Copula models of serial dependence with applications to economic and financial time series forecasting

Abstract

Almost all existing copula models use copula functions to capture cross-sectional dependence. Instead, this thesis explores using copula functions to capture serial dependence. The advantage of doing this, is that the parametric assumptions on the margins, employed to capture serial dependence, are no longer needed, making the margins fully flexible.

This thesis consists of three related papers. The first paper in Chapter 4 studies the properties of copulas of heteroskedastic processes, by retrieving the implied copula of the ARCH and SV models via simulation. The results show that the implied copula densities have an unusual cross shape, with mass concentratedat all four corners of the unit square. A major contribution of this paper is that it demonstrates that these copulas can be accurately approximated by a mixture of bivariate copulas. This mixture copula, when combined with a flexible marginal distribution, can be employed to model a wide range of heteroskedastic time series. In addition, because existing measures of dependence prove inadequate when heteroskdasticity is present, new measures of dependence are proposed. The model is also extended to multiple time series settings, and applied to foreign exchange rate return data.

The second paper in Chapter 5 focuses on extending the methods from Chapter 4 to discrete time series. This work contains two significant contributions. First, it employs a copula to capture cross-sectional and serial dependence for discrete and mixed time series data. This copula is initially employed for the single variable case, and then extended to the multivariate case with discrete and mixed data. Second, this paper proposes a new Variational Bayes estimation technique, applicable to all time series copula models with discrete or mixed data. This technique allows for faster estimation of the copula parameters compared to previous methods.

The last paper in Chapter 6 proposes a new copula created by the inversion of a multivariate unobserved component stochastic volatility model, and shows how to estimate it using Bayesian methods. This non-closed form copula is an alternative to the mixture copulas proposed in Chapter 4. An advantage of this inversion copula is that it is fast and simple to simulate from, and measures of dependence, filtered distributions, predictive distributions and other inference can all be computed readily and accurately via simulation. To illustrate the potential of this new copula, it is applied to a real-time U.S macroeconomic dataset. Out-sample predictions of GDP growth, inflation and an interest rate demonstrate that the copula model can improve upon predictions of the widely used VARSV model.