Quadratic Fractionally Integrated Moving Average Processes with Long-Range Dependence

Resumen

Stochastic processes with the long-range dependency (LRD) property are fundamental to modeling data that exhibit slow power decay of the covariance function. Such  behavior often appears in the analysis of financial data, telecommunications, and various natural phenomena. Thus, introducing new stochastic models and statistical methods that take the LRD into account is of great interest. Based on previous work, we introduce a  new stochastic process called quadratic fractionally integrated moving average, that arises from the Quadratic Ornstein-Uhlenbeck Type (QOUT) process, proposed in the literature. We consider Lévy noises of finite second-order moments and use a construction based on a moving average stochastic process whose kernel is that of a QOUT process. Then, using Riemann-Liouville fractional integrals, we propose a fractionally integrated moving average process, for which we highlight some results, including the LRD. We also propose the estimation of the parameters for this process for the case of fractional Brownian  noise, showing its efficiency through a Monte Carlo simulation. By an application based on Brazil’s stock market prices, we illustrate how this process can be used in practice with the Sao Paulo’s Stock Exchange Index data set, also known as the BOVESPA Index.