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19 Any time series (inhomogeneous)  Regularizing Operator  Homogeneous time series   of same variable     Any time series z (inhomogeneous)  Operator Q  Time series £2[z], new variable, same time points   (inhomogeneous) 
FIGURE 3.1 Different operator types to study time series: (a) Sampling an inhomogeneous time series at regular time intervals. The resulting homogeneous time series can be treated by standard methods of time series analysis. (b) Computing a new variable from the initial variable while keeping the initial (inhomogeneous) time points. Example: computing a series of local volatility values from the initial price series. An important distinction between operators has to be made. We introduce two operator types: Microscopic operators depend on the actual sampling of the inhomogeneous time series. Eliminating some random ticks leads to very different results. Macroscopic operators extract average behaviors of their time series argument. They are essentially immune to small variations of the individual ticks, including adding or eliminating few ticks. A possible technical definition of macroscopic operators is that they have a welldefined limit when the price quotes become infinitely dense. Practically, if the price quotes are sufficiently dense inside the range of the operator, we are close enough to this limit. For inhomogeneous time series, macroscopic operators are better behaved and more robust than microscopic operators. For homogeneous time series, this distinction is unnecessary because the sampling frequency is fixed and there is no reason to take a continuoustime limit or to formally add or remove ticks. Moreover, because homogeneous time series analysis is based on the backward shift operator (which is microscopic), most of the conventional time series analysis becomes unusable for inhomogeneous time series. The classification between operators is further explored by Zumbach and Muller (2001). Macroscopic operators can be represented by convolutions and are discussed in Section 3.3. The archetype of a macroscopic operator is the exponential moving average (EMA) that computes a moving average with an exponentially decaying weight of the past. Microscopic operators are presented in Section 3.4. Examples are the time difference St between ticks (e.g., in Stj = tj  f/ i) and the backward shift operator defined in Section 3.4.1.
3.2 VARIABLES IN HOMOGENEOUS TIME SERIES Basic variables such as the price, the return, the realized volatility, and the spread arc defined in Section 3.2. In order to capture the dynamics of the intraday market, some more variables such as the tick frequency are of interest. 3.2.1 Interpolation Before defining different variables, the generation of homogeneous time series has to be explained. A homogeneous time series, although taken for granted in time series analysis, is an artifact that has to be constructed from the raw data, which is an inhomogeneous series with times tj and values zj = z(tj). The index j refers to the irregularly spaced sequence of the raw series. By utilizing an interpolation method, we construct a homogeneous time series with values at times to + iAt, regularly spaced by At, rooted at a time to. The index i refers to the homogeneous series. The time fo + iAt is bracketed by two times tj of the raw series / = maxfj \tj < to +iAt) , tj < t0+iAt < fy+i (31) We interpolate between fy and tj+\. The two most important interpolation methods are linear interpolation to + iAt  tj> z(to + iAt) = zr + (zj+i  zr) (3.2) tj+i  tj and previoustick interpolation (taking the most recent value), z(to+iAt) = Zj (3.3) which was already proposed by Wasserfallen and Zimmermann (1985). Both methods which are illustrated by Figure 3.2 have their merits. Previoustick interpolation respects causality as it exclusively uses information already known at time fo + A?, whereas linear interpolation uses information from time fy+i, which lies in the future of time fo + iAt. When using previoustick interpolation over a gap (a long period of missing data) in the raw data, a spurious jump of z may be observed at the end of the gap, which may spoil a statistical analysis of extreme returns of z. In this example, linear interpolation would be the appropriate choice. As advocated in Miiller et al. (1990), linear interpolation is the appropriate method for a random process with identically and independently distributed (i.i.d.) increments. Many statistical studies and model estimations can be alternatively done with both interpolation methods, in practice. The difference between the results indicates the sensitivity to the choice of the interpolation method. The difference is often small, even negligible thanks to highfrequency data. In the empirical studies of the book, the choice of the interpolation method is discussed whenever it matters.
figure 3.2 interpolation methods to obtain a homogeneous time series: selectir\ values at equally spaced time points tj, indicated by dotted vertical lines. the inhomogeneous time sequence of raw observations is indicated by ticks below the horizontal time axis and by dashed vertical lines (only for the observations bracketing the time points tj two important interpolation methods are illustrated by empty circles: linear interpolatior (big circles) and previoustick interpolation (small circles). The transformation of an inhomogeneous time series to a homogeneous ont can also be understood as the result of a special microscopic time series operate which is discussed in Sections 3.3.1 and 3.4.2. 3.2.2 price Prices of assets are the most important variables explored in finance. Depending on the market structure and the data supplier, prices are available as quotes tr different forms: Bidask price pairs: and pask Transaction prices (which may or may not be former bid or ask quotes) Bid, ask, transaction prices in irregular sequence (not in pairs, not synchronous) Middle prices One individual observation at a time tj, also in the case of bidask pairs, is callei. a tick. Bidask price pairs are discussed first. FX prices and other asset prices, a: well as nonprice variables such as spot interest rates and implied volatility figure: from option markets, are quoted as bidask pairs. The most important variable
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