Time Series Models and Data Generation Processes

Transcription

1 Time Series Models and Data Generation Processes Athanassios Stavrakoudis 27/3/ / 39

2 Table of Contents 1 without drift with drift (1,1) 2 / 39

3 example T <- 100 y <- rnorm (T) y <- ts(t) plot (y, lwd =2, col =2) 3 / 39

4 Histogram of with noise example > mean (y) [1] > var (y) [1] > sd(y) [1] hist (y, xlim=c ( -3,3)) 4 / 39

5 Correlation of noise noise > library ( corrplot ) > d <- data. frame ( cbind (y [1:25], y [26:50], y [51:75], + y [76:100])) > cor <- cor (d) > corrplot (cor, method =" shade ") 5 / 39

6 Table of Contents without drift with drift 2 without drift with drift (1,1) 6 / 39

7 y example without drift with drift y t = y t 1 + ɛ t, t = 1, 2,..., T T <- 100 y <- cumsum ( rnorm (T)) y <- ts(y) plot (y, lwd =2, col =2) random walk Time 7 / 39

8 y example x10 without drift with drift random walk Time T <- 100 N <- 10 plot (y, xlim =c(0,t), ylim =c ( -20,20), type ="n") for (t in 1:N) { y1 <- cumsum ( rnorm (T)) lines (y1, col =t+2, lwd =2, lty =2) } 8 / 39

9 y example x1000 without drift with drift random walk Time T <- 100 N < plot (y, xlim =c(0,t), ylim =c ( -20,20), type ="n") for (t in 1:N) { y1 <- cumsum ( rnorm (T)) lines (y1, col =t+2, lwd =2, lty =2) } 9 / 39

10 without drift with drift Distribution of end value of random walk Y t = Y t 1 + e t, t = 1, 2, What is y 100, Var(y 100 ) or distribution of y 100? Histogram of x T <- 100 N < x <- rep (0, N) for ( i in 1: N) { y1 <- cumsum ( rnorm (T)) x[i] <- y1[t] } hist (x) Frequency x 10 / 39

11 correlation without drift with drift X1 X2 X3 X4 1 X X X X > library ( corrplot ) > d <- data.frame (cbind (y [1:25], y [26:50], y [51:75], y [76:100])) > cor <- cor (d) > corrplot (cor, method=" shade ") 11 / 39

12 y with drift Drift example without drift with drift y t = µ + y t 1 + e t, t = 1, 2, T <- 100 mu <- 0.1 y <- rep (0, T) y0 <- 0 e <- rnorm ( T) y [1] <- mu + y0 + e [1] for ( t in 2: T) { y[ t] <- mu + y[t -1] + e[ t] } y <- ts(y) random walk with drift Time 12 / 39

13 y with drift x1000 without drift with drift N < for ( i in 1: N) { y1 <- rep (0, T) y0 <- 0 e <- rnorm ( T) y [1] <- mu + y0 + e [1] for ( t in 2: T) { y1[t] <- mu + y1[t -1] + e[t] } } y1 <- ts(y1) lines (y1, col =i+2, lwd =2, lty =2) random walk with drift x Time 13 / 39

14 without drift with drift Distribution of end value of random walk with drift Y t = Y t 1 + e t, t = 1, 2, What is y 100, Var(y 100 ) or distribution of y 100? Histogram of x E(y t ) = y 0 + µt V (y t ) = tσe 2 Frequency / 39

15 Table of Contents (1,1) without drift with drift 3 (1,1) 15 / 39

16 y Autoregressive model (1,1) Autoregressive model Y t = µ + ρ 1 Y t 1 + ρ 2 Y t ρ p Y t p + ɛ t µ, ρ i constants, ɛ t white noise. random walk 1st order autoregressive model y t = y t 1 + ɛ t Time 16 / 39

17 (1,1) AR(1) Data Generation Process Autoregressive model µ, ρ constants, ɛ t white noise. Y t = µ + ρy t 1 + ɛ t 1st order autoregressive model > rho <- 0.8 > mu <- 0 > y <- NA > y <- rnorm (1) > for ( t in 2: T) { y[ t] <- mu + rho * y[t -1] + rnorm (1) } y <- ts(y) 17 / 39

18 (1,1) AR(1) Data Generation Process Autoregressive model µ, ρ constants, ɛ t white noise. Y t = µ + ρy t 1 + ɛ t 1st order autoregressive model > rho <- 0.8 > mu <- 0 > y1 <- mu + arima. sim ( n=t, list (ar=c( rho )) ) 18 / 39

19 AR(1) x100 (1,1) y t = 0.8y t 1 + ɛ t AR(1) x 100, rho=0.8 y Time 19 / 39

20 AR(1) various ρ (1,1) y t = ρy t 1 + ɛ t AR(1) x 100, rho=0.8 y Time 20 / 39

21 AR(2) (1,1) y t = 0.5y t y t 2 + ɛ t AR(2), rho1=0.5 rho2= y > rho1 <- 0.5 > rho2 <- 0.3 > y <- arima. sim ( n=t, list (ar=c (0.5, 0.3)) ) Time 21 / 39

22 y AR(2) x 100 (1,1) AR(2) x 100, rho1=0.5 rho2=0.3 AR(2) example y t = 0.5y t y t 2 + ɛ t Time 22 / 39

23 y Moving average model (1,1) Moving average model Y t = µ + ɛ t + θ 1 ɛ t 1 + θ 2 ɛ t θ p ɛ t p µ, θ i constants, ɛ t white noise. 1st order moving average model y t = ɛ t + 0.8θ t Time 23 / 39

24 (1,1) MA(1) Data Generation Process Moving average example model Y t = 0 + ɛ t + 0.8ɛ t 1 > mu <- 0 > theta <- 0.8 > y <- NA > e <- NA > e0 <- rnorm (1) > e <- rnorm ( T) > y [1] <- mu + e [1] - theta *e0 > for ( t in 2: T) { y[ t] <- mu + e[ t] + theta * e[t -1] } > y <- ts(y) 24 / 39

25 (1,1) MA(1) Data Generation Process with arima.sim Moving average example model Y t = 0 + ɛ t + 0.8ɛ t 1 > y <- 0 + arima. sim ( n=t, list (ma=c (0.8)) ) y Time 25 / 39

26 MA(1) x100 (1,1) Y t = ɛ t + 0.8ɛ t 1 MA(1) x 100, theta=0.8 y Time 26 / 39

27 MA(1) various θ (1,1) Y t = ɛ t + 0.8ɛ t 1 MA(1) various theta y Time 27 / 39

28 MA(1) estimation (1,1) J. Durbin, Biometrika, Vol. 46, (1959), pp / 39

29 MA(2) (1,1) y t = ɛ t + 0.5ɛ t ɛ t 2 + ɛ t AR(2), rho1=0.5 rho2= y > rho1 <- 0.5 > rho2 <- 0.3 > y <- arima. sim ( n=t, list (ar=c (0.5, 0.3)) ) Time 29 / 39

30 MA(2) step by step (1,1) y t = ɛ t + 0.5ɛ t ɛ t 2 T <- 100 theta1 <- 0.8 theta2 <- 0.3 mu <- 0.0 y <- NA e0 <- rnorm (1) e1 <- rnorm (1) e <- rnorm ( T) y [1] <- mu + e [1] + theta1 * e0 + theta2 * e1 y [2] <- mu + e [2] + theta1 *e1 + theta2 *e [1] for ( t in 3: T) { y[ t] <- mu + e[ t] + theta1 * e[t -1] + theta2 * e[t -2] } y <- ts(y) 30 / 39

31 y (1,1) (1,1) Data Generation Process with arima.sim Y t = µ + ρy t 1 + θɛ t 1 + ɛ t y t = 0 + ɛ t + 0.8y t 1 0.5ɛ t Time y <- arima. sim (n =100, list (ar=c (0.8), ma=c ( -0.5))) 31 / 39

32 (1,1) step by step (1,1) y t = 0 + ɛ t + 0.8y t 1 0.5ɛ t 1 T <- 100 rho <- 0.8 theta < mu <- 0.0 y <- NA y0 <- mu e <- rnorm ( T) e0 <- rnorm (1) y [1] <- mu + rho *y0 + theta *e0 + e [1] for ( t in 2: T) { y[t] <- mu + + rho *y[t -1] + theta *e[t -1] + e[t] } y <- ts(y) 32 / 39

33 Table of Contents without drift with drift (1,1) 4 33 / 39

34 Choose appropriate starting values y t = µ + ρy t + ɛ t E(y t ) = 0 V (y t ) = σ2 ɛ 1 ρ 2 y0 <- 0 y0 <- rnorm (1) y0 <- rnorm (1, mu/(1 - rho ), 1/(1 - rho ^2)) 34 / 39

35 Use efficient declaration of variables x <- NA for ( i in 1:100) { x[ i] = 0 } x <- rep (0, 100) 35 / 39

36 Avoid unnecessary loops x <- cumsum ( rnorm (100)) x <- rep (0, 100) x [1] <- rnorm (1) for ( i in 2:100) { x[ i] <- x[i -1] + rnorm } 36 / 39

37 Plot random walk with drift x1000 y Time 37 / 39

38 Test to be continued / 39

39 Σχόλια και ερωτήσεις Ωηιτε νοισε Ρανδομ ωαλκ ΑΡΜΑ ὃνλςυσιονς Σας ευχαριστώ για την προσοχή σας Είμαι στη διάθεσή σας για σχόλια, απορίες και ερωτήσεις 39 / 39