"Frontmatter". In: Analysis of Financial Time Series

212 HIGH-FREQUENCY DATAThe fitted model for D i isµ i = 0.049 − 0.840D i−1 − 0.004 ln(t i )ln(σ i ) = 0.244| D i−1 + D i−2 + D i−3 + D i−4 |,where standard deviations of the parameters in the mean equation are 0.129, 0.132,and 0.082, respectively, whereas that for the parameter in the variance equation is0.182. The price reversal is clearly shown by the highly significant negative coefficientof D i−1 . The marginally significant parameter in the variance equation isexactly as expected. Finally, the fitted models for the size of a price change areln(λ d,i ) = 1.024 − 0.327N i + 0.412 ln(t i ) − 4.474S i−1ln(λ u,i ) =−3.683 − 1.542N i + 0.419 ln(t i ) + 0.921S i−1 ,where standard deviations of the parameters for the “down size” are 3.350, 0.319,0.599, and 3.188, respectively, whereas those for the “up size” are 1.734, 0.976,0.453, and 1.459. The interesting estimates of the prior two equations are the negativeestimates of the coefficient of N i .AlargeN i means there were more transactions inthe time interval (t i−1 , t i ) with no price change. This can be taken as evidence of nonew information available in the time interval (t i−1 , t i ). Consequently, the size forthe price change at t i should be small. A small λ u,i or λ d,i for a Poisson distributiongives precisely that.In summary, granted that a sample of 194 observations in a given day may notcontain sufficient information about the trading dynamic of IBM stock, but the fittedmodels appear to provide some sensible results. McCulloch and Tsay (2000) extendthe PCD model to a hierarchical framework to handle all the data of the 63 tradingdays between November 1, 1990 and January 31, 1991. Many of the parameterestimates become significant in this extended sample, which has more than 19,000observations. For example, the overall estimate of the coefficient of ln(t i−1 ) in themodel for time duration ranges from 0.04 to 0.1, which is small, but significant.Finally, using transactions data to test microstructure theory often requires a carefulspecification of the variables used. It also requires a deep understanding of theway by which the market operates and the data are collected. However, ideas of theeconometric models discussed in this chapter are useful and widely applicable inanalysis of high-frequency data.APPENDIX A.REVIEW OF SOME PROBABILITY DISTRIBUTIONSExponential distributionA random variable X has an exponential distribution with parameter β > 0 if itsprobability density function (pdf) is given by