Taller 1 - MMF
Michele Roldan - Isabella Hernandez
21/8/2022
Ejercicio 1.3
\[x_{t} = -.9x_{t-2}+w_{t}\]
w = rnorm(100,0,1)
x = filter(w, filter=c(0,-.9), method="recursive")[-(1:50)]
# Moving average filter
v = filter(x,rep(1/4, 4), sides = 1)
par(mfrow=c(1,1))
plot(x, xlab = "Time", ylab = "Series", type="l")
lines(v, pch = 18, col = "red", type = "l")
lines(w, pch = 18, col = "green", type = "l")
legend("bottomleft", legend = c("Autoregression", "Moving average","White noise"),
col = c("black", "red","green"), lty = 1:2, cex = 0.7)
\[ x_{t} = cos(\frac{2t\pi}{4})\]
x <- c(cos(2*pi*1:100/4))
# Moving average filter
v = filter(x, rep(1/4, 4), sides = 1)
# Plot
par(mfrow=c(1,1))
plot(x, type="l")
lines(v, pch = 18, col = "red", type = "l")
legend("bottomleft", legend = c("Serie", "Moving average"),
col = c("black", "red"), lty = 1:2, cex = 0.7)
\[ x_{t} = cos(\frac{2t\pi}{4}) + w_t\]
w = rnorm(100,0,1)
x <- c(cos(2*pi*1:100/4)) + w
# Moving average filter
v = filter(x, rep(1/4, 4), sides = 1)
# Plot
par(mfrow=c(1,1))
plot(x, type="l")
lines(v, pch = 18, col = "red", type = "l")
legend("bottomleft", legend = c("Serie", "Moving average"),
col = c("black", "red"), lty = 1:2, cex = 0.7)
Ejercicio 1.7
\[ x_{t} = w_{t-1} + 2w_{t} + w_{t+1} \]
Plot the ACF as a function of \(h\).
w = rnorm(500) #generate random
x = filter(w, filter=c(1,2,1)) [2:499]
print(acf(x, type="correlation"))
##
## Autocorrelations of series 'x', by lag
##
## 0 1 2 3 4 5 6 7 8 9 10
## 1.000 0.648 0.125 -0.025 0.010 0.010 -0.006 0.019 0.073 0.090 0.059
## 11 12 13 14 15 16 17 18 19 20 21
## 0.038 0.026 0.014 0.056 0.123 0.137 0.126 0.122 0.095 0.056 0.046
## 22 23 24 25 26
## 0.044 0.010 -0.018 -0.011 -0.019Ejercicio 4
\[ X_{t} = Z_{t} + \theta Z_{t-2} \]
par(mfrow = c(2,1))
plot(arima.sim(list(order=c(0,0,2), ma=c(.1,.8)), n=100), ylab="x",
main=(expression(MA(2)~~~theta==+.8)))
plot(arima.sim(list(order=c(0,0,2), ma=c(.1,.8)), n=100), ylab="x",
main=(expression(MA(2)~~~theta==-.8)))