Diebold, F.X., Hahn, J. and Tay, A. (1999),
"Multivariate Density Forecast Evaluation and Calibration in Financial Risk Management: High-Frequency Returns on Foreign Exchange,"
Review of Economics and Statistics, 81, 661-673.
Real-Time Multivariate Density Forecast Evaluation and Calibration:
Monitoring the Risk of High-Frequency Returns on Foreign Exchange
Francis X. Diebold Jinyong Hahn Anthony S. Tay
University of Pennsylvania University of Pennsylvania National University
Stern School, NYU and M.I.T. of Singapore
and NBER
This Draft/Print: August 26, 1998
Copyright © 1998 F.X. Diebold, J. Hahn, and A.S. Tay. This paper is available on the World
Wide Web at http://www.ssc.upenn.edu/~diebold/ and may be freely reproduced for
educational and research purposes, so long as it is not altered, this copyright notice is
reproduced with it, and it is not sold for profit.
Abstract: We provide a framework for evaluating and improving multivariate density
forecasts. Among other things, the multivariate framework lets us evaluate the adequacy of
density forecasts involving cross-variable interactions, such as time-varying conditional
correlations. We also provide conditions under which a technique of density forecast
calibration can be used to improve deficient density forecasts. Finally, motivated by recent
advances in financial risk management, we provide a detailed application to multivariate high-
frequency exchange rate density forecasts.
Acknowledgments: Helpful comments were provided by Jose Campa, Neil Shephard, Charlie
Thomas, and Mike West, as well as seminar and conference participants at Stern, Wharton,
and Olsen & Associates. All remaining inadequacies are ours alone. For support we thank
the National Science Foundation, the Sloan Foundation, the University of Pennsylvania
Research Foundation, and the National University of Singapore.
2
1. Introduction
The forecasting literature has traditionally focused primarily on point forecasts.
Recently, however, attention has been given to interval forecasts (Chatfield, 1993) and
density forecasts (Diebold, Gunther and Tay, 1998). The reasons for the recent interest in
interval and density forecasts are both methodological and substantive. On the
methodological side, recent years have seen the development of powerful models of time-
varying conditional variances and densities (Bollerslev, Engle and Nelson, 1994; Ghysels,
Harvey and Renault, 1996; Hansen, 1994; Morvai, Yakowitz and Algoet, 1997). On the
substantive side, density forecasts and summary statistics derived from density forecasts have
emerged as a key part of the explosively-growing field of financial risk management (Duffie
and Pan, 1997; Jorion, 1997), as well in as more traditional areas such as macroeconomic
inflation forecasting (Wallis, 1998). Moreover, explicit use of predictive densities has long
been a prominent feature of the Bayesian forecasting literature (Harrison and Stevens, 1976;
West and Harrison, 1997), and recent advances in Markov Chain Monte Carlo (Gelman,
Carlin, Stern and Rubin, 1995) have increased the pace of progress. The closely related
prequential Bayesian literature (Dawid, 1984) also features density forecasts prominently.
Interest in forecasts of various sorts creates a derived demand for methods of
evaluating forecasts. In parallel with the historical emphasis on point forecasts, most
literature has focused on the evaluation of point forecasts (Diebold and Lopez, 1996), but
recent interest in interval and density forecasts has spurred development of methods for their
evaluation (Christoffersen, 1998; Diebold, Gunther and Tay, 1998). Diebold, Gunther and
Tay, in particular, motivate and approach the problem of density forecast evaluation from a
3
risk management perspective, drawing upon an integral transform dating at least to Rosenblatt
(1952), extended by Seillier-Moiseiwitsch (1993) and used creatively by Shephard (1994).
Here we extend the density forecast evaluation literature in three ways. First, we treat
the multivariate case, which lets us evaluate the adequacy of density forecasts involving
cross-variable interactions, such as time-varying conditional correlations, which are crucial in
financial settings. Second, we provide conditions under which a technique of density forecast
calibration can be used to improve deficient density forecasts. Finally, we provide a
detailed application to the evaluation of density forecasts of multivariate high-frequency
exchange rates, which is of direct substantive interest in addition to illustrating the ease with
which the methods can be implemented.
2. Density Forecast Evaluation and Calibration
We begin with a brief summary of certain key univariate evaluation results from
Diebold, Gunther and Tay (1998), in order to establish ideas and fix notation. We then
describe our methods of univariate calibration, after which we extend both the evaluation and
calibration methods to the multivariate case. Throughout, we intentionally ignore parameter
estimation uncertainty. We take density forecasts as primitives, and in particular, we do not
require that they be based on an estimated model.1 We take this approach because many
density forecasts of interest do not come from models with estimated parameters, as for
example when density forecasts are extracted from surveys (Diebold, Tay and Wallis, 1997)
or from prices of options written at different strike prices (Aït-Sahalia and Lo, 1998;
1
Some progress, however, is also being made at evaluating density forecasts produced
from estimated models; see Bai (1997) and Inoue (1997).
4
Soderlind and Svensson, 1997; Campa, Chang and Reider, 1998). This is the case for some
of the exchange rate density forecasts that we evaluate in section 4.
Univariate Evaluation
Consider one arbitrary period, t, and let yt be the variable of interest with conditional
density f(yt t 1), where represents the collection of past information available at time
t 1
t-1, and let pt 1(yt) be a density forecast of yt. Let zt be the probability integral transform of
yt with respect to pt 1(yt); that is,
yt
zt = pt 1(u)du Pt 1(yt).
Pt 1 (zt)
1
Then assuming that is continuous and non-zero over the support of zt, zt has
zt
support on the unit interval with density function
Pt 1 (zt) f(Pt 1 (zt) )
1 1
t 1
qt(zt) = f(Pt 1(zt) ) = .
1
t 1
zt
pt 1(Pt 1 (zt))
1
In particular, if pt 1(yt) = f(yt t 1), then
1 if 0 zt 1
qt(zt)
0 otherwise,
so that zt U(0,1).
Now consider a series of m density forecasts and realizations, rather than just one.
Diebold, Gunther and Tay (1998) show that if {yt}m1 is a series of realizations generated
t
5
from the series of conditional densities {f(yt| )}m1, and if a series of 1-step-ahead density
t 1 t
forecasts {pt 1 (yt)}m1 coincides with {f(yt| )}m1, then assuming a non-zero Jacobian
t t 1 t
with continuous partial derivatives, the series of probability integral transforms of {yt}m1
t
iid
with respect to {pt 1 (yt)}m1 is iid U(0,1). That is, {zt}m 1 U(0,1). Thus, to assess
t t
whether a series of density forecasts coincides with the corresponding series of true
conditional densities, we need only assess whether an observed series is iid U(0,1).
Univariate Calibration
An important practical question is how to improve suboptimal forecasts. In the point
forecast case, for example, we can regress the y's on the w's (the predicted values) and
potentially use the estimated relationship to construct an improved point forecast. Such a
regression is sometimes called a Mincer-Zarnowitz regression, after Mincer and Zarnowitz
(1969). In this paper we will use an analogous procedure, which we call density forecast
calibration, for improving density forecasts that produce an iid but non-uniform z series.2
Suppose that we are in period m and possess a density forecast of ym 1 . From our
earlier discussion,
f(ym 1 m) = pm (ym 1) qm 1(zm 1).
But if z is iid, then we can drop the subscript on q and write
f(ym 1 m) = pm (ym 1) q(zm 1).
2
We thank Charlie Thomas for pointing out that the basic idea of univariate density
forecast calibration traces at least to Fackler and King (1990). Shortly, however, we will
extend the calibration idea to the multivariate case. Moreover, we will emphasize that density
forecast calibration is not universally applicable and develop a sufficient condition for its
application in both the univariate and multivariate cases.
6
Thus if we knew q(z ), we would know the actual conditional distribution f(ym 1 m).3
m 1
Because q(z ) is unknown, we use an estimate q(z ) formed using {zt}m 1 to construct
Ć
m 1 m 1 t
Ć(ym 1 m). In finite samples, of course, there is no guarantee that the
an estimate f
improved forecast will actually be superior to the original, because it is based on an
estimate of q rather than the true q, and the estimate could be very poor. The practical
efficacy of our improvement methods is an empirical matter, which will be assessed shortly,
but with the large sample sizes typically available in financial applications we expect that
precise estimation of q will often be straightforward.
Multivariate Evaluation
The principles that govern the univariate techniques discussed thus far extend readily
to the multivariate case. Suppose that yt is now an Nx1 vector, and that we have a series of m
multivariate forecasts and their corresponding multivariate realizations. Further suppose that
we are able to factor each period s joint forecast into the product of the conditionals,
pt 1(y1t, y2t, ..., yNt) = pt 1(yNt| yN 1,t, ..., y1t) pt 1(y2t| y1t) pt 1(y1t).
Then for each period we can transform each element of the multivariate observation
( y1t, y2t,..., yNt ) by its corresponding conditional distribution. This procedure produces a set
of N z series that will be iid U(0,1) individually, and also when taken as a whole, if the
multivariate density forecasts are correct. The proof of this assertion is obtained by simply
arranging the multivariate series as a series of univariate observations comprising
3
Note that z is uniquely determined by y, so that for any given value of y, we can
always compute f(y) as p(y)q(z).
7
{... , y1t, y2t,..., yNt , ...}.4 The result for the univariate case can then be applied to show that
the resulting Nm vector z is iid U(0,1), and hence the z series corresponding to any particular
series of conditional forecasts is also iid U(0,1).
Note that N! z series can be produced, depending on how the joint density forecasts
are factored, giving us a wealth of information with which to evaluate the forecasts. To take
the bivariate case as an example, we can decompose the forecasts in two ways:
(i) pt 1(y1t, y2t) pt 1(y1t) pt 1(y2t| y1t) and
(ii) pt 1(y1t, y2t) pt 1(y2t) pt 1(y1t| y2t),
t = 1, ..., m, and label the respective z series z1, z2 |1, z2 and z1|2, where we obtain z by
i
taking the probability integral transform of yi with respect to pt 1(yi), and we obtain z by
i| j
transforming yi with respect to pt 1(yi| yj). Good multivariate forecasts will produce
z1, z2 |1, z2 and z1|2 that are each iid U(0, 1), as well as combined series {..., z , z , ...}
1, t 2 |1 , t
and {..., z2, t, z , ...} that are also iid U(0,1).
1 |2 , t
In many cases of economic and financial interest, such as the application to density
forecasting of DM/$ and Yen/$ returns in section 4 of this paper, the dimension N of the set
of variables being forecast is low. A classic example is density forecasting of returns across
different aggregate asset classes, such as equity, foreign exchange and fixed income. In such
cases, it is straightforward to examine and learn from each of the N! sets of z series.
Sometimes, however, N may be large, as for example when a density forecast is generated for
each of the stocks in a broad-based market index, such as the S&P 500, in which case
4
That is, we begin with had m observations on an N-variate variable, and we convert
them to a univariate series with Nm observations.
8
methods such as those of Clements and Smith (1998) may be used to aggregate the
information in the individual z series.
Multivariate Calibration
In parallel with the key formula underlying our discussion of univariate calibration, in
the multivariate case we have that
f(y1,m 1, y2,m 1,..., yN,m 1| ) pm(y1,m 1, y2,m 1,..., yN,m 1) q(z1,m 1, z2,m 1,..., zN,m 1).
m
Moreover, factoring both of the right-hand-side densities, we can write the formula in a way
that precisely parallels our multivariate evaluation framework,
N
f(y1,m 1, y2,m 1,..., yN,m 1| ) pm(yi,m 1| yi 1,m 1,..., y1,m 1) q(zi,m 1| zi 1,m 1,..., z1,m 1) .
m
i 1
As before, an estimate of q may be used to implement the calibration empirically, yielding
Ć
f(y1,m 1, y2,m 1,..., yN,m 1| ) pm(y1,m 1, y2,m 1,..., yN,m 1) q(z1,m 1, z2,m 1,..., zN,m 1),
Ć
m
or
N
Ć
f(y1,m 1, y2,m 1,..., yN,m 1| ) pm(yi,m 1| yi 1,m 1,..., y1,m 1) q(zi,m 1| zi 1,m 1,..., z1,m 1) .
Ć
m
i 1
Note that the multivariate calibrating density is the same (in population), regardless of which
of the possible N! factorizations is used.
3. More on Density Forecast Calibration
Here we elaborate on various aspects of density forecast calibration. First we give a
sufficient condition for an iid integral transform series, which is required for application of the
calibration method. Next we discuss subtleties associated with estimation of the density of an
integral transform, which by construction has compact support on the unit interval. Finally,
9
we show how the calibration method can be used to generate good density forecasts even
when the conditional density is unknown. In this section we use the generic notation z to
denote an integral transform; it is a scalar integral transform in the univariate case and a
vector of integral transforms in the multivariate case.
Conditions Producing an iid z Series
In empirical work, the legitimacy of the assumption that z is iid can be assessed in a
number of ways, ranging from examination of correlograms of various powers of z to formal
tests such as those discussed in Brock, Hsieh and LeBaron (1991). It is nevertheless desirable
to characterize theoretically the conditions under which z will be iid, in order to deepen our
understanding of whether and when we can reasonably hope for an iid z series. Here we
establish a sufficient condition: if the 1-step-ahead density f(yt t 1) belongs to a location-
scale family, and if the forecast pt 1(yt) adequately captures dynamics (in a sense to be made
precise shortly) but is mistakenly assumed to be in another location-scale family, then z will
be iid.
More precisely, suppose that f(yt t 1) belongs to a location-scale family; that is, the
yt µ( )
t 1
random variable is iid with unknown density f( ). It then follows that
( )
t 1
yt µ( )
1
t 1
f(yt t 1) f .
( ) ( )
t 1 t 1
Now suppose that we misspecify the model by another location-scale family, pt 1(yt), in
which we specify µ( ) and ( ) correctly, in spite of the fact that we use an incorrect
t 1 t 1
10
conditional density. Thus we issue the density forecast
yt µ( )
1
t 1
pt 1(yt) p .
( ) ( )
t 1 t 1
The probability integral transform of the realization with respect to the forecast is therefore
yt yt
yt µ( )
1
t 1
zt pt 1(yt)dyt p dyt.
( ) ( )
t 1 t 1
Now make the change of variable
yt µ( )
t 1
ut ,
( )
t 1
which yields
ut
1
zt p(ut) ( )dut P(ut).
t 1
( )
t 1
But this means that z is iid, as claimed, because zt P(ut) and ut is iid.
Estimation of q(z)
Estimation of q(z) requires care, because z has support only on the unit interval. One
could use a global smoother on the unit interval, such as a simple two-parameter beta
distribution, but the great workhorse beta family is unfortunately not flexible enough to
accommodate the multimodal shapes of q(z) that arise routinely, as documented in Diebold,
Gunther and Tay (1998). One could perhaps stay in the global smoothing framework by
using some richer parametric family or a series estimator, as discussed for example by Härdle
(1991).
In keeping with much of the recent literature, however, we prefer to take a local
11
smoothing approach, as for example with a kernel density estimator. But it has long been
recognized that standard kernel estimation of densities with bounded support suffers from a
boundary problem in that the density estimates at the boundaries are biased. One cause of
this problem is that near the boundary approximately half of the kernel mass falls outside the
range of the data (assuming a symmetric kernel). Therefore, kernel density estimates near
boundaries have expectations approximately equal to half the true underlying density.
Let us elaborate. For i.i.d. data z1, ..., zT with sufficiently smooth density q(z)on [0,
1], the kernel estimator q(z) of q(z)is
Ć
T
z zt
1
q(z) K ,
Ć
Tb b
t 1
where K( ) is a symmetric and sufficiently smooth kernel of choice and b>0 is bandwidth.
Begin by noting that
1 1 u z
E[q(z)] K q(u) du
Ć
b 0 b
1 z 1 z 1 z
1
b b 2 b 2 2
q(z) K(v)dv bq (z) K(v)vdv b q (z) K(v)v dv o(b ).
z z z
2
b b b
12
Now suppose as usual that the bandwidth is chosen such that b 0 and Tb as T . For z
1 x
b
K(v) f(x bv) dv
x
b
well in the interior of [0,1], we then have
2
E[q(z)] q(z) O(b ).
Ć
On the other hand, the kernel density estimates have expectations approximately equal to only
half of the true underlying density at boundaries:5
q(0)
E[q(0)] O(b)
Ć
2
q(1)
E[q(1)] O(b).
Ć
2
It can be shown that similar bias occurs at points near the boundaries.
There are a number of ways to address the boundary problem. One is to use a
modified kernel density estimator such as the one suggested by Müller (1993), which is
designed to have less bias near boundaries. Another local smoothing approach is to estimate
q(z) with a simple histogram. A histogram is a special type of kernel density estimator where
the kernel never exceeds the boundaries of the distribution. Therefore we expect that it would
not suffer from the problem associated with kernels that exceed the boundaries, and
5
See Härdle (1990, 130-131), for example, for related discussion in the context of
kernel regression, as opposed to density estimation.
13
experimentation revealed it to be superior in the present context; we make extensive use of
histograms as visual estimates of q(z) in the empirical application in section 5. Finally, we
can bypass the density estimation problem altogether by casting our evaluation and calibration
methods in terms of c.d.f. s rather than densities, and using empirical c.d.f. s instead of
estimated densities.
Alternatively, we may write the calibration formula in terms of joint c.d.f. s rather than
joint densities, as
F(y1,m 1,...,yN,m 1 m) Q(Pm(y1,m 1,...,yN,m 1)),
which we make operational by replacing population c.d.f. s with estimated (empirical)
c.d.f. s,
Ć Ć
F(y1,m 1,...,yN,m 1 m) Q(Pm(y1,m 1,...,yN,m 1)).
Although empirical c.d.f. s are not as visually digestible as estimated densities, they have a
number of attractive features: they are guaranteed to be 0 for z=0 and 1 for z=1, and there is
no need for bandwidth selection bandwidth selection in the estimation of Q. We make
extensive use of empirical c.d.f. s for doing the calibration transformations in the empirical
application in section 5.
Generating Density Forecasts When the Conditional Density is Unknown
It is interesting to note that our calibration methods can be used to generate density
forecasts even when the conditional density is unknown. Consider, for example, the problem
of 1-step-ahead density forecasting in models with unknown 1-step-ahead conditional density
and time-varying volatility. The problem is of intrinsic interest, and moreover, it is quite
general. In particular, it also covers h-step-ahead density forecasting, which in financial
14
contexts is equivalent to 1-step-ahead forecasting of h-period returns.6
As is well known, even under heroic assumptions such as conditionally Gaussian 1-
step-ahead returns, the h-step-ahead conditional density, or equivalently the h-period return,
will not be Gaussian and has no known closed form. Several earlier papers bear on the
problem, but all are lacking in certain respects. Baillie and Bollerslev (1992), for example,
study h-step-ahead prediction but must assume a Gaussian 1-step-ahead conditional density
and even in that case provide only Cornish-Fisher approximations to the multi-step densities.
Engle and González-Rivera (1991) use a different but closely related strategy; let us
contrast the two. Engle and González-Rivera exploit the Bollerslev-Wooldridge (1992) result
that GARCH volatility parameters are consistently estimated by maximum likelihood even
when the conditional density is misspecified. Hence (assuming for convenience a zero
conditional mean) one can assume a Gaussian conditional distribution (incorrectly, in
general), maximize the Gaussian likelihood, and standardize the data by the estimated series
of conditional standard deviations. In large samples the resulting series will be iid but not, in
general, Gaussian; but its density can be estimated using nonparametric techniques.
In tackling this problem, our calibration procedure also exploits the Bollerslev-
Wooldridge (1992) quasi-MLE result. We proceed by maximizing the likelihood, which we
assume to be Gaussian (incorrectly, in general). We then take the estimated volatility
parameters and use them to make conditionally Gaussian density forecasts (again incorrectly),
and we compute the probability integral transforms of the realizations with respect to those
forecasts. In large samples, under the conditions given earlier, the series of integral
6
See, for example, Duan (1995) and Duan, Gauthier and Simonato (1997).
15
transforms will be iid but not U(0,1); we therefore calibrate the forecasts using either a
histogram or a c.d.f. fit to the series of integral transforms.
The key difference between the Engle-González-Rivera approach and our calibration
approach is that the first involves nonparametric estimation of a density with infinite support,
whereas the second maps the problem into one of nonparametric estimation of a density with
bounded support. Given that in risk management contexts our interest centers on tail events,
the Engle and González-Rivera approach may be problematic, due to the well-known
bumpy-tail problem.7 The bumpy-tail problem may be overcome by applying larger
bandwidths at the tails, but instead of applying some ad hoc method of choosing variable
bandwidth, the integral transformation implicitly provides an empirically reasonable variable
bandwidth, so long as the density forecasts are close to the true density. In addition, we note
that our procedure is not limited to the GARCH paradigm, and is applicable to any situation
where the forecasts generate z series that are iid. As discussed previously, extensions to
multivariate density forecasting contexts are also straightforward.
The empirical application to which we now turn illustrates all of the ideas developed
thus far: we begin by evaluating a series of multivariate density forecasts that are revealed to
be poor by virtue of associated iid but non-uniform integral transform series, we estimate the
appropriate calibrating density, and we use it to transform the poor density forecasts into good
ones.
4. Evaluating and Calibrating Multivariate Density Forecasts of High-Frequency
Returns on Foreign Exchange
7
See, for example, Silverman (1986).
16
Here we present a detailed application of our methods of multivariate density forecast
evaluation and calibration to a bivariate system of asset returns. In particular, we study high-
frequency DM/$ and YEN/$ exchange rate returns, and we generate density forecasts from a
forecasting model in the tradition of JP Morgan s RiskMetrics (JP Morgan, 1996), which is a
popular benchmark in the risk management industry.
We use our multivariate density forecast evaluation tools to assess the adequacy of the
RiskMetrics approximation the dynamics in high frequency exchange rate returns, as well as
the adequacy of the conditional normality assumption, after which we attempt to improve the
forecasts using calibration methods. We begin with a description of the data and a discussion
of the statistical properties of the returns series. We then describe the forecasting model, after
which we evaluate and calibrate the density forecasts that it produces.
Data
The data, kindly provided by Olsen and Associates, are indicative bid and ask quotes
posted by banks, spanning the period from January 1, 1996, to December 31, 1996. The data
are organized around a grid of half-hour intervals; Olsen provides the quotes nearest the half-
hour time stamps. Foreign exchange trading occurs around the clock during weekdays, but
trading is very thin during weekends, so we remove them, as is customary.8 We consider the
weekend to be the period from Friday 21:30 GMT to Sunday 21:00 GMT, as this period
appears to correspond most closely to the weekly blocks of zero returns. Thus a trading week
spans Sunday 21:30 GMT to Friday 21:00 GMT, and each of the five trading days therein
spans 21:30 GMT on one day to 21:00 GMT the next day. We label the days Monday
8
See, for example, Anderson and Bollerslev (1997).
17
through Friday. This layout implies that we have one partial week of data, followed by 51
full weeks, followed by another partial week of data. Each full week of data has 5x48=240
data points. The first partial week has 233 observations (just less than five days of data), and
the sample ends with a partial week of 102 observations (about 2 days), for a total of 12,575
observations.
Computing Returns
We compute returns by computing bid and ask rates at each grid point as the linear
interpolation of the nearest previous and subsequent quotes, as in Andersen and Bollerslev
(1997). The rate of return on the exchange rate is then the difference between the mid-point
of the log bid and ask rate at consecutive grid points. That is, letting xt, ask/bid denote the log
ask / bid quote at the grid point t, xt, ask/bid, previous/next represent the nearest previous/next-
occurring bid or ask quote at time t, and timet, previous/next be the time between the grid point
and the previous/next quote, we obtain the exchange rate at time t, xt, as
xt 1 ( xt, ask xt, bid )
2
where
timet, previous
xt, ask/bid ln xt, ask/bid, previous (ln xt, ln xt, ask/bid, previous) .
timet, previous timet, next ask/bid, next
The exchange rate return at time t, rt, is then rt xt xt 1. It will sometimes be necessary
to refer to the return at a particular time of a particular day. In such cases, we will use r to
,i
represent the return at time of day i.
The returns exhibit the MA(1) conditional mean dynamics commonly found in asset
returns. Because our focus in this paper is on volatility dynamics and their relation to density
18
forecasting, we follow standard practice and remove the MA(1) dynamics from each return.
We do so by fitting MA(1) models and treating the residuals as our returns series. Hereafter,
rt will refer to returns series with the MA(1) component removed.
Properties of rt, rt2 and | rt |
There are strong intra-day calendar effects in both Yen/$ and DM/$ returns; Figure 1
displays the first 200 autocorrelations of rt, rt2 and |rt|. The calendar effects are particularly
pronounced in the absolute returns of both series; they occur because trading is more active at
certain times of day than at others. For instance, trading is much less active during the
Japanese lunch hour, and much more active when the U.S. markets open for trading. Figure
2, which plots volatility at the 48 half-hour intervals across an average day as measured by
mean absolute returns, clearly reveals these phenomena. The intra-day volatility pattern
corresponds closely to the pattern obtained by Andersen and Bollerslev (1997) for 5-minute
DM/$ returns.
As with the conditional mean of the returns process, intra-day calendar effects in
return volatility are not of primary concern to us; hence we remove them. Taking rt s Zt
, i
where Zt represents the nonseasonal portion of the process, we remove volatility calendar
effects by fitting time-of-day dummies to 2log|rt| 2logs 2logZt. The F tests of no
,i
time-of-day effects confirm that calendar effects are present in both series; the F(47, 12527)
statistics are 17.34 and 8.927 for the DM/$ and Yen/$ respectively and have zero p-values.
Suitably normalized so that s summed over the entire sample would be equal to one, we use
,i
these time-of-day dummies to standardize the returns.
Figure 3 displays the autocorrelations of the levels, absolute and squared standardized
19
returns of both currencies. The standardization removes the intraday calendar effects in
volatility, and as in Andersen and Bollerslev (1997), their removal highlights a feature of the
data that, at least in the case of the DM/$, was not obvious: the autocorrelations of absolute
returns decay very slowly.
Construction of Multivariate Density Forecasts
We construct density forecasts of the two exchange rates using the exponential
smoothing approach of RiskMetrics, which assumes that the returns are generated from a
multivariate normal distribution, yielding the bivariate density forecasts
y1t
0
11, t| t 1 12, t| t 1
N , ,
y2t 0
21, t| t 1 22, t| t 1
2
where (1 )y1, t 1
11, t| t 1 11, t 1 | t 2
2
(1 )y2, t 1
22, t| t 1 22, t 1 | t 2
(1 )y1, t 1 y2, t 1,
12, t| t 1 12, t 1 | t 2
and is the decay factor. RiskMetrics applies the same decay factor to each of the variances
and covariances, which ensures that the variance-covariance matrix is positive definite.
Density Forecast Evaluation
As discussed earlier, we evaluate the bivariate density forecasts generated by the
RiskMetrics approach decomposing the forecasts in two ways:
(i) p(y1t, y2t) p(y1t) p(y2t| y1t) and
(ii) p(y1t, y2t) p(y2t) p(y1t| y2t),
t = 1, ..., T. We label the respective z series z1, z2 |1, z2 and z1|2, where we obtain z by
i
20
taking the probability integral transform of yi with respect to p(yi), and we obtain z by
i| j
transforming yi with respect to p(yi| yj). Good multivariate forecasts will produce
z1, z2 |1, z2 and z1|2 that are each iid U(0, 1).
To illustrate the multivariate density forecast evaluation procedures we split the
sample in two, the first running from January 1, 00:30 GMT to June 30, 21:00 GMT (6234
observations) and the second running from June 30, 21:30 GMT to December 31, 23:30 GMT
(6431 observations); we call them the estimation sample and the forecast sample,
respectively. We first evaluate forecasts generated using a decay factor of 0.95, which is
typical of RiskMetrics implementations. Figures 4a and 4b display the histograms and
correlograms of z1, z2 |1, z2 and z1|2; the histograms show clearly that the normality
assumption is inappropriate, and the correlograms show clearly that the decay factor 0.95
does not produce forecasts that capture the dynamics in return volatilities and correlations.
Although the correlograms of levels of z s look fine, the correlograms of squares of z s
indicate strong serial dependence. The positive serial correlation suggests that the decay
factor of 0.95 produces volatilities that adapt too slowly. Thus it appears that a smaller decay
factor would be appropriate.
We next turn to forecasts generated with an optimal decay factor, which we obtain
from the estimation sample as the decay factor that produced integral transforms that visually
appeared closest to iid, as assessed in our usual way, via correlograms of z and its powers. As
expected, a smaller decay factor ( 0.83) turns out to be optimal. Figures 5a and 5b display
the histograms and correlograms of z1, z2 |1, z2 and z1|2 and their squares, based on the
estimation sample, each with its mean removed before the correlograms are computed. The
21
correlograms indicate that the forecasts capture adequately the dynamics in return volatilities
and correlations (that is, z1, z2 |1, z2 and z1|2 and their squares appear serially uncorrelated).
The histograms, in contrast, show clearly that the normality assumption is a poor one.
The smaller decay factor of 0.83 also produces much better out-of-sample joint
forecasts of the two exchange rate returns. Figures 6a and 6b display the histograms and
correlograms of z1, z2 |1, z2 and z1|2 and their squares, based on the forecast sample,
corresponding to density forecasts produced using 0.83. The histograms again show
clearly that the normality assumption is inadequate. The correlograms of squares, on the
other hand, all show clear improvement and indicate little, if any, serial correlation.
Density Forecast Calibration
We now attempt to improve the .83 RiskMetrics forecasts, which seemed to capture
dynamics adequately but which were plagued by an inappropriate normality assumption, by
calibrating. Recall that we use the transformation
Ć
f(y1,m 1, y2,m 1,..., yN,m 1| ) pm(y1,m 1, y2,m 1,..., yN,m 1) q(z1,m 1, z2,m 1,..., zN,m 1),
Ć
m
or equivalently,
Ć Ć
F(y1,m 1,...,yN,m 1 m) Q(Pm(y1,m 1,...,yN,m 1)).
Ć
We obtain Q( ) from the z series based on the estimation sample. Although we favor the
presentation of histograms for density forecast evaluation, calibration is facilitated by using
the empirical c.d.f. form. We present the histograms and correlograms of the four z series
corresponding to the calibrated forecasts in Figures 7a and 7b; clearly the histograms of the
calibrated forecasts are substantially improved relative to their non-calibrated counterparts,
22
and the correlograms remain good (that is, they are not affected by the calibration.)
5. Concluding Remarks and Directions for Future Research
We have proposed a framework for multivariate density forecast evaluation and
calibration. The multivariate forecast evaluation procedure is a generalization of the
univariate procedure proposed in Diebold, Gunther and Tay (1998) and shares its constructive
nature and ease of implementation. We illustrated the power of the procedure to detect and
remove defects in bivariate exchange rate density forecasts generated by a popular method.
An interesting direction for future research involves using recursive techniques for
real-time monitoring for breakdown of density forecast adequacy. Real-time monitoring
using CUSUM techniques is a simple matter in the univariate case, because under the
adequacy hypothesis the z series is iid U(0,1), which is free of nuisance parameters.
Appropriate boundary crossing probabilities for the CUSUM of the z series can be computed,
as in Chu, Stinchcombe and White (1996), using results on boundary crossing probabilities of
sample sums such as those of Robbins and Siegmund (1970). Multivariate CUSUM schemes
are also possible, as with the multivariate profile charts of Fuchs and Benjamini (1994).
23
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(a) DM / $ (b) YEN / $
0.3 0.3
0.2 0.2
0.1 0.1
0.0 0.0
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(c) DM / $ (abs) (d) YEN / $ (abs)
0.3 0.3
0.2 0.2
0.1 0.1
0.0 0.0
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(e) DM / $ (sq) (f) YEN / $ (sq)
0.3 0.3
0.2 0.2
0.1 0.1
0.0 0.0
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
Figure 1
A.c.f.s of Returns, Absolute Returns and Squared Returns
MA(1) Component Removed
DM / $
0.08
0.07
0.06
0.05
0.04
0.03
0.02
2130 0100 0500 0900 1300 1700 2100
GMT
YEN / $
0.08
0.07
0.06
0.05
0.04
0.03
0.02
2130 0100 0500 0900 1300 1700 2100
GMT
Figure 2
Mean Intraday Absolute Returns
(a) DM / $ (b) YEN / $
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0.0 0.0
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(c) DM / $ (abs) (d) YEN / $ (abs)
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0.0 0.0
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(e) DM / $ (sq) (f) YEN / $ (sq)
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0.0 0.0
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
Figure 3
A.c.f.s of Returns, Absolute Returns and Squared Returns
Conditional Mean MA(1) Component Removed
Volatility Intra-Day Calendar Effects Removed
(i) z1 (ii) z2|1
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(iii) z2 (iv) z1|2
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Figure 4a
Histograms of z (decay factor=0.95, forecast sample)
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
0.0
1.0
2.0
3.0
(i) z1 (ii) z1 (square)
0.2 0.2
0.2 0.2
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(iii) z2|1 (iv) z2|1 (square)
0.2 0.2
0.2 0.2
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(v) z2 (vi) z2 (square)
0.2 0.2
0.2 0.2
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(vii) z1|2 (viii) z1|2 (square)
0.2 0.2
0.2 0.2
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
Figure 4b
Correlograms of powers of z (decay factor=0.95, forecast sample)
(i) z1 (ii) z2|1
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(iii) z2 (iv) z1|2
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Figure 5a
Histograms of z (decay factor=0.83, estimation sample)
2.0
2.0
2.0
2.0
2.0
2.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
2.0
2.0
2.0
2.0
2.0
2.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
(i) z1 (ii) z1 (square)
0.1
0.1 0.1
0.0
0.0 0.0
-0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(iii) z2|1 (iv) z2|1 (square)
0.1 0.1
0.0 0.0
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(v) z2 (vi) z2 (square)
0.1
0.1 0.1
0.0
0.0 0.0
-0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(vii) z1|2 (viii) z1|2 (square)
0.1 0.1
0.0 0.0
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
Figure 5b
Correlograms of powers of z (decay factor=0.83, estimation sample)
(i) z1 (ii) z2|1
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(iii) z2 (iv) z1|2
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Figure 6a
Histograms of z (decay factor=0.83, forecast sample)
2.0
2.0
2.0
2.0
2.0
2.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
2.0
2.0
2.0
2.0
2.0
2.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
(i) z1 (ii) z1 (square)
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(iii) z2|1 (iv) z2|1 (square)
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(v) z2 (vi) z2 (square)
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(vii) z1|2 (viii) z1|2 (square)
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
Figure 6b
Correlograms of powers of z (decay factor=0.83, forecast sample)
(i) z1 (ii) z2|1
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
(iii) z2 (iv) z1|2
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Figure 7a
Histograms of z (decay factor=0.83, forecast sample, calibrated)
2.0
2.0
2.0
2.0
2.0
2.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
2.0
2.0
2.0
2.0
2.0
2.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
(i) z1 (ii) z1 (square)
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(iii) z2|1 (iv) z2|1 (square)
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(v) z2 (vi) z2 (square)
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
(vii) z1|2 (viii) z1|2 (square)
0.1 0.1
0.1 0.1
0.0 0.0
0.0 0.0
-0.1 -0.1
-0.1 -0.1
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
0 50 100 150 200 0 50 100 150 200
Figure 7b
Correlograms of powers of z (decay factor=0.83, forecast sample, calibrated)
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