Processos de markov ponderado-gama e modelagem de séries temporais inteiras multivariadas via cópulas

Abstract

In this work we propose two models for time series, motivated by financial market applications. First, we propose a Markov process for positive continuous series, driven by a gamma weight density, the weighted-gamma Markov process (PG). The PG process is stationary, time reversible, and is defined from its transition density. We investigated the GIG (Generalized Inverse Gaussian) distribution and the gamma distribution as potential marginal distributions for the process, which returned several explicit results. Parameter estimation of the PG-GIG and PG-Ga processes was performed via the maximum likelihood method. To evaluate the proposed inferential method, we performed a Monte Carlo simulation study. Additionally, we conducted an empirical study, regarding the adjustment of PG-GIG and PG-Ga processes to volatility data from the log-returns of the stocks that make up the FTSE 100 index, of the London Stock Exchange. We implement a \textit{pseudo prediction} exercise and evaluate the performance of the processes through a residuals analysis, by means of simulated envelopes, and verify the model calibration with the construction of PIT (Probability Integral Transform) histograms. The second proposal is a class of models for multivariate whole time series analysis, built by combining the INGARCH (INteger valued Generalized AutoRegressive Conditional Heteroskedastic) methodology with copulas. The proposed process is denoted as the copula-INGARCH (CINGARCH) process. Temporal and contemporaneous dynamics (cross-dependence of the model-fitted series) are incorporated into the process using the INGARCH methodology. As potential conditional distributions, we investigate the discrete Laplace (LD) and Skellam distributions. Parameter estimation of the LD-CINGARCH and Skellam-CINGARCH processes was performed via the two-steps maximum likelihood method. To evaluate the proposed inferential method, we performed a Monte Carlo simulation study. In addition, the proposed methodology was used to jointly model the change in the exchange rate ticks of the Euro to Pound Sterling (EUR/GBP) and the Euro to US Dollar (EUR/USD). We evaluate the goodness of fit of the LD-CINGARCH and Skellam-CINGARCH processes marginally by constructing PIT histograms and simulated envelopes for the Pearson residuals.