H.D. Vinod (Fordham University, Bronx, NY 10458)

E-mail: vinod@murray.fordham.edu

P.Samanta (Manhattan College, Riverdale, NY 10471)

E-mail: psamanta@manhattan.edu

Abstract

A stochastic differential equation is at the heart of short term interest rate dynamics. Various parameter restrictions on the unrestricted model lead to eight special cases. They are: Brennan and Schwartz, Cox-Ingersoll and Ross (CIR) square root process, Vasicek, CIR variable rate process, Dothan, constant elasticity of variance (CEV), geometric Brownian motion, and Merton. We apply these models to exchange rate dynamics and evaluate them from the viewpoint of out-of-sample forecasting. We estimate by the generalized method of moments (GMM) and suggest two new methods: (i) Kadane's (1970) small-sigma asymptotics in conjunction with Godambe-Durbin estimating function (SSA-EF) suggested in Vinod (1996), and (ii) a numerical conditional variance (NCV) method. The new methods avoid the considerable trial and error required by the GNM in choosing the instruments. We use daily, weekly, and monthly rates for the British pound and the Deutsche mark for the period 1975 to 1991. Our forecasts by the SSA-EF are very close to the GMM, and the signs of SSA-EF coefficients are self-consistent.

KEY WORDS: Stochastic differential equations; Small-sigma asymptotics; Short term interest rate models.

The models used to characterize short term interest rate movements capture the dynamics by a familiar stochastic differential equation (SDE). We claim that the same equation is applicable to short term exchange rate movements and verify this from an empirical standpoint. If true, this has implications for forecasting exchange rate changes, as well as for pricing currency derivatives and for hedging currency risk. The approach of this paper is straightforward. We test the above mentioned claim, by using the Generalized Method of Moments (GMM) technique of Hansen (1982) outlined in Chan et al (1992) for evaluating various short term interest rate models. We find that these interest rate models can be used to study exchange rates. Unfortunately, the out-of-sample forecasts by GMM yielded intuitively too large sum of squared errors (SSE) for some choices of the instruments mentioned in Table 0. Hence this paper proposes two new estimation methods, SSA-EF and NCV, that do not depend on any arbitrary choice of instruments. Before explaining the new estimation methods let us review the SDE, traditionally used to model short term interest rate dynamics.

A number of specifications have been proposed to capture the dynamic behavior of the short term interest rate^l. We use a general SDE to characterize the short term behavior of the exchange rate x. This differential equation specification is similar to that used to describe the short term interest rate processes and takes the following form:

(1) dx = (a + bx)dt + sxⁿdz

Remark I (Drift part): In our context, dx is the change in spot exchange rate, (a + bx)dt is the drift part. Now the conditional mean of dx (in a long-run equilibrium) is (a + bx)dt = k(m - x), which depends on x through b. The notation km = a and k = -b is intended to clarify its mean reversion toward m at the speed of adjustment k. This then represents mean reverting dynamics, if and only if k > O. Usually, statistically significant negative b values prevent explosive growth of the drift part.

Remark 2 LDiffusion part): Since the conditional variance of dx is (sxⁿ)², the volatility (measured by the standard deviation sxⁿ) obviously depends on x, the level of the exchange rate, through the important parameter n. When -n> 1, volatility is highly sensitive to the level. The dz in (1) represents a standard Wiener process or Brownian motion, Campbell, et al (1997, p. 344). The increments dz are normal random variables with E(dz)=O, and variance E(dz)²=dt.

Depending on the restrictions imposed on the parameters a, b, and n in equation (1) one obtains several nested models. We retain the names of these alternative models in accordance with their corresponding interest rate counterparts. Table I lists the nine models (including the unrestricted) considered here, and explicitly indicates parameter restrictions for each model. We estimate a discrete time specification of equation (1), using daily, weekly, and monthly values for the British pound and the German mark expressed in terms of one US dollar.

Table 1

Nine Models Considered for Short-Term Exchange Rate Dynamics With Explicit Listing of Parameter Restrictions on the Unrestricted Model: dx = (a + bx)dt + sxndz
ID. Model Name	a	b	n	s	Stochastic Differential Equation
1.Brennan-Schwartz			1.0		dx= (a + bx)dt + sxdz
2. CIR SR			0.5		dx = (a + bx)dt + sx1/2dz
3. Vasicek			0		dx = (a + bx)dt + sdz
4. CIR VR	0	0	1.5		dx = sx3/2dz
5. Dothan	0	0	1.0		dx = sxdz
6. CEV	0				dx = bxdt + sxndz
7. GBM	0		1.0		dx = bxdt + sxdz
8. Merton		0	0		dx = adt + sdz

The first three models impose no restrictions on either a or b. Models 4 and 5, set both a and b equal to zero, while Models 6, 7 and 8 set either a or b equal to zero. Model 1 used by Brennan and Schwartz (1980) implies that the conditional volatility of changes in the exchange rate is proportional to the level of the exchange rate x. Model 2 is the well known square root model of Cox Ingersoll and Ross (CIR) (1985). It implies that the conditional volatility of changes in the exchange rate (dx) is proportional to the square root of the level (x) of the exchange rate. Model 3 is the Omstein-Uhlenbeck diffusion process used by Vasicek (1977). The implication of this specification is that the conditional volatility of changes in the exchange rate x is constant. Model 4 used by CIR (1980) and by Constantinides and Ingersoll (1984) indicates that the conditional volatility of changes in the exchange rate is highly sensitive to the level of the exchange rate. Model 5 was used by Dothan (1978). Model 6 is the constant elasticity of variance (CEV) process proposed by Cox (1975) and Cox and Ross (1976). Model 7 is the famous geometric Brownian motion (GBM) process used by Black and Scholes (1973). Finally, Model 8 is used by Merton (1973) to represent Brownian motion with drift.

Estimating the parameters of many of these models by the maximum likelihood method requires closed forms of their likelihood functions, which are often either unavailable or too difficult to implement. For example, the CIR-SR implies a noncentral chi² and the GBM implies a lognormal density, for which closed forms are available. However, the closed form for the CEV model is difficult to implement as it involves the Bessel functions, and for some models no closed form is available. The appeal of the GMM for estimating the SDEs is that it does not require such closed forms.

Section 2 describes the three estimation methodologies starting with the GMM method. Sections 2.1 and 2.2 describe the newer SSA-EF and NCV methods. Section 3 describes the 17-year data and its source. Section 4 presents the estimation and forecasting results. Section 5 summarizes and concludes the paper.

2. THE GMM ESTIMATION METHODOLOGY

In this section we briefly review the traditional GMM methodology² and explain how one uses it to estimate the parameters of the nine SDEs. As several previous studies have done we estimate the parameters in equation (1) using a discrete-time specification³. The discrete-time specification of equation (1) can be written as

(2) x_t+1 - x_t = a + bx_t + z_t+1

(3) E[z_t+1] = 0

(4) E[(z_t+1)²] = s²(x_t)²ⁿ

Let theta be the vector of parameters a, b, s², and -n. The GMM technique requires the estimation of the parameters in theta using a set of instruments h_t that satisfy the orthogonality condition E[f_t(theta)] = 0, where f_t(theta) = z_t cross product of h_t. That is, f_t is a vector of cross-products of each instrument in h_t with each element of the residual vector z_t. Having obtained the GMM estimates of a, b, n, and s, we need to compute the out-of sample forecasts. It is convenient to discuss this in the following subsection where we explain the rationale used for the forecasting equation (11).

2.1. Small-Sigma Asymptotics - Estimating Functions (SSA-EF) Methods

Vinod (1996) proposes an alternative to the GMM using Godambe (1960) and Durbin's (1960) estimating functions (EFs). The EFs are defined as functions of data and parameters, g(y,theta)=O. The EF estimators are the roots of g=O, and need not be different from the familiar maximum likelihood, or instrumental variables estimators. Surveys by Godambe and Kale (1991), Dunlop (1994), Liang and Zeger (1995) explain that EF estimators can be used when the ML fails. For SDE estimation the EFs have the same appeal as the GMM. Furthermore, EFs satisfy an appealing small-sample property of Gauss consistency from early 19th century, which requires that the method of estimation should yield the correct estimates when all model equation errors are identically zero. Kadane's (1970) small-sigma asymptotics (SSA) explained in Vinod and Ullah (1981, p. 162) can be viewed as an implementation of Gauss's notion of consistency. Kadane writes the error as su, m, by injecting an additional constant s~ and finds limits as that constant s~ goes to 0. This constant s~ should not be confused with the s of the SDE in (1).

To illustrate SSA-EF, Vinod (1996) considers a representative agent model leading to consumption-based capital asset pricing (C-CAPM) model, see Tauchen (1986) and Ogaki (1993). The C-CAPM is traditionally estimated by GMM methods. The key idea is to rewrite the relevant expression from economic theory as E[g~(y,theta)]= 1, where E is the expectation operator and g~ represents a nonlinear expression. It is convenient to explain later, in our context, how a simplification arises from the fact that log of 1 is zero. The SSA-EF estimates of discounting parameter b and risk aversion parameter n of the C-CAPM model involve simple regressions. More importantly, they are economically more meaningful than the GMM estimates, since b < I and n < 2. To implement the key idea, we use (4) as the relevant expression from the theory of exchange rate dynamics, and write

(5) E [s^-2 (x_t)^-2n(z_t+1)²] = 1

When we remove the expectation operator from (5) we have individual observations, not their overall average. Hence we must introduce an error term (using the small-sigma notation) s~ u_t+1 , which should be zero when the expectation is evaluated E(s~ u_t+1)=O. Hence (5) becomes:

(6) [s^-2 (x_t)^-2n(z_t+1)²] = 1 + s~ u_t+1

Now take logs of both sides to yield:

(7) -2 ln(s) - 2n ln(x_t)+ 2 ln z_t+1 = ln(1 + s~ u_t+1)

where the right hand side can be written as s~ u_t+1 - (s~)²(u_t+1)²/2 + (s~)³(u_t+1)³/3 - .... The small sigma asymptotics means s~ 0, and a linear approximation means that we omit all terms with s~^j for j > 2. Then the right hand side of (7) becomes w_t+1 = s~ u_t+1, which is viewed as just another true unknown regression error w_t+1. Next, we replace the z_t+l in (7) by x_t+l - x_t - a - b x_t] from (2) to yield

(8)- ln(s^-2) - n (x_t)2 + ln[x_t+l - x_t - a - b x_t]² = W_t+1

This is SSA-EF nonlinear regression equation for the four unknowns a, b, n and s. A convenient iterative estimation uses two regressions. The first regression uses (2) to estimate a and b, while the second regression uses (8) to estimate n and s. For brevity, denote z_t+1= [x_t+l - x_t - a - b x_t]. To avoid the log of a zero in computing ln(z_t+1)² in (8), we "winsorize" with a tolerance constant⁴ (=O.0001, say) as follows. First initialize winsorized z_t+1 denoted by wins(z_t+1) = z_t+1 for all relevant values of t. Only if z_t+1 < , we do not compute its log. Instead, we replace the small number z_t+1 by , if z_t+1 > 0; and by (-), if z_t+1 <O. Our second SSA-EF regression involves rearranging (8) as

(9) ln(wins(x_t+l - x_t - a^ - b^ x_t))² = d + n ln(x_t)2 + W_t+1

where d=ln(s^-2) . Thus s^{-^}=exp(O.5 d^), and n is estimated as the slope coefficient.

Discrete Approximation to SDE: For the iterations we need to use these estimates of s and n in conjunction with those of a and b. Let us reconsider (5) without the E operator s² = (x_t)^-2n (z_t+1)². Since s^-2 > 0, only the positive square root of both sides is of interest. Hence we can substitute z_t+1 = s(x_t)ⁿ in (2) to read

(10) x_t+l - x_t - s(x_t)ⁿ = a + b x_t

Upon substituting the estimates of parameters we have a forecasting equation

(11) x_t+l = x_t + s^(x_t)^{n^} + a^ + b^ x_t + forecast error

This derivation represents a new way of implementing the diffusion process. It provides a convenient method of approximately discretizing the continuous SDE of (1). See Remark 2, above for relevant interpretations.

Substituting the estimates of s and n on the left side of (10) yields a regressand for each iteration. Denote the new estimates by a^(i)+ b^(i) for i-th iteration. Now let i=l, and compute the corresponding iterated estimates of s and n from a regression similar to (9):

(12) ln(wins(x_t+l - x_t + a^(1) + b^(1) x_t))² = d + nln(x_t)² + w_t+l

Hence we have s^(1)=exp( - 0.5 d^) and n^(1) from the intercept and the slope. Clearly for any i>l, one can compute the two simple linear regressions (10) and (12); thereby avoiding nonlinear estimation which is often sensitive to initial values of a^, b^, s^ and n^. The iterations can end when the absolute value of the change in parameter estimates is less than a tolerance constant

(0.001, say).

Since this paper emphasizes forecasting, we reserve a reasonable number (=N') of the original N observations for out-of-sample forecasting. Make all estimates by using only the initial (N - N') observations. Since the true values of the last x_t are known, we base the criterion for ending the iterations on the N' x I vector of forecast errors ^(i). For the i-th iterate, let us define the typical forecast error by:

(13) ^_t+1(i) = x_t+1 - [a^(i) + b^(i)x_t + s^(i)(x_t)^{n^(i)}]

The sum of squares of forecast errors for the i-th iterate is denoted by SSE = [^'(i) ^(i)]. At each iteration, we revise the current i-th estimates of ( a, b, s and n) only if the SSE is reduced. If SSE is increased by the (i+l)th iteration, we reject the (i+l)-th estimates and compute the next iteration until SSE is reduced. Since we are going to compare the out-of-sample forecasting performance of SSA-EF with GMM, we exclude N' extra observations in SSA-EF estimation. The N' observations are used in the choice of SSA-EF estimates based on reduction of the SSE over the set of N' observations. We make sure that both GMM and SSA-EF reserve the same observations for out-of-sample forecasting.

If both a and b are preassigned to be zero, (CIR-VR and Dothan models) there is no need to estimate them by the regression (10). Then the left side of (12) simplifies to be ln(wins(x_t+l - x_t))². When n is preassigned as in seven of nine models, (1O) simplifies to a version where n is simply replaced by its preassigned value. Obvious care is needed when parameters are preassigned. The computer program should (i) avoid taking logs of zeros, (ii) force the regression through the origin when the intercept is absent, and (iii) use the sample mean of the dependent variable as the regression coefficient when the regressor becomes a column of ones due to the preassignment of parameters.

In structural estimation, the usual F tests involve a comparison of the restricted residual sum of squares (RRSS) with the unrestricted (URSS). The choice of the model may be based on a ranking of the estimated F values for each of the eight models in comparison with the unrestricted model. We should also require that each parameter estimate be meaningful in terms of economic theory. These comparisons focus on in-sample fits only. Since we have chosen the out-of-sample forecasting as our criterion, we do not implement the traditional F tests. Instead, we use the SSE ranks for comparisons.

2.2. Numerical Conditional Variance (NCV) Method

Recall the discretization of the SDE (1) by (2) and (4). It is clear that at each date t, (4) involves a conditional variance based on all past information till date t-1, ignoring all subsequent data for t+l onward. Since numerical estimation is not difficult these days, we suggest directly computing conditional variances for all t, except for a few (say, five) initial observations. For each t we suggest a numerical brute force estimate of (z_t)² obtained from the variance of residuals of (2) using the data till date t- I only. At t=6, we start with five initial observations to compute the regression in (2) to estimate a^ and b^. There will be five regression residuals, some of which may well be zero. Next, we compute their variance as an estimate of (z₆)² associated with the 6th date. For each additional date, the NCV method involves recomputing the a^ and b^, a new sets of residuals and storing their variance. Thus, we create a time series of (z_t)² for t=6 to t=N of length N - 5 representing the conditional variances implicit in the left side of (4). Taking logs of (4) we have the NCV estimation equation for n and s.

(14) ln (z^_t+1)² = ln(s²) + nln((x_t)²)+ _t+1 (t=5, ..., N-1)

Having obtained the NCV estimates of a, b, n and s o, we compute the out-of-sample forecasts from the forecasting equation (11). Again, we make sure that GMM, SSA-EF and NCV reserve the same observations for out-of-sample forecasting.

3. THE DATA

The data set used in this analysis was obtained from Bekaert (1995). It consists of daily observations on the British pound and Deutsche mark for the period January 1, 1975 to July 19, 1991. All values are the averages of the bid and ask rates. For the weekly and monthly analysis the data was sampled each Wednesday. We present summary statistics for the data in Table 2 and Table 3. Note that we have 17 years of data and approximately one year of data (250 observations in daily data, 50 in weekly data and 12 in monthly data) were reserved for out-of-sample forecasting. The SSA-EF further excluded about six months of data for use in its iterative estimation of parameters. All iterations converged in less than nine iterations. For the NCV method we excluded initial five observations to start the numerical computation of conditional variances.

Table 2. Summary Statistics Means, Standard deviations and signs of autocorrelations of daily, weekly and monthly changes in the value of the British pound from January 1, 1975 to July 19, 1991.
Variable: xt+l-xt	obs.	mean	s.d.	Are p1 to p20 consistently + or -
Daily Changes	4144	.000042	.004389	No
Weekly Changes	828	.000212	.009460	No
Monthly Changes	206	.000979	.019643	No

Table 3. Summary Statistics Means, Standard deviations and signs of autocorrelations of daily, weekly and monthly changes in the value of the Deutsche Mark January 1, 1975 to July 19, 1991.
Variable: xt+l-xt	obs.	mean	s.d.	Are p1 to p20 consistently + or -
Daily Changes	4144	-.000153	.015853	No
Weekly Changes	828	-.000753	.033394	No
Monthly Changes	206	-.002785	.071338	No

4. THE ESTIMATION RESULTS

In this section we present the daily, weekly and monthly frequency results of the three estimation methods GMM, SSA-EF and NCV for both currencies. Tables 4 through 9 report the parameter estimates, their appropriate t-statistics, predicted out-of-sample SSE for the nine models and their SSE ranks. We divide all these tables into three sections for the three estimators compared here, with the GMM at the top, SSA-EF in the middle, and NCV at the bottom. In light of remark 1, our one-sided tests for b for the unrestricted model mostly have the negative sign and absolute t-values are mostly larger than 1.645, (one tail test at 5%). This suggests evidence of mean reversion for the majority of estimates. Very few cases have statistically insignificant b^ for the SSA-EF, namely Vasicek model for DM in Tables 7 to 9 and CIR-SR for DM in Table 9. The advantage of the SSA-EF is most apparent in the estimation of b. They are all negative, suggesting mean reversion. By contrast, the GMM method gives four significantly positive b^ for the DM in table 9, suggesting an unrealistic explosive behavior. The NCV has some positive b^'s, for the CEV and GBM models for BP, but they are statistically insignificant. All NCV b^'s are negative, but insignificant for the DM currency in tables 7 through 9. Although the NCV may be superior to the GMM, it is not superior to the SSA-EF.

For the GMM in section 1 of tables 6 and 9, high standard errors imply a failure to reject n=O in the unrestricted model. The magnitudes of GMM n^ do suggest a strong dependence of conditional volatility (of exchange rate changes) on the level of exchange rate. The GMM estimate n^ is negative in table 9 for the unrestricted model, but statistically insignificant. The GMM n^ is 0.437 in table 6, 0.96 in table 8, and larger than unity in all other cases where it is estimated (rather than preassigned). Similar strong dependence is also suggested by the SSA-EF and NCV estimators. For the SSA-EF, the t-values are all larger than 1.645 with only two exceptions (for CEV for BP using monthly data in table 6 and weekly DM data in table 8). The t-values associated with the NCV estimator n^ are all very large, except for the unrestricted model using DM monthly data (table 9). Generally large n^ for the unrestricted model from the SSA-EF estimation implies that the conditional volatility is highly sensitive to the level of exchange rates. Since a comparison between estimators may properly rely on the out-of-sample predicted SSE. By that criterion, the performance of GMM was extremely poor relative to the newer methods for some choices of instruments. For example, for the choice of instruments in Table 10 the SSE by GMM is about 8000 times that of the SSE by SSA-EF for the unrestricted case using daily British pound data. We needed several choices of instruments to get the performance of the GMM to be similar to that of the newer methods⁵.

Our tables report rankings as Rank= 1 for the model with the lowest SSE and the Rank=9 for the model with the largest SSE. However, the differences among the SSE values by different estimators are not large enough to conclusively suggest a preference for one over the others. If we must compare, an average rank across all models and all data sets may be considered. For the SSA-EF estimator, Vasicek method is the best with the average rank of 2.333. For the NCV estimator CIR-VR method has the average rank 1. For the GMM estimator, the lowest average rank is 3.667 for the Brennan-Schwartz model. We conclude that the average rank does not lead to a clear winner.

5. CONCLUSION

In this paper we show that the models used to characterize short term interest rate movements can also capture the dynamics of short term exchange rates. We consider eight distinct models nested within a general dynamic specification. The interpretation of the coefficients of the general model is given in remarks 1 and 2. We estimate the coefficients of these models using daily, weekly and monthly data on the British pound and the Deutsche mark.

The results of the tests for daily, weekly and monthly exchange rates using the new estimators indicate significant and negative b implying mean reversion. The GMM fails to support negative b in tables 4 and 6 for the CEV and the GBM models. A similar GMM failure occurs in table 5 for the CEV, and in table 9 for four out of nine models. The NCV fails to support negative b in tables 4 to 6 for the CEV and the GBM models. The positive n^>0.5 with large t-values from the GMM and the SSA-EF unrestricted models imply dependence of conditional volatility (of exchange rate changes) on the level of exchange rate. However, the NCV estimator supports less strong dependence. In short, the SSA-EF yields self-consistent dynamics. Also, our empirical evaluation techniques have distinct implications for pricing and hedging of currency dependent contingent claims, and deserve further attention.

This paper suggests two new methods as alternatives to the usual GMM method for estimating a fundamental stochastic differential equation. The new methods are flexible enough to estimate all nine models, whereas the GMM fails to converge for some models for some choices of instruments. Since a comparison between estimators may rely on the out-of-sample predicted SSE, the performance of the GMM relative to the newer methods can be poor. It took several GMM trials with alternative instruments to achieve the good out-of-sample forecasting performance reported here. Hence, for exchange rate models we should avoid total reliance on the GMM, unless we calibrate its out-of-sample forecasting performance, perhaps against the simpler SSA-EF.

E-mail: vinod@murray.fordham.edu
E-mail: psamanta@manhattan.edu

ACKNOWLEDGMENTS

We thank the seminar participants at Fordham University and at the 17th Annual International Symposium on Forecasting 97, June 20, 1997, in Barbados.

ENDNOTES

1. For a listing of some of these models the reader may refer to Chan et. al (1992) and Fabozzi and Fabozzi(1995).

2. The reader interested in a rigorous treatment of the GMM technique can refer to Hansen (1982), Hansen and Singleton (1988), Chamberlain (1987), Davidson and MacKinnon (1993), Hamilton (1994), Newey (1985), Newey and West (1987) and Pagan and Wickens (1989).

3. The discretized process is an approximation to the continuous process and we acknowledge the temporal aggregation issue associated with this.

4. Winsorization is a terminology familiar in statistical literature dealing with robust estimation. It is an alternative to trimming where instead of omitting extreme values they are kept at the farthest extreme value. We used a different tolerance constant =0.00001 for calibration and found that our results are not sensitive to its choice.

5. One way of verifying that the out of sample criterion is used is to note that the unrestricted model, which must have the lowest 'within sample' error sum of squares, need not give the best out of sample forecasts.

REFERENCES

Bekaert, Geert (1995) The time variation of expected returns and volatility in foreign exchange markets. Journal of Business and Economic Statistics 13, 397-408.

Black, Fisher and Myron Scholes (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81, 63 7-654.

Brennan, Michael J. and Eduardo S. Schwartz (1980) Analyzing convertible bonds. Journal of Financial and Quantitative Analysis 15, 907-929.

Campbell, J. Y., A. W. Lo and A. C. MacKinley (1997) The Econometrics of Financial Markets, Princeton University Press, Princeton, NJ.

Chamberlain, G. (1987) Asymptotic efficiency in estimation with conditional moment restrictions. Journal of Econometrics 34, 305-334.

Chan, K. C., G. Andrew Karolyi, Francis A. Longstaff and Anthony B. Sanders (1992) An empirical comparison of alternate models of the short-term interest rate. Journal of Finance 57, 1209-1027.

Constantinides, George M. and Jonathan E. Ingersoll (1984) Optimal bond trading with personal taxes. Journal of Financial Economics 13, 299-3 3 5.

Cox, John C. (1975) Notes on option pricing I: constant elasticity of variance diffusions. Working paper, Stanford University.

______, Jonathan E. Ingersoll and Stephen A. Ross (1980) An analysis of variable rate loan contracts. Journal of Finance 35, 3 89-403.

______, Jonathan E. Ingersoll and Stephen A. Ross (1985) A theory of the term structure of interest rates. Econometrica 53, 385-407.

______, and Stephen Ross (1976) The valuation of options for alternative stochastic processes. Journal of Financial Economics 3, 145-166.

Davidson, J. and J. MacKinnon (1993) Estimation and Inference in Econometrics. Oxford University Press, NY.

Dothan, Uri L. (I 978) On the term structure of interest rates. Journal of Financial Economics 6, 59-69.

Durbin, J. (I 960) Estimation of parameters in time-series regression models. Journal of the Royal Statistical Society, Ser. B, 22, 139-153.

Fabozzi, Frank J. and T. Dessa Fabozzi, eds. (1995) The Handbook of Fixed-Income Securities. Irwin Professional Publishing, Burr Ridge, Il.

Godambe V. P. (1960) An optimum property of regular maximum likelihood estimation. Annals of Mathematical Statistics 31, 1208-1212.

Godambe, V. P. and B. K. Kale (1991) Estimating functions: an overview, Ch. I in V. P. Godambe (ed.) Estimating functions, (Oxford: Clarendon Press).

Hamilton, J. (1994) Time Series Analysis. Princeton University Press, Princeton, NJ.

Hansen, Lars Peter (1982) Large sample properties of Generalized Method of Moments estimation. Econometrica 50, 1029-1054.

______ and K. Singleton (1988) Efficient estimation of asset pricing models with moving average errors. Mimeo, Department of Economics, Carnegie Mellon University.

Kadane, J. B. (1970) Testing overidentifying restrictions when disturbances are small. Journal of the American Statistical Association 65, 182-185.

Liang K. and S. L. Zeger (I 995) Inference based on estimating functions in the presence of nuisance parameters. Statistical Science 10, 158-173.

Merton, Robert C. (I 973) Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141-183.

Newey, W. (I 985) Generalized method of moments specification testing. Journal of Econometrics 29, 229-256.

______ and K. West (1987) Hypothesis testing with efficient method of moments estimation. International Economic Review 28, 777-787.

Ogaki, M. (1993). Generalized method of moments estimation. in G.S. Maddala, C.R. Rao, and H.D. Vinod (eds.) Handbook of Statistics: Econometrics, Vol. 11. North Holland, Elsevier, New York. Chap. 17, 455-488.

Pagan, A. and M. Wickens (1989) A survey of some recent econometric methods. Economic Journal 99, 962-1025.

Tauchen, G. (1987). Statistical properties of generalized method of moments. Journal of Business and Economic Statistics 4 (4), 397-425.

Vasicek, Oldrich (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177-188.

Vinod, H.D. (1996) Using Godambe-Durbin Estimating functions in Econometrics in I. Basawa, V.P. Godambe and R. Taylor (Eds.) Selected Proceedings of the Symposium on Estimating Functions, IMS Lecture Notes Series (in press).

Vinod, H.D. and Ullah, A. (1991) Recent Advances in Regression Methods. Marcel Dekker, New York

TABLE 4. British Pound Daily Data Parameter Estimates for GMM, SSA-EF and NCV and Forecast Errors
	a	b	n	s	SSE (Rank).(ID)
1) GMM BP-daily
Unrestricted	0.006643	-0.01115	-1.463	3.76E-06	0.005765
t-value	10.36	-10.56	-359.5	24.91	9
Brenn-Sch	0.002229	-0.00393	1	4.50E-05	0.005732
t-value	5.116	-5.095	-	25.57	1.1
CIR-SR	0.00223	-0.00397	0.5	2.49E-05	0.005734
t-value	5.064	-5.094	-	24.94	2.2
Vasicek	0.002245	-0.00404	0	1.34E-05	0.005737
t-value	5.054	-5.138	-	24.06	3.3
CIR-VR	0	0	1.5	7.88E-05	0.005745
t-value	-	-	-	25.99	7.4
Dothan	0	0	1	4.53E-05	0.005745
t-value	-	-	-	25.6	8.5
CEV	0	7.32E-05	2.02	0.000135	0.005741
t-value	-	0.7812	10.16	4.984	5.6
GBM	0	9.94E-05	1	4.54E-05	0.005741
t-value	-	1.059	-	25.49	4.7
Merton	5.70E-05	0	0	1.37E-05	0.005741
t-value	1.07	-	-	24.05	6.8
2) SSA-EF BP-daily
Unrestricted	0.00211	-0.00638	1.807	0.004281	0.005738
t-value	5.244	-9.391	16.21	15.38	4
Brenn-Sch	0.000718	-0.00389	1	0.002734	0.005737
t-value	1.785	-5.729	-	50.24	3.1
CIR-SR	-7.72E-05	-0.00249	0.5	0.002071	0.005737
t-value	-0.1918	-3.672	-	49.69	1.2
Vasicek	-0.00085	-0.00116	0	0.001568	0.005737
t-value	-2.113	-1.704	-	48.9	2.3
CIR-VR	0	0	1.5	0.003722	0.006168
t-value	-	-	-	51.66	9.4
Dothan	0	0	1	0.002776	0.006165
t-value	-	-	-	50.8	8.5
CEV	0	-0.00293	1.848	0.004446	0.005776
t-value	-	-24.21	16.84	15.62	7.6
GBM	0	-0.0027	1	0.002735	0.005746
t-value	-	-22.4	-	50.36	6.7
Merton	-0.00153	0	0	0.00157	0.005734
t-value	-21.39	-	-	48.96	5.8
3) NCV BP-daily
Unrestricted	0.00072	0.00072	0.40313	0.004465	0.008535
t-value	1.798	-1.7432	33.394	142.6	7
Brenn-Sch	0.00072	-0.00118	1	0.006215	0.008399
t-value	1.798	-1.7432	-	371.41	4.1
CIR-SR	0.00072	-0.00118	0.5	0.004711	0.008511
t-value	1.798	-1.7432	-	470.11	6.2
Vasicek	0.00072	-0.00118	0	0.003571	0.008642
t-value	1.798	-1.7432	-	417.85	9.3
CIR-VR	0	0	1.5	0.008199	0.0082
t-value	-	-	-	268.23	1.4
Dothan	0	0	1	0.006216	0.008289
t-value	-	-	-	371.35	2.5
CEV	0	1.84E-05	0.40353	0.004467	0.008436
t-value	-	0.15632	33.398	142.48	5.6
GBM	0	1.84E-05	1	0.006216	0.008306
t-value	-	0.15632	-	371.31	3.7
Merton	3.27E-05	0	0	0.003572	0.008576
t-value	0.46751	-	-	417.51	8.8
Note: Along the rows market t-value in the SSE column the number before the decimal point is the rank, the number after the decimal point is the model id (identity) from Table 1.

TABLE 5. British Pound Weekly Data Parameter Estimates for GMM, SSA-EF and NCV and Forecast Errors
	a	b	n	s	SSE (Rank).(ID)
1) GMM BP-weekly
Unrestricted	0.02761	-0.04891	1.556	0.000393	0.005424
t-value	15.79	-16.39	4.568	2.743	1
Brenn-Sch	0.02793	-0.0497	1	0.000228	0.005432
t-value	15.58	-16.23	-	14.92	2.1
CIR-SR	0.02811	-0.05029	0.5	0.000129	0.005443
t-value	15.32	-16.03	-	14.69	3.2
Vasicek	0.02818	-0.05068	0	7.16E-05	0.005456
t-value	15.06	-15084	-	14.26	5.3
CIR-VR	0	0	1.05	0.000365	0.005461
t-value	-	-	-	15.63	6.4
Doothan	0	0	1	0.000209	0.005464
t-value	-	-	-	15.37	7.5
CEV	0	3.17E-05	4.07	0.003882	0.005448
t-value	-	0.07159	2.422	0.7329	4.6
GBM	0	-0.00105	1	0.000203	0.005541
t-value	-	-2.475	-	14.75	9.7
Merton	-0.00029	0	0	6.23E-05	0.005507
t-value	-1.187	-	-	13.8	8.8
2) SSA-EF BP-weekly
Unrestricted	0.007014	-0.01874	1.833	0.01084	0.005427
t-value	3.631	-5.747	7.662	7.163	4
Brenn-Sch	0.003392	-0.01226	1	0.006815	0.005422
t-value	1.757	-3.761	-	23.26	3.1
CIR-SR	0.00141	-0.00878	0.5	0.00516	0.005422
t-value	0.7303	-2.693	-	22.95	1.2
Vasicek	-0.00052	-0.00545	0	0.003909	0.005422
t-value	-0.2678	-1.671	-	22.58	2.3
CIR-VR	0	0	1.5	0.009127	0.00574
t-value	-	-	-	23.19	8.4
Dothan	0	0	1	0.006896	0.005742
t-value	-	-	-	23.36	9.5
CEV	0	-0.00694	1.575	0.009439	0.005508
t-value	-	-11.91	6.511	7.083	7.6
GBM	0	-0.00656	1	0.006745	0.005465
t-value	-	-11.32	-	22.83	6.7
Merton	-0.00365	0	0	0.003868	0.005452
t-value	-10.65	-	-	22.24	5.8
3) NCV BP-weekly
Unrestricted	0.003508	-0.00571	0.87328	0.011517	0.007156
t-value	1.8086	-1.7473	20.677	40.846	6
Brenn-Sch	0.003508	-0.00571	1	0.012352	0.007138
t-value	1.8086	-1.7473	-	135.27	5.1
CIR-SR	0.003508	-0.00571	0.5	0.009369	0.007214
t-value	1.8086	-1.7473	-	129.66	8.2
Vasicek	0.003508	-0.00571	0	0.007106	0.007304
t-value	1.8086	-1.7473	-	109.17	9.3
CIR-VR	0	0	1.5	0.016439	0.00691
t-value	-	-	-	120.81	1.4
Dothan	0	0	1	0.0124	0.00691
t-value	-	-	-	137.19	2.5
CEV	0	0.00011	0.85748	0.011462	0.007015
t-value	-	0.193676	20.604	41.45	4.6
GBM	0	0.00011	1	0.012402	0.006997
t-value	-	0.19376	-	137.03	3.7
Merton	0.00017	0	0	0.007143	0.00721
t-value	0.50481	-	-	111.02	7.8
Note: Along the rows market t-value in the SSE column the number before the decimal point is the rank, the number after the decimal point is the model id (identity) from Table 1.

TABLE 6. British Pound Monthly Data Parameter Estimates for GMM, SSA-EF and NCV and Forecast Errors
	a	b	n	s	SSE (Rank).(ID)
1) GMM BP-monthly
Unrestricted	0.05483	-0.09061	0.4374	0.0007101	0.004495
t-value	6.019	-6.214	0.6154	1.361	1
Brenn-Sch	0.03491	-0.05803	1	0.0008429	0.004639
t-value	4.176	-4.175	-	8.219	2.1
CIR-SR	0.03534	-0.05942	0.5	0.0004786	0.004692
t-value	4.105	-4.1	-	8.036	4.2
Vasicek	0	-0.06085	0	0.0002657	0.00475
t-value	-	-4.078	-	7.735	5.3
CIR-VR	0	0	1.5	0.001448	0.004947
t-value	-	-	-	8.773	8.4
Doothan	0	0	1	0.000853	0.004968
t-value	-	-	-	8.711	9.5
CEV	0	0.001539	4.329	0.01945	0.004679
t-value	-	0.8905	7.248	2.145	3.6
GBM	0	0.00125	1	0.000854	0.004864
t-value	-	0.7125	-	8.638	7.7
Merton	0.0008578	0	0	0.0002719	0.004858
t-value	0.9094	-	-	8.175	6.8
2) SSA-EF BP-monthly
Unrestricted	0.01424	-0.03608	1.048	0.01343	0.004793
t-value	1.814	-2.72	1.952	3.187	6
Brenn-Sch	0.01389	-0.03547	1	0.01308	0.004792
t-value	1.77	-2.674	-	10.48	5.1
CIR-SR	0.01008	-0.02879	0.5	0.009922	0.004789
t-value	1.285	-2.717	-	10.48	3.2
Vasicek	0.006395	-0.02239	0	0.007496	0.004792
t-value	0.8149	-1.688	-	10.38	4.3
CIR-VR	0	0	1.5	0.01949	0.004265
t-value	-	-	-	12.25	1.4
Dothan	0	0	1	0.01458	0.004267
t-value	-	-	-	11.96	2.5
CEV	0	-0.01291	0.4823	0.01036	0.004916
t-value	-	-5.456	0.9624	3.413	8.6
GBM	0	-0.01326	1	0.01397	0.004981
t-value	-	-5.58	-	11.29	9.7
Merton	-0.00718	0	0	0.008037	0.004902
t-value	-5.112	-	-	11.33	7.8
3) NCV BP-monthly
Unrestricted	0.014977	-0.02452	0.80807	0.022682	0.00483
t-value	1.8121	-1.7653	9.0836	19.543	7
Brenn-Sch	0.014977	-0.02452	1	0.025203	0.004802
t-value	1.8121	-1.7653	-	64.892	5.1
CIR-SR	0.014977	-0.02452	0.5	0.019153	0.004877
t-value	1.8121	-1.7653	-	63.697	8.2
Vasicek	0.014977	-0.02452	0	0.014555	0.004963
t-value	1.8121	-1.7653	-	54.804	9.3
CIR-VR	0	0	1.5	0.033607	0.004546
t-value	-	-	-	57.889	1.4
Dothan	0	0	1	0.025539	0.004593
t-value	-	-	-	66.819	2.5
CEV	0	0.000277	0.76963	0.022518	0.004638
t-value	-	0.11556	8.9357	20.185	4.6
GBM	0	0.000277	1	0.025554	0.004612
t-value	-	0.11556	-	66.596	3.7
Merton	0.000604	0	0	0.01475	0.004803
t-value	0.42329	-	-	57.023	6.8
Note: Along the rows market t-value in the SSE column the number before the decimal point is the rank, the number after the decimal point is the model id (identity) from Table 1.

TABLE 7. Deutsche Mark Pound Daily Data Parameter Estimates for GMM, SSA-EF and NCV and Forecast Errors
	a	b	n	s	SSE (Rank).(ID)
1) GMM DM-daily
Unrestricted	0.007354	-0.003561	1.561	1 .90E-05	0.04748
t-value	6.369	-6.89	138.1	23.06	9
Brenn-Sch	0.00683	-0.003298	1	4.55E-05	0.04745
t-value	4.372	-4.522	-	25.23	7.1
CIR-VR	0.006307	-0.003097	0.5	9.56E-05	0.04738
t-value	4.008	-4.215	-	24.87	4.2
Vasicek	0.005834	-0.002921	0	0.0001947	0.04734
t-value	3.685	-3.951	-	24.22	1.3
CIR-VR	0	0	1.5	2.10E-05	0.04738
t-value	-	-	-	25.73	3.4
Doothan	0	0	1	4.60E-05	0.04737
t-value	-	-	-	25.66	2.5
CEV	0	-0.0001232	1.473	2.19E-05	0.04746
t-value	-	-1.825	115.1	22.71	8.6
GBM	0	-9.99E-05	1	4.59E-05	0.04746
t-value	-	-1.047	-	25.64	6.7
Merton	-0.000183	0	0	0.0001974	0.04739
t-value	-0.8895	-	-	24.57	5.8
2) SSA-EF DM-daily
Unrestricted	0.00253	-0.00423	1.213	0.002518	0.04732
t-value	1.671	-6.332	10.77	10.99	1
Brenn-Sch	0.001085	-0.00356	1	0.002979	0.04734
t-value	0.717	-5.33	-	51.08	3.1
CIR-SR	-0.00223	-0.00204	0.5	0.004421	0.04736
t-value	-1.475	-3.054	-	50.88	4.2
Vasicek	0.00547	-0.00058	0	0.006556	0.04734
t-value	-3.613	-0.8653	-	50.34	2.3
CIR-VR	0	0	1.5	0.002089	0.05054
t-value	-	-	-	53.13	8.4
Dothan	0	0	1	0.003056	0.05175
t-value	-	-	-	52.73	9.5
CEV	0	-0.00314	1.159	0.002638	0.04762
t-value	-	-27.02	10.36	11.07	7.6
GBM	0	-0.0031	1	0.002995	0.04746
t-value	-	-26.73	-	51.78	5.7
Merton	-0.00679	0	0	0.006587	0.04748
t-value	-25.8	-	-	50.96	6.8
3) NCV DM-daily
Unrestricted	0.000984	-0.00053	-0.08293	0.014544	0.092032
t-value	0.68303	-0.83339*	-5.1958	78.334	9
Brenn-Sch	0.000984	-0.00053	1	0.006251	0.069696
t-value	0.68303	-0.83339*	-	238.41	4.1
CIR-SR	0.000984	-0.00053	0.5	0.009231	0.078057
t-value	0.68303	-0.83339*	-	304.07	5.2
Vasicek	0.000984	-0.00053	0	0.013633	0.089698
t-value	0.68303	-0.83339*	-	351.21	7.3
CIR-VR	0	0	1.5	0.004244	0.063292
t-value	-	-	-	188.48	1.4
Dothan	0	0	1	0.006268	0.069252
t-value	-	-	-	240.03	3.5
CEV	0	-0.0001	-0.07617	0.014507	0.090146
t-value	-	-0.90647	-4.8082	78.923	8.6
GBM	0	-0.0001	1	0.006268	0.068482
t-value	-	-0.90647	-	240.04	2.7
Merton	-0.0002	0	0	0.01367	0.087845
t-value	-0.77046	-	-	353.98	6.8
Note: Along the rows market t-value in the SSE column the number before the decimal point is the rank, the number after the decimal point is the model id (identity) from Table 1.

TABLE 8. Deutsche Mark Weekly Data Parameter Estimates for GMM, SSA-EF and NCV and Forecast Errors
	a	b	n	s	SSE (Rank).(ID)
1) GMM DM-weekly
Unrestricted	0.01399	-0.007144	1.426	0.0001068	0.04768
t-value	2.455	-2.715	5.869	2.552	2
Brenn-Sch	0.01411	-0.007145	1	0.0002055	0.04766
t-value	2.513	-2.758	-	15.57	1.1
CIR-SR	0.01227	-0.006316	0.5	0.0004242	0.04769
t-value	2.151	-2.399	-	15.19	3.2
Vasicek	0.01048	-0.005512	0	0.0008491	0.4772
t-value	1.807	-2.059	-	14.5	4.3
CIR-VR	0	0	1.5	9.44E-05	0.04832
t-value	-	-	-	15.58	6.4
Doothan	0	0	0	0.0002052	0.04828
t-value	-	-	-	15.58	5.5
CEV	0	-0.0006761	0.9669	0.0001993	0.04867
t-value	-	-1.524	0.2445	0.1761	8.6
GBM	0	-0.0007192	1	0.0002052	0.0487
t-value	-	-1.779	-	15.59	9.7
Merton	-0.001252	0	0	0.0008511	0.04853
t-value	-1.409	-	-	14.54	7.8
2) SSA-EF DM-weekly
Unrestricted	0.002661	-0.00859	0.8946	0.007572	0.0482
t-value	0.3738	-2.739	3.602	4.984	3
Brenn-Sch	0.004326	-0.00936	1	0.006967	0.04818
t-value	0.6076	-2.983	-	23.18	2.1
CIR-SR	-0.00343	-0.00581	0.5	0.013034	0.04823
t-value	-0.4822	-1.852	-	23.15	4.2
Vasicek	-0.01101	-0.00239	0	0.01533	0.04818
t-value	-1.546	-0.7629	-	22.98	1.3
CIR-VR	0	0	1.5	0.004889	0.04997
t-value	-	-	-	24.39	8.4
Dothan	0	0	1	0.007024	0.0508
t-value	-	-	-	23.1	9.5
CEV	0	-0.00739	0.8637	0.007697	0.04845
t-value	-	-13.57	3.404	4.88	5.6
GBM	0	-0.00744	1	0.006928	0.0487
t-value	-	-13.67	-	22.88	6.7
Merton	-0.01626	0	-0	0.01524	0.0488
t-value	-13.16	-	-	22.69	7.8
3) NCV DM-weekly
Unrestricted	0.003795	-0.00213	-0.11952	0.030955	0.080929
t-value	0.5602	-0.70952	-3.4578	36.3176	9
Brenn-Sch	0.003795	-0.00213	1	0.012931	0.063018
t-value	0.5602	-.70952	-	105.93	4.1
CIR-SR	0.003795	-0.00213	0.5	0.019096	0.069327
t-value	0.5602	-.70952	-	136.72	5.2
Vasicek	0.003795	-0.00213	0	0.0282	0.078282
t-value	0.5602	-.70952	-	161.45	8.3
CIR-VR	0	0	1.5	0.008805	0.058269
t-value	-	-	-	83.935	1.4
Dothan	0	0	1	0.013003	0.062648
t-value	-	-	-	107.62	3.5
CEV	0	-0.00048	-0.10747	0.030832	0.078231
t-value	-	-0.88954	-3.181	37.013	7.6
GBM	0	-0.00048	1	0.013002	0.06137
t-value	-	-0.88954	-	107.63	2.7
Merton	-0.00093	0	0	0.028356	0.075556
t-value	-0.77548	-	-	165.31	6.8
Note: Along the rows market t-value in the SSE column the number before the decimal point is the rank, the number after the decimal point is the model id (identity) from Table 1.

TABLE 9. Deutsche Mark Monthly Data Parameter Estimates for GMM, SSA-EF and NCV and Forecast Errors
	a	b	n	s	SSE (Rank).(ID)
1) GMM DM-montly
Unrestricted	-0.1406	0.06791	-2.893	0.1968	0.05078
t-value	-4.475	4.901	-1.034	0.2905	1
Brenn-Sch	-0.09995	0.04434	1	0.0008244	0.07331
t-value	-3.797	3.829	-	7.466	9.1
CIR-VR	-0.1091	0.04969	0.5	0.001843	0.07308
t-value	-4.044	4.202	-	7.251	8.2
Vasicek	-0.1166	0.05437	0	0.003985	0.07171
t-value	-4.219	4.502	-	6.907	7.3
CIR-VR	0	0	1.5	0.0004209	0.05351
t-value	-	-	-	10.07	5.4
Doothan	0	0	1	0.0009832	0.05328
t-value	-	-	-	10.08	3.5
CEV	0	-0.0006588	1.283	0.0006087	0.05377
t-value	-	-0.08659	0.1539	0.06868	6.6
GBM	0	-0.0003087	1	0.0009833	0.05344
t-value	-	-0.2051	-	10	4.7
Merton	-0.000225	0	0	0.004291	0.05253
t-value	-0.06473	-	-	9.054	2.8
2) SSA-EF DM-montly
Unrestricted	0.02592	-0.02924	1.038	0.01551	0.05215
t-value	0.8553	-2.192	2.113	2.518	2
Brenn-Sch	0.02456	-0.02862	1	0.01596	0.05217
t-value	0.8105	-2.145	-	11.69	4.1
CIR-SR	0.006768	-0.02048	0.5	0.02362	0.0523
t-value	0.2233	-1.535	-	11.53	6.2
Vasicek	-0.01059	-0.01266	0	0.03515	0.05217
t-value	0.3494	-0.9488	-	11.55	3.3
CIR-VR	0	0	1.5	0.01069	0.05192
t-value	-	-	-	11.64	1.4
Dothan	0	0	1	0.01531	0.0522
t-value	-	-	-	10.71	5.5
CEV	0	-0.0182	1.534	0.01012	0.05681
t-value	-	-7.84	2.914	2.349	9.6
GBM	0	-0.01791	1	0.0159	0.05499
t-value	-	-7.731	-	11.75	7.7
Merton	-0.03926	0	0	0.0355	0.05527
t-value	-7.455	-	-	12.04	8.8
3) NCV DM-montly
Unrestricted	0.016467	-0.00936	0.077539	0.063079	0.079173
t-value	0.56834	-0.72794	1.6123	23.985	8
Brenn-Sch	0.016467	-0.00936	1	0.030719	0.064821
t-value	0.56834	-0.72794	-	67.901	4.1
CIR-SR	0.016467	-0.00936	0.5	0.045371	0.071347
t-value	0.56834	-0.72794	-	98.219	5.2
Vasicek	0.016467	-0.00936	0	0.067011	0.080902
t-value	0.56834	-0.72794	-	115.77	9.3
CIR-VR	0	0	1.5	0.021072	0.060231
t-value	-	-	-	51.281	1.4
Dothan	0	0