Analysis of Financial Time Series, Third Edition

5.6 Nonlinear Duration Models

Nonlinear features are also commonly found in high-frequency data. As an illustration, we apply some nonlinearity tests discussed in Chapter 4 to the normalized innovations of the WACD(1,1) model for the IBM transaction durations in Example 5.4; see Eq. (5.43). Based on an AR(4) model, the test results are given in part (a) of Table 5.8. As expected from the model diagnostics of Example 5.4, the Ori-F test indicates no quadratic nonlinearity in the normalized innovations. However, the TAR-F test statistics suggest strong nonlinearity.

Table 5.8 Nonlinearity Tests for IBM Transaction Durations from November 1 to November 7, 1990a

aOnly intraday durations are used. The number in parentheses of TAR-F tests denotes time delay.

Based on the test results in Table 5.8, we entertain a threshold duration model with two regimes for the IBM intraday durations. The threshold variable is xt−1 (i.e., lag-1 adjusted duration). The estimated threshold value is 3.79. The fitted threshold WACD(1,1) model is xi = ψiϵi, where

5.45

where w(α) denotes a standardized Weibull distribution with parameter α. The number of observations in the two regimes are 2503 and 1030, respectively. ...