# PROBLEM 7
# Generate n = 1000 observations from each of the models in the previous question and comment on the important patterns in the ACF and PACF plots
library(astsa)
ar = arima.sim( list(order = c( 1, 0, 0), ar = 0.6), n = 1000)
ma = arima.sim( list(order = c( 0, 0, 1), ma = 0.9), n = 1000)
arma = arima.sim( list(order = c( 1, 0, 1), ar = 0.6, ma = 0.9), n = 1000)
acf2(ar)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## ACF 0.61 0.36 0.19 0.11 0.06 0.05 0.06 0.06 0.03 0.02 -0.02 -0.07 -0.08
## PACF 0.61 -0.01 -0.04 0.02 0.00 0.03 0.04 0.00 -0.02 0.00 -0.05 -0.07 0.00
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## ACF -0.08 -0.04 -0.03 -0.03 -0.02 -0.02 -0.05 -0.07 -0.08 -0.05 -0.03 0.01
## PACF -0.01 0.03 -0.02 0.00 0.00 0.00 -0.05 -0.02 -0.02 0.02 0.00 0.03
## [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
## ACF 0.04 0.00 -0.03 -0.07 -0.08 -0.08 -0.05 -0.05 -0.07 -0.07 -0.07 -0.07
## PACF 0.02 -0.07 -0.01 -0.06 -0.01 -0.01 0.02 -0.05 -0.05 0.00 -0.02 -0.01
## [,38] [,39] [,40] [,41] [,42]
## ACF -0.06 -0.01 0.02 0.01 0.00
## PACF 0.01 0.06 0.01 -0.04 -0.01
acf2(ma)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## ACF 0.52 0.03 0.01 0.00 -0.02 0.03 0.04 0.00 0.01 0.02 0.01 0.02 0.01
## PACF 0.52 -0.33 0.25 -0.21 0.16 -0.06 0.06 -0.08 0.09 -0.08 0.08 -0.06 0.04
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## ACF -0.02 -0.01 0.01 -0.03 -0.05 -0.03 -0.04 -0.03 0.02 0.04 0.03 0.03
## PACF -0.07 0.08 -0.06 -0.01 -0.02 0.00 -0.04 0.05 0.00 0.03 0.02 0.00
## [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
## ACF -0.03 -0.07 -0.03 -0.04 -0.06 0.00 0.06 0.04 0.01 0.01 0.03 0.02
## PACF -0.08 0.02 -0.02 -0.05 -0.01 0.06 0.01 0.02 -0.03 0.05 0.01 0.00
## [,38] [,39] [,40] [,41] [,42]
## ACF -0.02 -0.06 -0.05 -0.02 -0.01
## PACF -0.06 -0.02 0.00 -0.01 0.01
acf2(arma)
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## ACF 0.82 0.54 0.38 0.26 0.17 0.12 0.09 0.04 0.02 0.01 0.00 0.01 0.00
## PACF 0.82 -0.40 0.30 -0.27 0.21 -0.11 0.05 -0.10 0.10 -0.09 0.11 -0.08 0.04
## [,14] [,15] [,16] [,17] [,18] [,19] [,20] [,21] [,22] [,23] [,24] [,25]
## ACF -0.01 -0.01 0.00 0.01 0.01 -0.01 -0.03 -0.04 -0.06 -0.08 -0.09 -0.09
## PACF -0.04 0.04 -0.02 0.04 -0.05 -0.02 0.00 -0.02 -0.02 -0.04 0.00 0.02
## [,26] [,27] [,28] [,29] [,30] [,31] [,32] [,33] [,34] [,35] [,36] [,37]
## ACF -0.07 -0.06 -0.04 -0.03 -0.01 0.01 0.02 0.02 0.00 -0.01 -0.01 -0.02
## PACF 0.00 0.02 -0.03 0.03 0.03 0.00 0.00 -0.03 -0.03 0.04 -0.02 -0.03
## [,38] [,39] [,40] [,41] [,42]
## ACF -0.05 -0.06 -0.06 -0.07 -0.06
## PACF -0.04 0.02 -0.02 0.00 -0.01