5.7 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 ith price change of an asset. As before, ti is measured in seconds from midnight of a trading day. Let be the transaction price when the ith price change occurred and Δti = ti − ti−1 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 ith price change measured in ticks. Under the new definitions, the price of a stock evolves over time by

5.46

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

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 normal trading hours, but only 195 trades resulted in a price change. Figure 5.16 shows the time plot of the price series for both cases. As expected, the price series are the same.

Figure 5.16 Time plots of intraday transaction prices of IBM stock on November 21, 1990: (a) all transactions and (b) transactions that resulted in price change.

The PCD model decomposes the joint distribution of (Δti, Ni, Di, Si) given Fi−1 as

5.47

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(Δti).

For the time duration between price changes, we use the model

5.48

where σ is a positive number and {ϵi} 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.

The conditional model for Ni is further partitioned into two parts because empirical data suggest a concentration of Ni at 0. The first part of the model for Ni is the logit model

5.49

where logit(x) = exp(x)/[1 + exp(x)], whereas the second part of the model is

5.50

where ∼ means “is distributed as,” and g(λ) denotes a geometric distribution with parameter λ, which is in the interval (0, 1).

The model for direction Di is

5.51

where ϵ is a N(0, 1) random variable, and

In other words, Di is governed by the sign of a normal random variable with mean μi and variance . A special characteristic of the prior model is the function for ln(σi). For intraday transactions, a key feature is the price reversal between consecutive price changes. This feature is modeled by the dependence of Di on Di−1 in the mean equation with a negative ω1 parameter. However, there exists an occasional local trend in the price movement. The previous variance equation allows for such a local trend by increasing the uncertainty in the direction of price movement when the past data showed evidence of a local trend. For a normal distribution with a fixed mean, increasing its variance makes a random draw have the same chance to be positive and negative. This in turn increases the chance for a sequence of all positive or all negative draws. Such a sequence produces a local trend in price movement.

To allow for different dynamics between positive and negative price movements, we use different models for the size of a price change. Specifically, we have

5.52

5.53

where p(λ) denotes a Poisson distribution with parameter λ, and 1 is added to the size because the minimum size is 1 tick when there is a price change.

The specified models in Eqs. (5.48)–(5.53) can be estimated jointly by either the maximum-likelihood method or the Markov chain Monte Carlo methods. Based on Eq. (5.47), the models consist of six conditional models that can be estimated separately.

Example 5.5

Consider the intraday transactions of IBM stock on November 21, 1990. There are 194 price changes within normal trading hours. Figure 5.17 shows the histograms of ln(Δti), Ni, Di, and Si. The data for Di are about equally distributed between “upward” and “downward” movements. Only a few transactions resulted in a price change of more than 1 tick; as a matter of fact, there were 7 changes with 2 ticks and 1 change with 3 ticks. Using Markov chain Monte Carlo (MCMC) methods (see Chapter 12), we obtained the following models for the data. The reported estimates and their standard deviations are the posterior means and standard deviations of MCMC draws with 9500 iterations. The model for the time duration between price changes is

where standard deviations of the coefficients are 0.415, 0.073, 0.384, and 0.073, respectively. The fitted model indicates that there was no dynamic dependence in the time duration. For the Ni variable, we have

where standard deviations of the estimates are 0.238 and 0.248, respectively. Thus, as expected, the number of trades with no price change in the time interval (ti−1, ti) depends positively on the length of the interval. The magnitude of Ni when it is positive is

where standard deviations of the estimates are 0.246 and 0.138, respectively. The negative and significant coefficient of ln(Δti) means that Ni is positively related to the length of the duration Δti because a large ln(Δti) implies a small λi, which in turn implies higher probabilities for larger Ni; see the geometric distribution in Eq. (5.27).

Figure 5.17 Histograms of intraday transactions data for IBM stock on November 21, 1990: (a) log durations between price changes, (b) direction of price movement, (c) size of price change measured in ticks, and (d) number of trades without price change.

The fitted model for Di is

where standard deviations of the parameters in the mean equation are 0.129, 0.132, and 0.082, respectively, whereas the standard error for the parameter in the variance equation is 0.182. The price reversal is clearly shown by the highly significant negative coefficient of Di−1. The marginally significant parameter in the variance equation is exactly as expected. Finally, the fitted models for the size of a price change are

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 negative estimates of the coefficient of Ni. A large Ni means there were more transactions in the time interval (ti−1, ti) with no price change. This can be taken as evidence of no new information available in the time interval (ti−1, ti). Consequently, the size for the price change at ti should be small. A small λu, i or λd, i for a Poisson distribution gives precisely that.

In summary, granted that a sample of 194 observations in a given day may not contain sufficient information about the trading dynamics of IBM stock, but the fitted models appear to provide some sensible results. McCulloch and Tsay (2000) extend the PCD model to a hierarchical framework to handle all the data of the 63 trading days between November 1, 1990, and January 31, 1991. Many of the parameter estimates become significant in this extended sample, which has more than 19,000 observations. For example, the overall estimate of the coefficient of ln(Δti−1) in the model 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 careful specification of the variables used. It also requires a deep understanding of the way by which the market operates and the data are collected. However, ideas of the econometric models discussed in this chapter are useful and widely applicable in analysis of high-frequency data.