Modelling conditional and unconditional heteroskedasticity with smoothly time-varying structure

Material from this paper has been presented at the International Symposium on Econometric Theory and Applications, Xiamen, April 2006; 5th Annual International Conference Forecasting Financial Markets and Economic Decision-making, Lodz, May 2006; 13th International Conference on Forecasting Financial Markets, Marseille, May-June 2006; 26th International Symposium on Forecasting, Santander, June 2006; Workshop Volatility day, Stockholm, November 2006; Nordic Econometric Meeting, Tartu, May 2007; Symposium on "Long Memory", Aarhus, June-July 2007; LACEA-LAMES, Bogotá, October 2007; and at the seminars at Banca dItalia, Rome, European University Institute, Florence, Humboldt University, Berlin, University of Minho, Braga, Stockholm School of Economics, Leonard N. Stern School of Business at New York University, and University of Vilnius. We would like to thank the articipants at these occasions for their comments, and Stefan Lundbergh, Mika Meitz, Anders Rahbek and Esther Ruiz for useful discussions and suggestions. The responsibility for any errors and hortcomings in this article remains ours.In this paper, we propose two parametric alternatives to the standard GARCH model. They allow the conditional variance to have a smooth time-varying structure of either additive or multiplicative type. The suggested parameterizations describe both nonlinearity and structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. A modelling strategy for these new time-varying parameter GARCH models is developed. It relies on a sequence of Lagrange multiplier tests, and the adequacy of the estimated models is investigated by Lagrange multiplier type misspecification tests. Finite-sample properties of these procedures and tests are examined by simulation. An empirical application to daily stock returns and another one to daily exchange rate returns illustrate the functioning and properties of our modelling strategy in practice. The results show that the long memory type behaviour of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance.This research has been supported by the Danish National Research Foundation. NIPE – Núcleo de Investigação em Políticas Económicas – is supported by the Portuguese Foundation for Science and Technology through the Programa Operacional Ciência e Inovação 2010 (POCI 2010) of the III Quadro Comunitário de Apoio (QCA III), which is financed by FEDER and Portuguese funds