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A stationary time series model is directional if it has properties that are not symmetric with respect to time, for example if, after the expected value is adjusted to 0, the expected value of the product of the variable squared and the following variable differs from the expected value of the product of the variable and the following variable squared. Marked directionality in time series is apparent from a comparison of time series plots in chronological order and in reverse order (time-to-go), but formal quantitative tests are needed to determine the strength of evidence against an assumption of reversibility (no directionality). Linear time series models with Gaussian noise (LGN models) are reversible whereas linear models with other noise distributions and non-linear models are directional. Evidence of directionality in time series can be found in many disciplines including financial time series. Examples of directional financial time series are presented, together with values of directionality statistics. A method of calculating the p-values of the directional statistics, corresponding to a null hypothesis of reversibility, using simulation is presented. The time series are then modelled by threshold autoregressive models, with noise distributions, which can be non-Gaussian, fitted to match the empirical distribution of the residuals. The fitting criterion is a weighted sum of the error sum of squares and a measure of discrepancy between values of the directionality statistics in the time series and long simulations from the potential model. Strategies for determining suitable weights for the criterion will be investigated. The directional time series models will be assessed in terms of the distribution of extreme values of the time series, which is a feature that was not used in the fitting process. The assessment will be made by comparing the distribution of extremes from: the observed time series; a long simulation from the fittted directional model; and a long simulation from a LGN model fitted to the time series on the basis of the Akaike information criterion (AIC).

Modelling and simulation of directional financial time series