Essays on the applications of distributional scaling in finance : estimation, forecasting and inference

This thesis addresses some of the current gaps in the literature on unifractal and multifractal processes in finance, through a combination of empirical and theoretical contributions spanning the key problems of estimation, forecasting and inference. In Chapter 2 a new method is proposed for producing density forecasts for daily financial returns from high-frequency intraday data, under the assumption that the return process possesses distributional scaling properties consistent with that of a unifractal process. In contrast to previous methods using intraday data to estimate and forecast daily return densities, the approach presented preserves information about both the sign and magnitude of the intraday returns and allows nonparametric specifications to be employed for the distribution of daily returns. The density forecasting performance of the method is shown to be competitive with existing methods based on intraday and daily returns for exchange rate and equity index data, particularly for shorter in-sample periods and during periods of high return volatility. However, as expected the performance of the method is stronger for return series with distributional scaling properties close to the unifractal scaling required by the method and poorer, though still competitive, for time series that exhibit larger deviations from unifractality. In response to the apparent limitations of the method proposed in Chapter 2, Chapter 3 develops an equivalent density forecasting method under the assumption that the return process belongs to the more general class of multifractal processes, thus permitting more flexible scaling behaviour than in Chapter 2. Whilst these distributional scaling laws are more problematic to apply in practice than those of Chapter 2, both the daily return variance and kurtosis can be estimated from the intraday data, providing additional flexibility over existing realised volatility based methods. The predictive ability of this alternative multifractal density forecasting approach is found to be competitive with existing density forecasting methods for both exchange rate and equity index data, but is outperformed by the unifractal approach of Chapter 2 for equity index data. Finally in Chapter 4, a formal testing framework is developed for determining whether a given sample of data is most consistent with a unifractal or multifractal data generating process. The testing methodology begins by proposing a set of possible statistics for testing the null hypothesis of unifractality against the alternative of multifractality, but due to the specific characteristics of the testing environment the distributions of the proposed test statistics are non-standard and the relevant rates of convergence are unknown. It is then shown that these difficulties can be overcome and test statistic distributions obtained using an appropriate model-based bootstrap resampling scheme. A series of Monte Carlo exercises demonstrate that the testing procedure possesses good empirical size and power properties in wide range of situations, being robust against various forms of multifractality under the alternative. Good performance for sample sizes that would be considered as small in the multifractality literature also confirms the suitability of the methodology for the study of both local and global scaling properties. This is demonstrated in an empirical exercise in which the testing methodology is applied to study the local scaling properties of the intraday dataset used in previous chapters.