Multivariate stochastic variance models

Changes in variance, or volatility, over time can be modelled using the approach based on autoregressive conditional heteroscedasticity (ARCH). However, the generalizations to multivariate series can be difficult to estimate and interpret. Another approach is to model variance as an unobserved stochastic process. Although it is not easy to obtain the exact likelihood function for such stochastic variance models, they tie in closely with developments in finance theory and have certain statistical attractions. This article sets up a multivariate model, discusses its statistical treatment and shows how it can be modified to capture common movements in volatility in a very natural way. The model is then fitted to daily observations on exchange rates.

[1]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[2]  W. R. Buckland,et al.  Distributions in Statistics: Continuous Multivariate Distributions , 1973 .

[3]  W. Dunsmuir.  A Central Limit Theorem for Parameter Estimation in Stationary Vector Time Series and its Application to Models for a Signal Observed with Noise , 1979 .

[4]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[5]  F. Diebold,et al.  The dynamics of exchange rate volatility: a multivariate latent factor ARCH model , 1986 .

[6]  T. Bollerslev,et al.  Generalized autoregressive conditional heteroskedasticity , 1986 .

[7]  H. Iemoto.  Modelling the persistence of conditional variances , 1986 .

[8]  Louis O. Scott.  Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application , 1987, Journal of Financial and Quantitative Analysis.

[9]  Alan G. White,et al.  Hedging the Risks from Writing Foreign Currency Options , 1987 .

[10]  C. Granger,et al.  Co-integration and error correction: representation, estimation and testing , 1987 .

[11]  T. Bollerslev,et al.  A CONDITIONALLY HETEROSKEDASTIC TIME SERIES MODEL FOR SPECULATIVE PRICES AND RATES OF RETURN , 1987 .

[12]  Stephen L Taylor,et al.  Modelling Financial Time Series , 1987 .

[13]  S. Johansen.  STATISTICAL ANALYSIS OF COINTEGRATION VECTORS , 1988 .

[14]  J. Wooldridge,et al.  A Capital Asset Pricing Model with Time-Varying Covariances , 1988, Journal of Political Economy.

[15]  M. Rothschild,et al.  Asset Pricing with a Factor Arch Covariance Structure: Empirical Estimates for Treasury Bills , 1988 .

[16]  A. Harvey,et al.  Continuous time autoregressive models with common stochastic trends , 1988 .

[17]  G. Schwert,et al.  Tests for Unit Roots: a Monte Carlo Investigation , 1988 .

[18]  G. Schwert,et al.  Heteroskedasticity in Stock Returns , 1989 .

[19]  M. Chesney,et al.  Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model , 1989, Journal of Financial and Quantitative Analysis.

[20]  T. Bollerslev,et al.  Modelling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model , 1990 .

[21]  S. Turnbull,et al.  Pricing foreign currency options with stochastic volatility , 1990 .

[22]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[23]  Daniel B. Nelson.  Stationarity and Persistence in the GARCH(1,1) Model , 1990, Econometric Theory.

[24]  Tim Bollerslev,et al.  A multivariate generalized ARCH approach to modeling risk premia in forward foreign exchange rate markets , 1990 .

[25]  S. Pantula.  Asymptotic Distributions of Unit-Root Tests When the Process Is Nearly Stationary , 1991 .

[26]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter. , 1991 .

[27]  Esther Ruiz Ortega.  Heteroscedasticity in financial time series. , 1992 .

[28]  Daniel B. Nelson,et al.  Inequality Constraints in the Univariate GARCH Model , 1992 .

[29]  R. Chou,et al.  ARCH modeling in finance: A review of the theory and empirical evidence , 1992 .

[30]  Tim Bollerslev,et al.  COMMON PERSISTENCE IN CONDITIONAL VARIANCES , 1993 .

[31]  E. Ruiz.  Quasi-maximum likelihood estimation of stochastic volatility models , 1994 .