# On the volatility of exchange rates: Tests of monetary and portfolio balance models of exchange rate determination

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On the Volatility of Exchange Rates: Tests of Monetary

and Portfolio Balance Models of

Exchange Rate Determination

By

Daniel Gros

C o n t e n t s : I. Introduction. - II. Methodology. - III. The Flexible Price Monetary Model. - IV. The Role of Sticky Prices. - V. Portfolio Balance Models. - VI. Con- clusions. - Appendix.

I. lntrodnction

T he highly erratic behaviour of floating exchange rates since the break- down of the fixed exchange rate system has been a puzzle for many observers. Some claim that the degree of volatility exhibited by ex-

change rates is excessive and thus undesirable; others claim that an efficient foreign exchange market is a better, or more efficient, outlet for many under- lying disturbances, rather than markets that might not be able to react as swiftly. Another aspect of this problem is that there seem to be tranquil and turbulent periods in the foreign exchange markets, that is the degree of volatility of exchange rates varies over time without corresponding changes in the behaviour of the fundamentals. It is widely claimed that the degree of this volatility is due to the volatility of the underlying policies - a claim which has not, as yet been substantiated.

The purpose of this paper is to contribute to the discussion on the volatility of exchange rates by analyzing a more specific question: is it possible to reject the joint hypothesis that (a) foreign exchange markets are efficient and (b) that the exchange rate is determined by a particular model? This procedure is an application of the so-called variance bound tests, developed in the finance literature, that examine the issue of excess volatility of stock prices. The basic idea behind this literature is that any asset price formed in an efficient market is a function of present and future fundamentals. The volatility of the asset price itself should thus not exceed the volatility of the fundamentals. Unfortunately,

Remark: This research was supported by grants from the Thyssen Foundation and Frau Von l.utteroti. The author wishes to thank Peter Borlo and Keller Hannah for excellent research assistance. C. Adams, D. Folkerts-Landau, J. Huss, P. lsard, and S. Ramachandran provided useful comments on an earlier version of this paper.

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however, there is no general agreement about the fundamentals that ought to determine exchange rates. Therefore, a number of the most widely used exchange rate models are analyzed to determine whether the volatility of exchange rates is larger than the volatility of fundamentals that ought to determine exchange rates. The purpose of this exercise is not to find the model that performs best, but to find out whether these different models yield similar conclusions.

The most widely used exchange rate models in empirical work are all variants of the monetary approach. They can be subsumed in a general specification that implies that the exchange rate is a function of five variables: money supplies, national incomes, interest rate differentials, inflation differ- entials and relative asset supplies.

The monetary approach has also been integrated with portfolio balance considerations which place the emphasis on differences in asset supplies. Accordingly, the tests performed in this paper are two variants of the monetary approach and one representation of the portfolio balance approaeh.

Assuming that the exchange rate is set in an efficient, forward-lo0king market, the fundamental determinants of exchange rates are the expected, discounted present values of future money supplies, incomes and asset sup- plies. The test consists in computing the variance of this present value and comparing it to the variance of the actual exchange rate.

However, since exchange rates are widely regarded as non-stationary variables, it is not possible to perform a straightforward test based on the variance of the level of the exchange rate. A solution adopted by a number of researchers [Bini-Smaghi, 1985; Huang, 1981; Wadwani, 1984] has been to use the first difference of the exchange rate, under the assumption that first differences are stationary. However, the results based on first differences have been inconclusive [see Bini-Smaghi, 1985; Wadwani, 1984], and different from the results that are based on levels. This paper, in contrast, uses a methodology that is based on levels but is not subject to the stationary problems because it does not use the variance but calculates a second moment around a different variable whose expectation exists even if the underlying process for exchange rates is non-stationary.

The procedure used for this variance bounds test here is quite different from the usual regression analysis because it asks whether there is enough variability in these fundamentals to justify the observed variability in exchange rates. In contrast, the usual regression analysis minimizes the variance of the difference between the estimated and the actual values. Thus, a good fit usually implies that the variance of the estimated exchange rate model is close to the variance of the actual exchange rate. But such a result, per se, does not answer the question whether exchange rates are too volatile to be compatible with efficient foreign exchange markets.

Gros: The Volatility of Exchange Rates 275

The tests performed in this paper are based on various DM exchange rates, this is in contrast to most empirical work on exchange rates which is usually based on the U.S. dollar. However, one of the results of the paper is that the behaviour of the U.S. dollar (that is the DM/U.S.$ exchange rate) is somehow different (from the other DM exchange rates); this implies that tests that are based on the U.S. dollar might yield similar results for most exchange rates because there is a common U.S. dollar factor. Another advantage of the DM as the base currency is that it allows one to analyze the experience of the EMS. The semi-fixed exchange rates in the EMS represent a useful contrast to the more freely floating currencies like the U.S. dollar or the Japanese yen. Indeed the main result of the test performed in this paper is that for the intra-EMS exchange rates, the variability of the fundamentals can account for the (re- duced) variability of the intra-EMS exchange rates. For the other currencies, considered here, the variability of the fundamentals appears much smaller than the variability of the exchange rates. This indicates that the EMS has created an environment in which the variability of the exchange rates has been reduced to the minimum that can be achieved given the variability in the fundamentals.

Section II discusses the general methodology to be followed in dealing with the issue of excess volatility. Section III presents the results of the application of this methodology to the flexible price monetary model. Section IV presents the results for the sticky price, Dornbusch model. Section V analyzes the implica- tions of the portfolio balance model and proposes three different ways to apply the volatility tests of the previous sections to this class of exchange rate models. Section VI contains some concluding remarks.

II. Methodology The view that exchange rates are asset prices can be represented by a

general model in which the (logarithm of the) exchange rate, st, is determined by:

st ---- Xt +/a [Et(st+,) - s,], (1)

where Et(St+l) denotes the expectation of the future exchange rate, st+t, formed and conditional upon information available at time t. Xt denotes the "fundamentals" that, according to the specific model, determine the exchange rate. The parameter # measures the sensitivity of the current exchange rate to its expected rate of change. Assuming that expectations are consistent with the application of( l) in all future periods ~ (1) can be solved by forward iteration to yield an expression for the current exchange rate in terms of present and future fundamentals:

That is, expectations are rational (and imposing a suitable boundary condition).

276 Wel tw i r t schaf t l i ches Arch iv

s, = (1 - a) j~0(ay E,(X,+j), (2)

where e~ -- #/(1 + #) < 1. This relationship suggests that the variance of the exchange rate should not

exceed the variance of the discounted sum of future fundamentals. 2 However, if the fundamentals follow a non-stationary process the variance,

that is the second moment around the mean, of both sides of(2) does not exist. The technique employed in this paper avoids this non-stationary problem by computing second moments around a different variable that is independent of the sample mean. This technique was proposed by Mankiw et al. [1985] and was applied by them to reexamine the issue of excess volatility of stock prices.

This technique avoids the use of sample means, or other sample statistics, by defining two additional variables. The first is the perfect foresight or ex-post rational exchange rate, s,*, which is equal to the discounted sum of the actual fundamentals:

s,* = (1 - a) jZ=0(ay X,+j. (3)

The second variable used in the tests of this paper is a so-called "naive forecast" exchange rate, st ~ based on some naive forecast of future fundamen- tals EN,(X,+j):

s, ~ = (1 - ~)j__:}0(~y EN,(X,+j). (4)

The naive forecast ENt (Xt+j) does not need to be rational, but it is assumed that rational agents have access to this naive forecast. The difference between the perfect forecast exchange rate, s,*, and the naive forecast exchange rate, s, ~ can be written as:

s , * - s, ~ = (s* - s , ) + (s , - s ,~ (5 )

Squaring both sides of (8) and taking expectations then implies:

E(s,* - s,~ 2 = E(s,* - st) 2 + E(s, - st~ 2 (6)

because the expectation of the cross product, E(s,* - s,) (s, - s, ~ is zero since s,* -

The difference between the actual exchange rate s,, and the perfect foresight exchange rate, s,*, (see equation (3)) is due to an exceptional error which is defined by: s,* = s, + Ut. A rational forecast error, like U,, must be uncorrelated with all information available at time t; and it must therefore also be uncorrelated with s, (i.e., the co-variance (s, U) = 0). This implies that the variance of s,*, denoted by Var (s,*), must exceed the variance of s, since: Var (s,*) = Var (st) + Var (U,), and thus: Var (s,*) >_ Var (s,). In an efficient foreign exchange market, the variance of the perfect foresight exchange rate should exceed the variance of the actual exchange rate if these variances exist.

Gros: The Volatility of Exchange Rates 277

s, = Ut is uncorrelated with any information available at time t or at the beginning of the sample period. The expectation in (6) is thus conditional upon information available at the beginning of the sample period. Equation (6) contains the two inequalities that are tested below:

E(st* - s,~ 2 _~ E(st* - s,) 2 (7)

E(st* - st~ 2 ~ E(st - st~ 2. (8)

The intuition behind (7) is that the mean square of the forecast errors is larger if the naive forecast is used instead of the efficient market forecast. The intuition behind (8) is that the naive forecast is closer to the market forecast (in terms of average squared distances) than to the perfect forecast. The conditio- nal expectations in (7) and (8) exist even if exchange rates follow a non-statio- nary process because they do not rely on a sample mean.

In actual tests, it is necessary to truncate the infinite series contained in (3) by using terminal values for the nominal exchange rate and the fundamentals. s,* can thus be redefined as:

T-t

st* = (1 - a) j~o(ay Xt+j + a T-' ST (9)

Since rationality implies st = Et(st*) for all t, this truncation does not affect the inequalities (7) and (8). Moreover, since (9) also includes the actual ex- change rate at T+I , (9) and thus (7) and (8) would hold even in the presence of speculative bubbles?

The most important problem in using the inequalities (7) and (8) for a volatility test on exchange rates is that there exist several different models that can be used to specify (1). The models differ not only regarding the fundamen- tals, X, but also regarding the discount factor, ~. However, by using several of the most "popular" models it should be possible to obtain results that are robust.

III. The Flexible Price Monetary Model

The models to be used in this paper, all Variants of the monetary approach, are those most frequently tested in the literature. The general results of the econometric tests performed so far are summarized in Meese and Rogoff [ 1983; 1985], Boughton [1986] and Isard [1986]. Meese and Rogoff, in particular, indicate that the in-sample fit of these models varies widely, depending on the

3 A speculative bubble is defined here as a situation in which the weight of the last term in (12) does not go to zero as T goes to infinity but all other relationships used so far continue to hold.

278 Weltwirtschaftliches Archiv

currency and the period of observation. However, the out-of-sample fit or predictive ability of these models is generally poor?

The starting point for all the models is a conventional money demand function of the form:

mtd -- k + ~byt 4- pt - oit, (10)

where mt d, y , and pt represent (the natural logarithm) of the quantity of money demands, income and general price level, th and o represent the income elasticity and interest semi-elasticity of money demand respectively; it repre- sents the nominal interest rate. Assuming purchasing power parity (PPP), interest parity, and equilibrium on the domestic money market (these three assumptions imply: pt ---- St -Jr- pt*, it - it* = Et(s,+~) - st and m, d = mts = mr) equation (10) can be rewritten as:

mt = k + ~byt + st + pt* - o[it* + E,(St+l) - st], (11)

where pt* and i,* represent the foreign price level and interest rate respectively. The three assumptions embedded in (11) will be relaxed subsequently. The assumption of continuous PPP is relaxed in the next section that will consider the role of sticky prices in a Dornbusch model. The version of the model that assumes continuous PPP is referred to as the flexible price version of the monetary model.

Strictly speaking, this flexible price version of the monetary model does not require PPP to hold in level form. It requires only that PPP holds in an expected sense. PPP holds in an expected sense if the real exchange rate follows a random walk. As documented by a number of empirical studies [Roll, 1979; Frenkel, 1981; Darby, 1981; Mishkin, 1981, p. 699; Hakkio, 1984] it is difficult to reject the hypothesis that the real exchange rate follows a random walk, the flexible price model can therefore not be rejected out of hand.

The assumption that desired money balances are always equal to actual balances, is relaxed in the tests that use quarterly data since the form of the adjustment assumed to govern the money market depends on the time horizon considered. For annual data it is assumed that the money market adjust...

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