# Short and long term exchange rate determinants

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## 1. Introduction

Recently there has been a revival of interest in modelling the long-run behaviour of nominal bilateral exchange rates using ‘fundamentals’ such as relative prices. In general, this line of research has established that for the recent floating period weak-form purchasing power parity (PPP) would seem to hold on a single currency basis, but strong-form PPP does not. Additionally, the adjustment to equilibrium in PPP-based equations is painfully slow. In order to obtain strong-form PPP results, and relatively rapid adjustment, researchers have used long runs of historical time series data or panel data sets defined for the recent float.

However, it is still of interest, from both an academic and a policy perspective, to define sensible long-run relationships for a single currency using only recent floating data. The key to resolving the failure of strong-form PPP lies in understanding the forces that keep a nominal exchange rate away from a PPP equilibrium. Undoubtedly, an element of this is related to the rigidity of prices in the face of nominal shocks, while the remainder reflects the impact of real disturbances. MacDonald and Marsh (1997) have demonstrated that proxying such real and nominal disturbances using interest rates produces sensible PPP-based equilibrium exchange rates and also impressive out-of-sample forecasts. The objective in the current paper may be seen as an attempt to specify the real factors proxied in the MacDonald and Marsh paper and also to empirically model their influence on the equilibrium effect exchange rates of the US dollar, German euro and Japanese yen over the period 1975, quarter 1 to 1993, quarter 2.

There have been a number of previous attempts at modelling equilibrium real exchange rates for the recent floating period and such work has not proved particularly fruitful. For example, modelling exercises which use single currency data fail to establish a significant long-run link between real exchange rates and fundamentals, such as real interest differentials. Given the rather negative conclusions to stem from this ‘behavioural’ literature, can another examination of this kind of modelling be justified? We believe it can. One key message to come from the literature on modelling nominal exchange rates is that the econometric methods used, and also the model specification, can have a crucial bearing on the findings of significant and sensible long-run relationships for single currencies. In this paper we use the method of Johansen (1988) and Johansen (1991) and report evidence of sensible and significant long-run relationships. An attractive feature of this econometric method is that it also facilitates computing the short-run dynamic behaviour of our chosen exchange rates. Although this is a secondary objective of our work it is, nevertheless, of interest to examine how an exchange rate returns to its equilibrium value after a disturbance and to pitch our dynamic models against the random walk paradigm.

In terms of the policy debate regarding equilibrium real exchange rates, much discussion has focused on the concept of a fundamental equilibrium exchange rate (FEER), an explicitly normative approach which offers an appealing way of thinking about the evolution of actual and equilibrium real exchange rates. However, although the FEER concept has a number of attractive features, the main difficulty associated with it is one of tractability in terms of the need to have a fully specified multilateral structural model and, further, it does not provide an empirical link between a real exchange rate and its determinants. A key attraction of behavioural time series methods for analysing single currency real exchange rates is the relative ease with which they may be computed and the fact that they do spell out the links with the underlying fundamental determinants.

The outline of the remainder of this paper is as follows. In the next section we use a decomposition of the real exchange rate which facilitates a discussion of the factors introducing systematic trends into the behaviour of the equilibrium real exchange rate. These factors are defined in Section 3, and are labelled the fundamentals exclusive of real interest rates (FERID); they include variables such as net foreign asset accumulation, productivity bias and fiscal balances. Using the real uncovered interest rate parity condition we go on, in Section 4, to define a static relationship for the current equilibrium exchange rate in terms of the FERID variables and the real interest differential (RID). We propose operationalising this model using a vector error correction framework. In Section 5 our data sources are given and the construction of the proxies for our variables, introduced in Section 4, defined. Our estimates of the long-run exchange rate relationships and short-run dynamic results are presented as well. The paper closes with a concluding section.

## 2. A real exchange rate decomposition

In this section we briefly discuss some real exchange rate decompositions which are useful in motivating our empirical tests. The real exchange rate, defined with respect to a general or overall price level, such as the CPI, is given by:

(1)Image

where qt denotes a real exchange rate, st denotes the nominal spot exchange rate, defined as the foreign currency price of a unit of home currency (this is the most convenient definition since in our empirical application we use effective exchange rates), pt denotes a price level and an asterisk denotes a foreign magnitude. In this context, therefore, a rise (fall) in qt denotes an appreciation (depreciation) of the general real exchange rate. A similar relationship may be defined for the price of traded goods as:

(2)Image

where superscript T indicates that the variable is defined for traded goods. If the prices in Eq. 2 are composite terms then, as we shall emphasise below, for Imageto be constant we have to assume that each of the goods prices which enters Imagehas an equivalent counterpart in Image, and the weights used to produce each of these composite price levels are the same.7

We assume that the general prices entering Eq. 1 may be decomposed into traded and non-traded components as:

(3)Image (3â€²)Image

where Î± denotes the share of non-tradeable goods sectors in the economy, assumed to be time-varying, and NT denotes a non-traded good. By substituting Eq. 2, Eq. 3 and Eq. 3â€² in Eq. 1 a general expression for the long-run equilibrium real exchange rate, Image, may be obtained as:

(4) Image

Eq. 4 is illuminating since it highlights three potentially important sources of long-run real exchange rate variability: non-constancy of the real exchange rate for traded goods, which will arise if the kinds of goods entering international trade are imperfect substitutes and there are factors (discussed below) which introduce systematic variability into Image; movements in the relative prices of traded to non-traded goods between the home and foreign country, due to say productivity differentials in the traded goods sectors; differing time-variability of the weights used to construct the overall prices in the home and foreign country. We do not consider the latter possibility in this paper. Let us consider the other sources of variability in a little more detail.

## 3 Sources of trends in the long-run real exchange rate

## 3.1. The traded non-traded price ‘ratio’

The first group of factors we consider relate to the relative price of traded to non-traded goods across countries, captured in Eq. 4 by the term Image. One way of interpreting this term is to think of it as capturing factors which impinge on the relative price of non-traded goods, without necessarily affecting the relative price of traded goods.

## 3.1.1. Balassa-Samuelson

Perhaps the best known source of systematic changes in the relative price of traded to non-traded goods is the Balassa-Samuelson effect. This presupposes that the nominal exchange rate moves to ensure the relative price of traded goods is constant over time; that is, Image. Productivity differences in the production of traded goods across countries can introduce a bias into the overall real exchange rate because productivity advances tend to be concentrated in the traded goods sector; the possibility of such advances in the non-traded sector is limited. If the prices of traded and non-traded products are linked to wages, wages linked to productivity and wages linked across non-traded and traded industries, then the relative price of traded goods will rise less rapidly over time for a country with relatively high productivity in the tradeable sector: the real exchange rate, defined using overall price indices, appreciates for fast growing countries, even when the law of one price holds for traded goods; in terms of Eq. 4, if the home country is a relatively fast growing country it will have a positive Imageterm, thereby pushing Image, above Image (remember the currency is defined here as the foreign currency price of a unit of home currency).

## 3.1.2. The demand side and non-traded goods

The existence of non-traded goods may allow a demand side bias which pushes an exchange rate away from its PPP level defined using traded goods prices. Assuming unbiased productivity growth, Genberg (1978) has demonstrated that if the income elasticity of demand for non-traded goods is greater than unity, the relative price of non-traded goods will rise as income rises (that is, as income rises households will spend a disproportionate amount of their income on services). This relative price change will be reinforced if, as seems likely, the share of government expenditure devoted to non-traded goods is greater than the share of private expenditure, and if income is redistributed to the government over time.

We may therefore think of the second term in Eq. 4 as having the following general functional form:

(5)Image

where PROD is a measure of productivity bias and DEM represents demand side bias. For the reasons noted above, a rise in the domestic value of either of these variables will, cetirus paribus, generate an appreciation of the overall real exchange rate.

## 3.2. Imperfect substitutability of traded goods prices

The factors in the last section can affect the real exchange rate even if traded goods are perfect substitutes across countries and Imageis constant. The constancy of the real exchange rate defined with respect to traded prices is not, however, uncontroversial. For example, there is now considerable evidence to suggest that the kinds of goods produced by industrial countries are not perfect substitutes, and therefore the idea that price differences are quickly arbitraged away is completely unrealistic. We now turn to some of the factors which may introduce systematic variability into Image.

## 3.2.1. National savings and investment and the real exchange rate

The relative price of traded goods, Image, is a major determinant of the goods and non-factor services component of the current account. The current account, in turn, is driven by the determinants of national savings and investment, and since one key component of national savings is the fiscal balance, it follows that the fiscal balance is a determinant of the Imagecomponent of the REER. Initial interest in the relationship between the government fiscal deficit and the real exchange rate was stimulated by the Reagen experiment in the 1980s and, more recently, by the desire on the part of the Clinton administration for fiscal consolidation. The effect of fiscal policy on the real exchange rate may be discussed by asking the question: will fiscal consolidation strengthen or weaken the external value of a currency?

Both outcomes are in fact potentially correct – it just depends on which particular view of the world is adopted. In the traditional Mundell-Fleming two countries model, a tightening of fiscal policy, which increases a country’s national savings, would lower the domestic real interest rate and generate a (permanent) real currency depreciation which, in turn, would produce a permanent current account surplus. The real currency depreciation would also occur in flexible price models. What we are picking up in all these models is the ‘crowding in’ effect of the exchange rate depreciation; the necessity for aggregate demand to equal aggregate supply forces this result irrespective of the class of model.

The basic Mundell-Fleming model, however, ignores the effects of the stock-flow implications of the initial current account imbalance. Models which account for the stock implications of the initial fiscal tightening are portfolio balance models and the asset market/balance of payments synthesis model. In the context of this class of model, the long-run is defined as a point at which the current account is balanced or, to put it slightly differently, any interest earnings on net foreign assets are offset by a corresponding trade imbalance. Hence, if the fiscal consolidation is permanent, it will imply a permanent increase in net foreign assets and an appreciation of the long-run real exchange rate. Other expenditure effects can be analysed in a similar fashion.

In terms of national savings and investment, the other key determinant of the Imagecomponent of the REER is private sector net savings. It is often assumed that such savings are relatively constant over time. This seems a reasonably sensible working assumption for the US, but is probably less so for a country like Japan. Given that there have been secular movements in the Japanese savings rate for the post-war period, the independent effect of this on the net foreign asset position should not be discounted. More generally, Masson et al. (1993) note: “demographic variables that reflect the age structure of the population seem to be important determinants of the cross country variations of saving ratesâ€¦ and hence should affect net foreign asset positions”.

## 3.2.2. The real price of oil

Changes in the real price of oil can also have an effect on the equilibrium real exchange rate, usually through their effect on the terms of trade. The importance of this variable was highlighted by the dramatic increases in the real price of oil in the 1970s (for example, in the early 1970s the real price of oil rose by approximately 65%) and the equally dramatic fall in the mid-1980s (by approximately 50%). In comparing a country that is self-sufficient in oil with one which requires to import oil, the former, ceteris paribus, would exhibit an appreciation as the price of oil rose in terms of the other country. More generally, countries which have at least some oil resources could find their currencies appreciating relative to countries which do not have oil resources.

The effect of the various variables discussed in this section on the real exchange rate may be summarised using the following relationship:

(6)Image

where FISC captures the effect of relative fiscal balances on the equilibrium real exchange rate, PS represents private sector savings and ROIL is the real price of oil. The signs above the variables summarise the long-run effects of these variables on the real exchange rate.

Combining Eq. 5 and Eq. 6 we obtain the following general relationship for the equilibrium real exchange rate, where the signs above the variables should be obvious from the above discussion:

(7)Image

In the next section we detail how Eq. 7 may be operationalised.

## 4. The adjustment of the real exchange rate to static equilibrium and econometric methods

In the last section we discussed the key determinants of the long-run equilibrium real rate. In this section we address the issue of how the actual exchange rate adjusts to the long rate. To tie up the short-run with the longer-run perspective we start by introducing the familiar uncovered interest parity (UIP) condition:

(8)Image

where it denotes a nominal interest rate, Î” is the first difference operator, Et is the conditional expectations operator, t+k defines the maturity horizon of the bonds and other symbols have the same interpretation as before. Eq. 8 may be converted into a real relationship by subtracting the expected inflation differential – Image-from both sides of the equation. After rearrangement this gives:

(9)Image

where Imageis the ex ante real interest rate. Expression (9) describes the current equilibrium exchange rate as being determined by two components, the expectation of the real exchange rate in period t+k and the real interest differential with maturity t+k. We assume that the unobservable expectation of the exchange rate, Et(qt+k), is the equilibrium exchange rate defined in the previous section, namely Image:

(9â€²)Image

In our model, therefore, the actual equilibrium exchange rate given by Eq. 9â€² comprises two components: the first component, Image, driven by the fundamentals exclusive of the real interest differential (FERID) discussed in the previous section, and the real interest differential (RID). The equilibrium condition represented by Eq. 9 is static and is unlikely to hold continuously. How then does the actual rate adjust to the rate given by Eq. 9?

Since the variables in the FERID and RID terms and qt are potentially I(1) processes (this issue is considered in the next section), and since there may exist cointegrating relationships amongst these variables, we propose using a cointegration framework to calculate the static relationship given by Eq. 9â€². Specifically, we define the (nÃ-1) vector of variables, consisting of the variables contained in the vector FERID and RID and qt as xt and assume that it has a vector autoregressive representation of the form:

(10)Image

where Î· is an (nÃ-1) vector of deterministic variables, and var

epsilonis an (nÃ-1) vector of white noise disturbances, with mean zero and covariance matrix Îž. Expression (10) may be reparameterised into the vector error correction mechanism (VECM) as:

(11)Image

where Î” denotes the first difference operator, Î¦i is an (nÃ-n) coefficient matrix (equal to Image) is an (nÃ-n) matrix (equal to Image) whose rank determines the number of cointegrating vectors. If Î is of either full rank, n, or zero rank, Î =0, there will be no cointegration amongst the elements in the long-run relationship (in these instances it will be appropriate to estimate the model in, respectively, levels or first differences). If, however, Î is of reduced rank, r (where r

We test for the existence of cointegration amongst the variables contained in xt using two tests proposed by Johansen. The likelihood ratio, or Trace, test statistic for the hypothesis that there are at most r distinct cointegrating vectors is:

(12)Image

where Imageare the Nâˆ’r smallest squared canonical correlations between xtâˆ’k and Î”xt series [where all of the variables entering xt are assumed I(1)], corrected for the effect of the lagged differences of the xt process [for details of how to extract the Î»s. Additionally, the likelihood ratio statistic for testing at most r cointegrating vectors against the alternative of r+1 cointegrating vectors – the maximum eigenvalue statistic – is given by:

(13)Image

Johansen (1988) shows that Eq. 17 has a non-standard distribution under the null hypothesis. He does, however, provide approximate critical values for the statistic, generated by Monte Carlo methods. It has been pointed out that these statistics may be subject to size distortions depending on the chosen DGP and sample size. To correct for the possibility of such, in this paper we follow Reimers (1992) and report, in addition to Eq. 12 and Eq. 13, the small sample corrected formulas:

(12â€²)Image(13â€²)Image

Although an examination of long-run exchange relationships is instructive, it can nevertheless be problematic since an interpretation of the coefficients in the long-run relationship as, say, elasticities is based on the (often implicit) ceteris paribus assumption that a unit shock does not have an effect on the other variables as well. For example, a fiscal shock will likely affect the real interest differential and perhaps also NFA (if it alters private sector savings). Since such interrelationships are summarised in our VAR model, we may use this to get a feel for these relationships. To do this, we employ an impulse response representation of the VAR. Such an approach has the more general benefit of illustrating the short-run dynamic responses of our group of three exchange rates with respect to the fundamentals.

Although impulse response methods have been used in a number of applications elsewhere, and therefore the method is well known, practically all previous applications ignore the implications of potential cointegrating relationships in the calculation of the impulse responses. In this paper we calculate the impulse responses with the long-run relationships imposed. The standard impulse response approach involves calculating the moving average (MA) representation of the VAR system (10) and examining the response of the exchange rate change to orthogonal impulses. More specifically, the approach involves the following. On the assumption that all of the variables in the vector xt are stationary (we return to this assumption below), then Wold’s decomposition theorem implies the following canonical MA representation for xt:

(14)Image

where, of the terms not previously defined, Î¨0=In and the infinite sum is defined as the limit mean square. This relationship may then be used to examine the effect of shocks, as represented by the white noise disturbances, var

epsilont, on the elements of the xt vector. However, a common problem with this is that since the covariance matrix Î£var

epsilon is unlikely to be diagonal, it is difficult to interpret the effects of a particular shock on, say, the exchange rate. This is because the shock will in all likelihood have a contemporaneous effect on other shocks which, in turn, will have an impact on the exchange rate, making it impossible to unravel the sole influence of the initial shock. A standard way of dealing with this problem is to use the MA representation with orthogonalised innovations. That is:

(15)Image

where the components of Ï‰ are uncorrelated and a matrix P is chosen so that Î£Ï‰ has unit variance [that is, Image]. The matrix P can be any solution of PPâˆ’1=Î£var

epsilon and perhaps the most popular assumption is that P is chosen, using a Choleski factorisation, as a lower triangular non-singular matrix with positive diagonal elements; other decompositions, such as the ‘structural’ decompositions of Bernanke (1986) and Blanchard and Quah (1989), also exist. In the (stable) case the Î¨i converge to zero as iâ†’âˆž and Î£x(h) converges to the covariance matrix of xt as hâ†’âˆž; however, this does not necessarily occur in the case of unstable, integrated or cointegrated VAR processes. Nevertheless, even for such processes it is still possible, as demonstrated by Lutkepohl (1993), to construct Î¨i and Î¸i. In this paper we follow the approach of Hendry and Mizon (1993), which involves reparameterising the error correction component of the VECM and then proceeding with the standard Choleski factorisation.

## 5. Data sources and definitions

We experimented with two measures of the real effective exchange rate. The first, LREER, is the multilateral CPI-based real effective exchange rate for the domestic country relative to its G7 partner countries, expressed in logarithms. The second, LREER1, is the equivalent ULC-based real effective exchange rate. We use a number of FERID variables to capture the influence of the fundamentals noted in Section 3. We experiment with two variables to proxy for PROD. The first, LTNT, is the ratio of the domestic consumer price index to the wholesale price index relative to the equivalent foreign (trade weighted) ratio, expressed in logarithms. The second is LPROD, constructed from rates of growth in real output in manufacturing at home relative to the trade weighted foreign equivalent. We capture the effect of fiscal deficits using the term FBAL, which is the domestic fiscal balance as a proportion of GDP relative to the weighted sum of the partner countries (where the weights are those used to construct the effective exchange rates). NFA is the ratio of the domestic country’s net foreign asset position to GDP and will also capture the effect of fiscal policy on the real exchange rate, but also other factors more closely associated with private sector savings, such as demographics.

Two variables are used to capture the effect of commodity shocks. The terms of trade, LTOT, are constructed as the ratio of domestic export unit value to import unit value as a proportion of the equivalent effective foreign ratio, expressed in logarithms. ROIL is the real price of oil defined as the ratio of the nominal price of oil to the domestic country’s wholesale price index, again expressed in logarithms. Finally, we use two relative real interest rate terms: RRL, which is the long-term real interest differential constructed using the domestic 10 year nominal bond yield minus a centered 12 quarter moving average of the inflation rate minus the equivalent foreign effective; and RRS, which is the equivalent short-term differential, where three month treasury bill rates and a centered four quarter moving average were used (we also experiment with unconstrained interest rates). The effect of all of these variables on the static equilibrium exchange rate [Eq. 9] is summarised in:

(16)Image

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## 5.1. Long-run relationships: the lock between real exchange rates and real interest rates

Testing the relationship between a real exchange rate and a real interest differential, conditional on a constant equilibrium rate, has proven to be a relatively popular, although unsuccessful, way of modelling real exchange rates. We re-examine the model here because it should serve as a useful benchmark with which to compare our more general model and also to establish if using more powerful econometric methods than those used by others produces satisfactory results. The model we estimate has the following form:

(17)Image

Eq. 17 may be derived from Eq. 9â€² by assuming Image, a constant; we let the real interest rates be unconstrained in our estimation and it is expected that Î²1>0 and Î²2<0; we test the symmetry restriction that estimated coefficients are of equal and opposite signs. Use the Engle-Granger two-step cointegration method to estimate Eq. 17, for a variety of currencies and time periods, and find no evidence of a long-run relationship. One reason for this may simply lie in the econometric technique used to estimate Eq. 17. Thus, Banerjee et al. (1986) have noted that the small sample properties of the Engle-Granger method are poor. Additionally, if the regressors in Eq. 17 are endogenous and (or) the errors exhibit serial correlation, then the asymptotic distribution of the coefficients will depend on nuisance parameters. Researchers have demonstrated that, in testing equilibrium relationships for the nominal exchange rate, econometric methods robust to simultaneity bias and potential endogeneity can make a significant difference to the outcome. Is the same true in the current application? We estimate Eq. 17 using the methods discussed in Section 4. These methods should produce asymptotically optimal estimates because they incorporate a parametric correction for serial correlation (which comes from the underlying VAR structure) and the systems nature of the estimator means the estimates should be robust to simultaneity bias.

Our results from Eq. 17 using, alternatively, short and long rates are reported in Table 1 and Table 2, respectively.

Table 1. Real exchange rates and real short term interest rates

Image

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Table 2. Real exchange rates and real long term interest rates

Image

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In Table 1, the LB, LM and NM statistics are multivariate residual diagnostic tests: LB is Hoskings multivariate Ljung-Box statistic, LM(1 and 4) are multivariate Godfrey (1988) LM-type statistics for first and fourth order autocorrelation, and NM(6) is a Doornik and Hansen (1994) multivariate normality test. Reported numbers are p-values and indicate, in general, an absence of serial correlation, although there is some evidence of non-normality in the Japanese yen and US dollar systems. In terms of the coefficients of determination, the explanatory power ranges from 0.16 for the dollar to 0.36 for the euro.

In Table 2 a similar set of results as those portrayed in Table 1 is presented for real exchange rates and real long-term interest rates. The picture here is broadly similar to that reported in Table 1. There is again evidence of significant long-run relationships for all three currencies, interest rate coefficients are generally correctly signed (apart from the coefficient on the foreign rate in the Japenese equation).

In sum, then, what is perhaps the simplest real exchange rate model does not do too badly relative to the metric set by other researchers, and also in terms of producing statistically significant long-run relationships which, in turn, produce dynamic equations that explain a reasonable percentage of the in-sample performance of an exchange rate change.

## 5.2. Long-run relationships: a non-constant real equilibrium exchange rate

For the general exchange rate model in which the equilibrium real exchange rate, Image, is time dependent and assumed to be a function of the variables contained in the vector FERID. As in the case of the simplest model, we experimented with both short and long interest rates. The Î»Max and Trace statistics for the systems containing short and long interest rates, respectively. On the basis of the standard set of significance values (that is, the values unadjusted for small sample bias), there is very strong evidence of cointegration for all three currencies, regardless of the interest rate measure. For both the dollar and yen systems we have let the interest rate terms be unconstrained. This is based partly on the pretesting noted in Table 1 and Table 2 and also on the fact that the relationships with the unconstrained interest rates produced the more appealing cointegrating vectors. We have therefore adjusted the Trace and Î»Max statistics using the Reimers (1992) small sample correction, reported in the columns labeled Tâˆ’np. With these adjusted statistics the picture changes – there is now one statistically significant vector for each currency. We therefore proceed on the basis of one significant cointegrating vector for each of the currencies.

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Estimates of the cointegrating vectors associated with the largest eigenvalues for each system for both short and long rates. Each of the vectors has been normalised on the exchange rate (that is, the LREER has a coefficient of âˆ’1, so the coefficients are written in equation format).

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