## Abstract

As described in Section II, the first two steps in the exchange rate assessment process generate estimates of underlying current account positions and of equilibrium saving-investment or current account balances. The underlying current account for country j depends on the country’s multilateral real exchange rate, while the equilibrium current account is assumed to be independent of the exchange rate. Exchange rates are perceived to be in equilibrium when the associated underlying current accounts coincide with equilibrium current account positions for every country.1 Thus, the equilibrium values of multilateral real exchange rates—one for each country—must satisfy the system of equations

As described in Section II, the first two steps in the exchange rate assessment process generate estimates of underlying current account positions and of equilibrium saving-investment or current account balances. The underlying current account for country j depends on the country’s multilateral real exchange rate, while the equilibrium current account is assumed to be independent of the exchange rate. Exchange rates are perceived to be in equilibrium when the associated underlying current accounts coincide with equilibrium current account positions for every country.1 Thus, the equilibrium values of multilateral real exchange rates—one for each country—must satisfy the system of equations

$\begin{array}{ccccc}c{{a}_{j}}^{e}-c{{a}_{j}}^{u}={\theta }_{j}\left({{r}_{j}}^{e}-rcu{r}_{j}\right);\text{}j=1,\dots ,n,& & & & \left(7.1\right)\end{array}$

where for each country j:

A set of first-stage estimates of equilibrium multi-lateral real exchange rates, denoted by ${{r}_{j}}^{e1}$, is obtained by inverting (7.1) as follows:

$\begin{array}{ccccc}\mathrm{\Delta }{{r}_{j}}^{1}=\mathrm{\Delta }\text{}c{{a}_{j}}^{1}/{\theta }_{j},j=1,\dots ,n,& & & & \left(7.2\right)\end{array}$

where $\mathrm{\Delta }\text{}{{r}_{j}}^{1}\equiv {{r}_{j}}^{e1}-rcu{r}_{j}$ and $\mathrm{\Delta }c{{a}_{j}}^{1}\equiv c{{a}_{j}}^{e}-c{{a}_{j}}^{u}$. These first-stage estimates are subject to two sorts of possible inconsistency. One is the possibility that underlying or equilibrium current accounts, or both, may not sum to zero across the world (after weighting by relative GDP levels);3 and beyond that, that the global sum of underlying current account estimates may differ from the global sum of equilibrium current account estimates. A second type of possible in-consistency arises from the fact that there are only n-1 independent bilateral exchange rates, so that it is not valid to estimate n multilateral real exchange rates independently, as is done in the first-stage calculations.

This section describes the model-based procedure that we use to obtain a set of equilibrium exchange rates that are internally consistent and to obtain underlying current accounts associated with these equilibrium exchange rates that approximately coincide with the equilibrium current accounts. Provided that the global discrepancy in the underlying current accounts associated with current exchange rates is (roughly) equal to the global discrepancy in the equilibrium current accounts, our methodology generates small approximation errors between the underlying current accounts associated with the equilibrium exchange rates and the equilibrium current accounts.4 In addition, it generates estimates of these internally consistent equilibrium exchange rates that approximately coincide with the first-stage estimates calculated from condition (7.2).

We also derive an alternative calibration of the coefficients in equation (7.1) for which there exists a set of internally consistent equilibrium exchange rates at which underlying current accounts coincide exactly with equilibrium current accounts. It is noted, however, that this “ideal” calibration or weighting scheme would lead to bizarre estimates of the “equilibrium” exchange rates for certain smaller industrial countries.

## A Model-Based Procedure for Calculating Equilibrium Exchange Rates

We consider a world with n countries and express each country’s multilateral real exchange rate as a trade-weighted average of the logarithms of n-1 bi-lateral real exchange rates vis-à-vis currency j. These expressions can be transformed into linear combinations of the logarithms of n bilateral real exchange rates vis-à-vis an arbitrary numeraire currency, where the nth bilateral rate (numeraire vis-à-vis itself) is fixed at zero (in logs). Specifically, in matrix notation, the (n × 1) column vector of multilateral exchange rates, R, can be expressed in terms of the (n × 1) vector of bilateral exchange rates, E, whose nth element is zero:

$\begin{array}{ccccc}R=\left(A-I\right)E,& & & & \left(7.3\right)\end{array}$

where A is an (n × n) matrix that has rows containing the trade weights (which sum to one) and has zeroes along its main diagonal (i.e., own-country trade weight is zero);5 I is the identity matrix of order n. Since E contains only n-1 independent exchange rates (and a constant), one of the n multilateral rates in R must be redundant. Conversely, the matrix AI is singular. Hence, one cannot solve for the n bilateral rates starting from the n effective rates.

It is useful to show precisely what the n-1 (redundancy) problem implies in terms of the linear dependence of the effective exchange rates. From equation (7.3), note that the trade weight matrix A has rows that respectively sum to unity. This property implies that the transpose of A has a unit eigenvalue, with corresponding eigenvector that we can call υ):

$\begin{array}{ccccc}{A}^{\prime }\upsilon =\upsilon .& & & & \left(7.4\right)\end{array}$

There is therefore a linear constraint across the effective exchange rates, with weights that correspond to the elements of the eigenvector υ) (whose elements can be normalized arbitrarily, since the eigenvector is not unique). Specifically, from equation (7.3) have:

$\begin{array}{ccccc}{\upsilon }^{\prime }R={\upsilon }^{\prime }\left(A-I\right)E=\left({\upsilon }^{\prime }-{\upsilon }^{\prime }\right)E=0.& & & & \left(7.5\right)\end{array}$

Thus, the condition that must be satisfied for multilateral exchange rate consistency is given by:

$\begin{array}{ccccc}{\upsilon }^{\prime }R=0,& & & & \left(7.6\right)\end{array}$

and given any n-1 effective exchange rates, the remaining (redundant) rate can be calculated from this relationship.

To derive internally consistent multilateral solutions satisfying equation (7.6) and to obtain the corresponding set of bilateral exchange rates, we proceed by reducing the system of n equations by discarding one country’s equation (say for j = n) and expressing the remaining n-1 first-stage multilateral changes (as calculated from equation (7.2)) relative to that reference country’s first-stage solution (i.e., $\mathrm{\Delta }{{r}_{j}}^{1}-\mathrm{\Delta }{{r}_{n}}^{1}$; for j = 1 to n–1). In matrix notation, the resulting set of relative solutions $\mathrm{\Delta }R{.}^{1}-1\mathrm{\Delta }{{r}_{n}}^{1}$, where – denotes that the nth currency term has been dropped and 1 is a conformable vector of 1’s, is a reduced dimension (n – 1 × 1) vector. The vector can be expressed using equation (7.3) in terms of the n - 1 equilibrium bilateral rate changes ΔE., dropping the nth currency here as well, as follows:

$\begin{array}{lll}\mathrm{\Delta }{R.}^{1}-1\mathrm{\Delta }{{r}_{n}}^{1}& =& B\mathrm{\Delta }E.-1\mathrm{\Delta }{{r}_{n}}^{1}\\ & =& \left(B-1\left[{a}_{n,1,}\dots ,{a}_{n,n-1}\right]\right)\mathrm{\Delta }E.\\ & =& C\mathrm{\Delta }E.& & & & \left(7.7\right)\end{array}$

where B is an (n - 1 × n - 1) matrix obtained by deleting the nth row and column of A - I, and where $\left\{{a}_{n,j}\right\}$ for j = 1 to n-1 are the trade weights in the nth currency multilateral real exchange rate. Multiplying both sides of equation (7.7) by the inverse of the nonsingular matrix C yields the solution for n – 1 equilibrium bilateral rate changes ΔE. as a function of the first-stage multilateral solutions. Under the normalization, the nth equilibrium bilateral rate associated with the numeraire currency remains fixed at zero, which gives us all n equilibrium bilateral real exchange rate changes, ΔE. Finally, to construct the (final-stage) consistent solutions for the n multilateral exchange rate changes, ΔR, we can apply equation (7.3) (expressed in differences) substituting in the equilibrium bilateral rates.

We could also get around the overdetermination problem by simply ignoring any one of the equations (7.2) in calculating the n multilateral rates. By considering relative solutions at the first stage, however, our technique solves the nth currency problem in a way that yields multilateral solutions that are invariant both to the choice of numeraire and to which corresponding equation is dropped. In this manner, our solution method—by addressing the redundant equation (nth currency) problem—allows us to obtain a consistent set of solutions for the bilateral exchange rates as well as reconciled effective rates that satisfy multilateral exchange rate consistency.

It is important to note that the resulting underlying current accounts associated with the equilibrium multilateral exchange rates no longer coincide exactly with the equilibrium current accounts embedded in the first-stage changes $\mathrm{\Delta }\text{}c{{a}_{j}}^{1}$, because we cannot hit n targets with only n-1 instruments. Moreover, whether the final current account changes sum to zero (i.e., satisfy current account consistency) will depend on the relation between current accounts and exchange rates, as summarized in the θj coefficients.

To gauge the magnitude of the discrepancies between the equilibrium current accounts and the underlying current accounts associated with the equilibrium exchange rates, we computed, at the end of October 1996, a set of underlying current account estimates associated with the average exchange rates that prevailed during October 1–15, 1996. A corresponding set of equilibrium current accounts was specified somewhat arbitrarily. The country coverage included each of 21 industrial countries plus six groups of developing and transition economies. The global discrepancies associated with these underlying and equilibrium current accounts were $111 billion and$97 billion, respectively. The global discrepancy in the underlying current accounts associated with the equilibrium exchange rates was \$102 billion, and for the individual countries and country groups, the maximum discrepancy between the latter underlying current accounts and the equilibrium current accounts was 0.06 percent of GDP.

Moreover, the maximum difference between the first-stage and final (or second-stage) estimates of equilibrium multilateral real exchange rates was 0.13 percent. Similar calculations at other points in time show similar orders of magnitude for the approximation errors. Thus, provided that we start from a “data set” with broadly similar global discrepancies in the underlying current accounts associated with current exchange rates and the equilibrium current accounts, we can infer, first, that our methodology for calculating equilibrium exchange rates (using the model-based θj coefficients) is globally consistent to a very close approximation, and second, that the first-stage, country-by-country estimates of equilibrium exchange rates provide close approximations to the final (second-stage) estimates.

## An Alternative Procedure

In general, with arbitrary coefficients θj in equation (7.2), resolving both exchange rate consistency and current account consistency at the same time is impossible, and approaches that succeed in one direction must fail in the other. Here, we focus on a particular definition of the coefficients that resolves both consistency issues simultaneously. The conceptual rationale for implementing these coefficients is not compelling, however, and illustrative calculations for some countries are bizarre. We therefore prefer the approach described above, which ensures exchange rate consistency and appears to involve an acceptably small degree of current account inconsistency.

To identify the set of coefficients that resolves both consistency issues simultaneously, we rewrite the equation linking changes in equilibrium current accounts and real effective exchange rates in terms of current account levels (denoted as uppercase CAj), expressed in a numeraire currency, such that equation (7.2) becomes:

$\begin{array}{ccccc}\mathrm{\Delta }C{A}_{j}=\left({\theta }_{j}{Y}_{j}\right)\mathrm{\Delta }{r}_{j}.& & & & \left(7.8\right)\end{array}$

For the general expression:

$\begin{array}{ccccc}\mathrm{\Delta }C{A}_{j}={\gamma }_{j}\mathrm{\Delta }{r}_{j},& & & & \left(7.9\right)\end{array}$

we can ask: What vector of coefficients γj will produce effective exchange rates that are consistent? It is easy to verify that if γj = υj (recall from equation (7.4) that υj is the jth component of the normalized eigenvector of the trade-weight matrix A), then consistency in initial current account changes implies consistency in (first-stage) effective exchange rate changes, without the need for any further multilateral reconciliation. In this case,

$\begin{array}{cccccc}{\mathrm{\Sigma }}_{j}\text{}{\upsilon }_{j}\text{}\mathrm{\Delta }{r}_{j}={\mathrm{\Sigma }}_{j}\text{}{\upsilon }_{j}\frac{\mathrm{\Delta }C{A}_{j}}{{\gamma }_{j}}={\mathrm{\Sigma }}_{j}\text{}\mathrm{\Delta }C{A}_{j}=0.& & & & & \left(7.10\right)\end{array}$

Viewed in the other direction, if γj = υj, then any set of changes of effective rates that satisfies exchange rate consistency is sufficient to generate current account changes that sum to zero. Moreover, the result that the final changes in current accounts (i.e., the differences between underlying current accounts evaluated at equilibrium versus current exchange rates) sum to zero holds regardless of whether the sum of initial current account changes (i.e., the differences between equilibrium current accounts and underlying current accounts evaluated at current exchange rates) is zero or not.

Table 7.1 shows the values of the model coefficients, θjYj and the eigenvector components, υj, for the individual industrial countries and developing country groups. As noted earlier, the trade matrix from which the eigenvector is derived reflects the IMF’s Information Notice System’s trade-weighting scheme. For many countries the two coefficients are similar, for others there are notable differences. Among the major industrial countries, Germany and France have model coefficients of 0.68 and 0.44 (the United States is normalized to unity), compared with eigenvector coefficients of 1.01 and 0.61. The developing country groups show the largest disparities, having model coefficients that are much larger than the eigenvector components—for instance, the weights for newly industrialized economies are 0.74 and 0.24, respectively.

Table 7.1.

Alternative Coefficients

It may also be noted that the eigenvector coefficients for Greece, New Zealand, Portugal, and the transition economies round to 0.00.6 For these cases, equation (7.9) would imply that unrealistically large changes in real effective exchange rates could be required to adjust underlying current accounts to their equilibrium levels; for example, the eigenvector coefficient for Portugal would suggest that a current account adjustment of 1 percent of GDP requires a 150 percent change in the real effective exchange rate, other things equal; and the orders of magnitude are larger still for the transition economies and Greece. The implausibility of such estimates, and the fact that the conceptual basis for INS weights, while reasonable, rests on some strong assumptions,7 suggests that reliance on the eigenvector coefficients is considerably less appealing than the model-based approach described earlier.

This includes six regional groups of developing and transition countries, for which underlying current account positions are calculated from the model described in Section V while equilibrium current account positions (S-I norms) are specified somewhat arbitrarily.

Recall from Section V, equation (5.2) that θ = (M/Y) - [(M/Ym + (X/Yx] where (M/Y), (X/Y) are the import and export shares of GDP, and βm, βx are the long-run elasticities of imports and exports with respect to the real exchange rate.

It is an unfortunate fact that recorded current account balances do not sum to zero across the world, as they should conceptually. See International Monetary Fund (1987).

In practice, there may be differences between the global discrepancies in the saving-investment norms and the underlying current accounts associated with current exchange rates—the two components of “the data set” for the calculations—perhaps suggesting that the saving-investment norms should be regarded as “initial estimates o f equilibrium current accounts that need to be adjusted to eliminate any differences between the two global discrepancies. The methodology described below does not require us to make an explicit adjustment in these initial saving-investment norms, and we thus refer to them, without qualification, as equilibrium current accounts.

The trade-weighted averages are based on the Fund’s Information Notice System (INS) “competitiveness” weights, as described in Zanello and Desruelle (1997) and McGuirk (1987).

The coefficients are 0.0036 for New Zealand, 0.0014 for Portugal, 0.0002 for the transition economies, and 0.00005 for Greece. The very small eigenvector components for these countries are associated with the fact that they comprise very small shares of other countries’ trade, as reflected in the matrix of INS trade weights. Because of their relatively small representation in other countries’ trade baskets, they require proportionately large changes in their real exchange rates to achieve a given adjustment of their net trade with the rest of the world, other things equal.

## References

• International Monetary Fund,1987, Report of the Working Group on the Global Current Account Discrepancy (Washington: International Monetary Fund).