The Exponential Model for the Spectrum of a Time Series: Extensions and Applications

Proietti, Tommaso and Luati, Alessandra (2013): The Exponential Model for the Spectrum of a Time Series: Extensions and Applications.

Abstract

The exponential model for the spectrum of a time series and its fractional extensions are based on the Fourier series expansion of the logarithm of the spectral density. The coefficients of the expansion form the cepstrum of the time series. After deriving the cepstrum of important classes of time series processes, also featuring long memory, we discuss likelihood inferences based on the periodogram, for which the estimation of the cepstrum yields a generalized linear model for exponential data with logarithmic link, focusing on the issue of separating the contribution of the long memory component to the log-spectrum. We then propose two extensions. The first deals with replacing the logarithmic link with a more general Box-Cox link, which encompasses also the identity and the inverse links: this enables nesting alternative spectral estimation methods (autoregressive, exponential, etc.) under the same likelihood-based framework. Secondly, we propose a gradient boosting algorithm for the estimation of the log-spectrum and illustrate its potential for distilling the long memory component of the log-spectrum.

• Item Type: MPRA Paper
• Language: English
• Keywords: Frequency Domain Methods; Generalized linear models; Long Memory; Boosting.
• Subjects: C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C52 - Model Evaluation, Validation, and Selection

References

Andrews, D.W.K. and Guggenberger P.(1973), A Bias-Reduced Log-Periodogram Regression Estimator for the Long-Memory Parameter, Econometrica, 71, 675–712.

Abramowitz, M., Stegun, I.A. (1972), Handbook of Mathematical Functions with Formulas, Courier Dover Publications.

Battaglia, F. (1983), Inverse Autocovariances and a Measure of Linear Determinism For a Stationary Process, Journal of Time Series Analysis, 4, 79-87.

Barndorff-Nielsen, O.E., Shou, G. (1973), On the Parametrization of Autoregressive Models by Partial Autocorrelations, Journal of Multivariate Analysis, 3, 408–419.

Beran, J. (1993), Fitting Long-Memory Models by Generalized Linear Regression, Biometrika, 80, 4, 817–822.

Bloomfield, P. (1973), An Exponential Model for the Spectrum of a Scalar Time Series, Biometrika, 60, 2, 217–226.

Bogert B.P., Healy M.J.R. and Tukey J.W., (1963) The quefrency alanysis of time series for echoes: cepstrum, pseudo-autocovariance, cross-cepstrum, and saphe cracking, in Rosenblatt M. Ed. Proceedings of the Symposium on Time Series Analysis Chapter 15, 209–243. Wiley, New York.

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 Autoregressive-Integrated Moving Average Time Series Models, Journal of the American Statistical Association, 65, 1509-1526.

Brillinger, D.R. (2001), Time Series. Data Analysis and Theory , SIAM, Holden Day, San Francisco.

Buhlmann, P. and Hothorn, T. (2007), Boosting Algorithms: Regularization, Prediction and Model Fitting, Statistical Science, 22, 477–505.

Cameron, M.A., Turner, T.R. (1987), Fitting Models to Spectra Using Regression Packages, Journal of the Royal Statistical Society, Series C, Applied Statistics, 36, 1, 47–57.

Carter, C.K. and Kohn, R. (1997), Semiparametric Bayesian Inference for Time Series with Mixed Spectra, Journal of the Royal Statistical Society, Series B, 36, 1, 47–57.

Childers, D.G. Skinner, D.P. Kemerait, R.C. (1977), The Cepstrum: A Guide to Processing, Proceedings of the IEEE, 65, 1428-1443.

Chow, Y. and Grenander, U. (1985), A Sieve Method for the Spectral Density, Annals of Statistics, 13, 998-1010.

Chiu, T.Y.M., Leonard, T. and Tsui, K-W. (1996), The Matrix-Logarithmic Covariance Model, Journal of the American Statistical Association, 91, 198-210.

Cleveland, W.S. (1972), The Inverse Autocorrelations of a Time Series and Their Applications, Technometrics, 14, 2, 277–293.

Dahlhaus, R. (1989), Efficient Parameter Estimation for Self Similar Processes. The Annals of Statistics, 17, 4, 1749-1766.

Dahlhaus, R. (2012). Locally Stationary Processes. In Handbook of Statistics, Time Series Analysis: Methods and Applications, Volume, 30, chapter 13, Elsevier, p. 351–408.

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.

Demmler, A. and Reinsch, C. (1975), Oscillation Matrices with Spline Smoothing. Numerische Mathematik, 24, 375–382.

Doob, J.L. (1953), Stochastic Processes, John Wiley and Sons, New York.

Efron, B., Hastie, T., Johnstone, I. and Tibshirani, R. (2004). Least angle regression (with discussion). The Annals of Statistics. 32, 407-499.

Eubank, R.L. (1999), Nonparametric Regression and Spline Smoothing, Marcel Dekker, New York.

Fan, J. and Kreutzberger, E. (1998), Automatic Local Smoothing for Spectral Density Estimation. Scandinavian Journal of Statistics, 25, 359–369.

Fay, G., Moulines, E. and Soulier, P. (2002), Nonlinear Functionals of the Periodogram, Journal of Time Series Analysis, 23, 523–551.

Fokianos, K., Savvides, A. (2008), On Comparing Several Spectral Densities, Technometrics, 50, 317–331.

Friedman, J., Hastie, T. and Tibshirani, R. (2000), Additive Logistic Regression: a Statistical View of Boosting, The Annals of Statistics, 28, 337- 407.

Fruewirth-Schnatter, S. and Fr¨uwirth-Schnatter, S. (2007), Auxiliary Mmixture Sampling with Applications to Logistic Models. Computational Statistics and Data Analysis, 51, 3509–3528.

Geweke, J. and Porter-Hudak, S. (1983), The Estimation and Application of Long-Memory Time Series Models, Journal of Time Series Analysis, 4, 221–238.

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.

Gray, H.L, Zhang, N.F. andWoodward,W.A. (1989), On Generalized Fractional Processes, Journal of Time Series Analysis, 10, 233–257.

Grenander, U. and Rosenblatt, M. (1957), Statistical Analysis of Stationary Time Series, John Wiley and Sons, New York.

Hastie, T., Tibshirani, R. and Friedman, J. (2009), The Elements of Statistical Learning, Second Edition, Springer, New York.

Hathaway, D.H. (2010), The Solar Cycle, Living Reviews in Solar Physics, 7, 1.

Hosking, J.R.M. (1982), Fractional Differencing, Biometrika, 68, 165–176.

Hsu N-J. and Tsai H. (2009), Semiparametric Estimation for Seasonal Long-Memory Time Series Using Generalized Exponential Models, Journal of Statistical Planning and Inference, 139, 1992–2009.

Hurvich, C.M. (2002), Multistep forecasting of long memory series using fractional exponential models, International Journal of Forecasting, 18, 167–179.

Janaceck, G.J. (1982), Determining the Degree of Differencing for Time Series via the Log Spectrum, Journal of Time Series Analysis, 3, 3, 177–183.

Koenker R. and Yoon J. (2009), Parametric Links for Binary Choice Models: A Fisherian-Bayesian Colloquy, Journal of Econometrics, 152, 120–130.

Koopmans, L.H. (1974), The Spectral Analysis of Time Series, Academic Press.

Li, L.M. (2005), Some Notes on Mutual Information between Past and Future, Journal of Time Series Analysis, 27, 309–322

Luati, A. and Proietti, T. (2010), Hyper-spherical and Elliptical Stochastic Cycles, Journal of Time Series Analysis, 31, 169–181.

Luati, A., Proietti, T. and Reale, M. (2012), The Variance Profile, Journal of the American Statistical Association, 107, 498, 607–621.

Mardia, K.V. and Jupp, P.E. (2000), Directional Statistics, Wiley, Chichester, UK.

McCullagh, T.S. and Nelder, J.A. (1989), Generalized Linear Models, Chapman & Hall Cambridge, UK.

McElroy, T.S. and Holan, S.H. (2012), On the Computation of Autocovariances for Generalized Gegenbauer Processes , Statistica Sinica, to appear.

Monahan, J.F. (1984), A Note Enforcing Stationarity in Autoregressive-Moving Average Models, Biometrika, 71, 2, 403–404.

Moulines, P. and Soulier, E. (1999), Broadband Log-Periodogram Regression of Time Series with Long-Range Dependence, Annals of Statistics, 27, 1415–1439.

Narukawa, M. and Matsuda, Y. (2011), Broadband Semi-Parametric Estimation of Long-Memory Time Series by Fractional Exponential Models, Journal of Time Series Analysis, 32, 175–193.

Oppenheimen A.V. and Schafer R.W. (2010) Discrete-Time Signal Processing, Third Edition. Pearson Eucation, Upper Saddle River.

Pawitan, Y. and O’Sullivan, F. (1994), Nonparametric Spectral Density Estimation Using Penalized Whittle Likelihood, Journal of the American Statistical Association, 89, 600–610.

Percival, D.B. and Walden, A.T. (1993), Spectral Analysis for Physical Applications, Cambridge University Press.

Pourahmadi M. (1983), Exact factorization of the spectral density and its application to forecasting and time series analysis Communications in Statistics, 12, 18, 2085–2094.

Pourahmadi M. (2001), Foundations of Time Series Analysis and Prediction Theory, Wiley Series in Probability and Statistics, John Wiley and Sons.

Proietti, T., and Luati , A. (2012), The Generalised Autucovariance Function, MPRA working paper, 47311.

Robinson, P. (1991). Nonparametric function estimation for long memory time series. In Nonparametric and Semiparametric Methods in Econometrics and Statistics: Proc. of the 5th Int. Symposium in Economic Theory and Econometrics, 437–457.

Robinson, P. (1995). Log-Periodogram Regression of Time Series with Long Range Dependence, Annals of Statistics, 23, 1048–1072.

Rosen, O., Stoffer, D.S., Wood, S. (2009). Local Spectral Analysis via a Bayesian Mixture of Smoothing Splines, Journal of the American Statistical Association, 104, 249–262.

Rosen, O., Stoffer, D.S., Wood, S. (2012). AdaptSPEC: Adaptive Spectral Estimation for Nonstationary Time Series, Journal of the American Statistical Association, 107, 1575–1589.

Wahba, G. (1980). Automatic Smoothing of the Log-Periodogram, Journal of the American Statistical Association, 75, 122–132.

Walker, A.M. (1964). Asymptotic Properties of Least Squares Estimates of Parameters of the Spectrum of a Stationary Non-Deterministic Time Series, Journal of the Australian Mathematical Society, 4, 363–384.

Whittle, P. (1953). Estimation and Information in Stationary Time Series, Arkiv f¨or Matematik, 2, 423–434.

Xia Y. and Tong, H. (2011), Feature Matching in Time Series Modeling. Statistical Science, 26, 1, 21-46.