Analyzing Strongly Periodic Series in the Frequency Domain: A Comparison of Alternative Approaches with Applications

Strongly periodic series occur frequently in many disciplines. This paper reviews one specific approach to analyzing such series viz. the harmonic regression approach. In this paper, the five major methods suggested under this approach are critically reviewed and compared, and their empirical potential highlighted via two applications. The out-of-sample forecast comparisons are made using the Superior Predictive Ability test, which specifically guards against the perils of data snooping. Certain tentative conclusions are drawn regarding the relative forecasting ability of the different methods.

[1]  F. Diebold,et al.  Comparing Predictive Accuracy , 1994, Business Cycles.

[2]  Gordon K. Smyth,et al.  Employing Symmetry Constraints for Improved Frequency Estimation by Eigenanalysis Methods , 2000, Technometrics.

[3]  Petre Stoica,et al.  Study of the statistical performance of the Pisarenko harmonic decomposition method , 1988 .

[4]  Michael R. Chernick,et al.  Wavelet Methods for Time Series Analysis , 2001, Technometrics.

[5]  B. G. Quinn,et al.  Estimating the frequency of a periodic function , 1991 .

[6]  H. Hartley,et al.  Tests of significance in harmonic analysis. , 1949, Biometrika.

[7]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[8]  B. Truong-Van.  A New Approach to Frequency Analysis with Amplified Harmonics , 1990 .

[9]  S. Lang,et al.  Frequency estimation with maximum entropy spectral estimators , 1980 .

[10]  David F. Findley,et al.  New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program , 1998 .

[11]  J. Magnus,et al.  Matrix Differential Calculus with Applications in Statistics and Econometrics , 1991 .

[12]  Peter C. Young,et al.  Recursive and en-bloc approaches to signal extraction , 1999 .

[13]  David R. Brillinger,et al.  Time Series: Data Analysis and Theory. , 1982 .

[14]  A. M. Walker,et al.  A Survey of statistical work on the Mackenzie River series of annual Canadian lynx trappings for the , 1977 .

[15]  A. Izenman,et al.  Fourier Analysis of Time Series: An Introduction , 1977, IEEE Transactions on Systems, Man and Cybernetics.

[16]  Hideaki Sakai,et al.  Statistical analysis of Pisarenko's method for sinusoidal frequency estimation , 1984 .

[17]  U. Grenander,et al.  Statistical analysis of stationary time series , 1958 .

[18]  R. J. Bhansali,et al.  A Mixed Spectrum Analysis of the Lynx Data , 1979 .

[19]  H. White,et al.  A Reality Check for Data Snooping , 2000 .

[20]  G. Stewart.  Introduction to matrix computations , 1973 .

[21]  William A. Gardner,et al.  Statistical spectral analysis : a nonprobabilistic theory , 1986 .

[22]  H. Akaike.  SEASONAL ADJUSTMENT BY A BAYESIAN MODELING , 1980 .

[23]  E. Hannan.  Testing for a Jump in the Spectral Function , 1961 .

[24]  A. G. Barnett,et al.  A time-domain test for some types of nonlinearity , 2005, IEEE Transactions on Signal Processing.

[25]  Ta-Hsin Li,et al.  Strong consistency of the contraction mapping method for frequency estimation , 1993, IEEE Trans. Inf. Theory.

[26]  E. Hannan.  The asymptotic theory of linear time-series models , 1973, Journal of Applied Probability.

[27]  Peter Reinhard Hansen,et al.  An Unbiased and Powerful Test for Superior Predictive Ability , 2001 .

[28]  Arthur Schuster,et al.  On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena , 1898 .

[29]  P. Whittle,et al.  TESTS OF FIT IN TIME SERIES , 1952 .

[30]  Víctor Gómez,et al.  Programs TRAMO and SEATS, Instruction for User (Beta Version: september 1996) , 1996 .

[31]  J. Cadzow,et al.  Spectral estimation: An overdetermined rational model equation approach , 1982, Proceedings of the IEEE.

[32]  Debasis Kundu,et al.  Constrained maximum likelihood estimators for superimposed exponential signals , 1997 .

[33]  S.M. Kay,et al.  Spectrum analysis—A modern perspective , 1981, Proceedings of the IEEE.

[34]  M. Waldmeier.  The sunspot-activity in the years 1610-1960 , 1961 .

[35]  James Durbin,et al.  The fitting of time series models , 1960 .

[36]  P. Hansen,et al.  A Forecast Comparison of Volatility Models: Does Anything Beat a Garch(1,1)? , 2004 .

[37]  M. B. Priestley.  Estimation of the Spectral Density Function in the Presence of Harmonic Components , 1964 .

[38]  R. Bhansali,et al.  Some properties of the order of an autoregressive model selected by a generalization of Akaike∘s EPF criterion , 1977 .

[39]  Joseph P. Romano,et al.  The stationary bootstrap , 1994 .

[40]  P. A. P. Moran,et al.  The statistical analysis of the Canadian Lynx cycle. , 1953 .

[41]  W. M. Carey,et al.  Digital spectral analysis: with applications , 1986 .

[42]  A. M. Walker.  On the estimation of a harmonic component in a time series with stationary dependent residuals , 1973, Advances in Applied Probability.

[43]  George M. Siouris,et al.  Applied Optimal Control: Optimization, Estimation, and Control , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[44]  R. Gencay,et al.  An Introduction to Wavelets and Other Filtering Methods in Finance and Economics , 2001 .

[45]  M. Hashem Pesaran,et al.  A Simple Nonparametric Test of Predictive Performance , 1992 .

[46]  Peter Whittle,et al.  Likelihood and cost as path integrals , 1991 .

[47]  S. Koopman,et al.  Disturbance smoother for state space models , 1993 .

[48]  G. Yule.  On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers , 1927 .

[49]  Dimitris Kugiumtzis,et al.  On the Reliability of the surrogate Data Test for nonlinearity in the Analysis of noisy Time Series , 2001, Int. J. Bifurc. Chaos.

[50]  E. J. Hannan,et al.  Non-linear time series regression , 1971, Journal of Applied Probability.

[51]  D. Brillinger.  Time series - data analysis and theory , 1981, Classics in applied mathematics.

[52]  Eugen Slutzky.  Summation of random causes as the source of cyclic processes , 1937 .

[53]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[54]  Sidney J. Yakowitz,et al.  Some contributions to a frequency location method due to He and Kedem , 1991, IEEE Trans. Inf. Theory.

[55]  V. Pisarenko.  The Retrieval of Harmonics from a Covariance Function , 1973 .

[56]  Shean-Tsong Chiu,et al.  Detecting Periodic Components in a White Gaussian Time Series , 1989 .

[57]  G. C. Tiao,et al.  Consistency Properties of Least Squares Estimates of Autoregressive Parameters in ARMA Models , 1983 .

[58]  P. Young,et al.  Dynamic harmonic regression. , 1999 .

[59]  K. West,et al.  Asymptotic Inference about Predictive Ability , 1996 .