Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T15:23:12.377Z Has data issue: false hasContentIssue false
  • English
  • Français

A LIMIT THEOREM FOR QUADRATIC FORMS AND ITS APPLICATIONS

Published online by Cambridge University Press:  14 May 2007

Wei Biao Wu  and
Xiaofeng Shao
Wei Biao Wu
Affiliation:
University of Chicago
Xiaofeng Shao
Affiliation:
University of Illinois at Urbana-Champaign
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider quadratic forms of martingale differences and establish a central limit theorem under mild and easily verifiable conditions. By approximating Fourier transforms of stationary processes by martingales, our central limit theorem is applied to the smoothed periodogram estimate of spectral density functions. Our results go beyond earlier ones by allowing a variety of nonlinear time series and by avoiding strong mixing and/or summability conditions on joint cumulants.We thank the two reviewers for their detailed comments, which led to substantial improvements. The work is supported in part by NSF grant DMS-0478704.

Type
Research Article
Information
Econometric Theory , Volume 23 , Issue 5 , October 2007 , pp. 930 - 951
Copyright
© 2007 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Anderson, T.W. (1971) The Statistical Analysis of Time Series. Wiley.
Andrews, D.W.K. (1984) Non-strong mixing autoregressive processes. Journal of Applied Probability 21, 930–934.Google Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–854.Google Scholar
Basalykas, A. (1994) Functional limit theorems for random multilinear forms. Stochastic Processes and Their Applications 53, 175–191.Google Scholar
Bentkus, R.Y. & R.A. Rudzkis (1982) On the distribution of some statistical estimates of a spectral density. Theory of Probability and Its Applications 27, 795–814.Google Scholar
Brillinger, D.R. (1969) Asymptotic properties of spectral estimates of second order. Biometrika 56, 375–390.Google Scholar
Brillinger, D.R. (1975) Time Series: Data Analysis and Theory. Holden-Day.
Brockwell, P.J. & R.A. Davis (1991) Time Series: Theory and Methods, 2nd ed. Springer-Verlag.
Chow, Y.S. & H. Teicher (1988) Probability Theory, 2nd ed. Springer-Verlag.
Dahlhaus, R. (1985) On a spectral density estimate obtained by averaging periodograms. Journal of Applied Probability 22, 592–610.Google Scholar
de Jong, R.M. & J. Davidson (2000) Consistency of kernel estimators of heteroskedastic and autocorrelated covariance matrices. Econometrica 68, 407–424.Google Scholar
de Wet, T. & J.H. Venter (1973) Asymptotic distributions for quadratic forms with applications to tests of fit. Annals of Statistics 1, 380–387.Google Scholar
Diaconis, P. & D. Freedman (1999) Iterated random functions. SIAM Review 41, 41–76.Google Scholar
Götze, F. & A.N. Tikhomirov (1999) Asymptotic distribution of quadratic forms. Annals of Probability 27, 1072–1098.Google Scholar
Hannan, E.J. (1970) Multiple Time Series. Wiley.
Hansen, B.E. (1992) Consistent covariance matrix estimation for dependent heterogenous processes. Econometrica 60, 967–972.Google Scholar
Henry, M. (2001) Average periodogram spectral estimation with long-memory conditional heteroscedasticity. Journal of Time Series Analysis 22, 431–459.Google Scholar
Hsing, T. & W.B. Wu (2004) On weighted U-statistics for stationary processes. Annals of Probability 32, 1600–1631.Google Scholar
Jansson, M. (2002) Consistent covariance matrix estimation for linear processes. Econometric Theory 18, 1449–1459.Google Scholar
Kallianpur, G. (1981) Some ramifications of Wiener's ideas on nonlinear prediction. In N. Wiener & P. Masani (eds.), Norbert Wiener, Collected Works with Commentaries, pp. 402–424. MIT Press.
Kiefer, N.M. & T.J. Vogelsang (2002) Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Econometric Theory 18, 1350–1366.Google Scholar
Lomnicki, Z.A. & S.C. Zaremba (1959) On some moments and distributions occurring in the theory of linear stochastic processes, part II. Monatshefte für Mathematik 63, 128–168.Google Scholar
Mikosch, T. (1991) Functional limit theorems for random quadratic forms. Stochastic Processes and Their Applications 37, 81–98.Google Scholar
Min, W. (2004) Inference on time series driven by dependent innovations. Ph.D. Dissertation, University of Chicago.
Phillips, P.C.B., Y. Sun, & S. Jin (2006a) Spectral density estimation and robust hypothesis testing using steep origin kernels without truncation. International Economic Review 47, 837–894.Google Scholar
Phillips, P.C.B., Y. Sun, & S. Jin (2006b) Long run variance estimation and robust regression testing using sharp origin kernels with no truncation. Journal of Statistical Planning and Inference 137, 985–1023.Google Scholar
Priestley, M.B. (1981) Spectral Analysis and Time Series, vol. 1. Academic Press.
Priestley, M.B. (1988) Nonlinear and Nonstationary Time Series Analysis. Academic Press.
Robinson, P.M. (1983) Review of various approaches to power spectrum estimation. In D.R. Brillinger & R.R. Krishnaiah (eds.), Time Series in the Frequency Domain, Handbook of Statistics 3, pp. 343–368. North-Holland.
Rosenblatt, M. (1971) Markov Processes. Structure and Asymptotic Behavior. Springer-Verlag.
Rosenblatt, M. (1984) Asymptotic normality, strong mixing and spectral density estimates. Annals of Probability 12, 1167–1180.Google Scholar
Shao, X. & W.B. Wu (2007a) Asymptotic spectral theory for nonlinear time series. Annals of Statistics, forthcoming.Google Scholar
Shao, X & W.B. Wu (2007b) Local Whittle estimation of fractional integration for nonlinear processes. Econometric Theory 23 (this issue).Google Scholar
ten Vregelaar, J.M. (1990) Note on the convergence to normality of quadratic forms in independent variables. Theory of Probability and Its Applications 35, 177–179.Google Scholar
Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford University Press.
Varberg, D.E. (1966) Convergence of quadratic forms in independent random variables. Annals of Mathematical Statistics 37, 567–576.Google Scholar
Whittle, P. (1964) On the convergence to normality of quadratic forms in independent variables. Theory of Probability and Its Applications 9, 103–108.Google Scholar
Wiener, N. (1958) Nonlinear Problems in Random Theory. MIT Press.
Wu, W.B. (2005a) Fourier transforms of stationary processes. Proceedings of the American Mathematical Society 133, 285–293.Google Scholar
Wu, W.B. (2005b) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences 102, 14150–14154.Google Scholar
Wu, W.B. (2007) Strong invariance principles for dependent random variables. Annals of Probability, forthcoming.Google Scholar
Wu, W.B. & J. Mielniczuk (2002) Kernel density estimation for linear processes. Annals of Statistics 30, 1441–1459.Google Scholar
Wu, W.B. & W. Min (2005) On linear processes with dependent innovations. Stochastic Processes and Their Applications 115, 939–958.Google Scholar
Wu, W.B. & X. Shao (2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425–436.Google Scholar
Wu, W.B. & M. Woodroofe (2000) A central limit theorem for iterated random functions. Journal of Applied Probability 37, 748–755.Google Scholar
Wu, W.B. & M. Woodroofe (2004) Martingale approximations for sums of stationary processes. Annals of Probability 32, 1674–1690.Google Scholar