Accounting for Exchange-Rate Variability in Present-Value Models When
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Accounting for Exchange-Rate Variability in Present-Value Models When the Discount Factor Is Near 1 By CHARLES ENGEL AND KENNETH D. WEST* A well-known stylized fact about nominal the variation in exchange-rate changes, then ex- exchange rates among low-inﬂation advanced change rates will nearly be random walks (even countries, particularly U.S. exchange rates, is if the standard “observed” fundamentals are that their logs are approximately random walks. not). Michael I. Mussa (1979) is most frequently In Engel and West (2003a) (hereinafter, EW), cited for observing this regularity. In a famous we propose an alternative explanation. We con- pair of papers, Richard A. Meese and Kenneth sider linear models of the exchange rate that are Rogoff (1983a, b) found that the structural in the “asset-market approach” to exchange models of the 1970’s could not “beat” a random rates. These models emphasize the role of ex- walk in explaining exchange-rate movements. pectations of future economic fundamentals in Recently some authors (Menzie Chinn and determining the current exchange rate. The ex- Meese, 1995; Nelson Mark, 1995; Mark and change rate (expressed as the home currency Donggyu Sul, 2001) have argued that the mod- price of foreign currency in this paper) can be els can outforecast the random walk at long written as a discounted sum of the current and horizons. But a comprehensive recent study by expected future fundamentals: Yin-Wong Cheung et al. (2003) documents that “no model consistently outperforms a random walk.” Why? One obvious explanation is that the (1) s t x tI 1 b b jE f t j zt j It j 0 macroeconomic variables that determine the ex- change rate themselves follow random walks. If 0 b 1 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 where ft and zt are economic fundamentals that property. The problem with this explanation is ultimately drive the exchange rate, such as that the economic “fundamentals” proposed in money supplies, money demand shocks, and the most popular models of exchange rates do productivity shocks. We differentiate between not, in fact, follow simple random walks. fundamentals observable to the econometrician, One resolution to this problem is that there ft, and those that are not observable, zt. E is the may be some other fundamentals, ones that expectations operator, and It is the information have been proposed in some models but are not set of agents in the economy that determine the easily measurable or ones that have not yet been exchange rate. proposed at all, that are important in determin- In EW we show that if the fundamentals are ing exchange rates. If these “unobserved” fun- I(1) (but not necessarily pure random walks), damentals follow random walks and dominate then as the discount factor approaches unity, the exchange rate will follow a process arbitrarily close to a random walk. Intuitively, we can * Engel: Department of Economics, University of Wis- decompose the I(1) fundamentals into the sum consin, 1180 Observatory Drive, Madison, WI, 53706-1393 of a random walk and a stationary component. (e-mail: email@example.com); West: Department of When the discount factor increases toward 1, Economics, University of Wisconsin (e-mail: kdwest@ more weight is being placed on expectations of facstaff.wisc.edu). We thank Mark Watson for helpful dis- cussion, and Camilo Tovar for excellent research assistance. the fundamentals far into the future. Transitory Both authors thank the National Science Foundation for components in the fundamentals become rela- support for this research. tively less important in determining exchange-rate 119 120 AEA PAPERS AND PROCEEDINGS MAY 2004 behavior. When the discount factor is near damentals, and Ut measures those fundamentals. unity, the variance of the change of the dis- Or, perhaps the exchange rate is driven in part by counted sum of the random-walk component in noise, in which case Ut represents that noise. If Ut fundamentals approaches a nonzero constant, is important in driving the exchange rate, then but the variance of the change of the stationary given the random-walk nature of exchange rates, component approaches zero. Therefore, the Ut must be a random walk.1 This in turn would variance of the change of the exchange rate is imply that st and xf are not cointegrated. tI dominated by the change of the random-walk Our task in this paper is to get a measure of component, and the exchange rate becomes in- the contribution of xf and Ut in driving ex- tI distinguishable from a random walk. change rates. We cannot say much about the In EW we argue that the theorem is a possible contribution of Ut, since it is not observed by us. explanation for the random-walk-like behavior But even measuring the contribution of xf maytI of exchange rates. In the standard models, the appear to be a quixotic goal: xf is also unob- tI fundamental typically is I(1), which is a condi- servable to the econometrician (even though ft is tion of the theorem. We show that empirical observable). That is because xf measures tI estimates of the discount factor are sufﬁciently agents’ expectations about future fundamentals, close to 1 so that, given the time-series behavior which are not perfectly observed by the econo- of observed fundamentals, the exchange rate metrician who only sees a subset of the infor- will appear to be a random walk if it is indeed mation that agents use in forming their determined as a discounted sum of the current expectations. For example, if the economic fun- and expected future fundamentals. damentals involve monetary policy, the econo- But is the EW result the most appealing ex- metrician might observe the time-series planation for the random walk behavior of ex- behavior of monetary-policy instruments and change rates? We can write might observe many of the macroeconomic variables that inﬂuence monetary policy. But (2) st f x tI Ut agents, in forecasting future monetary policy, have access to a wide variety of information that is difﬁcult to quantify (e.g., newspaper and where newswire reports, speeches by policymakers, etc.). Nonetheless, this paper demonstrates that we f can measure the variance of xf (the ﬁrst- tI (3) x tI 1 b b jE ft I . j t difference of xf ) when the discount factor, b, tI j 0 approaches 1. To be precise, deﬁne Here, xf is the discounted sum of current and tI expected future fundamentals that the econome- (4) f x tH 1 b b jE ft j Ht . trician observes (ft j). In this paper, we take ft j 0 to be the observable fundamental that emerges from one of two classes of asset-market Here, Ht is the information set used by the exchange-rate models: monetary models of ex- ˆf econometrician. An estimate xtH can be con- change rates developed in the 1970’s, and mod- structed from VAR’s that include ft and other els based on Taylor rules for monetary policy. observable macroeconomic variables that might The variable xf is the part of the exchange rate tI help forecast ft. This paper demonstrates that that can be explained from observed fundamen- f Var( xtH) approaches Var( xf ) when b ap- tI tals; Ut is the part of the exchange rate not determined by xf . We take an eclectic view on tI 1 what Ut might be. It might be the case that ex- Ut may be a random walk if the discounted sum of change rates are determined as in equation (1), in unobserved fundamentals, zt, and zt is I(1) and the discount factor is near 1. In that case, the EW theorem applies to the which case Ut is the expected discounted sum of discounted sum of expected current and future values of zt. current and future values of zt. Or, perhaps some However, Ut could be a random walk for any reason, not other type of model relates exchange rates to fun- just this one. VOL. 94 NO. 2 UNDERSTANDING EXCHANGE-RATE DYNAMICS 121 proaches 1. To be clear, this does not mean that change rate between t and t 1, based xf tI xtH as b 3 1, and for that reason we do not f on the information available at t, where look to the correlation between st and xtH to f f t z t represents the ordinary factors of gauge the EW explanation. Although xtI re- f supply and demand that affect the ex- mains unobservable to the econometrician, re- change rate on day t. These factors may include domestic and foreign money markably, the variance of xf can be estimated tI supplies, incomes, levels of output, etc. consistently. Equation (6) represents a sufﬁciently It follows from (2) that general relationship which may be f viewed as a “reduced form” that can be (5) Var st Var xtI Var Ut derived from a variety of models of exchange rate determination. f 2 Cov xtI, Ut . The two types of models we consider here If only observed fundamentals matter for the fall into this general form. The ﬁrst is the fa- exchange rate, then Var( st) Var( xf ). We tI miliar monetary model. Following Mark (1995) f will take Var( xtI)/Var( st) as a measure of the and others, we take the observable fundamental, importance of observed fundamentals in driving ft, to be mt yt (m* y*), where mt is the log t t the exchange rate, when the discount factor is of the domestic money supply, yt is the log of near 1. This satisﬁes our primary objective, domestic GDP, and m* and y* are the foreign t t which is to provide some insight into how ef- counterparts. Following the derivation in EW, fective the approach of EW is in accounting for the unobserved fundamental, zt, is a linear com- the random-walk behavior of exchange rates. bination of variables such as home and foreign The ability of the fundamentals to account for money-demand errors, a risk premium (multi- the variance of changes in the exchange rates plied by ), and real exchange-rate shocks aris- differs somewhat across measures of fundamen- ing from sources such as home and foreign tals and across exchange rates. Roughly, we ﬁnd productivity changes. In the monetary model, ˆf Var( xtH)/Var( st) to be around 0.4 when we the parameter represents the interest semi- draw the fundamentals from monetary models of elasticity of money demand (assumed to be exchange rates, and slightly lower when the fun- identical in the home and foreign country). damentals are derived from Taylor-rule models. The second model is less familiar and is based on Taylor-rules for monetary policy.2 In I. Asset-Market Models of Exchange Rates EW, we examine the implications of an interest- rate rule that has as one target (in either the In EW, we review the familiar models that home-country or foreign-country policy rule, or fall under the label of “the asset market ap- both) deviations of the exchange rate from its proach to exchange rates.” The simplest sum- purchasing-power-parity (PPP) level, st (pt mary comes directly from Jacob A. Frenkel’s p*), where pt is the log of the domestic price t (1981 pp. 674 –75) paper on “news” and ex- level and p*is the foreign counterpart. We show t change rates, which in many ways is a precursor that there are two different representations of of our work (here we have changed only the the model that fall into the class of models given notation to match ours): by (6). In the ﬁrst, ft pt p* and t, 1/ , where is the coefﬁcient on deviations from This view of the foreign exchange market (log) PPP in the Taylor rule. In this model, zt is can be exposited in terms of the following a linear combination of other variables targeted simple model. Let the logarithm of the by the Taylor rule as well as perhaps money- spot exchange rate on day t be determined demand errors and a risk premium. Intuitively, by: this model ﬁts neatly into the framework of equation (6) because the log of the exchange (6) st ft zt E st 1 It st where E(s t 1 I t ) s t denotes the ex- 2 Engel and West (2003b) explore the implications of pected percentage change in the ex- Taylor-rule models for real exchange-rate behavior. 122 AEA PAPERS AND PROCEEDINGS MAY 2004 rate is determined by its target, ft pt p* andt, For real seasonally adjusted GDP, the data the expected movement toward the target, come from the OECD with the exception that (1/ )[E(st 1 It) st]. Another representation of for Germany the data combine IFS data (1974: the same model adds the interest differential to 1–2001:1) with data from the OECD after the difference in the log of prices, so that the 2002:1. Interest rates are three-month Euro rates observed fundamental is given by ft pt p* t from Datastream. We take logs of all data but (it i*). In this case, t (1 )/ . In this interest rates, and we multiply all data by 100. alternative representation, zt is again a linear We use a measure of U.S. money supply that combination of other variables targeted by the adds “sweep account programs” to our measure Taylor rule, money demand errors, and a risk of M1 from the OECD. “Sweeps” refer to bal- premium. The exchange rate contains informa- ances that are moved by U.S. banks from tion not only about the long-run target, but also checking accounts to various interest-earning about the interest differential. The deviation of accounts by automated computer programs as a the exchange rate from its target helps markets way for banks to reduce their required reserve predict the path of interest rates set by monetary holdings. It has been argued that exclusion of policymakers. sweeps from the M1 data will lead to an under- Solving equation (6) forward for the ex- measurement of true transactions balances.5 change rate yields equation (1), where b The data on sweeps is obtained from the web /(1 ). Based on estimates of the interest site of the Federal Reserve Bank of St. Louis. semi-elasticity of money demand, we note in We examine, then, the behavior of three ob- EW that in quarterly data, for the monetary served fundamentals: mt yt (m* y*), pt t t model, b 0.97 or 0.98.3 The value of the p* and pt t, p* (it t i*), for six countries t discount factor is similar in the Taylor-rule relative to the United States. We performed model, based on estimates of the responsiveness augmented Dickey-Fuller tests (with four lags) of interest rates to exchange-rate targets in mon- with a constant and trend for all fundamentals etary policymaking rules. and exchange rates, and we failed to reject the null hypothesis of a unit root in almost all II. The Data cases.6 We proceeded to test for no cointegra- tion between the exchange rate and the corre- We use quarterly data, with most data span- sponding four fundamentals. In almost every ning 1973:1–2003:1.4 The United States is the case, we were unable to reject the null of no home country, and we measure exchange rates cointegration using Johansen’s max and trace and fundamentals relative to the other G7 coun- tests.7 This latter ﬁnding suggests that there tries: Canada, France, Germany, Italy, Japan, may be a role for unobserved unit-root variables and the United Kingdom. [the Ut from equation (2)] in driving exchange The exchange rates (end-of-quarter) and con- rates. sumer prices (CPI) come from the International Financial Statistics CD-ROM for all seven III. Accounting for the Variance countries. Seasonally adjusted money supplies of Exchange-Rate Changes come from the OECD’s Main Economic Indi- cators available on Datastream, (M4 for the If only observed fundamentals determined United Kingdom, M1 for the other countries). exchange rates, then we would have st xf , tI f where xtI is deﬁned in equation (3). As we have f 3 noted, we cannot measure xtI because we do not For example, the estimates of the semi-elasticity in have access to all of the information that mar- James H. Stock and Mark P. Watson (1993) are around 0.11. Stock and Watson express interest rates in percentages and use annual rates. To get the units correct for equation 5 (6), we want to express interest rates in decimal form, and We thank J. Huston McCulloch for pointing out this we are considering a quarterly frequency. So we multiply issue to us. 6 their estimate by 400, which implies an interest semi- The exceptions were for the fundamentals involving elasticity of 44, and b 44/45, or approximately 0.978. prices, for Japan and Italy. 4 7 For the precise data spans for each sample, see Engel The exceptions were for the United Kingdom, for the and West (2004). fundamentals involving prices. VOL. 94 NO. 2 UNDERSTANDING EXCHANGE-RATE DYNAMICS 123 kets use in forming their expectations of future in this example that as b nears 1, Var( xf ) 3 tI fundamentals. Here we show that we can, how- Var(e1,t e2,t) Var(e1,t e2,t 1) ever, measure the variance of xf , when the tI f Var( xtH). This equality holds even though xf tI discount factor, b, is close to 1. We ask whether xtH (even as b 3 1). In this example, the f the variance of xf is a substantial fraction of tI EW result completely explains the random walk the variance of st, so that observed fundamen- in st as b 3 1, but that does not mean the tals can account for much of the variance in the exchange-rate change can be completely ex- change of log exchange rates. plained by observable changes in ft. The corre- f f We can measure xtH as deﬁned in equa- lation between st and xtH [ corr(e1,t e2,t, tion (4)—the discounted sum of current and e1,t e2,t 1)] could be far less than 1 if the expected future fundamentals based on the variance of e2,t is large.8 econometrician’s information, Ht. Deﬁne the in- novation in xf as tI IV. Results f f f e tI xtI E xtI It 1 In this section, we report estimates of f Var( xtH)/Var( st) for our three measures of f and the innovation in xtH as observed fundamentals: mt yt (m* y*), t t pt p* and pt p* (it i*). In calculating t , t t f f f e tH xtH E xtH Ht 1 . this statistic, we take the econometrician’s in- formation set to be only the current and lagged Under the assumption that all the variables in It value of the fundamental in each case.9 f follow an ARIMA(q, r, s) process, q, r, s 0, To motivate our calculation of Var( xtH), let and that Ht is a subset of It that includes at least Wt be a (n 1) vector of observable variables, current and past values of ft, equation (6) in with ft a Wt. Assume that Wt follows a West (1988) shows that VAR of order d: Wt Wt Wt ... f 1 b2 f f f 1 1 2 2 Var e tH Var xtH x tI Var e . tI b2 d Wt d Wt . As b 3 1, Var(xtH f xf ) stays bounded, but tI Deﬁne (b) [I b 1 ... bd d] 1. Then (1 b )/b 3 0. It follows that for b near 1, 2 2 using equation (4), we can write the innova- f Var(etH) Var(ef ). tI f tion in x tH as: The EW theorem shows that when b is near 1, xftI ef , and xtH tI f f etH. Therefore, we can f use an estimate of Var( xtH) to measure f e tH a b Wt . f Var( xtI). A simple example may help develop intu- From the EW theorem, for b 1, we have f ition. Suppose ft ft 1 e1,t e2,t 1, where xtH a (b) Wt. e1,t and e2,t are mutually independent, indepen- Mechanically, then, we estimate an autore- dently and identically distributed, mean-zero gression (with four lags in all cases) on each processes. Assuming agents observe e1,t and e2,t measure of the fundamentals. We use estimates at time t, we can use (3) to solve and ﬁnd st ˆ (b) [I b ˆ 1 ... b4 ˆ 4] 1 and ˆ Wt to ( xf ) ft be2,t. Then, st ( tI xf ) tI ft construct xtH a ˆ (b) ˆ Wt. ˆ f b e2,t e1,t be2,t (1 b)e2,t 1. As b 3 1, st ( x f ) 3 e 1,t tI e 2,t . Note that, as in the EW theorem, when b approaches 1, s t 8 Mark Watson has pointed out to us that if Ut 0, then approaches a random walk. as the discount factor approaches 1, the long-run correlation f Now, continuing with the example, suppose between the change in xtH and the change in the exchange rate should approach 1. We do not implement this useful that Ht contains only current and lagged values observation here. of ft. Then, solving using equation (4), we ﬁnd 9 For additional empirical results, see Engel and West f f xtH ft, so xtH ft e1,t e2,t 1. We see (2004). 124 AEA PAPERS AND PROCEEDINGS MAY 2004 f TABLE 1—ESTIMATES OF Var( xtH)/Var( st) (CURRENT the fundamental is pt p* does the ratio exceed t, AND LAGGED FUNDAMENTALS ONLY IN Ht) 0.5. There are few previous studies that permit Fundamental comparison to these ﬁgures. The bounds on the m y p p* variance of s t and of s t E t 1 (s t ) of Roger Country b (m* y*) p p* i i* D. Huang (1981 p. 37) and Behzad T. Diba Canada 0.90 1.142 0.164 0.162 (1987 p. 106) use inequalities that are satis- 0.95 1.181 0.188 0.181 ﬁed by construction for b arbitrarily near 1. 0.99 1.213 0.211 0.199 1.00 1.221 0.218 0.204 Such inequalities unhelpfully guarantee val- ues greater than 1 for the ratio that we con- France 0.90 0.269 0.054 0.070 0.95 0.309 0.095 0.100 sider. Using the monetary model, West (1987 0.99 0.352 0.187 0.146 p. 70) ﬁnds a ratio of about 0.02– 0.08 for 1.00 0.365 0.233 0.163 the Deutschemark– dollar exchange rate. The Germany 0.90 0.257 0.050 0.054 present technique yields considerably higher 0.95 0.301 0.077 0.071 ﬁgures, suggesting there is rather more in the 0.99 0.349 0.127 0.095 monetary model than this previous volatility 1.00 0.364 0.148 0.103 test would suggest. Italy 0.90 0.316 0.146 0.143 We conclude that asset-market models in 0.95 0.360 0.245 0.226 which the exchange rate is expressed as a dis- 0.99 0.407 0.447 0.376 1.00 0.421 0.543 0.441 counted sum of the current and expected future values of these observed fundamentals can ac- Japan 0.90 0.364 0.039 0.020 count for a sizable fraction of the variance of 0.95 0.406 0.058 0.023 0.99 0.446 0.090 0.026 st when the discount factor is large. The EW 1.00 0.458 0.103 0.027 explanation for a random walk provides a ratio- United 0.90 0.444 0.139 0.152 nale for a substantial fraction of the movement Kingdom 0.95 0.540 0.201 0.206 in exchange rates. But there is still a role for 0.99 0.645 0.298 0.284 left-out forcing variables: perhaps money- 1.00 0.677 0.336 0.312 demand errors, a risk premium, mismeasure- ment of the fundamentals we have examined here, some other variables implied by other ˆf Table 1 reports Var( xtH)/Var( st). 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