TABLE 9.4 Significance of the forecast quality for 20 exchange rates.

The in-sample and out-of-sample forecast significance for 10 USD rates and 10 cross rates. The "+" sign indicates a forecast quality above the significance limits of all test criteria, otherwise the "-" sign is used. Example: "+/-" means a significant in-sample quality and an insignificant out-of-sample quality.

to assume that the markets need a finite time to adjust to any information and that this time depends on the nature of the information.

We think that these adjustments can be modeled and hence that a certain predictability of price movements exists. Our forecasting models, while a positive step in this direction, are nevertheless only a first one and there is still room for improvement through a better understanding and definition of intrinsic time and through the search for better indicators.

CORRELATION AND MULTIVARIATE RISK

10.1 INTRODUCTION y

Correlations and covariances between returns of different financial assets play an important role in fields such as risk management and portfolio allocation. This chapter addresses three problematic issues concerning linear correlation coefficients of returns, computed from high-frequency data:

1. The correlation of intraday, equally spaced time series derived from unevenly spaced tick-by-tick data deserves careful treatment if a bias resulting from the classical missing value problem is to be avoided, iffie propose a simple and easy-to-use method, which corrects for different data frequencies and gaps by updating the linear correlation coefficient calculation with the aid of covolatility weightaT This is a bivariate alternative to time scale transformations which treat neteroskedasticity by expanding periods of higher volatility while contracting periods of lower volatility.

2. It is generally recognized that correlations between financial time series vary over time. We probe the stability of correlation as a function of time for 7 years of high-frequency foreign exchange rate, implied forward interest rate, and stock index data. Correlations as functions of time in

10.2 ESTIMATING THE DEPENDENCE OF FINANCIAL TIME SERIES

turn allow for estimations of the memory that correlations have for their past values.

3. It has been demonstrated that there is a dramatic decrease in correlation, as data frequency enters the intrahour level (the "Epps effect"1). We characterize the Epps effect for correlations between a number of financiul time series and suggest its possible relation to tick frequency.

10.2 estimating the dependence of financial time series

Measuring the dependence or independence of financial time series is of increasing interest to those concerned with multivariate decision formation (e.g., in risk assessment or portfolio allocation). Often this is estimated quantitatively using the linear correlation coefficient,2 which is a basic measurement of the dependence between variables. Zumbach (1997) reviews many interesting measures of associations besides the linear correlation. The popularity of this measure stems from its simple definition, practical ease of use, and its straightforward results, which are easily interpreted, scale free, and directly comparable. Although the calculation of the correlation coefficient is well defined and rather simple, a number of unresolved issues exist with respect to application of the rule and interpretation of results in the high-frequency data domain.

The data input for the correlation coefficient calculation are two time series with equal (i.e., homogeneous) spacing between ticks. This necessity is easily satisfied for low frequency (< one tick per week) data. However, the intraday case deserves more careful treatment if a resulting data bias is to be avoided. A problem arises when the two time series of unevenly spaced tick-by-tick data have different frequencies or active hours within a day, which may or may not overlap. We propose a simple and easy-to-use normalization method, which corrects for frequency differentials and data gaps. This alternative formulation updates the correlation calculation only where data exists, ensuring that there is no measurement bias resulting from the classical missing value problem (see Krzanowski and Marriott, 1994, 1995) or from differences in the active hours of the financial time series. In addition, this formulation remains scale free and straightforward to understand and implement.

The linear correlation coefficient calculation largely discards the time variable. The variances of two time series and their covariance are constructed

1 Epps (1979).

2 The use of the linear correlation coefficient is appropriate not only for multivariate normal joint distributions but also for multivariate elliptical joint distributions. Many financial joint return distributions have been observed to fall into or close to this latter category. Also for fat-tailed return distributions, the linear correlation coefficient remains a useful and relevant measure of association; only the interpretation of results and, more specifically, the determination of accurate confidence limits is problematic. Correlations of squared returns from fat-tailed distributions are even more problematic.