The memory of stochastic volatility models

A valid asymptotic expansion for the covariance of functions of multivariate normal vectors is applied to approximate autocovariances of time series generated by nonlinear transformation of Gaussian latent variates, and nonlinear functions of these, with special reference to long memory stochastic volatility models, serving to identify the roles played by the underlying Gaussian processes and the nonlinear transformation. Implications for simple stochastic volatility models are examined in detail, with numerical and Monte Carlo calculations, and applications to cyclic behaviour, cross-sectional and temporal aggregation, and multivariate models are discussed.

[1]  P. Phillips.  BOOTSTRAPPING I(1) DATA BY PETER C. B. PHILLIPS COWLES FOUNDATION PAPER NO. 1310 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS , 2010 .

[2]  Jonathan H. Wright,et al.  Semiparametric estimation of long-memory volatility dependencies: The role of high-frequency data , 2000 .

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

[4]  T. Sun,et al.  A central limit theorem for non-instantaneous filters of a stationary Gaussian Process , 1987 .

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

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

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

[8]  C. Granger,et al.  Varieties of long memory models , 1996 .

[9]  M. Rosenblatt.  Independence and Dependence , 1961 .

[10]  R. K. Adenstedt,et al.  On Large-Sample Estimation for the Mean of a Stationary Random Sequence , 1974 .

[11]  P. Robinson.  Log-Periodogram Regression of Time Series with Long Range Dependence , 1995 .

[12]  M. G. Kendall,et al.  iv) Proof of Relations connected with the Tetrachoric Series and its Generalization , 1941 .

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

[14]  Peter M. Robinson,et al.  Modelling Nonlinearity and Long Memory in Time Series - (Now published in 'Nonlinear Dynamics and Time Series', C D Cutler and D T Kaplan (eds), Fields Institute Communications, 11 (1997), pp.61-170.) , 1997 .

[15]  M. S. Bartlett,et al.  Non-linear transformations of stochastic processes , 1966, The Mathematical Gazette.

[16]  H. L. Gray,et al.  ON GENERALIZED FRACTIONAL PROCESSES , 1989 .

[17]  C. Granger,et al.  A long memory property of stock market returns and a new model , 1993 .

[18]  N. Bingham.  INDEPENDENT AND STATIONARY SEQUENCES OF RANDOM VARIABLES , 1973 .

[19]  R. Davies,et al.  Tests for Hurst effect , 1987 .

[20]  C. Granger,et al.  Some Properties of Absolute Return, An Alternative Measure of Risk , 1995 .

[21]  C. Granger,et al.  Modeling volatility persistence of speculative returns: A new approach , 1996 .

[22]  F. Breidt,et al.  The detection and estimation of long memory in stochastic volatility , 1998 .

[23]  M. Magdalinos,et al.  A reinterpretation of the tests of overidentifying restrictions , 1996 .

[24]  M. Taqqu.  Weak convergence to fractional brownian motion and to the rosenblatt process , 1975, Advances in Applied Probability.

[25]  E. J. Hannan,et al.  Multiple time series , 1970 .

[26]  P. Robinson,et al.  Advances in Econometrics: Time series with strong dependence , 1994 .

[27]  Paolo Zaffaroni,et al.  Nonlinear time series with long memory: a model for stochastic volatility , 1998 .

[28]  P. Robinson,et al.  Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression , 1991 .

[29]  J. Geweke,et al.  THE ESTIMATION AND APPLICATION OF LONG MEMORY TIME SERIES MODELS , 1983 .