## Bivariate Models For Price Change And Duration

In this section, we introduce a model that considers jointly the process of price change and the associated duration. As mentioned before, many intraday transactions of a stock result in no price change. Those transactions are highly relevant to trading intensity, but they do not contain direct information on price movement. Therefore, to simplify the complexity involved in modeling price change, we focus on transactions that result in a price change and consider a price change and duration (PCD) model to describe the multivariate dynamics of price change and the associated time duration.

We continue to use the same notation as before, but the definition is changed to transactions with a price change. Let ti be the calendar time of the i th price change of an asset. As before, ti is measured in seconds from midnight of a trading day. Let Pti be the transaction price when the i th price change occurred and Ati = ti _ ti _i be the time duration between price changes. In addition, let Ni be the number of trades in the time interval (ti _1, ti) that result in no price change. This new variable is used to represent trading intensity during a period of no price change. Finally, let Di be the direction of the ith price change with Di = 1 when price goes up and Di = _1 when the price comes down, and let Si be the size of the i th price change measured in ticks. Under the new definitions, the price of a stock evolves over time by

and the transactions data consist of {Ati, Ni, Di, Si} for the i th price change. The PCD model is concerned with the joint analysis of (Ati, Ni, Di, Si).

Remark: Focusing on transactions associated with a price change can reduce the sample size dramatically. For example, consider the intraday data of IBM stock from November 1, 1990 to January 31, 1991. There were 60,265 intraday trades, but only 19,022 of them resulted in a price change. In addition, there is no diurnal pattern in time durations between price changes.

To illustrate the relationship among the price movements of all transactions and those of transactions associated with a price change, we consider the intraday tradings of IBM stock on November 21, 1990. There were 726 transactions on that day during the normal trading hours, but only 195 trades resulted in a price change. Figure 5.14 shows the time plot of the price series for both cases. As expected, the price series are the same.

The PCD model decomposes the joint distribution of (Ati, Ni, Di, Si) given Fi _1

as f (Ati, Ni, Di, Si | Fi_1) = f (Si | Di, Ni, Ati, Fi_1) f (Di | Ni, Ati, Fi_0 f (Ni | Ati, Fi_0 f (Ati | Fi_1).

This partition enables us to specify suitable econometric models for the conditional distributions and, hence, to simplify the modeling task. There are many ways to specify models for the conditional distributions. A proper specification might depend on the asset under study. Here we employ the specifications used by McCulloch and Tsay (2000), who use generalized linear models for the discrete-valued variables and a time series model for the continuous variable ln(Ati).

For the time duration between price changes, we use the model ln(Ati) = £0 + A ln(Ati _1) + £2 Si _1 + aei, (5.48)

where a is a positive number and {ei} is a sequence of iid N(0, 1) random variables. This is a multiple linear regression model with lagged variables. Other explanatory variables can be added if necessary. The log transformation is used to ensure the positiveness of time duration.

(a) All transactions

(a) All transactions

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