Nonparametric estimation of time-varying covariance matrix in a slowly changing vector random walk model

Feng, Yuanhua and Yu, Keming (2006): Nonparametric estimation of time-varying covariance matrix in a slowly changing vector random walk model.

Abstract

A new multivariate random walk model with slowly changing parameters is introduced and investigated in detail. Nonparametric estimation of local covariance matrix is proposed. The asymptotic distributions, including asymptotic biases, variances and covariances of the proposed estimators are obtained. The properties of the estimated value of a weighted sum of individual nonparametric estimators are also studied in detail. The integrated effect of the estimation errors from the estimation for the difference series to the integrated processes is derived. Practical relevance of the model and estimation is illustrated by application to several foreign exchange rates.

• Item Type: MPRA Paper
• Institution: Heriot-Watt University and Brunel University
• Language: English
• Keywords: Multivariate time series; slowly changing vector random walk; local covariance matrix; kernel estimation; asymptotic properties; forecasting
• Subjects: C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C32 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes ; State Space Models; G - Financial Economics > G0 - General > G00 - General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General

References

Ait-Sahalia, Y. (1996) Nonparametric pricing of interest rate derivative securities, Econometrica, 64, 527--560.

Beran, J. and Y. Feng (2002) SEMIFAR models -- a semiparametric approach to modelling trends, long-range dependence and nonstationarity. Comptat. Statist. and Data Anal., 40, 393--419.

Beran, J. and Ocker, D. (1999) SEMIFAR forecasts, with applications to foreign exchange rates. J. Statistical Planning and Inference, 80, 137--153.

Bollerslev, T., R. F. Engle and J. Wooldridge (1988) A capital asset-pricingmodel with time-varying covariances. Journal of Political Economy, 96, 116—131.

Csörgö, S. and J. Mielniczuk (1995) Nonparametric regression under long-range dependent normal errors. Annals of Statistics, 23, 1000--1014.

Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. Annals of Statistics, 25, 1--37.

Dahlhaus, R. (2000) A likelihood approximation for locally stationary processes. Annals of Statistics, 28, 1762--1794.

Efromovich, S. (1999) Nonparametric curve estimation: Methods, Theory, and Applications. New York: Springer.

Engle, R.F. (1982) Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of UK Inflation. Econometrica, 50, 987-–1008

Engle, R. (2002) Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics, 20, 339--350.

Fan, J. and I. Gijbels (1995) Data-driven bandwidth selection in local polynomial fitting: Variable bandwidth and spatial adaptation. J. Roy. Statist. Soc. Ser. B, 57, 371--394.

Fan, J. and Q. Yao (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika, 85, 645--60.

Feng, Y. (2004) Simultaneously modelling conditional heteroskedasticity and scale change. Econometric Theory, 20, 563--596.

Gasser, T., A. Kneip and W. Köhler (1991) A flexible and fast method for automatic smoothing. J. Amer. Statist. Assoc., 86, 643--652.

Gordon, A.H. (1991) Global warming as a manifestation of a random walk. Journal of Climate, 4, 589–-597.

Härdle, W., A.B. Tsybakov and L. Yang (1998) Nonparametric vector autoregression. J. Statist. Plann. Infer., 68, 221--245.

Härdle, W., H. Herwatz and V. Spokoiny (2003) Time inhomogeneous multiple volatility modelling. J. Financial Econometrics, 1, 55--99.

Hart, J. D. (1991) Kernel regression estimation with time series errors. J. Roy. Statist. Soc. Ser. B, 53, 173--187.

Harvey, A. (1989) Forecasting structural time series models and the Kalman filter. Cambridge: Cambridge University Press.

Harvey, A., Ruiz, E. and N. Shephard (1994) Multivariate stochastic variance models. Review of Economic Studies, 61, 247--264.

Herzel, S., C. Starica and R. Tutuncu (2006) A non-stationary multivariate model for financial returns. Forthcoming in Statistics for dependent data, Ed. Patrice Bertail, and P. Doukhan, Springer.

Kärner, O. (2002) On nonstationarity and antipersistency in global temperature series. Journal of Geophysical Research, 107, doi: 10.1029/2001JD002024.

Kijima, M. (2002) Stochastic Processes with Applications to Finance, Cambridge: Chapman and Hall.

Lovàsz, L. (1993) Random Walks on Graphs: A Survey, Mathematical Studies, 2, 1--46.

Neigel, J. E. and J.C. Avise (1993) Application of a random walk model to geographic distributions of animal mitochondrial DNA variation, Genetics, 135, 1209--20.

Ruppert, D. and M.P. Wand (1994) Multivariate locally weighted least squares regression. Annals of Statistics, 22, 1346--1370.

Wand, M.P. and M.C. Jones (1995) Kernel Smoothing, London: Chapman and Hall.

Weiss, G.H. (1994) Aspects and Applications of the Random Walk, Amsterdam: North Holland Press.

Wu, W. and M. Pourahmadi (2003) Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika, 90, 831--844.