Accounting for Exchange-Rate Variability in Present-Value Models When the Discount Factor Is near 1

by Charles Engel, Kenneth D. West
Accounting for Exchange-Rate Variability in Present-Value Models When the Discount Factor Is near 1
Charles Engel, Kenneth D. West
The American Economic Review
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Accounting for Exchange-Rate Variability in Present-Value
Models When the Discount Factor Is Near 1

A well-known stylized fact about nominal exchange rates among low-inflation advanced countries, particularly U.S. exchange rates, is that their logs are approximately random walks. Michael I. Mussa (1979) is most frequently cited for observing this regularity. In a famous pair of papers, Richard A. Meese and Kenneth Rogoff (1983a, b) found that the structural models of the 1970's could not "beat" a random walk in explaining exchange-rate movements. Recently some authors (Menzie Chinn and Meese, 1995; Nelson Mark, 1995: Mark and Donggyu Sul, 2001) have argued that the mod- els can outforecast the random walk at long horizons. But a comprehensive recent study by Yin-Wong Cheung et al. (2003) documents that "no model consistently outperforms a random walk."

Why? One obvious explanation is that the macroeconomic variables that determine the ex- change rate themselves follow random walks. If the log of the nominal exchange rate is a linear function of forcing variables that are random walks, then it will inherit the random-walk property. The problem with this explanation is that the economic "fundamentals" proposed in the most popular models of exchange rates do not, in fact, follow simple random walks.

One resolution to this problem is that there may be some other fundamentals, ones that have been proposed in some models but are not easily measurable or ones that have not yet been proposed at all, that are important in determin- ing exchange rates. If these "unobserved fun- damentals follow random walks and dominate

* Engel: Department of Economics, university of wis. consin, 1180 Observatory Drive, Madison, WI, 53706-1393; West: Department of

Economics, University of Wisconsin. We thank Mark Watson for helpful dis- cussion, and Camilo Tovar for excellent research assistance. Both authors thank the National Science Foundation for support for this research.

the variation in exchange-rate changes, then ex- change rates will nearly be random walks (even if the standard "observed" fundamentals are not).

In Engel and West (2003a) (hereinafter, EW), we propose an alternative explanation. We con- sider linear models of the exchange rate that are in the "asset-market approach" to exchange rates. These models emphasize the role of ex- pectations of future economic fundamentals in determining the current exchange rate. The ex- change rate (expressed as the home currency price of foreign currency in this paper) can be written as a discounted sum of the current and expected future fundamentals:


(1) Sr = Xri '(1 -b) CWE(fr+, + ZI+,I~)



where f,and z, are economic fundamentals that ultimately drive the exchange rate, such as money supplies, money demand shocks, and productivity shocks. We differentiate between fundamentals observable to the econometrician,

f,, and those that are not observable, z,. E is the expectations operator, and I, is the information set of agents in the economy that determine the exchange rate.

In EW we show that if the fundamentals are I(1) (but not necessarily pure random walks), then as the discount factor approaches unity, the exchange rate will follow a process arbitrarily close to a random walk. Intuitively, we can decompose the I(1) fUldamentals into the sum of a random walk and a stationary component. When the discount factor increases toward 1,

weight is being placed on expectations of

the fundamentals far into the future. Transitory components in the fundamentals become rela- tively less important in determining exchange-rate


behavior. When the discount factor is near unity, the variance of the change of the dis- counted sum of the random-walk component in fundamentals approaches a nonzero constant, but the variance of the change of the stationary component approaches zero. Therefore, the variance of the change of the exchange rate is dominated by the change of the random-walk component, and the exchange rate becomes in- distinguishable from a random walk.

In EW we argue that the theorem is a possible explanation for the random-walk-like behavior of exchange rates. In the standard models, the fundamental typically is I(1), which is a condi- tion of the theorem. We show that empirical estimates of the discount factor are sufficiently close to 1 so that, given the time-series behavior of observed fundamentals, the exchange rate will appear to be a random walk if it is indeed determined as a discounted sum of the current and expected future fundamentals.

But is the EW result the most appealing ex- planation for the random walk behavior of ex- change rates? We can write


Here, xfI is the discounted sum of current and expected future fundamentals that the econome- trician observes Cf,+j). In this paper, we takef, to be the observable fundamental that emerges from one of two classes of asset-market exchange-rate models: monetary models of ex- change rates developed in the 1970's, and mod- els based on Taylor rules for monetary policy. The variable x:I is the part of the exchange rate that can be explained from observed fundamen- tals; Ut is the art of the exchange rate not determined by .$. We take an eclectic view on what U, might be. It might be the case that ex- change rates are determined as in equation (I), in which case U, is the expected discounted sum of current and future values of z,. Or, perhaps some other type of model relates exchange rates to fun- damentals, and U, measures those fundamentals. Or, perhaps the exchange rate is driven in part by noise, in which case U,represents that noise. If Ut is important in driving the exchange rate, then given the random-walk nature of exchange rates, Ut must be a random walk.' This in turn would imply that st and xfI are not cointegrated.

Our task in this paper is to get a measure of the contribution of xfI and U, in driving exchange rates. We cannot say much about the contribution of U,, since it is not observed by us. But even measuring the contribution of xfI may appear to be a quixotic goal: xf, is also unob- servable to the econometrician (even thoughf, is observable). That is because xf, measures agents' expectations about future fundamentals, which are not perfectly observed by the econo- metrician who only sees a subset of the infor- mation that agents use in forming their expectations. For example, if the economic fun- damentals involve monetary policy, the econo- metrician might observe the time-series behavior of monetary-policy instruments and might observe many of the macroeconomic variables that influence monetary policy. But agents, in forecasting future monetary policy, have access to a wide variety of information that is difficult to quantify (e.g., newspaper and newswire reports, speeches by policymakers, etc.).

Nonetheless, this paper demonstrates that we can measure the variance of AxfI (the first- difference of x:,) when the discount factor, b, approaches 1. To be precise, define

Here, H, is the information set used by the econometrician. An estimate ~f, can be constructed from VAR's that include f, and other observable macroeconomic variables that might help forecast f,. This paper demonstrates that var(AxfH) approaches var(hxfI) when b ap

' U, may be a random walk if the discounted sum of unobserved fundamentals, z,, and z, is I(1) and the discount factor is near 1. In that case, the EW theorem applies to the discounted sum of expected current and future values of z,. However, U, could be a random walk for any reason, not just this one.

proaches 1. To be clear, this does not mean that xFI = x.:" as b + 1, and for that reason we do not look to the correlation between As, and Ax:" to gauge the EW explanation. Although xf, remains unobservable to the econometrician, re- markably, the variance of Ax:[ can be estimated consistently.

It follows from (2) that

+ 2 Cov(A$,, AU,).

If only observed fundamentals matter for the exchange rate, then Var(As,) = var(AxfI). We will take ~ar(Ax~,)~ar(As,)

as a measure of the importance of observed fundamentals in driving the exchange rate, when the discount factor is near 1. This satisfies our primary objective, which is to provide some insight into how ef- fective the approach of EW is in accounting for the random-walk behavior of exchange rates.

The ability of the fundamentals to account for the variance of changes in the exchange rates differs somewhat across measures of fundamen- tals and across exchange rates. Roughly, we find Var(A!&,)Nar(As,) to be around 0.4 when we draw the fundamentals from monetary models of exchange rates, and slightly lower when the fun- damentals are derived from Taylor-rule models.

I. Asset-Market Models of Exchange Rates

In EW, we review the familiar models that fall under the label of "the asset market ap- proach to exchange rates." The simplest sum- mary comes directly from Jacob A. Frenkel's (1981 pp. 674-75) paper on "news" and exchange rates, which in many ways is a precursor of our work (here we have changed only the notation to match ours):

This view of the foreign exchange market can be exposited in terms of the following simple model. Let the logarithm of the spot exchange rate on day t be determined by:

where E(s,+ ,I,) -s, denotes the ex-

pected percentage change in the exchange rate between t and t + 1, based on the information available at t, where f, + z, represents the ordinary factors of supply and demand that affect the ex- change rate on day t. These factors may include domestic and foreign money supplies, incomes, levels of output, etc. Equation (6) represents a sufficiently general relationship which may be viewed as a "reduced form" that can be derived from a variety of models of exchange rate determination.

The two types of models we consider here fall into this general form. The first is the fa- miliar monetary model. Following Mark (1995) and others, we take the observable fundamental, fr, to be m, -y, -(mT- yF), where m, is the log of the domestic money supply, y, is the log of domestic GDP, and mT and yT are the foreign counterparts. Following the derivation in EW, the unobserved fundamental, z,, is a linear com- bination of variables such as home and foreign money-demand errors, a risk premium (multi- plied by A), and real exchange-rate shocks aris- ing from sources such as home and foreign productivity changes. In the monetary model, the parameter h represents the interest semi- elasticity of money demand (assumed to be identical in the home and foreign country).

The second model is less familiar and is based on Taylor-rules for monetary policy.2 In EW, we examine the implications of an interest- rate rule that has as one target (in either the home-country or foreign-country policy rule, or both) deviations of the exchange rate from its purchasing-power-parity (PPP) level, st -(p, pT), where p, is the log of the domestic price level and pTis the foreign counterpart. We show that there are two different representations of the model that fall into the class of models given by (6). In the first, f, = p, -pT, and h = 110, where is the coefficient on deviations from (log) PPP in the Taylor rule. In this model, z, is a linear combination of other variables targeted by the Taylor rule as well as perhaps money- demand errors and a risk premium. Intuitively, this model fits neatly into the framework of equation (6) because the log of the exchange

Engel and West (2003b) explore the implications of Taylor-rule models for real exchange-rate behavior.

rate is determined by its target,f, = p, -pT, and the expected movement toward the target, (l/P)[E(s,+ ,lIr) -s,]. Another representation of the same model adds the interest differential to the difference in the log of prices, so that the observed fundamental is given by f,= p, -pT + (i, -i?). In this case, h = (1 -P)/P In this alternative representation, z, is again a linear combination of other variables targeted by the Taylor rule, money demand errors, and a risk premium. The exchange rate contains informa- tion not only about the long-run target, but also about the interest differential. The deviation of the exchange rate from its target helps markets predict the path of interest rates set by monetary policymakers.

Solving equation (6) forward for the exchange rate yields equation (I), where b = h/(l + A). Based on estimates of the interest semi-elasticity of money demand, we note in EW that in quarterly data, for the monetary model, b = 0.97 or 0.98.~ The value of the discount factor is similar in the Taylor-rule model, based on estimates of the responsiveness of interest rates to exchange-rate targets in mon- etary policymaking rules.

11. The Data

We use quarterly data, with most data span- ning 1973: 1-2003: 1.4 The United States is the home country, and we measure exchange rates and fundamentals relative to the other G7 coun- tries: Canada, France, Germany, Italy, Japan, and the United Kingdom.

The exchange rates (end-of-quarter) and con- sumer prices (CPI) come from the International Financial Statistics CD-ROM for all seven countries. Seasonally adjusted money supplies come from the OECD's Main Economic Indi- cators available on Datastream, (M4 for the United Kingdom, M1 for the other countries).

For example, the estimates of the semi-elasticity in James H. Stock and Mark P. Watson (1993) are around

0.1 1. Stock and Watson express interest rates in percentages and use annual rates. To get the units correct for equation (6),we want to express interest rates in decimal form, and we are considering a quarterly frequency. So we multiply their estimate by 400, which implies an interest semi-elasticity of 44, and b = 44/45, or approximately 0.978.

For the precise data spans for each sample, see Engel and West (2004).

For real seasonally adjusted GDP, the data come from the OECD with the exception that for Germany the data combine IFS data (1974: 1-2001:l) with data from the OECD after 2002: 1. Interest rates are three-month Euro rates from Datastream. We take logs of all data but interest rates, and we multiply all data by 100. We use a measure of U.S. money supply that adds "sweep account programs" to our measure of M1 from the OECD. "Sweeps" refer to bal- ances that are moved by U.S. banks from checking accounts to various interest-earning accounts by automated computer programs as a way for banks to reduce their required reserve holdings. It has been argued that exclusion of sweeps from the M1 data will lead to an under- measurement of true transactions balance^.^ The data on sweeps is obtained from the web site of the Federal Reserve Bank of St. Louis.

We examine, then, the behavior of three ob- served fundamentals: m, -y, -(mT -yT),p, pz and p, -pT + (it -iT), for six countries relative to the United States. We performed augmented Dickey-Fuller tests (with four lags) with a constant and trend for all fundamentals and exchange rates, and we failed to reject the null h pothesis of a unit root in almost all


cases. We proceeded to test for no cointegra- tion between the exchange rate and the corre- sponding four fundamentals. In almost every case, we were unable to reject the null of no cointegration using Johansen's A,,, and A,,,,, tests.' This latter finding suggests that there may be a role for unobserved unit-root variables [the U, from equation (2)]in driving exchange rates.

111. Accounting for the Variance of Exchange-Rate Changes

If only observed fundamentals determined exchange rates, then we would have sf = xfI, where xfl is defined in equation (3). As we have noted, we cannot measure xf, because we do not have access to all of the information that mar-

s We thank J. Huston McCulloch for pointing out this issue to us. The exceptions were for the fundamentals involving prices, for Japan and Italy. 'The exceptions were for the United Kingdom, for the fundamentals involving prices.

kets use in forming their expectations of future fundamentals. Here we show that we can, how- ever, measure the variance of when the discount factor, b, is close to 1. We ask whether the variance of Axf, is a substantial fraction of the variance of As,, so that observed fundamen- tals can account for much of the variance in the change of log exchange rates.

We can measure xf, as defined in equa- tion (4)-the discounted sum of current and expected future fundamentals based on the econometrician's information, H,. Define the in- novation in x:, as

and the innovation in xf, as

Under the assumption that all the variables in I, follow an ARIMA(q, r, s) process, q, r, s 2 0, and that H, is a subset of I, that includes at least current and past values off,, equation (6) in West (1988) shows that

As b -+ 1, var(x[, -xfI) stays bounded, but (1 -b2)lb2 +0. It follows that for b near 1, var(e:,) .= ~ar(e:~).

The EW theorem shows that when b is near I, Ax:, --eZI, and Axf, .= eefH. Therefore, we can use an estimate of var(AxfH) to measure


A simple example may help develop intu- ition. Supposef, = f,-, + e,,, + e2,,-,,where el,, and e,,, are mutually independent, indepen- dently and identically distributed, mean-zero processes. Assuming agents observe el,, and e,,,

in this example that as b nears 1, var(AxF1) + Var(e1.t + e,,,) = Var(e1.c + e2.t-1) = ~ar(h:~).

This equality holds even though Ax:,

f. hf,(even as b -+ 1). In this example, the EW result completely explains the random walk in s, as b -+ 1, but that does not mean the exchange-rate change can be completely explained by observable changes in 5.The corre- lation between As, and Ax:, [= corr(e,,, + e,,,, el,, + e,,,-,)] could be far less than 1 if the variance of e,, is large.'

IV. Results

In this section, we report estimates of ~ar(hxt,)/Var(As,) for our three measures of observed fundamentals: m, -y, -(mT -y3, pt -PE and p, -pT + (i, -i3.In calculating this statistic, we take the econometrician's in- formation set to be only the current and lagged value of the fundamental in each case.9

To motivate our calculation of v=(AxFH), let W, be a (n X 1) vector of observable variables, with f, = a'W,. Assume that AW, follows a VAR of order d:

AW, = @,Awl-, + @,AW,-, + ...

Define ((b) = . Then using equation (4), we can write the innova-


tion in x,, as:

From the EW theorem, for b .= 1, we have


Mechanically, then, we estimate an autore- gression (with four lags in all cases) on each peasure of the fundamentals. We use estimates

at time t, we can use (3) to solve and find s, ((6) = [I -b&, -... -b4&,]-'



(= Zl) =f, + be2,(. Then, As, (= Ax:,) = Af, + bhe,,, = el, + be,,, + (1 -b)e2,,-,. As b +1, As, (= -+ e,,, + e,,,. Note that, as in the EW theorem, when b approaches 1, sf approaches a random walk.

Now, continuing with the example, suppose that H, contains only current and lagged values off,. Then, solving using equation (4), we find xfH = fr, SO AxfH = AL = e,,, + e,,,-,.We see construct = a1t(b)&,,.

Mark Watson has pointed out to us that if U, = 0, then as the discount factor approaches 1, the long-run correlation between the change in x:, and the change in the exchange rate should approach 1. We do not implement this useful observation here.

For additional empirical results, see Engel and West (2004).




m-y-11 -p* + Country b (m* -y*) p -p* i -i*

Table 1 reports Var(A~~~)/Var(~s,).

When the fundamentals are m, -y, -(mT -4.;) from the monetary model, the notable result is that this ratio is fairly large, around 0.4 for most countries. Not surprisingly, the ratio rises as b increases toward 1. For one country, Canada, the results are troubling for both sets of funda- mentals, because the ratio exceeds 1 in all cases. From equation (5), that finding is sensible only when cov(AxR AU,) < 0. That is, there must be a negative correlation between the change in the discounted sum of current and expected future fundamentals with the unobserved AU,.


Table 1 also looks at the fundamentals p, -pT and p, -pT + (it -iT) from the Taylor-rule model. We find here that ~ar(h~:~)~ar(hs,)

is a bit lower than we found for the fundamentals from the monetary model. When b = 0.95 or 0.99, for most countries the ratio is around 0.20, though it is about half that size for Germany and Japan. In this case, all of the ratios are less than 1, but only in the case of Italy, when b = 1 and the fundamental is p, -p: does the ratio exceed


There are few previous studies that permit comparison to these figures. The bounds on the variance of As, and of st -E, ,(sf) of Roger

D. Huang (1981 p. 37) and Behzad T. Diba (1987 p. 106) use inequalities that are satis- fied by construction for b arbitrarily near 1. Such inequalities unhelpfully guarantee val- ues greater than 1 for the ratio that we con- sider. Using the monetary model, West (1987

p. 70) finds a ratio of about 0.02-0.08 for the Deutschemark-dollar exchange rate. The present technique yields considerably higher figures, suggesting there is rather more in the monetary model than this previous volatility test would suggest.

We conclude that asset-market models in which the exchange rate is expressed as a dis- counted sum of the current and expected future values of these observed fundamentals can ac- count for a sizable fraction of the variance of As, when the discount factor is large. The EW explanation for a random walk provides a ratio- nale for a substantial fraction of the movement in exchange rates. But there is still a role for left-out forcing variables: perhaps money-demand errors, a risk premium, mismeasure- ment of the fundamentals we have examined here, some other variables implied by other theories, or noise.


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