TimeSeriesBook.pdf

The spectral density is large in the neighborhood of

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the spectral density is large in the neighborhood of zero and small in the neighborhood of π . For a negative autocorrelation the picture is just reversed. AR(1): The spectral density of an AR(1) process X t = φX t - 1 + Z t with Z t WN(0 , σ 2 ) is: f ( λ ) = γ (0) 2 π 1 + X h =1 φ h ( e - ıhλ + e ıhλ ) ! = σ 2 2 π (1 - φ 2 ) 1 + φ e ıλ 1 - φ e ıλ + φ e - ıλ 1 - φ e - ıλ = σ 2 2 π 1 1 - 2 φ cos λ + φ 2 The spectral density for φ = 0 . 6 and φ = - 0 . 6 and σ 2 = 1 are plotted in Figure 6.1(b). As the process with φ = 0 . 6 exhibits a relatively large positive autocorrelation so that it is rather smooth, the spectral density takes large values for low frequencies. In contrast, the process with φ = - 0 . 6 is rather volatile due to the negative first order autocorrelation. Thus, high frequencies are more important than low frequencies as reflected in the corresponding figure.
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122 CHAPTER 6. SPECTRAL ANALYSIS AND LINEAR FILTERS 0 1 2 3 0.05 0.1 0.15 0.2 0.25 0.3 λ θ = 0.5 ρ = 0.4 θ = -0.5 ρ = -0.4 (a) MA(1) process 0 1 2 3 0 0.5 1 λ φ = 0.6 φ = -0.6 (b) AR(1) process Figure 6.1: Examples of spectral densities with Z t WN(0 , 1) Note that, as φ approaches one, the spectral density evaluated at zero tends to infinity, i.e. lim λ 0 f ( λ ) = . This can be interpreted in the following way. As the process gets closer to a random walk more and more weight is given to long-run fluctuations (cycles with very low frequency or very high periodicity) (Granger, 1966). 6.2 Spectral Decomposition of a Time Series Consider the simple harmonic process { X t } which just consists of a cosine and a sine wave: X t = A cos( ω t ) + B sin( ω t ) . (6.4) Thereby A and B are two uncorrelated random variables with E A = E B = 0 and V A = V B = 1. The autocovariance function of this process is γ ( h ) = cos( ω h ). This autocovariance function cannot be represented as R π - π e ıhλ f ( λ )d λ . However, it can be regarded as the Fourier transform of a discrete distribution function F : γ ( h ) = cos( ω h ) = Z ( - π,π ] e ıhλ d F ( λ ) , where F ( λ ) = 0 , for λ < - ω ; 1 / 2 , for - ω λ < ω ; 1 , for λ ω . (6.5)
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6.2. SPECTRAL DECOMPOSITION 123 The integral with respect to the discrete distribution function is a so-called Riemann-Stieltjes integral. 3 F is a step function with jumps at - ω and ω and step size of 1 / 2 so that the above integral equals 1 2 e - ı hω + 1 2 e - ı hω = cos( ). These considerations lead to a representation, called the spectral represen- tation , of the autocovariance function as the Fourier transform a distribution function over [ - π, π ]. Theorem 6.2 (Spectral representation) . γ is the autocovariance function of a stationary process { X t } if and only if there exists a right-continuous, nondecreasing, bounded function F on ( - π, π ] with the properties F ( - π ) = 0 and γ ( h ) = Z ( - π,π ] e ıhλ d F ( λ ) . (6.6) F is called the spectral distribution function of γ . Remark 6.2. If the spectral distribution function F has a density f such that F ( λ ) = R λ - π f ( ω )d ω then f is called the spectral density and the time series is said to have a continuous spectrum.
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