Introduction to Time Series Analysis. Lecture 16.

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1 Introduction to Time Series Analysis. Lecture Review: Spectral density 2. Examples 3. Spectral distribution function. 4. Autocovariance generating function and spectral density. 1

2 Review: Spectral density If a time series {X t } has autocovariance γ satisfying h= γ(h) <, then we define its spectral density as f(ν) = h= γ(h)e 2πiνh for < ν <. 2

3 Review: Spectral density 1. f(ν) is real. 2. f(ν) f is periodic, with period 1. So we restrict the domain off to 1/2 ν 1/2. 4. f is even (that is,f(ν) = f( ν)). 5. γ(h) = 1/2 1/2 e 2πiνh f(ν)dν. 3

4 Examples White noise: {W t },γ(0) = σ 2 w and γ(h) = 0 for h 0. f(ν) = γ(0) = σ 2 w. AR(1): X t = φ 1 X t 1 +W t,γ(h) = σwφ 2 h 1 /(1 φ2 1). f(ν) = σ 2 w 1 2φ 1. cos(2πν)+φ 2 1 If φ 1 > 0 (positive autocorrelation), spectrum is dominated by low frequency components smooth in the time domain. If φ 1 < 0 (negative autocorrelation), spectrum is dominated by high frequency components rough in the time domain. 4

5 Example: AR(1) 100 Spectral density of AR(1): X t = +0.9 X t 1 + W t f(ν) ν 5

6 Example: AR(1) 100 Spectral density of AR(1): X t = 0.9 X t 1 + W t f(ν) ν 6

7 Example: MA(1) X t = W t +θ 1 W t 1. σw(1+θ 2 1) 2 ifh = 0, γ(h) = σwθ 2 1 if h = 1, 0 otherwise. 1 f(ν) = h= 1 γ(h)e 2πiνh = γ(0)+2γ(1)cos(2πν) ( = σw 2 1+θ θ 1 cos(2πν) ). 7

8 Example: MA(1) X t = W t +θ 1 W t 1. f(ν) = σ 2 w ( 1+θ θ 1 cos(2πν) ). If θ 1 > 0 (positive autocorrelation), spectrum is dominated by low frequency components smooth in the time domain. If θ 1 < 0 (negative autocorrelation), spectrum is dominated by high frequency components rough in the time domain. 8

9 Example: MA(1) 4 Spectral density of MA(1): X t = W t +0.9 W t f(ν) ν 9

10 Example: MA(1) 4 Spectral density of MA(1): X t = W t 0.9 W t f(ν) ν 10

11 Introduction to Time Series Analysis. Lecture Review: Spectral density 2. Examples 3. Spectral distribution function. 4. Autocovariance generating function and spectral density. 5. Rational spectra. Poles and zeros. 11

12 Recall: A periodic time series k X t = (A j sin(2πν j t)+b j cos(2πν j t)) j=1 k = (A 2 j +Bj) 2 1/2 sin(2πν j t+tan 1 (B j /A j )). j=1 E[X t ] = 0 k γ(h) = σj 2 cos(2πν j h) j=1 γ(h) =. h 12

13 Discrete spectral distribution function ForX t = Asin(2πλt)+Bcos(2πλt), we have γ(h) = σ 2 cos(2πλh), and we can write γ(h) = 1/2 1/2 e 2πiνh df(ν), where F is the discrete distribution 0 ifν < λ, F(ν) = σ 2 2 if λ ν < λ, σ 2 otherwise. 13

14 The spectral distribution function For any stationary {X t } with autocovariance γ, we can write γ(h) = 1/2 1/2 e 2πiνh df(ν), where F is the spectral distribution function of {X t }. We can split F into three components: discrete, continuous, and singular. If γ is absolutely summable, F is continuous: df(ν) = f(ν)dν. If γ is a sum of sinusoids,f is discrete. 14

15 The spectral distribution function ForX t = k j=1 (A j sin(2πν j t)+b j cos(2πν j t)), the spectral distribution function isf(ν) = k j=1 σ2 j F j(ν), where 0 ifν < ν j, F j (ν) = 1 2 if ν j ν < ν j, 1 otherwise. 15

16 Wold s decomposition Notice that X t = k j=1 (A j sin(2πν j t)+b j cos(2πν j t)) is deterministic (once we ve seen the past, we can predict the future without error). Wold showed that every stationary process can be represented as X t = X (d) t +X (n) t, where X (d) t is purely deterministic andx (n) t is purely nondeterministic. (c.f. the decomposition of a spectral distribution function asf (d) +F (c).) Example: X t = Asin(2πλt)+ θ(b) φ(b) W t. 16

17 Introduction to Time Series Analysis. Lecture Review: Spectral density 2. Examples 3. Spectral distribution function. 4. Autocovariance generating function and spectral density. 17

18 Autocovariance generating function and spectral density SupposeX t is a linear process, so it can be written X t = i=0 ψ iw t i = ψ(b)w t. Consider the autocovariance sequence, γ h = Cov(X t,x t+h ) = E ψ i W t i ψ j W t+h j = σ 2 w i=0 ψ i ψ i+h. i=0 j=0 18

19 Autocovariance generating function and spectral density Define the autocovariance generating function as Then, γ(b) = γ h B h. γ(b) = σw 2 = σw 2 = σw 2 h= h= i=0 ψ i ψ i+h B h i=0 ψ i ψ j B j i j=0 ψ i B i ψ j B j = σwψ(b 2 1 )ψ(b). i=0 j=0 19

20 Autocovariance generating function and spectral density Notice that γ(b) = f(ν) = h= h= γ h B h. = γ ( e 2πiν) γ h e 2πiνh = σwψ ( 2 e 2πiν) ψ ( e 2πiν) = σw 2 ψ ( e 2πiν) 2. 20

21 Autocovariance generating function and spectral density For example, for an MA(q), we have ψ(b) = θ(b), so f(ν) = σwθ 2 ( e 2πiν) θ ( e 2πiν) = σw 2 θ ( e 2πiν) 2. For MA(1), f(ν) = σ 2 w 1+θ 1 e 2πiν 2 = σw 1+θ 2 1 cos( 2πν)+iθ 1 sin( 2πν) 2 = σw 2 ( 1+2θ1 cos(2πν)+θ1 2 ). 21

22 Autocovariance generating function and spectral density For an AR(p), we have ψ(b) = 1/φ(B), so f(ν) = = σ 2 w φ(e 2πiν )φ(e 2πiν ) σ 2 w φ(e 2πiν ) 2. For AR(1), f(ν) = = σ 2 w 1 φ 1 e 2πiν 2 σ 2 w 1 2φ 1 cos(2πν)+φ

23 Spectral density of a linear process If X t is a linear process, it can be written X t = i=0 ψ iw t i = ψ(b)w t. Then f(ν) = σw 2 ψ ( e 2πiν) 2. That is, the spectral density f(ν) of a linear process measures the modulus of the ψ (MA( )) polynomial at the point e 2πiν on the unit circle. 23

24 Introduction to Time Series Analysis. Lecture Review: Spectral density 2. Examples 3. Spectral distribution function. 4. Autocovariance generating function and spectral density. 24