Tommaso, Proietti and Alessandra, Luati (2012): The Generalised Autocovariance Function.

PDF
MPRA_paper_43711.pdf Download (813kB)  Preview 
Abstract
The generalised autocovariance function is defined for a stationary stochastic process as the inverse Fourier transform of the power transformation of the spectral density function. Depending on the value of the transformation parameter, this function nests the inverse and the traditional autocovariance functions. A frequency domain nonparametric estimator based on the power transformation of the pooled periodogram is considered and its asymptotic distribution is derived. The results are employed to construct classes of tests of the white noise hypothesis, for clustering and discrimination of stochastic processes and to introduce a novel feature matching estimator of the spectrum.
Item Type:  MPRA Paper 

Original Title:  The Generalised Autocovariance Function 
Language:  English 
Keywords:  Stationary Gaussian processes. Nonparametric spectral estimation. White noise tests. Feature matching. Discriminant Analysis 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C14  Semiparametric and Nonparametric Methods: General C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables > C22  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes 
Item ID:  43711 
Depositing User:  Tommaso Proietti 
Date Deposited:  11 Jan 2013 14:52 
Last Modified:  27 Sep 2019 04:46 
References:  Bartlett, M. S. (1954). Problemes de l’analyse spectral des series temporelles stationnaires. Publ. Inst. Statist. Univ. Paris, III–3 119–134. Battaglia, F. (1983), Inverse Autocovariances and a Measure of Linear Determinism For a Stationary Process, Journal of Time Series Analysis, 4, 7987. Battaglia, F. (1988), On the Estimation of the Inverse Correlation Function, Journal of Time Series Analysis, 9, 110. Battaglia, F., Bhansali, R. (1987), Estimation of the Interpolation Error Variance and an Index of Linear Determinism, Biometrika, 74, 4, 771–779. Beran, J. (1992), A GoodnessofFit Test for Time Series with Long Range Dependence, Journal of the Royal Statistical Society, Series B, 54, 3, 749–760. Bhansali, R. (1993), Estimation of the Prediction Error Variance and an R2 Measure by Autoregressive Model Fitting, Journal of Time Series Analysis, 14, 2, 125–146. Box, G.E.P., and Cox, D.R. (1964), An analysis of transformations (with discussion), Journal of the Royal Statistical Society, B, 26, 211–246. Box, G. E. P., and Pierce, D. A. (1970), Distribution of Residual Autocorrelations in AutoregressiveIntegrated Moving Average Time Series Models, Journal of the American Statistical Association, 65, 15091526. Brockwell, P.J. and Davis, R.A. (1991), Time Series: Theory and Methods, SpringerVerlag, New York. Chen, W.W. and Deo, R.S. (2004a), A Generalized Portmanteau GoodnessofFit Test for Time Series Models, Econometric Theory, 20, 382–416. Chen, W.W. and Deo, R.S. (2004b), Power transformations to induce normality and their applications. Journal of the Royal Statistical Society, Series B, 66, 117130. Cleveland, W.S. (1972), The Inverse Autocorrelations of a Time Series and Their Applications, Technometrics, 14, 2, 277–293. Davis, H.T. and Jones, R.H. (1968), Estimation of the Innovation Variance of a Stationary Time Series, Journal of the American Statistical Association, 63, 321, 141–149. Delgado, M.A., Hidalgo, J. and Velasco, C. (2005), Distribution Free GoodnessofFit Tests for Linear Processes, Annals of Statistics, 33, 2568–2609. Deo, R.S. (2000), Spectral Tests of the Martingale Hypothesis under Conditional Heteroscedasticity, Journal of Econometrics, 99, 291315. Doob, J.L. (1953), Stochastic Processes, John Wiley and Sons, New York. Durlauf, S.N. (1991). Spectral Based Testing of the Martingale Hypothesis, Journal of Econometrics, 50, 355–376. Efron, B., and Tibshirani R.J. (1993), An Introduction to the Bootstrap, Chapman&Hall. El Ghini, A., and Francq, C. (2006), Asymptotic Relative Efficiency of GoodnessOfFit Tests Based on Inverse and Ordinary Autocorrelations, Journal of Time Series Analysis, 27, 843–855. Erd´elyi A., Magnus W., Oberhettinger F., and Tricomi F.G. (1953), Higher Transcendental Functions, vol. II, Bateman Manuscript Project, McGraw and Hill. Fa¨y, G., Moulines, E. and Soulier, P. (2002), Nonlinear Functionals of the Periodogram, Journal of Time Series Analysis, 23, 523–551. Fuller, W.A. (1996), Introduction to Statistical Time Series, John Wiley & Sons. Gould, H.W. (1974), Coefficient Identities for Powers of Taylor and Dirichlet Series, The American Mathematical Monthly, 81, 1, 3–14. Gradshteyn I.S. and Ryzhik I.M. (1994) Table of Integrals, Series, and Products Jeffrey A. and Zwillinger D. Editors, Fifth edition, Academic Press. Graham, R.L., Knuth, D.E. and Patashnik, O., (1994), Concrete Mathematics, AddisonWesley Publishing Company. Gray, H. L., Zhang, N.F. and Woodward, W. A. (1989), On Generalized Fractional Processes. Journal of Time Series Analysis, 10, 233257. Hannan, E.J. and Nicholls, D.F. (1977), The Estimation of the Prediction Error Variance, Journal of the American Statistical Association, 72, 360, 834–840. Harvey, A.C., and J¨ager, A. (1993), Detrending, Stylized Facts and the Business Cycle, Journal of Applied Econometrics, 8, 231–247. Hong, Y. (1996), Consistent Testing for Serial Correlation of Unknown Form, Econometrica, 64, 4 837–864. Kl¨upperberg, C., Mikosch, T. (1996), Gaussian Limit Fields for the Integrated Periodogram, The Annals of Probability, 6, 3 969–991. Koopmans, L.H. (1974), The Spectral Analysis of Time Series, Academic Press. Ljung, G. M. and Box, G. E. P. (1978), On a Measure of Lack of Fit in Time Series Models, Biometrika, 65, 297303. Luati, A., Proietti, T. and Reale, M. (2012), The Variance Profile, Journal of the American Statistical Association, 107, 498, 607–621. Mardia, K., Kent, J. and Bibby, J. (1979). Multivariate Analysis, Academic Press. Milhøj, A. (1981), A Test of Fit in Time Series Models, Biometrika, 68, 1, 167–177. Miller, R.G. (1974), The Jackknife–A Review. Biometrika, 61, 1, 1–15. Percival, D.B. and Walden, A.T. (1993), Spectral Analysis for Physical Applications, Cambridge University Press. Pourahmadi, M. (2001), Foundations of Time Series Analysis and Prediction Theory, John Wiley and Sons. Priestley, M.B. (1981), Spectral Analysis and Time Series, John Wiley and Sons. Quenouille, M.H. (1949), Approximate Tests of Correlation in Time Series. Journal of the Royal Statistical Society, Series B, 11, 6884. Taniguchi, M. (1980), On Estimation of the Integrals of Certain Functions of Spectral Density, Journal of Applied Probability, 17, 1, 73–83. Taniguchi, M. and Kakizawa, Y. (2000), Asymptotic Theory for Statistical Inference of Time Series, Springer. Tieslau, M.A., Schmidt, P. and Baillie, R.T. (1996), A Minimum Distance Estimator for LongMemory Processes, Journal of Econometrics, 71, 249–264. Tukey, J. W. (1957), On the comparative anatomy of transformations, Annals of Mathematical Statistics, 28, 602–632. Walden, A. T. (1994), Interpretation of Geophysical Borehole Data via Interpolation of Fractionally Differenced WhiteNoise, Journal of the Royal Statistical Society, Series C, 43, 335–345. Whittle, P. (1961), Gaussian estimation in stationary time series, Bulletin of the International Statistical Institute, 39, 105–129. Woodward, W. A., Cheng, Q. C. and Gray, H. L. (1998), A kFactor GARMA Longmemory Model. Journal of Time Series Analysis, 19, 485504. Xia, Y., Tong, H. (2011), Feature Matching in Time Series Modeling. Statistical Science, 26, 1, 2146. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/43711 