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# Introduction to Time Series STAT 153

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This 20 page Class Notes was uploaded by Floy Kub on Thursday October 22, 2015. The Class Notes belongs to STAT 153 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/226731/stat-153-university-of-california-berkeley in Statistics at University of California - Berkeley.

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Date Created: 10/22/15

Introduction to Time Series Analysis Lecture 18 1 Review Spectral density Spectral distribution function 2 Autocovariance generating function and spectral density 3 Rational spectra Poles and zeros 4 Examples Review Spectral density and spectral distribution function If a time series X15 has autocovariance 7 satisfying ZZOO WUL lt 00 then we de ne its spectral density as 00 W Z me W h oo for oo lt 1 lt 00 We have 12 1 12 I 27rzuhfy dl 2713921fLCval7 12 12 Where dF1 fldV f measures how the variance of Xt is distributed across the spectrum Review Spectral density and spectral distribution function For any stationary X with autocovariance 7 we can write 12 1 wt emhdm 12 where F is the spectral distribution function of X15 If F has no singular part we can write F FC l FM where FC is continuous that is dFCl fldl and FM is discrete Autocovariance generating function and spectral density I Suppose X t is a linear process so it can be written Xt 20 WM WENVt Consider the autocovariance sequence Yh COVXt7Xth E wiwt q f i j Wthg 20 j0 0121 Zw h 7LO Autocovariance generating function and spectral density I De ne the autocovariance generating function as 73 Z VhBh h oo OO 00 Then 73 01211 Z ZTPWHhBh h oo 7LO OO 00 a Z Zw ijJ Z 70 90 03 ZmB i ijB aiwB 1gtwB 70 90 Autocovariance generating function and spectral density I Notice that 00 73 Z VhBh h oo 00 fV Z YhB 27rzuh h oo Y 6 27T7LV 03w 6 27T7LV w 27rilj w em 2 Autocovariance generating function and spectral density I For example for an MAq we have 1MB 6B so fl 036 6 2713923911 8 6271171 a 9 e W 2 For MA1 u 030 1 ale 27 2 0121 1 61 COS 27w 23961 SiIl 27Tl 2 012D 1 261 cos27w l 6 Autocovariance generating function and spectral density I For an ARp we have 1MB 1 B so 011 fV 6 27T7LV 27rilj 0 weW For AR1 0 fV 1 1 27m 2 02 U 1 2 1 cos27w l Spectral density of a linear process I If Xt is a linear process it can be written Xt 20 wildF7 wBWt Then m w em 2 That is the spectral density f 1 of a linear process measures the modulus of the 1p MAoo polynomial at the point 62 quot on the unit circle Spectral density of a linear process I For an ARMApq 1MB 6B B s0 2 66 27riu9 27r7 11 1 6 27T7LV 27rilj 96 27ri11 2 6 27T7LV This is known as a rational spectrum fV 0 2 O39w Rational spectra I Consider the factorization of 6 and d as 92 9qz z1z z2 z zq 502 pz p1z p2 z Pp Where 21 zq and p1 pp are called the zeros and poles 02 6C1 H31 2MV 2939 w 19 H 1 2MV 19939 7r7LI 2 293 q391 2 z9quot 039 j 2 w 129 IL e W m 2 f V Rational spectra I 2 q 27T ZV 2 6c j1 6 z a 2 w H 1 6 271211 pj As 1 varies from 0 to 12 6 27 moves clockwise around the unit circle from 1 to 6 7 1 And the value of f 1 goes up as this point moves closer to further from the poles pj zeros zj 2 fV Example ARMAI Recall ARl z 1 olz The pole is at 1 1 If 1 gt 0 the pole is to the right of 1 so the spectral density decreases as 1 moves away from 0 If 1 lt 0 the pole is to the left of 1 so the spectral density is at its maximum when 1 05 Recall MAl 6z 1 612 The zero is at 161 If 61 gt 0 the zero is to the left of 1 so the spectral density decreases as 1 moves towards 1 If 61 lt 0 the zero is to the right of 1 so the spectral density is at its minimum when 1 0 Example AR2 I Consider Xt 1Xt1 2Xt2 Wt Example 46 in the text considers this model with 1 1 2 09 and 0121 1 In this case the poles are at 91192 x 05555 i 208958 1054 01567 1054ei2m03916165 Thus we have 2 w p1 2 6 27T ZI p2 2 039 fV 3 6 27T ZI 27r7LI 27r7 016165 and this gets very peaked when 6 passes near 10546 fV Example AR2 Spectral density of AR2 XI Xt1 09 Xt 2Wt 05 Example Seasonal ARMA I Consider Xt 11Xt12 Wt 1 B 1M 1 1B127 1 2 0w1 1316 27ri121X1 1162m3912u 2 1 0w 1 2ltD1 cos247w 1 Notice that f 1 is periodic with period 1 12 Example Seasonal ARMA Spectral density of AR112 Xt 02 Xt12 W t 0 0 x x x 005 01 015 02 025 03 035 04 045 05 Example Seasonal ARMA I Another View 1 ltIgt1z120 ltgt zrew 7 T ID1 1127 62126 e targ bl For 11 gt 0 the twelve poles are at CID1 112 ikW6 for k 0i1i56 So the spectral density gets peaked as 6 27 passes near Dl 112 X 17 7L71396y 7L71393y 7L71392y 7L2713937 7L5713967 Example Multiplicative seasonal ARMA Consider 1 C131B121 1BXt Wt 2 f1 0w1 2D1 cos247ry 1 2 1 COS27TV This is a scaled product of the AR1 spectrum and the periodic AR112 spectrum The AR112 poles give peaks when 6 27 is at one of the 12th roots of 1 the AR1 poles give a peak near 6 27 1 Example Multiplicative seasonal ARMA Spectral density of AR1AR1 12 12 1o5 B102 B Xt vvt fV i 04 045 05 035 025 03 20

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