59
Forecasting using 10. Seasonal ARIMA models OTexts.com/fpp/8/9 Forecasting using R 1 Rob J Hyndman
Forecasting using
10. Seasonal ARIMA models
OTexts.com/fpp/8/9
Forecasting using R 1
Rob J Hyndman
Outline
1 Backshift notation
2 Seasonal ARIMA models
3 Example 1: European quarterly retail trade
4 Example 2: Australian cortecosteroid drugsales
5 ARIMA vs ETS
Forecasting using R Backshift notation 2
Backshift notationA very useful notational device is the backwardshift operator, B, which is used as follows:
Byt = yt−1 .
In other words, B, operating on yt, has the effect ofshifting the data back one period. Twoapplications of B to yt shifts the data back twoperiods:
B(Byt) = B2yt = yt−2 .
For monthly data, if we wish to shift attention to“the same month last year,” then B12 is used, andthe notation is B12yt = yt−12.
Forecasting using R Backshift notation 3
Backshift notationA very useful notational device is the backwardshift operator, B, which is used as follows:
Byt = yt−1 .
In other words, B, operating on yt, has the effect ofshifting the data back one period. Twoapplications of B to yt shifts the data back twoperiods:
B(Byt) = B2yt = yt−2 .
For monthly data, if we wish to shift attention to“the same month last year,” then B12 is used, andthe notation is B12yt = yt−12.
Forecasting using R Backshift notation 3
Backshift notationA very useful notational device is the backwardshift operator, B, which is used as follows:
Byt = yt−1 .
In other words, B, operating on yt, has the effect ofshifting the data back one period. Twoapplications of B to yt shifts the data back twoperiods:
B(Byt) = B2yt = yt−2 .
For monthly data, if we wish to shift attention to“the same month last year,” then B12 is used, andthe notation is B12yt = yt−12.
Forecasting using R Backshift notation 3
Backshift notationA very useful notational device is the backwardshift operator, B, which is used as follows:
Byt = yt−1 .
In other words, B, operating on yt, has the effect ofshifting the data back one period. Twoapplications of B to yt shifts the data back twoperiods:
B(Byt) = B2yt = yt−2 .
For monthly data, if we wish to shift attention to“the same month last year,” then B12 is used, andthe notation is B12yt = yt−12.
Forecasting using R Backshift notation 3
Backshift notation
First difference: 1− B.
Double difference: (1− B)2.
dth-order difference: (1− B)dyt.
Seasonal difference: 1− Bm.
Seasonal difference followed by a firstdifference: (1− B)(1− Bm).
Multiply terms together together to see thecombined effect:
(1− B)(1− Bm)yt = (1− B− Bm + Bm+1)yt= yt − yt−1 − yt−m + yt−m−1.
Forecasting using R Backshift notation 4
Backshift notation
First difference: 1− B.
Double difference: (1− B)2.
dth-order difference: (1− B)dyt.
Seasonal difference: 1− Bm.
Seasonal difference followed by a firstdifference: (1− B)(1− Bm).
Multiply terms together together to see thecombined effect:
(1− B)(1− Bm)yt = (1− B− Bm + Bm+1)yt= yt − yt−1 − yt−m + yt−m−1.
Forecasting using R Backshift notation 4
Backshift notation
First difference: 1− B.
Double difference: (1− B)2.
dth-order difference: (1− B)dyt.
Seasonal difference: 1− Bm.
Seasonal difference followed by a firstdifference: (1− B)(1− Bm).
Multiply terms together together to see thecombined effect:
(1− B)(1− Bm)yt = (1− B− Bm + Bm+1)yt= yt − yt−1 − yt−m + yt−m−1.
Forecasting using R Backshift notation 4
Backshift notation
First difference: 1− B.
Double difference: (1− B)2.
dth-order difference: (1− B)dyt.
Seasonal difference: 1− Bm.
Seasonal difference followed by a firstdifference: (1− B)(1− Bm).
Multiply terms together together to see thecombined effect:
(1− B)(1− Bm)yt = (1− B− Bm + Bm+1)yt= yt − yt−1 − yt−m + yt−m−1.
Forecasting using R Backshift notation 4
Backshift notation
First difference: 1− B.
Double difference: (1− B)2.
dth-order difference: (1− B)dyt.
Seasonal difference: 1− Bm.
Seasonal difference followed by a firstdifference: (1− B)(1− Bm).
Multiply terms together together to see thecombined effect:
(1− B)(1− Bm)yt = (1− B− Bm + Bm+1)yt= yt − yt−1 − yt−m + yt−m−1.
Forecasting using R Backshift notation 4
Backshift notation
First difference: 1− B.
Double difference: (1− B)2.
dth-order difference: (1− B)dyt.
Seasonal difference: 1− Bm.
Seasonal difference followed by a firstdifference: (1− B)(1− Bm).
Multiply terms together together to see thecombined effect:
(1− B)(1− Bm)yt = (1− B− Bm + Bm+1)yt= yt − yt−1 − yt−m + yt−m−1.
Forecasting using R Backshift notation 4
Backshift notation for ARIMA
ARMA model:yt = c + φ1yt−1 + · · ·+ φpyt−p + et + θ1et−1 + · · ·+ θqet−q
= c + φ1Byt + · · ·+ φpBpyt + et + θ1Bet + · · ·+ θqB
qet
φ(B)yt = c + θ(B)et
where φ(B) = 1− φ1B− · · · − φpBp
and θ(B) = 1 + θ1B + · · ·+ θqBq.
ARIMA(1,1,1) model:
(1− φ1B) (1− B)yt = c + (1 + θ1B)et
Forecasting using R Backshift notation 5
Backshift notation for ARIMA
ARMA model:yt = c + φ1yt−1 + · · ·+ φpyt−p + et + θ1et−1 + · · ·+ θqet−q
= c + φ1Byt + · · ·+ φpBpyt + et + θ1Bet + · · ·+ θqB
qet
φ(B)yt = c + θ(B)et
where φ(B) = 1− φ1B− · · · − φpBp
and θ(B) = 1 + θ1B + · · ·+ θqBq.
ARIMA(1,1,1) model:
(1− φ1B) (1− B)yt = c + (1 + θ1B)et
Forecasting using R Backshift notation 5
Backshift notation for ARIMA
ARMA model:yt = c + φ1yt−1 + · · ·+ φpyt−p + et + θ1et−1 + · · ·+ θqet−q
= c + φ1Byt + · · ·+ φpBpyt + et + θ1Bet + · · ·+ θqB
qet
φ(B)yt = c + θ(B)et
where φ(B) = 1− φ1B− · · · − φpBp
and θ(B) = 1 + θ1B + · · ·+ θqBq.
ARIMA(1,1,1) model:
(1− φ1B) (1− B)yt = c + (1 + θ1B)et↑
Firstdifference
Forecasting using R Backshift notation 5
Backshift notation for ARIMA
ARMA model:yt = c + φ1yt−1 + · · ·+ φpyt−p + et + θ1et−1 + · · ·+ θqet−q
= c + φ1Byt + · · ·+ φpBpyt + et + θ1Bet + · · ·+ θqB
qet
φ(B)yt = c + θ(B)et
where φ(B) = 1− φ1B− · · · − φpBp
and θ(B) = 1 + θ1B + · · ·+ θqBq.
ARIMA(1,1,1) model:
(1− φ1B) (1− B)yt = c + (1 + θ1B)et↑
AR(1)
Forecasting using R Backshift notation 5
Backshift notation for ARIMA
ARMA model:yt = c + φ1yt−1 + · · ·+ φpyt−p + et + θ1et−1 + · · ·+ θqet−q
= c + φ1Byt + · · ·+ φpBpyt + et + θ1Bet + · · ·+ θqB
qet
φ(B)yt = c + θ(B)et
where φ(B) = 1− φ1B− · · · − φpBp
and θ(B) = 1 + θ1B + · · ·+ θqBq.
ARIMA(1,1,1) model:
(1− φ1B) (1− B)yt = c + (1 + θ1B)et↑
MA(1)
Forecasting using R Backshift notation 5
Outline
1 Backshift notation
2 Seasonal ARIMA models
3 Example 1: European quarterly retail trade
4 Example 2: Australian cortecosteroid drugsales
5 ARIMA vs ETS
Forecasting using R Seasonal ARIMA models 6
Seasonal ARIMA models
ARIMA (p,d,q) (P,D,Q)m
where m = number of periods per season.
Forecasting using R Seasonal ARIMA models 7
Seasonal ARIMA models
ARIMA (p,d,q)︸ ︷︷ ︸ (P,D,Q)m
↑ Non-seasonalpart of themodel
where m = number of periods per season.
Forecasting using R Seasonal ARIMA models 7
Seasonal ARIMA models
ARIMA (p,d,q) (P,D,Q)m︸ ︷︷ ︸↑ Seasonal
part ofthemodel
where m = number of periods per season.
Forecasting using R Seasonal ARIMA models 7
Seasonal ARIMA modelsE.g., ARIMA(1,1,1)(1,1,1)4 model (without constant)
(1− φ1B)(1−Φ1B4)(1− B)(1− B4)yt = (1 + θ1B)(1 + Θ1B
4)et.
Forecasting using R Seasonal ARIMA models 8
Seasonal ARIMA modelsE.g., ARIMA(1,1,1)(1,1,1)4 model (without constant)
(1− φ1B)(1−Φ1B4)(1− B)(1− B4)yt = (1 + θ1B)(1 + Θ1B
4)et.
Forecasting using R Seasonal ARIMA models 8
Seasonal ARIMA modelsE.g., ARIMA(1,1,1)(1,1,1)4 model (without constant)
(1− φ1B)(1−Φ1B4)(1− B)(1− B4)yt = (1 + θ1B)(1 + Θ1B
4)et.
6
(Seasonaldifference
)
Forecasting using R Seasonal ARIMA models 8
Seasonal ARIMA modelsE.g., ARIMA(1,1,1)(1,1,1)4 model (without constant)
(1− φ1B)(1−Φ1B4)(1− B)(1− B4)yt = (1 + θ1B)(1 + Θ1B
4)et.
6(Non-seasonal
difference
)
Forecasting using R Seasonal ARIMA models 8
Seasonal ARIMA modelsE.g., ARIMA(1,1,1)(1,1,1)4 model (without constant)
(1− φ1B)(1−Φ1B4)(1− B)(1− B4)yt = (1 + θ1B)(1 + Θ1B
4)et.
6
(Seasonal
AR(1)
)
Forecasting using R Seasonal ARIMA models 8
Seasonal ARIMA modelsE.g., ARIMA(1,1,1)(1,1,1)4 model (without constant)
(1− φ1B)(1−Φ1B4)(1− B)(1− B4)yt = (1 + θ1B)(1 + Θ1B
4)et.
6(Non-seasonal
AR(1)
)
Forecasting using R Seasonal ARIMA models 8
Seasonal ARIMA modelsE.g., ARIMA(1,1,1)(1,1,1)4 model (without constant)
(1− φ1B)(1−Φ1B4)(1− B)(1− B4)yt = (1 + θ1B)(1 + Θ1B
4)et.
6
(Seasonal
MA(1)
)
Forecasting using R Seasonal ARIMA models 8
Seasonal ARIMA modelsE.g., ARIMA(1,1,1)(1,1,1)4 model (without constant)
(1− φ1B)(1−Φ1B4)(1− B)(1− B4)yt = (1 + θ1B)(1 + Θ1B
4)et.
6(Non-seasonal
MA(1)
)
Forecasting using R Seasonal ARIMA models 8
Outline
1 Backshift notation
2 Seasonal ARIMA models
3 Example 1: European quarterly retail trade
4 Example 2: Australian cortecosteroid drugsales
5 ARIMA vs ETS
Forecasting using R Example 1: European quarterly retail trade 9
European quarterly retail trade
Forecasting using R Example 1: European quarterly retail trade 10
Year
Ret
ail i
ndex
2000 2005 2010
9092
9496
9810
010
2
European quarterly retail trade
Forecasting using R Example 1: European quarterly retail trade 10
Year
Ret
ail i
ndex
2000 2005 2010
9092
9496
9810
010
2
> plot(euretail)
European quarterly retail trade
> auto.arima(euretail)ARIMA(1,1,1)(0,1,1)[4]
Coefficients:ar1 ma1 sma1
0.8828 -0.5208 -0.9704s.e. 0.1424 0.1755 0.6792
sigma^2 estimated as 0.1411: log likelihood=-30.19AIC=68.37 AICc=69.11 BIC=76.68
Forecasting using R Example 1: European quarterly retail trade 11
European quarterly retail trade
> auto.arima(euretail, stepwise=FALSE,approximation=FALSE)
ARIMA(0,1,3)(0,1,1)[4]
Coefficients:ma1 ma2 ma3 sma1
0.2625 0.3697 0.4194 -0.6615s.e. 0.1239 0.1260 0.1296 0.1555
sigma^2 estimated as 0.1451: log likelihood=-28.7AIC=67.4 AICc=68.53 BIC=77.78
Forecasting using R Example 1: European quarterly retail trade 12
European quarterly retail trade
Forecasting using R Example 1: European quarterly retail trade 13
Forecasts from ARIMA(0,1,3)(0,1,1)[4]
2000 2005 2010 2015
9095
100
Outline
1 Backshift notation
2 Seasonal ARIMA models
3 Example 1: European quarterly retail trade
4 Example 2: Australian cortecosteroid drugsales
5 ARIMA vs ETS
Forecasting using RExample 2: Australian cortecosteroid drug
sales 14
Cortecosteroid drug sales
Forecasting using RExample 2: Australian cortecosteroid drug
sales 15
Year
H02
sal
es (
mill
ion
scrip
ts)
1995 2000 2005
0.4
0.6
0.8
1.0
1.2
Year
Log
H02
sal
es
1995 2000 2005
−1.
0−
0.6
−0.
20.
2
Cortecosteroid drug sales
> fit <- auto.arima(h02, lambda=0)> fitARIMA(2,1,3)(0,1,1)[12]Box Cox transformation: lambda= 0
Coefficients:ar1 ar2 ma1 ma2 ma3 sma1
-1.0194 -0.8351 0.1717 0.2578 -0.4206 -0.6528s.e. 0.1648 0.1203 0.2079 0.1177 0.1060 0.0657
sigma^2 estimated as 0.004071: log likelihood=250.8AIC=-487.6 AICc=-486.99 BIC=-464.83
Forecasting using RExample 2: Australian cortecosteroid drug
sales 16
Cortecosteroid drug sales
Forecasting using RExample 2: Australian cortecosteroid drug
sales 17
residuals(fit)
1995 2000 2005
−0.
2−
0.1
0.0
0.1
0.2
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0 5 10 15 20 25 30 35
−0.
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00.
2
Lag
AC
F
0 5 10 15 20 25 30 35
−0.
20.
00.
2
Lag
PAC
F
Cortecosteroid drug sales
Training: July 91 – June 06
Test: July 06 – June 08
Forecasting using RExample 2: Australian cortecosteroid drug
sales 18
Model RMSE
ARIMA(3,0,0)(2,1,0)12 0.0661ARIMA(3,0,1)(2,1,0)12 0.0646ARIMA(3,0,2)(2,1,0)12 0.0645ARIMA(3,0,1)(1,1,0)12 0.0679ARIMA(3,0,1)(0,1,1)12 0.0644ARIMA(3,0,1)(0,1,2)12 0.0622ARIMA(3,0,1)(1,1,1)12 0.0630ARIMA(4,0,3)(0,1,1)12 0.0648ARIMA(3,0,3)(0,1,1)12 0.0640ARIMA(4,0,2)(0,1,1)12 0.0648ARIMA(3,0,2)(0,1,1)12 0.0644ARIMA(2,1,3)(0,1,1)12 0.0634ARIMA(2,1,4)(0,1,1)12 0.0632ARIMA(2,1,5)(0,1,1)12 0.0640
Cortecosteroid drug sales
Training: July 91 – June 06
Test: July 06 – June 08
Forecasting using RExample 2: Australian cortecosteroid drug
sales 18
Model RMSE
ARIMA(3,0,0)(2,1,0)12 0.0661ARIMA(3,0,1)(2,1,0)12 0.0646ARIMA(3,0,2)(2,1,0)12 0.0645ARIMA(3,0,1)(1,1,0)12 0.0679ARIMA(3,0,1)(0,1,1)12 0.0644ARIMA(3,0,1)(0,1,2)12 0.0622ARIMA(3,0,1)(1,1,1)12 0.0630ARIMA(4,0,3)(0,1,1)12 0.0648ARIMA(3,0,3)(0,1,1)12 0.0640ARIMA(4,0,2)(0,1,1)12 0.0648ARIMA(3,0,2)(0,1,1)12 0.0644ARIMA(2,1,3)(0,1,1)12 0.0634ARIMA(2,1,4)(0,1,1)12 0.0632ARIMA(2,1,5)(0,1,1)12 0.0640
Cortecosteroid drug sales
getrmse <- function(x,h,...){
train.end <- time(x)[length(x)-h]test.start <- time(x)[length(x)-h+1]train <- window(x,end=train.end)test <- window(x,start=test.start)fit <- Arima(train,...)fc <- forecast(fit,h=h)return(accuracy(fc,test)["RMSE"])
}
Forecasting using RExample 2: Australian cortecosteroid drug
sales 19
Cortecosteroid drug sales
getrmse(h02,h=24,order=c(3,0,0),seasonal=c(2,1,0),lambda=0)getrmse(h02,h=24,order=c(3,0,1),seasonal=c(2,1,0),lambda=0)getrmse(h02,h=24,order=c(3,0,2),seasonal=c(2,1,0),lambda=0)getrmse(h02,h=24,order=c(3,0,1),seasonal=c(1,1,0),lambda=0)getrmse(h02,h=24,order=c(3,0,1),seasonal=c(0,1,1),lambda=0)getrmse(h02,h=24,order=c(3,0,1),seasonal=c(0,1,2),lambda=0)getrmse(h02,h=24,order=c(3,0,1),seasonal=c(1,1,1),lambda=0)getrmse(h02,h=24,order=c(4,0,3),seasonal=c(0,1,1),lambda=0)getrmse(h02,h=24,order=c(3,0,3),seasonal=c(0,1,1),lambda=0)getrmse(h02,h=24,order=c(4,0,2),seasonal=c(0,1,1),lambda=0)getrmse(h02,h=24,order=c(3,0,2),seasonal=c(0,1,1),lambda=0)getrmse(h02,h=24,order=c(2,1,3),seasonal=c(0,1,1),lambda=0)getrmse(h02,h=24,order=c(2,1,4),seasonal=c(0,1,1),lambda=0)getrmse(h02,h=24,order=c(2,1,5),seasonal=c(0,1,1),lambda=0)
Forecasting using RExample 2: Australian cortecosteroid drug
sales 20
Cortecosteroid drug sales
Models with lowest AICc values tend to giveslightly better results than the other models.
AICc comparisons must have the same ordersof differencing. But RMSE test set comparisonscan involve any models.
No model passes all the residual tests.
Use the best model available, even if it doesnot pass all tests.
Forecasting using RExample 2: Australian cortecosteroid drug
sales 21
Cortecosteroid drug sales
Models with lowest AICc values tend to giveslightly better results than the other models.
AICc comparisons must have the same ordersof differencing. But RMSE test set comparisonscan involve any models.
No model passes all the residual tests.
Use the best model available, even if it doesnot pass all tests.
Forecasting using RExample 2: Australian cortecosteroid drug
sales 21
Cortecosteroid drug sales
Models with lowest AICc values tend to giveslightly better results than the other models.
AICc comparisons must have the same ordersof differencing. But RMSE test set comparisonscan involve any models.
No model passes all the residual tests.
Use the best model available, even if it doesnot pass all tests.
Forecasting using RExample 2: Australian cortecosteroid drug
sales 21
Cortecosteroid drug sales
Models with lowest AICc values tend to giveslightly better results than the other models.
AICc comparisons must have the same ordersof differencing. But RMSE test set comparisonscan involve any models.
No model passes all the residual tests.
Use the best model available, even if it doesnot pass all tests.
Forecasting using RExample 2: Australian cortecosteroid drug
sales 21
Cortecosteroid drug sales
Forecasting using RExample 2: Australian cortecosteroid drug
sales 22
Forecasts from ARIMA(3,0,1)(0,1,2)[12]
Year
H02
sal
es (
mill
ion
scrip
ts)
1995 2000 2005 2010
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Outline
1 Backshift notation
2 Seasonal ARIMA models
3 Example 1: European quarterly retail trade
4 Example 2: Australian cortecosteroid drugsales
5 ARIMA vs ETS
Forecasting using R ARIMA vs ETS 23
ARIMA vs ETS
Myth that ARIMA models are more general thanexponential smoothing.
Linear exponential smoothing models allspecial cases of ARIMA models.
Non-linear exponential smoothing models haveno equivalent ARIMA counterparts.
Many ARIMA models have no exponentialsmoothing counterparts.
ETS models all non-stationary. Models withseasonality or non-damped trend (or both)have two unit roots; all other models have oneunit root.
Forecasting using R ARIMA vs ETS 24
ARIMA vs ETS
Myth that ARIMA models are more general thanexponential smoothing.
Linear exponential smoothing models allspecial cases of ARIMA models.
Non-linear exponential smoothing models haveno equivalent ARIMA counterparts.
Many ARIMA models have no exponentialsmoothing counterparts.
ETS models all non-stationary. Models withseasonality or non-damped trend (or both)have two unit roots; all other models have oneunit root.
Forecasting using R ARIMA vs ETS 24
ARIMA vs ETS
Myth that ARIMA models are more general thanexponential smoothing.
Linear exponential smoothing models allspecial cases of ARIMA models.
Non-linear exponential smoothing models haveno equivalent ARIMA counterparts.
Many ARIMA models have no exponentialsmoothing counterparts.
ETS models all non-stationary. Models withseasonality or non-damped trend (or both)have two unit roots; all other models have oneunit root.
Forecasting using R ARIMA vs ETS 24
ARIMA vs ETS
Myth that ARIMA models are more general thanexponential smoothing.
Linear exponential smoothing models allspecial cases of ARIMA models.
Non-linear exponential smoothing models haveno equivalent ARIMA counterparts.
Many ARIMA models have no exponentialsmoothing counterparts.
ETS models all non-stationary. Models withseasonality or non-damped trend (or both)have two unit roots; all other models have oneunit root.
Forecasting using R ARIMA vs ETS 24
ARIMA vs ETS
Myth that ARIMA models are more general thanexponential smoothing.
Linear exponential smoothing models allspecial cases of ARIMA models.
Non-linear exponential smoothing models haveno equivalent ARIMA counterparts.
Many ARIMA models have no exponentialsmoothing counterparts.
ETS models all non-stationary. Models withseasonality or non-damped trend (or both)have two unit roots; all other models have oneunit root.
Forecasting using R ARIMA vs ETS 24
EquivalencesSimple exponential smoothing
Forecasts equivalent to ARIMA(0,1,1).Parameters: θ1 = α− 1.
Holt’s method
Forecasts equivalent to ARIMA(0,2,2).Parameters: θ1 = α + β − 2 and θ2 = 1− α.
Damped Holt’s method
Forecasts equivalent to ARIMA(1,1,2).Parameters: φ1 = φ, θ1 = α + φβ − 2, θ2 = (1− α)φ.
Holt-Winters’ additive method
Forecasts equivalent to ARIMA(0,1,m+1)(0,1,0)m.Parameter restrictions because ARIMA has m + 1parameters whereas HW uses only three parameters.
Holt-Winters’ multiplicative method
No ARIMA equivalenceForecasting using R ARIMA vs ETS 25
EquivalencesSimple exponential smoothing
Forecasts equivalent to ARIMA(0,1,1).Parameters: θ1 = α− 1.
Holt’s method
Forecasts equivalent to ARIMA(0,2,2).Parameters: θ1 = α + β − 2 and θ2 = 1− α.
Damped Holt’s method
Forecasts equivalent to ARIMA(1,1,2).Parameters: φ1 = φ, θ1 = α + φβ − 2, θ2 = (1− α)φ.
Holt-Winters’ additive method
Forecasts equivalent to ARIMA(0,1,m+1)(0,1,0)m.Parameter restrictions because ARIMA has m + 1parameters whereas HW uses only three parameters.
Holt-Winters’ multiplicative method
No ARIMA equivalenceForecasting using R ARIMA vs ETS 25
EquivalencesSimple exponential smoothing
Forecasts equivalent to ARIMA(0,1,1).Parameters: θ1 = α− 1.
Holt’s method
Forecasts equivalent to ARIMA(0,2,2).Parameters: θ1 = α + β − 2 and θ2 = 1− α.
Damped Holt’s method
Forecasts equivalent to ARIMA(1,1,2).Parameters: φ1 = φ, θ1 = α + φβ − 2, θ2 = (1− α)φ.
Holt-Winters’ additive method
Forecasts equivalent to ARIMA(0,1,m+1)(0,1,0)m.Parameter restrictions because ARIMA has m + 1parameters whereas HW uses only three parameters.
Holt-Winters’ multiplicative method
No ARIMA equivalenceForecasting using R ARIMA vs ETS 25
EquivalencesSimple exponential smoothing
Forecasts equivalent to ARIMA(0,1,1).Parameters: θ1 = α− 1.
Holt’s method
Forecasts equivalent to ARIMA(0,2,2).Parameters: θ1 = α + β − 2 and θ2 = 1− α.
Damped Holt’s method
Forecasts equivalent to ARIMA(1,1,2).Parameters: φ1 = φ, θ1 = α + φβ − 2, θ2 = (1− α)φ.
Holt-Winters’ additive method
Forecasts equivalent to ARIMA(0,1,m+1)(0,1,0)m.Parameter restrictions because ARIMA has m + 1parameters whereas HW uses only three parameters.
Holt-Winters’ multiplicative method
No ARIMA equivalenceForecasting using R ARIMA vs ETS 25
EquivalencesSimple exponential smoothing
Forecasts equivalent to ARIMA(0,1,1).Parameters: θ1 = α− 1.
Holt’s method
Forecasts equivalent to ARIMA(0,2,2).Parameters: θ1 = α + β − 2 and θ2 = 1− α.
Damped Holt’s method
Forecasts equivalent to ARIMA(1,1,2).Parameters: φ1 = φ, θ1 = α + φβ − 2, θ2 = (1− α)φ.
Holt-Winters’ additive method
Forecasts equivalent to ARIMA(0,1,m+1)(0,1,0)m.Parameter restrictions because ARIMA has m + 1parameters whereas HW uses only three parameters.
Holt-Winters’ multiplicative method
No ARIMA equivalenceForecasting using R ARIMA vs ETS 25
