Wolff, Christian C.P.

English

Research Papers in Economics

Vol. 50 (1989), No. 1, pp. 55-62 Jownol of Eeononuts
© by Springer-Verlag 1989
Exchange Rate Models and Parameter Variation:
The Case of the Dollar-Mark Exchange Rate
By
Christian C. P. Wolff, Maastricht, The Netherlands*
(Received August 22,1988; revised version received March 28,1989)
1. Introduction
Iti a recently published paper, Wolff (1987) used varying-
parameter estimation techniques, based on recursive application of
the Kalman filter, to improve the predictive performance of
monetary exchange rate models. Allowing estimated parameters to
vary over time was found to enhance the models' forecasting
performance for the Dollar-Pound, Dollar-Mark and Dollar-Yen
exchange rates. Contrary to earlier results in the literature (Meese
and RogofT, 1983), ex-post forecasts for the Dollar-Mark rate
compared favorably with those obtained from the naive random
walk forecasting rule.
In this paper the Wolff (1987) results are reexamined and
extended in two directions. First, test statistics are constructed to
test whether the monetary models do significantly better than the
random walk for the Dollar-Mark case. Second, a true ex-ante
forecasting experiment is performed in order to test whether the
earlier finding for the Dollar-Mark exchange rate remains valid in
a context where exchange rate forecasts are generated only on the
* This paper is drawn from Chapter 3 of the author's doctoral disser-
tation at the University of Chicago (Wolff, 1985). He is very grateful to the
members of his dissertation committee — Michael Mussa (Chairman),
Joshua Aizenman, Robert Aliber, Jacob Frenkel, David Hsieh, John
Huizinga, and Arnold Zellner — and to seminar participants at the
University of Chicago, the London Business School and INSEAD for
nian helpful comments.56 Chr. C. P. Wolff:
basis of infonnation that was available to economic agents at the
time. Such ex-ante forecasts based on monetary exchange rate
models have not been studied in the literature to date.
2. State-Space Versions of Monetary Exdiange Rate Models
Wolff (1987) studied exchange rate models of the following
form:
(1)
(2)
Here s (t) denotes the natural logarithm of the spot exchange rate
at time t, z it) is a vector of explanatory variables, c (t) is a vector of
time-varying coefficients and eit) and rjit) are uncorrelated white
noise disturbance terms. Equation (1) represents a dynamic
version of a class of monetary exchange rate models. The vector
zit) contains explanatory variables which are prescribed by
different versions of the monetary models. Two variants of the
monetary models were studied in Wolff (1987); the Frenkel-Bilson
model (due to Frenkel, 1976; and Bilson, 1978) and the Dom-
busch-Frankel model (due to Dombusch, 1976; and Frankel,
1979). For a more precise description of the models, see Wolff
(1987). In equation (2) parameter variation in the vector cit) is
allowed for to address possible parameter instability due to a
number of factors (on which, see Wolff, 1987). Equations (1) and
(2) are in so-called state-space format, so that the Kalman filter
can be employed for recursive estimation and prediction.
In Wolff (1987) an ex-post forecasting experiment was
performed based on equations (1)—(2). On the basis of a monthly
dataset it was found that allowing estimated parameters to vary
over time enhanced the models' forecasting performance for the
Dollar-Mark, Dollar-Yen and Dollar-Pound exchange rates. The
forecasting experiment was an ex-post experiment in the sense that
actual, realized values of tlw vector of explanatory variables z(t)
were used. Contrary to eariier results in the literature (Meese and
Rogoff, 1983) ex-post forecasts for the Dollar-Mark rate compared
favorably with those obtained from the naive random walk fore-
casting rule for all forecasting horizons that were studied.Exchange Rate Models and Parameter Variation 57
3. Significance Tests
In this section significance tests are constructed in order to
assess whether the state-space structural models for the
Dollar-Mark exchange rate predicted significantly better than the
random walk forecasting rule. Since the model forecasts are
generated for overlapping periods, the prediction errors for a given
horizon of n months (« > 1) will not be independently distributed.
This will require special care in the construction of a suitable test
statistic.
The approach that we take to the construction of test statistics
is the fallowing. First, we form the time series e^it, n) and e^it, n)
as follows;
e« (r, n) = Iw, (t, n)\-\w2(t, n)\ (3)
e, (t, n) = [w, (r, n)^] - [w^ (/, n)Y (4)
for all forecasting horizons n (n = 1, 3, 6, 12, 24, 36) and all points
in time f (/ = 1, 2, 3,..., TJ. In equations (3) and (4) w,(f, n) is the
forecast error resulting from model i's forecast, generated at time t,
of the logarithm of the spot exchange rate at time t + n. The
subscript i = 1 refers to the random walk model and / = 2 refers to
one of the state-space monetary models. T^ is the sample size of the
series [e^ (f, n)] or [e^ (f, n)] for a given horizon n. The number
e^ {t, n) is the difference in absolute forecast errors between the
random walk model and one of the state-space models for a given
combination of t and n, and e^ (t, n) is the difference in squared
forecast errors.
The Mean Absolute Error (MAE) of a state-space model's
forecast at horizon n is said to be significantly lower (higher) than
the MAE of the random walk forecasting rule if the mean of the
series [«„(/, n); t^-l, 2, 3,.. .,T„] is significantly greater (smaller)
than zero. Similarly, the mean square error (MSE) of a state-space
model's forecasts at horizon n is said to be significantly lower
(higher) than the MSE of the random walk forecast if the mean of
the series [e,(<,«); f = l, 2, 3, ..., T„] is significantly greater
(smaller) than zero.
Since the forecasts are overlapping, we use the Hannan effi-
cient estimator (Hannan, 1963) to compute asymptotic (-statistics.
Hannan efficient estimation is essentially a procedure for doing
Generalized Least Squares (GLS) in the frequency domain,
allowing for arbitrary patterns of serial correlation. The
construction ofthe test statistics involves the following steps;58 Chr. C. P. Wolff:
(a) The series [e^(/; n); / = 1, 2, 3, .... T„] or [e,(t, n); ( = 1, 2, 3,
..., r^] is regressed on a constant term, using Ordinary Least
Squares.
(b) The residuals from this regression and the dependent variable
are sent to the frequency domain.
(c) A consistent (i. e. smoothed) estimate of the spectrum of the
residuals is computed.
(d) The dependent variable is Fourier transformed, then filtered
[using the estimated spectrum from step (c)] and inverse trans-
formed.
(e) The filtered dependent variable is sent back to the time domain
and is regressed on a constant term. The large-sample
/-statistic on the constant term in this regression is the test
statistic we set out to compute.
The results of this procedure for the Dollar-Mark exchange rate
are presented in Table 1 (for MAEs) and Table 2 (for MSEs). Both
the Frenkel-Bilson and Dombusch-Frankel versions of the state-
space monetary models are considered. The monthly data that are
used are the same as those described in Wolff (1987). The esti-
mation and prediction procedure is also the same. The period over
which forecasts are generated ranges from November 1976
through April 1984. As in Meese and Rogoflf (1983) and Wolff
(1987), natural logarithms of exchange rates are predicted, rather
than their simple levels, in order to circumvent any problems
arising from Jensen's inequality.
Table 1. Comparison of Mean Absolute Prediction Errors of State-Space Monetary
Models and the Random Wcdk Forecasting Rule
Horizon (n)
(Montte)
1
3
6
12
24
36
MAE
(RW)»
2.52
4.52
6.65
10.58
19.1J
24.83
MAE
(FB)»
2.42
4.40
5.94
S.42
12.13
13.89
MAE
(DF)'
2.40
4.29
6.09
S.28
11.50
11.55
Test-
aatistic I""
0.62
0.12
0.61
1J2
2.57*
3.77**
Test-
Statistic T
0.S6
0.46
0.64
1.90
2.70**
3.75**
» MAE (RW) is the MAE of the random walk forecast. All MAEs are approxi-
mately in percentage terms.
•> MAE (FB) is the MAE ofthe state-space vetsion of tiw Frraikel-Bason model.
" MAE (DF) is the MAE of Ae state-space version of the Etombusch-FranKei
model.
<• Tests whether MAE (FB) is significantly different from MAE (RW).
' Tests whether MAE (DF) is signiftcantly difTerent from MAE (RW).
* Denotes statistical significance at the 5% level.
** Denotes statistical significance at the 1% level.Exchange Rate Models and Parameter Variation 59
Table 2. Comparison of Mean Square Prediction Errors of State-Space Monetary
Models and the Random Walk Forecasting Rule
Horizon (n)
(Months)
1
3
6
12
24
36
MSE
(RW)«
11.39
30.91
65.69
161.03
431.21
708.49
MSE
(FB)»>
10.86
27.68
51.71
109.57
223.31
283.95
MSE
(DF)'=
10.77
26.57
49.54
98.76
200.56
230.23
Test-
Statistic I**
0.54
0.61
0.90
1.54
2.26*
3.03**
Test-
Statistic 2=
0.72
0.97
1.28
2.13
2.64**
3.13**
^ MSE (RW) is the MSE of the random walk forecast. All MSEs are approxi-
mately in percent squared.
•> MSE (FB) is the MSE ofthe state-space version ofthe Frenkel-Bilson model.
' MSE (DF) is the MSE of the state-space version of the Dombusch-Frankel
model.
<" Tests whether MSE (FB) is significantly different from MSE (RW).
' Tests whether MSE (DF) is significantly difTerent from MSE (RW)-
* Denotes statistical significance at the 5% level.
** Denotes statistical significance at the 1% level.
The test statistics presented in Tables 1 and 2 indicate that the
state-space versions of both models did statistically significantly
better than the random walk forecasting rule for the longer
horizons (two years and up). The improvement was not significant
for the shorter horizons. These results should be interpreted with
caution, not only because the test statistics pertain to large
samples, but also because the overlapping nature of the forecasts
reduces the amount of information that is effectively present in our
sample, relative to a nonoverlapping sample with an equal number
of observations.
4. An Ex-Ante Forecasting Experiment
In the literature to date, only ex-post forecasts on the basis of
structural exchange rate models have been studied. Because Meese
and Rogoff (1983) showed that such ex-post forecasts were domi-
nated by simple random walk forecasts, it did not seem fruitful to
consider ex-ante forecasts (using predicted rather than realized
values of the explanatory variables) based on these models.
In the context of varying-parameter versions of the monetary
niodels, however, it is interesting to perform an ex-ante forecasting
experiment for the Dollar-Mark case in order to test whether the
finding that exchange rate forecasts based on state-space veraions
of the monetary models outperform the random walk forecast,
remains valid in a context where exchange rate forecasts are60 Chr. C. P. Wolff:
generated only on the basis of infonnation that was available to
economic agents at the time.
In order to be able to perform such an ex-ante forecasting
experiment, univariate time series models of the class advocated
by Box and Jenkins (1976) are used to forecast the variables that
the monetary models take to be exogenous. Tlie procedure is as
follows. Initially the time series models are estimated up through
the first forecasting period, November 1976, and predictions of the
exogenous variables up to 36 months ahead are generated. The
time series models are constructed for differences between
domestic and foreign variables rather than for individual variables.
The state-space models are also estimated up through November
1976 and exchange rate forecasts are then generated using the
predictions from the time series models as expectations of future
explanatory variables. Then December 1976 data are added to the
sample, the time series model are reestimated and again used for
the generation of predictions of the exogenous variables. The
parameters of the state-space models are then updated using
December 1976 data and exchange rate predictions are formed as
before. Tliis recursive process continues until forecasts are formed
using April 1984 data. The results are presented in Table 3.
Table 3. Summary Statistics on the Forecasting Performance of State-Space
Monetary Models for the Dollar-Mark Exchange Rate; Explanatory Variables are
Predicted on the Basis of Univariate Time Series Models
Horizon
(Months)
ME»
Frenkel-Bilson Model
1
3
6
12
24
36
0.27
0.89
1.66
3.44
7.02
2.10
MAE
2.48
4.48
6.28
9.54
17,79
22.66
Dombusch-Frankel Model
1
3
6
12
24
36
0.37
1.23
2.38
4.67
7.91
0.91
2.46
4.43
6.78
10.33
20.11
23.40
RMSE^"
3.36
5.45
7.63
11.57
20.65
27.06
3.35
5.45
8,03
12.53
22.98
27.53
U-Statistic'^
0.996
0.980
0.942
0.912
0,994
1.016
0.993
0.980
0.990
0.987
1.107
1.034
89
87
S4
78
66
54
89
87
84
78
66
54
.—
' ME denotes the Mean Error of toe Forecasts. The ME is approximately m
percentage tenns. .
•> RMSE denotes the Root Mean Square Error of the Forecasts. The RMSE is
approximately in percentage terms. ,
' Tlie U-Statistic is the ratio of the model's RMSB to the RMSE of the random
walk forecast.
'' A^denotes the number of observations.Exchange Rate Models and Parameter Variation 61
When comparing the summary statistics in Table 3 above with
those in Tables 4 and 5 in Wolff (1987), it is noted that the ex-ante
forecasts are on average considerably less accurate than the ex-post
forecasts: knowledge of the realizations of the explanatory vari-
ables of the monetary models is useful when generating forecasts.
The ex-ante forecasts are especially less accurate than the ex-post
forecasts for the longer horizons. The U-Statistics in Table 3 are
smaller than one in many cases, indicating that the state-space
monetary models did better than the random walk even on an
ex-ante basis (at least for horizons up to one year).
5. Conclusions
In this article a recent finding that ex-post exchange rate fore-
casts on the basis of state-space versions of monetary exchange
rate models outperform the simple random walk forecasting rule
for the Dollar-Mark exchange rate was reexamined in some detail.
Tests statistics were constructed to test whether the models did
significantly better than the random walk. These statistics indicate
that the models did significantly better than the random walk at
horizons of 2 years and up, but for the shorter horizons the
improvement was not significant.
A true ex-ante forecasting experiment was performed to test
whether the earlier finding remains valid in a context where
exchange rate forecasts are based only on information which was
available to economic agents at the time. The results indicate that
the state-space monetary models did better than the random walk
model, even on an ex-ante basis, for horizons up to about one year.
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Address of author: Prof. C. C. P. WolfF, Rijksuniversiteit Limburg,
Faculty of Economics and Business Administration, P. O. Box 616,
NL-6200 MD Maastricht, The Netherlands.

A Monetary Approach to the Exchange Rate: Doctrinal Aspects and Empirical Evidence.&quot;

Empirical Exchange Rate Models of the Seventies: Do They Fit Out of Sample?.&quot;

Exchange Rate Models, Parameter Variation and Innovations: A Study on the Forecasting Performance of Empirical Models of Exchange Rate Determination.&quot; Unpublished Ph.

Exchange Rates, Innovations and Forecasting.&quot; Joumal of Intemational Money and Finance 7: 49—61. Address of author: Prof.

Expectations and Exchange Rate Dynamics.&quot;

The Monetary Approach to the Exchange Rate.&quot;

Time Series Analysis: Forecasting and Control.

Time-Varying Parameters and the Out of Sample Forecasting Performance of Stmctural Exchange Rate Models.&quot;

Exchange rate models and parameter variation: The case of the dollar-mark exchange rate.

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Exchange rate models and parameter variation: The case of the dollar-mark exchange rate.

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