currency substitution and exchange rate determination

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Currency substitution and exchange ratedeterminationYijian He & Subhash C. SharmaPublished online: 06 Oct 2010.

To cite this article: Yijian He & Subhash C. Sharma (1997) Currency substitution and exchange ratedetermination, Applied Financial Economics, 7:4, 327-336, DOI: 10.1080/096031097333448

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An earlier version of this paper was presented at The Southern Economic Association Meeting, 18 Ð 20 November 1995, New Orleans.*The views expressed in this article are those of the author and not of the First National Bank of Chicago.

Applied Financial Economics, 1997, 7, 327 Ð 336

Currency substitution and exchange ratedetermination

YIJIAN HE1* and SUBHASH C. SHARMA2

1Capital Markets Department, First National Bank of Chicago, Chicago, IL 60670, USA,

2Department of Economics, Southern Illinois University, Carbondale, IL 62901, USA

The objectives of this paper are twofold. First, the monetary exchange rate model ofFrenkel Ð Bilson and Dornbusch Ð Frankel is extended to allow for the currency substi-tution between two countries, i.e. the domestic residents to hold the foreign moneyand the foreign residents to hold the domestic money. Second, by using the exchangerates from 1978.Q4 to 1991.Q2 between the US dollar and the Canadian dollar, USdollar and Japanese yen, US dollar and UK pound, and US dollar and German mark,we test if the new model is a long run exchange rate determination model and ifcurrency substitution is a signi® cant factor in in¯ uencing the long-run exchange rate.This is achieved by applying the techniques of cointegration and error correctionanalysis and the ten period out of sample forecasting performance of the extendedmodel. In general, the US Ð Canada, US Ð Germany, US Ð Japan and US Ð UK extendedmodels outperform the random walk model.

I . INTRODUCTION

One major research goal in the study of ¯ exible exchangerates is the desire to ® nd an acceptable model that explainsthe movement of the nominal spot exchange rates in termsof macroeconomic variables. Two main views of exchangerate determination, i.e. the monetary approach (in ¯ exible-price, sticky price) and the portfolio balance approach haveevolved since the early 1970s. In the monetary models,domestic and foreign securities are assumed to be perfectsubstitutes and hence gave identical risk; and the forwardrate is an unbiased expectation of the future spot rate. Theinterest parities hold to eliminate the riskless pro® ts fromthe interest rate di� erential. While the ¯ exible-price mone-tary model (Frenkel, 1976; Kouri, 1976; Mussa, 1976, 1979;Bilson, 1978) relies on the assumptions of purchasing powerparity (PPP) and the existence of a stable money demandfunctions for the domestic and foreign economies, the ex-change rate is the relative price of two monies. In the stickymonetary models of Dornbusch (1976a, 1976b) and Frankel(1979), di� erent speeds of adjustment for the goods andmoney markets were allowed. The Dornbusch Ð Frankelmodel is able to provide an explanation for the dynamic

adjustment of exchange rate towards a new equilibrium, andit follows that exchange rate behaviour re¯ ects the relativedemands for two monies. Monetary models also assumethat domestic money is demanded only by domestic resi-dents, and foreign money only by foreign residents. In theportfolio balance model of Branson (1977), the exchangerate is a principal determinant of the current account whichrepresents a rise or fall in net domestic holdings of foreignassets, which in turn a� ects the level of wealth. The modelsof exchange rate determination discussed above suggest thatexchange rates are jointly determined with macroeconomicvariables.

However, the empirical exchange rate models developedover the last two decades have done poorly in the in-sampleproperties and are unable to outperform the random walkmodel in out-of-sample forecasting ability (Meese andRogo� , 1983; MacDonald and Taylor, 1992). Meese (1990,p. 126 Ð 7) noted that, P̀otential estimation problems in-clude simultaneity, the reliance on limited informationestimation techniques, imposition of inappropriate con-straints or misspeci® ed dynamics, and small sample bias.’Attempts have been made to deal with potential estimationproblems in recent years. Engel and Hamilton (1990) and

0960 Ð 3107 Ó 1997 Routledge 327

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Diebold and Nason (1990) exploited nonlinearities in theexchange rate process but with limited success. Engel andHamilton (1990) found that forecasts from their model atthe 4-quarter horizon for the German mark and Frenchfranc exchange rates are outperformed by the random walkwith drift. Similarly, Diebold and Nason (1990) concludedthat their nominal exchange rate model, in general, cannotimprove upon the random walk without drift for weeklydata.

The recent work of MacDonald and Taylor (1994) andMark (1995) has shown some success. By using the cointeg-ration procedure of Johansen (1988) and Johansen andJuselius (1990), MacDonald and Taylor (1994) demon-strated that the ¯ exible-price monetary model is a validlong-run exchange rate determination model for theUK Ð US exchange rate in the UK. Furthermore, they foundthat the short-run error correction model outperforms therandom walk and other models in an out-of-sample fore-casting contest. Their work has suggested that the previousstudies of exchange rate models were unable to capture thedynamic data generating process.

On the other hand, Mark (1995) accounted for problemsof small sample bias and slow convergence to the asymp-totic distribution by drawing inference from bootstrapdistributions generated under the null hypothesis ofexchange-rate unpredictability. Mark (1995) presentedevidence that long-horizon changes in the logarithm ofspot exchange rates are predictable for the US Ð Canada,US Ð German mark, US Ð Swiss franc, and US Ð yen exchangerates based on the estimated projections of 1-, 4-, 8-,12- and 16-quarter changes in the log exchange rate on thederivation of the current log exchange rate from itsfundamentals. The root mean square error (RMSE)of the forecasts based on regressions for the Swiss francand the yen are smaller than that of the driftless randomwalk at every horizon and at the 12- and 16-quarter hor-izons for the German mark. But the Canadian-dollar regres-sion cannot outperform the random walk after 1-quarterhorizon.

In this paper, we ® rst extend the monetary exchange ratemodels (Frenkel, 1976; Bilson, 1978; Dornbusch, 1976a;1976b; Frankel, 1979) to allow for the currency substitutionbetween two countries. Our extended model allows thedomestic residents to hold the foreign money and foreignresidents to hold the domestic money. We then test if theproposed model is a long-run exchange rate determinationmodel and if currency substitution is a signi® cant factor inin¯ uencing the long-run exchange rate. The US dollar Ð Ca-nadian dollar, US dollar Ð German mark, US dollar Ð Ja-panese yen, and US dollar Ð UK pound exchange rates forthe period from 1978:Q4 to 1993:Q4 were studied. Finally,we use currency substitution factors and the other macro-economic variables (in the reduced rank model) and evalu-ate the out-of-sample forecasting performance of the ex-change rate models from 1991:Q3 to 1993:Q4.

II . CURRENCY SUBSTITUTION ANDTHE EXCHANGE RATE MODEL

We assume that in two-country model, the demand formoney consists of demand by domestic residents and de-mand by foreign residents, and that money markets are inequilibrium. Then the money demand functions for thehome and foreign countries can be expressed as

m = m*1 + p + ky - l r (1)and

m* = m1 + p* + ky* - l r* (2)

where m, p, y and r represent the money supply, price level,industrial production and interest rate of home country; m*,p*, y* and r* are the counterparts of the foreign country;m1 is the demand for foreign money by home residents andm*1 is the demand for home money by foreign residents.k and l are constant income elasticity and interest elasticityof the demand for money, respectively.

Let s denote the spot exchange rate (home currency priceof foreign currency) and we assume that the long-run pur-chasing power parity holds, i.e.

s = p - p* (3)

Further, by substituting Equations 1 and 2 into Equation 3,we obtain the long-run ¯ exible price exchange rate model

s1 = m - m* + m1 - m*1 - k (y - y*) + l (r - r*) (4)

where s1 is the long-run exchange rate. Note that this

long-run ¯ exible price exchange rate model is the extensionof Frenkel Ð Bilson’s model (see Frenkel, 1976; Bilson, 1978)to incorporate currency substitution into the model. Dorn-busch (1976a, 1976b) and Frankel (1979) allowed for short-run deviations from PPP caused by sticky domestic prices.Frankel (1979) demonstrated that the augmented regressiveexpectations are rational, i.e.

set+ 1 - st = u (s1 - s)t + (p e - p *e )t (5)

where set+ 1 is the expected exchange rate in the next period

conditional on the current information, (p e - p *e)t is theexpected in¯ ational di� erential and u is the adjustmentparameter. Further, we assume that the uncovered interestparity holds, i.e.

set+ 1 - st = r - r* (6)

Substituting Equations 6 and 4 into Equation 5, we obtainthe exchange rate model similar to the quasi-reduced formof the Dornbusch Ð Frankel Model (see Meese and Rogo� ,1983, p. 85), i.e.

s = m - m* + m1 - m*1 - k (y - y*) + (l - 1/ u ) (r - r*)

+ 1/ u (p e - p *e ) (7)

Finally, by substituting the realized in¯ ation di� erential forthe expected one we obtain a monetary exchange rate re-duced form, which is an extension of Meese’s (1990) model,

328 Y . He and S. C. Sharma

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to incorporate currency substitution:

st = a + b1 mt + b2 m*t + b3 m1 t + b4 m*1 t + b5 (yt - y*t )

+ b6 (rt - r*t ) + b7 (p t - p *t ) (8)

where st is the spot exchange rate (home currency price offoreign currency); mt represents the demand for homemoney by home residents; m1 t represents the demand forforeign money by home residents; m*t represents the demandfor foreign money by foreign residents; m*1 t represents thedemand for home money by foreign residents; the funda-mentals in parentheses represent the industrial productiondi� erential (yt - y*t ), interest rate di� erential (rt - r*t ), andin¯ ation di� erential (p t - p *t ), respectively. The spot ex-change rate and money quantities are expressed in naturallogarithms. The industrial production, interest rates andin¯ ation di� erential are indices and are equal to 100 in thebase year 1990.

In the absence of currency substitution factors, theFrenkel Ð Bilson ¯ exible-price monetary model posits coe� c-ient restriction: b1 > 0, b2 < 0, b5 < 0, b6 > 0 and b7 = 0. Incontrast, The Dornbusch Ð Frankel sticky-price monetarymodel hypothesizes that b1 > 0, b2 < 0, b5 < 0, b6 < 0 andb7 > 0. The parameters on the holdings of foreign monies,b3 and b4 , denote the channels through which a change inthe holdings of foreign monies in¯ uence the exchange rate,thus indicating the e� ect of currency substitution on ex-change rate. If b3 or b4 is signi® cantly di� erent from zero,currency substitution should be viewed as a valid funda-mental variable in analysing the long-run exchange ratemovement. We expect b3 to be positive and b4 to be nega-tive. An increase in the demand for domestic currency bydomestic residents and one by foreign residents have di� er-ent e� ects on the exchange rate. An increase in the demandfor domestic money by foreign residents is met throughtransaction in the foreign exchange market while an in-crease in the demand for domestic currency by domesticresidents is primarily met by the credit expansion in thedomestic money market. Therefore, an increase in the de-mand for domestic money by foreign residents may havea signi® cant e� ect on exchange rate, although the holdingsof domestic money by foreign residents are relatively smallcompared with the holdings of domestic money.

One novel feature of our model is that it incorporates thecurrency substitution factors, and does not express themoney variables in terms of di� erences. Meese and Rogo�(1983, p. 96) noted that, `Whether or not money demandinstability and/or misspeci® cation is responsible for theexchange rate results, it is certainly true that the conven-tional money demand equation does not work well whenexpressed in terms of US minus foreign variables’ . Meeseand Rogo� (1983) also investigated the possibility that theirresult were generated solely by money demand instability inthe US by performing ex post forecast experiments. Theyconcluded that money demand instability or misspeci® ca-

tion is an important potential explanation for their results.Thus, our model can be viewed as an attempt to improve theunderlying money demand speci® cations.

III . THE LONG-RUN EXCHANGE RATEDETERMINATION

Quarterly data from 1978:Q4 to 1993:Q4 is used in thisstudy. The money supply (mt and m*t ) is represented bymonetary aggregate M1 for each country. The US demandfor foreign money (m1 t) is the foreign currency-denominateddeposit claims on foreigners reported by banks and non-banking ® rms in the US from the department of the USTreasury. The demand for domestic money by foreign resi-dents (m*1 t) is represented by the US banks on demanddeposit liabilities payable in US dollars to private non-bankforeign residents from the US Treasury Bulletin (Table CM-I-4). The home country is the US and foreign countriesconsidered are Canada, Germany, Japan and the UnitedKingdom. The industrial production index (1990 as a baseyear) is used for the industrial production and the three-month treasury bill rate is used for the interest rate. Data onmoney (mt and m*t ), industrial production, prices and inter-est rates are obtained from the International Financial Stat-istics (CD ROM). The spot exchange rates are taken fromthe International Money Market.

Order of integration in series

First, the order of integration is determined in each timeseries by using Dickey and Fuller (1979, 1981), Perron (1988)and Phillips and Perron (1988) tests. For this the followingregression is estimated

Xt = m + b (t - n/2) + a Xt ± 1 + e t (9)

where Xt is the series being tested. We test the hypothesisH

10 : b = 0, a = 1 using the test statistics F 3 and Z (F 3 ), and

H0 : a = 1 using the test statistics ta, t t , Z (ta ) and Z (a ). t

aand

Zt are the Dickey Ð Fuller statistics allowing for a constantmean and the trend in mean. Test statistics Z ( F 3 ), Z (a ) andZ (ta

) are given in Perron (1988, pp. 308 Ð 9). Since there is noconsensus among econometricians on the unit root tests, sixdi� erent tests are used on each series to test for a unit root.

The unit root test statistics for all the series in levels andin ® rst di� erences are reported in Table 1. From Table 1, we® nd that unit root hypothesis is rejected for US holdings ofCanadian money, m1 , Canadian holdings of US money, m*1 ,Germany money supply, m*, German holdings of USmoney, m*1 , UK holdings of US money, m*1 . However, for allother series, we fail to reject the unit root hypothesis inlevels. But the unit root hypothesis is rejected in the ® rstdi� erence of each series (the ® rst di� erences are the ® rstdi� erences of the logs of series for money series m, m*,m1 and m*1 .) Thus, in the US Ð Canada exchange rate

Currency substitution and exchange rate determination 329

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Table 1. Unit root test statistics for level and Þ rst di¤ erence of series time period 1978:Q4 Ð 1993:Q4

Level First di� erence

Series F 3 Z(F 3 ) t a t t Z (a ) Z (t a ) F 3 Z (F 3 ) t a t t Z (a ) Z (t a )

US Ð Canada exchange rate

s 0.78 1.05 - 1.23 - 1.23 - 1.48 - 1.49 28.3 28.7 - 7.6 - 7.5 - 7.8 - 7.7m 3.34 3.13 - 0.57 - 2.41 - 0.96 - 2.45 66.4 64.4 - 11.6 - 11.5 - 11.5 - 11.5m* 1.30 1.08 - 0.61 - 1.60 - 0.58 - 1.54 76.8 68.6 - 12.4 - 12.4 - 11.7 - 11.9m1 8.49 8.41 - 3.93 - 4.09 - 4.08 - 4.23 55.2 61.7 - 10.5 - 10.5 - 11.2 - 11.3m*1 22.10 21.60 - 6.08 - 6.65 - 6.16 - 6.78 57.9 111.0 - 10.9 - 10.8 - 15.2 - 15.2y - y* 2.00 2.80 0.05 - 1.86 - 0.32 - 2.36 9.8 9.6 - 4.4 - 4.4 - 4.5 - 4.5r - r* 4.18 4.01 - 2.81 - 2.82 - 2.82 - 2.90 33.7 36.1 - 8.2 - 8.2 - 8.6 - 8.7p - p * 1.65 2.56 - 1.75 - 1.72 - 2.30 - 2.29 13.5 12.7 - 5.2 - 5.2 - 5.1 - 5.0

USÐ Germany exchange rate

s 2.13 2.20 - 1.23 - 1.91 - 1.39 - 1.03 25.7 25.6 - 7.2 - 7.2 - 7.3 - 7.3m* 22.70 23.80 - 4.19 - 6.73 - 4.37 - 7.08 127.1 182.1 - 15.3 - 15.9 - 16.0 - 19.3m1 4.27 3.96 - 1.23 - 2.83 - 0.98 - 2.79 53.3 59.5 - 10.4 - 10.3 - 11.0 - 11.1m*1 22.60 23.80 - 4.19 - 6.73 - 4.37 - 7.08 80.4 148.0 - 12.8 - 12.7 - 17.4 - 17.5y - y* 1.50 2.26 - 0.98 - 1.60 - 1.48 - 2.13 21.0 21.7 - 6.4 - 6.5 - 6.6 - 6.7r - r* 3.34 2.92 - 1.33 - 2.58 - 1.12 - 2.54 26.8 28.3 - 7.4 - 7.3 - 7.8 - 7.8p - p * 0.96 1.68 - 1.38 - 1.04 - 1.88 - 1.39 14.7 14.7 - 5.4 - 5.5 - 5.0 - 5.3

US Ð Japan exchange rate

s 4.07 4.18 - 0.23 - 2.58 - 0.38 - 2.75 24.3 24.0 - 6.9 - 6.9 - 7.1 - 7.1m* 19.60 19.60 - 0.14 - 6.20 - 0.74 - 6.39 65.5 118.0 - 11.5 - 11.4 - 15.4 - 15.7m1 2.61 2.70 - 1.16 - 2.28 - 1.09 - 2.44 44.7 44.8 - 9.5 - 9.4 - 7.1 - 7.1m*1 4.47 4.33 - 1.69 - 2.98 - 1.56 - 3.07 34.1 36.5 - 8.3 - 8.3 - 9.6 - 9.7y - y* 2.84 2.21 - 1.58 - 0.10 - 2.05 - 1.01 15.4 16.4 - 5.0 - 5.4 - 5.4 - 5.8r - r* 4.97 5.48 - 2.79 - 3.09 - 2.97 - 3.39 16.8 15.3 - 5.8 - 5.8 - 5.7 - 5.8p - p * 1.84 2.24 - 1.89 - 1.59 - 2.19 - 1.71 25.5 25.7 - 7.1 - 7.1 - 7.3 - 7.2

USÐ UK exchange rates 1.31 1.48 - 1.58 - 1.61 - 1.72 - 1.78 21.3 20.7 - 6.5 - 6.5 - 6.6 - 6.6m* 1.56 1.64 - 0.48 - 1.77 - 0.48 - 1.90 37.1 36.9 - 8.7 - 8.6 - 8.7 - 8.7m1 1.64 1.36 - 1.48 - 1.53 - 1.53 - 1.35 39.9 42.9 - 8.8 - 8.9 - 9.1 - 9.4m*1 19.80 13.30 - 3.66 - 6.30 - 3.68 - 6.57 89.3 139.0 - 13.5 - 13.4 - 16.9 - 16.9y - y* 1.98 2.79 0.04 - 1.85 - 0.33 - 2.35 16.5 16.1 - 4.4 - 4.4 - 4.5 - 4.5r - r* 3.52 3.46 - 2.38 - 2.63 - 2.42 - 2.75 26.7 26.5 - 6.9 - 6.8 - 7.0 - 7.0p - p * 5.55 5.90 - 3.13 - 2.70 - 3.39 - 3.05 29.2 29.21 - 7.6 - 7.6 - 7.8 - 7.7

(i): s, m, m*, m1 , m*1 are the spot exchange rate (US dollars per unit of foreign currency), US money supply M1, foreign money supply M1, US holdings of foreign money andforeign holdings of US money, respectively, and are expressed in logarithms. y - y*, r - r* and p - p * are the di� erences of the industrial productions, short-term interest ratesand in¯ ation rates between US and foreign countries.(ii): The critical value at the 5% level for t a , and Z (a ) is - 2.93 and for t t , Z (ta ) is - 3.50 (Fuller, 1976), and the critical value at the 5% level for F 3 and Z (F 3 ) is 6.73 (Dickey andFuller, 1981, p. 1063).

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1 We did include those series which are integrated of order 0 but the results are unsatisfactory, in the sense that many basic restrictions oncoe� cients of variables are violated. These results are not reported here. However, with the exclusion of those series integrated of order 0,the cointegrating vectors have meaningful economic interpretation.2 Since the tests for cointegrations are sensitive to the lag lengths, we tried lag lengths equal to 2 and 3. Although, as expected, the laglengths 2 and 3 change the statistics and cointegrating vectors slightly, basic conclusions about the existence of cointegrating vectors andsigns of the estimated coe� cients in cointegrating vectors remain the same. Thus the results are reported only for lag length equal to 2.

Table 2. Tests for the number of cointegrating vectors

Test Number of cointegrating vectorsModel statistic r < 0 r < 1 r < 2 r < 3 r < 4 r < 5 r < 6 r < 7

US Ð Canada l -max 40.81 36.31 26.76 19.93 16.65 8.49 1.48l -Trace 150.4 109.6 73.32 46.56 26.62 9.97 1.48

US Ð Germany l -max 49.91 24.76 12.99 6.84 5.19 0.34l -Trace 100.0 50.13 25.37 12.38 5.53 0.34

US Ð Japan l -max 54.37 44.48 43.22 34.10 18.70 15.24 9.21 0.75l -Trace 220.1 165.7 121.2 78.02 43.91 25.21 9.96 0.75

US Ð UK l -max 43.24 37.64 21.57 19.59 8.57 7.42 5.37l -Trace 143.4 100.2 62.52 40.95 21.36 12.79 5.37

10% critical values l -max 48.91 43.25 37.45 31.66 25.56 19.77 13.75 7.52l -Trace 159.5 126.6 97.18 71.86 49.65 32.00 17.85 7.52

Note: The 10% critical values of the maximum eigenvalue and the Trace statistics are taken from Osterwald-Lenum (1992).

determination model, s, m, m*, m1 , y - y*, r - r*, p - p * areintegrated of order 1, I(1); m*1 is integrated of order 0, I (0).Similarly, in the US Ð Germany exchange rate determinationmodel, s, m, m1 , y - y*, r - r*, p - p * are I (1), and m* andm*1 are I (0); in the US Ð Japan exchange rate determinationmodel, all series are I (1); and in the US Ð UK exchange ratedetermination model, s, m, m*, m1 , y - y*, r - r* andp - p * are I(1) and m*1 is I (0).

The long-run relationship

The time series data in levels are probably non-stationary .However, linear combinations of non-stationary series maybe stationary (Engle and Granger, 1987). When this occursthe time paths of the individual variables are ultimatelyconstrained to an equilibrium relationship and are said tobe cointegrated. Next, we test whether there is any long runrelationship among the currency substitution and the macrovariables.

In this analysis, for testing the number of cointegratingvectors, some of the series are integrated of order 0 andothers are integrated of order 1. We exclude those seriesintegrated of order 0 in cointegrating vectors.1

Hence the long run relationship is tested in the followingmodels

US-C = (s, m, m*, m1 , y - y*, r - r*, p - p *)

US-G = (s, m, m1 , y - y*, r - r*, p - p *)

US-J = (s, m, m*, m1 , m*1 , y - y*, r - r*, p - p *)

US-UK = (s, m, m*, m1 , y - y*, r - r*, p - p *)

Note that US-C, US-G, US-J and US-UK represents themodels for US Ð Canada, US Ð Germany, US Ð Japan andUS Ð United Kingdom, respectively.

The Johansen (1988) and Johansen and Juselius (1990)l -trace and l -max statistic are used to test for the number ofcointegrating vectors in the above models. First, the l -tracestatistics are used to test for r < r0 cointegrating vectors andthen l -max statistics are used to test for r = r0 cointegratingvectors. These statistics are reported in Table 2.2

For the US Ð Canada cointegrating vector, we fail to rejectthe hypothesis of zero cointegrating vector from the l -tracestatistics at the 10% level. However, from the l -max statis-tic, we cannot reject the hypothesis of the zero cointegratingvector at the 10% level. Hence we conclude that there is nocointegrating vector for US Ð Canada. For the US Ð Germanycointegrating vector, we fail to reject the hypothesis of thezero cointegrating vector but cannot reject the hypothesis ofone cointegrating vector for the trace statistics. Thus, thereexists at most one cointegrating vector for the trace statis-tics. Then, from the l -max statistic we accept the hypothesisof one cointegrating vector against the alternative of thezero cointegrating vector. Similarly, for US Ð Japan andUS Ð UK, we fail to reject the hypothesis that there is nocointegrating vector. Actually, from the l -max statistics weconclude that there are four cointegrating vectors forUS Ð Japan and two cointegrating vectors for US Ð UK.

Thus, we ® nd that there exists a long run relationshipbetween the spot exchange rate and macroeconomic vari-ables for the exchange rate models investigated here, exceptfor the US Ð Canada exchange rate model. This shows thatour exchange rate determination model given by Equation 8


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Table 3. The normalized cointegrating vectors

(i) US Ð Canada: (s, m, m*, m1 , y - y*, r - r*, p - p *)

No cointegrating vectors exist.

(ii) USÐ Germany: (s, m, m1 , y - y*, r - r*, p - p *)

(1.00, - 2.92, - 0.173, 0.135, - 0.368, 0.022).

(iii) US Ð Japan: (s, m, m*, m1 , m*1 , y - y*, r - r*, p - p *)

(1.00, 0.834, - 5.396, 0.371, - 0.95, 0.021, - 0.388, - 0.098)

(1.00, - 3.447, 3.024, - 0.058, 0.152, 0.032, 0.004, - 0.138)

(1.00, - 0.059, 4.998, - 0.413, - 0.675, 0.018, 0.039, 0.080)

(1.00, 2.360, - 4.366, - 0.071, - 0.106, - 0.028, - 0.014, - 0.071).

(iv) US Ð UK: (s, m, m*, m1 , y - y*, r - r*, p - p *)

(1.000, - 16.449, 2.255, 3.118, 0.121, 0.350, - 0.169)

(1.000, 326.553, 17.70, - 67.86, - 4.132, 8.952, - 12.773).

Table 4. The likelihood ratio test statistics

Null LR test CriticalModel hypotheses statistics 10% value

US Ð Germany b 3 = 0 x21 = 3.565 2.71

b 7 = 0 x21 = 0.68 2.71

US Ð Japan b 3 = b 4 = 0 x28 = 36.71 13.36

b 7 = 0 x24 = 25.12 7.78

US Ð UK b 3 = 0 x22 = 18.75 4.61

b 7 = 0 x22 = 13.02 4.61

Under the assumption of existence of cointegration relation, thelinear hypotheses on the cointegrating relations can be tested bythe likelihood ratio test given by Johansen and Juselius (1990), i.e.

- 2ln(Q) = Tr

+i= 1

ln1 - l 0 i

1 - l i

The asymptotic distribution of this statistic is x2r(p ± s) where r (p - s)

are degrees of freedom, r is the number of cointegrating vectors andp - s is the number of restrictions.

is a valid long run exchange rate determination model forthe US Ð Germany, US Ð Japan and US Ð UK.

The normalized estimates of cointegrating vectors arereported in Table 3. Since there is only one cointegratingvector for the US Ð Germany model, with the normalizedcointegrating vector, we obtain a long-run exchange ratedetermination model

s = 2.92 m + 0.173 m1 - 0.135 (y - y*) + 0.368 (r - r*)

- 0.022 (p - p *) (10)

For the US Ð Japan model, there are multiple cointegratingvectors. The ® rst cointegrating vector obviously violates thebasic restrictions on coe� cients of variables m and m* insigns. However, the second cointegrating vector does havea meaningful economic interpretation. Thus, we choose theexchange rate model corresponding to the second cointe-grating vector i.e.

s = 3.447 m - 3.024 m* + 0.058 m1 - 0.152 m*1

- 0.032( y - y*) - 0.004 (r - r*) + 0.138 (p - p *) (11)

Since, in the US Ð UK model, there are two cointegratingvectors and only the ® rst cointegrating vector has economicmeaning, we obtain the long-run exchange rate model cor-responding to the ® rst cointegrating vector as follows

s = 16.449 m - 2.255 m* - 3.118 m1 - 0.121 (y - y*)

- 0.35 (r - r*) + 0.169 (p - p *). (12)

From Equations 10 to 12, we observe that in all threemodels, parameters b1 , b2 and b6 are consistent in signs withthe monetary exchange rate determination model, i.e.b1 > 0, b2 < 0 and b6 < 0, and the signs of parametersb3 and b4 , are consistent with our hypothesis, i.e. b3 < 0,b4 > 0. The estimated parameter b5 is consistent with the

Frenkel Ð Bilson ¯ exible-price monetary model (b5 < 0) forthe US Ð German model, whereas it is consistent with theDornbusch Ð Frankel sticky-price monetary model (b5 < 0)for the US Ð Japan and US Ð UK models.

The existence of cointegration allows us to proceed to testfor the signi® cance of currency substitution factors and forthe hypothesis posited in the FrenkelÐ Bilson model. Thus,in the US Ð Japan model we formulate the hypotheses thatb3 = b4 = 0 and that b8 = 0. But in the US Ð Germany andUS Ð UK models, we formulate the hypotheses that b3 = 0and that b8 = 0 since we do not include the holding of UScurrency by foreign residents in the model due to the sta-tionarity of series m*1 for Germany and UK. The likelihoodratio test statistics are reported in Table 4.

From the likelihood ratio test statistics, x2 , we reject the

hypotheses that b3 = b4 = 0 (or b3 = 0) at 10% for all threemodels, indicating that currency substitution is an impor-tant factor in the long-run exchange rate determinationmodel. Similarly, from the likelihood ratio test statistics, x

2 ,we reject the hypothesis that b8 = 0 for the US Ð Japanmodel and US Ð UK model but cannot reject the hypotheth-esis that b8 = 0 for the US Ð Germany model. This result,combined with the signs of b5 , indicates that the US ÐGermany model is consistent with the FrenkelÐ Bilson ¯ ex-ible-price monetary model whereas the US Ð Japan andUS Ð UK models are consistent with the Dornbusch ÐFrankel sticky-price monetary model.

IV. OUT-OF-SAMPLE FORECASTINGPERFORM ANCE

In this section, we use the long-run cointegration relation-ship to obtain the error correction term. Then the error


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correction term is used together with the variables in Equa-tion 8 to form three s̀hort-run’ exchange rate forecastingmodels for the US Ð Germany, US Ð Japan and US Ð UK spotexchange rates. These short-run equations are estimatedand used to construct out-of-sample forecasts.

For the US Ð Germany, US Ð Japan and US Ð UK modelsthe error correction terms are obtained from Equations 10,11 and 12. Although we fail to reject the hypothesis of thezero cointegrating vector at the 10% level from the l -maxstatistics for the US Ð Canada cointegrating vector, we can-not reject the hypothesis of the zero cointegrating vector atthe 15% level. For the forecasting purpose we still choosethe ® rst cointegrating vector and use it to obtain the errorcorrection term, since the error term in this case can beinterpreted to be the deviation of the exchange rate from themacroeconomic variables, which is as important as the errorcorrection term in the presence of the cointegration rela-tionship.

Thus, the error correction term for US Ð Canada is

ecmc = s - 0.367 m + 0.394 m* + 0.220 m1 + 0.013 (y - y*)

+ 0.024 (r - r*) + 0.019 (p - p *),

and ecm for Germany, Japan and UK are obtained fromEquations 10, 11 and 12 respectively. These error correctionterms and variables in Equation 8 are used to form thefollowing three variations of exchange rate forecasting mod-els for each of four exchange rates

Model I: st = a + b3 m1 t + b4 m*1 t + b8 ecmt + e t

Model II: st = a + b1 mt + b2 m*t + b3 m1 t + b4 m*1 t

+ b5 (yt - y*t ) + b6 (rt - r*t )

+ b7 (p t - p *t ) + e t

Model III: st = a + b1 mt + b2 m*t + b5 (yt - y*t )

+ b6 (rt - r*t ) + b7 (p t - p *t ) + e t

Model I contains currency substitution factors (the USholdings of foreign currency and foreign holdings of the UScurrency) and the error correction term. Model II containsmacroeconomic variables and currency substitution factors.Model III is the typical monetary exchange rate modelwithout currency substitution factors. We believe that addi-tional emphasis on the currency substitution factors and theerror correction term will enhance the explanatory power ofthe monetary exchange rate model. This will also help usdistinguish the role of the currency substitution factors inthe exchange rate forecasts.

To formulate an econometric model for the exchangerate forecasting models we add a stochastic component toModels I, II and III. The estimation is done in two stages. Inthe ® rst stage, each model is estimated by the OrdinaryLeast Square (OLS) method. The estimation is carried outby using SAS/ETS (1993). The autocorrelation function(ACF) and partial autocorrelation function (PACF) of the

residual for each equation is analysed to determine theautocorrelation. In each equation, ACF and PACF suggestthat the residual follows an autoregressive process oforder 1, AR(1), i.e.

e t = r e t ± 1 + ut , | r | < 1 (13)

where the random errors, ut , are assumed to be independentand identically distributed as normal variates. Thus, in the® nal stage, each equation is corrected for AR(1) errors and isestimated by OLS again. The quarterly data from 1978:Q4to 1991:Q2 are used in the estimation and to forecast 10-period out of sample forecasts, and the data from 1991:Q3to 1993.Q4 are used as a test set to evaluate the forecast.

The ® nal parameter estimates are presented in Table 5. R2

is reasonably high for all models. In Model I, we observethat b3 is signi® cant at the 10% level for all four two-country exchange rate models but b4 is signi® cant at the10% level only for the US Ð Japan exchange rate model. b8 issigni® cant for the US Ð Canada and US Ð Germany exchangerate models. This shows the importance of the currencysubstitution factors and the error correction term. It alsoshows that the holdings of foreign money by US residents isa more important factor than the holdings of US money byforeign residents in the determination of the exchange ratein the US. In Model II, there is no signi® cant coe� cient atthe 10% level for the US Ð Canada model; a, b3 and b7 aresigni® cant at the 10% level for the US Ð Germany model; a,b1 and b7 are signi® cant at the 10% for the US Ð Japanmodel; and b2 is signi® cant at the 10% level for the US Ð UKmodel. In Model III, the signi® cant parameters remain thesame as those in Model II, except that Model III excludesb3 and b4 .

Finally, we compare the out-of-sample performance ofour three models with the random walk model. The root-mean-square-error (RMSE) is used as a measure of thedeviation of the forecasted value from its actual time path,which is given as follows

RMSE = 1h

+i= 1

(xft+ i - xt+ i)

2

h 21 /2

where h is the forecasting horizon and xf and x are forecas-ted and actual values, respectively. The RMSE of the out-of-sample forecasts is reported in Table 6. For the US Ð Canadaexchange rate forecast, Model I outperforms the randomwalk model except for 1- and 2-quarter horizons. Thegreater RMSE of the 2-quarter forecast horizon is due togreater 1-quarter forecast error. Moreover, the RMSE ofour model is less than half of that of the random walk modelon average. Models II and III also outperform the randomwalk model for 3-quarter to 10-quarter horizons. Amongthe three models, Model I appears to be superior to ModelsII and III. For the US Ð Germany exchange rate forecast,Model I clearly beats the random walk model and domin-ates two other models for all forecast horizons. For the


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Table 5. Parameter estimates of the short-run exchange rate model 1978:Q4 Ð 1991:Q2

Countries

Parameters Canada Germany Japan UK

Model I

a - 0.06194 - 0.06202 - 1.67118 0.381995( - 1.76) ( - 0.35) ( - 5.36) (1.69)

b3 - 0.07239 0.093339 0.074905 - 0.06082( - 4.65) (4.08) (3.58) ( - 1.87)

b4 - 0.00976 0.041208 0.086829 0.028408( - 0.73) (0.85) (1.84) (0.7)

b8 0.31913 0.060517 - 0.047998 0.008173(5.04) (2.01) ( - 0.67) (0.68)

R2 0.889 0.886 0.911 0.843DW 1.311 1.148 1.541 0.88

Model II

a - 0.28089 - 1.16417 - 4.13127 0.352442( - 1.06) ( - 3.59) ( - 11.06) (1.01)

b1 0.20626 0.284631 0.883427 - 0.30619(0.91) (0.93) (3.44) ( - 1.15)

b2 - 0.13806 0.148879 - 0.01525 0.140354( - 1.01) (0.61) ( - 0.05) (1.79)

b3 0.00122 0.048241 0.009881 - 0.0676(0.11) (1.98) (0.39) ( - 1.42)

b4 - 0.00631 - 0.05493 0.033531 0.030267( - 0.31) ( - 1.15) (0.81) (0.75)

b5 0.0022 - 0.00442 0.004206 - 0.00397(0.87) ( - 0.75) (0.55) ( - 0.67)

b6 - 0.0027 - 0.01061 - 0.01061 - 0.00505( - 0.54) ( - 0.97) ( - 1.34) ( - 0.59)

b7 0.00633 0.044701 0.03316 - 0.01022(1.68) (3.76) (2.71) ( - 1.01)

R2 0.846 0.914 0.943 0.864DW 1.165 1.292 1.508 1.091

Model III

a - 0.24684 - 1.00592 - 3.80384 0.793195( - 1.05) ( - 3.71) ( - 14.22) (2.37)

b1 0.17085 0.267895 0.967037 - 0.52713(0.88) (0.85) (4.25) ( - 2.28)

b2 - 0.11791 0.199196 0.00478 0.139932( - 1.00) (0.8) (0.02) (1.78)

b5 0.00225 - 0.00151 0.003193 - 0.00319(0.93) ( - 0.27) (0.43) ( - 0.54)

b6 - 0.00329 - 0.0176 - 0.00986 - 0.00544( - 0.76) ( - 1.8) ( - 1.28) ( - 0.60)

b7 0.00621 0.043227 0.030239 - 0.01051(1.7) (3.59) (2.51) ( - 0.98)

R2 0.846 0.909 0.943 0.841DW 1.167 1.502 1.557 1.092

Asymptotic t-statistics are reported in parentheses.Model I: st = a + b3 m1 t + b4 m*1 t + b8 ecmt + e t

Model II: st = a + b1 mt + b2 m*t + b3 m1 t + b4 m*1 t + b5 (yt - y*t ) + b6 (rt - r*t ) + b7 (p t - p *t ) + e t

Model III: st = a + b1 mt + b2 m*t + b5 (yt - y*t ) + b6 (rt - r*t ) + b7 (p t - p *t ) + e t

US Ð Japan exchange rate forecast, Model I cannot beat therandom walk model for 1-, 2- and 3-quarter forecast hor-izons, but the greater RMSE in 2- and 3-quarter forecasthorizons is attributed to 1-quarter forecast error. Models II

and III beat the random walk model for 2- to 10-quarterforecast horizons. For the US Ð UK exchange rate forecast,our three models outperform the random walk model for 2-to 10-quarter horizons but not for the 1-quarter horizon.


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Table 6. The root-mean-square error of out of sample forecasts

Forecast Randomhorizon Model I Model II Model III model

US Ð Canada

1-Quarter 0.011870 0.034300 0.033110 0.0092302-Quarter 0.011758 0.026937 0.025278 0.0105653-Quarter 0.018183 0.030798 0.029161 0.0249294-Quarter 0.020983 0.027253 0.025652 0.0325465-Quarter 0.025078 0.032888 0.030920 0.0489406-Quarter 0.023332 0.035645 0.033932 0.0625837-Quarter 0.021700 0.033365 0.031584 0.0685648-Quarter 0.022105 0.035503 0.033151 0.0760459-Quarter 0.027970 0.038065 0.036017 0.08846710-Quarter 0.027101 0.039058 0.036685 0.096053

US Ð Germany


US Ð Japan


US Ð UK


Model I: st = a + b3 m1 t + b4 m*1 t + b8 ecmt + e t

Model II: st = a + b1 mt + b2 m*t + b3 m1 t + b4 m*1 t + b5 (yt - y*t )

+ b6 (rt - r*t ) + b7 (p t - p *t ) + e t

Model III: st = a + b1 mt + b2 m*t + b5 (yt - y*t ) + b6 (rt - r*t )

+ b7 (p t - p *t ) + e t

It can also be seen that Models I and II are superior toModel III. This again shows the role and importance ofcurrency substitution factors in the models.

IV . CONCLUSIONS

We extended the monetary exchange rate determinationmodel (e.g. Frenkel, 1976; Bilson, 1978; Dornbush, 1976a,1976b; Frankel, 1979) to incorporate currency substitution,i.e. to allow the domestic residents to hold the foreignmoney and foreign residents to hold the domestic money.Based on the Johansen cointegration analysis we concludethat the new extended exchange rate model is a valid long-run exchange rate determination model for the US Ð Germanexchange rate, US Ð Japan exchange rate and US Ð UK ex-change rate, although not for the US Ð Canada exchange rate.

Next, we test for the signi® cance of currency substitutionfactors and the hypotheses posited in the monetary ex-change rate model. Based on the likelihood ratio test statis-tics, we observe that currency substitution is an importantfactor in the long-run in, US Ð Germany, US Ð Japan andUS Ð UK exchange rate determination. Our analysis alsoreveals that the US Ð Germany model is consistent with theFrenkel Ð Bilson ¯ exible-price monetary model whereasthe US Ð Japan and US Ð UK models are consistent with theDornbusch Ð Frankel sticky price monetary model.

Finally, we compare the out-of-sample forecasting perfor-mance of our three models with the random walk model.Model I beats the random walk model for every forecasthorizon for the US Ð Germany exchange rate forecast, for3 to 10-quarter horizons for the US Ð Canada exchange rateforecast, for 4 to 10-quarter horizons for the US Ð Japanexchange rate forecast, and for 2 to 10-quarter horizons forthe US Ð UK exchange rate forecast. Model II outperformsthe random walk model for 3 to 10-quarter horizons for theUS Ð Canada exchange rate forecast, for 2 to 10-quarterhorizons for the US Ð Germany, US Ð Japan and US Ð UKexchange rate forecasts. Model III outperforms the randomwalk model for 4 to 10-quarter horizons for the US Ð Canadaexchange rate forecast, for 2 to 10-quarter horizons for theUS Ð Germany, US Ð Japan and US Ð UK exchange rate fore-casts. Among three models, Model I appears to be superiorto Models II and III except for the US Ð Japan exchange rateforecast. Moreover, the RMSE of Model I is much smallerthan that of the random walk model for forecast horizonsgreater than 1-quarter (except for the US Ð Japan exchangerate forecast).

The evidence that currency substitution is an importantfactor for analysing both long-run and short-run exchangerates provides strong support for the point of view ofMeese and Rogo� (1983) that money demand misspeci® ca-tion is an important potential explanation for the poorperformances of the structural exchange rate determinationmodels.


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REFERENCES

Bilson, F. O. (1978) Rational expectations and the exchange rate, ineds J. A. Frankel and H. G. Johnson, The Economics ofExchange Rates, AddisonÐ Wesley, Reading, MA.

Branson, W. H. (1977) Asset markets and relative prices in ex-change rate determination, Institute for International Eco-nomic Studies, Reprint Series no. 98.

The Bulletin of the US Treasury Department, Department of USTreasury, various issues.

Dickey, D. A. and Fuller, W. A. (1979) Distribution of the es-timators for autoregressive time series with a unit root, Jour-nal of the American Statistical Association, 74, 427 Ð 32.

Dickey, D. A. and Fuller, W. A. (1981) Likelihood ratio statisticsfor autoregressive time series with a unit root, Econometrica ,49, 1057 Ð 72.

Diebold, F. X. and Nason, J. A. (1990) Nonparametric exchangerate prediction? Journal of International Economics, 28(3 Ð 4),315 Ð 32.

Doan, T.A. (1994) RATS User’s Manual, Version 4.10, 1994.Dornbusch, R. (1976a) Expectations and exchange rate dynamics,

Journal of Political Economy, 84, 1161 Ð 76.Dornbusch, R. (1976b) Exchange rate expectations and monetary

policy, Journal of International Economics, 6, 231 Ð 44.Engel, C. and Hamilton, J. D. (1990) Long swings in the dollar: are

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Fuller, W. A. (1976) Introduction to Statistical T ime Series (Wiley:New York).

Frankel, J. A. (1979) On the mark: a theory of ¯ oating exchangerate based on real interest di� erentials, American EconomicReview, 64, 610 Ð 22.

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Johansen, S. (1988) Statistical analysis of cointegration vectors,Journal of Economic Dynamic and Control, 12, 231 Ð 54.

Johansen, S. and Juselius K. (1990) The full information maximumlikelihood procedure for inference on cointegration with ap-plication, Preprint, Institute of Mathematical Statistics, Uni-versity of Copenhagen.

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MacDonald, R. and Tayor, M. P. (1994) The monetary model ofthe exchange rate: long-run relationships, short run dynamicsand how to beat a random walk, Journal of InternationalMoney and Finance, 13, 276 Ð 90.

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Meese, R. (1990) Currency ¯ uctuations in the post-Bretton Woodsera, Journal of Economic Perspectives, 4, 117 Ð 34.

Meese, R. A. and Rogo� , D. (1983) Empirical exchange rate modelsof the seventies: do they ® t out of sample? Journal of Interna-tional Economics, 14, 3 Ð 74.

Mussa, M. (1976) The exchange rate, the balance of payments, andmonetary policy under a regime of controlled ¯ oating, Scandi-navian Journal of Economics, 78, 229 Ð 78.

Mussa, M. (1979) Our recent experience with ® xed and¯ exible exchange rates, Carnegie Rochester Supplement, no. 3,1 Ð 50.

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Perron, P. (1988) Trends and random walks in macroeconomictime series: further evidence from a new approach, Journal ofEconomic Dynamics and Control, 12, 297 Ð 332.

Phillips, P. C. B. and Perron, P. (1988) Testing for a unit root intime series regression, Biometrika, 75, 335 Ð 46.

SAS/ETS User’s Guide, Version 6 (1993).

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currency substitution and exchange rate determination

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