Luati, Alessandra and Proietti, Tommaso and Reale, Marco (2011): The Variance Profile.
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The variance profile is defined as the power mean of the spectral density function of a stationary stochastic process. It is a continuous and non-decreasing function of the power parameter, p, which returns the minimum of the spectrum (p → −∞), the interpolation error variance (harmonic mean, p = −1), the prediction error variance (geometric mean, p = 0), the unconditional variance (arithmetic mean, p = 1) and the maximum of the spectrum (p → ∞). The variance profile provides a useful characterisation of a stochastic processes; we focus in particular on the class of fractionally integrated processes. Moreover, it enables a direct and immediate derivation of the Szego-Kolmogorov formula and the interpolation error variance formula. The paper proposes a non-parametric estimator of the variance profile based on the power mean of the smoothed sample spectrum, and proves its consistency and its asymptotic normality. From the empirical standpoint, we propose and illustrate the use of the variance profile for estimating the long memory parameter in climatological and financial time series and for assessing structural change.
|Item Type:||MPRA Paper|
|Original Title:||The Variance Profile|
|Keywords:||Predictability; Interpolation; Non-parametric spectral estimation; Long memory.|
|Subjects:||C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General
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 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes
|Depositing User:||Tommaso Proietti|
|Date Deposited:||24. Apr 2011 13:03|
|Last Modified:||16. Feb 2013 00:31|
Atkinson, A. C. (1985), Plots, Transformations and Regression: An Introduction to Graphical Methods of Diagnostic Regression Analysis, Oxford University Press, Oxford, UK.
Baillie, R.T. (1996), Long memory processes and fractional integration in econometrics, Journal of Econometrics, 73, 5, 1–59.
Battaglia, F., Bhansali, R. (1987), Estimation of the Interpolation Error Variance and an Index of Linear Determinism, Biometrika, 74, 4, 771–779.
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.
Chatfield, C.S. (1972), Inverse Autocorrelations, Journal of the Royal Statistical Society, Series A, 142, 3, 363–377.
Choudhuri, N., Ghosal, S., Roy, A. (2004), Bayesian Estimation of the Spectral Density of a Time Series, Journal of the American Statistical Association, 99, 468, 1050–1059.
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.
Diebold, F.X. and Kilian, L. (2001), Measuring Predictability: Theory and Macroeconomic Applications, Journal of Applied Econometrics, 16, 657–669.
Ding, Z., Granger, C.W.J. and Engle, R.F. (1993), A long memory property of stock market returns and a new model, Journal of Empirical Finance, 1, 1, 83–106.
Eicker F. (1963), Central Limit Theorems for Families of Sequences of Random Variables, The Annals of Mathematical Statistics, 34, 2, 439–446.
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.
Gleser L.J. (1965), On the Asymptotic Theory of Fixed-Size Sequential Confidence Bounds for Linear Regression Parameters, The Annals of Mathematical Statistics, 36, 2, 463–467.
Gnedenko, B.V. and Kolmogorov, A.N. (1954), Limit Distributions for Sums of Random Variables, Addison-Wesley Publishing Company.
Janacek, G. (1975), Estimation of the Minimum Mean Square Error of Prediction, Biometrika, 62, 1, 175–180.
Johnson N.L. and Kotz, S. (1972), Power Transformations of Gamma Variables, Biometrika, 59, 1, 226–229.
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.
Hannan, E.J., Terrell, R.D., and Tuckwell, N.E. (1970), The seasonal adjustment of economic time series, International Economic Review, 11, 24–52.
Harvey, A.C. (1989), Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press.
Haywood, J. and Tunnicliffe-Wilson, G. (2000), An Improved State Space Representation for Cyclical Time Series, Biometrika, 87, 3, 724–726.
Kensahara, Y., Pourahmadi, M. and Inoue, A. (2009), Duals of Random Vectors and Processes with Applications to Prediction Problems with Missing Values, Statistics and Probability Letters, 79, 1637–1646.
Kohli, P. and Pourahmadi, M. (2011), Nonparametric estimation of the innovation variance and judging the fit of ARMA models, forthcoming in Bell, W.R., Holan, S.H. and McElroy T. (eds) Economic Time Series: Modeling and Seasonality, Chapman and Hall/CRC.
Kolmogorov, A.N. (1941), Interpolation and Extrapolation of Stationary Random Sequences (in Russian), Izv. Akad. Nauk SSSR Sere Mat., 5, 3–14.
Kolmogorov, A.N. (1992), Interpolation and Extrapolation of Stationary Random Sequences, in Selected works of A.N. Kolmogorov, Volume II, Probability and Mathematical Statistics, 272–280.
Koopmans, L.H. (1974), The Spectral Analysis of Time Series, Academic Press.
Luati, A. and Proietti, T. (2010), Hyper-spherical and Elliptical Stochastic Cycles, Journal of Time Series Analysis, 31, 169–181.
Masani, P. (1960), The prediction theory of multivariate stochastic processes, III Unbounded spectral densities, Acta Mathematica, 104, 1-2, 141–162.
McConnell, M.M. and Perez-Quiros, G. (2000), Output Fluctuations in the United States: What Has Changed since the Early 1980’s?, American Economic Review, 90, 1464–1476.
Nelson, C.R. (1976), The Interpretation of R2 in Autoregressive-Moving Average Time Series Models, The American Statistician, 30, 175–180.
Pukkila, T. and Nyquist, H. (1985), On frequency domain estimation of the innovation variance of a stationary univariate time series. Biometrika, 72, 317-323.
Salehi, H. (1979), Algorithms for Linear Interpolator and Interpolation Error for Minimal Stationary Stochastic Processes, The Annals of Probability, 7, 5, 840–846.
Szego, G. (1920), Beitrage zur Theorie der Toeplitzen Formen (Ersten Mit- teilung), Mathematische Zeitschrift, 6, 167-202.
Szego, G., Askey, R. (1982), Gabor Szego: Collected Papers, Vol. 1: 1915-1927, Birkauser.
Stock, J.H., and Watson, M.W. (2002), Has the Business Cycle Changed and Why? NBER Macroeconomics Annual, 17, 159–230, National Bureau of Economic Research.
Taniguchi, M. (1980), On Estimation of the Integrals of Certain Functions of Spectral Density, Journal of Applied Probability, 17, 1, 73–83.
Tong, H. (1979), Final Prediction Error and Final Interpolation Error: A Paradox?, IEEE Transactions on Information Theory, It-25, 6, 758-759.
Walden, A. T. (1995), Multitaper estimation of innovation variance of a stationary time series, IEEE Transactions on Signal Processing, 43, 181-187.
Walden, A. T. (2000), A unified view of multitaper multivariate spectral estimation. Biometrika, 87, 767-788.
West, M.and Harrison, J. (1989), Bayesian Forecasting and Dynamic Models, 1st edition, New York, Springer-Verlag.
West, M.and Harrison, J. (1997), Bayesian Forecasting and Dynamic Models, 2nd edition, New York, Springer-Verlag.
Wiener, N. (1949), Extrapolation, Interpolation and Smoothing of Stationary Time Series, The Technology Press of MIT and John Wiley and Sons.