A Non-Stationary Paradigm for the Dynamics of Multivariate Financial Returns

Abstract

A simple non-stationary paradigm for the dynamics of multivariate returns is discussed.Unlike most of the multivariate econometric models for financial returns, our approach supposes the volatility to be exogenous and non-stationary. The vectors of returns are assumed to be animated by a slowly changing unconditional covariance structure. The methodological frame is that of non-parametric regression with fixed, equidistant design points. The regression function is the time evolving unconditional covariance. Special attention is payed to the accurate description of the extremal dependence of the vector of returns. The non-stationary paradigm is first applied to describe the changing dynamics of a multivariate data set of returns on three financial risk factors: a foreign exchange rate, an index and an interest rate. Then, its one-day-ahead multivariate distributional forecast performance is evaluated. We show through an out-of sample simulation experiment that our methodology is superior to the plain-vanilla specification of the industry standard RiskMetrics in forecasting the distribution of returns on portfolios of the three risk factors over horizons of one day, ten days and twenty days.JEL classification: C14, C16, C32.

Shortcomings of a Parametric VaR Approach and Nonparametric Improvements Based on a Non-Stationary Return Series Model

“…, n) according to the modelled exposure type. 43 We take up the conceptual framework of Herzel et al (2005) and Drees and Starica (2002) (univariate case) for analysing the return dynamics via classical nonparametric regression with fixed equidistant design points. The vectors of financial returns are assumed to have a time-varying unconditional covariance matrix that evolves smoothly through time.…”

“…Regarding the included return information we have to distinguish later between the two-sided (symmetrical) and the one-sided (historical) estimation. Herzel et al (2005) motivate the application of nonparametric regression by theoretical results of Müller and Stadtmüller (1987) in an asymptotic context (compare section 3.2 in Herzel et al (2005)). That way, they additionally derive propositions on confidence intervals for (Σ i,j (t)) i,j .…”

“…In this article, we will support the non-stationary paradigm of Herzel et al (2005) and Drees and Starica (2002) for modelling financial time series. The aims and scope of our research article are the following: From the starting point of the aforementioned, opposite basic risk model, which is in detail a widespread parametric VaR approach, we will motivate the necessity of an advanced non-stationary model.…”

“…Drees and Starica (2002) show on a twelve-year S&P 500 example that their non-stationary model fits the data adequately and gets better short-term forecasts on the return distribution than conventional GARCH models. Herzel et al (2005) extend these ideas to a multivariate, non-stationary framework. Vectors of financial returns are assumed to have a time-varying unconditional covariance matrix that evolves smoothly, captured by classical nonparametric regression.…”

“…While a symmetric volatility estimatorσ(t) is employed to describe heteroscedasticity in the historical sample, a one-sided, historical versionσ (1) (t), that includes only past and present data, is employed for forecasting exercises. The statistical properties of nonparametric regression are executed for the symmetric case in Herzel et al (2005) and Mikosch and Starica (2004a) referring on the results of Müller and Stadtmüller (1987). Gürtler et al (2009) provide self-contained full proofs for both the two-sided and one-sided kernel estimators, we outline the consistency and asymptotic normality results here.…”