The Out-of-Sample Failure of Empirical Exchange Rate Models: Sampling Error or Misspecification?

International Finance Discussion Papers Number 204

March 1982

TEE OUT-OF-SAMPLE FAILURE OF EMPIRICAL EXCHANGE RATE MODELS: SAMPLING ERROR OR MISSPECIFICATION?

by

Richard Meese and Kenneth Rogoff

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors.

I. Introduction!’

A companion study [Meese-Rogoff (1981)] compares the out-of-sample fit of various structural and time series exchange rate models, and finds that the random walk mode12/ performs as well as any estimated model at one to twelve month horizons for 1970's dollar/mark, dollar/pound, dollar/yen and trade-weighted dollar exchange rates 2! The structural models perform . poorly ever though their forecasts are purged of all uncertainty concerning the future paths of their explanatory variables by using actual realized values.

The present study demonstrates that the dismal performance of the structural models is not attributable to the sample distribution of the coefficient. estimates. We rule out that explanation by showing that the models (with autoregressive error terms) perform poorly at one to twelve month forecast horizons over a wide range of coefficient values. These values are based on the theoretical and empirical literature on money demand and purchasing power parity. Since the coefficent-constrained models only require estimation of the intercept terms, it is possible to look at longer forecast horizons here than in our other study. There the relative superiority of the random walk model over the structural models diminishes as the forecast horizon approaches twelve months. The present study explores the possibility that the structural models may improve on the random walk model forecasts at horizons of twelve to thirty-six months.

The main part of the paper is contained in section 3, which discusses the coefficient-constrained experiments. In section 2, vector autoregressions (VAR) are used to identify the factors that influence the exchange rate over short versus long horizons. The results from the VAR

experiments also highlight the difficulties in finding legitimate instruments

with which to estimate the structural models, thus motivating the constrained-coefficient approach of section 3. Section 4 asks which of the common building blocks of the structural models is most likely to have failed. It appears that the breakdown of empirical exchange rate equations is the international counterpart of the breakdown of money demand

specifications.

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2. Decomposing the Forecast Error Variance of the Exchange Rate at Long and Short Horizons

Before proceeding to tests of the representative structural exchange rate models, we first examine a vector autoregression consisting of the exchange rate and the explanatory variables of these models: relative money

supplies, relative outputs ,~/

relative short-teria and long-term interest rates, and trade balances. The VAR is a tool for analyzing the relative importance of the explanatory variables in exchange rate model forecasts at both short and long horizons. As a by-product, the VAR also provides information on whether the conventional exogeneity assumptions used in estimation of the structural models are appropriate.

A convenient normalization for estimation of the VAR is one in which

the contemporaneous value of each variable is regressed against lagged values

of all the variables; e.g., the exchanye rate equation is given by

(1) Sp * AyySe_y t ByaSpig + eee@GnSten + BG tid * Biakt-2 * °° Binkten

+ ite

where s; 15 the (logarithm of the) exchange rate at time t and Xj is a vector of lagged values of the other included variables (listed above). Expressing the VAR system in the fonn of equation (1) facilitates estimation, as ordinary least squares equation by equation is an efficient estimation strategy. This normalization does not, however, preclude contemporaneous

interactions between the variables, as these effects are captured in the

covariance matrix of the disturbance terns Uses The uniform lag length

n across all (seven) equations is estimated using Parzen's (1975) lag length selection criterion! We estimate the VAR model for the dollar/inark, dollar/pound, and

dollar/yen exchange rates over the floating rate period; the data consist of monthly observations for March 1973 through June 1981 (our seasonal adjustment procedures are described in the data appendix). Once having obtained the coefficient estimates, the dynamic interactions among the variables are most easily studied with the use of the moving average (MA) representation, which is derived by inverting the autoregressive (AR) representation to express each of the endogenous variables in terms of the disturbances or innovations rthe

u., in (1) for example]. Studying the MA representation is complicated

it by the fact that the disturbance terms in the MA (or AR) representation are in general contemporaneously correlated; see Sims (1980) or Fischer (1981). So in order to simulate a "typical" shock to a given variable it is necessary to recognize that the expectation of other disturbances in the system, conditional on the particular shock of interest, are usually nonzero. Unfortunately, for two correlated disturbances z;(t) and Zo(t), if the

E[z}(t)| zo(t) = 1] = a, a an arbitrary constant, it is not in general

true that ELzo(t)| z(t) = a] = 1. Because of this fact, there is no

unique way to simulate “typical” shocks to these systems of endogenous variables when contemporaneous variable interactions are present. (In cther words, when the covariance matrix of the disturbance terms is nondiagonal.)

In order to identify a typical shock to the VAR system with a particular variable, we will follow the Sims' (1980) procedure of specifying a variable ordering a priori. The variable ordering essentially specifies that the first

variable is predetermined with respect to all other variables, that the second

variable is predetermined with respect to all but the first, etc. The identification of the VAR systems is pursued in greater detail in the technical appendix.

The multi-horizon forecast error variance decompositions listed in tables 1-3 are based on a variable ordering with the logarithm of U.S. to foreign relative money supplies m=i first, followed by the logarithm of relative outputs y-y, the short-term interest differential rom Tes the long-term interest differential roe rs the U.S. and foreign trade balances T8 and TB, and the loyarithm of the dollar price of foreign currency s. In tables 4-6 the variable ordering is reversed. In the U.S.-German system, the laryest estimated contemporaneous correlation is 44% between the short and long-term interest differential equations. The other estimated contemporaneous correlations range frow 5 to 20%. These results suggest that the variable ordering is potentially important in the U.S.-German VAR system. And indeed there are some differences between the U.S.-Gernan VAR systems, regular versus reverse order, at both short and long forecast horizons. Hote that in the regular (reverse) order system, exchange rate and long-term interest rate innovations account for 78.6% (93.7%) and 12.8% (4.9%) of the one-ionth-ahead forecast error variance of the exchanye rate, and 48.1% (60.1%) and 15.4% (10.5%) of the 36-month-ahead forecast error variance of thie exchange rate. However, the significance of these differences cannot be ascertained from tables 1 and 4. We have not yet performed the requisite (expensive) stochastic Simulations to obtain estimates of the dispersion of these forecast error variance decompositions. Of course, the data necessarily contain less

information about long-run variable interactions than short-run. Similar

observations apply to the U.S.-U.K. and U.S.-Japanese VAR systems.

A second important observation to be made from tables 1-6 is that no variable appears to be exogenous to the VAR system. Abstracting froin coefficient uncertainty, an exogenous variable would manifest itself as follows: at all horizons a variable's own innovations would account for all of its forecast error variance, so there would be a one in the column corresponding to a variable's own innovation and zeros elsewhere. (Block exoyeneity is the obvious jultivariate generalization. )2/ In the U.S.-German VAR the exchange rate, relative incomes, the long-term interest differential, and tine German and U.S. trade balances all appear to have large exogenous components, since for both variable orderings and all horizons (1-36 months) own innovations in these variables explain at least 48%, 55%, 50%, 59%, and 65% of their respective forecast error variances. For the U.S.-U.K. VAR, own innovations in the exchange rate, the U.K. and U.S. trade balances, and the long-term interest differential account for most of “he forecast error variance uf these variables. In the U.S.-Japanese system, it is the exchange rate, the Japanese and U.S. trade balances, and relative incomes that have this property.

The last feature of tables 1-6 that we wish to emphasize concerns the difference between those factors which appear to explain the forecast error variance of the three bilateral exchanye rates at snort horizons (1-3 sionths) as opposed to longer horizons (1-3 years). Based on the numbers reported in these tables it is clear that own innovations in exchange rates explain a large fraction of the exchanye rate forecast error variance at one and three

month forecast horizons, while innovations in the other variables becoine

relatively more important at horizons of one and thre years. This result is

not atypical of VARs estimated on macro-economic data; see Fischer (1981). All of the features of tables 1-6 noted above suggest both (1) the

difficulty in specifying the menu of variables to include in a structural

exchange rate equation, and (2) the problems associated with finding

” legitimate instruinents with which to consistently estimate the parameters of

these mode's. The latter difficulty has led to the constrained-coefficient

methodology of the next section.

3. Predicting and Explaining the Exchange Rate Qut of Sample Using — Structural Models with Constrained Coefficients

Elsewhere [Meese-Rogoff (1981)] we employ rolling regressions t:o construct out-of-sample forecasts of the exchange rate using three structural models: a flexible-price monetary model (Frenkel-Bilson), a sticky*p”ice monetary model (Dornbusch-Frankel), and a sticky-price asset model which incorporates the trade balance (Hooper-Morton) / The fact that these structural models do not outperfonn the random walk model at horizons of one to twelve months cannot be attributed to the inherent unpredictability of the explanatory variables; this uncertainty is purged from the forecasts by using realized explanatory variable values. Still, there remains the possibility that our small-sanuple results can be attributed to poor parameter estimates rather than specification error. This possibility is especially worrisome in light of the estimated VAR models presented in the previous section. They indicate that it is difficult to find legitimate exogenous variables in the three structural exchange rate models. If this is the case, then consistent coefficient estimation becomes problematic and requires a priori knowledge of the serial correlation process of the error terms. These possible estimation problems may explain why the instrumental variables techniques implemented in our other study do not yield better results than ordinary least squares.

Here we explore a range of constrained-coefficient models and present evidence that our other results concerning one to twelve month forecast horizons cannot be explained by coefficient uncertainty. In addition, since the coefficient-constrained models do not require a significant portion of the limited floating-rate data set for estimation, we are able to look at

longer forecast horizons.

3a. The representative structural models

All three of the structural exchange rate models we consider are based on a common money demand specification, thereby allowing us to impose coefficient constraints on a consistent basis across models. The quasi-reduced form specification of each of the models is subsumed in the general

specification below:

9 _ * * * e *e (2) S = a + ay (i -m) + aa(y - y) + a,(r -r.) + ay (m - a)

S) ) *

+ a (TB - TB) +t uy

where (1° - n°) is the expected lony-tenn inflation differential, Tb and TB are the cumulated U.S. and foreign trade balances, u is a disturbance term, and the other variables are as defined in section 2 above. In (2) we have imposed the usual constraint that domestic and foreign variables affect the exchange rate with coefficients of equal but opposite sign; this constraint is relaxed in a limited number of experiments both nere and in our earlier

study .2/

We choose not to specify an ad-hoc lagged adjustinent tiechanisi in (2), preferring to model the dynamics using an autoregressive error term as described below.

All three models hypothesize first-deyree homogeneity of the exchange rate with respect to relative imoney supplies, or a; = 1. The Frenkel-Bilson, or flexible-price monetary model, formed by differencing two identical ioney demand specifications while imposing purchasing power parity (PPP), posits the

additional coefficient restrictions ay < 0, a3 > 0, aq = a5 = 0.

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The Dornbusch-Frankel, or sticky-price monetary inodel also hypothesizes that the coefficient on relative incomes a> > 0, but in contrast to the Frenkel-bilson model hypothesizes that the coefficient on the short-term interest differential a3 < O, and that the coefficient on tie long-term expected inflation differential aq > 0. The derivation of tyese coefficient restrictions is exposited in Frankel (1979). The principal theoretical difference between the Frenkel-Bilson :nodel and the Dornbusch-Frankel model is that the latter allows for short-run deviations from purchasing power parity due to sticky domestic prices. Prices adjust only gradually, in response tuo both excess demand, which depends on the terins of trade, and to secular inflation differentials [n® - n° in equation (2)]. The long-run or flexible price exchange rate § is derived in the same manner as s in the Frenkel-Bilson model except that it depends on n® - n°, which is equal to the long-run short-term interest

differential:

(3) = -a+ (m- ia) - Ply - y) + alae - 4°),

Using money demand functions of the fora (4a) m-peza- Ar, + Dy

* * * *

(4.) m-pe=a- Ar. - Dy,

and a price adjustment equation of tine fuorn

(5) (D = Day - (P= De = O(s- pt p), + (n° - 4°),

-jj-

Frankel demonstrates that augmented regressive expectations are

9/

rational :=

* + (ne = ne

(6) Seay - Se = OS - 8)

t t?

where See] is the exchange rate expected to prevail at time ttl

based on period t information. Substituting (3) into (6) for §, and also

) r

imposing uricovered interest parity by substituting r, - Vr.

for Sta] - Sz, one arrives at the quasi-reduced form of the

Dornbusch-Frankel iodel:

(7) s = at (m= in) = Oy ~ 9) ~ Eley - Fe) + pt Ade ~ 8°) OY

So in the Dornbusch-Frankel model a3, the coefficient on the short-tenna interest differential Po 7 Tes does not depend on the nominal

interest rate semi-elasticity of the demand for real balances A. Rather it depends on the negative of the inverse of 9, tne coefficient on excess demand in the price adjustment equation. The coefficient on the expected long-run inflation differential, ag, is the sui of 1/6 and i.

The Hooper-Morton trade-weighted dollar model imposes the same constraints as the Dornbusch-Frankel model, except that it allows unanticipated shocks to the U.S. trade balance to affect the PPP or long-run real level of the exchanye rate. In our bilateral version of their model, incipient trend U.S. trade balance surpluses require an appreciation of the lony-run real exchanye rate, while incipient trend foreign surpluses require a depreciation. Thus, as < 0. It should be noted that the random walk

riodel is also subsumed in the general specification (2). That model is given

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by ay = ao = a3 = aq = as = 0, and Ut = Ur] + et» where

Qe, is a white noise process.

3b. A description of the coefficient constraints

The least controversial constraint we impose is that a), the coefficient on the logarithm of relative money supplies, is unity. While we shall not consider other values for a), we do experiment with different definitions of the money supply; the reserve adjusted base, MI-B, and Mz (in conjunction with their respective foreign counterparts)

Widespread agreement is lacking on the values of the other parameters. For example, there are a range of theoretical and empirical estimates of the interest and income elasticities of money demand. The quantity theory puts the income elasticity at one, and the interest elasticity at zero. Alternatively, the Baumol (1952) - Tobin (1956) inventory theoretic approach, in its simplest form, can be used to derive an income elasticity of .5 and an interest elasticity of -.5. Taking into account integer constraints raises the income elasticity towards one and the interest elasticity towards zero; see Barro (1976). The Miller-Orr (1966) model of a firm's optimal caSh-management procedures yields an interest elasticity of -.33. Tne “ncome elasticity suggested by that model ranges from .33 to .67, depending on whether a rise in income brings a rise in the number of transactions or in the average size of transactions. The Whalen (1966) model of the precautionary demand for money also suggests an interest elasticity of ~-.33. In addition it yields an income elasticity which depends on how the size versus frequency of transactions changes as income rises, ranging from .33 to 1. Finally, we consider empirical estimates of the demand for money, for

which Goldfeld's (1973) paper is a standard reference. He estimates the

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income elasticity of money demand to be .19 in the short run, and .65 in the long run; his short-run and long-run interest elasticities are -.064 and -.23. Since the present study takes the approach of modeling the serial correlation properties of the error term rather than specifying an ad-hoc lagged adjustment mechanism, it follows that the higher long-run elasticities are more relevant for our purposes.

We are now ready to specify a complete grid of constraints for the Frenkel-Bilson model. The income elasticity constraints considered are .5, -65, .75, .85, and 1. This grid excludes the lowest ranges of income elasticities obtained in the theoretical models; we implicitly assume that income yrowth is accompanied by some yrowth in the size of transactions. The interest rate semi-elasticity constraints include -3, -4.5, -6, -7.5, and -10. The latter grid encompasses interest rate elasticity priors ranging from somevhere between -.18 and -.2] to -.60 and -.70, depending on the bilateral exchange rate. The sewi-elasticity priors are obtained by dividing the interest elasticity priors by the average prevailing level of short-terin interest rates during the sainple.

The grids of constraints for the Dornbusch-Frankel and Hooper-Morton models incorporate the same range of income elasticity and interest-rate semi-elastiicity constraints as the Frenkel-Bilson model yrid. The two sticky-price models also require the specification of a grid for 0, the speed of adjustment parameter in the goods market. We choose a ranye of constrain:s for 6 using the fact that it also represents the speed at which

deviations from the long-run real exchanye rate are damped. The grid for 6

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is based on the assumption that between 33% and 100% of today's deviation from PPP is expected to be eliminated one year hence. This range encompasses Genberg's (1978) estimates as well as those of Frankel (1979), both of whicn are based on data for Germany. Since in the Dornbusch-Frankel inodel the coefficient on relative short-term interest rates, a3, is equal to -1/8,

our grid of priors for a3 in that model includes -1, -2, and -3.. (The values for @ and therefore -1/6 are conceived on an annual basis, since the short-term interest differentials and expected long-term inflation differentials in the data set are annualized.) The coefficient on the expected long-run inflation differential aq is equal to A + 1/6, where -A

is the interest semi-elasticity of money demand. The grid of constraints for ay is 4,7, 9, 11, 13, which includes the minimum and maximum possible values of + 1/6 given the individual grids of constraints for A and 1/6. For consistency, we exclude from our overall grid for the Dornbusch-Frankel model combinations of aq and a3 such that aq - a3 is less than 3 or

greater than 10, the bounds on the yrid for 4.

The coefficients on the cumulative monthly trade balances (taken as deviations from trend) in the Hooper-Morton model are based primarily on Hooper and Horton's work. We assume that a billion dollar U.S. trade balance surplus above trend level leads, ceterus paribus, to an offsetting .3 to .5 percent appreciation of the dollar; that is, ag is -003 or -.005.

The results reported below are robust to using values of as of .01 or .J2. For simplicity, and in order to limit the size of the large grid of coefficient constraints for the Hooper-Morton model, we assune that a foreign trade balance surplus has an effect on the exchange rate of equal magnitude

but opposite siyn.

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“he final variable for which it is necessary to specify a grid of constraints is one which we logically know nothing about-- the error tena

Up. We assume that uy follows a first order autoregressive process:

(8) Up = Ub) +e = e,/(] - pl),

where @, is wnite noise and L is the lay operator. The yrid for the autoregressive parameter p is 0, .2, “4, -6, -8, and 1.0 , so both the no serial correlation case and the first-difference case are covered.-¢/ The decision to analyze only a first-order autoregressive process is tiade in part to limit the size of the parameter grids, but it is also in part because of the results of our other study. There, optimal linear combinations of structural models and very general autoregressive tiie series models are analyzed. A wide variety of optimal lag length selection criteria are used in developing the time series components of the forecasts; these criteria generally select a lag lenyth of one for the univariate models. Given the ranye of constraints we have selected, the grid for the Frenke!-Bilson model contains 150 different combinations of parameters, the grid for the Dornbusch-Frankel mode] has 330 elements, and the Hooper-tiorton

model grid has 660 elements 2/

The grid of paraneter values developed above is now used to perfomnn two basic experiments, designed to compare the structural models to the random walk model at forecast horizons of one, three, six, twelve, eighteen,

twenty-four, thirty, and thiry-six months. The "ex-post" and "ex-ante"

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forecasting experiments differ mainly in whether forecasts are generated using realized values of the explanatory variables (ex-post), or using predictions of the explanatory variables based on information available at the time of the forecast (ex-ante) 4/ The other difference is that

ex-ante forecasting begins in June 1975 while ex-post forecasting ccvers the entire sample period. The ex-ante experiment requires enough observations for first-round estimation of the VAR that generates predictions of the explanatory variables. (Because the ex-ante experiment is quite expensive to conduct, it is performed only for the Dornbusch-Frankel model.) Otherwise, the experiments are conducted in identical fashion. Constant terms corresponding to each constellation of parameter values are estimated using rolling regressions. The autoregressive component of forecasts made at time t are based on the period t error term.

The results of the ex-post forecastiny experiment are broadly characterized in table 7, where the structural model "forecasts" are compared with the random walk model forecasts on the basis of RMSE and mac .19/.16/ For each model and exchange rate, table 7 reports the shortest forecast horizon, in months, at which 0.1%, 10%, 25% and 50% of each model's parameter grid outpredicts the random walk model when realized values of the explanatory variables are used. Table 7 demonstrates that the results of Meese-Rogoff (1981) cannot be explained by parameter uncertainty. For the entire parameter yrid and for all three exchanye rates, the structural models never improve at all, much less significantly, on the random walk model in

MAE or RMSE at forecast horizons less than twelve months. llowever, at

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horizons of twelve months or more--longer than we could examine in our study based on estimated ccefficients--the RMSE and MAE of the models do sometimes improve on the random walk model. This result is tempered by the fact that the minimum RMSE or MAE coefficient configurations bounce around at different forecasting horizons. Still the percentage of the parameter grids which improve on the random walk model does increase with forecast horizon.

Overall these essentially in-sample results--in-sample, because not all coefficient configurations improve on the random walk model--must be interpreted with caution.

Table 8 presents best representative parameter values for each of the models, toyether with their corresponding RMSE and mag L// These two statistics are also given for the random walk model. At 36 months, the best representative coefficient values for the Dornbusch-Frankel and Hooper-Morton models clo about 50% better than the random walk model in RMSE and MAE; the Frenkel-Bilson model only does about 30% better.

Since the models do not forecast well at short horizons in the ex-post experiment, it is not surprising that the one model considered in the ex-ante experiment does poorly at short horizons as wel —8/ Tables 9 and 10 present results for ex-ante forecasting experiment with the Dornbusch-Frankel model. No parameterization of that model ever improves on the random walk model in MAE for horizons under 12 months; the threshold horizon is even longer when RMSE is the metric. Furthermore, for the dollar/pound and dollar/yen exchange rates, over 90% of the parameter grids fail to beat the random walk model in MAE or RMSE at any horizon. It is true, however, that

at 36 months the best representative Dornbusch-Frankel model performs almost

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as well in the ex-ante experiment as the best representative Dornbusch-Frankel model in the ex-post experiment; compare tables 8 and 10. Again, we should emphasize that the evidence presented here on the possible forecasting superiority of the structural tiodels is essentially in-samiple, since not all confiyurations of the parameter constraints improve on the random walk model.

Also reported in Table 10 are the forecasting properties of the seven-variable VAR system of section 2. This model, estimated by rolling regressions, is a true ex-ante forecaster. The VAR outforecasts the random walk model at three-year horizons for the dollar/DM rate. It does worse at one-year horizons for that exchange rate, though, and worse at all horizons for the dollar/pound and dollar/yen exchange rates. It is possible that these results can improved by imposing probabilistic priors on the VAR; see Litterman (1979). (An identified structural inodel such as the Dornbusch-Frankel] model can be thought of as a VAR with a priori

restrictions.)

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4. The Poor Performances of the Structural Models: Possible Causes

The constrained-coefficient experiments of section 3 reinforce the results of our earlier study. The selected structural models (with autoregressive error terms) fail to forecast or even explain out of sample as well as the random walk model at horizons of up to twelve months. The models do sometimes produce better forecasts than the random walk model at longer horizons, but in an unstable fashion. As noted in section 2, the limited floating rate data set necessarily contains more information about short than about lang forecast horizons.

In this section we try to trace the instability or misspecification of these empirical exchange rate equations to their building blocks, such as uncovered interest parity?’ , the particular money demand specification, the proxies for inflationary expectations, and the goods markets specifications. These building blocks are not, of course, strictly indepencient.

The assumption of uncovered interest parity has been strongly challenged by recent work on exchanye rate risk premia22/ However, while some authors find evidence of risk premia, the weight of the evidence is that the maynitudes involved are not larye. Nevertheless, volatile time-varying risk premia remain a possible explanation of the results.

The goods market specifications of the three representative structural models are relatively simple. The flexible-price Frenkel-Bilson monetary model imposes purchasing power parity, even in the short run. The sticky-price Dornbusch-Frankel monetary model allows for short-run deviations

from PPP. The Hooper-Horton model is similar except that it attempts to

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incorporate movements in the long-run PPP level of the exchange rate by assuming that these movements take place in response to unanticipated trade balance (current account) deficits or surpluses. While short-run PPP does not provide an accurate characterization of the 1970's 2 there is no strong evidence that the long-run PPP level of the exchange rate changed significantly. The results in Meese and Rogoff (1982) suggest also that although the deviations from PPP damp quite slowly, the rate at which they damp is relatively stable.

The performance of the Dornbusch-Frankel and Hooper-Morton models are potentially quite sensitive to the use of a variable other than the long-term interest differential as a proxy for the long-run expected inflation - differential. Although we did not find a proxy which yielded better results [see footnote (15)], this issue merits further attention.

However, the major problem with the structural models considered here may be the instability of the underlying money demand specifications. The recent breakdown of U.S. money demand relationships was first noted by Goldfeld (1976) and is documented extensively by Simpson and Porter (1980). Conventional empirical money demand specifications such as equations (4) of section 3 have consistently underpredicted U.S. Ml velocity since mid-1974. For this reason, the present study uses MI-B, for which the systematic bias over the sample period is much smaller, and the new definition of M2, for which the bias is negligible. But equations (4) still fail to predict these agyregates or the reserve-adjusted base with any notable deyree of precision.

As reported above, our exchanye rate experiment results are not sensitive to.._

at

which of these agyregates (together with their respective foreign _

counterparts) we employ.

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Whether or not money demand instability and/or misspecification is responsible for the exchange rate results, it is certainly true that the conventional money demand equation does not work well when expressed in terms of U.S. ininus foreign variables. That equation [(4(a) minus 4(b)] fails Chow (1960) tests for the stability of the intercept tern at four different breaks in the sample. It also fails Goldfeld-Quandt (1965) tests of homoscedastic disturbance terms over the same sample breaks 24/

To investigate the possibility that our results are generated solely ‘by money demand instability in the U.S., we perforined ex-post forecasting experiments using the Dornbusch-Frankel model on the pound/mark , pound/yen and yen/mark cross exchange rates. For the case of the yen/mark rate, we found coefficient values for which the model pulled even with the random walk model as early as six months. But the subsequent improvement at longer horizons never exceeded 30%. (The pound/mark and pound/yen cross-rate results are no better than the results for the various dollar exchange rates.)

In sum, money demand instability is an important potential explanation of our results, but further work is needed to demonstrate that time-varying risk premia, volatile long-run real exchange rates, or poor measurenent of

inflationary expectations are not the dominant problems.

5. Conclusions

The unimpressive out-of-sample performance of the Frenkel-Bilson, Dornbusch-Frankel and Hooper-Morton empirical exchange rate models cannot be attributed to inconsistent or inefficient parameter estimates. These models fail to yield any improvement over the random walk model in mean absclute or root mean squared error one to twelve iwunths out of sample for a broed range of theoretically plausible coefficient values even when autoregressive error terns are introduced. Thus it is unlikely that tore efficient estimation techniques, such as imposing all the cross-equation rational expectations restrictions, will yield parameter estimates which do better.42/ The coefficient-constrained models do prevail at longer horizons but in an unstable fashion; the best coefficient values bounce around dependiny on the forecast horizon.&4/

Wnile the breakdown of empirical exchange rate models may be due to volatile time-varying risk premia, volatile long-run real exchange rates, or poor weasurement of inflationary expectations, the main problems appear to lie in their money demand specifications. If this is the case, then the same improvements which resuscitate domestic empirical money demand equations should lead to great improvements in empirical excilange rate equatiors as

well.

~-23- Footnotes

the authors have benefited from the coments and suggestions of

Jeffrey Frankel, Robert Flood, Robert Hodrick, Peter Hooper, Peter Isard, and Julio Rotemberg. We are indebted to Julie Withers and Tamara McKann for excellent research assistance. This paper represents the views of the authors and sitould not be interpreted as reflecting the views of the Board of

Governors of the Federal Reserve Systei or other members of its staff.

2/the structural inodels are described here in section 3 below. It

should be noted that all the todels considered are derivatives of the monetary or asset approach in that they specify real money demand at hone and abruad as a function of real income, short-term interest rates and possibly wealth. The random walk iodel predicts that today's exchange rate will

obtain at all future dates.

in a study of the dollar/pound rate, Hacche and Townend (1981) use different methods to arrive at a similar conclusion; that the models do a very poor job of explaining the dollar/pound rate. The present study

examines the three bilateral dollar rates and also cross-rates.

4 the asSuuption that U.S. and foreign variables enter exchange rate

equation systens with equal but opposite signs is relaxed later in a limited

number of experiments on tne structural models. Economizing on variables in

the otherwise highly paraneterized VAR systeus is quite important, so we only

estimate the VAR models with the relative variables.

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5/

~Parzen's (1975) criterion selects an order g* which iinimizes

yz!

N -1 _CAT(2) = trace(= . -V ), f=, 2, ...L, T 1 J g

i! 1 2

J where N is the number of variables in the VAR, T is sample size, L is the waximal order considered, and V5 is an estimate (adjusted for deyrees of freedom) of the covariance matrix of disturbances for the inodel with 2 lags of each variable. Asymptotically, the order selected is never less than the true order, assuming the true order is finite.

S/n Meese-Rogoff (1981), the block exogeneity assumptions of the

Dornbusch-Frankel model are formally tested.

see Bilson (1978, 1979), Frenkel (1976), Dornbusch (1976b), Frankel

(1979, 1981), and Hooper and Morton (1982). The identification of particular empirical models with authors who contributed significantly to tneir development follows one conventional nomenclature. It is relevant to note, however, that several of these saime authors have analyzed more than one of the three models. For example, Frenkel (1981b) discusses a sticky-;rice model, and emphasizes that the flexible-price model is a limiting approximation which is applicable in a highly inflationary environment. Dornbusch (1976a) examines a flexible-price monetary model with traded and nontraded yoods.

8/aynes and Stone (1981) suggest .that a problei with the

representative structural models we consider is the restriction of equal but opposite coefficients on domestic and foreiyn variables. In Meese-Rogoff

(1981), relaxing this restriction by lettiny certain domestic and foreign

~25-

variables--incomes, money supplies and cumulated trade balances--enter eyuation (2) separately yielded no forecasting improvement. Here we tried separating the incomes and trade balances, but again found no forecasting

improvement.

Wie is also straightforward to show that deviations from purchasing

power parity caused by monetary shocks are expected to damp at rate 0.

10/Frankel uses both long-term interest differentials and past inflation differentials as proxies for 1 - ne, the flexible price or

long-run expected inflation differential.

Ue oy the dollar/pound rate we use M3's, since there is no data on Ne for the U.K. The results presented later in this section are based on M1 (M1-B) data. However, we obtain very similar results witi the different

monetary aggregates.

12/5 oy the Dornbusch-Frankel model, we also experimented with a range of constraints on p concentrated between .8 and 1. The lowest end of this

range produced tne best results.

13/ Recall that the Dornbusch-Frankel and Hooper-Morton imodel grids exclude combinations of ay and ay incompatible with the range of constraints specified for the interest rate semi-elasticity of real ioney

deinand A.

Wty the absence of greater knowledge about the true underlying structure than is inherent in equation (2), it is not possible to take advantaye of any currelation between the error term and the explanatory

variables in generating the ex-post forecasts. Such correlation is likely,

~26-

though, given the endogeneity of the explanatory variables indicated by the VAR's in section 2. In fact, if the variance of the error tena is larye and its (unknown) covariance with the relevant linear combination of the explanatory variables (the "fundaientals") negative, our ex-post forecaster need not dominate optimal ex-ante forecasters. In this perverse case, knowing that the realized fundamentals suyyest a higner exchange rate means

that you should guess a lower excnanye rate.

L/the results for the Dornbusch-Frankel and Hooper-Morton models

reported in Tables 7-10 are obtained using long-teri interest differentials as a proxy for expected long-run inflation differentials. It is important to recognize that these models are potentially quite sensitive to this variable. However, uSiny instead current-period inflation differentials, a moving averaye of past inflation differentials, or future inflation differentials, yields yualitatively similar results in the ex-post forecasting experi ijents (Tables 7 and 8). We did not try these other proxies in the expensive ex-ante experiments (Tables 9 and 10).

1/i ot k = 1, ..., 36 denote tne forecast step, Ny the total number

of forecasts in the projection period for which the actual value A(t) is known, F(t) denote the forecast value, and let forecasting beyin in period

(t+1). Define

Nv -1

Mean absolute error = kK {F(t +s +k) - A(t +s +k) | /My s=0

Nol 2. 91/2 Root mean square error = ¢ x LF(t +s +k) - A(t +s + k)J in, s=0

-2/-

Our use Of mean absolute error covers problems that might arise if, as sugyested by Westerfield (1977), exchange rate changes are drawn from a stable Paretian distribution with infinite variance. The mean errors of the models (not reported) are small relative to mean absolute errors in almost all cases where p > .2, indicating that the structural models are not siinply

systematically over or underpredicting.

/ The "best" representative set of parameter constraints for each

model in Table 8 is chosen in an ad hoc fashion as the one which comes in first (also ahead of the random walk model) at the yreatest number of horizons. The maximum improvements over the random walk model in MAE or RMSE at 36 month horizons exhibited by these representative inodels are as larye as those exhibited by any other parameter confiyurations.

18/vhile only one model is considered in the ex-ante experiment, note

that all three models yield qualitatively equivalent results for the ex-post experiuent. Also, since the Dornbusch-Frankel model predicts that the exchange rate will return in the lony run to its flexible-price or Frenkel-Bilson model value, we should a priori expect the perforiance of both

models at lony forecast horizons to be quite similar.

Wty later versions of the Hooper-Morton model this assumption is

relaxed.

20/ See for example Hansen and Hodrick (1980a,b), Cumby and Ubstfeld (1981), Hakkio (1981), Tryon (1979), Bilson (1981), Meese and Singleton (1980), or Geweke and Feige (1979). Hansen and Hodrick study this issue in

tneir paper contained in this volume.

-28-

21/tsard (1976), Genbery (1978), and Frenkel (1981a,b) provide

evidence on this point. In this context, it would be useful to remind the reader that our identification of particular models with particular authors oversimplifies the history, development, and application of these models.

[See footnote (7)].

22/The breaks in the sample at which these stability tests are

conducted are chosen arbitrarily and correspond to (1) June 1974 - the start

of the mature float, (2) November 1976 - the approximate sample midpoint, (3) November 1978 - the dollar support program, and (4) October 1979 - the change in Federal Reserve operating procedures. The tests were conducted using all

parameter configurations with the grids for (A, @) reported in section 3b.

23/ cee Driskell and Sheffrin (1981) or Glaessner (1982). These more sophisticated statistical techiques may provide superior expectations

proxies, however.

24/6 the true structural model were known, and combined with an accurate representation of the serial correlation process of the error tern,

then such a model would produce mininum MSE forecasts at all norizons.

-29-

References

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Bilson, John F.0. 1978. Rational expectations and the exchange rate. In J. Frenkel and H. Johnson, eds., The economics of exchange rates. Reading: Addison-Wesley Press. . 1979. Tne deutsche mark/dollar rate -- A monetary analysis. In K. Brunner and A. Meltzer, eds., Policies for employment, prices and exchange rates. Carnegie-Rochester conference volume 11. Amsterdaia:

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. 1981. The "speculative efficiency" hypothesis. Journal of

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Chow, Gregory C. 1960. Tests of equality between sets of coefficients in

two linear reyressions. Econometrica 28: 591-605.

Cumby, Robert and Obstfeld, Maurice. 1931. A note on exchange-rate expectations and nominal interest differentials. The Journal of

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Dornbusch, Rudiyer. 1976a. The tineory of flexible exchange rate regimes and facroeconomic policy. Scandinavian Journal of Economics 78: 255-75. . 1976b. Expectations and exchange rate dynamics.

Journat of Political Economy 84: 1161-76.

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Driskell, Robert A. and Sheffrin, Steven M. 1981. On the mark: Comment.

American Economic Review 71: 1068-74.

Fischer, stanley. 1981. Relative price variability and inflation in the United States and Germany. Forthcoming in the European Economic

Review.

Frankel, Jeffrey A. 1979. Un the mark: A theory of floating exchanye rates based on real interest differentials. American Economic %eview

69: 610-22.

- 1981. "On the inark: reply." American Economic Review 71: 1075-82. Frenkel, Jacob A. 1976. A monetary approach to the exchange rate:

Doctrinal aspects and enipirical evidence. Scandanavian Journal of

Economics 78: 200-24.

- 198la. The collapse of purchasing power parities during

the 1970's. European Economic Review 16: 145-65.

- 1981b. Flexible exchange rates, prices, and the role of news: Lessons from the 1970's. Journal of Political Economy 39:

905-705.

Genbery, llans. 1978. Purchasiny power parity under fixed and flexible

exchange rates. Journal of International Economics 8: 247-76.

Geweke, John and Feige, Edward. 197S. Soe joint tests of the efficiency of markets for forward foreign exchange. Review of Econoniics and _

Statistics 61: 334-41.

-3]-

Glaessner, Thomas. 1982. Theoretical and empirical essays on spot and forward exchange rate determination. Ph.D Diss., University of Virginia.

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homuscedasticity. Journal of the American Statistical Association 60: 539-47.

Goldfeld, Stephen N. 1973. The demand for money revisited. In Artnur Ckun and George Perry, eds., Brookings papers on economic activity 3. hashington: The Brookings Institution.

. 1976. The case of the missing money. In Arthur Okun and George Perry, eds., Brookings papers on economic activity 3. Washington: The Brookings Institution.

Hacche, Graham and Townend, John. 1981. Exchange rates and inonetary policy: Modeling sterling's effective exchange rate, 1972-80. In W. Eltis and P. Sinclair, eds., The money supply and the excianye vate. Oxford: Oxford University Press.

Hakkio, Craig S. 1981. Expectations and the forward exchange rate.

International Economic. Review 22: 663-78.

Hansen, Lars P. and Hodrick, Robert J. 1980a. Forward exchange rates as optimal predictions of future spot rates: an econometric analysis. Journal of Political Economy 88: 829-53.

. 1980b. Speculation and forward foreign exchange rates: Some tests of cross-currency restrictions. Mimeoyraph.

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Haynes, Stephen E. and Stone, Joe A. 1981. On the mark: Comment.

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Hooper, Peter and Morton, John E. 1982. Fluctuations in the dollar:

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Economic Review 67: 942-48.

Meese, Richard A. and Sinyleton, Kenneth J. 1980. Rational expectations, | risk premia, and the market for spot and forward exchange. International Finance Discussion Papers no. 165. Board of Governors of the Federal Reserve System. Meese, Richard A. and Rogoff, Kenneth S. 1981. Empirical exchange rate models of the seventies: Are any fit to survive? International _

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Reserve Systen.

Miller, Merton and Orr, Daniel. 1966. A model of the demand for money by firms. Quarterly Journal of Economics 80: 413-35.

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Sinpson, Thomas D. and Porter, Richard D. 1980. Some issues involving the definition and interpretation of the monetary aggregates. In

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Sims, Christopher A. 1980. Macroeconomics and reality. Econometrica 48:

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Tobin, James. 1956. The interest-elasticity of transactions demand for

cash, Review of Economics and Statistics 38: 241-47.

Tryon, Ralph. 1979. Testing for rational expectations in foreign exchange

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Westerfield, Janice M. 1977. An examination of foreign exchange risk under fixed and floating rate regimes. Journal of International

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-34-

Technical Appendix

In this appendix we describe the triangularization of the VAR system used in section 2 to analyze the dynanic effects of an innovation to a particular variable. First suppose the t-th observation of the VAR is

represented by

where LIy - A(L)J is a matrix polynowial in the lag operator L, Yt is the N x 1 vector of variables in the system, E(u) = 0, and Var(uy) = V, positive definite. Using the Cholesky factorization V = WW', where W is lower triangular, we can transform (Al) to the system

1

-] ye _ (A2) WwW (I, - A(L) Jy, = W Up = eps

where E(e,) = 9 and VAR(e¢) = Iy, the order N identity matrix. Since wr! is also lower triangular, the system (A2) is recursive as described in the text. The moving averaye representation of (A2) is

(A3) yy, = LIy - A(L)I We,,

aid in this expression the contemporaneous value of the first component of

€ enters all N equations, the contemporaneous value of the second component of e enters the last N-1 equations, etc. Because the decomposition of V is not unique, studying the effect of the uncorrelated innovations €¢ On ve

will depend on the variable ordering unless V is diagonal, i.e. unless the

-35-

system (2) has na contemporaneous interactions among variables.

Expression (A3) is also used to construct the variance decompositions uf tables 1-6. Since all components of e have unit variance, the variance of Vit (the i-th element of the vector Yt) is tne sun of squares of the elements in the i-th row of [Iy - A(L) Tw. ‘The percentage of the forecast error variance of Vit explained by the j-th innovation eit (the j-th element in the vector e;) is calculated as the ratio of the sum

of squares of the (i, j) element of Lly- A(L) Iw to the variance of

Vit’

Data Appendix

The data set consists of seasonally unadjusted monthly observations over the period March 1973 to June 1981. All the raw data are seasonally adjusted using dummy variables (the results reported in the text are

described in Meese-Rogoff (1981)). insensitive to the use of more sophisticated seasonal adjustment procedures/\ In the U.K. data set, the spot and forward exchange rates, short-term interest rate, and long-term bond rate are always drawn from the same date. Because daily bond series are not readily available for Japan and Germany, only the exchange and interest rates correspond in these data sets. All other series are monthly data, and all data are taken from publicly available sources.

The bilateral data sets draw exchange rate data from identical

sources, as follows: One, Six, and Twelve-Month Forward Exchange Rates

Data Source: Data Resources, Inc. data base.

Series: One, six, and twelve-month forward bid rates in U.S.

dollars per local currency unit.

Description: Daily data based on 10:00 a.m. opening New York market

rates. Three-Month Forward and Spot Exchange Rates Data Source: Federal Reserve Board data base. Series: Three-month forward and spot bid rates in U.S. dollars

per local currency unit.

Description: Daily data based on 12:00 noon New York market rates.

~37-

Sources of the other data series are discussed below by country. German Bond Yields Data Source: Deutsche Bundesbank, Statistical Supplement to the

Monthly Reports of the Deutsche Bundesbank, Series 2, Securities Statistics, Table 7b.

Series: Yields in percent per annum on fully taxed outstanding bonds of the Federal Republic of Germany.

Description: Monthly data. Data are calculated as averages of four bank-week return dates including the end-of-month yield of the preceding month.

Consumer Prices

Data Source: Deutsche Bundesbank, Monthly Report of the Deustche Bundesbank, Table VIII-7.

Series: Total cost of living index for all households.

Description: Monthly index.

Industrial Production

Data Source: O.E.C.D., Main Economic Indicators.

Series: Total industrial production.

Description: Monthly index.

‘Interesit Rates (Three-Month)

Data Source: Frankfurter Allegemeine Zeitung.

Series: "Geldmarkt Vierteljahresgeld" in percent per annum. (3-month interbank rate).

Description: Daily data.

~38-

Monetary Base (Reserve-Adjusted)

Data Source:

Series:

Description:

Money Supplies

Data Source:

Series:

Description:

Deutsche Bundesbank, Monthly Report of the Deutsche

Bundesbank, Table II-1 (components of the unadjusted

monetary base) and Table IV (average reserve ratio). The unadjusted base is’.calculated in millions of DM as total Bundesbank assets less the reserve adjustment balancing asset, foreign and domestic public authority deposits, SDR allocations, EMCF gold contributions, liquidity paper liabilities, and "other" liabilities. The reserve adjustment is made by multiplying the unadjusted base statistic by [.631 + 3.2/total average reserve ratio] where .631 = the currency percentage

of the unadjusted base in the base period (January 1980) and 3.2 = [1-.631] [the total average reserve ratio in the base period].

Monthly data. Data for components of the unadjusted base refer to the last banking day of the month. The

average reserve ratio is a monthly average statistic.

Deutsche Bundesbank, Monthly Report of the Deustche

Bundesbank, Table I-2.

Money stock Ml and money stock M2 in millions of DM. Monthly data. Data refer to the last banking day of

the month.

Adjustment:

Trade Balance Data Source: Series: Description:

Japan

Bond Yields

Data Source:

Series:

Description:

Consumer Prices Data Source: Series:

Description:

wet gee

~39-

A break in the series, caused by the introduction of

oye ol

a new method of computation, occurs in December 1973. The 1973 statistics are adjusted using the ratio of

the new to the old statistic for December 1973.

O.E.C.D., Main Economic Indicators. Trade balance (f.0.b. - c.i.f.) in billions of DM.

Monthly data.

Data prior to 1981 are taken from Bank of Japan, Economic Statistics Monthly, Table 71(2). 1981 data are taken

from Planning and Research Department, Tokyo Stock Exchange, Monthly Statistics Report, Table 8-1.

Yields in percent per annum on listed government bonds (Tokyo Stock Exchange).

Monthly data.

Data refer to the last banking day of the

month.

Bank of Japan, Economic Statistics Monthly, Table 119(1).

General consumer price index for all Japan.

Monthly index.

Industrial Production

Data Source: Series:

Description:

0.E.C.D., Main Economic Indicators. Total industrial production.

Monthly index.

~40-

Interest Rates (Three-Month)

Data Source:

Series:

Description: Money Supplies

Data Source:

Series:

Description:

Trade Balance Data Source: Series: Description: United Kingdom Bond Yields Data Source:

Series:

Description: Consumer Prices

Data Source:

Series:

Description:

Federal Reserve Board data base. "Over two-month ends" bill discount rate (Tokyo Stock Exchange) in percent per annum.

Daily data based on Reuters quotes.

Bank of Japan, Economic Statistics Monthly, Table 4.

Ml and M2+CD in 100 million yen. Monthly data. Data refer to the last banking day of the

month.

O.E.C.D., Main Economic Indicators. Trade Balance (f.0.b. - c.i.f.) in billions of yen.

Monthly data.

Financial Times "British funds, Undated, War loans 34s" in percent per annum.

Daily data.

Department of Employment, Employment Gazette, Table 6.4. General index of retail prices, all items.

Monthly index.

-41-

Industrial Production

Data Source: O.E.C.D., Main Economic Indicators.

Series: Total industrial production.

Description: Monthly data.

Interest Rates (Three-Month)

Data Source: Financial Times.

Series: Three-month local authority deposits (London money rates) in percent per annum.

Description: Daily data.

Monetary Base (Reserve-Adjusted) and Money Supplies

Tata Source: Bank of England, Quarterly Bulletin, Table 1 (monetary base components) and Table II (money supplies).

Series: Money stock Ml and money stock sterling M3 in millions of pounds. The reserve-adjusted monetary base is calculated in millions of pounds as total currency in circulation plus bankers' deposits.

Description: Monthly data. Data refer to the third Wednesday of the month (second in December).

Trade Balance

Data Source: O.E.C.D., Main Economic Indicators.

Series: Trade balance (f.o.b. - c.i.f.) in millions of pounds.

Description: Monthly data.

United States

With the exception of the trade balance statistics, all data are taken

from the Federal Reserve Board data base. Many of these series are published

in the Federal Reserve Bulletin, and all are available to the public.

Bond Yields Series: Description: Consumer Prices Series:

Description:

~42-

Government bonds with at least 10 years to maturity.

Daily data.

Consumer Price Index.

Monthly index.

Industrial Production

Series:

Description:

Total industrial production.

Monthly index.

Interest Rates (Three-month)

Series:

Description:

Treasury bill rates.

Daily data.

Monetary Base (Reserve-Adjusted) and Money Supplies

Series: Description: Trade Balance Data Through 1978:

Data Source:

Series:

Description:

Reserve-adjusted monetary base, M1-B, M2, and M3.

Weekly Wednesday data.

Department of Commerce, Highlights of U.S. Export and

Import Trade, Exports Table E-1; Imports Table I-1. Domestic and foreign exports, excluding Department of Defense shipments, in millions of $ on a F.A.S. value basis; General imports in millions of $ on a Customs Valuation basis changing to a F.A.S. basis in 1974.

Monthly data.

-43-

Adjustment: 1973 statistics are adjusted to a F.A.S. vaiue basis using the 1974 average ratio of Customs Valuation to F.A.S. value. 1979-1981 Data:

Data Source: Department of Commerce, Summary of U.S. Export and Import Merchandise Trade, December 1980 (advance statistics for Highlights of U.S. Export and Import Trade), Exports Table 3; Imports Table 5.

Series: Total domestic exports, excluding Department of Defense grant-aid, in millions of $ on a F.A.S. value basis; General imports in millions of $ on a F.A.S. value basis.

Description: Monthly data.

-44- Table 1

Unconstrained U.S.-German VAR, March 1973-June 1981, regular variable order

Proportions of forecast error variance k months ahead attributable to each innovation—

Innovation in: Forecast error

i i k nm y r ie TB TB variance in n-m y-y. Fos: Ot, s * mm 1.843 .005 .132 .004 .001 .904 .011 3.629 .003 259 029 .019 =.940 = .020 12.293 .007 295 .100 .035 .249 .016 36.252 .007 . 268 .122 .037 = .254.— 061 * y-y 1 006 942 — .000 035 .011 .905~=—.001 3.006 856 .002 .010 .008 .927 .001 12.014 646 004 223 .029 .970 .014 36 =. 015 622 004 224 .031 .972 .032 * r-r 1 007 024 .838 .073 .030 .907 .021 ss 3 011 .017 641 .080 .097. .124 ~=—.929 12 008 .016 491 081 .091 .282 .030 36 = 011 017 469 .088 .088 .276 .051 * ryt, 1.024 .000 161 .781 .002 .002 .030 3.056 054 .126 .683 .007. -.028 ~=— 044 12.152 099 | 101 524 .036 .048 .040 36.147 .096 .108 .510 .039 = .052—s «048 TB 1 .080. .111 009 .005 .769 001.025 3.078 144 011 .005 .721 = -.014 026 12 ~—.068 .167 .010 .063 .617. 024.050 36 ~—-.068 .163 .010 | 071 .596 .028 .063 TB 1 021 034 .063 °° .008 .042 .332 .001 3.024 061 047 012 .036 .313 .006 12. = 023 101 037 061 .032.-.676 069 36 ~—-.028 096 040 .069 .030 .649 .086 s 1.041 011 022 .128 .005 .006 .786 3.077 .008 022 .163 .032. .010 »=—-.687 12.155 034 .018 .125 .064 .050 .553 36155 .033 .030 154 .058 .090 .481 al

Notes for Columns of the table correspond to innovations in a particular variable tables 1-6: for the specified forecast horizon k = 1, 3, 12, and 36. ‘The rows add

to one because the total forecast error variance attributable to each variable on the left of the table is allocated across the seven innovations. Abstracting from coefficient uncertainty, an exogenous variable would manifest itself as follows: At all horizons a variable's own innovations would account for all of its forecast error variance, so there would be

a one in the column corresponding to a variable's own innovation and

zeros elsewhere.

-45-- Table 2

Unconstrained U.S.-Japan VAR, March 1973-June 1981, regular variable order

Proportions of forecast error variance k months ahead attributable to each innovation

Innovation in:

Forecast error * x k x * variance in k m-m y-y Tots rot TB TB s

* m-m 1 -943 -003 -001 002 012 024 -014 3 . 866 -045 -004 010 024 021 029 12 551 298 021 -009 045 039 038 36 424 319 .070 .013 -048 -045 -080

* y-y 1 -044 929 -009 001 009 -000 -008 3 038 915 009 .008 005 020 004 12 -111 637 086 019 042 -090 014 36 -120 531 111 019 .048 072 099

* Tok. 1 142 -076 .770 -006 002 -001 003 3 .183 092 581 022 -030 -058 034 12 .110 -058 - 284 -058 .069 348 073 36 115 054 238 -076 -120 .320 -076

* ToT 1 128 014 294 -520 .000 003 -041 3 252 039 207 342 -024 022 .113 12 219 -046 108 -184 -079 092 172 36 224 -071 104 172 . 186 085 157 TB 1 -O11 -016 -032 -000 904 .005 .031 3 010 022 059 O11 858 -008 -031 12 -011 032 -078 017 726 .097 -040 36 012 .036 .079 024 -671 -111 -068

~

TB 1 011 -036 031 -064 030 823 005 3 .010 945 .023 095 -040 . 780 -007 12 015 .O51 935 105 .103 -626 066 36 019 -041 039 .093 -130 -523 154 s 1 O11 007 007 -020 -O71 055 828 3 016 004 -004 014 139 -080 741 12 .016 -004 -003 -057 285 127 507 36 026 006 003 .060 313 .138 455

-46- Table 3

Unconstrained U.S.-U.K. VAR, March 1973-June 1981, regular variable order

Proportions of forecast error variance k months ahead attributable to each innovation

Innovation in: Forecast error

i i k mn y r r B B variance in n-m y-y To's ty T T

* n-m 1 £911 .023 059 001 006 001 3 .789 .041 142 004 004 019 12 398 058 292 052 004 .140 36 270 .032 200 .076 028 148

* y-y 1 .009 .820 039 .012 026 .027 3 034 .600 029 .087 059 .073 12 .058 431 019 .138 066 .183 36 058 407 .021 145 .068 177 r-r 1 .047 .018 880 034 .001 019 ss 3 .034 .012 830 .031 .001 089 12 .136 018 641 050 000 144 36 159 017 600 .060 002 .139

* rot, 1 £027 007 209 725 003 018 3 026 .032 .190 637 005 066 12 018 .091 149 528 025 050 36 022 086 .160 496 036 £051 TB 1 .013 .041 .113 016 £755 .018 3 014 .050 126 069 634 £057 12 .018 052 .117 .078 .511 —-.082 36 036 .047 105 074 420 6115

x ,

TB 1 028 069 052 002 .008 .830 3 054 .106 049 .038 034 .710 12 .118 .128 044 068 030 596 36 .119 .116 052 079 033 555 s 1 000 021 014 026 .007 005 3 .001 .018 008 038 008 035 12 012 006 002 .088 048 .168

36 -066 -007 .058 .099 059 -150

-000 001 057 ~245

-066 123 -104 124

-000 .003 -009 024

-O11 -043 -138 -148

044 .049 143 - 203

-010 .008 -015 045

-926 891 -675 - 560

-47- Table 4

Unconstrained U.S.-German VAR, March 1973-June 1981, reverse variable order

Proportions of forecast error variance k months ahead attributable to each innovation

Innovation in:

Forecast error * + R * x variance in k m-m y-y ToT. Toth TB TB * mn-m 1 703 017 111 .047 042 .017. 062 3 522 011 138 191 .025 -058 .054 12 281 024 081 .298 .020 .264 .032 36 243 022 .067 .287 026 .264 .089 *

y-y 1 011 868 007 .018 068 .020 .009 3 .009 £779 .014 .068 .057 .059 = .013 12 £014 580 .049 .048 .075 .094 .096 36 .015 558 050 146 .076 .095 .059 r-t 1 £011 .016 .522 350 013 .085 .002 ss 3 008 013 343 .333 083 .115 004 12 .014 .018 232 279 .085 .353 019 36 .015 018 222 272 084 .343 047 r- 1 .000 .000 .002 963 .008 .002 .024 LoL 3 .013 068 .023 848 .014 .017. 026 12 .080 145 014 663 048 .023. .027 36 081 140 £015 647 .051 .028 .038 TB 1 011 036 .000 006 918 .005 .023 3 014 .053 .008 006 .873 .022 023 12 017 .070 .024 032 .757 .032 068 36 018 .069 025. 036 .733 .036 .083 TB 1 001 014 .001 .019 .028 .932 .004 3 003 .025 .009 016 .023 .906 .017 12 003 050 .034 .023 024 .750 117 36 008 047 .035 .028 .024 .721. «137 s 1 .001 005 001 .049 004 .003. .937 3 013 007 001 068 £045 .007. .859 12 056 .013 .001 060 024 .054 .690 36 .073 013 002 105 116 .091 =.601

Forecast error variance in

* m-m

TB

~48- Table 5

Unconstrained U.S.-Japan VAR, March 1973-June 1981 reverse variable order

Proportions of forecast error variance k months ahead attributable to each innovation

Innovation in:

k * * * .m-m y-y a ae rot, TB TB 1 -749 - 106 .028 -060 -013 -036 3 - 686 -178 -020 -039 -034 -024 12. -394 396 -023 045 064 -049 36 271 - 469 045 - 066 -051 -051 1 -000 -898 -058 -002 -015 .019 3 -003 857 -050 -016 014 044 12 014 693 -073 -088 -016 -092 36 014 -658 -068 .090 -028 -O71 1 -O11 -001 -505 417 -016 -049 3 -056 -001 -503 274 022 124 12 -046 -017 308 134 -052 -320 36 052 -020 -273 e117 -107 -302 1 -009 -003 -000 -826 -012 122 3 134 002 -017 -599 -026 .118 12 -151 .029 -032 -304 -133 -105 36 -193 -068 -035 . 283 -148 .097 1 -006 -016 -005 -007 842 .009 3 -006 -017 041 -006 -800 .010 12 -009 .017 072 -010 -672 114 36 -010 -029 075 -012 -622 .131 1 -012 -008 -000 -001 -033 .928 3 -008 .013 -008 .003 -046 . 906 12 -017 -035 -058 -006 -119 1739 36 -019 -038 -053 -012 -131 .601 1 -001 -000 -002 -020 -000 001 3 004 -000 -004 014 -024 012 12 -009 001 024 .017 -137 »042 36 -016 002 -027 .016 2175 088

-010 -019 -024 -045

-009 015 -022 -071

001 -020 124 - 130

-026 .109 -246 -226

e115 -119 -105 -120

-018 -016 -026 -146

-976 942 739 -676

-49- Table 6

Unconstrained U.S.-U.K. VAR, March 1973-June 1981, reverse variable order

Proportions of forecast error variance k months ahead attributable to each innovation

Innovation in:

Forecast error * * ke x x variance in k n-m y-y a tT ry TB TB s * n-m 1 .749 .106 .028 .060 .013 .036 .010 3 .686 .178 020 .039 .034 .024 .019 12 394 .396 .023 .049 .064 £049 024 36 .271 .470 .045 .066 £051 £051 = .045 * y-y 1 .000 .898 .058 .002 £015 .019 +=.009 3 .003 .857 .050 .016 .014 .044 .015 12 .014 "693 .073 .088 .016 .092. .022 36 .014 .658 .068 .090 .028 .071 .072 * rot. 1 .011 .001 505 .417 .016 £049. 001 3 .056 .001 503 .274 .022 .024 .020 12 .046 .017 .308 134 .052 6320 .124 36 .052 .020 .273 117 .107 .302 .130 r- 1 .009 .003 .000 826 (012.122 ~©.026 LoL 3 134 .002 .017 .594 .026 .118 .109 12 151 .029 .032 304 133 .105 = .246 36 .143 .068 .035 283 .148 .097 .226 TB 1 .006 .016 .005 .007 842 £009 «115 3 .006 .017 .041 .006 .800 .010 .119 12 .009 .014 .072 .010 .672 6114 -.105 36 .010 .029 .075 .012 622 £131 =.120 na 1 .012 .008 .000 .001 033. -.928 ~=—«.018 3 .008 .013 .008 .003 046 .906 .016 12 .017 .035 .058 .006 119 .739 .026 36 .019 .038 .053 .012 131 .601 .146 8 1 .001 .000 | .002 .020 .000 .001 .976 3 .004 .001 .004 .014 .024 .012 .942 12 .009 .001 .024 .017 137 .072 739 36 .016 .002 .027 .016 175 .088 .676

-50-

Table 7

Shortest forecast horizon (in months) for which at least x% of each model's parameter grid improves on the random walk model in MAE/RMSE

when realized values of the explanatory variables are used.

Exchange rate: . $/DM S/E $/Yen Metric: MAE RMSE MAE RMSE MAE RMSE Model Threshold Months ahead

0-1% 24 30 18 24 12 12

Frenkel-Bilson 10% 30 30 18 24 18 18 (Grid size = 150) 25% 30 30 . 24 30 24 24 50% 36 36 30 36 36 30

0-17 12 18 18 18 12 12

Dornbusch-Frankel model 10% 18 18 24 24 12 12 (Grid size = 330) 25% 30 30 30 36 12 12 50% - - - - 24 18

0-1% 12 18 18 18 12 12

Hooper-Morton model 10% 18 18 24 24 12 12 (Grid size = 660) 25% 30 30 30 36 | 12 18 50% - - - - 24 18

MAE is mean absolute error, RMSE is root mean square error. The text contains

a description of the parameter grids. To read table 7: Consider the entry in the second row, first column (30). The earliest forecast horizon at which at least 10% of the monetary model parameter grid can improve on the random walk model in

MAE for the $/DM rate is 30 months out.

when realized values of the explanatory variables are used,

Model

(with best parameter configurati

Random Walk model

Frenkel-Bil

(ays a,» a,

Dornbusch-F model

(ay> ay» ay,

Hooper-Mort model

-51-

Table 8

Comparing the random walk model and the structural models

(with their best representative parameter configurations)

Horizon

ons)

12 36

son 1

» Pp)

rankel

12 36

> aes ce)

on 1 3 12 36

$/DM MAE —-—sORMSE 2.4 3.2 4.8 6.2 9.4 10.9 18.1 21.0 (-.5, 4.5, .4) 9.1 11.4 11.5 14.2 12.2 15.2 12.6 17.0 (-.85, -l, 6, 5.5 6.9 8.1 9.7 8.8 10.8 8.2 10.5

.005, 0) 8.3 10.4 8.8 11.0 9.2 11.6 9.3. 11.6

$/E MAE RMSE 2.9 2.5 3.2 5.1 9.8 11.5 23.4 25.4 (-1, 3, .8) 4.2 6.1 8.7 11.1 13.5 16.6 15.5 18.8 -4) (-.5, -1, 4, 0) 8.1 10.0 8.6 10.5 10.4 12.3 8.3 10.0 (-.5, -1, 4, .005, 0)

4.0 10.0 8.5 10.5 10.0 12.0 10.5 12.5

$/Yen MAE RMSE 2.1: 3.0 4.2 5.77 10.6 13.8 19.4 23.3 (-.5, 4.5, .8) 4.5 6.4 7.9 11.5 9.7 13.3 10.2 14.5 (-.5, -l, 9, .8) 4.4 8.4. 7.4 9.4 7.0 8.5 8.8 10.2 (-1, -1, 9, .003, 8) 4.9 6.7 8.3 10.9 8.8 11.7 9.3 12.2

MAE (mean absolute: error) and RMSE (root mean squared error) are approximately

in percentage terms, since "forecasts" are for the logarithm of the exchange

rate.

Forecasts are compared over the period March 1973-June 1981.

-52-

Table 9

Shortest forecast horizon (in months) for which at least x% of the

Dornbusch-Frankel model's parameter grid improves on the random walk model

in MAE/RMSE when predicted values of the explanatory variables are used,2/ $/DM $/& $/Yen MAE RMSE MAE RMSE MAE RMSE Threshold Months ahead 0-1% 12 24 30 36 12 18 Dornbusch-Frankel model 10% - 12 24 - - - - (Grid size = 330) 25% 36 36 - - - - 50% - - - - - -

al the description of table 9 is essentially the same as given for Table 8. However, in contrast to the experiment of Table 8, in'which realized values of the explanatory variables are used to generate forecasts of the exchange rate, table 9 is based on an experiment in which the explanatory variables

are forecast with a VAR.

-53- Table 10 Comparing the random walk model, a VAR model estimated by roliing

regressions, and the best representative Dornbusch-Frankel model when

predicted values of the explanatory variables are used.

Model $/DM $/& $/Yen Horizon MAE RMSE MAE RMSE MAE RMSE

Random Walk model 1 2.2 3.0 2.0 2.6: 2.3 3.2 3 4.0 5.2 4.1 5.2 4.7 6.2 12 10.1 11.7 10.1 11.5 13.2 16.1 36 24.2 26.2 18.8 21.2 24.8 28.1 {a,> Ags Bays 0) (-5, -1, 4, 1.0) (-.5, -l1, 4, 0) (-5., -3, ll, Dornbusch-Frankel 1 2.4 3.2 14.0 17.3 2.9 4.2 model 3 4.6 5.7 14.7 18.0 6.6 8.6 (with best 12 7.3 10.9 17.5 20.1 12.4 16.0 parameter con- 36 12.4 19.1 9.0 10.8 17.0 18.7

figuration)

Unconstrained

VAR model 1 5.2. 6.3 5.4 6.3 4.9 6.4 3 7.9. 9.5 8.0 9.6 7.4 9.5 12 11.1 13.2 17.3 19.3 15.9 19.6 36 16.9 18:5 39.5 44.8 37.5 40.6

MAE (mean absolute error) and RMSE (root mean squared error) are approximately in percentage terms, since forecasts are for the logarithm of the exchange rate.

Forecasts are compared over the period June 1975-June 1981.