Estimation and inference of the vector autoregressive process under heteroscedasticity
Authors:
T. Bodnar and T. Zabolotskyy
Journal:
Theor. Probability and Math. Statist. 83 (2011), 27-45
MSC (2010):
Primary 62H12, 62M15; Secondary 62H10
DOI:
https://doi.org/10.1090/S0094-9000-2012-00839-9
Published electronically:
February 2, 2012
MathSciNet review:
2768846
Full-text PDF Free Access
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Additional Information
Abstract: In this paper we derive the asymptotic distribution of the estimator for the parameters of the vector autoregressive process of order $p$ with an unconditionally heteroscedastic error process. The covariance matrix of the error process is modeled as a deterministic matrix function and it is estimated nonparametrically at each time point. This estimator is used for deriving inference procedures for the parameters of the vector autoregressive process.
References
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References
- D. W. K. Andrews, Laws of large numbers for dependent nonidentically distributed random variables, Econometric Theory 4 (1988), 458–467. MR 985156 (90c:60013)
- T. Breusch and A. Pagan, A simple test of heteroscedasticity and random coefficient variation, Econometrica 47 (1979), 1287–1294. MR 545960 (81b:62081)
- P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, Springer, New York, 1991. MR 1093459 (92d:62001)
- G. Cavaliere, Unit root tests under time-varying variance shifts, Econometric Reviews 23 (2004), 259–292. MR 2089641 (2005d:62138)
- S. Darolles, C. Gourieroux, and J. Jasiak, Structural Laplace transform and compound autoregressive models, Journal of Time Series Analysis 27 (2006), 477–503. MR 2245710
- J. Davidson, Stochastic Limit Theory: An Introduction for Econometricians, Oxford, Oxford University Press, 1994. MR 1430804 (97k:60002)
- H. Drees and C. Stărică, A Simple non-Stationary Model for Stock Returns, Working paper, Chalmers University of Technology, 2002.
- R. F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50(4) (1982), 987–1007. MR 666121 (83j:62158)
- R. F. Engle and J. G. Rangel, The Spline GARCH Model for Unconditional Volatility and its Global Macroeconomic Causes, Working paper, New York University and University of California, San Diego, 2004.
- E. F. Fama, Stock return, real activity, inflation and money, American Economic Review 71 (1981), 545–565.
- L. G. Godfrey, Testing against general autoregressive and moving average error models when the regressors include lagged dependent variables, Econometrica, Econometric Society 46(6) (1978), 1293–1301.
- C. Gourieroux, J. Jasiak, and R. Sufana, The Wishart autoregressive process of multivariate stochastic volatility, Journal of Econometrics 150 (2009), 167–181. MR 2535514 (2011a:62289)
- W. H. Greene, Econometric Analysis, Pearson/Prentice Hall, New Jersey, 2008.
- B. E. Hansen, Autoregressive conditional density estimation, International Economic Review 35(3) (1994), 705–730.
- D. A. Harville, Matrix Algebra: Exercises and Solutions, Springer, Berlin, 1997. MR 1874239
- D. A. Hsu, R. Miller, and D. Wichern, On the stable Paretian behavior of stock-market prices, Journal of American Statistical Association 69 (1974), 108–113.
- C. S. Kwon and T. S. Shin, Cointegration and causality between macroeconomic variables and stock market returns, Global Finance Journal 10(1) (1999), 71–81.
- R. Merton, On estimating the expected return on the market: an exploratory investigation, Journal of Financial Economics 8 (1980), 323–361.
- P. C. B. Phillips and K.-L. Xu, Inference in autoregression under heteroscedasticity, Journal of Time Series Analysis 27(2) (2006), 289–308. MR 2235847 (2007g:62092)
- J. Polzehl and V. Spokoiny, Varying Coefficient GARCH Versus Local Constant Volatility Modeling: Comparison of Predictive Power, Working paper, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany, 2006.
- M. Rockinger and E. Jondeau, Entropy densities with an application to autoregressive conditional skewness and kurtosis, Journal of Econometrics 106 (2002), 119–142. MR 1875530 (2003a:62024)
- C. Stărică, Is GARCH (1,1) as Good a Model as the Nobel Prize Accolades would Imply?, Working paper, Chalmers University of Technology, 2003.
- H. White, A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica 48(4) (1980), 817–838. MR 575027 (81k:62097)
- H. White, Nonlinear regression with dependent observations, Econometrica 52(1) (1984), 143–161. MR 729213 (86b:62172)
- W. H. Wong, On the consistency of cross validation in kernel nonparametric regression, Annals of Statistics 11 (1983), 1136–1141. MR 720259 (85k:62094)
- K.-L. Xu and P. C. B. Phillips, Adaptive estimation of autoregressive models with time-varying variances, Journal of Econometrics 142 (2008), 265–280. MR 2408736
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Additional Information
T. Bodnar
Affiliation:
Department of Statistics, European University Viadrina, PO Box 1786, 15207 Frankfurt (Oder), Germany
Email:
bodnar@euv-frankfurt-o.de
T. Zabolotskyy
Affiliation:
Department of Statistics, European University Viadrina, PO Box 1786, 15207 Frankfurt (Oder), Germany
Email:
zabolotskyy@euv-frankfurt-o.de
Keywords:
Heteroscedasticity,
inference procedure,
parameter estimation,
vector autoregressive process
Received by editor(s):
October 5, 2009
Published electronically:
February 2, 2012
Article copyright:
© Copyright 2012
American Mathematical Society