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47 smaller, and there are no monotonically drifting or excessively repeated quotes. This latter observation can be made for the Swiss Franc (CHF) interest rate futures of Table 4.6, where all the rejected ticks are true outliers. However, the outlier rate is rather high, about 8.5%. A closer look shows that most of these rejected ticks are empty ticks with formally quoted values of zero. Whatever the reason of the data supplier to post these empty ticks, the filter rightly rejects them as outliers. The rejection rates of Table 4.6 have been computed for the filter running in realtime mode. The corresponding rejection rates of historical filtering (see Section 4.7.2) are similarusually slightly lower. A data filter needs a testing environment to analyze its statistical behavior. Table 4.6 presents a simple example of results produced by such a testing environment. A good general method to test the effects of filtering in practice is a sensitivity analysis of the following kind. The data application, whatever it is, is implemented twice, using two different filters. Both filters may follow the same algorithm, but one of them is weak with more tolerant parameters, leading to a lower rejection rate, perhaps only half the rejection rate of the other filter. Then the results of both applications are compared to each other. The deviations between analogous results directly reflect the sensitivity or robustness of the analysis against changes in the data cleaning algorithm, and indirectly the possible degree of distortion by the filter. This has been done, for example, in the case of an extreme value study of FX returnsa type of analysis very sensitive to outliers (which naturally lead to extreme return observations). Fortunately, the results for both filters are very similar, which means that both filters successfully eliminate the true outliers. The doubtful ticks that are accepted by one filter and rejected by the other one have little influence on the final results.
BASIC STYLIZED FACTS 5.1 introduction Gathering basic stylized facts on the behavior of financial assets and their returns is an important research activity. Without such facts it is not possible to design models that can explain the data. Highfrequency data opened up a whole new field of exploration and brought to light some behaviors that could not be observed at lower frequencies. In this chapter we review the main stylized facts for foreign exchange (FX) rates, interbank money market rates, and Eurofutures contracts. These stylized facts can be grouped under four main headings: autocorrelation of return, distributional issues, scaling properties, and seasonality. We find a remarkable similarity between the different asset types. Hence, we shall examine each of the properties first for FX rates and then show how they are present or modified for the others. FX rates have been the subject of many studies. However, these studies do not present a unified framework of the return distributions of the datagenerating process. Most of the earlier literature analyzed daily time series, but, more and more, recent publications deal with intraday prices. They essentially confirm the findings of this chapter. Here, we use a set of intraday time series covering a worldwide 24 hr market,1 and we present a study of fundamental statistical For a full description of the data, we refer the reader to Chapter 2.
properties of the intraday data. More specifically, this chapter demonstrates the following: At the highest frequency, the middle price is subject to microstructure effects (e.g., the bouncing of prices between the bid and ask levels). The price formation process plays an important role and overshadows some of the properties encountered at lower frequencies. The distributions of returns are increasingly fattailed as data frequency increases (smaller interval sizes) and are hence distinctly unstable. The second moments of the distributions most probably exist while the fourth moments tend to diverge. Scaling laws describe mean absolute returns and mean squared returns as functions of their time intervals (varying from a few minutes to one or more years). We find that these quantities are proportional to a power of the interval size. There is evidence of seasonal heteroskedasticity in the form of distinct daily and weekly clusters of volatility. This effect may partly explain the fattailedness of the returns and should be taken into account in the study of the return distributions. Daily and weekly patterns also exist in quote frequency. Daily and weekly patterns are also found for the average bidask spread, which is negatively correlated to the volatility. The trading activity in terms of price quoting frequency has a positive correlation to the volatility and a negative one to the spread. These findings imply that the trading volume is also positively correlated to the volatility. The daily patterns of all these variables may be explained by the behavior of three main markets America, Europe, and East Asiawhose active periods partially overlap. Our intraday and intraweek analysis shows that there are systematic variations of volatility, even within what are generally considered business hours. The literature presents a number of views regarding the distributions of FX returns and the corresponding datagenerating process. Some papers claim FX returns to be close to Paretian stable ones, for instance, (McFarland et al., 1982; Westerfield, 1997); some to Student distributions that are not stable (Rogalski and Vinso, 1978; Boothe and Glassman, 1987); some reject any single distribution (CalderonRossel and BenHorim, 1982). Most researchers now agree that a better description of the data generating process is in the form of a conditional heteroskedastic model rather than being from an unconditional distribution. Among the earliest to propose this for the FX rates were Friedmann and Vandersteel (1982); Wasserfallen and Zimmermann (1985); Tucker and Scott (1987) and Diebold (1988). On distributional issues, the only agreement seems to be that daily returns are fattailed and that there are substantial deviations from a Gaussian random walk model. Moreover, all of the literature on GARCH agrees that the distribution is not stable. Many of the studies of the late 1980s have been limited to daily or even weekly data except for
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