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1 Kano Lab. Yuchi MATSUOKA December 22, / 32

2 ARMA 2.1 ARMA 2 / 32

3 ARMA 2.1 ARMA 3 / 32

4 / 32

5 / 32

6 1.1.3 y t µ t = E(y t ), V ar(y t ) = E(y t µ t ) 2. V ar(y t ) γ kt = Cov(y t, y t k ) = E[(y t µ t )(y t k µ t k ]. - - k 6 / 32

7 ρ kt = Corr(y t, y t k ) = Cov(y t, y t k ) V ar(yt ) V ar(y t k ) = γ kt γ0t γ 0,t k - - ρ 0t = 1. - k - 7 / 32

8 t t {y t } T t=1 {y t} t= t= 8 / 32

9 ARMA 2.1 ARMA 9 / 32

10 - Definition ( ) t k E(y t ) = µ Cov(y t, y t k ) = E[(y t µ)(y t k µ)] = γ k. 10 / 32

11 Definition ( ) t k (y t, y t+1,..., y t+k ) T i.e, t,k 11 / 32

12 ARMA 2.1 ARMA 12 / 32

13 Definition (iid ) i.i.d. iid iid 0 iid Definition ( ) t σ 2, k = 0 E(ϵ t ) = 0, γ k = E(ϵ t ϵ t k ) = 0, k 0 ϵ t 13 / 32

14 ARMA 2.1 ARMA 14 / 32

15 1 ȳ = 1 T T y t, ˆγ k = 1 T t=1 T t=k+1 (y t ȳ)(y t k ȳ), k = 0, 1, 2,... ˆρ k = ˆγ k ˆγ 0, k = 1, 2, 3,... 2 H 0 : ρ k = 0 vs H ρ k N(0, 1/T ) 15 / 32

16 H 0 : ρ 1 = ρ 2 = = ρ m = 0 vs H 1 : k [1, m] ρ k 0. Q(m) = T (T + 2) m k=1 ˆρ 2 k T k χ 2 (m) 16 / 32

17 ARMA 2.1 ARMA 17 / 32

18 ARMA ARMA 18 / 32

19 ARMA 2.1 ARMA 19 / 32

20 - y t y t 1 { y t = a + b y t 1 = b + c - y t y t 1 y t = ay t 1 + b (MA) (AR) 20 / 32

21 2.1.1 MA MA MA(1) y t = µ + ϵ t + θ 1 ϵ t 1, ϵ t W.N.(σ 2 ) y t MA(1) - y t 1 = µ + ϵ t 1 + θ 1 ϵ t 1 ϵ t 1 θ 1 21 / 32

22 mu=0, theta1=0.8, sig=1 mu=2, theta1=0.5, sig= mu=2, theta1=0.3, sig=2 mu=0, theta1=0.3, sig= mu=2, theta1=0.5, sig=0.5 mu=2, theta1=0.8, sig= / 32

23 MA(1) µ E(y t ) = E(µ + ϵ t + θ 1 ϵ t 1 ) = µ θ 1 0 = µ. ϵ γ 0 = V ar(y t ) = V ar(µ + ϵ t + θ 1 ϵ t 1 ) = V ar(ϵ t + θ 1 ϵ t 1 ) = V ar(ϵ t ) + θ1v 2 ar(ϵ t 1 ) + 2θ 1 Cov(ϵ t, ϵ t 1 ) = σ 2 + θ1σ = (1 + θ1)σ 2 2. θ 2 1σ 2 ϵ 23 / 32

24 MA(1) γ 1 = Cov(y t, y t 1 ) = Cov(µ + ϵ t + θ 1 ϵ t 1, µ + ϵ t 1 + θ 1 ϵ t 2 ) = Cov(ϵ t + θ 1 ϵ t 1, ϵ t 1 + θ 1 ϵ t 2 ) = Cov(ϵ t, ϵ t 1 ) + Cov(ϵ t, θ 1 ϵ t 2 ) + Cov(θ 1 ϵ t 1, ϵ t 1 ) + Cov(θ 1 ϵ t 1, θ 1 ϵ t 2 ) ( 0) = θ 1 Cov(ϵ t 1, ϵ t 1 ) = θ 1 σ 2. ρ 1 = γ1 γ 0 = θ1 1+θ t MA(1) 24 / 32

25 MA(q) q y t = µ + ϵ t + θ 1 ϵ t θ q ϵ t q, ϵ t W.N(σ 2 ). 1 E(y t ) = µ. 2 γ 0 = V ar(y t ) = (1 + θ θ2 q)σ 2. 3 { (θ k + θ 1 θ k θ q k θ q )σ 2, 1 k q γ k = 0, k q MA 5 ρ k = { θk +θ 1 θ k+1 + +θ q k θ q 1+θ θ2 q, 1 k q 0, k q / 32

26 AR MA(q) AR AR AR(1) y t = c + ϕ 1 y t 1 + ϵ t, ϵ t W.N(σ 2 ) 26 / 32

27 y y y y y y c=2,phi=0.8,sigma=1 c= 2,phi=0.3,sigma = 0.5 c=0,phi= 0.3,sigma= Index Index Index c= 2,phi= 0.8,sigma=1 c=2,phi=1,sigma=1 c=2,phi=1.1,sigma= Index Index Index 27 / 32

28 MA AR AR(1) ϕ 1 < 1 MA AR c µ = E(y t ) = E(c + ϕ 1 y t 1 + ϵ t ) = c + ϕ 1 E(y t 1 ) = c + ϕ 1 µ µ = c 1 ϕ 1 MA σ 2 γ 0 = V ar(y t ) = V ar(c + ϕ 1 y t 1 + ϵ t ) = V ar(ϕ 1 y t 1 + ϵ t ) = ϕ 2 1V ar(y t 1 ) + V ar(ϵ t ) + 2Cov(y t 1, ϵ t ) = ϕ 2 1 = ϕ 2 1γ 0 + σ 2 γ 0 = σ 2 /(1 ϕ 2 1). 28 / 32

29 AR(1) k γ k = Cov(y t, y t k ) = Cov(ϕ 1 y t 1 + ϵ t, y t k ) = Cov(ϕ 1 y t 1, y t k ) + Cov(ϵ t, y t k ) = ϕ 1 γ k 1 γ 0 ρ k = ϕ 1 ρ k 1 ( ) - AR ρ 0 = 1 (AR(1) ρ k = ϕ k 1) 29 / 32

30 AR(p) p AR AR(p) y t = c + ϕ 1 y t ϕ p y t p + ϵ t, ϵ t W.N.(σ 2 ). AR(p) 1 µ = E(y t ) = c 1 ϕ 1 ϕ 2 ϕ p. 2 γ 0 = V ar(y t ) = σ 2 1 ϕ 1 ρ 1 ϕ pρ p. 3 y t AR p γ k = ϕ 1 γ k ϕ p γ k p, k 1 ρ k = ϕ 1 ρ k ϕ p ρ k p, k 1 4 AR 30 / 32

31 2.1.3 ARMA (ARMA) AR MA ARMA(p,q) y t = c + ϕ 1 y t ϕ p y t p + ϵ t + θ 1 ϵ t 1 + θ p ϵ t p, ϵ t W.N(σ 2 ). - AR MA ARMA Ex. MA AR ARMA 31 / 32

32 ARMA ARMA(p,q) 1 µ = E(y t ) = c 1 ϕ 1 ϕ 2 ϕ p. 2 q+1 p γ k = ϕ 1 γ k ϕ p γ k p, k q + 1 ρ k = ϕ 1 ρ k ϕ p ρ k p, k q MA 3 ARMA 32 / 32

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