Contributions to time series modeling under first order moment assumptions

For time series modeling in science, engineering, industry, government and military, a key conceptual and methodological tool is the autocovariance function. Heavy tailed distributions and data are important for time series settings in economics, finance, and actuarial science, for example. However, the use of covariance presupposes that the variables have finite variances, a restriction that excludes accommodation of heavy tailed distributions and data. Incorrect modeling based on invalid 2nd order assumptions sometimes has severe negative impacts on inference, prediction, and simulation. It is desirable to allow for heavy tailed situations by developing concepts and methods that impose only 1st order (or lower) moment assumptions on the variables. Hence, they are valid in the heavy tailed setting, conceptually meaningful in a population model and effective in practical applications.;As a solution to this problem, we introduce a "Gini autocovariance function" defined under merely first order moment assumptions. Conceptually, it plays a similar role as the usual Pearson autocovariance function, and thus represents a new fundamental tool in time series modeling.;Estimators for AR models based on the sample Gini autocovariance function are linear, easily interpreted, and have closed form expressions. We study their performance via simulation studies allowing a wide range of typical innovation and outlier scenarios. Comparisons are made with the Least Squares and Robust Least Squares approaches. It is seen that the "Gini" approach competes very well with standard methods and provides a new reliable tool in time series modeling in heavy tailed settings or situations with outliers.;Also, the nonlinear structure is becoming increasingly of interest. We also present contributions toward fitting a nonlinear heavy tailed autoregressive time series model of Pareto type (ARP(1)). We focus on the role of the "Gini autocovariance function" that is well-defined under just first-order moment assumptions. Further, estimators of the ARP(1) parameters are discussed. Finally, a diagnostic using Brock, Dechert, Scheinkman (BDS) test is presented for deciding whether to fit a standard linear autoregressive model or an ARP(1) model to any given time series data set.