Three models are estimated: a univariate autoregressive conditional heteroskedasticity (ARCH) model, an extended ARCH model using the random coefficient ...
ELSEVIER
International Journal of Forecasting 12 (1996) 483-493
Modelling optimal strategies for the allocation of wealth in • 1 multicurrency mvestments C o s t a s C h r i s t o u a, P . A . V . B . S w a m y b, G e o r g e
S. T a v l a s c'*
aRoom 8-548, International Monetary Fund, 700 19th Street, N.W., Washington, DC 20431, USA "Federal Reserve Board and Office of the Comptroller of the Currency, 250 E Street, S.W., Bank Research Division, Mail Stop 6-5, Office of the Comptroller of the Currency, Washington, DC 20219, USA "Room 7-210, International Monetary Fund, 700 19th Street, N.W., Washington, DC 20431, USA
Abstract
This paper analyzes rates of return on financial assets denominated in five major currencies and provides a framework for the determination of optimal strategies for the allocation of wealth in multicurrency investments. Three models are estimated: a univariate autoregressive conditional heteroskedasticity (ARCH) model, an extended ARCH model using the random coefficient (RC) procedure, and a pure RC model. A comparison of the forecasts of these models with those generated by a random walk model demonstrates that forecasts based on the RC/extended ARCH procedure are superior to those based on the random walk model and those based on direct ARCH estimation. These results could be useful for both international investors for the allocation of their wealth among fixed-income investment securities and central banks for the management of their external reserve assets. Keywords: ARCH models; Random coefficient models; Forecasting; Optimal portfolios
1. I n t r o d u c t i o n
Statistical estimation of a regression model is simplest if it satisfies all the classical linear least squares assumptions. (A regression model is the operational counterpart of a behavioral or technological equation of economic theory.) One can begin by estimating a model under these assump* Corresponding author. ~We have benefitted from helpful comments by Brian Henry and two referees. The views expressed are the authors' own and are not to be interpreted as those of the International Monetary Fund, the Board of Governors or staff of the Federal Reserve System, the Office of the Comptroller of the Currency, or of the US Department of the Treasury.
tions, but it is necessary to validate the model by comparing its performance in terms of prediction and explanation with other models, unless the classical linear least squares assumptions are thought to be physical truths. This situation is not the case in most instances, however, including those of models analyzed in this paper. Some special conditions need to be satisfied for linear or nonlinear regression models to coincide with stochastic laws. These conditions will not be satisfied if the functional form of an economic relationship is misspecified or if the regression coefficients are misinterpreted (see Swamy and Tavlas, 1995). The ability of theoretical models to perform well in prediction and explanation depends on how closely they coincide with
0169-2070/96/$15.00 ~ 1996 Elsevier Science B.V. All rights reserved PII S0169-2070(96)00673-5
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C. Chrbtou et al. I International Journal of Forecasting 12 (1996) 483-493
stochastic laws. Accordingly, appropriate estimation procedures under alternative sets of assumptions are necessary in order to deal with the fact that the true physical process is unknown. In what follows, we provide a method of dealing with this issue. This paper analyzes the results of alternative estimation procedures applied to the rates of return on financial assets denominated in five major currencies; the US dollar, the deutsche mark, the French franc, the Japanese yen, and the pound sterling. Such an analysis, however, cannot use the linear least squares assumption that the variance of the error term of a regression is constant, but should take into account the time-varying nature of asset-return volatility, as suggested in the finance literature (see Bodie et al., 1993, p. 349). Even this situation is not difficult to analyze if the variance of the error term of the equation for every rate of return is related with a known functional form to one or more observable variables or if knowledge of the functional form and the arguments of the variance function for every rate of return is a physical truth. A particular form for the variance function, which may or may not be true, is given by the autoregressive conditional heteroskedasticity ( A R C H ) model. This model is based on the insight that a natural way to vary the error variance over time is to view it as a conditional variance and make it a linear function of the squares of the given past values of the error itself, with positive coefficients. A less restrictive alternative to the A R C H model is known as a random coefficient (RC) model, which assumes that all of the coefficients of a regression equation change over time. The A R C H model specifies one type of variance function and the R C model specifies another. By relaxing some of the assumptions underlying the A R C H model, it can be brought closer to RC models. We derive such models and call them extended A R C H models. The purpose of this paper is to examine whether the A R C H , extended A R C H , and RC approaches give better predictions of the out-ofsample rates of return on financial assets denominated in the five major reserve currencies
than a naive (i.e. random walk) model. We should point out that this paper does not address the issue of whether the models suggested by economic theory are adequate for purposes of obtaining useful forecasts; previous studies have typically found that theoretical models often provide less accurate forecasts than random walk models. 2 The rationale used in this paper for evaluating models on the basis of forecast comparisons is provided by the cross-validation approach, which consists of splitting the data sample into two subsamples; the choice of a model, including any necessary estimation, is based on one subsample and its performance is assessed by measuring its prediction against the other subsample. The premise of this approach is that the validity of statistical estimates should be judged by data different from those used to derive them (see Mosteller and Tukey, 1977, pp. 36-40). To empirically implement the cross-validation approach, we first construct an A R C H model of the rates of return from short-term investments and then use that model to generate extended A R C H models. Further, we use the RC procedure to estimate both an RC model and an extended A R C H model. In order to evaluate the ability of the A R C H process to forecast out-of-sample rates of return, this paper compares predictions of an A R C H , an extended A R C H , and an RC model with those of a random walk model by calculating the root mean 2 In this connection, it is noteworthy that previous work comparing forecasts based on a r a n d o m walk model with those of a variety of conventional models has often found that the forecasts of the former model are superior to those based on the latter models. In this regard, perhaps the most f a m o u s results are those of Meese and Rogoff (1983, 1984) showing that a simple r a n d o m walk model has superior forecasting power to any of the conventional models of exchange rate determination in out-of-sample forecasting periods of 1 year or less. 3 We use the conditional expectations of future values given the sample values which are the optimal predictors with m i n i m u m m e a n square errors (MMSEs). Consequently, the use of R M S E is appropriate here. T h e reason is that the M M S E s imply quadratic loss functions and the conditional expectations are optimal relative to these loss functions. We cannot use any other loss function because the functional forms of predictors which are optimal relative to other loss functions are u n k n o w n (see Schinasi and Swamy, 1989).
c. Christou et al. / International Journal of Forecasting 12 (1996) 483-493
square forecast error ( R M S E ) corresponding to each model. 3 The r e m a i n d e r of this p a p e r is divided into three sections. Section 2 describes the A R C H model and its connections with the R C model; the section also describes the data used in the estimation and explains how to appraise the estimated models by using a form of cross-validation, notably, RMSEs. Section 3 provides both the within sample empirical results and the postsample forecasts of rates of return. Concluding r e m a r k s are presented in Section 4.
2. The models and data 2.1. The A R C H m o d e l
T h e A R C H model expresses the current conditional variance of total returns as a linear function of the squares of its past errors. 4 We utilize the A R C H model by assuming that the conditional mean, E ( y t l Y , _ l , Y,-2 . . . . , Y,-p), of each asset return, y,, is a linear function of Y , - l , Y,-2 . . . . . y,_p, while its conditional variance, h,, has an A R C H form shown below. The p t h order autoregressive model, A R ( p ) , for y,, c o m b i n e d with A R C H ( q ) errors, can be written as y,=~o+fl~y,
1+" " ' + f l p Y t _ p + e t ,
(1)
where y,_p) = 0
E(e, l y , _ l , y , _ z . . . . .
var(e, l Yt 1, Y,-z . . . . . =E(,~ly,_l,y,
with
and
Yt-p)
z.....
yt_p)=h,
2 h ~ = O o + O l e ~ _ , + . . • -I- Oqe,_q
(2)
As in previous studies, we impose the constraint 0j->0 for j = 0, 1,2 . . . . , q to ensure that h t is positive. Intuitively, 00 can be interpreted as a ' n o r m a l ' stationary element in the series on asset returns and the ~ for j ¢ 0 as the heteroskedastic 4 See, for example, Engle (1982, 1983), Engle and Kraft (1983), Engle and Chowdhury (1989), Engle and Rothschild (1992), and Diebold and Nerlove (1989).
485
coefficients. While an A R ( 1 ) model means that Yt depends on Y,-1 but not on earlier (Y,-2, Yt-3 . . . . ) observations, an A R C H ( I , 1 ) model means that y, depends on y,_l but not on earlier (Yt-2, Yt-3 . . . . ) observations and that h t depends on e,_~, 2 but not on earlier squared ~2 2 errors ( ,-2 . e,-3, . . . ). The standard identification and statistical tests that are generally used for selecting the orders of p and q of Model (1), which may be carried out prior to its use, are discussed briefly in G r e e n e (1993, pp. 441, 568-577). Model (1) gets a fair chance to be a winner in our forecast comparisons only when any of these statistical tests are not used because these tests can fail to reject false hypotheses. More importantly, it is difficult to discern whether the failure of these tests to reject an hypothesis implies that the hypothesis is true, as shown in Swamy and Tavlas (1995, p. 171, Footnote 7). In Section 2.3 below, we show that the coefficients of Eq. (1) may not completely incorporate the effects of excluded variables such as the interaction a m o n g the rates of return across countries. 2.2. The R C m o d e l
The A R C H model assumes that the conditional variance (2) of the error term of Eq. (1) changes over time in a specific way and that the coefficients (/30,/31 . . . . . tip) of the conditional mean of the equation are constants. The assumption that the coefficients of the A R C H process are constants imposes a restriction on the data that may not, in fact, be true. In order to eliminate this constancy assumption, the following R C procedure is adopted: {y,} is said to be a nonstationary process of a general n o n h o m o g e neous type if it can be expressed in the following form: 5 y,
t
= Ollt ~- x 2 t o l 2 t
e, = q~x2, + v , ,
!
-~-
e t8l
,
(3) (4)
5Any sequence {y,) satisfying Eqs. (3) and (4) has a distribution that is not the same as that of (y,+~} for every integer s ~ 0. Hence it is nonstationary.
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where the scalar al, represents the time-varying intercept, x2, is a p-vector of lagged dependent variables included in Eq. (1), a2, is a p-vector of coefficients, e t is an r-vector of variables which, in conjunction with x, = (1, x'2t)', completely determine Yt hut are excluded from Eq. (1), 8, is an r-vector of coefficients, ~ is an [r x p] matrix and v t is an r-vector of errors. Eq. (4) postulates a relationship between excluded and included variables. Substituting this equation for e t in (3) gives the following RC model: yt
= (o/it +
P
t
v't~t) + x2t(ol2t q- ~ t ~ t ) = xrt~t ,
(5)
where x, = (1, x'2t )' , 13t = [(o/It +
and
Dtt~t) t' (Ol2t "F l I ~ t ) ' ] t
Regarding /3t, it is assumed that 13, = IIzt + R u t
(6)
and U t = (J~Ut_ 1 + a t ,
(7)
w h e r e / / i s a [(p + 1) x K] matrix of coefficients, zt = (1, z'2t)' is a K-vector of variables which
determine 13, R is a known matrix of order [(p + 1) × m] and u t is an m-vector of errors, is an m × m matrix, E ( a t ) = O, E ( a t a ' t ) = o'2Aa, and E ( a t a ' s ) = 0 for t ~ s. Substituting the right-hand side of Eq. (6) for in Eq. (5) gives y, = x'j-lzt + x ' t R u t ,
(8)
which is the RC Model (5) in another form. 6 It follows from Eqs. (3) and (4) that each coefficient of Eq. (5) is the sum of two effects: a direct effect of an element of x2t on Yt and an indirect effect due to the fact that the element of x2, affects excluded variables which, in turn, affect y,. The z2t in Eq. (6) are called concomitant variables which are introduced to separate direct effects from indirect effects (see Swamy and Tavlas, 1995, p. 174). Within the framework of
Eq. (8), the most general specification is obtained by setting R = I and not restricting q~ and Ao to be diagonal. The assumption that q~ is nondiagonal means that each element of u t is correlated not only with its own lagged value but also with the lagged values of the other elements of u t. A subset of the nonzero coefficients on z2~ in Eq. (6) introduces interactions between x2~ and z2, into Eq. (5), as in Eq. (8). Also, note that the matrix R is introduced in Eq. (6) to impose restrictions on the covariance matrix of u, and hence is known. For example, if the jth row of R is restricted to be zero (or null), then the jth element of/3, becomes an exact function of zt. These restrictions will not be present if R = I. This shows that it is not correct to say that Model (8) with R = I, • ~ 0, and nondiagonal Aa is little more than an automatic interactive ordinary least squares model. In empirical applications of Model (8) in Section 3, we set R = I so thatp+l=m and s e t p = l and z 2 t - O for all five currencies. The richness of Eq. (8) includes the possibility of using concomitant variables (the z2~ in Eq. (6)) to explain the time profile of the varying coefficients of Eq. (5). In contrast to RC models, the random walk and A R C H models do not have this capability. It is for this reason that the RC model is to be preferred a priori. In Section 3 below, we did not attempt this richer specification with the RC model because we did not want to give this model an unfair advantage over A R C H and random walk models in our out-of-sample forecast comparisons. This means that z2,~-0 in this paper's applications. 2.3. C o n n e c t i o n s b e t w e e n A R C H models
The connection between the A R C H Eq. (1) and the RC Eq. (8) will be obvious if we write the former as P
Yt - [30 -
6For a detailed discussion of the advantages of the RC procedure, see Swamy and Tavlas (1995). Chang et al. (1992) show how to estimate the RC model.
and RC
~
i=1
q
[3iYt-i = Et = U,o + ~-~ ut/e,-/ , j=l
(9)
where, given the lagged values Et_j, the u,j for j = O, 1 , . . . , q and all t are mutually and serially
C. Christou et al. / International Journal o f Forecasting 12 (1996) 4 8 3 - 4 9 3
489
Table 1 Descriptive statistics of total returns a
Mean Variance Skewness Kurtosis
US dollar
Deutsche mark
French franc
Japanese yen
P o u n d sterling
0.132 0.001 -0.677 2.198
0.247 2.570 -0.494 3.053
0.276 2.380 -0.542 4.209
0,160 2.125 0.275 2.039
0.310 2.504 -0.671 1.977
" The sample period is July 1 9 8 8 - A u g u s t 1993.
investments.) We define dollar-based returns as follows: Yt.k
= [(1 + i,_l.k) × (1 + e,.k) -- 1] × 100
(15)
where e,. k = ( E t , k - E t _ I . k ) / E , _ I .
~
and Yt,k =
The US dollar-based returns on financial assets denominated in currency k at time
tion. A left-skewed distribution is characterized by less likely but extreme negative deviations from its mean and more likely but small positive deviations. This pattern is reversed in rightskewed distributions. Furthermore, only two out of five currency returns considered exhibit kurtosis values greater than 3, indicating that these two returns follow distributions that are more peaked in the neighborhood of their modes than a normal distribution.
t.
Three-month treasury bill rate on a weekly basis for the country corresponding to currency k, at time t - 1. et, k = Rate of appreciation/depreciation of currency k against the US dollar, at time t. Et, k = Spot rate defined as US dollars per unit of local currency k, at time t. k = The US dollar, the deutsche mark, the pound sterling, the French franc, or the Japanese yen. In calculating returns, we have omitted capital gains or losses resulting from changes in the prices of bills, on the assumption that such shortterm securities generally have relatively small price variations over the course of a week. Table 1 displays the values of descriptive statistics (mean, variance, skewness, and kurtosis) for the returns of investment in the five currencies used in the analysis. 9 These statistics indicate that the currency returns, except for the Japanese yen, display negative skewness. The skewness of the distribution will be positive (or negative) for a right- (or left-) skewed distribui t - l,k =
T h e source for exchange rates and interest rates for the various countries employed is the IMF Treasurer's Departm e n t database.
3. Estimation results
As we have already indicated in Section 2.3, Eq. (1) with p = 1 and q = 1 fits our data well. It should be noted that the optimal lag lengths, p and q, are determined as the values, less than 6, that maximize the log-likelihood function. T o maximize the likelihood function, an iterative procedure based on the Brendt et al. (1974) algorithm is used. The LM statistics for the presence of A R C H effects are then calculated and they are asymtotically distributed a s x2(q) under the null hypothesis of no A R C H effects. In this estimation, Yt denotes total returns in US dollars obtained by investing in five currency assets and only the data for the weekly period July 1988 through June 1992 were used. The results of this estimation are presented in Table 2. These estimates are subsequently used in Eq. (10) to obtain estimates of the parameters of Eqs. (6) and (7) with R = I, Zz, = 0, p = 1, and K = 1 for the above period. These are called the estimates of extended A R C H models. Estimates of the RC Model (8) with R = I, p = 1, z2, = 0, ~ 0, K = 1, and zla ~ 0 were also obtained for the same period. It should be pointed out that
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Table 2 A R C H estimation results for total returns" (t-ratios in parentheses)
/3o /3~ 0o 01 LL LM statistic
US dollar
Deutsche mark
French franc
Japanese yen
Pound sterling
-0.000 (_)b 1.000 (_)b 0.000 (_)b 0.169 (_.)b -~ -"
0.265 (2.211 ) * 0.144 (4.290)* 2.635 (10.046)* 0.008 (0.097) - 197.070 0.029
0.291 (2.508) * 0.041 (2.190)* 2.424 (9.131)* 0.002 (0.023) - 189.290 0.001
0.153 ( 1.471 ) 0.057 (3.156)* 1.880 (10.091)* 0.120 (1.555) - 175.270 9.244"
0.318 (2.525)* 0.027 (3.125)* 2.509 (9.907)* 0.007 (0.114) - 193.670 0.041
a LL denotes the log-likelihood and LM the Langrange multiplier statistic, which is used to test the null hypothesis of no A R C H effects in total returns. * and ** denote statistically significant LM- or t-statistics at 0.05 and 0.10 levels, respectively. b The t-ratios, the LL, and the LM statistic are undefined because the y, are essentially equal to a constant. Therefore, the likelihood function is an infinite spike at a fixed point.
Eqs. (8) and (10) were estimated by employing the most general specification of the random coefficient procedure, i.e. both q~ and zaa were assumed to be nondiagonal. Post sample forecasts based on all these estimates were generated over the weekly period July 1992 through August 1993. Root mean square forecast errors (RMSEs) based on each model were calculated by taking the square root of the following MSE 1
F
MSE = -ff~] (~3T+~ -- yr+~) 2 ,
(16)
s=l
where F is the forecast horizon July 1992 through August 1993 (i.e. 61 weeks), ~3T+s is the forecast of YT+s made in period T, and Yr+s is the actual return on each asset in period T + s. Note that C V E ( j ) in (14) reduces to the MSE defined above if )~i(j) = )~r+~, Yi = Yr+~, and T = F. Comparison of these RMSE values could provide some direction about how the estimated
A R C H , extended A R C H , RC, and random walk models compare in terms of their ability to forecast out-of-sample rates of return on the reserve currencies we considered. Table 3 presents the RMSEs associated with the predictions generated by the above four models, namely: (i) the random walk model (Eq. (11)); (ii) the direct A R C H model (Eqs. (1) and (2) with p = l and q = 1); (iii) an extended A R C H model (Eq. (10) estimated on the basis of the RC procedure); and (iv) the RC model (Eq. (8) with R = I, p = 1, z2, = 0, q~ ~ 0, K = 1, and Aa ~ 0) estimated directly by using the RC procedure. Several points are worth noting. First, forecasts generated by the models estimated on the basis of the RC procedure (see columns 3 and 4 in the table) are superior to those based on the random walk model (column 1) or the pure A R C H estimation (column 2) in four out of five cases. The RMSE using the RC procedure is
Table 3 Root mean-square forecast errors a
French franc Deutsche mark Japanese yen Pound sterling US dollar
Random walk (1)
Direct A R C H (2)
Extended A R C H (3)
RC (4)
2.8528 2.6976 1.9893 3.2290 0.0018
2.0922 2.0024 1.3769 2.3953 0.0019
1.9218 1.8109 1.3685 2.2856 0.0018
2.0103 1.9824 1.3770 2.3495 0.0018
a The estimation period is from July 1988 through June 1992, and the postsample forecasts were generated for the period from July 1992 through August 1993.
c. Christou et al. / International Journal of Forecasting 12 (1996) 483-493
independent with means zero and variances E(u~j) = Oj. It should be noted that the us in Eq. (9) are different from those in Eq. (6). From this specification it follows that the conditions under which the A R C H model coincides with the RC model are: (i) (y, - /30 - Ei=l P /3iY,-i) appears in place of y, in Eq. (5), (ii) p + l = q + l , (iii) (1, E,_I . . . . . e,_q) appears in place of x', in Eq. (5), (iv) R = I, (v) H = 0, (vi) q~ = 0, (vii) 0-2 = 1, and (viii) A a is diagonal having 0j as its jth diagonal element. 7 Since there is no correspondence between the/], of Eq. (5) and the/3s of Eq. (1), the latter coefficients, unlike the former, cannot completely capture the effects of excluded variables, particularly when these effects are time varying. Using an RC estimation program, if we estimate Eq. (5) subject to the restrictions (i)-(viii), we then obtain the estimates of the A R C H model. In this estimation, if we do not impose some or all of the restrictions (vi)-(viii), we then obtain the estimates of extended A R C H models. For example, changing the restriction (viii) to the restriction that A, is nondiagonal, generalizes the variance h, to 00 +
2 + zql In addition, serial correlation is also introduced into the error terms of A R C H models, if the restriction (vi) is relaxed by assuming that 4) ~ 0. The usefulness of extended A R C H models is that they allow the variances and covariances of E, to evolve over time in much more general manner than the pure A R C H model. The RC Model (8) is even more general than an extended A R C H model because it does not assume that the /3i of Eq. (1) are constants. This constancy assumption is incorrect if the indirect effects contained in each/3i are not constants. We now show that the connections between A R C H and RC models may be exploited to validate the former models. We fit Model (1) to data on asset returns (to be described in Section 2.6 below) and show, using some usual statistical tests, that an A R C H ( I , 1 ) model fits the data well. It follows from Friedman and Schwartz's (1991, p. 47) discussion that a persuasive test of 7These conditions were previously noted by Swamy and von zur Muehlen (1988, p. 130).
487
this model must be based on data not used in its estimation. That might mean using the estimated A R C H ( I , 1 ) model to predict the same kind of phenomena for a future period. If these forecasts can be improved by fitting a RC model to the residuals, ~, = y , - / 3 0 -/31y,_~, where/30 and /3~ are our estimates from the A R C H ( I , 1 ) model, or Eq. (1) with p = 1 and q = 1, then the estimated A R C H ( I , 1 ) model may be judged to be inadequate for estimating asset returns. Consequently, we consider the model 6, =/3,0 +/3,,6r , ,
(10)
where the vector (/3t0,/3,~)' satisfies Eqs. (6) and (7) with z:, -= 0 and R = I. It should be noted that the /3s in Eq. (10) are different from those in Eq. (1). 2.4. The r a n d o m w a l k m o d e l
The random walk process is based on the following specification: Y, =Y,-I + E,,
(11)
where e, is a stochastic error term following a white noise process, whereby the error term has a constant variance, is serially independent, and has a mean equal to zero. The random walk model implies that each period's 'best' forecast of y, is equal to its actual value in the previous period. To demonstrate this outcome, suppose, we wish to.make a 'best' forecast for such a random walk process. The 'best' forecast of Yt+ 1, given Yt is given by ~,+, = E ( y , + , [ y , ) .
(12)
But according to Model (11), ~,+ t is independent of y,. Thus, the 'best' forecast of Y,+I is simply )3,+1 =Y, + E(E,+t
l Y,) = Y , .
This result follows since an assumption of the white noise process is that the mean of the error term is zero. Note that Model (11) has the property that even though the conditional expectation (12) is finite, the unconditional expectation of (y,+~ _ f ( y , ) ) z is not finite if y, ¢
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f(Yt). In this case, it is not strictly correct to call E(Yt+l [Y,)=Yt a minimum mean square error (MMSE) forecast. The quotes 'best' are there precisely for this reason.
2.5. A method of choosing among ARCH, extended ARCH, and RC models We use a technique called cross-validation to choose among the models covered in Sections 2.1-2.4. This technique has been provided with a suitable theory and successfully applied in many areas of research.8 Let the variables that appear in ARCH, extended ARCH, RC, and random walk models be observable for the period t = 1 , 2 , . . . , T. Then we compute the conditional expectation
Yi~j) = E(yi l model = J and data without the ith data point)
(13)
from the jth estimated model which is any one of the models given by ARCH, extended ARCH, RC, and random walk, and which is estimated by omitting the ith data point. The cross-validation error (CVE) in prediction based on the jth estimated model is T
C V E ( j ) = T -1 ~ ()3,(j) - yi) 2
(14)
i=l
For different choices of models we compute (14) and choose that model for which it is a minimum. It should be noted that testing the A R C H model on the data which are used to choose its numerical coefficients is almost certain to overestimate performance, for use of statistical tests leads to false models if both the null and alternative hypotheses considered for these tests are false, as shown by Swamy and Tavlas (1995, p. 171 and Footnote 7). There is no guarantee that either a null or an alternative hypothesis will be true if an AR C H model is specified. It will be true if the model is broad enough to cover the 8 See, for example, Mosteller and Tukey (1977), Mosteller and Wallace (1964), Stone (1974), Swamy and Tavlas (1992, 1995), Swamy and Schinasi (1989), Schinasi and Swamy (1989), and Swamy et al. (1990).
true model as a special case. Indeed, the idea of extending the A RCH model is to make it broader so that this broader model will have a better chance of covering the true model as a special case than the A RCH model itself. It should also be noted that in the above method, one need not take the summation over all T data points, as we have done in Eq. (14). If future prediction is needed in a specified region of the independent variables, one need only take the summation over those data points for which the observed independent variables are close to the specified region.
2.6. Calculation of asset returns in different currencies Weekly observations on asset returns were used in order to estimate the above models over the sample period beginning in July 1988 and ending in June 1992, and forecast their dependent variables over the period beginning in July 1992 and ending in August 1993. This choice of the estimation and forecast periods allows for the turbulence experienced in foreign exchange markets beginning in August 1992 and extending into August 1993 to be captured in the forecast period. In particular, pressures within the exchange rate mechanism (ERM) of the European Monetary System in the summer of 1992 led to the withdrawal of the pound sterling and the Italian lira from the ERM in September 1992. Subsequent pressures culminated in the widening of the ERM bands to -+15% in August 1993. Consequently, the forecast period provides a rather firm test with which to gauge the predictive accuracy of models estimated on the basis of a more tranquil data set. The data set contains returns on short-term investment securities denominated in the US dollar, the deutsche mark, the Japanese yen, the pound sterling, and the French franc. For the sake of convenience, since exchange rates are expressed bilaterally against the US dollar, the analysis is carried out on a dollar basis. (The same procedure could, however, be repeated for the yen-, mark-, franc-, or pound sterling-based
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equal to that using the random walk model in the case of the US dollar, 30% lower in the case of the French franc, 26% lower for the deutsche mark, 31% lower for the Japanese yen, and 27% lower for the pound sterling. Moreover, the RC procedure gives lower RMSEs than those generated by the direct ARCH estimation, with the exception of the Japanese yen for which the two procedures provide comparable results. The RMSE using the RC procedure is equal to that using the direct ARCH model in the case of the Japanese yen, 5% lower in the case of the US dollar, 4% lower in the case of French franc, 1% lower for the deutsche mark, and 2% lower for the pound sterling. Second, forecasts generated by the extendedA R C H model are superior to the ones estimated on the basis of the RC procedure (see columns 3 and 4 in the Table) in four out of five cases. The RMSE using the extended-ARCH model is equal to that using the RC procedure in the case of the US dollar, 4% lower in the case of French franc, 9% lower for the deutsche mark, 1% lower for the Japanese yen, and 3% lower for the pound sterling. Third, the result that the RMSEs are much lower for all four models in the case of the US dollar arises as a direct consequence of the fact that the variability of returns on US investments comprises only interest earnings on the respective assets, since the investments are dollarbased. For all other currency investments, the variability of returns includes both an interest rate component and an exchange rate component (against the US dollar). Over the course of the estimation period, the US dollar was essentially unchanged; the nominal effective exchange rate (as computed by the IMF [°) appreciated from 64.9 at the beginning of 1988 to 65.4 at the end of 1992. There were, however, considerable swings in the dollar's real effective value within the period; it peaked at 71.9 in September 1989 (a 10% appreciation) and reached a low of 59.8 (a 9% depreciation) in August 1992. The RMSEs using the RC or the extended ARCH model are below those of the "~See InternationalMonetaryFund (1993).
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random walk model for the four non-US dollar currencies, whereas for the US dollar an RC model performs in prediction as well as the random walk model. This result indicates that the RC procedures together with the extended ARCH specification are better able to account for both the interest component and the exchange rate valuation component of investment returns. Monte Carlo and analytical results presented in Yokum et al. (1994) and Swamy and Schinasi (1989) show that the superior forecasting performance of RC and extended ARCH models observed for the sample data utilized in this paper can be replicated frequently in repeated sampling procedures.
4. Conclusions
This paper analyzed the rates of returns on financial assets denominated in five major reserve currencies and provided a framework for the determination of optimal strategies for the allocation of wealth in multicurrency investments. By using weekly data for returns on five major currency investments for the period July 1988 through June 1992, three models were estimated for the returns on a US dollar-based investment: (i) a univariate ARCH model; (ii) an extended ARCH model using the RC procedure; and (iii) a pure RC model. The estimated models were then used to generate forecasts on weekly rates of return for the period July 1992 through August 1993, which were compared with the forecasts generated by a random walk model. This comparison demonstrated that forecasts based on the RC/extended ARCH procedure are superior to those based on the random walk model and those based on the direct ARCH estimation. It has also been shown that the superiority of the RC procedure relative to the direct ARCH procedure is based on theoretical foundations. Specifically, in the presence of excluded variables, each regression coefficient can be interpreted as the sum of a direct effect of a right-hand-side (RHS) variable on a left-hand-
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side (LHS) variable, and an indirect effect due to the fact that the RHS variable affects excluded variables, which, in turn, affect the LHS variable. This interpretation of regression coefficients implies that such coefficients change over time if indirect effects change, even if direct effects are constant. The RC procedure accounts for such time-varying effects, but the ARCH procedure does not. The results could be useful for both international investors and central banks. The former group could utilize such procedures in order to determine the optimal allocation of their wealth among fixed-income investment securities, while the latter group could determine more accurately a number of 'best' strategies for the management of their external reserve assets.
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Biographies: Costas CHRISTOU is an economist with the African Department of the International Monetary Fund. He holds a Ph.D. from the University of Maryland at College Park. He has worked at the Center for Regional Economic Issues at Case Western Reserve University, the Interindustry Forecasting Group at the University of Maryland and the World Bank. His research interests include macroeconomics, monetary economics, international finance, and applied econometrics.
C. Christou et al. / International Journal of Forecasting 12 (1996) 483-493 P.A.V.B. SWAMY is a financial economist in the Bank Research Division of the Office of the Comptroller of the Currency. He received a Ph.D. from the University of Wisconsin, Madison, and has taught at the State University of New York at Buffalo and the Ohio State University, Columbus. He is the author (or co-author) of one book and over 75 published research papers. His research interests include foundations of econometrics, econometric theory in estimation and forecasting. George S. TAVLAS is Deputy Chief of the Financial Relations Division of the International Monetary Fund. He has
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previously worked at the US Department of State and the Organization of Economic Cooperation and Development, and has been a Guest Scholar at the Brookings Institution. He has published numerous articles on macroeconomic issues, international finance, and econometric modelling. He is on the board of editors of the Greek Economic Review and the Journal of Policy Modeling, and has been a guest editor of Open Economies Review. He is a research associate of the Athens Institute of Policy Studies and is an affiliated scholar of the Center for the Study of Central Banks, New York University School of Law.
