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ARE EXCHANGE RATES EXCESSIVELY VOLATILE?
AND WHAT DOES "EXCESSIVELY VOLATILE" MEAN, ANYWAY?
by
Leonardo Bartolini and Gordon M. Bodnar
Federal Reserve Bank of New York
Research Paper No. 9601
January 1996
This paper is being circulated for purposes of discussion and comment only.
The contents should be regarded as preliminary and not for citation or quotation without
perrni.•sion of the author. The views expressed are those of the author and do not necessarily
reflect those of the Federal Reserve Bank of New York or the Federal Reserve System.
Single copies are available on request to:
Public Information Department
Federal Reserve Bank of New York
New York, NY 10045
Are Exchange Rates Excessively Volatile?
And What Does "Excessively Volatile" Mean, Anyway?
Leonardo Bartolini
Research Department
Federal Reserve Bank of New York
and
Gordon M. Bodnar
The Wharton School
University of Pennsylvania
January 1996 •
Abstract
Using data for the major currencies from 1973 to 1994, we apply recent tests of asset price volatility
to reexamine whether exchange rates have been "excessively" volatile with respect to the predictions
of the monetary model of the exchange rate and of standard extensions that allow for sticky prices,
sluggish money adjustment, and time-varying risk premia. Consistent with previous evidence from
regression-based tests, most of the models that we examine are rejected by our volatility-based tests.
In general, however, we find that exchange rates have not been excessively volatile relative to
movements of their determinants, with respect to the predictions of even the most restrictive version
of the monetary model. Alternative measures of volatility, however, may disguise the cause of
rejection as excessive exchange rate volatility.
JEL Classification Numbers: F3 l, E44.
• Research for this paper, which is to be published in IMF Staff Papers, was conducted when the first author was affiliated
with, and the second author was visiting, the Research Department of the IMF. The authors thank E. Prasad for providing
them a Gauss code of the Hodrick-Prescott filter, P. Isard, P. Clark, C. Osler, and participants in seminars at the IMF and
at the Federal Reserve Bank of New York for comments, and S. Mursula for technical assistance.
Are exchange rate fluctuations justified by changes in their "fundamental" determinants? In an
efficient market with rational investors, ei,change rates are forward-looking prices that reflect
anticipated changes of relative demands and supplies of two monies. Hence, their volatility should
reflect investors' expectations of changes in the determinants of money stocks, such as incomes and
interest rates. Given a model of exchange rate determination, market efficiency places restrictions
on the relative volatility of exchange rates and of their determinants, which can be tested to yield
insight on the validity of the underlying model. A family of such tests is applied in this paper, to
assess whether some popular exchange rate models are capable of matching observed patterns of
exchange rate volatilities over the post-Bretton Woods period.
Our interest in exchange rate volatility is motivated by both analytical and policy concerns.
Exhibit A in policy discussions of exchange rate volatility is often represented by a chart such as
Figure I, pointing to the dramatic increase in exchange rate volatility for major currencies since the
breakdown of the Bretton Woods system. This evidence is often accompanied by an expression of
concern that "private markets may not always anchor their behavior to economic fundamentals, thus
making their responses susceptible to contagion and bandwagon effects that may be disruptive and
detrimental to economic performance" (Mussa et al., 1994, pg. 18). Excessive exchange rate volatility
is also often advocated as ground for sand in the wheels of currency markets (see, for instance,
Eichengreen, Tobin, and Wyplosz, 1995).
Be:cre policies to inhibit market response be advocated, however, one ought to verify that the
volatility of exchange rates does not simply reflect that of their underlying determinants. Also, as
these tests are typically joint tests of market efficiency and of a specific exchange rate model, the
results must be conditioned on the particular model adopted as their basis.
I
Given the prominence in research on exchange rates of the monetary model and of its variants,
early studies of exchange rate volatility, including Huang (1981), Vander Kraats and Booth (1983),
and Wadhwani (1987), followed Shiller's (1981) work on stock price volatility to construct "variance
bounds" tests of the monetary model of the exchange rate.
Invariably, these studies found the
volatility of exchange rates since the breakdown of the Bretton Woods system to exceed the model's
predictions, leading their authors and several commentators (see, for instance, Levich, 1985), to assert
either the inefficiency of foreign exchange markets, or the invalidity of the underlying model, or both.
Later studies, however, questioned those results. Diba (I 987), for instance, showed that the
tests of Huang (1981) and Vander Kraals and Booth (1983) were vitiated by an erroneous
transformation of annual semi-elasticities of money demand to interest rates into their high-frequency
counterparts. When correctly calibrated, those tests failed to reject the monetary model's predictions,
although that failure was likely to reflect the weak restrictions placed by those tests on the model.
Early volatility tests of asset price models were also shown to suffer from serious statistical biases,
leading them to reject the underlying model too often in finite samples (see, for instance, Kleidon,
1986, and Marsh and Merton, 1986). To address some of these concerns, Gros (1989) used improved
volatility inequalities to present new evidence of excessive exchange rate volatility. His analysis,
however, was not based on a formal statistical test, was subject to the same calibration problems
pointed out by Diba (1987), and involved measuring exchange rate volatility in a way that was likely
to overstate his finding of excessive volatility. In summary, after fifteen years of research, evidence
on the ab.,ity of the most popular exchange rate models to match the observed variability of exchange
rates rests largely on mis-specified and weak statistical tests. Not surprisingly, several authors have
expressed a perception of futility when reviewing the inconclusive state of research on exchange rate
volatility (see, for instance, Frankel and Meese, 1987, pp. 134-36).
2
In this paper we use recent volatility-based tests of asset price models to provide new, clearer
evidence on the ability of some popular exchange rate models to match observed patterns of exchange
rate volatilities over the post-Bretton Woods period.
We apply the methodology developed by
Mankiw, Romer and Shapiro (1991) in the context of stock price models, and use data for the eight
major currencies from 1973 to 1994, to test the textbook flex-price version of the monetary model,
as well as more general models including some with sticky prices, sluggish adjustment of money
stocks, and time-varying risk premia.
Our tests show these models to be broadly rejected by the data, but that this rejection cannot--in
general--be attributed to excessive exchange rate volatility, but rather to the inconsistency of these
models with the assumed efficiency of currency markets. Hence, our tests confirm the wisdom from
standard regression-based tests of these models (see Hodrick, 1987, for a survey). However, since
our tests possess better statistical properties than previous regression-based tests (in particular, they
are unbiased in small samples), they provide even stronger evidence against the joint hypotheses of
the monetary model (or of its extensions) and of market efficiency. Furthermore, because they are
explicitly constructed as volatility tests, our tests help clarify the role played by exchange rate
volatility in the models' rejection.
Specifically, our tests point to the importance of choosing an economically meaningful
definition of exchange rate volatility as a basis for the tests, so as to avoid disguising a model's
rejection as evidence of excessive volatility. The tests show that when exchange rate volatility is
defined--t:aditionally--as the average of conditional or unanticipated exchange rate changes, then there
is no evidence that exchange rates may have been excessively volatile with respect to the predictions
of even the most restrictive version of the monetary model. Evidence of excessive exchange rate
volatility, however, may emerge on the basis of alternative definitions of volatility.
3
Our findings, we hope, will contribute to future research on exchange rate volatility being
more
clearly defined in its scope and, perhaps, to greater caution when formulating claims of
"excessive"
exchange rate volatility. Our imposing on the data the straightjacket of the monetary model
should
even strengthen our conclusions: if exchange rates do not appear to be excessively volatile
even with
respect to a framework predicting their close movement with money and income alone, the
likelihood
that evidence of excess volatility may be uncovered based on more flexible models appears
even more
remote.
I. Volatility tests of the monetary model
1.1. The monetary model of the exchange rate. The standard specification of the monetar
y model
involves two equations describing domestic and foreign money demands, a purchasing power
parity
equation, and an uncovered interest parity equation. Assuming, as is standard, that the domesti
c and
foreign money demand parameters are the same, the model can be written as:
m, - P,
= ~Y,
- ai1
,
. .= y,.- az,.
~
m, - p,
P,
i,
where
= s,
'
+ P,
.. =E,[s,.,
- 1,
(I)
(2)
•
(3)
- s,
l.
(4)
m,, P,, and y denote (log) domestic money supply, prices, andreal income, respectively, andi,
1
denotes the interest rate at time t on deposits maturing at time t+ I. Foreign variables
are denoted
with an asterisk. The (log) exchange rate, defined as the domestic price of a unit of foreign
currency,
4
is denoted by s,, while E, [. ] denotes the rational expectation operator conditional on information
available at time t.
The exogenous terms Y, and m, may incorporate a stochastic component;
alternatively, a stochastic ("velocity") term can be added to introduce ucertainty in the right-hand side
of (I). For ease of exposition, we shall adhere to the convention that m, already incorporates an
exogenous stochastic shock.
Equations (I)-(4) can be collapsed into
s, =
f,
I +a
+
a
[ s, .. 1
-1--E,
+a
l
,
(5)
where
(6)
Equations (5) and (6) express the exchange rate as the sum of current "fundamentals", f,, plus
a linear function of its own value at time t+ 1. Solving (5) forward up to time t+h, yields
1
s
' =
1 (•I +a
~
(
I a+a )' E,[J,.,]
J + ( I a+a )• E,[S,•• ].
(7)
The standard assumption in the literature is to assume the absence of exchange rate bubbles,
i.e., that Hm ( ~ ) ' E,[J,.,]=0, and solve equation (7) forward solely in terms of fundamentals:
i ➔ = I +a
(8)
The tests considered in this paper, however, are robust to the presence of bubbles, and can be
performed directly on (7): if s"• incorporates a bubble term (i.e., a capital gain that reflects solely
5
the anticipation of a future currency transaction), so does s,, and equation (7) remains valid. In most
of our tests, the investment's holding period, h, is set at 3, as forward markets tend to be most liquid
for 3-month maturities, and the power of our tests falls when h increases much above 3. Tests for
different values of h are discussed below.
1.2. Volatility tests of the monetary model.
Let us now define the perfect-foresight (or
"fundamental") exchange rate, s, ·, as the value that the exchange rate would take if investors could
predict with certainty future fundamentals, as well as the exchange rate at t+h. The fundamental
exchange rate s, • is obtained from equation (7) by dropping the expectation operator:
s1
. = -I- h-1 (-a- J; f. .
I +a ( ~
I +a
,.,
J ( I a a Jh
+
__
+
s
1
·•
(9)
,
so that, by definition, s,=E,[s,•] under the assumptions of the monetary model.
We must now choose a "benchmark" exchange rate, denoted by s,0 , about which to measure
the volatility of both market and fundamental exchange rates. The need for a benchmark rate reflects
analytical and statistical considerations. First, there is simply not a unique way to define exchange
rates' volatility, except with respect to a reference benchmark (be it its sample mean or other
variables). Second, the notorious non-stationarity of exchange rates requires measuring exchange rate
movements with respect to a specific stochastic trend.
Thac are two main requirements for the choice of s,0 : that it be known to investors at time
t, and that the differences s, -s,0 and s,• -s,0 be stationary. There are many alternative ways to
0
choose s,0 , however, so as to satisfy these requirements. For instance, s, could be defined as the
model's prediction at time t-1 of the exchange rate at time t, which can be obtained by lagging
6
equation (5) once and solving for E,_ 1[s,]:
f,_I
(10)
a
When s,0 is chosen in this fashion, the series s,'-s,° and s, -s,0 describe fundamental and
actual exchange rate surprises, based on the predictions of the monetary model.
0
Alternatively, exchange rate volatility could be measured by setting s, =s,~L• where L is a
suitable lag. For instance, it is customary to focus on conditional volatilities, defined (with L=l) as
the volatility of the first difference of exchange rates, s,-s,_ 1 • Our task, in this case, would be to
assess the consistency of the model's predictions of market and "fundamental" exchange rate changes
from the last known realization of the exchange rate.
Yet another possibility would be to choose s,0 as some "naive" exchange rate forecast, for
instance as the value that the exchange rate would take if investors expected fundamentals to evolve
as a random walk. Under this assumption (and, in this case, under the no-bubble condition that
_Jim
1 ➔ 00
(-5:._J'
E,[J,.,]=O), Equation (7) can be solved forward with E,[J,.,] =f,, as implied by the
I +a
random walk hypothesis, to obtain s, =f,: the exchange rate itself should follow a random walk under
the assumptions of the monetary model. Note that this "naive" forecast need not be a rational one
(although there is, indeed, a considerable amount of evidence that exchange rates may be well
0
0
approximated by random walks): as long ass, =f, is known to investors at time t, and s, -s, ands,• -s,
are static,:ary, then s,0=!, is an acceptable benchmark for our volatility tests. Indeed, Mankiw,
Romer, and Shapiro (1985, 1991) have suggested a very similar benchmark when testing for excess
stock price volatility, by assuming stock dividends to follow a random walk. Gros (1989) followed
their lead, measuring exchange rates with respect to a random walk benchmark. These issues, and
7
their implications for the volatility tests performed in this paper, are further discussed in the
following.
Now, with rational expectations, the f9recast error
s,'-E,[s,·]=s,'-s,
should be uncorrelated
with variables known at time t, including s, and s,0 • This implies that we should have
E,[ (s,• -s,) (s, -s,
0
)]
=
O.
(11)
Therefore, squaring both sides of the identity
(12)
taking expectations, and using equation (11 ), yields
(13)
or
(14)
Therefore, the monetary model implies the testable restriction q, =O, and hence the restriction
E[q,]=O: the sample mean of the q,'s,
q, should be close to zero if the model describes correctly
the tlynar:.ics of exchange rates. As noted above, this test is robust to the presence of exchange rate
bubbles, since if the market exchange
s, incorporates a bubble term, so do s,•
and
s,
0
,
and equation
( 14) remains valid.
Thus, we can construct a test of the monetary model as follows. First, we can compute the
8
fundamental exchange rate, s, •, and the benchmark exchange rate, s,0 , from (9) and (I 0), after
calibrating the money demand parameters a and ~- Then we can use an asymptotic distribution of
the sample mean
q and reject the model when q is significantly different from zero.
Method of Moments distribution of
q (see Bollerslev and Hodrick,
A Generalized
1992) is a normal distribution
(15)
whose variance is estimated by
1
V= C(O)+ 2 "'L.,,,,1
C(j), with C(j)= h - j "
h
T
L,~1+1
(q,.,q,.,-;),
T
which is
robust to heteroskedasticity and serial correlation in the error terms.'
Equation (13) also implies
E[(s,•-s,)2]:, E[(s, -s,•)'],
(16A)
E[(s,-s,
(16B)
0
0
E[(s, -s,·rl,
0
)']:,
where the expectations are now taken unconditionally. Should either one of these inequalities be
violated, then E,[q,]<0 (although the opposite is not necessarily true), thus suggesting to use (16A-B)
as diagnostics for the model. Inequality (I 6A), in particular, states that the market exchange rate, s,,
should forecast the behavior of the fundamental rate, s, •, better than the benchmark rate, s,0 , in terms
of the usual mean-square error criterion. Inequality (16B) states that the market exchange rate should
be less volatile around the benchmark rate than the fundamental rate. This latter inequality provides
an "excess volatility" test of the model, and its intuition is simple: the fundamental exchange rate
1
Serial correlation is often a problem when using overlapping observations. In our case, in particular, observations
overlap over the holding period h.
9
should deviate from the benchmark rate by as much as the market rate does, plus a forecast error
s, • - s,. If markets are efficient and the monetary model correctly describes exchange rate dynamics,
this forecast error should not be systematically related to information available to investors. The
volatility of the fundamental exchange rate around the benchmark rate, therefore, should exceed that
of the market rate.
1.3, Relation to alternative tests of the model. Two points should be noted about the test procedure
just outlined. First, there is a close relationship between the volatility-based tests presented here and
previo_us regression-based tests (see Hodrick, 1987, for a survey). Recall that the starting point of our
analysis is the condition that s,· -s, and s,-s,
0
should be uncorrelated if exchange rates reflect
information available to market participants and the monetary model correctly describes the.dynamics
of exchange rates. Using regression analysis, the natural test of this hypothesis would be to regress s, • -s,
on s,-s,
0
,
and test if the regression coefficient
(17)
is zero. A non-zero ft 1 implies that the prediction error s, •-s, can be forecast at time t. This, among
other things, implies that s,. 1- E ,[ s,, 1] is forecastable, since
s, ·- s,=
:a (
1
s,,i - E ,[ s,,i ]), i.e., that there
is a systematic bias between exchange rate predictions and realizations.
In ,;,omparison with (17), the volatility tests implemented in this paper reject the model when
the mean of the q 1 'sis zero. Now, since
ii can be written as
ii= -2.. L (s,-s, )(s,' -s,) ·2
0
T,
10
(18)
a link between the regression-based and the volatility-based methods is apparent. Both methods
involve testing whether
~~
( s,- s,
0
) (
s, •-s,), suitably standardized, is significantly different from
zero, and both tests provide (when statistically significant) evidence of excess returns' forecastability,
i.e., that either markets are inefficient, or the assumed exchange rate model is invalid, or both. There
are two main differences between the two methods, however.
The first difference is that the test used in this paper is directly linked to measures of exchange
rate volatility and to corresponding volatility inequalities. These inequalities allow testing whether
"excessive" volatility (suitably defined) is indeed a cause of failure of the model.
The second difference is more technical: the volatility-based tests used here have been shown
to exhibit better statistical properties than the corresponding regression-based tests. Specifically,
Mankiw and Shapiro (1986) and Mankiw, Romer, and Shapiro (1991) present Montecarlo simulations
showing that the finite-sample distribution of a sample-mean
q is well approximated by its asymptotic
distribution, namely, that the volatility-based test tends to reject the underlying model on average the
right number of times. In contrast, the finite-sample distribution of the regression coefficient ft, is
poorly approximated by its asymptotic distribution.
In particular, if the dependent and the
independent variables are correlated, and the independent variable is itself serially correlated--which
is certainly the case in our tests, where the same "fundamentals" drive both the dependent and
independent variables--, then regression-based tests are systematically biased, tending to reject the
underlying model too often in finite samples.
Ne
A~,
the ability of the test to reject the model, as well as the interpretation of the test results,
depend crucially on the particular choice of s,0 • The usefulness of the rejection of the model, in
particular, depends on s,0 being a useful benchmark for the measurement of exchange rate volatility.
Establishing the best benchmark is a task that goes well beyond the scope of this paper, particularly
11
since different benchmarks are bound to be useful in different contexts. In welfare analysis, for
instance, the appropriate definition of "exchange rate volatility" depends on agents' attitudes toward
risk, the nature of adjustment costs, etc. Nevertheless, it is clear that the usefulness of a particular
benchmark is likely to depend on whether exchange rates tend to gravitate, in some loose sense,
around it. Few scientists seem to find useful a description of stars' and planets' position in space
with respect to the moon, nor to find useful the (formally correct) statement that the sun is more
volatile than the earth about the moon.
As the issue of choosing a suitable benchmark cannot be resolved in a statistical context--only
on the basis of a well specified economic model--, in our statistical analysis we follow an eclectic
approach. We present results for a variety of choices of s,0 , drawing from common usage in research
and policy analysis, in order to highlight the implications of alternative definitions of "exchange .rate
volatility".
We focus, in particular, on the conditional volatility of exchange rates and on the
volatility of their unanticipated movements. We present tests that use the model's one-period-ahead
prediction of the exchange rate as a benchmark (see equation (IO)), to test whether unanticipated
changes in exchange rates are consistent with movements in their determinants. We also present tests
that use the lagged exchange rate as a benchmark, to test whether total changes in exchange rates are
justified by movements in their determinants. Finally, we present tests where the benchmark rate is
set at the value that the exchange rate would take if investors believed fundamentals to behave as
random walks. The tests can be applied in a straightforward way to other definitions of s,0 •
Th0::gh our tests' dependence on the benchmark
s,
0
may seem unfortunate, this dependence
simply reflects the intrinsic ambiguity of the concept of exchange rate volatility: there is simply no
unique way to measure exchange rates' volatility--just as there is no unique way to measure planets'
movements--, except with respect to a specific benchmark. This is the main point we wish to
12
emphasize in this paper, that this ambiguity should be explicitly recognized.
II. More general specifications of the model
The test procedure described in the previous section can be applied to a variety of exchange
rate models. We discuss here some extensions of the monetary model that allow for slow adjustment
of prices and money holdings, and for violations of the interest parity conditions. The point here is
not to insist on the empirical accuracy of these specifications, although we draw from the empirical
literature in calibrating our models.
The aim, rather, is to illustrate the flexibility of the testing
procedure, and to provide preliminary evidence of the robustness of our results.
Our qualitative
findings tum out to be rather robust, suggesting that greater effort to incorporate empirically accurate
models, while welcome, is unlikely to overturn our main results.
II.I. Sticky prices. Following Mussa (1985) and Gros (1989), we adopt a sticky-price version of
Purchasing Power Parity (PPP), much along the lines of the standard Dornbusch model.
In this
model, the real exchange rate s, + p, •-p, moves toward its PPP value at the rate 0, as in
(19)
In (,.:jUation (19), when 0=0 the real exchange rate follows a random walk with no tendency to
revert to PPP. When 0= 1 the exchange rate is expected to converge to PPP in a single period.
Replacing Equation (3) with Equation (19) in the model, some tedious but straightforward
algebra yields the "fundamental" sticky-price exchange rate, solved forward up to time t+h:
13
s,
where y= ( l - 9 )a, and
1+a
s;
. s,-· =
(20)
denotes the fundamental exchange rate for the flex-price model (see
equation (9)). The expectation at t-1 of the market exchange rate at time t is
s,
o
=
s,_ 1 -
f,_ I
_
.
+
P,-1 -P,-1
()(
(21)
()(
For 0=1, the sticky-price model degenerates into the flex-price model; for smaller values of
0, the sticky-price model predicts greater volatility of exchange rates in response to changes in their
determinants.
Except for this redefinition of variables, the test procedure remains the same as
described in the previous section.
11.2. Sluggish money adjustment. We can also relax the assumption that real money balances adjust
instantaneously to their equilibrium value. Partial adjustment equations of the form
m, - P, = A( ~Y, - ai,) + (1-1,,)(m,_ 1
m, - p,' = A( ~y,· - ai,')
+ (
P,) ,
-
1 -1,,)(m,•_ 1
-
p,') ,
(22)
(23)
have beer'. widely used in the literature on money demand functions. The terms ( l-1,,)(m,_ -p,) and
1
(1-A)(m,~ 1 -p,') include lagged money terms and contemporaneous price terms, reflecting an
adjustment process of money balances specified in nominal terms. Alternatively, the adjustment term
could have been specified as ( 1 -A)(m,_ -p,_ ) (and similarly for the foreign equation), if the
1
1
14
assumed slow response were of real money balances. We use the former specification because it
allows the familiar forward solution of the exchange rate in terms of fundamentals, and because of
its superior empirical performance (see Fair,. I 987). 2
Thus, the coefficients~ and a in (22) and (23) describe the long-run income elasticity and longrun interest semi-elasticity of money demand, respectively; 'J. , measures the speed of adjustment of
real money balances to their steady state, and takes values between zero and one: when 'A= I, real
money balances adjust immediately to their steady state; when 'A=O, money demand does not respond
to interest rates and income, and the model has no steady-state solution.
Replacing (I) and (2) with (22) and (23) yields
(m,-m,')-( I -'J..,)(m,_ 1 -m,'_ 1 )
f,=-------,,------'J. ,
(24)
Except for this redefinition, the test procedure remains the same as described above.
11.3. Time-varying risk premia. Finally, we can allow for a time-varying residual, x,, in the interest
parity equation (4):
(25)
Fm ~ur purposes, an ex post estimate of the residual x, can be obtained by de-trending
We limited our exploration of money demand equations to specifications with instantaneous or partial adjustment
of money stocks. Recent literature has considered buffer-stock and error-correction models that imply even greater
departure from the standard specification of the monetary model (see Boughton, 1992, for a survey). These specifications
are difficult to integrate in equilibrium exchange rate models, due to their data-dependent parameterization, and their
tendency to involve lags of interest rates at different maturities and to predict long-run price non-homogeneity.
2
15
i,-ii'-(s ,.,-s,), that is, (l+a)s, -as,. -(m,-m ,')+~(y ,-y,') with a Hodrick-Prescot
t filter. The test
1
procedure then remains the same as that described in Section 3, with "fundamentals" redefine
d as
f, = (m,-m, ') - ~(y,-y, ')
+
x,.
(26)
One possible interpretation of x, is as a risk-premium and, for simplicity, we shall use this label
in our discussion. Nevertheless, our simple treatment must be viewed only as an ad hoc way
to relax
the interest parity equation, not as a structural model of a risk premium. Our aim, once again,
is not
to provide a satisfactory exchange rate model, but only to verify the robustness of our
results to
relaxation of the basic assumptions of the monetary model. 3 Hence, in our tests we have
used a
variety of values for the Hodrick-Prescott smoothing coefficient, p, summarized by the
choices
p=14,400 (which, following the literature, equals 100 times the square of the sample frequenc
y), and
p= 1,600, a lower value that allows the model to better fit the data.
III. Empirical results
III.I. Data and calibration of the tests. The full sample covers data for the United States,
Japan,
Germany, France, the United Kingdom, Italy, Canada, and Switzerland during the period January
I 973
- September 1994. We use the same proxys for the monetary model's variables used in
previous
studies: r:ionthly data for narrow and broad money supply, GDP, industrial production,
consumer
price indices, and exchange rates against the U.S. dollar (end-of-period data). All data are
from the.
3 In particular
, since we use all the model's equations to obtain an ex post estimate of x,, the model
would be
identically satisfied (i.e., s; &s ) if we set Hodrick-Prescott smoothing coefficient, p, at zero.
The model specified with
1
(25) imposes restrictions on the data only to the extent that the filtered residual is required
to be smooth over time.
16
IFS of the IMF, except for a few incomplete IFS series that were obtained from the Current Economic
Indicator data-base of the IMF.
We calibrated our tests using parameters estimated in previous empirical studies, choosing wide
ranges to encompass both estimated and plausible values for each parameter.
Most estimates of the annualized semi-elasticity of money demand to short term interest rates,
for money demand functions with instantaneous adjustment, fall in the range 1 to 4 (see, for instance,
Bilson, I 978, and Laidler, 1993). Estimated exchange rate models tend to yield smaller values of a
(see, for instance, Flood, Rose, and Mathieson, 1991), while money demand equations with partial
adjustment tend to yield somewhat higher values of a (see, for instance, Fair, 1987, and Goldfeld and
Sichel, 1990). We took the values a=0.l and a=6 (to be multiplied by 12 in monthly tests) as
spanning the plausible range of a, using a= 1 as a baseline.
Most estimates of the income elasticity of money demand, ~. range from about 0.5 to about 1.5,
independently of the assumed speed of money adjustment (see, for instance, Fair, 1987, and Goldfeld
and Sichel, 1990). We begin by considering the values ~=0.2 and ~=2, and then fix
~
at unity for
the rest of the analysis.
The speed of adjustment of money stocks to their steady state, 11., is more difficult to calibrate.
Annualized estimates of 11. from pre-1974 (quarterly) data typically range around 0.3 - 0.5, suggesting
a half-life of money shocks between six and eighteen months, already a surprisingly slow response.
In fact, estimates from post-1974 data typically yield near-zero estimates A., implying no response of
money dun;md to changes in its determinants, and a breakdown of the steady-state solution of the
model. It is now well understood that the estimated unstable behavior of money balances from post1974 data reflects more the difficulty of capturing with simple dynamic specifications the financial
innovations and institutional changes that occurred since 1974, than an implausibly low speed of
17
portfolio adjustment (see Goldfeld and Sichel, 1990, and Boughton, 1992, for a discussion). We
report results for the baseline case of instantaneous adjustment, A-=I, and for the cases of 1c=0.S and
1c=0. I (corresponding to monthly values of about A.=.06 and A.=.01 ). We also present results of tests
conducted over split samples, as a preliminary check of robustness of our inference to structural
breaks.
Estimated half-lives of price shocks in PPP equations range anywhere from one to six years,
depending on currency and sample (see, for instance, Hakkio, 1992, and Lothian and Taylor, 1993).
Hence, (annualized) values of 0 should range between 0.1 and 0.5. We report tests with 0 set at 0.3;
results for tests calibrated with 0=0.1 and 0=0.5 were very similar.
Finally, "fundamental" data for output, money, and prices must be normalized. First, data are
reported only in index number form.
Second, econometricians can only guess which macro-
aggregates best capture the theoretical concepts of "output", "money", and "prices", suggesting
allowance for a degree of freedom in scaling raw data. Upon allowing for a multiplicative factor in
raw data (i.e., for an additive constant in log-transformed data), the model suggests a logical way to
normalize the series: by scaling.raw data so that the model holds on average over the sample (that
is, so that 'i,' =
s,).
111.2. Results. Tables 1-3 report a sample of the volatility tests that we implemented.4
Table I reports results of tests of the textbook flex-price instantaneous-money-adjustment
version of the monetary model, for different values of the elasticities ~ and ex. Several important
features are apparent from these results.
'A more complete set of tests is available from the authors, together with the data and a copy of the GAUSS
program
used for the tests, upon receipt of a stamped self-addressed envelope and a formatted 3½" high-densit
y disk.
18
First, the model is strongly rejected in almost all cases, with test statistics often exceeding their
99 percent critical values of ±2.57. The rejection is strong when s, es E,_,[ s,] is used as a benchmark,
0
and is due to the occurrence of a positive test statistic.
This positive sign reflects a positive
correlation between the forecast error, s,·-s,, and the exchange rate surprise at time t,
That is, on average, either
s,-E,_,[s,].
s,'>s,>E,_,[s,] or s,'<s,<E,_,[s,]. Hence, these tests indicate that market
exchange rates tend to stay closer to their one-period-ahead forecasts than predicted by the monetary
model, confirming evidence from regression-based tests that the joint hypothesis of the monetary
model and of market efficiency are mutually exclusive, as systematic profit opportunities would have
persisted in the major currency markets over the sample period. The tests also show, however, that
excessive exchange rate volatility is not a cause of rejection, as inequality (16B) (a violation of which
is marked "B" in Tables 1-3) is never violated when
s, =E,_,[s,] is used as a benchmark: the sample
0
volatility of market exchange rates around the model's prediction at time t-1,
than that of fundamental rates around the same benchmark,
E[(s,• -s,
0
e[(s,-s,0 ) 2 ], is smaller
)2 ]. Market exchange rates at
time t also appear to forecast fundamental rates better than the model's predictions at time t-1, as
required by inequality (16A) (a violation of which is marked "A" in the tables).
Thus, both
inequalities (16A) and (16B) are satisfied. These inequalities are only necessary--not sufficient-conditions for model acceptance, however, and their fulfillment does not prevent the model from
being rejected on grounds of forecastability of excess returns.
There is also no evidence of excessive exchange rate volatility when the lagged exchange rate
is used as a benchmark (s,0 -=s,_ 1): conditional volatilities of major exchange rates do not exceed the
predictions of even the most restrictive version of the monetary model over the post-Bretton Woods
period. Inequality (16A) is violated for the Japanese yen at low values of a, and for the Italian lira
at low values of a and large values of ~- Interestingly, it is much more difficult to reject the model
19
0
when s, =s,_ 1 than when s,0=Ei-1[s,]: exchange rate innovations, s,-s,_ , appear as largely random
1
in the data (particularly at short horizons) and only for high values of a is the model rejected with
confidence. These are intuitive results: as a falls, the discount rate implicit in the model rises, so
that the weights attached by the test to events in the near future rise. As exchange rates are usually
difficult to distinguish from random walks at very short horizons, the test cannot reject the hypothesis
that s, -s,_ 1 is pure noise.
Setting s,0 =/, causes inequality (16B) to be violated for all currencies and parameter values,
and
q to
be significantly negative:
market exchange rates are too volatile around s,0 =f, to be
consistent with the predictions of the monetary model. Inequality ( 16A) is also violated, though only
for very low values of a. Violation of (16B) is consistent with the evidence presented by Gros
(1989) who, following Mankiw, Romer, and Shapiro's (1985) work on stock prices, stationarized
exchange rates around a random walk benchmark. Indeed, it is visually apparent from plots of
market, fundamental, and benchmark exchange rates (available upon request from the authors), thats,0
tracks s, more closely than s,• when s,0 =E,_ [s,] or s,0 =s,_ • In contrast, s,0 tracks s,• more closely
1
1
than s, when s,0 =f,. As a result, s, is not excessively volatile when s,0 =E,_,[ s,] or s,0 =s,_ , but is
1
excessively volatile when s,0 =f,, for all currencies and acceptable parameter values. Evidence of
excessive volatility when stationarizing exchange rates around a "random walk" benchmark is
therefore formally correct, though most researchers (including ourselves) would argue this to be a
finding of little usefulness. Loosely speaking, what would be viewed as "excessively volatile", for
most purr~~es, would be the benchmark itself, not the exchange rate. Indeed, when exchange rates
are measured with respect to common benchmarks (the value predicted by the underlying model based
on last period's information, or the last value in the public's information), then there is no evidence
that major exchange rates may have been too volatile in the post-Bretton Woods period, with respect
20
to the predictions of even the most restrictive version of the monetary model.
Tables 2 and 3 confirm the previous discussion for alternative specifications of the model.
Table 2 presents results for the baseline set of parameters (a=~=A=0= I), for the model with
partial adjustment of money stocks (A=O.5 and A.=O.1 ), and for the sticky price model. Predictably,
the models with sluggish money and price adjustment can be rejected with somewhat less confidence
than the baseline model. 5 More careful modelling of the money and price processes is likely to
weaken the evidence against the model even further. For the purpose of the present study, however,
suffice to note that these extensions tend to make the model's rejection marginally less significant,
while not affecting the volatility inequalities: there is no evidence of excess volatility when exchange
rates are measured with respect to their lagged values or with respect to the trends anticipated by the
model; there is evidence of excess volatility, however, when exchange rates are measured with
respect to a random walk benchmark.
Table 3 presents test results for the model with time-varying risk premia and for the baseline
model over two half-samples. Allowing for violations of the interest parity condition sharply weakens
the overall evidence against the model, particularly when the Hodrick-Prescott parameter p is set to
such a low value (p= 1,600), that much of the interest parity residuals are incorporated into
fundamentals. While these tests are clearly skewed in favor of the model (little structure is imposed
on the premia x,, except that they should be smooth over time), our previous discussion of the
volatility inequalities (16A-B) requires little change. Indeed, these results suggest that more accurate
modellin., ;::f the interest parity condition is likely to weaken the evidence of excess volatility even
based on a random walk benchmark.
' Given the existing econometric evidence on money demand equations, it is also not surprising that the data favor
an extremely low value of'/,.. See the discussion in Section III.
21
Finally, overall evidence against the model remains strong when the tests are performed over
split samples, and the qualitative aspects of the excess volatility tests are also unchanged.
A variety of alternative specifications qf the tests, for which results are available upon request,
gave very similar results. We relaxed the assumption of long-run PPP by allowing for a time-varying
trend in the real exchange rate (estimated, as in the case of a time-varying risk premium, by passings,+ p 1• -p 1
through a Hodrick-Prescott filter); we performed tests with narrow money instead of broad money
as the monetary variable, using moving averages rather than the Hodrick-Prescott filter to filter the
data, using quarterly (instead of monthly) data and, in this case, using real GDP instead of industrial
production data; we measured exchange rates with respect to the Deutsche mark rather than the
dollar; we considered different holding periods, h, in the definition of the fundamental exchange rate,
and different lags, L, in the definition of the benchmark exchange rate. Our volatility-based tests
continued to provide evidence against the model as a whole (more precisely: evidence of excess
returns forecastability), but no evidence of excess volatility--except with respect to a random walk
benchmark. 6
IV. Concluding remarks
The volatile behavior of exchange rates since the breakdown of the Bretton Woods system has
led scholars and policymakers to question the consistency of exchange rate fluctuations with
movemer::s in their underlying determinants, with complaints of unjustified (or irrational) market
6
Exceptions included moving to quarterly data, which caused most tests to become insignificant, likely as a result
of a drastic fall in the degrees of freedom; and increasing the holding period beyond 6 months and the Jag L in
s,0 ss beyond 3 months, which also caused many of the tests to become insignificant. Replacing broad money with
narro~-inoney tended to increase the confidence of rejection.
22
turbulence rising loud during episodes of exchange rate instability. Several previous studies have
presented evidence that exchange rates since I 973 have been excessively volatile with respect to the
predictions of a variety of popular exchange rate models.
However, for reasons ranging from
calibration errors, small-sample biases, and lack of a proper testing procedure, these studies failed to
present convincing evidence of the inability of these models to match the observed pattern of
exchange rate volatilities over the post-Bretton Woods period.
This paper has tried to provide more solid e_vidence on the ability of a popular family of
exchange rate models to match observed patterns of exchange rate changes, by applying recent tests
of asset price volatility. Two main points have emerged from the analysis.
First, consistent with previous regression-based tests of the monetary model, our volatility-based
tests strongly reject the monetary model in its textbook flex-price format and in more general versions
that allow for sluggish price and money adjustments: excess returns in currency markets appear to
be too forecastable for these markets to be efficient and exchange rates being governed by the rules
of the monetary model. As the volatility-based tests that we employ have been shown to possess
better statistical properties that previous regression-based tests, our evidence against the monetary
model should be viewed as stronger than that previously available in the literature. Nevertheless,
rejection of the monetary model is hardly a novelty, and this negative outcome would be, on its own
account, of secondary interest.
Second, and more interesting, our tests highlight the ambiguity of the concept of exchange rate
volatility, .t.-id the implications of this ambiguity for claims of "excess" exchange rate volatility.
Measuring exchange rate deviations with respect to different benchmarks leads to different
conclusions on whether major currencies' exchange rates have been excessively volatile over the postBretton Woods period. While we have made no attempt to resolve this ambiguity, we have showed
23
that based on certain (odd, in our view) definitions of exchange rate volatility, claims that majpr
exchange rates may have been too volatile over the post-Bretton Woods period may be formally
justified. However, based on definitions of exchange rate volatility common in the exchange rate
literature (which focus on the volatility of exchange rate surprises, and on the conditional volatility
of exchange rates), major exchange rates over the post-Bretton Woods period do not appear to violate
the predictions of even the most restrictive version of the monetary model:
their volatility is
consistent with the anticipated volatility of their determinants.
Our results should lead to greater skepticism against future claims of "excessive" exchange rate
volatility, and our focus on a model as restrictive as the monetary model should reinforce this
skepticism: models that do not tie down exchange rate movements to those of such a narrow set of
macroeconomic variables (for instance, portfolio balance models with imperfect substitutability, assetprice models with variable rates of discount, etc.) should provide even less evidence of excess
. volatility. (Formally, for a perfectly-fitting model,
s,•=s,, q=O, and both (16A) and (16B) would be
satisfied.) The technique applied in this paper to test for excess exchange rate volatility is simple and
general, and future research is bound to subject this conjecture to a direct test.
24
&&·-......
Germany, Japan, and the United States:
Volatility of Nominal Exchange Rates, January 1962 - September 1994
(Percentage clumges from previous month)
16r-------------,------------------------~16
Japanese yen/U.S. dollar
10
IO
'
0
.
-10
-10
15r-------------,------------------------~15
Deutsche mark/U.S. dollar
10
IO
-10
-10
"~-----------------------------------~"
Deutsche mark/Japanese yen
10
-10
Dotted line indicates approximate dale or breakdown or the Bretlon lfoods System.
Source: IMP, International Financial Statistics.
10
-10
Table I. Flexible Price Model
,;;., = E1_ 1[s,J
a=6
a.--0.1
s: = fc
0
st = s,_ 1
a=0.1
0:=6
o:--0.1
o:~
13=0.2
13=2
!3=0.2
P=2
ll=0.2
13=2
P=o.2
P=2
Pound sterling
4.56
4.70
2.52
2.58
-0.39
-0.25
2.29
2.24
-4.60
AB
-4.74
AB
-2.07
French franc
3.62
4.22
2.83
2.84
-0.25
0.33
2.76
2.79
-3.69
AB
-4.36
AB
-2.19"
-2.28
9
Deutsche mark
4.09
4.44
2.76
2.96
-0.30
-0.24
2.58
2.63
-4.14
AB
-4.47
AB
-2.23
B
-2.43
B
Italian lira
3.98
4.33
2.53
2.42
-0.37
-0.86
2.37
2.32
-4.01
AB
-4.65
AB
-1.38
B
-1.12
B
Swiss franc
5.41
4.74
2.78
2.93
-0.18
0.09
2.59
2.61
-5.46
AB
-4.79
AB
-1.87
B
-2.40
B
Canadian dollar
4.88
4.95
2.41
2.75
0.94
1.17
1.76
1.83
-4.92
AB
-5.01
AB
-3.()()
B
-3.65
Japanese yen
5.09
4.14
2.63
2.45
-1.25
2.05
2.07
-5.07
AB
-4.]5
AB
-2.14
B
-1.64 "
A
-1.43
A
A
P=0.2
P=2
p=0.2
9
P=2
-2.52 "
Table Notes: The reported values are the standard normal statistics for the null hp. that q, =0, where q I is defined by equation (14)
in the text. Newey-West standard errors, with a lag truncation of 6·(n/255)u4 , where n is the number of observations, were used. A
superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data
for broad money and industrial production are used in these tests. The sample goes from April 1973 to September 1994. The number
of observations is 255.
9
Table 2. Baseline, sluggish money adjustment, and sticky prices
:;,· =
s; = I,
0
E,.,[s,]
st = s,-1
baseline
baseline
baseline
,o=l
A.=0.5
A.----0.1
sticky prices
N=l
A.=0.5
A.----0.1
sticky prices
,o=I
A.----0.5
A.----0.1
sticky prices
Pound sterling
3.80
4.00
3.48
3.60
0.88
0.71
-0.14
0.84
-4.35 '
-4.51 '
-4.11 '
-4.23 '
French franc
2.90
2.96
2.78
2.51
1.81
2.14
2.52
2.04
-3.33 '
-3.65 '
-4.60'
-2.97 '
3.74
3.76
3.43
3.59
l.28
I.JI
0.16
1.36
-4.11 '
-4.07
8
Deutsche mark
-2.44
B
-3.68
3.17
2.37
1.33
l.72
2.46
1.73
-3.60 •
-3.93
-4.74
8
2.91
3.17
8
Italian lira
-3.07 '
l.63
2.23
2.41
1.49
-4.82
-4.13
-2.21
8
-4.32
8
4.14
3.13
8
4.13
3.02
8
Swiss franc
3.86
2.46
l.44
1.17
0.33
1.21
-4.81 '
-4.55
-4.43 '
-4.90
8
4.37
4.32
8
Canadian dollar
Japanese yen
4.54
4.64
3.02
2.98
0.18
0.44
l.04
0.47
-4.57
-4.49'
-3.77 '
8
-4.75 '
8
Table Notes: The reported values are the standard normal statistics for the null hp. that q, =0, where q, is defined by equation (14) in the text
Newey-West standard errors, with a lag truncation of 6·(n/255) 114 , where n is the number of observations, were used. A superscript A indicates tha
inequatity ( l6A)is violated, while a s-HpeFSct°ipt-B mdi€aws that ~ ¥ { UiB-)-is violated. Monthl:i- data for broad money and industrial pmductio1
are used in these tests. The baseline model uses a= I, ~=I, A= I, 0= I; the sticky price model uses 0=.3. The sample goes from April I 973 t<
September 1994. The number of observations is 255.
Table 3. Time-Varying Risk Premia and Various Subsamples
.;; =
variable risk premia
p:14.400
p=l,600
E,_,[s,]
s: = I,
0
s, = s:-1
1973.04
1984.01
to
to
1983.12
1994.09
variable risk premia
p=l4,400
p=l,600
1973.04
1984.01
to
to
1983.12
1994.09
variable risk premia
p=l4,400
8
p=l,600
1984.01
to
to
1983.12
1994.09
-2.62
8
-3.38
8
-1.66
-3.36
8
-2.59
8
8
-2.31
-3.53
8
-3.41
8
1.32
-2.87 '
-1.48
-3.20
8
-2.98
8
1.07
1.32
-2.88'
-1.96
-3.55 •
-2.93 '
-1.94
0.11
1.58
-3.20
-2.65 •
-3.68
8
-4.09 •
-0.40
0.44
0.76
-2.35 •
-1.44
-2.68
8
-3.39 •
Pound sterling
2.18
1.53
2.22
2.83
1.07
0.36
1.17
0.88
-3.53
French franc
2.58
1.53
2.35
2.18
1.29
0.41
0.98
1.33
-3.13 •
Deutsche mark
2.51
1.20
2.73
3.05
0.91
-0.13
0.78
1.31
-3.37
Italian lira
2.18
I.II
2.28
2.46
1.07
0.36
0.29
Swiss franc
2.51
1.20
3.12
2.45
1.09
0.13
Canadian dollar
0.61
-0.75
3.03
3.85
-1.26
Japanese yen
2.17
0.79
2.09
2.89
0.71
8
-2.63
8
1973.04
Table Notes: The reported values are the standard normal statistics for the null hp. that ii,=O, where q 1 is defined by equation (14)
in the text. Newey-West standard errors, with a lag truncation of 6·(n/255)"4 , where n is the number of observations, were used. A
superscript A indicates that inequality (16A) is violated, while a superscript B indicates that inequality (16B) is violated. Monthly data
for broad money and industrial production are used in these tests. The coefficient pis the smoothing coefficient for the Hodrick-Prescott
filter. The sample goes from April 1973 to September 1994 unless mentioned. The number of observations in the four samples is 255,
255, 128, and 126, respectively.
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26
