10.3 Forecasting exchange rate densities

The preceding section illustrated how the GARCH model may be employed effectively to describe empirical price variations of foreign exchange rates. For practical purposes, as for instance scenario analysis, VaR estimation (Chapter 1), option pricing (Chapter 16), one is often interested in the future joint density of a set of asset prices. Continuing the comparison of the univariate and bivariate approach to model volatility dynamics of exchange rates it is thus natural to investigate the properties of these specifications in terms of forecasting performance.

We implement an iterative forecasting scheme along the following lines: Given the estimated univariate and bivariate volatility models and the corresponding information sets $ {\cal F}_{t-1},t=1,\ldots,T-5$ (Figure 10.2), we employ the identified data generating processes to simulate one-week-ahead forecasts of both exchange rates. To get a reliable estimate of the future density we set the number of simulations to 50000 for each initial scenario. This procedure yields two bivariate samples of future exchange rates, one simulated under bivariate, the other one simulated under univariate GARCH assumptions.

A review on the current state of evaluating competing density forecasts is offered by Tay and Wallis (1990). Adopting a Bayesian perspective the common approach is to compare the expected loss of actions evaluated under alternative density forecasts. In our pure time series framework, however, a particular action is hardly available for forecast density comparisons. Alternatively one could concentrate on statistics directly derived from the simulated densities, such as first and second order moments or even quantiles. Due to the multivariate nature of the time series under consideration it is a nontrivial issue to rank alternative density forecasts in terms of these statistics. Therefore, we regard a particular volatility model to be superior to another if it provides a higher simulated density estimate of the actual bivariate future exchange rate. This is accomplished by evaluating both densities at the actually realized exchange rate obtained from a bivariate kernel estimation. Since the latter comparison might suffer from different unconditional variances under univariate and multivariate volatility, the two simulated densities were rescaled to have identical variance. Performing the latter forecasting exercises iteratively over 3714 time points we can test if the bivariate volatility model outperforms the univariate one.

To formalize the latter ideas we define a success ratio $ SR_J$ as

$\displaystyle SR_J=\frac{1}{\vert J\vert}\sum_{t \in J }\boldsymbol{1}\{\hat{f}_{biv}(R_{t+5})> \hat{f}_{uni}(R_{t+5})\},$ (10.9)

where $ J$ denotes a time window containing $ \vert J\vert$ observations and $ \boldsymbol{1}$ an indicator function. $ \hat{f}_{biv}(R_{t+5})$ and $ \hat{f}_{uni}(R_{t+5})$ are the estimated densities of future exchange rates, which are simulated by the bivariate and univariate GARCH processes, respectively, and which are evaluated at the actual exchange rate levels $ R_{t+5}$. The simulations are performed in 21095 XFGmvol04.xpl .


Table 10.1: Time varying frequencies of the bivariate GARCH model outperforming the univariate one in terms of one-week-ahead forecasts (success ratio)
Time window $ J$ Success ratio $ SR_J$
1980 1981 0.744
1982 1983 0.757
1984 1985 0.793
1986 1987 0.788
1988 1989 0.806
1990 1991 0.807
1992 1994/4 0.856


Our results show that the bivariate model indeed outperforms the univariate one when both likelihoods are compared under the actual realizations of the exchange rate process. In 81.6% of all cases across the sample period, $ SR_J=0.816, \quad J=\{t:t= 1,...,T-5\},$ the bivariate model provides a better forecast. This is highly significant. In Table 10.1 we show that the overall superiority of the bivariate volatility approach is confirmed when considering subsamples of two-years length. A-priori one may expect the bivariate model to outperform the univariate one the larger (in absolute value) the covariance between both return processes is. To verify this argument we display in Figure 10.4 the empirical covariance estimates from Figure 10.2 jointly with the success ratio evaluated over overlapping time intervals of length $ \vert J\vert=80$.

Figure: Estimated covariance process from the bivariate GARCH model ( $ 10^4\hat{\sigma}_{12}$, blue) and success ratio over overlapping time intervals with window length 80 days (red).
\includegraphics[width=1.5\defpicwidth]{successfig.ps}

As is apparent from Figure 10.4 there is a close co-movement between the success ratio and the general trend of the covariance process, which confirms our expectations: the forecasting power of the bivariate GARCH model is particularly strong in periods where the DEM/USD and GBP/USD exchange rate returns exhibit a high covariance. For completeness it is worthwhile to mention that similar results are obtained if the window width is varied over reasonable choices of $ \vert J\vert$ ranging from 40 to 150.

With respect to financial practice and research we take our results as strong support for a multivariate approach towards asset price modeling. Whenever contemporaneous correlation across markets matters, the system approach offers essential advantages. To name a few areas of interest multivariate volatility models are supposed to yield useful insights for risk management, scenario analysis and option pricing.