"Frontmatter". In: Analysis of Financial Time Series

190 HIGH-FREQUENCY DATATable 5.4. Parameter Estimates of the Ordered-Probit Model in Eq. (5.19) and Eq. (5.20)for the 1988 Transaction Data of IBM, Where t Denotes the t Ratio.(a) Boundary partitions of the probit modelPar. α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8Est. −4.67 −4.16 −3.11 −1.34 1.33 3.13 4.21 4.73t −145.7 −157.8 −171.6 −155.5 154.9 167.8 152.2 138.9(b) Equation parameters of the probit modelPar. γ 1 γ 2 β 1 : ti ∗ β 2 : y −1 β 3 β 4 β 5 β 6Est. 0.40 0.52 −0.12 −1.01 −0.53 −0.21 1.12 −0.26t 15.6 71.1 −11.4 −135.6 −85.0 −47.2 54.2 −12.1Par. β 7 β 8 β 9 : β 10 β 11 β 12 β 13Est. 0.01 −1.14 −0.37 −0.17 0.12 0.05 0.02t 0.26 −63.6 −21.6 −10.3 47.4 18.6 7.75.4.2 A Decomposition ModelAn alternative approach to modeling price change is to decompose it into three componentsand use conditional specifications for the components; see Rydberg andShephard (1998). The three components are an indicator for price change, the directionof price movement if there is a change, and the size of price change if a changeoccurs. Specifically, the price change at the ith transaction can be written aswhere A i is a binary variable defined asy i ≡ P ti − P ti−1 = A i D i S i , (5.21)A i ={ 1 if there is a price change at the ith trade0 if price remains the same at the ith trade.(5.22)D i is also a discrete variable signifying the direction of the price change if a changeoccurs—that is,{ 1 if price increases at the ith tradeD i | (A i = 1) =−1 if price drops at the ith trade,(5.23)where D i | (A i = 1) means that D i is defined under the condition of A i = 1, and S iis size of the price change in ticks if there is a change at the ith trade and S i = 0ifthere is no price change at the ith trade. When there is a price change, S i is a positiveinteger-valued random variable.Note that D i is not needed when A i = 0, and there is a natural ordering in thedecomposition. D i is well defined only when A i = 1andS i is meaningful when