- Email: [email protected]
Nonparametric exchange rate prediction?
Journal of International
Economics 28 (1990) 315-332. North-Holland
Francis X. DIEBOLD* University of Pennsylvania, Philadelphia, PA 19104-6297,
USA
James A. NASON University of British Columbia, Vancouver, British Columbia V6T I W5, Canada
Received June 1989, revised version received August 1989 Conditional heteroskedasticity is frequently found in the prediction errors of linear exchange rate models. It is not clear whether such conditional heteroskedasticity is a characteristic of the true data-generating process, or whether it indicates misspecification associated with linear conditional-mean representations. We address this issue by estimating nonparametrically the conditional-mean functions of ten major nominal dollar spot rates, 1973-1987, which are used to produce in-sample and out-of-sample nonparametric forecasts. Our findings bode poorly for recent conjectures that exchange rates contain nonlinearities exploitable for enhanced point prediction.
1. Introduction It is widely agreed that a variety of high-frequency asset returns are well described as linearly unpredictable, conditionally heteroskedastic, and unconditionally leptokurtic. Documentation of linear unpredictability may be traced at least to early work on efficient markets, such as Cootner (1964) and Fama (1965); similarly, leptokurtosis has been appreciated at least since Mandelbrot (1963). The early writers were also aware of the apparent occurrence of volatility clustering in asset returns, and the work of Engle (1982) provided a tool for its formal study. It is now agreed that many time series of asset returns, while approximately uncorrelated, are not temporally independent; dependence arises through persistence in the conditional variance and perhaps in other conditional moments. *Diebold gratefully acknowledges financial support from the National Science Foundation and the University of Pennsylvania Research Foundation. An earlier version of this paper was presented at the 1988 North American Winter meetings of the Econometric Society, New York. We thank the participants there, as well as an anonymous referee, Buz Brock, Dee Dechert, Blake LeBaron, Andy Rose, and seminar participants at Boston College, Carnegie-Mellon, Wisconsin, Washington, LSU, and Stonybrook for useful comments. We especially thank William S. Cleveland, who generously provided his software, and Roberto Sella, who provided outstanding research assistance. 22- 1996/90/%3.50 c] 1990,
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F.X. Diebold and J.A. Nason, Nonparametric
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rate prediction?
All of these results are manifest in exchange rates, which are the focus of the present study. The well-known work of Meese and ogoff (1983a, b) provides graphic illustration of the failure of a variety of economic models to outperform a simple random walk in out-of-sample predi:tion.’ Extensive reviews of linear unpredictability and leptokurtosis in exci:ange rates are contained in Westerfield (1977), Boothe and Glassman (1987), and Diebold and Nerlove (1989a). Conditional heteroskedasticity, in the form of ARCH and related effects, has also been repeatedly documented in exchange rates. Diebold (1988), for example, examines seven nominal dollar spot rates and finds little linear predictability but strong ARCH effects in all of them. A number of explanations have been advanced for the leptokurtosis and volatility clustering, as well as reduction of excess kurtosis under temporal aggregation. The phenomena may be jointly explained, for example, by subordinating exchange rates to a process dictating information arrival. The seminal work of Clark (1973) shows that subordination to an i.i.8. information-arrival process produces leptokurtic returns. As mentioned by Diebold (1986) and formali: Y! by Gallant, Hsieh and Tauchen (1988), Clark’s analysis may be extended by allowing for persistence in the information-arrival process, which produces ARCH-like movements in higher-order conditional moments. Finally, Diebold (1988) uses central limit theorems for dependent observations to show that, under quite general conditions (not requiring, for example, existence of the fourth unconditional moment), the unconditionally fat-tailed behavior associated with stationary ARCH effects diminishes with temporal aggregation. If the above characterization of exchange rate dynamics (linear conditional mean with nonlinearities working through the conditional variance) is correct, then the nonlinearities cannot be exploited to generate improved point predictions relative to linear models. It is not clear, however, that the ARCH effects are structural, i.e. that they are a characteristic of the true data-generating process (DGP). Instead, ARCH may indicate misspecitication, serving as a proxy for neglected nonlinearities in the conditional mean.2 A finding of significant conditional-mean nonlinearity would be important for both theoretical and empirical work: theoretically, a substantial challenge to our understanding of asset-price dynamics would be posed, and empirically, a source of improved point prediction relative to linear models would be provided. Interestingly, recent empirical and theoretical results are consistent with
‘Their candidate models include a flexible price monetary model, a sticky price monetary model, a sticky price monetary model with current account effects, six univariate time series models, a vector autoregressive model, and the forward rate. 2For an illustration of the difficulties involved in separating conditional-mean from conditional-varia.nce dynamics, see Weiss (1986), who discusses ARCH and bihnearity.
F.X. Diebold and J.A. Nason, Nonparametric
exchange
rate prediction?
3 tIr
the conjecture that nonlinearities may be present in asset-return conditional means. The empirical results may be categorized into two groups: (1) those using ideas from the theory of stochastic nonlinear time series, and (2) those using ideas from the theory of deterministic chaotic systems. In the nonlinear time series area, a number of studies, including Domowitz and Hakkio (1985), Hinich and Patterson (1985, 1987), Weiss (1986), Engle, Lillien and Robins (1987), Diebold and Pauly tl988), and others, appear to detect statistically significant nonlinearity in conditional means of various asset prkes and other economic aggregates. Recent work on regime switching, including Flood and Garber (1983), Engel and Hamilton (1988), and Froot and Obstfeld (1989), is also squarely in the nonlinear tradition. Similar results have been obtained in the chaos literature, using tests based on estimated Lyaponov exponents and correlation dimensions, as developed in Brock, Dechert and Scheinkman (1987), inter alia. Scheinkman and LeBaron (1989), for example, find strong evidence of nonlinearity in common stock returns and suggest that it could be exploited for improved point prediction. Similarly, Gallant, Hsieh and Tauchen (1988) and Hsieh (1989) report evidence of residual nonlinearity in exchange rates, after controlling for conditional heteroskedasticity. T6:ese empirical results are provocative, because they challenge us to take seriously the possible existence of nonlinear conditional-mean dynamics in asset prices. A number of recent theoretical developments are beginning, implicitly or explicitly, to address this challenge. Sims (1984), for example, shows that general-equilibrium asset-pricing models imply martingale asset-price behavior only at arbitrarily short horizons. Thus, economic theory cannot rule out the possibility of nonlinear dependence in conditional means (as well as in higher-order conditional moments) of asset returns. Moreover, substantial progress has been made in general equilibrium models with time-varying risk, such as Abel (1988), Hodrick (1987), Baldwin and Lyons (1988), and Nason ( 1988). In summary, there appears to be strong evidence, consistent with rigorous economic theory, that important nonlinearities may be operative in exchange rate determination. Upon further consideration, however, it becomes clear that the literature is not in satisfactory condition, owing to a puzzle that immediately arises: Why is it that while statistically significant rejections of linearity in exchange rates routinely occur, no nonlinear model has been found that can significantly outperform even the simplest linear model in ay be operative, a out-of-sample forecasting? Because a number urse, is that the number of explanations may be offered. nonlinearities present may be in even-ordered conditional moments, and -sample nonlinearities therefore are not useful for point prediction. d may cause various such as outliers and structural shifts may be linearity tests to reject, while nevertheless being of n!c use for out-of-sa
F.X. Diebold and J.A. Nason, Nonparametric
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exchange
rate prediction?
forecasting. Third, very slight conditional-mean nonlinearities rAght be truly present and be detectable with large datasets, while nevertheless yielding negligible ex ante forecast improvement.3 inally, even if conditional-mean nonlinearities are present and are important, the overwhelming variety of plausible candidate nonlinear models makes determination of a good approximation to the DGP a difficult task. The seemingly large variety of parametric nonlinear models that have received attention lately (e.g. bilinear, threshold, and exponential autoregressive) is in fact a very small subset of the class of plausible nonlinear DGPs. In this paper we contribute to a resolution of this puzzling behavior of exchange rates by estimating conditional-mean functions nonparametrically. By so doing, we avoid the parametric model-selection problem; the class of potential models entertained is expanded greatly. In section 2 we discuss various aspects of nonparametric functional estimation, and the locally weighted regression procedure, which we use extensively, is highlighted. Section 3 contains empirical results; in particular, both in-sample LWR fits and out-of-sample LWR forecasts are compared to those arising from linear models. Section 4 concludes.
etric
iction
Nonparametric techniques may be used for estimation of a variety of densities and econometric functionals, including regression functions, first and higher-order derivatives of regression functions, conditional-variance functions, hazard and survival functions, etc.4 We shall generallv be concerned with nonparametric estimation of conditional expectation, or regression, functions,
which we use for nonparametric prediction? This is achieved by (explicit or implicit) substitution of nonparametric estimates of the underlying joint and marginal densities into (1). In our dynamic models, the stochastic conditioning vector x is composed of lagged dependent variables: x, = {Yt--l,***, %
other
Yt-,I*
words, significance
(2) of nonlinearity
does not necessarily imply its economic
importance.
4For surveys of various aspects of r;onparametric and semiparametric estimation, see Ullah (1 bvious from the context, we use lower-case letters for both random variables and their realizations, and we use f to denote all probability density functions.
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
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e shall work with the very general nonlinear autoregressive structure:
t=l
, . . . ,1: so that (1) may be rewritten more specifically as: wYt+llYt9*Y
Yt-p+l)=SYt+lf(Yt+llYtmYt-p+lb-lYt+l
The regression function estimates may be obtained by a variety of interrelated nonparametric methods, including kernel, series, and nearest-neighbor (NN) techniques, consistency results for which have ken obtained in time series environments by Robinson (1983), Gallant and Nychka (1987), and Yakowitz (1987), respectively. In this study we make use of a NN technique, known as locally-weighted regression (LWR). NN methods proceed by estimating g(x), at an arbitrary point X=X* in p-dimensional Euclidean space, via a weight function: T
ii(
c W&t)Yts
t=l
where w&)= l/kT if x, is one of the kT nearest neighbors of x*, and w&t) = 0 otherwise. R estimator, as proposed by Cleveland (1979) and refined by Cleveland and evlin (1988) and Clevelan (1988), is an important generalization f the NN estimator. Lik estimator, LWR fits the surface at a point X* as a function of the y values corresponding to the kT nearest neighbors of x*. Unlike 9 ho ther does not take &u*) as a simple average of those y value fitted value from a regression surface. This corresponds to a simple average only in the very unlikely case that the constant term is the sole regressor with explanatory power. We now discuss the procedur estimate of the surface at a point constant such that O
nearest
nei
OTS
F.X. Diebold and J.A. Nason, Nonparametric
320
excha
these xf, x;, . . . , x& Thus, XT is closest to x*, x; is second closest to x so on. Let &a, b) measuie Euclidean distance; then I(x*, x&J is the Euclidean distance from x* to its k,th closest neighbor:
I(X*,X&.)=
E
1 l/2
C
(Xk*Tj-xi*)2
j=l
l
(6)
Form the neighborhood weight function: u,(x,,x*,x&J= W(x,, x*)/m*,
x&)19
(7)
where C( .) is the tricube function: for r.44, otherwise.
’
(8)
The value of the regression surface at x* is then computed as: j”
z&x*)
=x*//j,
(9)
where i v,(y,-~j/?)~
pl=argmin
[ t=1
1 .
(10)
R procedure, exactly as described above, is used in our subsequent empirical work. 0bviously, ii reflects a number of judgmental decisions, such as use of the Euclidean norm and tricube neighborhood weighting, as well as locally linear (as opposed to higher-order, such as quadratic) fitting. The Euclidean norm has obvious geometric appeal, as does the tricube weight function, which produces a smooth, gradual decline in weight with distance from x*. Locally linear fitting is also highly reasonable (and computationally le) in the present context.’ f greater interest is the choice of <, which determines the number of the degree of smoothing. Consistency of nearest neighbors used, and he NN estimators (and hence L ) requires that the number of nearest neighbors used go to infinity with sample size. but at a slower rate, i.e. lim kr=OO, T+CQ
'See Clevelan
lim (k,/T) = 0. T+GO
evlin and Grosse (1988) for further discussion.
(11)
F.X. Diebold and J.A. Nason, Nonpararnetrrc exchange r
This implicitly creates a ‘window’ whose width be sample size goes to infinity, but at a slower rate. In t window nevertheless contains progressively more neigh reduced along with variance. Similar issues arise in kernel and series estimation. In window width corresponds to the bandwidth, which mus size but at a slower rate. the series case, the windo (inversely) to the number ‘ncluded series terms; ag that the truncation point increases with sample size, While the asymptotic mechancs of LWR estimators kernel and series estimators, ;t is interesting to note properties are likely to be superior to those of kernel Series estimators produce global, as opposed to local, is likely to make them inflexible in all but the la estimators do produce local approximations, but the constant, as opposed to locally linear, which may int in finite samples. It is interesting to note that the earlier-discussed L selecting the number of nearest neighbors does n regularity condition for consistency. In any finite however, this is of no consequence, since there exists a Ta, a c 1, which dses satisfy the regularity condition an an identical number of nearest neighbors. For exam 800, the c T rule with t = 0.5 selects approximately 400 does the T” rule with a =0.9.*
redict ion?
321
smaller as the ay the shrinking s, so that bias is
ilar to those of series estimators.
substantial bias
l
3. Empirical analysis We study ten nominal New York interbank dollar spot rates, 12.30 p.m., from the first week of 1973 (3 January) through the thirty-eighth week of 1987 (23 September). Sample size is therefore equal to 769. The rates are Canadian Dollar (CD), French Franc (FF), German Lira (LIR), Japanese Ytln (YEN), Swiss Franc (SF), Belgian Franc (BF), Danish Kroner (DK) and Dutch Gui measured in cents per unit of foreign currency and were obtained from the Federal Reserve Board database. Throughout, we use S, as notation for a generic exchange rate at time t. Our interest centers on percent exchange rate changes, A Ins,, on which we have 768 o nd week of 1973 through thirty-eighth week of 1987); st additional benefit of avoiding theoretical *Moreover, in the empirical work that follows, we explore a wide range of c values.
322
F.X. Diebold and J.A. Nason, Nonparavnetric exchange rate prediction? Table 1 In-sample analysis, p = 1.
;r).lO 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2.00 3.00 10.00 RW
CD
FF
DM
LIR
YEN
SF
BP
DG
DK
BF
2.7095 3.6230 2.6906 3.6260 2.6900 3.6379 2.6923 3.6429 2.6904 3.6418 2.6848 3.6407 2.6786 3.6419 2.6728 3.6445 2.6606 .---3.6409 2.6645 3.6557 2.6925 3.6712 2.7089 3.6820 2.7305 3.6964 2.9402 3.8568
1.9321 0.9407 1.9384 0.9328 1.9379 0.9350 1.9364 0.9375 1.9336 0.9386 1.9308 0.9398 1.9262 0.9419 1.9168 0.9427 1.9055 0.9472 1.9307 0.9714 1.9421 0.9757 1.9453 0.9769 1.9486 0.9780 2.0736 1.0174
2.1849 1.0192 2.1857 1.0118 2.1845 1.0133 2.1787 1.0158 2.1737 1.0179 2.1674 1.0189 2.1572 1.0193 2.1445 1.0194 2.1315 1.0206 2.1488 1.0393 2.1864 1.0489 2.2029 1.0538 2.2230 1.0599 2.3174 1.0879
1.6824 0.8469 1.6748 0.8411 1.6731 0.8402 1.6715 0.8390 -1.6712 0.8408 1.6694 0.8429 1.6653 0.8443 1.6571 0.8468 1.6501 0.8535 1.6980 0.8833 1.7189 0.8900 1.7286 0.8935 1.7409 0.8979 1.8046 0.9117
1.7298 0.8512 1.7310 0.8379 1.7291 0.8326 1.7311 0.8316 1.7312 0.8328 1.7288 0.8357 1.7218 0.8385 1.7034 0.8397 1.6719 0.8388 1.6663 -0.8546 1.6794 0.8575 1.6858 0.8592 1.6938 0.8612 1.9367 0.9153
2.7994 1.1669 2.7898 1.1713 2.7886 1.1737 2.7850 1.1719 2.7778 1.1700 2.7653 1.1683 2.7477 1.1680 2.7245 1.1684 2.6974 1.1693 2.6769 1.1838 2.7086 1.1921 2.7233 1.1956 2.7410 1.1997 2.9265 1.2348
1.9205 0.9329 1.9200 0.9269 1.9170 0.9242 1.9143 0.9230 1.9115 0.9237 1.9074 0.9250 1.8974 0.9256 1.8841 0.9266 1.8669 ---0.9274 1.8717 0.9491 1.8892 0.9556 1.8975 0.9584 1.9073 0.9616 1.9741 0.9819
1.9462 0.9725 1.9398 0.9680 1.9332 0.9674 1.9292 0.9684 1.9221 0.9674 1.9135 0.9665 1.9026 0.9666 1.8929 0.9680 1.8755 0.9700 1.8755 0.9902 1.8890 0.9962 1.8944 0.9983 1.9003 1.0004 2.0519 1.0500
3.3442 1.0139 3.3383 1.0047 -3.3331 1.0065 3.3292 1.0069 3.3226 1.0062 3.3156 1.0070 3.3082 1.0077 3.2983 1.0083 3.2899 1.0087 3.2905 1.0190 3.2974 1.0218 3.2998 1.0227 3.3025 1.0238 3.4965 1.0903
1.8685 0.9372 1.8517 0.9343 1.8481 0.9343 1.8444 0.9355 1.8361 0.9370 1.8292 0.9399 1.8225 0.9423 1.8085 0.9444 1.7937 0.9450 1.8063 0.9629 1.8200 0.9681 1.8242 0.9697 1.8286 0.9713 2.0176 1.0343
Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are x lo’, except for the CD, which is x 1 .O! MAPEs are x lo*, except for the CD, which is x 103. Smallest MSPE and MAPE for each currency are underlined.
of nonstationary nonparametric regression functions, as well as numerical problems with highly collinear regressors.g For each exchange rate we examine both in-sample ‘fit’ and out-of-sample predictive performance of the LWR nonparametric conditional-mean estimator. Observations 6 through 701 are used for in-sample analysis, while observations 702 through 768 are reserved for out-of-sample forecast comparison. Our out-of-sample forecasts are completely ex ante, using LWR estimates formed recursively in real time, using only information actually available. The in-sample results for p = 1, 3 and 5 appear in tables 1, 2 and 3, respectively. e perform a sensitivity analysis with respect to c, exploring a ‘Furthermore, as pointed out by Meese and Rogoff (1983a,b) the use of logs eliminates prediction problems arising from knsen’s inequality. For a discussion of issues related to unit iebold and Nerlove (1989b).
RX. Diebold and LA. Nason, Nonparametric exchange rate prediction?
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Table 2 In-sample analysis, p = 3. 5 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 la0 2.00 3.00 10.00 RW
CD
FF
D&i
LIR
YEN
SF
BP
DG
DK
BF
2.8388 3.7433 2.7539 3.6637 2.7174 3.6316 2.6894 3.6160 2.6629 3.6072 2.6378 3.5998 2.6114 3.5905 2.5782 3.5769 2.5341 3.5589 2.5437 3.5812 2.5961 3.6166 2.6204 3.6336 2.6502 3.6525 2.9402 3.8568
1.9753 0.9338 1.9291 0.9275 1.9073 0.9236 1.8854 0.9199 1.8596 0.9153 1.8286 0.9108 1.7972 0.9063 1.7589 0.9013
2.1892 0.9990 2.0951 0.9886 2.0613 0.9847 2.0417 0.9824 2.0153 0.9794 1.9893 0.9771 1.9670 0.9751 1.9441 0.9728
1.6725 0.8450 1.6455 0.8390 1.6395 0.8362 1.6343 0.8345 1.6263 0.8329 1.6151 0.8313 1.6010 0.8306 1.5896 0.8325
1.7711 0.8799 1.7262 0.8604 1.7149 0.8524 1.7088 0.8489 1.7019 0.8466 1.6818 0.8419 1.6501 0.8349 1.6114 0.82M 1.5760 0.8196 ---1.5791 0.8271 1.5987 0.8337 1.6065 0.8363 1.6155 0.8393 1.9367 0.9153
2.8763 1.1642 2.8388 1.1655 2.7990 1.1634 2.7652 1.1620 2.7340 1.1601 2.6952 1.1567 2.6574 1.1529 2.6249 1.1504 2.5904 1.1493 2.5764 1.1607 2.6201 1.1728 2.6391 1.1771 2.6607 1.1818 2.9265 1.2348
2.2728 0.9975 2.1893 0.9848 2.1431 0.9739 2.1126 0.9669 2.0797 0.9625 2.0416 0.9583 2.0036 0.9526 1.9662 0.9482 1.9316 0.9463 1.9327 0.9644 1.9753 0.9742 1.9949 0.9785 2.0174 0.9836 1.9741 0.9819
1.9677 0.9613 1.9236 0.9592 1.8986 0.9561 1.8804 0.9545 1.8572 0.9516 1.8330 0.9483 1.8093 0.9452 1.7734 0.9403 1.7321 0.9351 1.7146 0.9478 1.7371 0.9569 1.7454 0.9598 1.7544 0.9626 2.0519 1.osOO
3.4300 1.0200 3.3713 1.0102 3.3451 1.0051 3.3251 1.0016 3.3043 0.9991 3.2788 0.9966 3.2548 0.9937 3.2319 0.9911 3.209?
1.8790 0.9377 1.8240 0.9335 1.8060 0.9336 1.7948 0.9343 1.7810 0.9338 1.7617 0.9322 1.7363 0.9278 1.7008 0.9211 1.6631 0.9139 1.6392 0.9201 1.6648 0.9291 1.6733 0.9318 1.6824 0.9346 2.0176 1.0343
1.5804 0.8361 1.7326 0.9242 1.7422 0.9271 1.7521 0.9299 2.0736 1.0174
2.0139 1.0077 2.0251 1.0108 2.0367 1.0140 2.3174 1.0879
1.6117 0.8570 1.6444 0.8682 1.6582 0.8729 1.6749 0.8785 1.8046 0.9117
0.9888 3.1941 -0.9947 3.2131 1.0008 3.2195 1.0027 3.2264 l.OQ47 3.4965 1.0903
Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are x 104, except for the CD, which is x 10s. MAPEs are x 102, except for the CD, which is x 103. Smallest MSPE and MAPE for each currency are underlined.
wide range of values from 0.10 through 10.O.loIn table 1 (p= 1) we find that mean squared prediction error (MSPE) is minimized for all currencies at either { =0.9 or c = 1.0. The MSPE associated with the optimal e choice is always lower, often substantially so, than the random Mean absolute prediction error (MAPE) is minimized generally smaller { values ranging from 0.1 to 0.6. Like associated with the optimal { choice is always smaller tha MAPE.” Similar results are obtained in table 2 (p= 3) and table 3 (p= 5): loNote that t = 1 does not correspond to a linear autoregression, because the observations are still weighted. Rather, our algorithm is such that as c approaches infinity, the !inear autoregression emerges. (In practice, t = 10 produces an approximately linear autor “It is interesting to note that the MAPE-optimal t values are smaller than optimal counterparts, reflecting t willingness to accept h er variance in return bias under MAPE as opposed to
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rate prediction?
Table 3 In-sample analysis, p = 5.
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
1.00 2.00 3.00 10.00 RW
LIR
CD
FF
DM
3.2387 3.9357 3.0368 3.7960 2.9215 3.7221 2.8357 3.6693 2.7724 3.6371 2.7133 3.6116 2.6589 3.5932 2.6056 3.5766 2.5511 3.5617 my2.5333 -3.5891 2.5874 3.6239 2.6107 3.6384 2.6390 3.6575 2.9402 3.8568
2.1601 0.9764 1.9977 0.9489 1.9425 0.9359 1.8983 0.9249 1.8602 0.9175 1.8194 0.9103 1.7851 0.9047 1.7479 0.8998 1.7120 0.8969 1.7005 0.9138 1.7312 0.9428 1.7420 0.9279 1.7535 0.93 10 2.0736 1.0174
2.3172 1.7498 1.0377 0.8762 2.1687 1.6525 1.0187 0.8606 2.1218 1.6190 1.OOSO 0.8487 2.0836 1.5856 0.9995 0.8361 2.0433 1.5577 0.992 1 0.8254 2.0043 1.5409 0.9848 0.8201 1.9668 1.5337 0.978 1 0.8200 1.9328 1.5372 0.9726 0.8240 1.9070 1.5408 0.9701 0.8287 1.9544 1.6059 0.9899 0.8563 2.0027 1.6410 1.0031 0.8682 2.0177 1.6548 1.0077 0.8728 2.0339 1.6707 1.0124 0.8752 2.3174 1.8046 1.0879 0.9117
YEN
SF
BP
DG
DK
BF
2.0348 0.9522 1.8473 0.9011 1.7864 0.8785 1.7430 0.8634 1.7010 0.85 16 1.6586 0.8399 1.6153 0.8282 1.5745 0.8 175 1.5424 0.8102 1.5623 0.8236 1.5854 0.8307 1.5946 0.8336 1.6054 0.8370 1.9367 0.9153
3.1651 1.2282 2.9725 1.1976 2.8847 1.1859 2.8227 1.1787 2.7698 1.1723 2.7157 1.1660 2.6658 1.1583 2.6223 1.1513 2.582 J. 1.1471 2.5638 1.1560 2.623 1 1.1713 2.6457 1.1765 2.6703 1.1822 2.9265 1.2348
2.4893 1.0314 2.3119 1.OO53 2.2363 0.9928 2.1787 0.9808 2.1174 0.9690 2.0631 0.9606 2.0208 0.9559 1.9817 0.9522 1.9461 0.9482 -1.9605 0.9614 2.0085 0.9742 2.0263 0.9792 2.0464 0.9849 1.9741 0.9819
2.1542 0.9913 2.0008 0.9734 1.9402 0.965 1 1.8947 0.9573 1.8554 0.95 17 1.8183 0.9460 1.7794 0.9400 1.7409 0.9333 1.7019 0.9275 1.6932 0.9389 1.7265 0.9523 1.7388 0.9566 1.7523 0.96 12 2.0519 1.0500
3.5454 1.0348 3.4173 1.0188 3.3744 1.0122 3.3377 1.0074 3.3026 1.0028 3.2692 0.9968 3.2408 0.9916 3.2132 0.9864 3.1853 0.9818 3.1805 0.9888 3.2074 0.9980 3.2166 1.0010 3.2263 1.0042 3.4965 1.0903
2.1070 0.9927 1.9375 0.9655 1.8672 0.9530 1.8166 0.9441 1.7765 0.9365 1.7388 0.9298 1.7031 0.9222 1.6715 0.9147 1.6398 0.9085 1.6276 0.9157 1.6583 0.9268 1.6688 0.9304 1.6802 0.9342 2.0176 1.0343
Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are x 104, except for the CD, which is x 10’. MAPEs are x 102, except for the CD, which is x 103. Smallest MSPE and MAPE for each curvncy are underlined.
nonparametric conditional-mean estimation leads to substantial reductions in loss, whether measured by MSPE or MAPE, relative to the random walk. Both MSPE- and MAPE-optimal { are almost always at 0.9 or 1.0. Finally, it is interesting to note the general tendency for both MSPE and MAPE to decrease and then level off as more lags are included. Although this need not happen (different nearest neighbors are used, in general, for different p), it is intuitively reasonable by analogy to the fact that inclusion of additional regressors in an OLS regression must lower (or, at worst, leave unchanged) the sum of squared residuals. We now turn to the one-step-ahead out-of-sample analysis. Again, we estimate nonparametric autoregressions of order 1 through 5 using the LWR procedure, with values of the smoothing parameter < ranging from 0.1 through 10.0 for each p, corresponding to the use of roughly 70 nearest neighbors (with neighborhood weighting) through ‘all’ nearest neighbc s
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
325
Table 4 One-step-ahead out-of-sample forecast analysis, p = 1. 5
CD
FF
DM
LIR
YEN
SF
0.10
BP
DG
DK
2.7119 2.0596 2.1725 2.1474 1.9619 3.0868 1.5960 2.2309 2.1069 3.8859 1.1435 1.1572 1.1895 1.0444 1.3759 1.0067 1.1638 1.1575 0.20 2.5998 2.0606 2.2107 2.0917 1.9457 2.9372 1.5877 2.1936 2.1059 3.8181 1.1424 1.1656 1.1662 1.0365 1.3506 1.0005 1.1660 1.1635 0.30 2.5515 2.0301 2.2238 2.0509 1.9351 2.8978 1.5960 2.1926 2.1209 3.7797 1.1348 1.1679 1.1546 1.0287 1.3420 1.0027 1.1660 1.1635 0.40 2.5373 2.0097 2.2135 2.03 17 1.9235 2.8984 1.6034 2.1740 2.1181 3.7603 1.1300 1.1680 1.1490 1.0252 1.3422 1.0070 1.1613 1.1657 2.5322 1.9944 2.1985 2.0197 1.9104 2.9007 1.6074 2.1547 2.1163 0.50 3.7525 1.1276 1.1669 1.1460 1.0226 1.3439 1.0082 1.1577 1.1665 0.60 2.5282 1.9795 2.1856 2.0080 1.9031 2.9092 1.6076 2.1467 2.1119 3.7518 1.1240 1.1650 1.1443 1.0204 1.3466 1.0079 1.1568 1.1655 2.5269 1.9653 2.1711 1.9885 1.8989 2.9142 1.6078 2.1407 2.1091 0.70 3.7514 1.1211 1.1621 1.1423 1.0191 1.3488 1.0081 1.1566 1.1649 2.528 1 1.9473 2.1609 1.9646 1.8975 2.9069 1.6123 2.1343 2.1097 0.80 3.7471 1.1184 1.1603 1.1401 1.0204 1.3493 1.0090 1.1567 1.1661 0.90 2.5392 1.9217 2.1466 1.9360 1.9070 2.8911 1.6155 2.1221 2.1097 3.7459 1.1147 1.1586 1.1367 1.0253 1.3497 1.0092 1.1565 1.1677 2.5552 1.8533 2.0900 1.8717 1.9327 2.8195 1.5968 2.0785 2.0890 1.00 3.7549 1.1054 1.1621 1.1220 1.0369 1.3424 1.0006 1.1588 1.1683 2.5458 1.8430 2.0725 1.8536 1.9271 2.7962 1.5933 2.0717 2.0809 2.00 3.7517 1.1030 1.1608 1.1147 1.0344 1.3385 0.9985 1.1588 1.1654 2.5423 1.8430 2.0725 1.8536 1.9271 2.7962 1.5933 2.0717 2.0809 3.00 3.7501 1.1030 1.1608 1.1165 1.0352 1.3398 0.9852 1.1591 1.1661 2.5387 1.8377 2.0557 1.8360 1.9228 2.7777 1.5888 2.0655 2.0756 10.00 3.7480 1.1014 1.1596 1.1126 1.0334 1.3371 0.9990 1.1584 1.1647 RW -2.4368 -1.7398 2.0431 1.7402 1.9433 2.7569 1.5483 1.9463 2.0183 1.1514 1.1253 -3.6497 -1.0703 1.1617 1.0847 1.0218 1.3407 0.9889 -Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are except for the C&D,which is x 10‘. MAPEs are x 102, except for the CD, which is Smallest MSPE and MAPE for each currency are underlined.
BF 2.423 1 1.2047 2.4041 1.2038 2.3805 1.2038 2.3543 1.1997 2.3351 1.1951 2.3 193 1.1951 2.3028 1.1930 2.2789 1.1883 2.2632 1.1867 2.2167 1.1833 2.2036 1.1798 2.2036 1.1807 2.1952 1.1788 2.0754 1.1506 x 104, x 10’.
(with no neighborhood weighting). For each c and each p, out-of-sample forecasts are computed by recursively estimating E(d In S,02 1LI In S7O1,.. . , A 1nS 702 _,,), and then E(d In S70&4 In §To2,. . . , A In S703_J and so forth, in real time. This is continued until the sample is exhausted, resulting in a sequence of 67 ex ante one-step-ahead forecasts. The one-step-ahead results for p= 1, 3 and 5 are contained in tables 4, 5 and 6, respectively. Were the random ajvalkfares much better, indicating that result of overfitting. the in-sample loss reduction may be the sp (with the best-performing t sample loss reductions due to the tise of value) generally do not exist, and on the few occasions w n they do, they hese qualitative are much smaller than those of the in-sam conclusions hold regardless of the choice of p. ante forecasting, in which even the 6 value must be chosen by the investigator (based upon a l
326
F.X. Diebold and J.A. Nason, ifbnpa?afWric exchange rate prediction? Table 5
r
One-stepahead
out-of-sample forecast analysis, p = 3.
CD
FF
LIR
2-3486 __- --3.5420 2.3839 3.5467 2. 3.5747 2.4280 3.6070 2.4464 3.6291 2.4609 3.6429 2.4656 3.6381 2.4581 3.6200 2.4589 3.6152 2.4971 3.6842 2.5106 3.7047 2.5167 3.7147 2.5244 3.7270 2.4368 3.6497
2.1847 ---- -1.1641 2.0176 1.1115 1.9693 1.0960 1.9409 1.0890 1.9132 1.0857 1.8863 1.0842 1.8607 1.0813 1.8380 1.0788 1.8211 1.0774 1.7642 1.0746 1.7638 1.0752 1.7657 1.0756 1.7685 1.0762 1.7398 1.0703
DM
YEN
SF
BP
DG
D
2.2729 2.2872 1.1977 1.1986 2.1936 2.2267 0.20 1.1858 1.1878 2.1478 2.2334 0.30 1.1747 1.1851 2.1217 2.2253 0.40 1.1676 1.1797 2.1079 2.2232 0.50 1.1650 1.1769 2.0945 2.2181 0.60 1.1618 1.1748 2.0773 2.2055 0.70 1.1592 1.1725 2.0605 2.1791 0.80 1.1578 1.1674 2.0508 2.1495 0.90 1.1574 1.1625 2.0209 2.0940 1.00 1.1525 1.1595 2.60 2.0225 2.0977 1.1510 1.1603 3.00 2.0231 2.0998 1.1502 1.1605 10.00 2.0242 2.1026 1.1493 1.1607 RW 2.0183 2.0754 -1.1253 1.1506 -Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are x 104, except for the CD, which is x 10s. MAPEs are x 102, except for the CD, which is x 103. Smallest MSPE and MAPE for each currency are underlined. 0.10
2.4820 ~ -1.2178 2.4108 1.2086 2.~436 1.1970 2.2890 1.1873 2.2492 1.1799 2.2137 1.1711 2.1764 1.1656 2.1403 1.1603 2.1085 1.1554 2.0120 1.1464 1.9934 1.1438 1.9884 1.1430 1.9837 1.1423 2.0431 1.1617
2.0735 1.1652 2.0080 1.1425 1.9825 1.1350 1.9668 1.1350 1.9505 1.1346 1.9360 1.1331 1.9210 1.1299 1.9032 1.1250 1.8804 1.1182 1.8221 1.1071 1.8121 1.1061 1.8091 1.1063 1.8063 1.1066 1.7402 1.0847
1.9106 1.0188 1.8601 0.9984 1.8386 0.9902 1.8367 0.9884 1.8227 0.9872 1.8073 0.9900 1.8018 0.9951 1.8042 1.0004 1.8132 1.0043 1.8503 1.0166 1.8500 1.0153 1.8493 1.0148 1.8486 1.0141 1.9433 1.0218
3.0651 1.3710 2.9110 1.3622 2.8567 1.3506 2.8191 1.3405 2.7993 1.3336 2.7911 1.3298 2.7815 1.3249 2.7725 1.3220 2.7666 1.3221 2.7628 1.3343 2.7547 1.3340 2.7523 1.3336 2.7505 1.3330 2.7569 1.3407
1.4823 0.9738 1.4627 0.9706 1.4962 0.9805 1.5299 0.9854 1.5427 0.9847 1.5533 0.9867 1.5602 0.9880 1.5714 0.9907 1.5851 0.9983 1.6047 1.0025 1.6107 1.0009 1.6125 1.0007 1.6144 1.0007 1.5483 0.9889
2.2189 1.1508 2.1570 1.1520 2.1485 1.1556 2.1435 1.1577 2.1328 1.1565 2.1215 1.1521 2.1015 1.1457 2.0751 1.1419 2,0473 1.1393 1.9973 1.1414 1.9940 1.1426 1.9928 1.1425 1.9918 1.1423 1.9463 1.1514
combination of prior information and previous sample information), the scope for improved prediction appears very limited. The negative results uncovered thus far are for one-step-ahead conditional expectations; the possibility of nonlinear dependence at longer horizons remains. Such does not appear to be the case; the results for four-, eight- and twelve-step-ahead forecasts reported in an appendix (available from the authors on request) are qualitatively identkal. f the results are summarized graphically in figs. 1 ( n-sample loss, rel forecasts based upon pth order t-of-sample loss relative to the random walk for k-step ahea forecasts (k=1,4,8,12) from pth order
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
327
Table 6 One-step-ahead out-of-sample forecast analysis, p = 5. CD 0.10
FF
DM
LIR
YEN
SF
BP
DG
DK
2.9123 2.5456 2.9236 2.6215 2.3325 3.3799 1.9555 2.4573 2.83 17 4.1417 I.2889 1.3795 1.3076 1.1520 1.4933 1.0928 1.2349 1.3886 0.20 2.6505 2.3 117 2.6230 2.293 1 2.0809 3.1792 1.8207 2.2870 2.5481 3.9257 1.2175 1.2974 1.2310 1.0697 1.4507 1.0469 1.1923 1.3113 2.5028 2.1942 2.4743 2.1955 2.0016 3.0785 1.7494 2.2235 2.39 13 0.30 3.8199 1.1829 1.2550 1.1972 1.0453 1.4162 1.0328 1.1720 1.2566 2.4139 2.1025 2.3675 2.1214 1.9477 2.9925 1.6962 2.1836 2.2810 0.40 3.7456 I.1553 1.2190 1.1758 1.0285 1.3889 1.0209 1.1579 1.2200 2.3949 2.0272 2.27 14 2.0396 1.9088 2.9060 1.6593 2.1508 2.2026 0.50 3.7195 1.1310 1.1891 1.1497 1.0148 1.3605 1.0135 1.1422 1.1918 2.4004 1.9606 2.11931 1.9760 1.8806 2.8413 1.6336 2.1036 2.1355 0.60 3.7092 1.1092 1.1688 1.1277 1.0048 1.3379 1.0106 1.1256 1.1670 2.4216 1.9108 2.1267 1.9296 1.8656 2.7974 1.6164 2.0670 2.0902 0.70 3.6996 1.0930 1.1505 1.1150 1.0038 1.3218 1.0095 1.1124 1.1542 2.4438 1.8816 2.0867 1.8902 1.8950 2.7652 1.6101 2.0403 2.0628 0.80 3.6950 1.0859 1.1404 1.1047 1.0048 1.3144 1.0079 1.1098 1.1465 2.44609 1.8527 2.0615 1.8582 1.8530 2.7403 1.6154 2.0173 2.0542 0.90 3.6899 1.0826 1.1365 1.1000 1.0049 1.3161 1.0087 1.1163 1.1487 2.4860 1.8015 2.0285 1.8303 1.8695 2.7587 1.6526 2.0041 2.0398 1.00 3.7133 1.0840 1.1498 1.1057 1.0172 1.3356 1.0082 1.1375 1.1534 2.4636 1.7981 2.0087 1.8209 1.8860 2.768 1 1.6526 2.0070 2.0363 2.00 3.7305 1.0846 1.1473 1.1067 1.0197 1.3394 1.0060 1.1434 1.1529 2.4996 1.7987 2.0031 1.8182 1.8924 2.7705 1.6491 2.0070 2.0349 3.00 3.7407 1.0851 1.1465 1.1070 1.0199 1.3401 1.0052 1.1444 1.1523 2.5083 1.8001 1.9980 1.8157 1.9000 2.7733 1.6443 2.0070 2.0334 10.00 3.7545 1.0855 1.1456 1.1074 1.0196 1.3410 1.0045 1.1453 1.1515 1.5483 1.9463 2.0183 RW 2.4368 1.7398 2.0431 1.7402 1.9433 2.7569 --1.3407 0.9889 1.1514 1.1253 3.6497 i.0703 1.1617 1.0847 1.0218 -Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are except for the CD, which is x 105. MAPEs are x lo*, except for the CD, which is Smallest MSPE and MAPE for each currency are underlined.
BF 2.6892 1.2547 2.4742 1.2089 2.3782 1.1892 2.3299 1.1813 2.2857 1.1716 2.2363 1.1588 2.1902 1.1461 2.1536 1.1378 2.1186 1.1351 2.1037 1.1525 2.1097 1.1575 2.1119 1.1588 2.1146 1.1602 2.0754 1.1506 x 104, x 10’.
models (p= 1,3,5). The MSPE and MAPE values, indicated with diamonds, correspond to use of the best t value.12 Clear and distinct patterns emerge. In-sample, MSPE and tions in the neighborhood of 10-20 percent relative to the random walk are pervasive. 13 Out-of-sample, however, the nonparametric predictor fares much less well. Typically, the loss associated with the nonparametric predictor is ‘*To make matters concrete, consider the leftmost diamond of the Canada graph in fig. 1. It corresponds to in-sample analysis with p = 1, and its height of approximately - 11 ir&ates that the nonparametric model fit by locally weighted regression has mean squared error approximately 11 percent smaller than that of the random walk. Now consider the first diamond to the right of the vertical dashed line. It corresponds to out-of-sample analysis with k= p= 1, and its height of approximately +4 indicates that the nonparametric model fit by lscally weighted regression has mean squared error approximately 4 percent larger than that of the random walk. 131t is interesting to note that the marginal benefit of including more lags (i.e. increasing p) is diminishing.
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
328
10
0
0
-11
-11
-22
-22
11
11
0
-11
, . ,
-11
-22
-22
11
11
0
-11
I ‘ ,
-11
-22
,
I
I I
5135
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I 4 I * I I2 OUT OF SAMPLE
II" i
(135/I35135135135~
I
I
m
4
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4
1
Il2(lK
OUT O? SAMPLE
Fig. 1, 1MSFE comparisons: in-sample vs. out-of-sample performance. I
8
I
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
6
329
. , I
I
6
-10 -14
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I12
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330
F.X. Diebo1dand J.A. Nason, Nonparametric exchatige rate prektion?
larger than that of the random walk, regardless of the number of lags included (p) or the forecast horizon (k). When loss reduction does occur it is t the loss typically very small, and must be tempered by the which are reductions graphed correspond to the ex post optimal not known in real time.
Using a powerful nonparametric prediction technique, we are generally unable to improve upon a simple random walk in out-of-sample prediction of ten major dollar spot rates in the post-1973 float. Our exchange rate results therefore corroborate those of others using different techniques and different asset prices, such as Prescott and Stengos (1988), who find that kernel techniques deliveT @loimprovement upon the random walk in the prediction of gold prices, 2nd White (1988), who is unable to improve forecasts of IBM stock returns using a neural network. Taken together, these results constitute fairly strong evidence against the existence of asset price nonlinearities that are exploitable for improved point prediction. Our reduced-form results carry important implications for structural models of exchange rate determination. In particular, it would appear unlikely that significant structural nonlinearities, whether of functional form or of regime-switching, have been operative in the post-1973 float. In complementary and independent work, Meese and Rose (1989) argue forcefully for this position. The research could of course be extended in a number of directions. The analysis could be made completely ex ante by choosing c in real time by cross validation, and multivariate generalizations might be undertaken. Computational considerations render some of these extensions infeasible at the present time.
Abel, A.B., 1988, Stock prices under time varying dividend risk: An exact solution in an inllnite horizon general equilibrium model, Journal of Monetary Economics 22,375-393. Baldwin, R. and R. Lyons, 1988,The mutual amplification effect of exchange rate volatility and unresponsive trade prices, NBER Working Paper no. 2677. Boothe, P. and D. Glassman, 1987, The statistical distribution of exchange rates, Journal of International Economics 22, 297-319. Brock, W.A., W.D. Dechert and J.A. Scheinkman, 1987, A test for independence based on the correlation dimension, SSRI Working Paper no. 8702 (Department of Economics, University of Wisconsin, Madison, WI). Clark, PK., 1973, A subordinated stochastic process model with finite variance for speculative prices, Econometrica 41, 135-155.
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
331
Cleveland, W.S., 1979, Robust locally weighted regression and smoothing _seatterpIots,Journal of the American Statistical Association 74, 829-836. Cleveland, W.A. and S.J. Devlin, 1988, Locally weighted regression: An approach to regression analysis by local fitting, Journal of the American Statistical Association 83, 596-610. Cleveland, W.S., S.J. Devlin and E. Grosse, 1988, Regression by local fitting: Methods, properties and computational algorithms, Journal of Econometrics 37, 87-l 14. Cootner, P., ed., 1964, The random character of stock market prices ( IT Press, Cambridge, MA). Diebold, F.X., 1986, Modeling persistence in conditional variances: A comment, Econometric Reviews 5, 51-56. Diebold, F.X., 1988, Empirical modeling of exchange rate dynamics (Springer-Verlag, New York). Diebold, F.X. and M. Nerlove, 1989a, The dynamics of exchange rate volatility: A multivariate latent-factor ARCH model, Journal of Applied Econometrics 4, l-22. Diebold, F.X. and M. Nerlove, 1989b, Unit roots in economic time series: A selective survey, in: T.B. Fomby and G.F. Rhodes, eds., Advances in econometrics: Cwintegration, spurious regressions, and unit roots (JAI Press, Greenwich, CT) forthcoming. Diebold, F.X. and P. Pauly, 1988, Endogenous risk in a rational-expectations portfolio-balance model of the deutschemark/dollar rate, European Economic Review 3% no. 1, 27-54. Domowitz, I. and C.S. Hakkio, 1985, Conditional variance and the risk premium in the foreign exchange market, Journal of International Economics 19,47-66. Engel, C, and J.D. Hamilton, 1988, Long swings in exchange rates: Are they in the data and do markets know it?, Manuscript (Department of Economics, University of Virginia, VA). Engle, R.F., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation, Econometrica 50,987-1008. Engle, R.F., D.M. Lillien and R.P. Robbins, 1987, Estimating time-varying risk premia in the term structure: The ARCH-M model, Econometrica 55,391-408. Fama, E.F., 1965, The behavior of stock market prices, Journal of Business 38,34-105. Fama, E.F., 1976, Foundations of finance (Basic Books, New York). Flood, R. and P. Garber, 1983, A model of stochastic process switching, Econometrica 51, 537-564. Froot, F.A. and M. Obstfeld, 1989, Exchange rate dynamics under stochastic regime shifts: A unified approach, NBER Working Paper no. 2835. Gallant, A.R. and D.W. Nychka, 1987, Semi-nonparametric maximum likelihood estimation, Econometrica 55,363390. Gallant, A.R., D. Hsieh and G. Tauchen, 1988, On fitting a recalcitrant series: The pound/dollar exchange rate, 1974-1983, Manuscript (Graduate School of Business, University of Chicago, Chicago, IL). Hinich, M. and D. Patterson, 1985, Evidence of nonlinearity in daily stock returns, Journal of Business and Economic Statistics 3, 69-77. Hinich, M. and D. Patterson, 1987, Evidence of nonlinearity in the trade-by-trade stock market return generating process, Manuscript (Department of Government, University of Texas at Austin, TX). Hodrick, R.J., 1987, Risk, uncertainty and exchange rates, NBER Working Paper no. 2429, forthcoming in Journal of Monetary Economics. Hsieh, D.A., 1988, A nonlinear stochastic rational expectations model of exchange rates, Manuscript (Graduate School of Business, University of Chicago. Chicago, IL). Hsieh, D.A., 1989, Testing for nonlinear dependence in foreign exchange rates: 19741983, Journal of Business 62,339-368. Department of Economics, LeBaron, B., 1987, Nonlinear puzzles in stock returns, University of Chicago, Chicago, IL). Mandelbrot, B., 1963, The variation of certain speculative prices, Journal of Business 36, 394419. Meese, R.A. and K. Rogoff, 1983a, Empirical exchange rate models 01 the seventies: out of sample?, Journal of International Economics 14, 3-24. Meese, R.A. and K. Rogoff, 1983b, The out of sample failure of empirical exchange rate models: ed., Exchange rates and international Sampling error or misspecification?, in: J. Fren economics (University of Chicago Press, Chicago,
332
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
Meese, R.A. and A.K. Rose, 1989, An empirical assessment of nonlinearities in models of exchange rate determination, Manuscript (School of Business, University of California, Berkeley, CA). Nason, J.M., 1988, The equity premium and time-varying risk behavior, Finance and Economics Discussion Series no. 8, Federal Reserve Board. Prescott, D.M. and T. Stengos, 1988, Do asset markets overlook exploitable nonlinearities? The case of gold, Manuscript (Department of Economics, University of Guelph). Priestley, M.B., 1980, Statedependent models: A general approach to nonlinear time series analysis, Journal of Time Series Analysis 1,47- Il. Robinson, P.M., 1983, Nonparametric estimators for time series, Journal of Time Series Analysis 4, 185-207. Robinson, P.M., 1988, Semiparametric econometrics: A survey, Journal of Applied Econometrics 3, 35-51. Scheinkman, J.A. and B. LeBaron, 1989, Nonlinear dynamics and stock returns, Journal of Business 64, 3 1l-338. Sims, C.A., 1984, Martingale-like behavior of prices and interest rates, Discussion Paper no. 205 (Department of Economics, University of Minnesota, Minneapolis, MN). Ullah, A., 1988, Nonparametric estimation of econometric functionals, Canadian Journal of Economics 2 1,625-658. Weiss, A.A., 1984, ARMA models with ARCH errors, Journal of Time Series Analysis 5, 129-143. Weiss, A.A., 1986, ARCH and bilinear time series models: Comparison and combination, Journal of Business and Economic Statistics 4, 59-70. Westerlield, J.M., 1977, An examination of foreign exchange risk under fixed and floating rate regimes, Journal of International Economics 7, 181-200. White, pi., 1988, Economic prediction using neural networks: The case of IBM daily stock returns. Working Paper no. 88-20 (Department of Economics, UCSD). Yakcwltz, S.J., 1987, Nearest neighbor methods for time series analysis, Journal of Time Series Analysis 8, 235-247.
Economics 28 (1990) 315-332. North-Holland
Francis X. DIEBOLD* University of Pennsylvania, Philadelphia, PA 19104-6297,
USA
James A. NASON University of British Columbia, Vancouver, British Columbia V6T I W5, Canada
Received June 1989, revised version received August 1989 Conditional heteroskedasticity is frequently found in the prediction errors of linear exchange rate models. It is not clear whether such conditional heteroskedasticity is a characteristic of the true data-generating process, or whether it indicates misspecification associated with linear conditional-mean representations. We address this issue by estimating nonparametrically the conditional-mean functions of ten major nominal dollar spot rates, 1973-1987, which are used to produce in-sample and out-of-sample nonparametric forecasts. Our findings bode poorly for recent conjectures that exchange rates contain nonlinearities exploitable for enhanced point prediction.
1. Introduction It is widely agreed that a variety of high-frequency asset returns are well described as linearly unpredictable, conditionally heteroskedastic, and unconditionally leptokurtic. Documentation of linear unpredictability may be traced at least to early work on efficient markets, such as Cootner (1964) and Fama (1965); similarly, leptokurtosis has been appreciated at least since Mandelbrot (1963). The early writers were also aware of the apparent occurrence of volatility clustering in asset returns, and the work of Engle (1982) provided a tool for its formal study. It is now agreed that many time series of asset returns, while approximately uncorrelated, are not temporally independent; dependence arises through persistence in the conditional variance and perhaps in other conditional moments. *Diebold gratefully acknowledges financial support from the National Science Foundation and the University of Pennsylvania Research Foundation. An earlier version of this paper was presented at the 1988 North American Winter meetings of the Econometric Society, New York. We thank the participants there, as well as an anonymous referee, Buz Brock, Dee Dechert, Blake LeBaron, Andy Rose, and seminar participants at Boston College, Carnegie-Mellon, Wisconsin, Washington, LSU, and Stonybrook for useful comments. We especially thank William S. Cleveland, who generously provided his software, and Roberto Sella, who provided outstanding research assistance. 22- 1996/90/%3.50 c] 1990,
316
F.X. Diebold and J.A. Nason, Nonparametric
exchange
rate prediction?
All of these results are manifest in exchange rates, which are the focus of the present study. The well-known work of Meese and ogoff (1983a, b) provides graphic illustration of the failure of a variety of economic models to outperform a simple random walk in out-of-sample predi:tion.’ Extensive reviews of linear unpredictability and leptokurtosis in exci:ange rates are contained in Westerfield (1977), Boothe and Glassman (1987), and Diebold and Nerlove (1989a). Conditional heteroskedasticity, in the form of ARCH and related effects, has also been repeatedly documented in exchange rates. Diebold (1988), for example, examines seven nominal dollar spot rates and finds little linear predictability but strong ARCH effects in all of them. A number of explanations have been advanced for the leptokurtosis and volatility clustering, as well as reduction of excess kurtosis under temporal aggregation. The phenomena may be jointly explained, for example, by subordinating exchange rates to a process dictating information arrival. The seminal work of Clark (1973) shows that subordination to an i.i.8. information-arrival process produces leptokurtic returns. As mentioned by Diebold (1986) and formali: Y! by Gallant, Hsieh and Tauchen (1988), Clark’s analysis may be extended by allowing for persistence in the information-arrival process, which produces ARCH-like movements in higher-order conditional moments. Finally, Diebold (1988) uses central limit theorems for dependent observations to show that, under quite general conditions (not requiring, for example, existence of the fourth unconditional moment), the unconditionally fat-tailed behavior associated with stationary ARCH effects diminishes with temporal aggregation. If the above characterization of exchange rate dynamics (linear conditional mean with nonlinearities working through the conditional variance) is correct, then the nonlinearities cannot be exploited to generate improved point predictions relative to linear models. It is not clear, however, that the ARCH effects are structural, i.e. that they are a characteristic of the true data-generating process (DGP). Instead, ARCH may indicate misspecitication, serving as a proxy for neglected nonlinearities in the conditional mean.2 A finding of significant conditional-mean nonlinearity would be important for both theoretical and empirical work: theoretically, a substantial challenge to our understanding of asset-price dynamics would be posed, and empirically, a source of improved point prediction relative to linear models would be provided. Interestingly, recent empirical and theoretical results are consistent with
‘Their candidate models include a flexible price monetary model, a sticky price monetary model, a sticky price monetary model with current account effects, six univariate time series models, a vector autoregressive model, and the forward rate. 2For an illustration of the difficulties involved in separating conditional-mean from conditional-varia.nce dynamics, see Weiss (1986), who discusses ARCH and bihnearity.
F.X. Diebold and J.A. Nason, Nonparametric
exchange
rate prediction?
3 tIr
the conjecture that nonlinearities may be present in asset-return conditional means. The empirical results may be categorized into two groups: (1) those using ideas from the theory of stochastic nonlinear time series, and (2) those using ideas from the theory of deterministic chaotic systems. In the nonlinear time series area, a number of studies, including Domowitz and Hakkio (1985), Hinich and Patterson (1985, 1987), Weiss (1986), Engle, Lillien and Robins (1987), Diebold and Pauly tl988), and others, appear to detect statistically significant nonlinearity in conditional means of various asset prkes and other economic aggregates. Recent work on regime switching, including Flood and Garber (1983), Engel and Hamilton (1988), and Froot and Obstfeld (1989), is also squarely in the nonlinear tradition. Similar results have been obtained in the chaos literature, using tests based on estimated Lyaponov exponents and correlation dimensions, as developed in Brock, Dechert and Scheinkman (1987), inter alia. Scheinkman and LeBaron (1989), for example, find strong evidence of nonlinearity in common stock returns and suggest that it could be exploited for improved point prediction. Similarly, Gallant, Hsieh and Tauchen (1988) and Hsieh (1989) report evidence of residual nonlinearity in exchange rates, after controlling for conditional heteroskedasticity. T6:ese empirical results are provocative, because they challenge us to take seriously the possible existence of nonlinear conditional-mean dynamics in asset prices. A number of recent theoretical developments are beginning, implicitly or explicitly, to address this challenge. Sims (1984), for example, shows that general-equilibrium asset-pricing models imply martingale asset-price behavior only at arbitrarily short horizons. Thus, economic theory cannot rule out the possibility of nonlinear dependence in conditional means (as well as in higher-order conditional moments) of asset returns. Moreover, substantial progress has been made in general equilibrium models with time-varying risk, such as Abel (1988), Hodrick (1987), Baldwin and Lyons (1988), and Nason ( 1988). In summary, there appears to be strong evidence, consistent with rigorous economic theory, that important nonlinearities may be operative in exchange rate determination. Upon further consideration, however, it becomes clear that the literature is not in satisfactory condition, owing to a puzzle that immediately arises: Why is it that while statistically significant rejections of linearity in exchange rates routinely occur, no nonlinear model has been found that can significantly outperform even the simplest linear model in ay be operative, a out-of-sample forecasting? Because a number urse, is that the number of explanations may be offered. nonlinearities present may be in even-ordered conditional moments, and -sample nonlinearities therefore are not useful for point prediction. d may cause various such as outliers and structural shifts may be linearity tests to reject, while nevertheless being of n!c use for out-of-sa
F.X. Diebold and J.A. Nason, Nonparametric
’ 318
exchange
rate prediction?
forecasting. Third, very slight conditional-mean nonlinearities rAght be truly present and be detectable with large datasets, while nevertheless yielding negligible ex ante forecast improvement.3 inally, even if conditional-mean nonlinearities are present and are important, the overwhelming variety of plausible candidate nonlinear models makes determination of a good approximation to the DGP a difficult task. The seemingly large variety of parametric nonlinear models that have received attention lately (e.g. bilinear, threshold, and exponential autoregressive) is in fact a very small subset of the class of plausible nonlinear DGPs. In this paper we contribute to a resolution of this puzzling behavior of exchange rates by estimating conditional-mean functions nonparametrically. By so doing, we avoid the parametric model-selection problem; the class of potential models entertained is expanded greatly. In section 2 we discuss various aspects of nonparametric functional estimation, and the locally weighted regression procedure, which we use extensively, is highlighted. Section 3 contains empirical results; in particular, both in-sample LWR fits and out-of-sample LWR forecasts are compared to those arising from linear models. Section 4 concludes.
etric
iction
Nonparametric techniques may be used for estimation of a variety of densities and econometric functionals, including regression functions, first and higher-order derivatives of regression functions, conditional-variance functions, hazard and survival functions, etc.4 We shall generallv be concerned with nonparametric estimation of conditional expectation, or regression, functions,
which we use for nonparametric prediction? This is achieved by (explicit or implicit) substitution of nonparametric estimates of the underlying joint and marginal densities into (1). In our dynamic models, the stochastic conditioning vector x is composed of lagged dependent variables: x, = {Yt--l,***, %
other
Yt-,I*
words, significance
(2) of nonlinearity
does not necessarily imply its economic
importance.
4For surveys of various aspects of r;onparametric and semiparametric estimation, see Ullah (1 bvious from the context, we use lower-case letters for both random variables and their realizations, and we use f to denote all probability density functions.
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
319
e shall work with the very general nonlinear autoregressive structure:
t=l
, . . . ,1: so that (1) may be rewritten more specifically as: wYt+llYt9*Y
Yt-p+l)=SYt+lf(Yt+llYtmYt-p+lb-lYt+l
The regression function estimates may be obtained by a variety of interrelated nonparametric methods, including kernel, series, and nearest-neighbor (NN) techniques, consistency results for which have ken obtained in time series environments by Robinson (1983), Gallant and Nychka (1987), and Yakowitz (1987), respectively. In this study we make use of a NN technique, known as locally-weighted regression (LWR). NN methods proceed by estimating g(x), at an arbitrary point X=X* in p-dimensional Euclidean space, via a weight function: T
ii(
c W&t)Yts
t=l
where w&)= l/kT if x, is one of the kT nearest neighbors of x*, and w&t) = 0 otherwise. R estimator, as proposed by Cleveland (1979) and refined by Cleveland and evlin (1988) and Clevelan (1988), is an important generalization f the NN estimator. Lik estimator, LWR fits the surface at a point X* as a function of the y values corresponding to the kT nearest neighbors of x*. Unlike 9 ho ther does not take &u*) as a simple average of those y value fitted value from a regression surface. This corresponds to a simple average only in the very unlikely case that the constant term is the sole regressor with explanatory power. We now discuss the procedur estimate of the surface at a point constant such that O
nearest
nei
OTS
F.X. Diebold and J.A. Nason, Nonparametric
320
excha
these xf, x;, . . . , x& Thus, XT is closest to x*, x; is second closest to x so on. Let &a, b) measuie Euclidean distance; then I(x*, x&J is the Euclidean distance from x* to its k,th closest neighbor:
I(X*,X&.)=
E
1 l/2
C
(Xk*Tj-xi*)2
j=l
l
(6)
Form the neighborhood weight function: u,(x,,x*,x&J= W(x,, x*)/m*,
x&)19
(7)
where C( .) is the tricube function: for r.44, otherwise.
’
(8)
The value of the regression surface at x* is then computed as: j”
z&x*)
=x*//j,
(9)
where i v,(y,-~j/?)~
pl=argmin
[ t=1
1 .
(10)
R procedure, exactly as described above, is used in our subsequent empirical work. 0bviously, ii reflects a number of judgmental decisions, such as use of the Euclidean norm and tricube neighborhood weighting, as well as locally linear (as opposed to higher-order, such as quadratic) fitting. The Euclidean norm has obvious geometric appeal, as does the tricube weight function, which produces a smooth, gradual decline in weight with distance from x*. Locally linear fitting is also highly reasonable (and computationally le) in the present context.’ f greater interest is the choice of <, which determines the number of the degree of smoothing. Consistency of nearest neighbors used, and he NN estimators (and hence L ) requires that the number of nearest neighbors used go to infinity with sample size. but at a slower rate, i.e. lim kr=OO, T+CQ
'See Clevelan
lim (k,/T) = 0. T+GO
evlin and Grosse (1988) for further discussion.
(11)
F.X. Diebold and J.A. Nason, Nonpararnetrrc exchange r
This implicitly creates a ‘window’ whose width be sample size goes to infinity, but at a slower rate. In t window nevertheless contains progressively more neigh reduced along with variance. Similar issues arise in kernel and series estimation. In window width corresponds to the bandwidth, which mus size but at a slower rate. the series case, the windo (inversely) to the number ‘ncluded series terms; ag that the truncation point increases with sample size, While the asymptotic mechancs of LWR estimators kernel and series estimators, ;t is interesting to note properties are likely to be superior to those of kernel Series estimators produce global, as opposed to local, is likely to make them inflexible in all but the la estimators do produce local approximations, but the constant, as opposed to locally linear, which may int in finite samples. It is interesting to note that the earlier-discussed L selecting the number of nearest neighbors does n regularity condition for consistency. In any finite however, this is of no consequence, since there exists a Ta, a c 1, which dses satisfy the regularity condition an an identical number of nearest neighbors. For exam 800, the c T rule with t = 0.5 selects approximately 400 does the T” rule with a =0.9.*
redict ion?
321
smaller as the ay the shrinking s, so that bias is
ilar to those of series estimators.
substantial bias
l
3. Empirical analysis We study ten nominal New York interbank dollar spot rates, 12.30 p.m., from the first week of 1973 (3 January) through the thirty-eighth week of 1987 (23 September). Sample size is therefore equal to 769. The rates are Canadian Dollar (CD), French Franc (FF), German Lira (LIR), Japanese Ytln (YEN), Swiss Franc (SF), Belgian Franc (BF), Danish Kroner (DK) and Dutch Gui measured in cents per unit of foreign currency and were obtained from the Federal Reserve Board database. Throughout, we use S, as notation for a generic exchange rate at time t. Our interest centers on percent exchange rate changes, A Ins,, on which we have 768 o nd week of 1973 through thirty-eighth week of 1987); st additional benefit of avoiding theoretical *Moreover, in the empirical work that follows, we explore a wide range of c values.
322
F.X. Diebold and J.A. Nason, Nonparavnetric exchange rate prediction? Table 1 In-sample analysis, p = 1.
;r).lO 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 2.00 3.00 10.00 RW
CD
FF
DM
LIR
YEN
SF
BP
DG
DK
BF
2.7095 3.6230 2.6906 3.6260 2.6900 3.6379 2.6923 3.6429 2.6904 3.6418 2.6848 3.6407 2.6786 3.6419 2.6728 3.6445 2.6606 .---3.6409 2.6645 3.6557 2.6925 3.6712 2.7089 3.6820 2.7305 3.6964 2.9402 3.8568
1.9321 0.9407 1.9384 0.9328 1.9379 0.9350 1.9364 0.9375 1.9336 0.9386 1.9308 0.9398 1.9262 0.9419 1.9168 0.9427 1.9055 0.9472 1.9307 0.9714 1.9421 0.9757 1.9453 0.9769 1.9486 0.9780 2.0736 1.0174
2.1849 1.0192 2.1857 1.0118 2.1845 1.0133 2.1787 1.0158 2.1737 1.0179 2.1674 1.0189 2.1572 1.0193 2.1445 1.0194 2.1315 1.0206 2.1488 1.0393 2.1864 1.0489 2.2029 1.0538 2.2230 1.0599 2.3174 1.0879
1.6824 0.8469 1.6748 0.8411 1.6731 0.8402 1.6715 0.8390 -1.6712 0.8408 1.6694 0.8429 1.6653 0.8443 1.6571 0.8468 1.6501 0.8535 1.6980 0.8833 1.7189 0.8900 1.7286 0.8935 1.7409 0.8979 1.8046 0.9117
1.7298 0.8512 1.7310 0.8379 1.7291 0.8326 1.7311 0.8316 1.7312 0.8328 1.7288 0.8357 1.7218 0.8385 1.7034 0.8397 1.6719 0.8388 1.6663 -0.8546 1.6794 0.8575 1.6858 0.8592 1.6938 0.8612 1.9367 0.9153
2.7994 1.1669 2.7898 1.1713 2.7886 1.1737 2.7850 1.1719 2.7778 1.1700 2.7653 1.1683 2.7477 1.1680 2.7245 1.1684 2.6974 1.1693 2.6769 1.1838 2.7086 1.1921 2.7233 1.1956 2.7410 1.1997 2.9265 1.2348
1.9205 0.9329 1.9200 0.9269 1.9170 0.9242 1.9143 0.9230 1.9115 0.9237 1.9074 0.9250 1.8974 0.9256 1.8841 0.9266 1.8669 ---0.9274 1.8717 0.9491 1.8892 0.9556 1.8975 0.9584 1.9073 0.9616 1.9741 0.9819
1.9462 0.9725 1.9398 0.9680 1.9332 0.9674 1.9292 0.9684 1.9221 0.9674 1.9135 0.9665 1.9026 0.9666 1.8929 0.9680 1.8755 0.9700 1.8755 0.9902 1.8890 0.9962 1.8944 0.9983 1.9003 1.0004 2.0519 1.0500
3.3442 1.0139 3.3383 1.0047 -3.3331 1.0065 3.3292 1.0069 3.3226 1.0062 3.3156 1.0070 3.3082 1.0077 3.2983 1.0083 3.2899 1.0087 3.2905 1.0190 3.2974 1.0218 3.2998 1.0227 3.3025 1.0238 3.4965 1.0903
1.8685 0.9372 1.8517 0.9343 1.8481 0.9343 1.8444 0.9355 1.8361 0.9370 1.8292 0.9399 1.8225 0.9423 1.8085 0.9444 1.7937 0.9450 1.8063 0.9629 1.8200 0.9681 1.8242 0.9697 1.8286 0.9713 2.0176 1.0343
Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are x lo’, except for the CD, which is x 1 .O! MAPEs are x lo*, except for the CD, which is x 103. Smallest MSPE and MAPE for each currency are underlined.
of nonstationary nonparametric regression functions, as well as numerical problems with highly collinear regressors.g For each exchange rate we examine both in-sample ‘fit’ and out-of-sample predictive performance of the LWR nonparametric conditional-mean estimator. Observations 6 through 701 are used for in-sample analysis, while observations 702 through 768 are reserved for out-of-sample forecast comparison. Our out-of-sample forecasts are completely ex ante, using LWR estimates formed recursively in real time, using only information actually available. The in-sample results for p = 1, 3 and 5 appear in tables 1, 2 and 3, respectively. e perform a sensitivity analysis with respect to c, exploring a ‘Furthermore, as pointed out by Meese and Rogoff (1983a,b) the use of logs eliminates prediction problems arising from knsen’s inequality. For a discussion of issues related to unit iebold and Nerlove (1989b).
RX. Diebold and LA. Nason, Nonparametric exchange rate prediction?
323
Table 2 In-sample analysis, p = 3. 5 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 la0 2.00 3.00 10.00 RW
CD
FF
D&i
LIR
YEN
SF
BP
DG
DK
BF
2.8388 3.7433 2.7539 3.6637 2.7174 3.6316 2.6894 3.6160 2.6629 3.6072 2.6378 3.5998 2.6114 3.5905 2.5782 3.5769 2.5341 3.5589 2.5437 3.5812 2.5961 3.6166 2.6204 3.6336 2.6502 3.6525 2.9402 3.8568
1.9753 0.9338 1.9291 0.9275 1.9073 0.9236 1.8854 0.9199 1.8596 0.9153 1.8286 0.9108 1.7972 0.9063 1.7589 0.9013
2.1892 0.9990 2.0951 0.9886 2.0613 0.9847 2.0417 0.9824 2.0153 0.9794 1.9893 0.9771 1.9670 0.9751 1.9441 0.9728
1.6725 0.8450 1.6455 0.8390 1.6395 0.8362 1.6343 0.8345 1.6263 0.8329 1.6151 0.8313 1.6010 0.8306 1.5896 0.8325
1.7711 0.8799 1.7262 0.8604 1.7149 0.8524 1.7088 0.8489 1.7019 0.8466 1.6818 0.8419 1.6501 0.8349 1.6114 0.82M 1.5760 0.8196 ---1.5791 0.8271 1.5987 0.8337 1.6065 0.8363 1.6155 0.8393 1.9367 0.9153
2.8763 1.1642 2.8388 1.1655 2.7990 1.1634 2.7652 1.1620 2.7340 1.1601 2.6952 1.1567 2.6574 1.1529 2.6249 1.1504 2.5904 1.1493 2.5764 1.1607 2.6201 1.1728 2.6391 1.1771 2.6607 1.1818 2.9265 1.2348
2.2728 0.9975 2.1893 0.9848 2.1431 0.9739 2.1126 0.9669 2.0797 0.9625 2.0416 0.9583 2.0036 0.9526 1.9662 0.9482 1.9316 0.9463 1.9327 0.9644 1.9753 0.9742 1.9949 0.9785 2.0174 0.9836 1.9741 0.9819
1.9677 0.9613 1.9236 0.9592 1.8986 0.9561 1.8804 0.9545 1.8572 0.9516 1.8330 0.9483 1.8093 0.9452 1.7734 0.9403 1.7321 0.9351 1.7146 0.9478 1.7371 0.9569 1.7454 0.9598 1.7544 0.9626 2.0519 1.osOO
3.4300 1.0200 3.3713 1.0102 3.3451 1.0051 3.3251 1.0016 3.3043 0.9991 3.2788 0.9966 3.2548 0.9937 3.2319 0.9911 3.209?
1.8790 0.9377 1.8240 0.9335 1.8060 0.9336 1.7948 0.9343 1.7810 0.9338 1.7617 0.9322 1.7363 0.9278 1.7008 0.9211 1.6631 0.9139 1.6392 0.9201 1.6648 0.9291 1.6733 0.9318 1.6824 0.9346 2.0176 1.0343
1.5804 0.8361 1.7326 0.9242 1.7422 0.9271 1.7521 0.9299 2.0736 1.0174
2.0139 1.0077 2.0251 1.0108 2.0367 1.0140 2.3174 1.0879
1.6117 0.8570 1.6444 0.8682 1.6582 0.8729 1.6749 0.8785 1.8046 0.9117
0.9888 3.1941 -0.9947 3.2131 1.0008 3.2195 1.0027 3.2264 l.OQ47 3.4965 1.0903
Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are x 104, except for the CD, which is x 10s. MAPEs are x 102, except for the CD, which is x 103. Smallest MSPE and MAPE for each currency are underlined.
wide range of values from 0.10 through 10.O.loIn table 1 (p= 1) we find that mean squared prediction error (MSPE) is minimized for all currencies at either { =0.9 or c = 1.0. The MSPE associated with the optimal e choice is always lower, often substantially so, than the random Mean absolute prediction error (MAPE) is minimized generally smaller { values ranging from 0.1 to 0.6. Like associated with the optimal { choice is always smaller tha MAPE.” Similar results are obtained in table 2 (p= 3) and table 3 (p= 5): loNote that t = 1 does not correspond to a linear autoregression, because the observations are still weighted. Rather, our algorithm is such that as c approaches infinity, the !inear autoregression emerges. (In practice, t = 10 produces an approximately linear autor “It is interesting to note that the MAPE-optimal t values are smaller than optimal counterparts, reflecting t willingness to accept h er variance in return bias under MAPE as opposed to
324
F.X. Diebold and J.A. Nason, Nonparametric
exchange
rate prediction?
Table 3 In-sample analysis, p = 5.
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
1.00 2.00 3.00 10.00 RW
LIR
CD
FF
DM
3.2387 3.9357 3.0368 3.7960 2.9215 3.7221 2.8357 3.6693 2.7724 3.6371 2.7133 3.6116 2.6589 3.5932 2.6056 3.5766 2.5511 3.5617 my2.5333 -3.5891 2.5874 3.6239 2.6107 3.6384 2.6390 3.6575 2.9402 3.8568
2.1601 0.9764 1.9977 0.9489 1.9425 0.9359 1.8983 0.9249 1.8602 0.9175 1.8194 0.9103 1.7851 0.9047 1.7479 0.8998 1.7120 0.8969 1.7005 0.9138 1.7312 0.9428 1.7420 0.9279 1.7535 0.93 10 2.0736 1.0174
2.3172 1.7498 1.0377 0.8762 2.1687 1.6525 1.0187 0.8606 2.1218 1.6190 1.OOSO 0.8487 2.0836 1.5856 0.9995 0.8361 2.0433 1.5577 0.992 1 0.8254 2.0043 1.5409 0.9848 0.8201 1.9668 1.5337 0.978 1 0.8200 1.9328 1.5372 0.9726 0.8240 1.9070 1.5408 0.9701 0.8287 1.9544 1.6059 0.9899 0.8563 2.0027 1.6410 1.0031 0.8682 2.0177 1.6548 1.0077 0.8728 2.0339 1.6707 1.0124 0.8752 2.3174 1.8046 1.0879 0.9117
YEN
SF
BP
DG
DK
BF
2.0348 0.9522 1.8473 0.9011 1.7864 0.8785 1.7430 0.8634 1.7010 0.85 16 1.6586 0.8399 1.6153 0.8282 1.5745 0.8 175 1.5424 0.8102 1.5623 0.8236 1.5854 0.8307 1.5946 0.8336 1.6054 0.8370 1.9367 0.9153
3.1651 1.2282 2.9725 1.1976 2.8847 1.1859 2.8227 1.1787 2.7698 1.1723 2.7157 1.1660 2.6658 1.1583 2.6223 1.1513 2.582 J. 1.1471 2.5638 1.1560 2.623 1 1.1713 2.6457 1.1765 2.6703 1.1822 2.9265 1.2348
2.4893 1.0314 2.3119 1.OO53 2.2363 0.9928 2.1787 0.9808 2.1174 0.9690 2.0631 0.9606 2.0208 0.9559 1.9817 0.9522 1.9461 0.9482 -1.9605 0.9614 2.0085 0.9742 2.0263 0.9792 2.0464 0.9849 1.9741 0.9819
2.1542 0.9913 2.0008 0.9734 1.9402 0.965 1 1.8947 0.9573 1.8554 0.95 17 1.8183 0.9460 1.7794 0.9400 1.7409 0.9333 1.7019 0.9275 1.6932 0.9389 1.7265 0.9523 1.7388 0.9566 1.7523 0.96 12 2.0519 1.0500
3.5454 1.0348 3.4173 1.0188 3.3744 1.0122 3.3377 1.0074 3.3026 1.0028 3.2692 0.9968 3.2408 0.9916 3.2132 0.9864 3.1853 0.9818 3.1805 0.9888 3.2074 0.9980 3.2166 1.0010 3.2263 1.0042 3.4965 1.0903
2.1070 0.9927 1.9375 0.9655 1.8672 0.9530 1.8166 0.9441 1.7765 0.9365 1.7388 0.9298 1.7031 0.9222 1.6715 0.9147 1.6398 0.9085 1.6276 0.9157 1.6583 0.9268 1.6688 0.9304 1.6802 0.9342 2.0176 1.0343
Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are x 104, except for the CD, which is x 10’. MAPEs are x 102, except for the CD, which is x 103. Smallest MSPE and MAPE for each curvncy are underlined.
nonparametric conditional-mean estimation leads to substantial reductions in loss, whether measured by MSPE or MAPE, relative to the random walk. Both MSPE- and MAPE-optimal { are almost always at 0.9 or 1.0. Finally, it is interesting to note the general tendency for both MSPE and MAPE to decrease and then level off as more lags are included. Although this need not happen (different nearest neighbors are used, in general, for different p), it is intuitively reasonable by analogy to the fact that inclusion of additional regressors in an OLS regression must lower (or, at worst, leave unchanged) the sum of squared residuals. We now turn to the one-step-ahead out-of-sample analysis. Again, we estimate nonparametric autoregressions of order 1 through 5 using the LWR procedure, with values of the smoothing parameter < ranging from 0.1 through 10.0 for each p, corresponding to the use of roughly 70 nearest neighbors (with neighborhood weighting) through ‘all’ nearest neighbc s
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
325
Table 4 One-step-ahead out-of-sample forecast analysis, p = 1. 5
CD
FF
DM
LIR
YEN
SF
0.10
BP
DG
DK
2.7119 2.0596 2.1725 2.1474 1.9619 3.0868 1.5960 2.2309 2.1069 3.8859 1.1435 1.1572 1.1895 1.0444 1.3759 1.0067 1.1638 1.1575 0.20 2.5998 2.0606 2.2107 2.0917 1.9457 2.9372 1.5877 2.1936 2.1059 3.8181 1.1424 1.1656 1.1662 1.0365 1.3506 1.0005 1.1660 1.1635 0.30 2.5515 2.0301 2.2238 2.0509 1.9351 2.8978 1.5960 2.1926 2.1209 3.7797 1.1348 1.1679 1.1546 1.0287 1.3420 1.0027 1.1660 1.1635 0.40 2.5373 2.0097 2.2135 2.03 17 1.9235 2.8984 1.6034 2.1740 2.1181 3.7603 1.1300 1.1680 1.1490 1.0252 1.3422 1.0070 1.1613 1.1657 2.5322 1.9944 2.1985 2.0197 1.9104 2.9007 1.6074 2.1547 2.1163 0.50 3.7525 1.1276 1.1669 1.1460 1.0226 1.3439 1.0082 1.1577 1.1665 0.60 2.5282 1.9795 2.1856 2.0080 1.9031 2.9092 1.6076 2.1467 2.1119 3.7518 1.1240 1.1650 1.1443 1.0204 1.3466 1.0079 1.1568 1.1655 2.5269 1.9653 2.1711 1.9885 1.8989 2.9142 1.6078 2.1407 2.1091 0.70 3.7514 1.1211 1.1621 1.1423 1.0191 1.3488 1.0081 1.1566 1.1649 2.528 1 1.9473 2.1609 1.9646 1.8975 2.9069 1.6123 2.1343 2.1097 0.80 3.7471 1.1184 1.1603 1.1401 1.0204 1.3493 1.0090 1.1567 1.1661 0.90 2.5392 1.9217 2.1466 1.9360 1.9070 2.8911 1.6155 2.1221 2.1097 3.7459 1.1147 1.1586 1.1367 1.0253 1.3497 1.0092 1.1565 1.1677 2.5552 1.8533 2.0900 1.8717 1.9327 2.8195 1.5968 2.0785 2.0890 1.00 3.7549 1.1054 1.1621 1.1220 1.0369 1.3424 1.0006 1.1588 1.1683 2.5458 1.8430 2.0725 1.8536 1.9271 2.7962 1.5933 2.0717 2.0809 2.00 3.7517 1.1030 1.1608 1.1147 1.0344 1.3385 0.9985 1.1588 1.1654 2.5423 1.8430 2.0725 1.8536 1.9271 2.7962 1.5933 2.0717 2.0809 3.00 3.7501 1.1030 1.1608 1.1165 1.0352 1.3398 0.9852 1.1591 1.1661 2.5387 1.8377 2.0557 1.8360 1.9228 2.7777 1.5888 2.0655 2.0756 10.00 3.7480 1.1014 1.1596 1.1126 1.0334 1.3371 0.9990 1.1584 1.1647 RW -2.4368 -1.7398 2.0431 1.7402 1.9433 2.7569 1.5483 1.9463 2.0183 1.1514 1.1253 -3.6497 -1.0703 1.1617 1.0847 1.0218 1.3407 0.9889 -Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are except for the C&D,which is x 10‘. MAPEs are x 102, except for the CD, which is Smallest MSPE and MAPE for each currency are underlined.
BF 2.423 1 1.2047 2.4041 1.2038 2.3805 1.2038 2.3543 1.1997 2.3351 1.1951 2.3 193 1.1951 2.3028 1.1930 2.2789 1.1883 2.2632 1.1867 2.2167 1.1833 2.2036 1.1798 2.2036 1.1807 2.1952 1.1788 2.0754 1.1506 x 104, x 10’.
(with no neighborhood weighting). For each c and each p, out-of-sample forecasts are computed by recursively estimating E(d In S,02 1LI In S7O1,.. . , A 1nS 702 _,,), and then E(d In S70&4 In §To2,. . . , A In S703_J and so forth, in real time. This is continued until the sample is exhausted, resulting in a sequence of 67 ex ante one-step-ahead forecasts. The one-step-ahead results for p= 1, 3 and 5 are contained in tables 4, 5 and 6, respectively. Were the random ajvalkfares much better, indicating that result of overfitting. the in-sample loss reduction may be the sp (with the best-performing t sample loss reductions due to the tise of value) generally do not exist, and on the few occasions w n they do, they hese qualitative are much smaller than those of the in-sam conclusions hold regardless of the choice of p. ante forecasting, in which even the 6 value must be chosen by the investigator (based upon a l
326
F.X. Diebold and J.A. Nason, ifbnpa?afWric exchange rate prediction? Table 5
r
One-stepahead
out-of-sample forecast analysis, p = 3.
CD
FF
LIR
2-3486 __- --3.5420 2.3839 3.5467 2. 3.5747 2.4280 3.6070 2.4464 3.6291 2.4609 3.6429 2.4656 3.6381 2.4581 3.6200 2.4589 3.6152 2.4971 3.6842 2.5106 3.7047 2.5167 3.7147 2.5244 3.7270 2.4368 3.6497
2.1847 ---- -1.1641 2.0176 1.1115 1.9693 1.0960 1.9409 1.0890 1.9132 1.0857 1.8863 1.0842 1.8607 1.0813 1.8380 1.0788 1.8211 1.0774 1.7642 1.0746 1.7638 1.0752 1.7657 1.0756 1.7685 1.0762 1.7398 1.0703
DM
YEN
SF
BP
DG
D
2.2729 2.2872 1.1977 1.1986 2.1936 2.2267 0.20 1.1858 1.1878 2.1478 2.2334 0.30 1.1747 1.1851 2.1217 2.2253 0.40 1.1676 1.1797 2.1079 2.2232 0.50 1.1650 1.1769 2.0945 2.2181 0.60 1.1618 1.1748 2.0773 2.2055 0.70 1.1592 1.1725 2.0605 2.1791 0.80 1.1578 1.1674 2.0508 2.1495 0.90 1.1574 1.1625 2.0209 2.0940 1.00 1.1525 1.1595 2.60 2.0225 2.0977 1.1510 1.1603 3.00 2.0231 2.0998 1.1502 1.1605 10.00 2.0242 2.1026 1.1493 1.1607 RW 2.0183 2.0754 -1.1253 1.1506 -Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are x 104, except for the CD, which is x 10s. MAPEs are x 102, except for the CD, which is x 103. Smallest MSPE and MAPE for each currency are underlined. 0.10
2.4820 ~ -1.2178 2.4108 1.2086 2.~436 1.1970 2.2890 1.1873 2.2492 1.1799 2.2137 1.1711 2.1764 1.1656 2.1403 1.1603 2.1085 1.1554 2.0120 1.1464 1.9934 1.1438 1.9884 1.1430 1.9837 1.1423 2.0431 1.1617
2.0735 1.1652 2.0080 1.1425 1.9825 1.1350 1.9668 1.1350 1.9505 1.1346 1.9360 1.1331 1.9210 1.1299 1.9032 1.1250 1.8804 1.1182 1.8221 1.1071 1.8121 1.1061 1.8091 1.1063 1.8063 1.1066 1.7402 1.0847
1.9106 1.0188 1.8601 0.9984 1.8386 0.9902 1.8367 0.9884 1.8227 0.9872 1.8073 0.9900 1.8018 0.9951 1.8042 1.0004 1.8132 1.0043 1.8503 1.0166 1.8500 1.0153 1.8493 1.0148 1.8486 1.0141 1.9433 1.0218
3.0651 1.3710 2.9110 1.3622 2.8567 1.3506 2.8191 1.3405 2.7993 1.3336 2.7911 1.3298 2.7815 1.3249 2.7725 1.3220 2.7666 1.3221 2.7628 1.3343 2.7547 1.3340 2.7523 1.3336 2.7505 1.3330 2.7569 1.3407
1.4823 0.9738 1.4627 0.9706 1.4962 0.9805 1.5299 0.9854 1.5427 0.9847 1.5533 0.9867 1.5602 0.9880 1.5714 0.9907 1.5851 0.9983 1.6047 1.0025 1.6107 1.0009 1.6125 1.0007 1.6144 1.0007 1.5483 0.9889
2.2189 1.1508 2.1570 1.1520 2.1485 1.1556 2.1435 1.1577 2.1328 1.1565 2.1215 1.1521 2.1015 1.1457 2.0751 1.1419 2,0473 1.1393 1.9973 1.1414 1.9940 1.1426 1.9928 1.1425 1.9918 1.1423 1.9463 1.1514
combination of prior information and previous sample information), the scope for improved prediction appears very limited. The negative results uncovered thus far are for one-step-ahead conditional expectations; the possibility of nonlinear dependence at longer horizons remains. Such does not appear to be the case; the results for four-, eight- and twelve-step-ahead forecasts reported in an appendix (available from the authors on request) are qualitatively identkal. f the results are summarized graphically in figs. 1 ( n-sample loss, rel forecasts based upon pth order t-of-sample loss relative to the random walk for k-step ahea forecasts (k=1,4,8,12) from pth order
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
327
Table 6 One-step-ahead out-of-sample forecast analysis, p = 5. CD 0.10
FF
DM
LIR
YEN
SF
BP
DG
DK
2.9123 2.5456 2.9236 2.6215 2.3325 3.3799 1.9555 2.4573 2.83 17 4.1417 I.2889 1.3795 1.3076 1.1520 1.4933 1.0928 1.2349 1.3886 0.20 2.6505 2.3 117 2.6230 2.293 1 2.0809 3.1792 1.8207 2.2870 2.5481 3.9257 1.2175 1.2974 1.2310 1.0697 1.4507 1.0469 1.1923 1.3113 2.5028 2.1942 2.4743 2.1955 2.0016 3.0785 1.7494 2.2235 2.39 13 0.30 3.8199 1.1829 1.2550 1.1972 1.0453 1.4162 1.0328 1.1720 1.2566 2.4139 2.1025 2.3675 2.1214 1.9477 2.9925 1.6962 2.1836 2.2810 0.40 3.7456 I.1553 1.2190 1.1758 1.0285 1.3889 1.0209 1.1579 1.2200 2.3949 2.0272 2.27 14 2.0396 1.9088 2.9060 1.6593 2.1508 2.2026 0.50 3.7195 1.1310 1.1891 1.1497 1.0148 1.3605 1.0135 1.1422 1.1918 2.4004 1.9606 2.11931 1.9760 1.8806 2.8413 1.6336 2.1036 2.1355 0.60 3.7092 1.1092 1.1688 1.1277 1.0048 1.3379 1.0106 1.1256 1.1670 2.4216 1.9108 2.1267 1.9296 1.8656 2.7974 1.6164 2.0670 2.0902 0.70 3.6996 1.0930 1.1505 1.1150 1.0038 1.3218 1.0095 1.1124 1.1542 2.4438 1.8816 2.0867 1.8902 1.8950 2.7652 1.6101 2.0403 2.0628 0.80 3.6950 1.0859 1.1404 1.1047 1.0048 1.3144 1.0079 1.1098 1.1465 2.44609 1.8527 2.0615 1.8582 1.8530 2.7403 1.6154 2.0173 2.0542 0.90 3.6899 1.0826 1.1365 1.1000 1.0049 1.3161 1.0087 1.1163 1.1487 2.4860 1.8015 2.0285 1.8303 1.8695 2.7587 1.6526 2.0041 2.0398 1.00 3.7133 1.0840 1.1498 1.1057 1.0172 1.3356 1.0082 1.1375 1.1534 2.4636 1.7981 2.0087 1.8209 1.8860 2.768 1 1.6526 2.0070 2.0363 2.00 3.7305 1.0846 1.1473 1.1067 1.0197 1.3394 1.0060 1.1434 1.1529 2.4996 1.7987 2.0031 1.8182 1.8924 2.7705 1.6491 2.0070 2.0349 3.00 3.7407 1.0851 1.1465 1.1070 1.0199 1.3401 1.0052 1.1444 1.1523 2.5083 1.8001 1.9980 1.8157 1.9000 2.7733 1.6443 2.0070 2.0334 10.00 3.7545 1.0855 1.1456 1.1074 1.0196 1.3410 1.0045 1.1453 1.1515 1.5483 1.9463 2.0183 RW 2.4368 1.7398 2.0431 1.7402 1.9433 2.7569 --1.3407 0.9889 1.1514 1.1253 3.6497 i.0703 1.1617 1.0847 1.0218 -Notes: The first entry of each cell is MSPE, while the second is MAPE. MSPEs are except for the CD, which is x 105. MAPEs are x lo*, except for the CD, which is Smallest MSPE and MAPE for each currency are underlined.
BF 2.6892 1.2547 2.4742 1.2089 2.3782 1.1892 2.3299 1.1813 2.2857 1.1716 2.2363 1.1588 2.1902 1.1461 2.1536 1.1378 2.1186 1.1351 2.1037 1.1525 2.1097 1.1575 2.1119 1.1588 2.1146 1.1602 2.0754 1.1506 x 104, x 10’.
models (p= 1,3,5). The MSPE and MAPE values, indicated with diamonds, correspond to use of the best t value.12 Clear and distinct patterns emerge. In-sample, MSPE and tions in the neighborhood of 10-20 percent relative to the random walk are pervasive. 13 Out-of-sample, however, the nonparametric predictor fares much less well. Typically, the loss associated with the nonparametric predictor is ‘*To make matters concrete, consider the leftmost diamond of the Canada graph in fig. 1. It corresponds to in-sample analysis with p = 1, and its height of approximately - 11 ir&ates that the nonparametric model fit by locally weighted regression has mean squared error approximately 11 percent smaller than that of the random walk. Now consider the first diamond to the right of the vertical dashed line. It corresponds to out-of-sample analysis with k= p= 1, and its height of approximately +4 indicates that the nonparametric model fit by lscally weighted regression has mean squared error approximately 4 percent larger than that of the random walk. 131t is interesting to note that the marginal benefit of including more lags (i.e. increasing p) is diminishing.
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
328
10
0
0
-11
-11
-22
-22
11
11
0
-11
, . ,
-11
-22
-22
11
11
0
-11
I ‘ ,
-11
-22
,
I
I I
5135
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I3Sl3Sl
I 4 I * I I2 OUT OF SAMPLE
II" i
(135/I35135135135~
I
I
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4
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Fig. 1, 1MSFE comparisons: in-sample vs. out-of-sample performance. I
8
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F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
6
329
. , I
I
6
-10 -14
!-
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I
I I
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Fig. 2.
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I12
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330
F.X. Diebo1dand J.A. Nason, Nonparametric exchatige rate prektion?
larger than that of the random walk, regardless of the number of lags included (p) or the forecast horizon (k). When loss reduction does occur it is t the loss typically very small, and must be tempered by the which are reductions graphed correspond to the ex post optimal not known in real time.
Using a powerful nonparametric prediction technique, we are generally unable to improve upon a simple random walk in out-of-sample prediction of ten major dollar spot rates in the post-1973 float. Our exchange rate results therefore corroborate those of others using different techniques and different asset prices, such as Prescott and Stengos (1988), who find that kernel techniques deliveT @loimprovement upon the random walk in the prediction of gold prices, 2nd White (1988), who is unable to improve forecasts of IBM stock returns using a neural network. Taken together, these results constitute fairly strong evidence against the existence of asset price nonlinearities that are exploitable for improved point prediction. Our reduced-form results carry important implications for structural models of exchange rate determination. In particular, it would appear unlikely that significant structural nonlinearities, whether of functional form or of regime-switching, have been operative in the post-1973 float. In complementary and independent work, Meese and Rose (1989) argue forcefully for this position. The research could of course be extended in a number of directions. The analysis could be made completely ex ante by choosing c in real time by cross validation, and multivariate generalizations might be undertaken. Computational considerations render some of these extensions infeasible at the present time.
Abel, A.B., 1988, Stock prices under time varying dividend risk: An exact solution in an inllnite horizon general equilibrium model, Journal of Monetary Economics 22,375-393. Baldwin, R. and R. Lyons, 1988,The mutual amplification effect of exchange rate volatility and unresponsive trade prices, NBER Working Paper no. 2677. Boothe, P. and D. Glassman, 1987, The statistical distribution of exchange rates, Journal of International Economics 22, 297-319. Brock, W.A., W.D. Dechert and J.A. Scheinkman, 1987, A test for independence based on the correlation dimension, SSRI Working Paper no. 8702 (Department of Economics, University of Wisconsin, Madison, WI). Clark, PK., 1973, A subordinated stochastic process model with finite variance for speculative prices, Econometrica 41, 135-155.
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
331
Cleveland, W.S., 1979, Robust locally weighted regression and smoothing _seatterpIots,Journal of the American Statistical Association 74, 829-836. Cleveland, W.A. and S.J. Devlin, 1988, Locally weighted regression: An approach to regression analysis by local fitting, Journal of the American Statistical Association 83, 596-610. Cleveland, W.S., S.J. Devlin and E. Grosse, 1988, Regression by local fitting: Methods, properties and computational algorithms, Journal of Econometrics 37, 87-l 14. Cootner, P., ed., 1964, The random character of stock market prices ( IT Press, Cambridge, MA). Diebold, F.X., 1986, Modeling persistence in conditional variances: A comment, Econometric Reviews 5, 51-56. Diebold, F.X., 1988, Empirical modeling of exchange rate dynamics (Springer-Verlag, New York). Diebold, F.X. and M. Nerlove, 1989a, The dynamics of exchange rate volatility: A multivariate latent-factor ARCH model, Journal of Applied Econometrics 4, l-22. Diebold, F.X. and M. Nerlove, 1989b, Unit roots in economic time series: A selective survey, in: T.B. Fomby and G.F. Rhodes, eds., Advances in econometrics: Cwintegration, spurious regressions, and unit roots (JAI Press, Greenwich, CT) forthcoming. Diebold, F.X. and P. Pauly, 1988, Endogenous risk in a rational-expectations portfolio-balance model of the deutschemark/dollar rate, European Economic Review 3% no. 1, 27-54. Domowitz, I. and C.S. Hakkio, 1985, Conditional variance and the risk premium in the foreign exchange market, Journal of International Economics 19,47-66. Engel, C, and J.D. Hamilton, 1988, Long swings in exchange rates: Are they in the data and do markets know it?, Manuscript (Department of Economics, University of Virginia, VA). Engle, R.F., 1982, Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation, Econometrica 50,987-1008. Engle, R.F., D.M. Lillien and R.P. Robbins, 1987, Estimating time-varying risk premia in the term structure: The ARCH-M model, Econometrica 55,391-408. Fama, E.F., 1965, The behavior of stock market prices, Journal of Business 38,34-105. Fama, E.F., 1976, Foundations of finance (Basic Books, New York). Flood, R. and P. Garber, 1983, A model of stochastic process switching, Econometrica 51, 537-564. Froot, F.A. and M. Obstfeld, 1989, Exchange rate dynamics under stochastic regime shifts: A unified approach, NBER Working Paper no. 2835. Gallant, A.R. and D.W. Nychka, 1987, Semi-nonparametric maximum likelihood estimation, Econometrica 55,363390. Gallant, A.R., D. Hsieh and G. Tauchen, 1988, On fitting a recalcitrant series: The pound/dollar exchange rate, 1974-1983, Manuscript (Graduate School of Business, University of Chicago, Chicago, IL). Hinich, M. and D. Patterson, 1985, Evidence of nonlinearity in daily stock returns, Journal of Business and Economic Statistics 3, 69-77. Hinich, M. and D. Patterson, 1987, Evidence of nonlinearity in the trade-by-trade stock market return generating process, Manuscript (Department of Government, University of Texas at Austin, TX). Hodrick, R.J., 1987, Risk, uncertainty and exchange rates, NBER Working Paper no. 2429, forthcoming in Journal of Monetary Economics. Hsieh, D.A., 1988, A nonlinear stochastic rational expectations model of exchange rates, Manuscript (Graduate School of Business, University of Chicago. Chicago, IL). Hsieh, D.A., 1989, Testing for nonlinear dependence in foreign exchange rates: 19741983, Journal of Business 62,339-368. Department of Economics, LeBaron, B., 1987, Nonlinear puzzles in stock returns, University of Chicago, Chicago, IL). Mandelbrot, B., 1963, The variation of certain speculative prices, Journal of Business 36, 394419. Meese, R.A. and K. Rogoff, 1983a, Empirical exchange rate models 01 the seventies: out of sample?, Journal of International Economics 14, 3-24. Meese, R.A. and K. Rogoff, 1983b, The out of sample failure of empirical exchange rate models: ed., Exchange rates and international Sampling error or misspecification?, in: J. Fren economics (University of Chicago Press, Chicago,
332
F.X. Diebold and J.A. Nason, Nonparametric exchange rate prediction?
Meese, R.A. and A.K. Rose, 1989, An empirical assessment of nonlinearities in models of exchange rate determination, Manuscript (School of Business, University of California, Berkeley, CA). Nason, J.M., 1988, The equity premium and time-varying risk behavior, Finance and Economics Discussion Series no. 8, Federal Reserve Board. Prescott, D.M. and T. Stengos, 1988, Do asset markets overlook exploitable nonlinearities? The case of gold, Manuscript (Department of Economics, University of Guelph). Priestley, M.B., 1980, Statedependent models: A general approach to nonlinear time series analysis, Journal of Time Series Analysis 1,47- Il. Robinson, P.M., 1983, Nonparametric estimators for time series, Journal of Time Series Analysis 4, 185-207. Robinson, P.M., 1988, Semiparametric econometrics: A survey, Journal of Applied Econometrics 3, 35-51. Scheinkman, J.A. and B. LeBaron, 1989, Nonlinear dynamics and stock returns, Journal of Business 64, 3 1l-338. Sims, C.A., 1984, Martingale-like behavior of prices and interest rates, Discussion Paper no. 205 (Department of Economics, University of Minnesota, Minneapolis, MN). Ullah, A., 1988, Nonparametric estimation of econometric functionals, Canadian Journal of Economics 2 1,625-658. Weiss, A.A., 1984, ARMA models with ARCH errors, Journal of Time Series Analysis 5, 129-143. Weiss, A.A., 1986, ARCH and bilinear time series models: Comparison and combination, Journal of Business and Economic Statistics 4, 59-70. Westerlield, J.M., 1977, An examination of foreign exchange risk under fixed and floating rate regimes, Journal of International Economics 7, 181-200. White, pi., 1988, Economic prediction using neural networks: The case of IBM daily stock returns. Working Paper no. 88-20 (Department of Economics, UCSD). Yakcwltz, S.J., 1987, Nearest neighbor methods for time series analysis, Journal of Time Series Analysis 8, 235-247.