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Spectral Analysis for Bivariate Time Series with Long Memory

Published online by Cambridge University Press:  11 February 2009

J. Hidalgo
J. Hidalgo
Affiliation:
London School of Economics
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Abstract

This paper provides limit theorems for spectral density matrix estimators and functionals of it for a bivariate covariance stationary process whose spectral density matrix has singularities not only at the origin but possibly at some other frequencies and, thus, applies to time series exhibiting long memory. In particular, we show that the consistency and asymptotic normality of the spectral density matrix estimator at a frequency, say λ, which hold for weakly dependent time series, continue to hold for long memory processes when λ lies outside any arbitrary neighborhood of the singularities. Specifically, we show that for the standard properties of spectral density matrix estimators to hold, only local smoothness of the spectral density matrix is required in a neighborhood of the frequency in which we are interested. Therefore, we are able to relax, among other conditions, the absolute summability of the autocovariance function and of the fourth-order cumulants or summability conditions on mixing coefficients, assumed in much of the literature, which imply that the spectral density matrix is globally smooth and bounded.

Type
Research Article
Information
Econometric Theory , Volume 12 , Issue 5 , December 1996 , pp. 773 - 792
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Andrews, D.W.K. (1991) Hetcroscedastic and autocorrelation consistent covariance matrix estimation. Econometrica 59, 817–858.CrossRefGoogle Scholar
Brillinger, D.R. (1979) Confidence intervals for the crossvariance function. Selecla Statistica Canadian 5, 3–16.Google Scholar
Brillinger, D.R. (1981) Time Series. Data Analysis and Theory. New York: Holden-Day.Google Scholar
Brillinger, D.R. & Hatanaka, M. (1970) A permanent income hypothesis relating to the aggregate demand for money (an application of spectral and moving spectral analysis). Economic Studies Quarterly 21, 44–71.Google Scholar
Dahlhaus, R. (1989) Efficient parameter estimation for self-similar processes. Annals of Statistics 17, 1749–1766.CrossRefGoogle Scholar
Fox, R. & Taqqu, M.S. (1986) Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of Statistics 14, 517–532.CrossRefGoogle Scholar
Fuller, W.A. (1976) Introduction to Statistical Time Series. New York: John Wiley and Sons.Google Scholar
Giraitis, L. & Surgailis, D. (1990) A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle's estimate. Probability Theory and Related Fields 86, 87–104.CrossRefGoogle Scholar
Granger, C.W.J. (1969) Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424–438.CrossRefGoogle Scholar
Granger, C.W.J. (1980) Long memory relationships and the assumptions of dynamic models. Journal of Econometrics 14, 227–238.CrossRefGoogle Scholar
Granger, C.W.J. & Hatanaka, M. (1964) Spectral analysis of economic time series. Princeton, New Jersey: Princeton University Press.Google Scholar
Granger, C.W.J. & Joyeux, R. (1980) An introduction to long-memory time series and fractional differencing. Journal of Time Series Analysis 1, 15–30.CrossRefGoogle Scholar
Granger, C.W.J. & Rees, H.J.B. (1968) The spectral analysis of the term structure of interest rates. Review of Economic Studies 35, 67–76.CrossRefGoogle Scholar
Gray, H.L., Zhang, N.-F., & Woodward, W.A. (1989) On generalized fractional processes. Journal of Time Series Analysis 10, 233–257.CrossRefGoogle Scholar
Hannan, E.J. (1963a) Regression for time series. In Rosenblatt, M. (ed.), Time Series Analysis. New York: Wiley.Google Scholar
Hannan, E.J. (1963b) Regression for time series with errors of measurement. Biometrika 50, 293–302.CrossRefGoogle Scholar
Hannan, E.J. (1970) Multiple Time Series. New York: John Wiley and Sons.CrossRefGoogle Scholar
Hannan, E.J. & Robinson, P.M. (1973) Lagged regression with unknown lags. Journal of the Royal Statistical Society Series B, 252–267.Google Scholar
Hannan, E.J. & Thomson, P.J. (1971) Spectral inference over narrow bands. Journal of Applied Probability 8, 157–169.CrossRefGoogle Scholar
Hosking, J. (1981) Fractional differencing. Biometrika 68, 165–176.CrossRefGoogle Scholar
Hosoya, Y. (1991) The decomposition and measurement of the interdependency between secondorder stationary processes. Probability Theory and Related Fields 88, 429–444.CrossRefGoogle Scholar
Hosoya, Y. (1993) A Limit Theory in Stationary Processes with Long-Range Dependence and Related Statistical Models. Preprint, Tohoku University.Google Scholar
Hosoya, Y. & Taniguchi, M. (1982) A central limit theorem for stationary processes and the parameter estimation of linear processes. Annals of Statistics 10, 132–153.CrossRefGoogle Scholar
Nerlove, M., Grethel, D.M., & Carvalho, J.L. (1979) Analysis of Economic Time Series. San Diego: Academic Press.Google Scholar
Robinson, P.M. (1978) Statistical inference for a random coefficient autoregressive model. Scandinavian Journal of Statistics 5, 163–168.Google Scholar
Robinson, P.M. (1979) Distributed lag approximation to linear time-invariant system. Annals of Statistics 7, 507–515.CrossRefGoogle Scholar
Robinson, P.M. (1986) On the errors-in-variables problem for time series. Journal of Multivariate Analysis 19, 240–250.CrossRefGoogle Scholar
Robinson, P.M. (1994a) Time series with strong dependence. In Sims, C.A. (ed.), Advances in Econometrics: Sixth World Congress, vol. 1, pp. 47–95. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Robinson, P.M. (1994b) Rates of convergence and optimal bandwidth in spectral analysis of processes with long range dependence. Probability Theory and Related Fields 99, 443–473.CrossRefGoogle Scholar
Robinson, P.M. (1995) Log-periodogram regression for time series with long range dependence. Annals of Statistics 23, 1048–1072.CrossRefGoogle Scholar
Sargent, T.J. (1968) Interest rates in the nineteen-fifties. Review of Economics and Statistics 50, 164–172.CrossRefGoogle Scholar
Sims, C.A. (1971) Discrete approximations to continuous time distributed lags in econometrics. Econometrica 39, 545–563.CrossRefGoogle Scholar
Sims, C.A. (1972) The role of approximate prior restrictions in distributed lag estimation. Journal of the American Statistical Association 67, 164–175.CrossRefGoogle Scholar
Sims, C.A. (1974) Distributed lags. In Intrilligator, M.D. & Kendrick, D.A. (eds.), Frontiers of Quantitative Economics, vol. II, Amsterdam: North Holland.Google Scholar
Soulier, P. (1993) Pointwise Estimation of the Spectral Density of a Strongly Dependent Stationary Gaussian Field. Preprint, University of Paris, Orsay Cedex.Google Scholar
Yajima, Y. (1989) A central limit theorem of Fourier transforms of strongly dependent stationary processes. Journal of Time Series Analysis 10, 375–383.CrossRefGoogle Scholar
Yong, C.H.. (1974) Asymptotic Behaviour of Trigonometric Series. Hong Kong: Chinese University of Hong Kong.Google Scholar