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ARIMA MODELING WITH R
ARIMA - Integrated ARMA
ARIMA Modeling with R
Identifying ARIMA● A time series exhibits ARIMA behavior if the differenced
data has ARMA behavior> # Simulation ARIMA(p = 1, d = 1, q = 0)
> x <- arima.sim(list(order = c(1, 1, 0), ar = .9), n = 200) > plot(x, main = "ARIMA(p = 1, d = 1, q = 0)") > plot(diff(x), main = "ARMA(p = 1, d = 0, q = 0)")
ARMA(p = 1, d = 0, q = 0)
Time
diff(x)
0 50 100 150 200
−4−2
02
4ARIMA(p = 1, d = 1, q = 0)
Time
x
0 50 100 150 200
−100
−50
0
ARIMA Modeling with R
Identifying ARIMA● A time series exhibits ARIMA behavior if the differenced
data has ARMA behavior> # Simulation ARIMA(p = 1, d = 1, q = 0)
> x <- arima.sim(list(order = c(1, 1, 0), ar = .9), n = 200) > plot(x, main = "ARIMA(p = 1, d = 1, q = 0)") > plot(diff(x), main = "ARMA(p = 1, d = 0, q = 0)")
ARIMA(p = 1, d = 1, q = 0)
Time
x
0 50 100 150 200
−200
−100
0
ARMA(p = 1, d = 0, q = 0)
Time
diff(x)
0 50 100 150 200
−6−4
−20
24
6
ARIMA Modeling with R
Identifying ARIMA● A time series exhibits ARIMA behavior if the differenced
data has ARMA behavior> # Simulation ARIMA(p = 1, d = 1, q = 0)
> x <- arima.sim(list(order = c(1, 1, 0), ar = .9), n = 200) > plot(x, main = "ARIMA(p = 1, d = 1, q = 0)") > plot(diff(x), main = "ARMA(p = 1, d = 0, q = 0)")
ARIMA(p = 1, d = 1, q = 0)
Time
x
0 50 100 150 200
−200
−100
−50
0
ARMA(p = 1, d = 0, q = 0)
Time
diff(x)
0 50 100 150 200
−6−4
−20
24
6
ARIMA Modeling with R
ACF and PCF of an Integrated ARMA> x <- arima.sim(list(order = c(1, 1, 0), ar = .9), n = 200) > acf2(x)
5 10 15 20 25
0.0
0.4
0.8
Series: x
LAG
ACF
5 10 15 20 25
0.0
0.4
0.8
LAG
PACF
ARIMA Modeling with R
ACF and PCF of a Differenced ARIMA> x <- arima.sim(list(order = c(1, 1, 0), ar = .9), n = 200) > acf2(diff(x))
5 10 15 20 25
−0.2
0.2
0.6
1.0
Series: diff(x)
LAG
ACF
5 10 15 20 25
−0.2
0.2
0.6
1.0
LAG
PACF
Indicates an AR(1) for the differenced data
ARIMA Modeling with R
Weekly Oil PricesTails
Tails
● Looks like ARIMA(1, 1, 1)
Random Walk
Stationary
ARIMA MODELING WITH R
Let’s practice!
ARIMA MODELING WITH R
ARIMA Diagnostics
ARIMA Modeling with R
Weekly Oil Prices ARIMA(1, 1, 1)?
ARIMA Modeling with R
Weekly Oil Prices ARIMA(1, 1, 1)?> oil <- window(oil, end = 2006) > x <- sarima(oil, p = 1, d = 1, q = 1) > x$ttable
Estimate SE t.value p.value ar1 -0.4987 0.0995 -5.0131 0.0000 ma1 0.7316 0.0734 9.9732 0.0000 constant 0.1091 0.0936 1.1664 0.2443
ARIMA Modeling with R
Weekly Oil Prices ARIMA(1, 1, 1)!Standardized Residuals
Time
2000 2001 2002 2003 2004 2005 2006
−4−2
01
2
0.0 0.1 0.2 0.3 0.4 0.5
−0.2
0.0
0.2
0.4
0.6
ACF of Residuals
LAG
ACF
−3 −2 −1 0 1 2 3
−4−2
01
2
Normal Q−Q Plot of Std Residuals
Theoretical Quantiles
Sam
ple
Qua
ntile
s
5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
p values for Ljung−Box statistic
lag
p va
lue
ARIMA Modeling with R
Overfit: ARIMA(2, 1, 1) and ARIMA(1, 1, 2)> oil_fit1 <- sarima(oil, p = 2, d = 1, q = 1) > oil_fit1$ttable
Estimate SE t.value p.value ar1 -0.4704 0.1117 -4.2121 0.0000 ar2 -0.0738 0.0652 -1.1319 0.2586 ma1 0.6771 0.0986 6.8696 0.0000 constant 0.1088 0.0878 1.2391 0.2163
> oil_fit2 <- sarima(oil, p = 1, d = 1, q = 2) > oil_fit2$ttable
Estimate SE t.value p.value ar1 -0.3664 0.1816 -2.0178 0.0445 ma1 0.5777 0.1818 3.1777 0.0016 ma2 -0.0836 0.0837 -0.9989 0.3186 constant 0.1088 0.0884 1.2306 0.2194
ARIMA MODELING WITH R
Let’s practice!
ARIMA MODELING WITH R
Forecasting ARIMA
ARIMA Modeling with R
Forecasting ARIMA Processes● The model describes how the dynamics of the time
series behave over time
● Forecasting simply continues the model dynamics into the future
● Use sarima.for() to forecast in the astsa-package
ARIMA Modeling with R
Forecasting ARIMA Processes> oil <- window(astsa::oil, end = 2006) > oilf <- window(astsa::oil, end = 2007) > sarima.for(oil, n.ahead = 52, 1, 1, 1) > lines(oilf)
Time
oil
2004.0 2004.5 2005.0 2005.5 2006.0 2006.5 2007.0
4050
6070
8090
ARIMA MODELING WITH R
Let’s practice!
