# If y t and y t τ are statistically independent then

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Ifytandyt-τare statistically independent, then their joint probabilitydensity function is the product of their individual probability density functionsso thatf(yt, yt-τ) =f(yt)f(yt-τ). It follows that(v)γτ=E(yt-μ)E(yt-τ-μ) = 0for allτ6= 0.Ify0, . . . , yTis a sample from the process, and ifτ < T, then we may estimateγτfrom the sample autocovariance or empirical autocovariance of lagτ:(vi)cτ=1TT-1Xt=τ(yt-¯y)(yt-τ-¯y).The periodogram and the autocovariance functionThe periodogram is defined by(vii)I(ωj) =2T"T-1Xt=0cos(ωjt)(yt-¯y)2+T-1Xt=0sin(ωjt)(yt-¯y)2#.16
D.S.G. POLLOCK : THE METHODS OF TIME-SERIES ANALYSISThe identitytcos(ωjt)(yt-¯y) =tcos(ωjt)ytfollows from the fact that,by construction,tcos(ωjt) = 0 for allj. Hence the above expression has thesame value as the expression in (2). Expanding the expression in (vii) gives(viii)I(ωj) =2TXtXscos(ωjt) cos(ωjs)(yt-¯y)(ys-¯y)+2TXtXssin(ωjt) sin(ωjs)(yt-¯y)(ys-¯y),and, by using the identity cos(A) cos(B) + sin(A) sin(B) = cos(A-B), we canrewrite this as(ix)I(ωj) =2TXtXscos(ωj[t-s])(yt-¯y)(ys-¯y).Next, on definingτ=t-sand writingcτ=t(yt-¯y)(yt-τ-¯y)/T, we canreduce the latter expression to(x)I(ωj) = 2T-1Xτ=1-Tcos(ωjτ)cτ,which appears in the text as equation (15).References[1] Alberts, W. W., L. E. Wright and B. Feinstein (1965), “PhysiologicalMechanisms of Tremor and Rigidity in Parkinsonism.”Confinia Neuro-logica, 26, 318–327.[2] Beveridge, Sir W. H. (1921), “Weather and Harvest Cycles.”EconomicJournal, 31, 429–452.[3] Beveridge, Sir W. H. (1922), “Wheat Prices and Rainfall in Western Eu-rope.” Journal of the Royal Statistical Society, 85, 412–478.[4] Box, G. E. P. and D. R. Cox (1964), “An Analysis of Transformations.”Journal of the Royal Statistical Society, Series B, 26, 211–243.[2] Box, G. E. P. and G. M. Jenkins (1970), Time Series Analysis, Forecastingand Control. Holden–Day: San Francisco.[6] Buys–Ballot, C. D. H. (1847), “Les Changements Periodiques de Temper-ature.” Utrecht.[7] Cooley, J. W. and J. W. Tukey (1965), “An Algorithm for the MachineCalculation of Complex Fourier Series.” Mathematics of Computation, 19,297–301.17
THE METHODS OF TIME-SERIES ANALYSIS[8] Granger, C. W. J. and A. O. Hughes (1971), “A New Look at Some OldData: The Beveridge Wheat Price Series.”Journal of the Royal StatisticalSociety, Series A, 134, 413–428.[9] Groves, G. W. and E. J. Hannan, (1968), “Time-Series Regression of SeaLevel on Weather.” Review of Geophysics, 6, 129–174.[10] Gudmundson, G. (1971), “Time-Series Analysis of Imports, Exports andother Economic Variables.”Journal of the Royal Statistical Society, SeriesA, 134, 383.[11] Hassleman, K., W. Munk and G. MacDonald, (1963), “Bispectrum ofOcean Waves.”In Time Series Analysis, M. Rosenblatt, (ed.)125–139.

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