a t e s of U K Short T e r m Interest Rate Models: 1-Month Model
~
~
CKLS
0.0003 (0.0002)
Merton
-0.0001 (0.0001)
Vasicek
-0.0037 (0.0034) 0.0
132 0.0055 (0.0026)
"~
1.4316 (0.0954)
Log Likelihood
X2 Test
df
3506.5602
<0.0001 (<0.0001 )
0.0
3406.6791 199.7622
2
0.0003 (0.0003)
-0.0037 <0.0001 (0.0033) (<0.0001)
0.0
3407.3061 198.5082
1
CIRSR
0.0003 (0.0003)
-0.0036 0.0001 (0.0032) (<0.0001)
0.5
3462.0745 88.9714
1
Dothan
0.0
0.0007 (.a).OOOl)
1.0
3495.9327 21.2550
3
GBM
0.0
-0.0004 (0.0011)
0.0007 (<0.0001)
1.0
3 4 9 5 . 9 9 5 2 21.1300
2
BS
0.0003 (0.0002)
-0.0036 0.0007 (0.0033) (<0.0001)
1.0
3496.5595 20.0014
1
CIRVR
0.0
1.5
3505.6510
1.8184
3
CEV
0.0
3505.9332
1.2540
1
0.0
0.0 -0.0001 (0.0010)
0.0077 (0.00O4) 0.0055 (0.0004)
1.4298 (0.0947)
3.27%; one month rate mean is 9.25% with standard deviation 3.19%; three month rate mean is 9.30% with standard deviation 3.14%; six month rate mean is 9.31% with standard deviation 3.06%; twelve month rate mean is 9.37% with standard deviation 2.90%. For the U.S. we have the following average levels and standard deviations of the rates: seven day rate mean is 6.01% with standard deviation 1.85%; one month rate mean is 6.07% with standard deviation 1.81%; three month rate mean is 6.15% with standard deviation 1.79%; six month rate mean is 6.26% with standard deviation 1.78%; twelve month rate mean is 6.49% with standard deviation 1.75 %.
IV.
E M P I R I C A L AND F O R E C A S T I N G RESULTS
We now present the empirical results of the models for the U.K. and U.S. interest rates. In each of the Tables 2-11 the Gaussian estimates (standard errors in brackets) and maximized log likelihoods of the different continuous time models of the short-term interest rate are presented. Following Chan, Karolyi, Longstaff, and Sanders (1992b) and Nowman (1997) the explanatory power compared to the unrestricted model is compared using the maximized Gaussian log likelihood function values.
U.K. Results For the U.K. seven day rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV model performs the best followed by the CIRVR, BS, GBM,
196
I N T E R N A T I O N A L R E V I E W O F F I N A N C I A L A N A L Y S I S / Vol. 7(3)
Table 4 Estimates of UK Short Term Interest Rate Models: 3-Month Model
0¢
[3
o2
"[
CKLS
0.0002 (0.0002)
-0.0031 (0.0028)
0.0024 (0.0012)
1.3578 (0.1030)
Merton
-0.0OOl
0.0
(0.0001)
<0.0001
Log Likelihood
)~2 Test
df
3648.5757
0.0
3567.7554
161.6406
2
(<0.0001)
Vasicek
0.0002 (0.0002)
-0.0024 (0.0026)
<0.0001 (<0.0001 )
0.0
3568.1943
160.7628
1
CIRSR
0.0002 (0.0002)
-0.0027 (0.0025)
<0.0001 (<0.0001)
0.5
3615.5955
65.9604
1
0.0004
1.0
3642.2430
12.6654
3
Dothan
0.0
0.0
(<0.OO01 ) GBM BS
0.0 0.0002 (0.0002)
CIRVR
0.0
CEV
0.0
-0.0005 (0.0008)
0.0004 (
1.0
3642.4311
12.2892
2
-0.0030 (0.0026)
0.0004 (<0.OO01)
1.0
3642.9327
11.2860
l
0.0048 (0.0O03)
1.5
3647.1649
2.8216
3
3648.2308
0.6898
1
0.0 -0.0003 (0.OO08)
0.0023 (0.OO 12)
1.3531 (0.1030)
Table 5 Estimates of UK Short Term Interest Rate Models: 6-Month Log
Z2
Model
O~
~
02
~t
Likelihood
Test
CKLS
0.0003 (0.0002)
-0.0036 (0.0030)
0.0018 (0.0009)
1.2656 (0.0997)
3595.1866
Merton
-0.OO01 (0.0002)
Vasicek
0.0002 (0.0003)
CIRSR
0.0002 (0.0002)
Dothan
0.0
0.0
df
<0.OO01 (<0.0001 )
0.0
3520.1661
150.0410
2
-0.0029 (0.0029)
<0.0001 (<0.0001)
0.0
3520.6677
149.0378
1
-0.0031 (0.0029)
0.0001 (<0.0001 )
0.5
3567.0343
56.3046
1
0.0005
1.0
3591.2052
7.9628
3
0.0
(<0.0001 ) GBM BS
0.0 0.0003 (0.0002)
CIRVR
0.0
CEV
0.0
-0.0004 (0.0009)
0.0005 (<0.0001)
1.0
3591.3306
7.7120
2
-0.0034 (0.0029)
0.0005 (<0.0001)
1.0
3591.9262
6.5208
1
0.0/157 (0.00O3)
1.5
3591.8008
6.7716
3
3594.7164
0.9404
1
0.0 -0.0003 (0.0009)
0.0018 (0.0008)
1.2606 (0.0998)
197
Forecasting U.K. and U.S. Interest Rates
Table 6 Estimates of UK Short Term Interest Rate Models: 12-Month Model
OL
~
CKLS
0.0003 (0.0003)
Merton
-0.0001 (o.oool)
Vasicek
0.0003 (0.0003)
CIRSR
0.0003 (0.0003)
Dothan
0.0
GBM
0.0
-0.0004 (0.0009)
BS
0.0003 (0.0003)
CIRVR
0.0
CEV
0.0
-0.0042 (0.0034)
~2
Log
~2
Likelihood
Test
df
1.3500 (0.1001)
3533.2704
0.0
3450.7572 165.0264
2
-0.0039 <0.0001 (0.0034) (<0.0001)
0.0
3451.3842 163.7724
1
-0.0040 0.0001 (0.0033) (<0.0001)
0.5
3 4 9 9 . 2 8 7 0 67.9668
1
0.0006 (
1.0
3526.4988
13.5432
3
0.0006 (<0.0001)
1.0
3526.5928
13.3552
2
-0.0041 0.0006 (0.0033) (<0.0001)
1.0
3527.2825
11.9758
1
1.5
3531.2953
3.9502
3
3532.4553
1.6302
1
0.0
0.0
0.0 -0.0001 (0.0009)
0.0032 (0.0015)
'~
<0.0001 (<0.0001)
0.0066 (0.0OO4) 0.0031 (0.0015)
1.3472 (0.1009)
Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate of y is 1.8090 and highly significant. The unrestricted model also implies only weak evidence of mean reversion in the short rate; the parameter [3 is insignificant. Based on the Z 2 likelihood ratio test under the null hypothesis that the nested models restrictions are valid the results imply that we can reject all models except the CEV model at the five percent level. For the U.K. one month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model the CEV and C I R V R models perform the best followed by the BS, GBM, Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate of y is 1.4316 and significant. This compares to Dahlquist's (1996) estimate of 0.1562 using the monthly one month Eurocurrency rate; N o w m a n ' s (1997) estimate using the monthly one month interbank rate of 0.2898 and N o w m a n and Sorwar's (in press) estimate of 1.0461 using the monthly one month Eurocurrency rate. In Ball and Torous (1998) they report the stochastic volatility model estimates of 0.398 using weekly one month Eurocurrency rate over the period 1985-1995 and using monthly and weekly interbank data estimates of 0.261 and 0.486 respectively. Based on the )~2 likelihood ratio test implies that we can reject all models except the CIRVR and CEV models. For the U.K. three month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV and CIRVR models perform the best followed by the BS, GBM, Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model,the Gaussian estimate o f y i s 1.3578 and significant. This estimate compares to the estimate of Tse (1995) on monthly three month money market data of 0.1132 and N o w m a n and Sorwar' s (1998) estimate of 1.3564 using the monthly three month Eurocurrency rate. Based on the ~2 likelihood ratio test implies that we can reject all models except the CIRVR and CEV models.
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Table 7 Estimates of US Short Term Interest Rate Models: 7-Day Log
X2
Model
~
~
~2
It
Likelihood
Test
df
CKLS
0.0001 (0.0003)
0.0003 (0.0078)
0.8746 (0.4730)
1.9458 (0.0940)
3219.6136
Merton
-0.0001 (0.0002)
Vasicek
0.0027 (0.0007)
CIRSR
0.0014 (0.0005)
Dothan
0.0
GBM
0.0
BS
0.0006 (0.0004)
CIRVR
0.0
CEV
0.0
0.0
<0.0001 (<0.0001 )
0.0
2993.3137
452.5998
2
-0.0467 (0.01131
<0.0001 (<0.000l)
0.0
3002.0321
435.1630
1
-0.0242 (0.0092)
0.00113 (<0.00011
0.5
3096.0695
247.0880
1
0.0047 (0.001)3)
1.0
3164.0928
111.0416
3
0.0009 (0.0027)
0.0047 (0.0/)/)3)
1.0
3164.1555
110.9162
2
-0.0111 (0.0070)
0.0047 (0.00113)
1.0
3165.3781
108.4710
1
0.0712 (0.O040)
1.5
3206.3526
26.5220
3
3219.4255
0.3762
1
0.0
0.0 0.0031 (0.0022)
0.8930 (0.4459)
1.9498 (0.0867)
Table 8 Estimates of US Short Term Interest Rate Models: 1-Month Log
X2
Model
O~
~
~2
It
Likelihood
Test
CKLS
0.0004 (0.0002)
-0.0082 (0.0038)
0.0008 (0.0003)
0.9544 (0.0614)
3615.0939
Merton
-0.0001 (0.0001)
Vasicek
0.0009 (0.0003)
CIRSR
0.0006 (0.0002)
Dothan
0.0
0.0
df
<0.0001 (<0.0001)
0.0
3490.1014
249.9850
2
-0.0160 (0.0050)
<0.0001 (<0.0001 )
0.0
3495.0234
240.1410
1
-0.0107 (0.0043)
0.0001 (<0.0001)
0.5
3583.4931
63.2016
1
0.0011
1.0
3610.6108
8.9662
3
0.0
(<0.0001 ) GBM BS
0.0 0.0004 (0.0002)
CIRVR
0.0
CEV
0.0
-0.0005 (0.0013)
0.0011 (<0.0001)
1.0
3610.7049
8.7780
2
-0.0081 (0.0039)
0.0011 (<0.0001 )
1.0
3612.7113
4.7652
1
0.0213 (0.0012)
1.5
3573.4297
83.3284
3
3610.7989
8.5900
1
0.0 -0.0006 (0.0013)
0.0009 (0.0003)
0.9714 (0.0606)
199
Forecasting U.K. and U.S. Interest Rates
Table 9 E s t i m a t e s o f US S h o r t T e r m I n t e r e s t R a t e M o d e l s : 3 - M o n t h Model
(X
CKLS
0.0003 (0.0001)
Merton
-0.0001 (0.0001)
Vasicek
~
Log
X2
Likelihood
Test
df
3761.3416
<0.0001 (<0.0001)
0.0
3552.2058 418.2716
2
0.0008 (0.0003)
-0.0146 <0.0001 (0.0048) (<0.0001)
0.0
3557.1277 408.4278
1
CIRSR
0.0005 (0.0002)
-0.0093 <0.0001 (0.0038) (<0.0001)
0.5
3674.7843 173.1146
1
Dothan
0.0
1.0
3742.7511
37.1810
3
GBM
0.0
-0.0008
1.0
3 7 4 3 . 0 3 3 2 36.6168
2
BS
0.0003 (0.0002)
(0.0010) (<0.0001) -0.0066 0.0007 (0.0032) (<0.0001)
1.0
3744.7261
33.2310
1
1.5
3755.4478
11.7876
3
6.7088
1
0.0
CEV
0.0
0.0
0.0
0.0 -0.0003 (0.0010)
0.0044 (0.0017)
~t 1.3367 (0.0654)
CIRVR
-0.0055 (0.0031)
(y2
0.0007 (<0.0001) 0.0007
0.0113 (0.0006) 0.0049 (0.0018)
1.3532 (0.0649)
3757.9872
For the U.K. six month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV model performs the best followed by the BS, CIRVR, G B M , Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate of ~' is 1.2656 and significant. Based on the ~2 likelihood ratio test implies that we can reject all models except the C I R V R and CEV models. For the U.K. twelve month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV model performs the best followed by the CIRVR, BS, GBM, Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate o f 7 is 1.3500 and significant. Based on the ~2 likelihood ratio test implies that we can reject all models except the CIRVR and CEV models.
U.S. Results For the U.S. seven day rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the CEV model performs the best followed by the CIRVR, BS, GBM, Dothan, CIRSR, Vasicek and Merton models. Regarding the unrestricted model, the Gaussian estimate of 7 is 1.9458 and significant. The unrestricted model also implies no evidence of mean reversion. Based on the Z 2 likelihood ratio test implies that we can reject all models except the C E V model. For the U.S. one month rate based on the maximized Gaussian log likelihood values compared to the unrestricted model, the BS, CEV, G B M , and Dothan models perform the best followed by the CIRSR, CIRVR, Vasicek and Merton models. Regarding the unre-
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I N T E R N A T I O N A L R E V I E W O F F I N A N C I A L A N A L Y S I S / Vol. 7(3)
Table 10 Estimates of US Short Term Interest Rate Models: 6-Month Model
(X
~
•2
~t
CKLS
0.0003 (0.0002)
-0.0062 (0.0035)
0.0044 (0.0017)
1.3096 (0.0702)
Merton
-0.0001 (0.0001)
Vasicek
0.0009 (0.0003)
C1RSR
0.0005 (0.0002)
Dothan
0.0
GBM
0.0
BS
0.0004 (0.0002)
CIRVR
0.0
CEV
0.0
0.0
Log Likelihood
X2 Test
df
3699.2686
<0.0001 (
0.0
3520.3228
357.8916
2
-0.0153 (0.0049)
<0.0001 (<0.0001)
0.0
3525.1194
348.2984
1
-0.0101 (0.0042)
0.0001 (<0.0001)
0.5
3628.1355
142.2662
1
0./)008 (<0.0001)
1.0
3684.7222
29.0928
3
-0.0007 (0.00 11 )
0.0008 (<0.1)001)
1.0
3684.9103
28.7166
2
-0.0072 (0.00 11 )
0.00(/8 (<0.0001 )
1.0
3686.6973
25.1426
1
0.0131 10.0/)/)7)
1.5
3692.9673
12.6026
3
3696.0082
6.5208
1
0.0
0.0 -0.0002 10.0011 )
0.0(149 (0.00 191
1.3269 (0.0698)
Table 11 Estimates of US Short Term Interest Rate Models: 12-Month Log
Model
O~
~
~2
y
CKLS
0.0004 (0.0002)
-0.0079 (0.0039)
0.0020 (0.0008)
1.1577 10.0714)
Merton
-41.0001 (0.0001 )
Vasicek
0.0009 (0.0003)
CIRSR
0.0006 (0.0003)
Dothan
0.0
GBM
0.0
BS
0.0005 (0.0002)
CIRVR
0.0
CEV
0.0
0.0
Likelihood
X2
Test
df
3643.0581
<0.0001 (<0.0001 )
0.0
3504.0208
278.0746
2
-0.0156 (0.0051 )
<0.0001 (<0.0001 )
0.0
3508.5979
268.9204
1
-0.0109 (0.0044)
0.1101/1 (<0.0001)
0.5
3595.5315
95.0532
1
0.00119 (<0.0001 )
1.0
3635.7535
14.6092
3
-0.0006 (0.0012)
0.0009 (<0.00011
1.0
3635.9103
14.2956
2
-0.0084 (0.0040)
0.0009 (<0.0001 )
1.0
3637.9480
10.2202
1
0.0142 (0.0008)
1.5
3628.6057
28.9048
3
3639.0453
8.0256
I
0.0
0.0 -0.0004 10.0012)
0.0023 (0.00119)
1.1760 (0.0703)
201
Forecasting U.K. and U.S. Interest Rates
T a b l e 12
UK Summary Statistics for the Forecasts Model
CKLS
Merton
Vasicek
CIRSR
Dothan
GBM
BS
C1RVR
CEV
7D RMSE MAE ME U
0.0010 0.0009 0.0006 0.0072
0.0018 0.0015 --0.001 5 0.0125
0.0048 0.0042 0.0042 0.0318
0.001 0.001 0.001 0.011
0.0012 0.0010 -0.000 9 0.0085
0.0008 0.0006 -<.000 1 0.0053
0.0010 0.0009 0.0006 0.0070
0.0012 0.0010 -0.000 9 0.0085
0.0013 0.0011 0.0009 0.0091
1M RMSE MAE ME U
0.0008 0.0007 0.0006 0.0053
0.0007 0.0006 ~).000 1 0.0048
0.0008 0.0007 0.0006 0.0053
0.000 8 0.000 7 0.000 7 0.005 5
0.0007 0.0005 0.0005 0.0047
0.0006 0.0005 0.0003 0.0042
0.0008 0.0007 0.0007 0.0055
0.0007 0.0005 0.0005 0.0047
0.0007 0.0005 0.0005 0.0045
3M RMSE MAE ME U
0.0005 0.0005 0.0004 0.0036
0.0004 0.0003 <.0000 1 0.0026
0.0008 0.0007 0.0007 0.0055
0.000 7 0.000 6 0.000 6 0.004 6
0.0007 0.0006 0.0006 0.0047
0.0005 0.0005 0.0004 0.0034
0.0006 0.0005 0.0005 0.0038
0.0007 0.0006 0.0006 0.0047
0.0006 0.0005 0.0005 0.0039
6M RMSE MAE ME U
0.0014 0.0014 0.0014 0.0095
0.0008 0.0007 0.0006 0.0055
0.0012 0.0011 0.0011 0.0078
0.001 0.001 0.001 0.007
1 0 0 3
0.0013 0.0012 0.0012 0.0086
0.0011 0.0010 0.0010 0.0075
0.0015 0.0015 0.0015 0.0101
0.0013 0.0012 0.0012 0.0086
0.0012 0.0011 0.0011 0.0077
12M RMSE MAE ME U
0.0014 0.0012 0.0012 0.0091
0.0010 0.0008 0.0007 0.0069
0.0015 0.0013 0.0013 0.0098
0.001 4 0.001 3 0.001 3 0.009 6
0.0014 0.0013 0.0013 0.0097
0.0013 0.0011 0.0011 0.0087
0.0014 0.0012 0.0012 0.0093
0.0014 0.0013 0.0013 0.0097
0.0014 0.0013 0.0013 0.0094
6 3 3 2
stricted m o d e l , t h e G a u s s i a n e s t i m a t e o f T is 0 . 9 5 4 4 . T h i s c o m p a r e s to S h o j i a n d O z a k i ' s ( 1 9 9 6 ) e s t i m a t e o f 1.1473 for m o n t h l y o n e m o n t h T.bill rate a n d C K L S ' s e s t i m a t e o f 1.4999 u s i n g m o n t h l y o n e m o n t h T . b i l l rate. In N o w m a n ( 1 9 9 8 a ) u s i n g m o n t h l y o n e m o n t h E u r o c u r r e n c y rate a n e s t i m a t e o f 1.0519 u p to M a r c h 1995 is r e p o r t e d . B a l l a n d T o r o u s ( 1 9 9 8 ) r e p o r t a n e s t i m a t e o f 0 . 7 5 4 b a s e d o n w e e k l y o n e m o n t h E u r o c u r r e n c y rate o v e r the p e r i o d 1 9 8 5 - 1 9 9 5 . B a s e d o n the ~2 l i k e l i h o o d ratio test i m p l i e s t h a t w e c a n r e j e c t all m o d e l s . F o r the U.S. t h r e e m o n t h rate b a s e d o n the m a x i m i z e d G a u s s i a n l o g l i k e l i h o o d v a l u e s c o m p a r e d to the u n r e s t r i c t e d m o d e l , the C E V a n d C I R V R m o d e l s p e r f o r m t h e b e s t f o l l o w e d b y t h e B S , G B M , D o t h a n , C I R S R , V a s i c e k a n d M e r t o n m o d e l s . R e g a r d i n g the u n r e s t r i c t e d m o d e l , the G a u s s i a n e s t i m a t e o f T is 1.3367. T h i s c o m p a r e s to T s e ' s ( 1 9 9 5 ) e s t i m a t e for m o n t h l y t h r e e m o n t h m o n e y m a r k e t d a t a o f 1.7283. B a s e d o n the ~2 l i k e l i h o o d ratio test i m p l i e s t h a t w e c a n r e j e c t all m o d e l s . F o r the U.S. six m o n t h rate b a s e d o n the m a x i m i z e d G a u s s i a n log l i k e l i h o o d v a l u e s c o m p a r e d to the u n r e s t r i c t e d m o d e l , the C E V a n d C I R V R m o d e l s p e r f o r m the b e s t f o l l o w e d b y the B S , G B M , D o t h a n , C I R S R , V a s i c e k a n d M e r t o n m o d e l s . R e g a r d i n g t h e u n r e s t r i c t e d m o d e l , the
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Table 13 US Summary
Statistics for the
Forecasts
Model
CKLS
Merton
Vasicek
CIRSR
Dothan
GBM
BS
CIRVR
CEV
7D RMSE MAE ME U
0.0011 0.0010 0.0008 0.0098
0.0007 0.0005 -0.0005 0.0063
0.0009 0.0008 0.0006 0.0080
0.0007 0.0006 0.0004 0.0063
0.0005 0.0004 0.0001 0.0047
0.0007 0.0006 0.0004 0.0064
0.0005 0.0004 <0.0001 0.0044
0.0005 0.0004 0.0001 0.0047
0.0014 0.0013 0.0012 0.0129
1M RMSE MAE ME U
0.0044 0.0044 0.0044 0.0382
0.0044 0.0043 0.0043 0.0379
0.0046 0.0046 0.0046 0.0395
0.0047 0.0047 0.0047 0.0404
0.0050 0.0049 0.0049 0.0427
0.0048 0.0048 0.0048 0.0412
0.0044 0.0044 0.0044 0.0384
0.0050 0.0049 0.0049 0.0427
0.0047 0.0047 0.0047 0.0409
3M RMSE MAE ME U
0.0016 0.0015 0.0015 0.0145
0.0012 0.0011 0.0010 0.0107
0.0015 0.0014 0.0014 0.0134
0.0015 0.0014 0.0014 0.0136
0.0017 0.0016 0.0016 0.0153
0.0015 0.0014 0.0013 0.0131
0.0013 0.0012 0.0012 0.0116
0.0017 0.0016 0.0016 0.0153
0.0016 0.0015 0.0015 0.0144
6M RMSE MAE ME U
0.0015 0.0014 0.0014 0.0134
0.0013 0.0012 0.0011 0.0117
0.0019 0.0018 0.0018 0.0168
0.0014 0.0013 0.0012 0.0124
0.0018 0.0017 0.0017 0.0161
0.0016 0.0015 0.0015 0.0142
0.0017 0.0016 0.0016 0.0153
0.0018 0.0017 0.0017 0.0161
0.0018 0.0016 0.0016 0.0155
12M RMSE MAE ME U
0.0021 0.0018 0.0016 0.0179
0.0019 0.0017 0.0014 0.0168
0.0023 0.0020 0.0019 0.0195
0.0022 0.0019 0.0018 0.0187
0.0024 0.0021 0.0020 0.0203
0.0022 0.0019 0.0018 0.0189
0.0024 0.0021 0.0021 0.0205
0.0024 0.0021 0.0020 0.0203
0.0022 0.0020 0.0019 0.0194
G a u s s i a n e s t i m a t e o f 7 i s 1.3096. B a s e d o n the ~2 l i k e l i h o o d ratio test i m p l i e s t h a t w e c a n reject all m o d e l s . F o r the U.S. t w e l v e m o n t h rate b a s e d o n the m a x i m i z e d G a u s s i a n log l i k e l i h o o d v a l u e s c o m p a r e d to the u n r e s t r i c t e d m o d e l , the C E V , B S , G B M , a n d D o t h a n m o d e l s p e r f o r m the b e s t f o l l o w e d b y the C I R V R , C I R S R , V a s i c e k a n d M e r t o n m o d e l s . R e g a r d i n g t h e u n r e s t r i c t e d m o d e l , t h e G a u s s i a n e s t i m a t e o f T is 1.1577. B a s e d o n the X 2 l i k e l i h o o d r a t i o test u n d e r the null h y p o t h e s i s t h a t the n e s t e d m o d e l s r e s t r i c t i o n s are v a l i d the results i m p l y that w e c a n r e j e c t all m o d e l s .
Forecast Analysis W e n o w c o m p a r e the f o r e c a s t i n g p e r f o r m a n c e o f the d i f f e r e n t s i n g l e f a c t o r c o n t i n u o u s t i m e t e r m s t r u c t u r e m o d e l s . In T a b l e s 12-13 w e r e p o r t a n u m b e r o f s u m m a r y f o r e c a s t statistics (see P i n d y c k & R u b i n f e l d , 1998). In p a r t i c u l a r w e c o m p u t e t h e r o o t m e a n s q u a r e error (RMSE), mean absolute error (MAE), mean error (ME) and Theil's inequality measure (U). I f w e let the f o r e c a s t v a l u e o f r(t) b e r * ( t ) the f o u r m e a s u r e s are t h e n g i v e n b y
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Forecasting U.K. and U.S. Interest Rates
~/[ T-1 ]~ (r* (t) - r(t))2l
RMSE
=
MAE
=
T-I~I
r*(t) -
r(t)l
ME
=
T-1]~(r*(t) -
r(t))
U
=
,J[ T-1 ]~(r* (t) - r(t)) 2] J [ T - l ~ ( r * ( t ) ) 2]
+
~[T-I~-" (r(t)) 2]
and the summations run from t = T + 1 to T + K. If the forecasts are perfect then each of the above measures is zero. If U = 1, the predictive performance of the model is as bad as it could possibly be. It should be noted that the forecasts are dynamic K step-ahead forecasts. We take the forecast period as K = 11 starting beginning of 1998. Turning to the U.K. forecasting results in Table 12. For the seven day rate based on the RMSE's the GBM model performs the best followed by CKLS, BS, Dothan, CIRVR, CEV, CIRSR, Merton and Vasicek models. For the one month rate based on the RMSE's the GBM model performs the best followed by Merton, Dothan, CIRVR, CEV, Vasicek, CIRSR, CKLS, and BS models. For the three month rate based on the RMSE's the Merton model performs the best followed by CKLS, GBM, BS, CEV, CIRSR, Dothan, CIRVR, and Vasicek models. For the six month rate the Merton model performs the best followed by the CIRSR, GBM, CEV, Vasicek, Dothan, CIRVR, CKLS and BS models. For the twelve month rate based on the RMSE's the Merton model performs the best followed by GBM, CKLS, CIRSR, Dothan, BS, CIRVR, CEV and Vasicek models. Turning to the U.S. forecasting results in Table 13. For the seven day rate based on the RMSE's the BS, Dothan and CIRVR models perform the best followed by Merton, CIRSR, GBM, Vasicek, CKLS, and CEV models. For the one month rate based on the RMSE's the CKLS, Merton, and BS models perform the best followed by Vasicek, CIRSR, CEV, GBM Dothan and CIRVR models. For the three month rate based on the RMSE's the Merton model performs the best followed by the BS, Vasicek, CIRSR, GBM, CKLS, CEV, Dothan and CIRVR models. For the six month rate based on the RMSE's the Merton model performs the best followed by the CIRSR, CKLS, GBM, BS, Dothan, CIRVR, CEV and Vasicek models. For the twelve month rate based on the RMSE's the Merton model performs the best followed by the CKLS, CIRSR, GBM, CEV, Vasicek, Dothan, BS and CIRVR models. The reader is refereed to Tables 12 - 13 for conclusions based on the other summary statistics. Generally the results imply that the models forecasting performance differs across models and currencies.
V.
CONCLUSIONS
In this paper we have compared the forecasting performance of different one factor interest rate models commonly used in the financial markets. In particular we have estimated the general CKLS model and the special cases of Merton, Vasicek, CIRSR, Dothan, GBM, Brennan-Schwartz, CIRVR, and CEV models using the approach of Nowman (1997). The models were estimated using weekly Euro-currency data for the U.K. and U.S. over a range of maturities. We have compared the forecasting performance of the models and our results imply that the forecasting performance varies across models and markets.
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ACKNOWLEDGMENT The second author is grateful to Professor Rex Bergstrom for advice and his continued support in his continuous time research.
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