AR(1) model...
model {
# AR(1):
for (t in 1:n) {
y[t] ~ dnorm(m[t], tau)
yr[t] <- 1769 + t
}
for (t in 2:n) {
m[t] <- c + theta*y[t-1]
eps[t] <- y[t] - m[t]
}
m[1] <- y[1] - eps[1]
eps[1] ~ dnorm(0, 0.0001)
theta ~ dnorm(0, 0.0001)
c ~ dnorm(0, 0.0001)
tau <- 1/pow(sigma, 2)
sigma ~ dunif(0, 100)
}
Inits:
list(c = 50, theta = 0, sigma = 10)
Data:
list(n = 100, y = c(100.8, 81.6, 66.5, 34.8, 30.6, 7, 19.8, 92.5,
154.4, 125.9, 84.8, 68.1, 38.5, 22.8, 10.2, 24.1, 82.9,
132, 130.9, 118.1, 89.9, 66.6, 60, 46.9, 41, 21.3, 16,
6.4, 4.1, 6.8, 14.5, 34, 45, 43.1, 47.5, 42.2, 28.1, 10.1,
8.1, 2.5, 0, 1.4, 5, 12.2, 13.9, 35.4, 45.8, 41.1, 30.4,
23.9, 15.7, 6.6, 4, 1.8, 8.5, 16.6, 36.3, 49.7, 62.5, 67,
71, 47.8, 27.5, 8.5, 13.2, 56.9, 121.5, 138.3, 103.2,
85.8, 63.2, 36.8, 24.2, 10.7, 15, 40.1, 61.5, 98.5, 124.3,
95.9, 66.5, 64.5, 54.2, 39, 20.6, 6.7, 4.3, 22.8, 54.8,
93.8, 95.7, 77.2, 59.1, 44, 47, 30.5, 16.3, 7.3, 37.3,
73.9))
node mean sd MC error 2.5% median 97.5% start sample
c 8.539 3.506 0.03699 1.436 8.567 15.35 501 9500
sigma 21.81 1.597 0.01637 18.97 21.72 25.26 501 9500
theta 0.8113 0.05886 5.718E-4 0.6976 0.8103 0.9295 501 9500
ARMA(2,1) model...
model {
# ARMA(2,1):
for (t in 1:n) {
y[t] ~ dnorm(m[t], tau)
yr[t] <- 1769 + t
}
for (t in 3:n) {
m[t] <- c + theta[1]*y[t-1] + theta[2]*y[t-2]
+ phi*eps[t-1]
eps[t] <- y[t] - m[t]
}
m[1] <- y[1] - eps[1]
m[2] <- y[2] - eps[2]
eps[1] ~ dnorm(0, 0.0001)
eps[2] ~ dnorm(0, 0.0001)
for (i in 1:2) {
theta[i] ~ dnorm(0, 0.0001)
}
phi ~ dnorm(0, 0.0001)
c ~ dnorm(0, 0.0001)
tau <- 1/pow(sigma, 2)
sigma ~ dunif(0, 100)
}
Inits:
list(c = 50, theta = c(0, 0), phi = 0, sigma = 10)
Data:
list(n = 100, y = c(100.8, 81.6, 66.5, 34.8, 30.6, 7, 19.8, 92.5,
154.4, 125.9, 84.8, 68.1, 38.5, 22.8, 10.2, 24.1, 82.9,
132, 130.9, 118.1, 89.9, 66.6, 60, 46.9, 41, 21.3, 16,
6.4, 4.1, 6.8, 14.5, 34, 45, 43.1, 47.5, 42.2, 28.1, 10.1,
8.1, 2.5, 0, 1.4, 5, 12.2, 13.9, 35.4, 45.8, 41.1, 30.4,
23.9, 15.7, 6.6, 4, 1.8, 8.5, 16.6, 36.3, 49.7, 62.5, 67,
71, 47.8, 27.5, 8.5, 13.2, 56.9, 121.5, 138.3, 103.2,
85.8, 63.2, 36.8, 24.2, 10.7, 15, 40.1, 61.5, 98.5, 124.3,
95.9, 66.5, 64.5, 54.2, 39, 20.6, 6.7, 4.3, 22.8, 54.8,
93.8, 95.7, 77.2, 59.1, 44, 47, 30.5, 16.3, 7.3, 37.3,
73.9))
node mean sd MC error 2.5% median 97.5% start sample
c 15.89 3.473 0.04195 9.321 15.85 22.83 4001 6000
phi 0.4013 0.1329 0.005149 0.1205 0.41 0.6383 4001 6000
sigma 15.11 1.135 0.01632 13.12 15.03 17.51 4001 6000
theta[1] 1.204 0.1189 0.003421 0.9601 1.207 1.429 4001 6000
theta[2] -0.5408 0.1139 0.003274 -0.7561 -0.5444 -0.3036 4001 6000