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18 2.7 EQUITY MARKETS of an economy rather than individual companies and their behavior is less erratic than that of individual equities. Highfrequency data for the main indices are available and are interesting objects of research. Due to their mathematical definition, they often show a positive autocorrelation of returns at a lag of up to 15 minutes. This may be a consequence of a lag structure between leading main equities and the less liquid equities of the basket. Individual equity futures or equity index futures are liquid instruments with highquality, highfrequency data that have been studied by many researchers. Options also exist for individual equities as well as equity indices. Their implied volatility figures can be investigated by time series methods.
TIME SERIES OF INTEREST An adequate analysis of highfrequency data relies on explicit definitions of the variables under study. Tn this chapter, we study the common mathematical framework used to analyze these variables. Some aspects of preparing and preprocessing a time series are rather technical. Readers interested in economic results may prefer to skip the technical Chapters 3 and 4 and continue their reading in Chapter 5. Researchers conducting their own highfrequency studies may profit from Chapters 3 and 4. If they have no access to preprocessed time series (i.e., cleaned time series with regular spacing in time), they will need the techniques described in these chapters. The literature often ignores technicalities of dealing with irregularly spaced highfrequency data, so we have a good reason to discuss them in two chapters of this book. 3.1 TIME SERIES AND OPERATORS Many types of time series data can be obtained at high frequency, often intraday, at market tickbytick frequency. For most methods, these raw time series are not suitable to work with, because market ticks arrive at random times.1 The There are models for the stochastic nature of these times, such as Engle and Russell (1997, 1998).
time series operator formalism developed by Zumbach (1996) and Zumbach and Miiller (2001) offers a powerful way to deal with irregularly spaced data. Section 3.3 is based on this formalism and the notations of Zumbach and Miiller (2001). In time series analysis, a first important classification is done according to the spacing of data points in time. Regularly spaced time series are called homogeneous, whereas irregularly spaced series are called inhomogeneous. The concept of inhomogeneous time series also has to be distinguished from two other concepts, which are the concept of missing observations (where a series is essentially homogeneous with few gaps) and the concept of continuoustime finance (which belongs to theory rather than data sampling). When considering the spacing of data in time, a discussion of the time scale is necessary. Many time series of daily data in finance, for example, have only five observations per week; there are no observations on Saturdays and Sundays. Such a time series is homogeneous only if using a special "business time" scale, which omits weekends (and holidays). Even more sophisticated business time scales can be introduced in order to cope with some characteristics of intraday data such as the seasonality of volatility, (see Chapter 6 and Dacorogna et al, 1993), the heteroskedasticity (Zhou, 1993), or both seasonality and heteroskedasticity (Guillaume et al, 1997; Miiller et al., 1993a). Time is denoted by t in Chapter 3, but t may stand for any choice of time scale, not only physical time or clock time. The terms "homogeneous" and "inhomogeneous" have to be understood in the context of the chosen time scale. Inhomogeneous time series by themselves are conceptually simple. The difficulty lies in efficiently extracting and computing information from them. Time series operators are a major tool used to transform a raw, inhomogeneous time series to the (homogeneous or inhomogeneous) time series of the variable to be analyzed. In most books on time series analysis, the field of time series is restricted to homogeneous time series.2 In Section 3.2, we follow this restriction, which induces numerous simplifications, both conceptually and computationally. There, we need only one time series operator type: an operator to transform an inhomogeneous time series to a homogeneous one, type (a) of Figure 3.1. In Sections 3.3 and 3.4, we follow Zumbach and Miiller (2001) and build a computational toolbox for directly and efficiently treating inhomogeneous time series. In practice, this toolbox is attractive enough to be applied to any time series, including homogeneous ones. Given a time series z, such as an asset price, the general point of view is to compute another time series, such as the volatility of the asset, by the application of an operator Q[z] where the resulting series stays inhomogeneous with the same time points as the original series. This operator type is called type (b) in Figure 3.1. Classical textbooks on homogeneous time series are Granger and Newbold (1977); Priestley (1989); Hamilton (1994).
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