Extension of permutation entropy and ordinal pattern analysis with application to financial time series analysis

This thesis explores and extends the use of Permutation Entropy (PE), a complexity measure that has been extensively used in the biomedical and physical fields but less so in the area of financial time series. I make both methodological and empirical contributions to the literature, specifically in relation to financial time series analysis. First, I establish the connection of PE with the mainstream financial time series models by demonstrating the relation of PE with the Autocorrelation Function (ACF) in the Gaussian process and the parametrizations of the Autoregressive-moving-average (ARMA) and the Generalized Autoregressive Conditional Heteroskedastic (GARCH) models. Additionally, I examine the way that PE responds to a number of commonly observed features of financial data, such as high-kurtosis and non-stationarity, in order to provide appropriate interpretation of this measure when it is used in empirical applications. Second, I develop a new temporal dependence measure, Permutation Dependence (PD) by adjusting the specification of PE. PD is proposed as a remedy to the major drawbacks of PE, such as its insensitivity to slowly diminishing temporal structures and its non-monotonic corresponding to the strength of temporal dependence at the chosen delay/lag. I show that the PD measure indicates the temporal dependence of the observed time series at the selected delay/lag and it resembles the ACF in linear processes but is not grounded in detecting linear structures. Just like the ACF, by plotting PD against increasing delays, it can be used to reflect the evolution of temporal dependence relations as the lag between entries increases. Additionally, with the help of the new PD measure, I develop a visualization plot for the time series investigated to reveal the deterministic structures captured by different models. The visualization plot provides a universal framework to facilitate comparison between different deterministic relations postulated by both parametric and non-parametric models. Therefore it helps to explain the superior and suboptimal prediction performance of various models when applied to data with different properties. My third methodological contribution is proposing a model sufficiency test using the ordinal pattern concept in PE to study a given model's point prediction accuracy. Compared to some classical model sufficiency tests, such as the Broock et al. (1996) test, our proposal does not require a sufficient model to eliminate all structures exhibited in the estimated residuals. When the additive innovations in the investigated data's underlying dynamics show a certain structure, such as higher-moment serial dependence, the Broock et al. (1996) test can lead to erroneous conclusions about the sufficiency of point predictors. Due to the structured innovations, inconsistency between the model sufficiency tests and prediction accuracy criteria can occur. Our proposal overcomes this lack of cohesion between model and prediction evaluation approaches and remains valid when the underlying process has non-white additive innovation. This thesis also provides critical empirical contributions. By making use of PE, PD and the new model sufficiency test on 10-minute EUR/USD exchange rate return and volatility series, I successfully apply the ordinal based analysis to real-world financial data. In particular, I investigate the existence and nature of the temporal dependence underlying the observed return and volatility dynamics. Additionally, by applying the newly proposed sufficiency test on high-frequency exchange rate return and volatility series, I study the sufficiency of a number of commonly used models in point forecasting the real-world financial series. The model sufficiency evaluation studies also lead me to uncover the main reasons behind each prediction model's sufficient/insufficient performance. My empirical results have profound implications for future studies as they identify the main obstacles to accurate prediction and replication of the dynamics of financial time series, thereby pointing out how to construct more capable and better performing models. I also present a brief analysis using the ordinal pattern based tools on monthly sea surface temperature data to demonstrate the potential applications of the PE and PD measures in a wider range of disciplines.