A Model of Currency Substitution in Exchange Rate Determination, 1973–78

At the outset of the current floating exchange rate regime, expectations were widespread that the new system would allow countries a high degree of policy independence yet reduce strains on the free international movement of goods, services, and capital. Even long-standing critics of exchange rate flexibility seemed unable to oppose meaningfully the advent of flexible exchange rates in the face of the severe strains of reserve changes and international payments imbalances. Domestic financial policies and economic performance in the major industrial countries had diverged to such an extent that fixed relationships among currencies were obviously unfeasible. After several years’ experience with floating exchange rates, however, it is far from clear that exchange rate flexibility enhances the ability of countries to pursue their own chosen policies without contributing to severe disruptions in the international monetary system. In fact, as experience with the regime grows, a number of the fundamental tenets of flexible exchange rate advocates are being called into question.

Abstract

At the outset of the current floating exchange rate regime, expectations were widespread that the new system would allow countries a high degree of policy independence yet reduce strains on the free international movement of goods, services, and capital. Even long-standing critics of exchange rate flexibility seemed unable to oppose meaningfully the advent of flexible exchange rates in the face of the severe strains of reserve changes and international payments imbalances. Domestic financial policies and economic performance in the major industrial countries had diverged to such an extent that fixed relationships among currencies were obviously unfeasible. After several years’ experience with floating exchange rates, however, it is far from clear that exchange rate flexibility enhances the ability of countries to pursue their own chosen policies without contributing to severe disruptions in the international monetary system. In fact, as experience with the regime grows, a number of the fundamental tenets of flexible exchange rate advocates are being called into question.

At the outset of the current floating exchange rate regime, expectations were widespread that the new system would allow countries a high degree of policy independence yet reduce strains on the free international movement of goods, services, and capital. Even long-standing critics of exchange rate flexibility seemed unable to oppose meaningfully the advent of flexible exchange rates in the face of the severe strains of reserve changes and international payments imbalances. Domestic financial policies and economic performance in the major industrial countries had diverged to such an extent that fixed relationships among currencies were obviously unfeasible. After several years’ experience with floating exchange rates, however, it is far from clear that exchange rate flexibility enhances the ability of countries to pursue their own chosen policies without contributing to severe disruptions in the international monetary system. In fact, as experience with the regime grows, a number of the fundamental tenets of flexible exchange rate advocates are being called into question.

The long-standing case for floating exchange rates has centered principally around the argument that exchange rate flexibility allows individual countries to choose financial policies independently while removing the burden of intervention on those countries whose policies deviate from the average. This argument has been summarized forcefully by Professor Harry G. Johnson:

The adoption of flexible exchange rates would have the great advantage of freeing governments to use their instruments of domestic policy for the pursuit of domestic objectives, while, at the same time, removing pressures to intervene in international trade and payments for balance-of-payments reasons. Both of these advantages are important in contemporary circumstances. On the one hand, a great rift exists between nations like the United Kingdom and the United States, which are anxious to maintain high levels of employment and are prepared to pay a price for it in terms of domestic inflation, and other nations, notably the West German Federal Republic, which are strongly averse to inflation. Under the present fixed exchange-rate system, these nations are pitched against each other in a battle over the rate of inflation that is to prevail in the world economy, since the fixed rate system diffuses that rate of inflation to all the countries involved in it. Flexible rates would allow each country to pursue the mixture of unemployment and price trend objectives it prefers, consistent with international equilibrium, equilibrium being secured by appreciation of the currencies of “price-stability” countries relative to the currencies of “full-employment” countries.1

The essential point made by flexible exchange rate advocates, and reflected in the above passage, is that exchange rates that are allowed to adjust freely to market forces provide an adjustment mechanism that insulates each economy from external influences that would, under fixed exchange rates, dominate its own policy decisions. Under flexible exchange rates, it is assumed that each domestic monetary authority controls the supply of an independent currency that is not a substitute for others. Each monetary authority can set a particular money growth rate for its currency, and excess demands for each currency will be eliminated by exchange rate and price level changes, with purchasing power parity (PPP) establishing the equilibrium relationship between the two. This process is in obvious contrast to a fixed rate system, which can be understood most easily in the context of a stylized model with one world money and one price level for goods in terms of money. In this case, differences among countries’ excess demands for money are eliminated through the balance of payments, which changes the physical distribution of world money among countries. There is, then, only one rate of inflation, and that is determined by the world excess demand for money.

This simple view of the distinction between fixed and flexible exchange rates leaves out a critical aspect of the differences between them. In the case of a fixed exchange rate regime, the concept of the world demand for money is clear. Yet, just as in the case of the world price level under fixed exchange rates, prices or exchange rates under a flexible exchange rate system are determined by the world excess demand for each particular currency. Thus, the fundamental distinction between the monetary approach to the balance of payments and the monetary approach to exchange rates lies in the fact that the former should be concerned with national demands for a world currency while the latter should be concerned with the world demand for a national currency.

This distinction between national and global demands for a currency takes on startling importance when the concept of the world demand for national currencies is examined more closely. Residents of any country may want to hold a variety of currencies in their portfolios, both to facilitate transactions in different currencies and to earn the rate of appreciation of a particular currency vis-à-vis others. As any one currency becomes less attractive as a store of value or medium of exchange, it is reasonable for portfolio holders to replace it with other stronger currencies. In addition, as the decline in the real value of a currency makes the losses involved in holding it larger, its role as a medium of exchange is likely to be taken over by stronger currencies. In the extreme case when currencies are highly substitutable and expectations of the continuing depreciation of a currency are held with certainty, the relative attractiveness of a strong currency will eliminate demand for a weak currency, and the exchange rate between the two will cease to exist.2

The lesson of this admittedly stylized conclusion is that, when currencies are substitutes, monetary authorities face similar types of constraints under flexible rates and under fixed rates. Under fixed exchange rates and unimpeded trade in capital and goods, excessive domestic credit creation results in a balance of payments deficit that eventually leads to a reversal of the expansionary policy. When exchange rates are permitted to change, monetary authorities may have more flexibility in the short run. In the long run, however, a continuing attempt to expand the money supply faster than demand for it grows will steadily erode demand and increase the rate of depreciation of the currency as money holders attempt to switch into other currencies. Thus, even with flexible exchange rates, there are limits to the policies available to monetary authorities. In the long run, excessively expansionary policies must be reversed or capital and trade restrictions will have to be imposed.3

In the short run, when an expansionary monetary policy is unlikely to drive a currency out of existence, a more complex relationship occurs among the different currencies which is, perhaps, equally limiting to the pursuit of an independent monetary policy. In particular, while the substitution between strong and weak currencies will still be important, complementarity or sub-stitutability with respect to third currencies may be equally influential. In this case, an expansionary monetary policy may not only weaken a country’s own currency but also weaken those currencies that have in the past tended to follow the weakening currency and strengthen those that have tended to diverge from it.

In modeling third-currency effects, the importance of uncertainty and its consequences on exchange rate determination should be emphasized. Indeed, uncertainty plays a central role in the development of an asset view of exchange rate determination, as it is the uncertainty of asset returns that induces a wealth holder to invest in a variety of assets with different expected returns. In particular, a wealth holder will hold cash as well as bonds even though the former has a lower expected rate of return because it has the offsetting advantage of being less risky. This principle of risk diversification can be extended in a more general model that includes various currencies as well as bonds and other financial instruments among the available assets. By aggregating across individual portfolios and assuming that domestic capital markets are well integrated in international capital markets, one can postulate a market portfolio that includes a number of national currencies. As in the case of an individual, the market holds a variety of currencies in order to diversify risk so that the prices of currencies will reflect their expected rates of return as well as their expected covariances. Thus, assuming that the supply of each currency is exogenously determined, changes in demand will normally affect the price of the currency or, in other words, its exchange rate. Since the demand for any one currency is tied to the demand for all other currencies by investors’ desire to diversify risk, changes in the demand for one currency and, hence, its exchange rate, will affect the demand for and exchange rate of all other currencies. This interdependence among currencies can produce quite interesting patterns of exchange rate movements. For example, policy changes in one country may induce an appreciation of another country’s currency, which in turn may induce a third currency to appreciate with it and a fourth to depreciate.

Such complex reactions to a policy shift are particularly likely to occur when investors use information from past behavior or policy announcements to anticipate particular relationships among exchange rate movements. While it may be difficult to predict which of several similarly behaving currencies will, say, appreciate, it is often easier to anticipate exchange rate movements for the group as a whole. Similarly, if the economic forces that tend to strengthen one currency weaken another, and one expects the first to appreciate, then it may be reasonable to expect the second to depreciate.

Since these third-currency effects are likely, even in the short run, to prevent floating exchange rates from insulating an economy from foreign shocks, it is important for policymakers to be aware of them. This paper is an attempt to measure the importance and nature of relationships between major currencies by examining them within the general framework of asset portfolio allocation decisions. From the general framework, an empirically manageable model is derived and estimated for eight currencies during the period March 1973-June 1978. The results show that currency substitution is an empirically significant phenomenon that must be accounted for in any modeling of exchange rate determination.

I. Portfolio Model of Exchange Rate Determination

This section presents a general framework formalizing the relationships among all assets, including currencies. The model views these relationships in the context of national capital markets that are sufficiently integrated to be viewed in the aggregate as a world capital market, of which the market for national currencies is a subset.4 It postulates an international portfolio with three types of assets—currencies, bonds, and titles to ownership of commodities. From this general framework, a model specifically suited to the task of testing the significance of substitutability among currencies is easily derived.

The general portfolio model portrays the portfolio allocation process of individual investors attempting to maximize utility. The utility derived from wealth is assumed to depend on the rate of return and the riskiness of the total portfolio. Investors, therefore, allocate an exogenously given amount of real wealth among all available assets in order to maximize the real return they receive for a particular level of risk. Risk is measured in terms of the variances and covariances of asset returns. By assuming that the variance/covariance matrix is constant throughout the period, it can simply be subsumed in the parameters of the portfolio allocation function.

It is assumed that the investment horizon for portfolio holders is one month, implying that expectations are formed and portfolio allocation choices are revised on a monthly basis. A general disequilibrium framework, as suggested by Brainard and Tobin (1968), is used to characterize the dynamics of adjustment to general portfolio equilibrium. In this framework, the change in holdings of each asset is determined by a weighted average of the discrepancies between last period’s actual holdings and this period’s desired holdings of each asset in the portfolio. By fully accounting for the impact of asset-holding disequilibria on the demand for each asset, this sort of general disequilibrium specification avoids the possibility of producing conclusions that inadvertently imply inconsistent behavior.5

The general portfolio model postulates that the percentage change in the real stock of the kth asset (Δak) is equal to the weighted average of the discrepancies between the desired (aj*) and actual (aj (–1)) levels for each asset in the portfolio plus a proportion of the percentage change in real worldwide wealth (Δw); the weights are given by the speed of adjustment of holdings of the k th asset to the desired change in holdings of each of the j assets (Δkj) weighted by the share of asset j in total wealth (θj,).6,7

Δak=Σjqθjλkj(aj*aj(1))+ψkΔwk=1q(1)

The desired holdings of each asset (aj*) depend on the vectors of expected real rates of return on currencies (r˜c), bonds (r˜), and nontraded goods (Δp˜).8 It is assumed that, ceteris paribus, an increase in wealth induces a proportional increase in desired holdings of each asset. These relationships are expressed as follows for each asset, j:

aj*=Aj(r˜c,r˜,Δp˜)+w(2)

To evaluate any portfolio, the value of all stocks of assets must be expressed in terms of common accounting units. In the case of portfolio holders residing in a single country, the domestic price level is an obvious deflator. For an international portfolio, the domestic price level can still be the deflator, provided PPP among currencies holds during the observation period. Then, each domestic price level (and its constituent parts) is systematically linked with the price level of the numeraire currency. Considerable evidence, however, that systematic deviations from PPP can be expected on a monthly basis militates against the assumption that PPP constrains the short-term exchange rate determination process.

Once PPP and its implicit assumption of a fixed relationship between the price of traded and nontraded goods are dropped, it is no longer possible to aggregate over the two types of goods, so that a distinction must be made between them. This distinction allows the assumption that international arbitrage continuously maintains PPP in terms of traded goods, while different demand and supply pressures in each country move the relative price of nontraded and traded goods from its long-run equilibrium level. This property suggests that traded goods are a suitable numeraire which can be used by international portfolio holders, regardless of residence, to evaluate assets.9 The model, therefore, translates the value of all stocks of assets into a numeraire currency (the U. S. dollar) and deflates these stocks by the U. S. dollar price index of traded goods. Similarly, all asset returns are expressed in terms of the return on traded goods.

Given this method of defining real assets, it is a simple step to convert the subsection of the general portfolio model that describes holdings of currencies into an exchange rate determination model. First the percentage change in real holdings of each of the n currencies (Δai) is written as the rate of increase of the money supply (Δmi), plus the appreciation of the exchange rate vis-à-vis the U. S. dollar (Δei), less the rate of inflation of traded goods in dollar terms (ΔpT).

Δai=Δmi+ΔeiΔPTi=1n(3)

Assuming that money supplies are exogenously determined, each of the n equations of the form (3) is substituted into equation (1), which can then be rearranged to arrive at the following exchange rate determination model:

Δei=ΔpTΔmi+Σjqθjλij(aj*aj(1))+ψiΔwfori=1n1(4)

Since the nth exchange rate (for the U. S. dollar against itself) is equal to one, the nth equation determines the change in the price level of traded goods:

ΔpT=ΔmnΣjqθjλnj(aj*aj(1))+ψnΔw(5)

In order to estimate the exchange rate determination model given by equations (4) and (5), empirical proxies must be found for the unobservable rates of return on assets in equation (2). A brief description is given here of the proxies used in the empirical work. A fuller explanation of the assumptions behind the choice of proxies is given in Appendix I.

Currencies yield both a pecuniary and a nonpecuniary rate of return. The pecuniary return on the ith currency is equal to the expected appreciation of the ith currency in terms of dollars, plus a proportion of the yield on bonds denominated in the ith currency. The expected appreciation of the ith currency is proxied by its forward premium (fi), which is assumed to be an unbiased estimate of the expected appreciation. Regardless of currency denomination, bonds are perfect substitutes for each other, so that by implication interest parity holds continuously and the nominal return on the ith country’s bond can be proxied by the one-month Eurodollar deposit rate (in), plus the one-month forward premium of the ith currency in terms of U. S. dollars. The nonpecuniary return is proportional to the volume of transactions in the ith currency (for which real income in the ith country (in) is a proxy) and a time trend (T). The difference between the rate of increase of consumer and wholesale price indices in the ith country is the proxy for the expected real return on the ith country’s nontraded goods. A version of the Fisher effect that suggests an equality between the Eurodollar rate and the real interest rate plus the expected rate of inflation of traded goods prices in terms of U. S. dollars is used to derive a measure of the expected change in the price level of traded goods. The expected rate of inflation of the price of traded goods in terms of U. S. dollars equals the Eurodollar rate less the real interest rate, which is assumed constant. An index of industrial countries’ exports is the proxy for the actual price level of traded goods (pT).10

The portfolio model developed thus far is highly general in structure and, as such, presents prohibitive problems in estimation. Therefore, a number of restrictions have been imposed on the model to make it amenable to the task of testing the importance of substitutability and complementarity among currendes. Two objectives have determined the choice of restrictions. First, an attempt has been made to reduce to a minimum the number of parameters that do not directly influence portfolio holders’ choices among currencies, as opposed to among other assets included in the model. Second, the estimated model has been forced to conform to certain theoretical properties of the underlying process of utility maximization.

Since the focus of this paper is on the determination of exchange rates and price levels that clear the markets for eight currencies, the empirical work deals exclusively with the subset of the portfolio model that determines the demands for these eight currencies. In describing the restrictions on the model, therefore, only those imposed on the structure of the currency subset of the model will be discussed.

To put the portfolio allocation process into a manageable perspective, the allocation of wealth among currencies is viewed as a two-step decision.11 First, a decision is made on the desired ratio of money (defined as the sum of all currencies) to nonmoney assets to be held in the portfolio. Second, total money holdings resulting from this decision are allocated among the available currencies. This approach implies various restrictions on the general model developed above. Changes in the expected returns of nonmoney assets affect only the first decision on the ratio of money to nonmoney assets in the portfolio; they do not affect the distribution of money holdings among currencies. As a further simplification, it is assumed that all the semielasticities of money with respect to each of the nonmoney asset returns are identical, implying that all nonmoney assets may be viewed as one aggregate asset. In keeping with the two-step approach to portfolio allocation, it is assumed that the speed of adjustment of the money/nonmoney asset composition of the portfolio differs from that of the currency composition within the money subportfolio. Specifically, it is assumed that it takes time to adjust the money/nonmoney asset composition to its desired ratio, but that shifts among currencies occur instantaneously.

Finally, symmetry conditions, derived from the process of maximizing the utility of wealth subject to a wealth constraint, are imposed on the matrix of coefficients, ɑkj. This restriction requires that the partial elasticity of the ith currency with respect to the jth currency’s return is equal to that of the jth currency with respect to the ith currency’s return.

The model can be estimated using the empirical proxies outlined above and imposing the restrictions as described. Alternatively, the model can be rewritten to highlight the two-stage decision-making process with no substantial change to its implicit structural form or the implications of the restrictions. The first step of the portfolio allocation process, in which investors allocate their wealth between currencies and nonmoney assets, is captured by equation (4) summed over each currency in the model weighted by its share in total world currency plus equation (5) weighted by the share of the numeraire in total world currency. After rearranging, this aggregate demand for money equation determines the price level of traded goods (pT) as the difference between the actual aggregate money supply (m) and short-term aggregate money demand. The latter, as described earlier, depends on the vector of proxies for rates of return on currencies (r˜c), the increase in the real price of nontraded goods in each country in the model (Δp˜j), world wealth (w), the change in world wealth (Δw), a time trend (T), and the lagged value of the real aggregate money supply (m(– 1) – pT (–1)). The speed of adjustment coefficient (λ) determines the time it takes to alter the composition of the portfolio between money and nonmoney assets.12,13

pT=mλ(Σjnαmjr˜jc+ηΣjnΔp˜j+w)+γmT+ψΔw+(1λ)(m(1)pT(1))(6)m=Σinωi(mi+ei)worldwidemoneysupplyintermsofU.S.dollars
ωi=shareofcurrencyiinthesumofncountriesmoneysuppliesr˜jʹc=fi+ϕj(infj)+βjYjin14W=ΣinYi

The second step of the portfolio allocation process, distributing total money holdings among individual currencies, is characterized in n – 1 equations determining the excess demand for each currency as a proportion of world currency holdings. These n – 1 equations determine n – 1 exchange rates, ei, which are equal to the difference between the excess demand for currency i and the excess demand for worldwide money. Exchange rates are, therefore, affected only by rates of return on currencies and not by rates of return on nonmoney assets, as suggested by the two-step portfolio allocation process.

ek=mmk+Σjn(αkjαmj)r˜jʹc+γkʹTk=17(7)

Symmetry restrictions on the ɑkj require that ɑkjωk = ɑjkωj.

Estimating an overall demand for money equation (6) makes the eighth equation of the form (7) redundant. Consequently, a share equation for the numeraire, the U. S. dollar, was not estimated, although estimates of the implicit parameters required for consistency with the rest of the model can be derived. It should also be noted that, although six of the eight ϕ’s could be identified in estimation, calculated values for ϕj’s were imposed on the model to avoid the arbitrary decision of which ones to estimate and which ones to fix. Imposed values were derived from the formula, one minus the ratio of the average value of high-powered money to the average value of M2 for each currency.15 In addition, note that yk could not be identified and, therefore, time trends are included with an aggregate coefficient, γk.

II. Empirical Results

The model described in the previous section is designed to examine empirically the role of currency substitution in the exchange rate determination process. This section presents the results of estimating the model using monthly data for eight countries during the period March 1973–June 1978. To test the importance of the cross effects between currencies, a model that restricts all these cross effects to zero is also estimated. One can make formal tests of this restriction, as well as examine its cost in terms of a goodness-of-fit criterion, by comparing the results from the two models. These tests are useful not only because they provide a means of assessing the usefulness of the general portfolio model but also because they examine the validity of the restricted model, which is the one most frequently found in literature on exchange rate determination.

The model was estimated with a full-information maximum likelihood estimation procedure that allows cross-equation restrictions to be imposed.16 The initial estimation produced significant first-order serial correlation in a number of the residuals. Therefore, a correction was made by introducing a single correlation coefficient with a value restricted to lie between -1 and 1 for the entire model.17

The parameter estimates, given in Table 1, generally conform to prior expectations. One surprise, however, is the rather high value of ρ, the estimated autocorrelation coefficient, of 0.948, which indicates a very high degree of first-order serial correlation. In the equation describing the aggregate money demand, coefficients on six of the rates of return on currencies are, as expected, positive—three at the 95 per cent confidence level. The negative coefficient on the rate of return on nontraded goods implies that money and goods are substitutes in portfolios. The coefficient on the change in wealth, which is negative and significant at the 95 per cent confidence level, indicates, somewhat surprisingly, that an increase in the rate of saving reduces the quantity of money held. The estimated rate of adjustment of actual to desired total money holdings is considerably higher than those obtained in other empirical work on demand for money. Moreover, the fact that λ is not significantly different from one indicates that portfolio adjustments take place quickly, so that the assumption of rapid adjustment in the portfolio of currencies seems not to be unreasonable.

Table 1.

FIML Estimates of Portfolio Models of Exchange Rate Determination, March 1973–June 19781

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Denotes significance at the 95 per cent confidence level.

Denotes significance at the 10 per cent significance level.

Numbers in parentheses represent t-statistics.

The dependent variable for the U. S. dollar equation is the U. S. price level; all others are the respective exchange rates.

R-squared is calculated as one minus the ratio of variance of residual to that of dependent variable.

Box-Pierce statistic calculated with 12 degrees of freedom.

Root-mean-square error (RMSE) of dynamic in sample simulations.

The focus of interest is, however, the matrix of semielasticities on the rates of return on currencies. Inspection of Table 1 reveals that it was difficult to get precise estimates of these semielasticities—particularly of the cross or off-diagonal ones. The own semielasticities (along the diagonal of the matrix) are, with the exception of that for the Japanese yen, positive, as expected, and significantly different from zero in half the cases. The own semielasticity for the Japanese yen is negative, but small, and has a relatively large standard error. Of the off-diagonal terms of the matrix, only about one fifth are significantly different from zero at the 95 per cent confidence level. More than half of these coefficients, however, have t- statistics greater than one, indicating that these variables do contribute significantly to the explanatory power of the model; a more formal test of this proposition is given below.

The pattern of coefficients on off-diagonal rates of return reveal several interesting characteristics of the relationships between the currencies. The continental European currencies exhibit strong complementarity, with half the cross semielasticities being significantly different from zero. Both the U. S. dollar and the Canadian dollar are indicated to be substitutes for several of the European currencies. The relationships of the pound sterling with other currencies seem to be difficult to estimate with much precision, although complementarity vis-à-vis the Italian lira and U.S. dollar is suggested. The surprising feature of the results for the Japanese yen is the small size of most of the semielasticities estimated.

To get an idea of the importance of the complementarity among continental European currencies, consider the result of aggregating the equations for these currencies using money stock shares as weights. The resulting equation is representative of the type of demand function that would be applicable to the group if all of the members attempted to fix their exchange rates. The resulting own semielasticity is 2.15—a value greater than the highest own semielasticity in the individual equations (1.72) and more than twice the average own semielasticity in individual equations. In contrast, the response of this aggregate to changes in the rate of return on U. S. dollars is relatively small (–0.37). While such an aggregation is admittedly a rough approximation to behavior that would characterize a European currency bloc, it is suggestive of the potential gains from complementarity that those currencies could enjoy in a currency union.

One way to gain perspective on the impact of currency substitution on exchange rate determination is to compare the size of the own semielasticity with the cross semielasticities in the same equation. Of the 56 such comparisons, 14 cross semielasticities are at least as large as the own semielasticity. These relatively large coefficients occur most frequently in the equations for the pound sterling, the Canadian dollar, and the Swiss franc. By contrast, the own semielasticity for the U. S. dollar clearly dominates the off-diagonal terms. This, of course, indicates that the demand for the U.S. dollar is relatively unaffected by changes in the rates of return on other currencies as opposed to changes in the rate of return on U. S. dollars. Since the pecuniary return on U. S. dollars has only an interest rate component, reflecting inflationary expectations, and no exchange rate component, this result implies that the demand for the U. S. dollar is more sensitive to expected price level changes than to expected exchange rate changes.

The coefficients relating income to the nonpecuniary return on each currency (βj) produced some surprises. A priori, βj’s are expected to be positive since an increase in the volume of transactions in a currency should increase its nonpecuniary return. Only three of the eight estimates of βj’s are positive, and two of the ones that are negative (those for the French franc and the deutsche mark) are significantly different from zero. These negative values may be an indication that income is a poor determinant of the nonpecuniary return on money.

Since the full model with a correction of serial correlation contains more variables than there are observations, it is not possible to estimate a fully unconstrained version of the model and, consequently, not possible to test the validity of the total set of restrictions at one time. It is possible, by estimating a series of nested models, to test each type of restriction individually. Although a thorough analysis of the model would require that each set of restrictions be tested in this way, the procedure is beyond the scope of this paper. Since the interest here is primarily in examining the importance of third-currency effects in the exchange rate determination process, their importance is formally tested and, in addition, the loss in forecasting power in neglecting these effects is examined. The test is conducted by comparing the results from the full model with those from a model (henceforth called the restricted model) that restricts all the cross semielasticities (αkj where k ≠ j) in equations (6) and (7) to equal zero.18 The similarity between the restricted model and many models that have been used recently to investigate the monetary approach to exchange rates makes this test particularly useful.

In general, the parameter estimates from the restricted model (presented in the right-hand columns of the table) resemble those from the full model. All but two of the coefficients on own rates of return have positive signs, with the rate of return on the Japanese yen having a negative coefficient in both versions. Coefficients on trends also have the same pattern of signs and approximately the same size in the two models. One marked difference between the two models, however, is the estimated values of βj’s, the coefficients on income in the functions expressing nonpecuniary returns. Not only are three of the eight different in sign between the two models but also the size of several of the estimates differs widely. Parameter estimates in the overall demand equation are quite similar for η and ψ. Although the speed of adjustment in the restricted model is slower than that in the full model, neither adjustment coefficient is significantly different from one.

To test whether third-currency effects embodied in the full portfolio model have a significant influence on the determination of exchange rates, the following question is posed: Is the explanatory power of the full portfolio model enough greater than that of the restricted model to offset the loss of degrees of freedom in the unrestricted version? A formal test designed to answer this type of question is the log-likelihood ratio test. Applying this test, the chi-square value of the ratio of likelihood values of the two models of 68.60 with 28 degrees of freedom rejects the restricted model in favor of the full model at any confidence level. The implication, therefore, is that, taken together, the off-diagonal rates of return in the money demand functions yield a significant improvement over the restricted version in precision of exchange rate determination.

A comparison of dynamic simulations of seven exchange rates vis-à-vis the U.S. dollar and the price level of traded goods from the two models over the entire sample period produces more ambiguous results. In four of the eight simulations, the full portfolio model outperforms the restricted model by a root-mean-square error criterion (see Table 1). Interestingly, three of the four currencies with exchange rates that are more accurately simulated by the full model are European currencies, for which third-currency effects were generally found to be more prevalent and stronger than average. Broadly speaking, however, the pattern of simulation errors is similar for each model, with both picking up and in several cases anticipating the major movements of exchange rates. There is, however, a tendency for both simulations to vary considerably less than the actual exchange rates and price level. (See Chart 1 for the dynamic simulation of the full model.) This is probably a reflection of the fact that models of the type estimated here may not be able to pick up factors which contribute to overshooting or variations around the major movements of exchange rates.

Chart 1.
Chart 1.

Dynamic Simulation Results from Full Portfolio Model of Exchange Rates and Traded Goods Price Level

(U. S. dollar/domestic currency)

Citation: IMF Staff Papers 1979, 003; 10.5089/9781451972597.024.A004

III. Summary and Concluding Remarks

Before the recent period of exchange rate flexibility, conventional wisdom among monetary economists held that floating exchange rates would insulate the domestic economy from external shocks. Although experience in the early 1970s confirmed that exchange rate changes cannot protect an economy from external real shocks, the hope remained that flexible exchange rates could contain monetary shocks and thereby enable monetary authorities to pursue independent policies. Underlying this view is the assumption of a stable demand function for each national currency that is largely unaffected by developments in other countries and is, therefore, amenable to manipulation by the domestic monetary authority. Increasing experience with flexible exchange rates and the deepening of international integration of national financial markets call for a critical examination of this assumption.

National currencies, in a world of perfect certainty, have been viewed as competing assets in an international market for money services. The competitive position of each currency determines its share of the international market; a deteriorating competitive position, for example, causes a currency’s share to decline. In a world of uncertainty investors are induced to hold a portfolio of currencies, so that relationships among currencies are considerably more complex. In this case, complementarity and sub-stitutability have an influence on the demand for currencies requiring that third-currency effects be accounted for in modeling exchange rate behavior.

To capture these third-currency effects, the demand for national currencies has been treated as a worldwide demand integrated in a larger framework of portfolio allocation among several types of assets—currencies, bonds, and titles to traded and nontraded goods. In this framework, a dynamic model of portfolio allocation along the lines suggested by Brainard and Tobin (1968) was built and tested empirically. In an effort to streamline the model to the task of examining exchange rate behavior and to minimize the problem of limited data availability, a number of restrictions were imposed on the general framework. In particular, the focus of the model is on intracurrency relationships, while the attention given to movements between currencies and other assets is minimized. Drawing on theoretical specifications of portfolio models, symmetry is imposed on the partial derivatives of currency demands with respect to rates of return. In addition, the dynamic adjustments in the model are constrained by assuming that intracurrency adjustments are completed within one observation period (i.e., one month). Testing and correcting for serial correlation of the residuals is accomplished by estimating one autocorrelation parameter.

An interesting feature of the model is the role played by income in the demand for currencies. Traditionally, income is a proxy for both the transaction demand for money and wealth. The model clearly distinguishes these two roles and thereby severely restricts the way income enters the currency demand equations.

Full-information, maximum-likelihood estimates of the demand equations for currencies of seven major industrial countries plus Switzerland over the period March 1973-June 1978 clearly indicate important interrelationships among the currencies. Generally, close complementarity among continental European currencies is found, while the U.S. dollar tends to be a substitute for these currencies. The behavior of the remaining three currencies (pound sterling, Japanese yen, and Canadian dollar) proves somewhat more difficult to estimate or categorize. In order to test the third-currency effects formally, a model restricting them to zero is estimated. A chi-square test on the ratio of the likelihood values of the two models rejects the restricted specification. This test is interesting not only because it sheds light on the role of currency substitution but also because the restricted model is isomorphic to many recent models expounding the monetary approach to exchange rates.

Although the model estimated in this paper is strictly a technical exercise, it suggests some interesting conclusions for policymakers. First, the complementarity revealed among European currencies implies that investors tend to view these currencies as a group. It also implies that any coordinated actions on the part of European countries would be expected to have a larger impact on each currency than the same action taken by an individual country. This expectation would, in itself, be an incentive for policy coordination. Second, although rates of return on other currencies significantly affect the demand for the U.S. dollar, these effects are small compared with that of its own rate of return. This suggests that in determining the effect of their own policies, the U.S. monetary authorities need not be as concerned as European authorities about the effect of foreign monetary shocks on the demand for their currency. Finally, portfolio adjustments occur rather quickly, so that policy effects on exchange rates are likely to be felt quickly.

This model is clearly a first step in providing an integrated view of the exchange rate determination process. It is designed to examine a particular proposition rather than to forecast exchange rates or give policy simulations. To extend the model for these purposes, it would be necessary for interest rates, the money supply process, and, perhaps, even the parameter estimates themselves to be endogenous. Further work in this direction might also refine the specification of the model by testing the validity of the various restrictions imposed for estimation.

APPENDICES

I. Empirical Measures of Rates of Return

Since empirical measures of the rate of return on each asset included in the model are not directly available, those used are based on a particular view of the characteristics of the rate of return on each of the three types of asset. The purpose of this Appendix is to describe, in more detail than in the text, the characteristics of the rates of return used in this paper. The rate of return on each currency, ijc, is constructed to reflect the assumption that money is held because (a) it is a temporary store of value and yields a pecuniary rate of return and (b) it is useful in facilitating transactions and as such yields a nonpecuniary return. This approach goes beyond much of the literature dealing with the demand for money, in which it is common to assume that money does not yield a pecuniary rate of return. Even in a single-currency model ignoring the pecuniary return on money holdings, which may take an explicit form such as interest paid on savings deposits or an implicit form such as services rendered for checking accounts, is likely to lead to a significant understatement of the return on money.19 In a framework where portfolio holders can invest in a variety of currencies, the error becomes considerably more serious since substantial nominal returns on each currency in terms of a numeraire can be expected.

In this paper, it is therefore assumed that each money, broadly defined, yields a nominal rate of return proportional to the interest rate on domestic nonmoney financial assets plus the change of its value in terms of U. S. dollars. Specifically, each country’s domestic currency is expected to yield ϕjij, where ϕj is a (constant) proportion of the yield on nonmoney financial assets and ij is the nominal interest rate on country j’s nonmoney financial assets. The second component of the return on currency j is simply the expected appreciation of the value of currency j in terms of U. S. dollars, Δe˜j. Combining these two components, expressing the first in terms of its expected dollar value, gives the total pecuniary return on currency j, ijP, as:

i˜jP=ϕjij+Δe˜j(8)

In addition to its pecuniary return, money provides services in facilitating transactions in goods and asset markets. These services can be viewed as a nonpecuniary return which is typically captured in demand for money functions via a scale variable such as income or expenditure. Following this practice it is assumed that the nonpecuniary return (ijn) on currency j can be described by a linear function of real income (Y¡) in country j and a time trend (T).20 The former is a proxy for the volume of transactions made with that currency and the latter for the technological changes which affect the services of money.

ijn=βjYj+γjT(9)

The expected rate of return on currency j (r˜jc) is therefore given as follows:

r¯jc=i˜jP+ijrΔp˜T

Although ij in equation (8) in principle should be the rates of return on bonds issued by the jth country, it is assumed that all noncurrency financial assets are perfect substitutes, so that there is effectively only one rate of return common to all of them; that is, the assumption of perfect substitutability implies that interest arbitrage equalizes the nominal yields on financial assets issued in any currency in terms of the numeraire. The yield on the jth country’s financial asset, ij, plus the expected rate of appreciation of the currency, Δẽj therefore, is assumed to equal the yield on financial assets in the numeraire currency, in.

in=ij+Δe¯j(10)

Perfect substitutability among financial assets implies that the demand for financial assets denominated in any individual currency is indeterminate, so that it is possible only to determine the demand for the world aggregate of financial assets. In the model, therefore, the returns on individual financial assets are expressed collectively in terms of the yield on the financial asset of the numeraire currency, ĩn, deflated by the expected inflation rate of traded goods, also in terms of the numeraire.

r˜=i˜nΔp˜T(11)

Without the assumption of interest rate parity, it would be necessary to account for rates of return on comparable assets for each of the national capital markets—a difficult and cumbersome procedure. As a justification for the approach taken here, it should be noted that the assumption of interest parity is not unreasonable for most countries. It seems to fail only for currencies of countries that impose taxes or controls on capital transactions or that present sovereign risk in some other capacity.21 Even then, as long as such measures do not change over time, returns on assets should move together in a fixed relationship.

The rates of return on both money and nonmoney financial assets include components that cannot be directly observed but that can be determined by using another variable as proxy without a great loss of accuracy. Several recent papers provide evidence that the market forward premium for the jth currency (fj) is an unbiased forecast of the rate of appreciation of the currency, at least for the short maturities considered in this paper.22 Therefore, the forward premium has been used as a proxy for the expected rate of appreciation of a currency.

Δe¯j=fj(12)

Finding a proxy for the expected rate of return on traded goods, ΔρT, is more difficult, although the relationship between expectations about the future inflation rate and rates of return on financial assets suggests a possible proxy. Given evidence that the interest rate is an unbiased forecast of the underlying inflation rate in the economy, 23 it is assumed that when interest parity holds the underlying inflation rate, expressed in a numeraire currency, is common to all countries. Since the inflation rate of nontraded goods is not equalized among all countries in the short run, the underlying inflation rate forecast by the interest rate must reflect only the inflation on traded goods. The expected change in the price of traded goods, ΔpTis, therefore, equal to the interest rate in the numeraire currency, in, less the real interest rate, which is assumed to be constant, ro.

Δp˜T=inr0(13)

Relaxation of the purchasing power parity assumption for the short run implies that the relative price of nontraded and traded goods is not constant. Portfolio holders, therefore, face different returns on holdings of traded and nontraded goods. Although the nontraded goods of any one country, by definition, cannot be consumed in another country, it is possible to divorce the residence of a wealth holder from the storage place of a commodity that is purchased solely as a store of value. This is achieved simply by issuing titles to ownership of commodities, such as warehouse receipts and deeds. This separation of residence of the wealth holder from storage place of commodity stocks through the use of titles allows nontraded goods to enter the international portfolio, even though only the titles and not the goods themselves are internationally traded. Without titles to ownership the only real asset in the international portfolio would be traded goods, since nontraded goods could be included only in domestic asset portfolios.

While individual price indices of nontraded and traded goods are not available, the expression for the rate of return on nontraded goods requires only the expected differential rate of inflation between nontraded and traded goods. It is, therefore, possible to use as a proxy the difference between changes in two price indices, such as the consumer price index and wholesale price index, which include both traded and nontraded goods but with different weights, the former giving more weight to nontraded goods and the latter to traded goods. It is assumed that the expected differential change is equal to the current value, which is exogenous to the model.

II. Restrictions on General Portfolio Model

Since many of the restrictions are imposed directly on the functions expressing the equilibrium level of demand for each currency, it is convenient to rewrite the equilibrium equations (equation (2) in Section I) for the currency subset of the model in explicit functional form:

ak*=Σjnαkjr¯jc+ΣjnηkjΔp¯j+w+μkk=1…nn=8(14)

where

r˜jc=r˜jc+γjT

μk = constant term

All other variables are defined in Appendix I. The coefficient on real wealth is constrained to equal one, reflecting the assumption that the demand for real balances denominated in any currency is homogeneous of degree one in wealth.

The first set of restrictions results from the symmetric properties of the matrix of coefficients, αkj. These properties can be derived directly from utility maximization of wealth subject to a wealth constraint.24 The maximization process produces a term for the substitutability between any two assets that can be interpreted in the same way as a Slutsky equation in consumer demand theory, i.e., the effect on the demand for currency k of a change in the return on currency j when the investor is compensated for the change in expected wealth resulting from the new configuration of expected returns on his assets. Given the assumption that the covariance matrix of asset returns is symmetrical, it can be shown that these substitution effects are also symmetrical.

In practice, symmetry restrictions are imposed by requiring that the partial derivative of currency k with respect to the return on currency j is identical to the partial derivative of currency j with respect to the return on currency k. In terms of the notation,

αkjωk=αjkωj(15)

where ωj,(ωk) is the mean share of currency j (k) in total currency holdings. Second, the coefficients on rates of return on nonmoney assets are restricted to be identical in each equation. In terms of the notation,

η1j==ηuj=η2j==ηun=η(16)

The implication of this restriction can be understood best by viewing the allocation of wealth among currencies as a two-step decision. In the first step, portfolio holders allocate their wealth among money (the aggregate of all eight currencies) and other assets (bonds and stocks of commodities). In the second step they allocate their money holdings among the eight currencies. Imposing this process on equation (14) results in an equation with the following form to describe the desired demand for total money holdings denominated in each of the eight currencies in the model:

m*=Σjnαmjr˜jc=ΣjnηΔp˜j+w+μm(17)

where

m=Σknωkak=realworldwidemoneysupply
αmj=Σknωkαkj

μm = constant term

Given the total demand for money as a function of wealth and the rates of return on all assets, the restriction ηij implies that the desired share of each currency k in total money holdings is a function only of rates of return on currencies.

ak*m*=Σjn(αkjαmj)r˜jc+μkμm(18)

The rest of the restrictions are imposed on the dynamic structure of the model which, as developed in Section I of the text for the entire model, is easily applied to the currency subsection as follows:

Δak=Σjqλkjθjdj+ψkΔwfork=1…n(19)

dj = deviation of desired holdings of asset j from last period’s actual holdings, aj*aj(1)

Corresponding to the equilibrium demand for total money holdings defined in equation (17), an additional adjustment function is defined as follows:

Δm=λmθmdm+Σj=n+1qλmjθjdj+ψmΔw(20)

where

dm=m*m(1)=Σjnωjdj

θm = the mean share of money in total wealth

λm = rate of adjustment of actual to desired money holdings

We begin with the assumption that changes in currency holdings are not affected by disequilibria in holdings of nonmoney assets. Lack of stock data for these assets makes this assumption unavoidable. In terms of the model, the adjustment coefficients on all nonmoney assets in the money demand subset of the full model are set equal to zero.

λkj=0forallk<n<j(21)

The second set of restrictions on the dynamic structure requires that adjustments to the desired shares of currencies in total money holdings take place instantaneously. In other words, for any given level of total money holdings, the desired and actual shares of each currency are always equal. Algebraically, equality between desired and actual shares of currencies implies:

akm=ak*m*(22)

In estimating the model with these dynamic restrictions, two approaches are feasible. First, it is possible to derive the restrictions implied by (22) on the individual dynamic money demand equations, (19). This is done by subtracting ak(–1) and m (–1) from both sides of equation (22), rearranging, and substituting for Am in equation (20) to get

Δak=dj(1θmλm)dm+ψmΔw(23)

Since m has been defined as Σjnθjaj(and correspondingly m* as Σjnωja*j), equations (23) is simply

Δak=Σjnϵkjdj+ψmΔw(24)

where

ϵkk=1(1θmλm)ωk
ϵkj=(1θmλm)ωjforjk

In terms of the original equation (19), these restrictions imply

λkk=λmθmθkθmθk(25)
λkj=λm1θmforjk(26)
ψk=ψm(27)

One approach to estimation, then, is to estimate money demand equations of the form (19) with the restrictions given by (25), (26), and (27) imposed directly. A second approach, which was used here, is to estimate an overall money demand equation of the form (20) and seven currency share equations of the form (22). The two approaches obviously yield identical results in terms of the structural equation (19), and the choice was made solely for convenience.

III. Data Sources

Exchange rates are defined as U.S. dollars per domestic currency units. The U.S. interest rate is the rate on 30-day Eurodollar deposits. The forward premiums and discounts are the rates on 30-day contracts. The data for these rates are taken from Weekly Review, International Money Markets and Foreign Exchange Rates, Harris Bank, Chicago, Illinois. The observations used correspond to the rate available on the last Friday of the month.

The nominal stock of each currency is measured by the stock of broad money in each country.25 For most countries, the data are constructed by adding seasonally adjusted money (line 34 … b) and quasi-money (line 35, seasonally adjusted), both taken from International Monetary Fund, International Financial Statistics (IFS). For the United Kingdom, a broader concept, M3, taken from the Bank of England, Quarterly Bulletin and seasonally adjusted, is used.

Real income is measured as gross national or gross domestic product in 1975 prices, depending on the country. These data are obtained from IFS For Italy and Switzerland, 1978 data are Fund staff estimates of gross domestic product. Since the data are available only on a quarterly basis (with the exception of Switzerland), monthly estimates were made by interpolating the quarterly data using monthly industrial production indices as benchmarks. For Switzerland, the yearly data were interpolated to a quarterly frequency using industrial production and to a monthly frequency using a quadratic interpolation of the quarterly series. Real incomes in national currencies were converted into U.S. dollars using the average 1975 exchange rate.

Wholesale and consumer price indices were taken from IFS. The index of industrial country export unit values (line Ml1075 … d from IFS), seasonally adjusted, was used as a proxy for the price index of traded goods.

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*

Mr. Brillembourg, economist in the Special Studies Division of the Research Department, is a graduate of Harvard University and of the University of Chicago.

Ms. Schadler, economist in the Special Studies Division of the Research Department, holds degrees from Mount Holyoke College and the London School of Economics and Political Science.

2

In a simple theoretical model of the demand for two currencies, Girton and Roper (1976) develop the implications of different degrees of currency substitution for the stability of the system and for policy independence.

3

Kareken and Wallace (1978) in fact argue that insofar as currencies are intrinsically useless they are necessarily perfect substitutes. They argue that perfect substitutability among currencies, while not always evident, implicitly requires that countries choose between the options of harmonizing financial policies and imposing prohibitive controls on capital and trade flows.

4

It is assumed that either there are no legal impediments to the exchange of assets across national borders, or changes in any impediments that exist are negligible over the observation period. In principle, taxes or nonprohibitive controls could be included in the model, but the process of accounting for them would add considerably to the complexity of the model, perhaps without proportionate rewards.

5

Smith (1975) gives a concise explanation of the necessity for the general disequilibrium specification and of its correct functional form.

6

Throughout the paper, notations are as follows: Upper and lower case letters in the Roman alphabet denote levels and logarithms, respectively, of variables (with the exception of interest rates). Upper and lower case letters in the Greek alphabet denote functions and parameters, respectively. Upper case deltas (Δ) denote first differences. Stars (*) and tildes (˜;) as superscripts denote desired and expected variables, respectively. Nominal and real interest rates are denoted by i and r, respectively. All assets are evaluated in terms of the numeraire currency (U. S. dollars) and deflated by an index of traded goods prices (industrial country export prices), also in terms of the numeraire. The real rates of return are nominal rates less the expected inflation rate of traded goods prices.

7

Because a logarithmic rather than a linear functional form is used, the speeds of adjustment (Δkj) must be weighted by θj in order to preserve the usual constraint that Σkλkj=1.

8

As explained below, all rates of return are evaluated in terms of the return on traded goods, so that the real return on traded goods is its rate of depreciation, which is neglected here.

9

For a discussion of this issue in the context of the capital asset pricing model, see Ross (1978).

10

It should be noted that the assumptions of a constant real rate of interest on bonds and continuous interest rate parity imply that (r), the vector of real rates of return on bonds issued by each country, is a scalar. In the formulation of the model for estimation, therefore, the term to the real rate of return on bonds is subsumed in the constant term.

11

Tobin (1958) uses a similar type of portfolio allocation process in which wealth holders first allocate their liquid asset holdings between the general categories of cash and interest-bearing financial assets and then allocate the latter among the various interest-bearing assets. The two-step process is made simpler in his analysis than it is here because he assumes cash to be riskless. Nevertheless, he defends the general idea of dividing the portfolio allocation process into decisions at different levels of aggregation as a seemingly “permissible and perhaps even indispensible simplification both for the theorist and for the investor himself.”

12

The time trend is included as an independent term although theoretically it constitutes part of the proxy for the rate of return on currencies. This specification is necessary because the structural coefficient relating the time trend to the rate of return on money cannot be identified when the time trend is specified directly as a part of that term.

13

Note that for notational convenience θm λm ≡ λ.

14

Recall from the discussion of the rate of return on currencies that the proxy is composed of the expected rate of appreciation of the jth currency, measured by the forward rate (fj), plus a fixed proportion of the jth country’s treasury bill rate, which given interest arbitrage is equal to the difference between the Eurodollar (in) and the forward rates, a fixed proportion of the volume of transactions in the jth currency proxied by real income (Yj) less the expected rate of inflation of traded goods prices, proxied by the Eurodollar rate.

15

See Klein (1974) for a derivation of this formula.

16

The program was developed by Clifford Wymer, and its general properties are described fully in Wymer (1977).

17

This procedure may also be viewed as a proxy for the savings process; see Brillembourg (1978).

18

The model is therefore similar in structure to models estimated by Bilson (1978), Frenkel (1977), Frenkel and Clements (1978), Hodrick (1978), and Miles (1978).

19

See Klein (1974) for a discussion of this problem.

20

The advantage of formulating an explicit nonpecuniary return is that it clearly divorces the role of income as a proxy for transactions from that of income as a proxy for wealth. Furthermore, this formulation imposes strong restrictions on the income parameters. Income enters the functions for non-pecuniary returns in levels rather than logarithms because the level formulation gives a marginally better fit in estimation. Real income is measured in 1975 prices and converted into U.S. dollars using the average 1975 exchange rate.

21

Aliber (1973) and Dooley (1976) develop explanations for deviations from interest rate parity based on these ideas.

22

See Brillembourg (1978), Cornell (1977), Grauer, Litzenberger, and Stehle (1976), and Levich (forthcoming) for the theory behind this assumption and empirical evidence in support of it.

24

See Royama and Hamada (1967) for a rigorous derivation of symmetry restrictions in the context of a portfolio allocation model of the type developed in this paper. Parkin (1970) has applied these restrictions in a similar way to an empirical model of U. K. discount house portfolio selection.

25

Although it would have been desirable to introduce some measure of Eurodeposits in the money supply figures to account for what is probably an important channel of currency substitution, no reliable data were available.

IMF Staff papers: Volume 26 No. 3
Author: International Monetary Fund. Research Dept.