Journal of Business Finance & Accounting, 16(1) Spring 1989, 0306 686X $2.50

SPOT AND FORWARD EXCHANGE RATES: ACAUSALITY ANALYSIS

JEFFREY L. CALLEN, M . W . LUKE CHAN AND CLARENCE C . Y. KWAN*

INTRODUCTION

The purpose of this paper is to establish empirically the causality relationshipsbetween forward and spot exchange rates using the notion of Granger (1969)causality. Intuitively, spot rates cause forward rates in the sense of Grangerif past spot rate information is useful in predicting future forward rates. Simi-larly, forward rates cause spot rates if past forward rates have predictive con-tent with respect to future spot rates.

The rationale for estimating the causality relationships between spot and for-ward exchange rates is twofold. First, there is an extensive literature in inter-national finance which is concerned with predicting future exchange rates forexchange rates management purposes.' In the context of this literature, wewant to see if causality analysis yields superior exchange rate predictions atleast in comparison to standard univariate time-series analysis. Indeed, fromthe theory of exchange rate determination (see, for example, Grauer,Litzenberger and Stehle, 1976) we know that forward rates — or at least thecurrent forward rate — contain information about future spot rates.^Therefore, we should expect to find that forward rates cause spot rates.^ Thatis, forward rates are useful in predicting future spot rates.

Second, there exists a related literature which is concerned with the efficiencyof foreign exchange markets.* According to this literature, in an efficientmarket, the information conveyed by past spot rates should be fully impoundedin future spot and forward rates. In particular then, spot rates should not proveuseful in predicting forward rates, or alternatively, spot rates should not causeforward rates. The remainder of this paper is devoted to using causality analysisto examine these two issues empirically, namely, do forward rates cause spotrates and do spot rates not cause forward rates.

The following section defines Granger causality more rigorously, and thethird section introduces the models and procedures for determining causalityrelationships. The fourth section describes the data and empirical results. Thefinal section concludes the paper.

*The first author is from the Jerusalem School of Business Administration, the Hebrew Univer-sity of Jerusalem, and the Faculty of Business, McMaster U.iiversity. The second and third authorsare from the Faculty of Business, McMaster University. They wish to acknowledge the helpfulcomments of Cheng Hsiao. (Paper received October 1985, revised May 1987)

105

106 CALLEN, CHAN AND KWAN

DEFINING CAUSALITY

The concept of Granger causality stems from the simple observation that thefuture cannot cause the present or the past. Gonsider the bivariate stochasticprocess [S,, Fi] where S; denotes the spot rate and F, the forward rate at timet. Suppose that at time n we are trying to predict i?, + i. Then forward ratesare said to cause spot rates provided S„ ^ i is better predicted by adding pastforward rates to past spot rates than by using the past spot rate series alone.Similarly, spot rates cause forward rates if /%, . j is better predicted by thebivariate series of past spot rates and forward rates than by using past forwardrates alone. Nothing in this definition precludes forward and spot rates fromcausing each other and, in that case, the two time series are said to exhibitfeedback.

Gausality can be defined more rigorously in terms of information sets. LetX, denote a stationary stochastic process with components [Fi, S, ]. Let X,, F,,SI denote the set of past values oi X;, F,, S, before time t, respectively. Let X^— Fi denote the set of elements in Xi and not in i ,. Define oi^{F,IXi) to be theminimum mean square linear prediction error of F, given the information setX,. We can now define Granger causality formally.

Definition: (Gausality) U a\F,/Xi) < Oi\F,IX, - 'S,) — that is, the infor-mation set which includes past spot rate data yields a more accurate predictionof forward rates than in the same information set without past spot rate data— in the mean squared error sense, then spot rates are said to cause forwardrates.

Definition: (Feedback) If o^iFilX,) < a^{F,IX, - 'S,) and a^iSifx,) <{SIX — F,) then feedback between forward rates and spot rates is said to

The relationship between market efficiency and Granger causality isreasonably obvious from the above definition. If markets are efficient (semi-strong) then knowledge of past spot rates should have no bearing on predictingtoday's forward rate. In fact, if markets are efficient, a {Fj/Xi) should be equalto a {Fj/Xi — Si) since the information sets are the same except for the publiclyavailable information contained in past spot rates. But, at the same time, ac-cording to exchange rate theory, past forward rates will incorporate relevantinformation about market expectations of future spot rates. Thus, one shouldexpect that adding past forward rates to the information set will yield betterpredictions of future spot rates, i.e., a^(Si/Xi) < a^{S,/Xi — F,).

MODEL

It is well known (e.g., Masani, 1966) that a regular full rank stationary stochasticprocess {St, Ft] can be modelled, under fairly general conditions, by the

SPOT AND FORWARD EXCHANGE RATES 107

autoregressive representation

F, = ^P,,iL)Fl+^,,{L)St + v„ (1)

S,= ^P2l{L)F,+^|^22{L)S,+u,. (2)

Here, L denotes the lag operator L''FI = F,_i, and i/'y(Z.) is the lag polynomial2°°i = i ^ijk^''- The [vi, Ui} are zero mean white noise innovations with cons-tant variance-covariance matrix.

Gausal relationships enter the model in a very natural way. If i/'i2(Z-) = 0(i.e., \l/i2k = 0 for all k), then it is clear from equation (1) that spot rates haveno effect on predicting forward rates, that is, spot rates do not cause forwardrates. Similarly, if i/'2i(L) = 0, forward rates do not cause spot rates. Thus, intheory, one could determine the causal relationships between spot and forwardrates by first of all fitting equations (1) and (2) by least squares — yieldingestimates which are consistent and asymptotically normally distributed — andthen testing to see if i/',-,(L) = O for i V / This approach is problematic,however, because the test of i/'y(Z,) = 0 is generally quite sensitive to the orderof the lags of the ^ij{L). (See Hsiao, 1979a and 1979b). Instead, followingHsiao we will use the data itself to determine the lag structure. Specifically,the optimal order to the lag for each l y^L) in each of equations (1) and (2)is determined using Akaike's (1969a and 1969b) Final Prediction Error (FPE)criterion as described below.

The FPE Criterion

The FPE {of Fi) is defined to be the (asymptotic) mean squared predictionerror

E(F, - F,f (3)

where F, is the predictor of F, given by the least square estimates

^ (4)

The superscripts m and n denote the order lags of i/'ii(Z') and ^i2(-^) and^i2(L), respectively, where m and n are bounded above by the maximum lagorder investigated, say Q^. i^n (^). " 12 (^) and B are the least squaresestimates of i/'n(L), ^I2(^). and the constant term b, respectively. Akaikeestimates the Firial Prediction Error by

FPE,im,n) = I! " " " " " j f I {Fi-F^flT] (5)T-m-n-1 L J

By choosing the lag structure with minimum FPE, Akaike's criterion tries tobalance the bias from choosing too small a lag order against the increasedvariance from a higher lag order specification. More specifically, Akaike hasshown that, if the dependence between i/',-, and recent values of the variables

108 ' CALLEN, CHAN AND KWAN

(in our case spot and forward rates) decreases as the length of past history in-creases, the FPE is comprised of two components. The first component is dueto the FPE of the best linear prediction forgiven m and n while the second com-ponent is due to the statistical deviations of ^n(L) and 4'"2{L) from V'n(L)and \1/\2{L). Generally, as m and n are increased, the first componentdecreases whereas the .second component increases for a finite length of obser-vations of spot and forward rates.

The FPE approach yields a number of distinct benefits in terms of identi-fying the model. Firstly, as we already pointed out, the data itself is used todetermine the lag structure rather than presupposing some arbitrary lag orderspecification. Secondly, the FPE criterion does not constrain the lag structureof each variable to be identical, i,e,. in general, m ^ n. Thirdly, it has beenshown that the FPE criterion is equivalent to choosing the model specificationon the basis of an F test with varying significance levels.^ Thus, rather thanspecifying an adhoc significance level of 5 or 10 percent, the choice of whetherto include a variable is determined by an explicit optimality criterion, namely,minimizing the mean square prediction error.^ Specifically, suppose thatFPE,.(rrt,n) < FPEy{m + p, n + q) so that {m,n) is chosen as the optimal lagstructure for ^n(L) and 4'i2(L).'' This is equivalent to choosing (m,n) on thebasis of an approximate F statistic. A formal discussion of this point can befound in Hsiao (1981, pp. 89-90).

Hsiao's Approach

In addition to identifying the model, the FPE criterion can be used to deter-mine causal relationships directly. Suppose we are to test if spot rates causeforward rates. Then Hsiao (1979a) has suggested the following sequential pro-cedure. First, the FPE criterion is used to determine the optimal order of theunivariate autoregressive process for forward rates alone. Gall this order q, sothat the resulting FPE is FPE,..(q,O). Second, fix the lag structure for the for-ward rate at q and use the FPE criterion to specify equation (1). Let r be theresulting order for the lag operator \l/i2{L) so that the FPE is FPEy(q,r). Third,holding the order of the lag operator ^ \2{i-) at r, let the order of lag operatorV'ii(I) vary from 0 to q. Ghoose the order of lAii(Z-) that gives the smallestFPE, sa.y s — s is not necessarily equal to q — thereby yielding FPE^-{s,r). Thisthird step is a check to see whether the lag structure for forward rates (whichwas orginally q) is sensitive to the lag structure of the manipulated variable,that is, the spot rates.** Finally, \iFPE^.{q,Q) < FPE^{s,r) then forward ratesare best represented by a one-dimensional autoregressive process so that spotrates do not cause forward rates. Gonversely, li FPE^'{q,O) > FPEp{s,r) thenwe would conclude that spot rates cause forward rates. A similar approach wouldbe used on equation (2) to test if forward rates cause spot rates.

It is worth noting that this FPE procedure for testing causality directly isvery much consistent with the Granger definition of causality. In both cases.

SPOT AND FORWARD EXCHANGE RATES 109

the criterion for causality is to minimize the mean square prediction error. Fur-thermore, as we have already noted, minimizing the FPE involves a tradeoffbetween the size and power of the test. The usual alternative approach, in-volving standard hypothesis testing, arbitrarily specifies the size (i.e., thesignificance level) of the test. The latter need bear no relationship to the under-lying Granger criterion of minimizing the mean square prediction error.However, if one is interested primarily in the true model structure of the rela-tionships between spot and forward exchange rates, an alternative criterionbased on determining and identifying which of the models has the highest proba-bility of being the true model structure would be more appropriate. Yet fortesting causality in the Granger sense, the minimization of the forecast errorshould be the yardstick in choosing the most appropriate model. Therefore,the FPE criterion which is based explicitly on minimizing the forecast erroris an appropriate choice here.

THE EMPIRICAL RESULTS

Causality Relationships

The data base is comprised of month-end non-overlapping spot rates and onemonth forward rates for six foreign currencies against the US dollar. All datawere collected from the Wall Street Journal. The data period for model estima-tion is from September 1977, to August 1981. An additional six data pointsfor each currency, from September 1981, to February 1982, were held backfor testing forecasting accuracy. To reduce the problem of serial correlationin the time series, first differences in the logarithms of both spot and forwarddata were taken. The model [Equations (1) and (2)] was then estimated foreach foreign currency. Q. = 14 was the a priori specified maximum possible lagfor each variable.

Table 1 lists the FPE for spot {S) and forward {F) rates of the pound, theyen, the French franc, the Ganadian dollar, the Swiss franc, and the West Ger-man marc, where each variable is treated as a one-dimensional autoregressiveprocess. The smallest FPE for each currency and each variable is underlined.Thus, for example, a univariate autoregressive model of lag 2 in the spot ratesyields the smallest FPE of 0.2448 X 10~^ in the case of the Ganadian dollar.

Table 2 presents the FPE of the spot and forward rates using the bivariateprocesses. The term controlled variable refers to the variables on the left handside of equations (1) and (2). These variables are controlled in the sense thattheir lag structure is fixed at the optimum lag determined in Table 1 (Step2 of the Hsiao procedure). That optimum lag is indicated in parentheses incolumn 2 beside each controlled variable. The FPE of the controlled variableis computed by varying the order of the manipulated (independent) variablefrom one to fourteen. Golumn 4 in Table 2 specifies the lag for the manipulated

no CALLEN, CHAN AND KWAN

Table 1

FPE of Fitting a One-Dimensional Autoregressive Process for Spot andForward Rates

The OrderOf Lags

1

2345

678

CT

l

1011121314

The OrderOf Lags

1

2

345

6789

1011121314

PoundFPE

5x102

0.1455

0.15030.15970.16950.1428

0.14910.15810.16950.18120.18880.16500.17420.18420.1982

CanadianFPE

5x10^

0.2571

0.2448

0.25050.24820.2465

0.25900.26310.27790.29830.30580.32900.32680.35170.3673

of

0.1385

0.14220.15090.16000.1361

0.14260.15060.16140.17220.17960.15880.16800.17610.1903

Dollar

of

0.2514

0.2363

0.24310.23940.2344

0.24690.25470.26590.29550.29170.31360.31710.33820.3464

YerFPE

5x102

0.1507

0.16070.17150.17420.1804

0.18770,19200.20540.21760.22930.24540.25680.21210.2193

of

0.1477

0.15760.16810.16980.1769

0.18380.18660.19960.21160.22250.23850.24680.20350.2097

Swiss FrancFPE

FxlO2

0.1705

0.1818

0.19190.20480.2180 .

0.23200.24840.26280.28090.30190.27720.28660.31030.3230

ofFxlO2

0.1652

0.1761

0.18480.19740.2098

0.22370.23940.25190.26890.28900.26640.27250.29510.3079

FrenchFPE

5x10^

0.1347

0.13750.14280.15130.1551

0.16520.17680.1893

. 0.20250.21770.19420.20740.21620.2326

Francof

0.1314

0.13320.13760.14470.1474

0.15730.16810.18010.19270.20710.18010.19070.19870.2149

MarkFPE

5x102

0.1510

0.1574

0.16460.17460.1818

0.19440.20810.22200.23730.24610.20540.21490.23040.2334

FxlO2

0.1452

0.1507

0.15680.16570.1726

0.18460.19770.21110.22550.23470.19340.19820.21220.2176

variable which yields the minimum FPE for the controlled variable. Theminimum FPE is given in the last column of Table 2.

Tables 1 and 2 correspond to the first two steps of the Hsiao sequential pro-cedure for testing causality. The third step — determining if the lag structure

SPOT AND FORWARD EXCHANGE RATES

Table 2

111

The Optimum Lags of the Manipulated Variable and the FPE of theControlled Variable

Currency

(1)

Pound

Yen

French Franc

Canadian Dollar

Swiss Franc

Mark

ControlledVariable

(2)

S(5)F(5)

S(l)F(l)

S(l)F(l)

S(2)F(2)

S(l)F(l)

S(l)F(l)

ManipulatedVariable

(3)

FS

FS

FS

FS

FS

FS

The Optimum Lagof Manipulated

Variable(4)

11

11

11

22

11

11

(5)

0.14640.1402

0.15940.1565

0.14370.1402

0.022840.02316

0.16200.1575

0.13750.1339

of the controlled variable is sensitive to the lag structure of the manipulatedvariable — changed the minimum FPE only in the case of the Canadian dollar.Specifically, when F is the controlled variable and 5 is fixed at a lag of 2, theoptimum lag structure for F is reduced from 5 to 2.

Causal relationships are determined by comparing the minimum FPE of theunivariate process (Table 1) with the appropriate minimum FPE for thebivariate process which, with the exception of the Canadian dollar noted above,is listed in Table 2. For example, in the case of the pound, the minimum FPEfor the univariate spot process is 0.1428 X lO"'^. However, by adding the for-ward rate (which enters optimally with a one period lag) to the spot rate pro-cess, the minimum FPE of the spot rate is 0.1464X 10~^. Since the univariateprocess yields a lower FPE, we conclude that forward rates do not cause spotrates in the case of the pound. Taking another example, consider the Swissfranc. The univariate process for the forward rate yields a minimum FPE of0.1652 X 10~^ at a lag of one. The bivariate process for the forward rate, onthe other hand, gives a minimum FPE of 0.1575 X 10~^ where spot and for-ward rates enter the equation with a lag of one each. Since the overall minimumFPE is obtained by the bivariate process, we conclude that spot rates causeforward rates in the case of the Swiss franc.

A similar analysis for each currency provides the following overall results.

112 CALLEN, CHAN AND KWAN

For three currencies, the pound, the yen, and the French franc there are nocausal relationships, i.e., forward and spot rates are unrelated. This is in contra-distinction to the remaining three currencies, the Canadian dollar, Swiss franc,and mark for which feedback is exhibited, i.e., forward and spot rates causeeach other.

The upshot of these results is that foreign exchange markets do not appearto be efficient in general. This conclusion is clear at least for the Canadiandollar, Swiss franc, and mark since in these cases spot rates cause forward rates.In an efficient market, the information conveyed by past spot rates shouldalready be incorporated in the forward rate so that no causality from spot ratesto forward rates should be observed.

While foreign exchange markets do not appear to be inefficient in the caseof the pound, yen and French franc, they do not yield expected results. Wehave expected from the theory of forward exchange that forward rates shouldcause spot rates. Perhaps, we obtain these anomalous results because forwardrates are unsystematically biased estimates of future spot rates so that causa-lity from forward rates to spot rates may be difficult to observe.^

It is also worth noting that, for the yen, the French franc, the Swiss francand the mark, the univariate spot and univariate forward rate processes appearto be random walks.'" But, in at least two cases — the Swiss franc and themark — the random walk characteristic of spot rates is deceptive. Rather, theunderlying model is basically bivariate in nature with spot and forward ratesentering with a one period lag each." Therefore, the commonly found resultthat exchange rates are random walks may simply reflect, at least for somecurrencies, misspecification of the underlying model which is bivariate and notunivariate.

Forecast Accuracy

As we noted earlier, some of the data — from September 1981 to February1982 — were held back to test the forecasting accuracy of the causality models.The models are described more fully in Table 3 for the pound, yen, and Frenchfranc and in Tables 4 and 5 for the Canadian dollar, Swiss franc and mark.Since the currencies in Table 3 did not exhibit any causality relationships, eachequation for spot and forward rates represents a univariate autoregressive modelwhich was estimated by ordinary least squares.

The currencies in Table 4 are far more interesting because they exhibitedfeedback. Therefore, two potential models could be estimated for each of theselatter currencies. The first model [equations (12)—(17)] as shown in Table 4is simply the univariate autoregressive scheme determined in Table 1 andestimated by ordinary least squares. The second model [equations (18)—(23)]as shown in Table 5 is based on the causality relationships described in Table2 (except for the Canadian dollar which required the additional step mentionedabove). Civen the causality relationships, this second model is simultaneous

SPOT AND FORWARD EXCHANGE RATES

Table 3

113

Univariate Autoregressive Time Series Models Fstimated by OLS for thePound, Yen and French Franc

(6)

Pound

St =

SER =

Ft =

SER =

Yen

St =

SER =

Ft =

SER =

French Franc

St =

-0.00532(0.0060)

- 0 .(0.

+ 0.08447 St-3(0.1767)

0.03329

-0.00527(0.0059)

+ 0.09033 }(0.1772)

0.03249

-0.0045(0.0065)

0.03649

-0.004668(0.0070)

0.03603

-0.01030(0.0063)

0.03441

-0.01046(0.0062)

- 0 .(0.

03022 S, _ 11627)

+ 0.21865 5,_(0.1841)

DW =

026111 F ,_ ,1627)

+ 0.22515 F,.(0.1850)

DW =

-0.00274 5, _(0.1644)

DIV =

-0.01389 F,_(0.1700)

DW =

-0.13981 5,_(0.1611)

DW =

-0.1188 Fi_i(0.1613)

+ 0.13659 S,_2(0.1613)

4 +0.48641 5, ..5(0.1808)

2.0003

+ 0.14666 F,_2(0.1610)

.4 +0.47487 F,_r(0.1805)

1.9754

1

1.7774

• 1 .

1.7822

1

1.5979

1

(7)

(8)

(9)

(10)

(11)

SER 0.03398 DW = 1.5923

Note: Numbers in parentheses are standard errors of the estimate;SER = Standard Error of the Regression;DW = Durbin-Watson

in nature so that the parameters were re-estimated by a full information maxi-mum likelihood technique.'^

Table 6 presents summary statistics of the forecast accuracy of each of themodels described in Tables 3, 4 and 5. Forecast accuracy is measured by both

114 CALLEN, CHAN AND KWAN

Table 4

Univariate Autoregressive Time Series Models Estimated By OLS for theCanadian Dollar, Swiss Franc, and Mark

Canadian DollarS, = 0.00134

(0.0025)

SER =

Ft =

SER =

Swiss Franc

St =

SER =

Fi =

SER =

Mark

St =

SER =

< =

0.01443

-0.00164(0.0025)

-0.10816(0.1969)

0.01349

-0.01026(0.0060)

0.03933

-0.01050(0.0068)

0.03811

-0.00955(0.0066)

0.03643

-0.00975(0.0064)

0.03572

-0.19185 S,_i(0.1802)

DW =

-0.22643 Fi_i(0.1863)

F,_3 - 0 .23605 /(0.1822)

DW =

-0.32821 6',_|(0.1400)

DW =

-0.34617 F , _ ,(0.1409)

DW =

-0.21299 ^^.i(0.1614)

DW =

-0.22756 Fi_i(0.1602)

DW =

-0.35461 S,_2(0.1863)

2.0709

-0,45284 F,_2(0.2009)

^,_4 +0.28404 F^_5(0.1807)

1.9198

0.7973

1.7719

1.7043

1.6927

(12)

(13)

(14)

(15)

(16)

(17)

the root mean square error and Theil's U coefficient.'"* There are three ten-tative conclusions to be derived from Table 6. First, predictive accuracy doesnot vary substantially from one model to another or even from one currencyto another. Second, on a root mean square error basis, the forecast accuracyof the forward rate is better than that of the spot rate. However, this resultdoes not generalize to Theil's [/coefficient. Third, and most importantly, therelative forecast accuracy of the two models in the case of the causally relatedcurrencies seems to be a direct function of the strength of the causal relation-ship. Consider spot rates first. The minimum FPE of the univariate model isseven percent, five percent and nine percent higher than the overall minimum

SPOT AND FORWARD EXCHANGE RATES

Table 5

115

Bivariate Autoregressive Time Series Models Estimated by FIML for theCanadian Dollar, Swiss Franc and Mark

Canadian

St =

Ft =

Dollar

: -0.00194(0.0022)

-3.40411(1.7059)

: -0.00194(0.0022)

-3.42265(1.7115)

Swiss Franc

St =

F, =

Mark

St =

Ft =

: -0.01050(0.0063)

•• -0.01055(0.0062)

-0.01044(0.0059)

-0.01043(0.0058)

+ 3.05205 S',.,(1.6882)

F ,_ , -3.19665(1.6345)

+ 3.12845 S,_i(1.6938)

F,_^ -2.91920(1.6399)

+ 3.83902 S,_^(2.0696)

+ 4.05326 S,_i(2.0408)

+ 7.03253 j " , . ,(3.0433)

+ 6.84643 5 ,_ ,(3.0032)

+ 2.93648 5', _2(1.6160)

Ft-2

+ 2.67168 S,_2(1.6214)

-4.19928 i^,_,(2.0814)

-4.41446 F ,_ ,(2.0524)

-7.34492 F,_^(3.0816)

-7.15228 F^_^(3.0409)

(18)

(19)

(20)

(21)

(22)

(23)

FPE for the Canadian dollar, Swiss franc, and mark, respectively.'* Thus,since the mark exhibits the strongest causal relationship, the bivariate modelpredicts better for the mark spot rate than does the univariate model in termsof both error metrics. On the other hand, since causality is somewhat less evi-dent for the dollar and Swiss franc, this could explain why the univariate modelis more accurate than the bivariate model for these two currencies.

A similar result holds in the case of forward rates. For forward rates, theminimum FPE of the univariate model is one percent, five percent and eightpercent greater than the overall minimum FPE for the Canadian dollar, Swissfranc, and mark, respectively. Thus, the bivariate model predicts better forthe Swiss franc and mark which show the stronger causal relationships. Onthe other hand, the univariate model is more accurate for the Canadian dollarwhich exhibits the weakest causad relationship. This last conclusion is only sug-gestive, however, since the caus2Llity relationships are somewhat marginal evenin the case of the mark.

116 CALLEN, CHAN AND KWAN

Table 6

Summary Statistics of Forecast Accuracy

Univartate

RMSE

Theil's U

Univariate

RMSE

Theil's U

Btvariate i

RMSE

Theil's U

Model

SFSF

Model

SFSF

Models

SFSF

Found

0.129580.120461.707471.64211

Canadian Dollar

0.042230.040071.174421.15566

Canadian Dollar

0.075240.073132.092432.10915

French Franc

0.077340.072630.992981.00729

Swiss Franc

0.140780.137261.069891.07162

Swiss Franc

0.144900.134971.101201.05374

Yen

0.102180.068960.994440.96837

Mark

0.092090.090101.076941.07798

Mark

0.089480.087301,046421.04433

Nute: RM.SE = Root Mean Square Krror

CONCLUSION

We argued from the theory of e.xchange rate determination and the theory ofmarket efficiency that forward rates should cause spot rates but not the reverse.Utilizing monthly spot and forward rates for six foreign currencies, we wereable to test for causal relationships using the Hsiao methodology. Specifically,the underlying models were assumed to be bivariate autoregressive. The lagstructure of the models were estimated and causal relationships determinedby Akaike's Final Prediction Error criterion. The overall results indicate thatforeign exchange markets are apparently inefficient. In particular, feedbackwas exhibited for the Canadian dollar, Swiss franc, and mark. Yet, in an effi-cient market, spot rates should not cause forward rates since this implies thatthe publicly available information conveyed by past spot rates is not impoundedin the forward rate.

Another important conclusion to be derived from the causality analysis isthat exchange rates, spot and forward, do not necessarily follow a random walk.For three currencies, the Canadian dollar, the Swiss franc and the mark, bothspot and forward rates are better described by bivariate rather than univariateprocesses.

In addition to determining the causal nature of the underlying models, wealso tested these models for forecast accuracy. We found that predictive ac-curacy did not vary substantially from one model, or even one currency, toanother. However, there was some evidence that forward rates are easier to

SPOT AND FORWARD EXCHANGE RATES 11 7

predict than spot rates. Finally, there was some suggestive evidence that thestronger the causal relationship, the more likely that the bivariate (simultaneous)model predicts better than the univariate model as would be expected.

NOTES

1 See, for example, Goodman (1979).2 Formally, Grauer, Litzenberger and Stehle (1976) have shown that

where S, is the spot rate at time /, F, _ , is the one period forward rate at time / — I, L, \sthe liquidity premium at time./, and E, _^(Si) is the market expectation at time / — 1 of thefuture spot rate 5,.

3 In essence, we will test the joint hypothesis that exchange rate theory is correct and that marketparticipants are correct on average in predicting future spot rates. However, as in all studiesof this type, we will assume that the theory is correct and focus on the predictive ability ofthe market.

4 See Fama (1970) for a review of market efficiency concepts in general and Kolhagen (1978)and Levich (1979a and 1979b) for market efficiency in the context of foreign exchange markets.

5 See Hsaio (1981).6 It could be argued that the F/'E objective is itself arbitrary. However, as we argue below,

the FPE criterion is consistent with the Granger definition of causality since both are basedon minimizing the mean square prediction error. On the other hand, standard significancetests bear no relationship whatsoever to the Granger definition.

7 If the direction of the inequality is reversed then (m + p,n + q) would be chosen.8 As we shall see this third step is usually unnecessary. The alternative to this sequential pro-

cedure i.s to use all possible combinations of lags for forward and spot rates and then to choosethe one combination which yields the minimum FPE. Glearly the sequential procedure isfar more efficient.

9 Evidence of liquidity premia in foreign exchange markets was found by Jacobs (1982) andmore recently by Fama (1983).

10 The rarndom walk characteristic of exchange rates is well documented. See Kolhagen (1978)and Levich (1979a and 1979b).

11 The fact that forward rates enter only in the form of a one period lag is consistent with thetheory. See note 2 above.

12 Unfortunately, a proper technique for determining causai relationships in a true simulaneoussystem has yet to be developed. Nevertheless, once having determined causality by OLS,better parameter estimates should be obtainable by utilizing a FIML approach.

13 Theil's coefficient is defined as

RMSE

u = —:, . , ,1/2, AJn)1

where RMSE is the root mean square error, /f, denotes the actual (as opposed to predicted)value, and n is the number of data points.

14 Take the mark, for example. The minimum univariate FPE (or the spot rate isO.151OX 10 .The overall (bivariate) minimum FPE for the mark is O.1375X 1 0 ' , a difference of 9 per-cent (with the univariate spot rate as base).

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