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1 W707 1 / 37

2 TOPIX30 2 / 37

3 1 2 TOPIX30 3 / 37

4 x Time 4 / 37

5 t {X t } t i.i.d. t 5 / 37

6 Definition ( ) {X t } t. t 1,..., t N X t1,..., X tn 6 / 37

7 Definition ( ) {X t } t t 1,..., t N, h X t1,..., X tn X t1 +h,..., X tn +h 7 / 37

8 z Index (X t = 0.7X t + 0.1ϵ t, AR(1) ) 8 / 37

9 x z Index Index ( ) 8 / 37

10 : µ t := E[X t ] ( t) µ t t µ = µ t : γ(t, s) := Cov(X t, X s ) = E[(X t µ t )(X s µ s )] γ(t, s) t s γ(h) = γ(t, t + h) 9 / 37

11 ( ) Definition ( ) {X t } t µ t γ(t, s) t s 10 / 37

12 11 / 37

13 : m <- decompose(co2) # co2 timeseries plot(m) # Decomposition of additive time series random seasonal trend observed Time 12 / 37

14 (stl) stl decompose stllc <- stl(co2, "periodic") plot(stllc) remainder trend seasonal data time 13 / 37

15 xsmooth <- kernapply(x,kernel("daniell", 10)) # Daniell x Time 14 / 37

16 : ρ(t, s) := γ(t, s) γ(t, t)γ(s, s). t s acf(stllc$time.series[,"remainder"]) CO2 Series stllc$time.series[, "remainder"] ACF Lag 15 / 37

17 acf(stllc$time.series[,"remainder"], type = "covariance") Series stllc$time.series[, "remainder"] ACF (cov) Lag 16 / 37

18 AR 3 17 / 37

19 AR 3 AR X t = p ϕ i X t i + ϵ t. i=1 ϵ t N(0, σ 2 ) (i.i.d.). p AR 17 / 37

20 R AR ar(x, aic = TRUE, order.max = NULL, method = c("yule-walker", "burg", "ols", "mle", "yw"), AR AIC order.max AR method yule-walker 18 / 37

21 Yule-Walker X t = X t 1 ϕ X t p ϕ p + ϵ t X t h X t = X t h X t 1 ϕ X t h X t p ϕ p + X t h ϵ t γ(h) = γ(h 1)ϕ γ(h p)ϕ p. ( ) Yule-Walker : γ(1) γ(0) γ(1) γ(p 1) ϕ 1 γ(2) γ(1) γ(0) γ(p 2) ϕ 2 = γ(p) }{{} γ(p 1) } γ(p 2) {{ γ(0) ϕ p }}{{} γ Γ ϕ γ Γ : ϕ = Γ 1 γ. 19 / 37

22 AR : AIC ar.co2$aic Index > ar.co2 <- ar(stllc$time.series[,"remainder"]) > ar.co2$aic AIC 20 / 37

23 AR(1) X t = ϕ 1 X t 1 + ϵ t AR(1) ϕ 1 ϕ 1 1 AR 21 / 37

24 R Dickey-Fuller > adf.test(ukgas) Augmented Dickey-Fuller Test data: UKgas Dickey-Fuller = , Lag order = 4, p-value = alternative hypothesis: stationary 22 / 37

25 (VAR) X t R d AR (VAR): X t = A 1 X t A p X t p + ϵ t. A i R d d 23 / 37

26 R VAR AR VAR library(vars) varsel <- VARselect(tsx,lag.max=5) # var.topix <- VAR(tsx,p=varsel$selection[1]) #AIC 24 / 37

27 1 2 TOPIX30 25 / 37

28 TOPIX CORE 30 TOPIX CORE (2013/6/ /7/4) ( ) 26 / 37

29 ts(topix30[, 1:10]) & Time Time 27 / 37

30 : R t = log(x t /X t 1 ). 28 / 37

31 Series ts(logrt[, 20]) Series ts(logrt[, 1]) ACF ACF Lag UFJ Lag 29 / 37

32 AR UFJ ufjts <- ts(logrt[,20]) # ar.ufj <- ar(ufjts) #AR 30 / 37

33 AIC ar.ufj$aic :(length(ar.ufj$aic) 1) UFJ AIC 31 / 37

34 AR > ar.ufj$ar #AR [1] / 37

35 AR Normal Q Q Plot Sample Quantiles Theoretical Quantiles 33 / 37

36 > shapiro.test(ar.ufj$resid[5:230]) Shapiro-Wilk normality test data: ar.ufj$resid[5:230] W = , p-value = / 37

37 (R t > 0 R t < 0 ) 3 L1 35 / 37

38 tedata$y Index / 37

39 37 / 37

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²�ËÜËÜ¤Ç»þ·ÏÎó²òÀÏÊÙ¶¯²ñ - Âè£±¾Ï¤ÈÂè£²¾ÏÁ°È¾ Kano Lab. Yuchi MATSUOKA December 22, 2016 1 / 32 1 1.1 1.2 1.3 1.4 2 ARMA 2.1 ARMA 2 / 32 1 1.1 1.2 1.3 1.4 2 ARMA 2.1 ARMA 3 / 32 1.1.1 - - - 4 / 32 1.1.2 - - - - - 5 / 32 1.1.3 y t µ t = E(y t ), V