Lecture 10 – Seasonal ARIMA Models
Lecture 10 – Seasonal ARIMA Models
MAS 640 – Times Series Analysis and Forecasting
February 14, 2018
Seasonal ARIMA Models
I Stationarity
I Constant mean
I Constant variance
I No seasonality
Dealing with nonstationarity
I We have covered two ways for removing trends from a series
1. Model it and study the residuals
2. Difference it
I We have covered how to correct for nonconstant variance
I Transformation
I Box-Cox for guidance
I Seasonality is a common problem that needs to be handled as
well
Seasonality nonstationarity
Time
m
ilk
1994 1996 1998 2000 2002 2004 2006
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Seasonality causing nonstationarity
I 1st order differencing generally used for removing trends
I But seasonality persists. . .
Time
m
ilk
1994 1996 1998 2000 2002 2004 2006
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1
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5
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Seasonal Lag Differencing
I To remove seasonality (hopefully), we can try taking
seasonal-lagged differences
I Try both seasonal lag on the differences, and only a seasonal lag
I Remember, we are just looking for something that looks
stationary
I Seasonal-lag difference (with monthly data)
∇12Yt = Yt − Yt−12
Seasonal Differencing
Seasonal Difference
Time
m
ilk
1996 1998 2000 2002 2004 2006
−
5
0
0
5
0
1
0
0
Seasonal and First order Difference
Time
m
ilk
1996 1998 2000 2002 2004 2006
−
4
0
0
2
0
Guidance on Differencing
I Plot the ACF of the original data
I If it tails TOO slow at recent lags, need first order difference
I If it tails TOO slow at seasonal lags, need seasonal difference
Finding Seasonal Correlation
I As before, after converting our series to stationary we hope to
model that process with some ARMA model
I Unlike before, we can now incorporate seasonal ARMA terms
Detrending and Studying Residuals
Time
g
n
p
1950 1960 1970 1980 1990 2000
2
0
0
0
4
0
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0
6
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Detrending and Studying Residuals
0 50 100 150 200
−
4
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Index
fit
$
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si
d
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a
ls
5 10 15 20
−
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0
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.6
1
.0
Series fit$residuals
Lag
A
C
F
Seasonal Autocorrelations
Time
b
e
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rs
a
le
s
1975 1980 1985 1990
1
0
1
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1
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1
6
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Seasonal Autocorrelations
Time
b
e
e
rs
a
le
s
1975 1980 1985 1990
1
0
1
2
1
4
1
6
J
F
M
A
M
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A
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N
D
JFM
A
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A
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A
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OND
J
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M
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J
A
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N
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JF
M
A
M
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J
A
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ND
J
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A
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JJ
A
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Plot and ACF of Residuals
0 50 100 150
−
2
−
1
0
1
Index
fit
$
re
si
d
u
a
ls
5 10 15 20
−
0
.1
0
.1
0
.3
Series fit$residuals
Lag
A
C
F
Seasonal Autocorrelations
Estimate Std. Error
(Intercept) -71497.79 8791.41
t 71.96 8.87
I(t^2) -0.02 0.00
monthFebruary -0.16 0.21
monthMarch 2.05 0.21
monthApril 2.35 0.21
monthMay 3.54 0.21
monthJune 3.78 0.21
monthJuly 3.68 0.21
monthAugust 3.51 0.21
monthSeptember 1.46 0.21
monthOctober 1.13 0.21
monthNovember -0.19 0.21
monthDecember -0.58 0.21
Consider February. . .
From the regression results, the estimated equation for the month of
February is
Yt = (−71497.79− 0.16) + 71.96t − 0.02t2
Time
b
e
e
rs
a
le
s
1975 1980 1985 1990
1
0
1
2
1
4
1
6
Seasonal Autocorrelation
1975 1980 1985 1990
−
2
−
1
0
1
February Residuals Highlighted
Time
R
e
si
d
u
a
l
1975 1980 1985 1990
−
0
.5
0
.0
0
.5
Time
R
e
si
d
u
a
l
Modeling Process
1. Plot time series
I Observe trends, variance, seasonality, abrupt changes, outliers
2. Convert to stationary
2.1 Transform for constant variance if needed
I Box-Cox procedure for guidance
2.2 Remove trend if needed
I Differencing (lag 1, seasonal lag, both)
I Build model, use residuals
Modeling Process
3. Investigate autocorrelations via ACF/PACF plots
I Starting point for p, q, P, and Q
4. Fit model and diagnose
I Normality, independence
I After fitting time series model, no correlation should exist in
residuals
I Residual plots, ACF of residuals, Ljung-Box p-values
5. Overfit until you’re happy with model
6. Forecast
I Always provide interval with your estimate