Annastiina Silvennoinen

Multivariate GARCH models

This article contains a review of multivariate GARCH models. Most common GARCH models are presented and their properties considered. This also includes nonparametric and semiparametric models. Existing specification and misspecification tests are discussed. Finally, there is an empirical example in which several multivariate GARCH models are fitted to the same data set and the results compared.

Modelling Multivariate Autoregressive Conditional Heteroskedasticity with the Double Smooth Transition Conditional Correlation GARCH Model

In this paper, we propose a multivariate GARCH model with a time-varying conditional correlation structure. The new double smooth transition conditional correlation (DSTCC) GARCH model extends the smooth transition conditional correlation (STCC) GARCH model of Silvennoinen and Terasvirta (2005) by including another variable according to which the correlations change smoothly between states of constant correlations. A Lagrange multiplier test is derived to test the constancy of correlations against the DSTCC-GARCH model, and another one to test for another transition in the STCC-GARCH framework. In addition, other specification tests, with the aim of aiding the model building procedure, are considered. Analytical expressions for the test statistics and the required derivatives are provided. Applying the model to the stock and bond futures data, we discover that the correlation pattern between them has dramatically changed around the turn of the century. The model is also applied to a selection of world stock indices, and we find evidence for an increasing degree of integration in the capital markets.

Modeling Conditional Correlations of Asset Returns: A Smooth Transition Approach

In this paper we propose a new multivariate GARCH model with time-varying conditional correlation structure. The time-varying conditional correlations change smoothly between two extreme states of constant correlations according to a predetermined or exogenous transition variable. An LM–test is derived to test the constancy of correlations and LM- and Wald tests to test the hypothesis of partially constant correlations. Analytical expressions for the test statistics and the required derivatives are provided to make computations feasible. An empirical example based on daily return series of five frequently traded stocks in the S&P 500 stock index completes the paper.

Parameterizing Unconditional Skewness in Models for Financial Time Series

In this paper we consider the third-moment structure of a class of nonlinear time series models. Empirically it is often found that the marginal distribution of financial time series is skewed. Therefore it is of importance to know what properties a model should possess if it is to accommodate for unconditional skewness. We consider modelling the unconditional mean and variance using models which respond nonlinearly or asymmetrically to shocks. We investigate the implications these models have on the third moment structure of the marginal distribution and different conditions under which the unconditional distribution exhibits skewness as well as nonzero third-order autocovariance structure. With this respect, the asymmetric or nonlinear specification of the conditional mean is found to be of greater importance than the properties of the conditional variance. Several examples are discussed and, whenever possible, explicit analytical expressions are provided for all third order moments and cross-moments. Finally, we introduce a new tool, shock impact curve, that can be used to investigate the impact of shocks on the conditional mean squared error of the return.

Testing constancy of unconditional variance in volatility models by misspecification and specification tests

Abstract The topic of this paper is testing the hypothesis of constant unconditional variance in GARCH models against the alternative that the unconditional variance changes deterministically over time. Tests of this hypothesis have previously been performed as misspecification tests after fitting a GARCH model to the original series. It is found by simulation that the positive size distortion present in these tests is a function of the kurtosis of the GARCH process. Adjusting the size by numerical methods is considered. The possibility of testing the constancy of the unconditional variance before fitting a GARCH model to the data is discussed. The power of the ensuing test is vastly superior to that of the misspecification test and the size distortion minimal. The test has reasonable power already in very short time series. It would thus serve as a test of constant variance in conditional mean models. An application to exchange rate returns is included.