## Models For Price Changes

The discreteness and concentration on "no change" make it difficult to model the intraday price changes. Campbell, Lo, and MacKinlay (1997) discuss several econometric models that have been proposed in the literature. Here we mention two models that have the advantage of employing explanatory variables to study the intraday price movements. The first model is the ordered probit model used by Hauseman, Lo, and MacKinlay (1992) to study the price movements in transactions data. The second model has been considered recently by McCulloch and Tsay (2000) and is a simplified version of the model proposed by Rydberg and Shephard (1998); see also Ghysels (2000).

### 5.4.1 Ordered Probit Model

Let y* be the unobservable price change of the asset under study (i.e., y* = P*. _ Pt*_ 1), where P. is the virtual price of the asset at time t. The ordered probit model assumes that y* is a continuous random variable and follows the model where xi is a p-dimensional row vector of explanatory variables available at time ti_i, 3 is a k x 1 parameter vector, E(ei | xi) = 0, Var(ei | xi) = of, and Cov(ei, ) = 0 for i = j. The conditional variance of is assumed to be a positive function of the explanatory variable wi —that is, where g(.) is a positive function. For financial transactions data, wi may contain the time interval ti _ ti_i and some conditional heteroscedastic variables. Typically, one also assumes that the conditional distribution of ei given xi and wi is Gaussian.

Suppose that the observed price change yi may assume k possible values. In theory, k can be infinity, but countable. In practice, k is finite and may involve combin y* = xi 3 + €i,

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