A
additive outliers
about 115
Danish fertility example 139–140
Danish hotel overnight stays by US citizens example 121–122
e-commerce example 117
aggregation of time series
about 19, 161
aggregation to monthly averages 168–173
aggregation to series observed every second day 179–181
aggregation to weekly averages 173–179
aggregation to yearly averages 161–168
analysis of daily series 181–185
TIMESERIES procedure supporting 19–22
AIC (Akaike information criterion) 77
Airline Model 80
Akaike information criterion (AIC) 77
ALPHA= option
FORECAST statement (ESM) 45
FORECAST statement (UCM) 138, 163
ANYDTDTE informat 12–13
ANYDTDTE20. informat 13, 16
ANYDTDTM informat 13
ANYDTTME informat 13
Arctic ice coverage example
about 185–186
aggregation to monthly averages 168–173
aggregation to series observed every second day 179–181
aggregation to weekly averages 173–179
aggregation to yearly averages 161–168
analysis of daily series 181–185
ARIMA (Box-Jenkins) models
about 79–82
seasonal adjustments and 116–118, 123–127
unobserved components models and 132, 134
ARIMA procedure
about 5
ARIMA (Box-Jenkins) models and 81
ARMA models and 172
ARIMA statement, X12 procedure 118
ARMA models
Arctic ice coverage example 172
Danish fertility example 82–85
Swiss Business Indicator example 90–95
autocorrelations 78, 81
AUTOMDL statement, X12 procedure 117–118, 123
AUTOREG procedure
about 5
ARIMA (Box-Jenkins) models and 81
ARMA models and 172
AUTOREG statement, UCM procedure 164
autoregressive models
about 76–77
Arctic ice coverage example 171–173
exponential smoothing and 75–76
fitting 77–78
B
BACK= option, ESM procedure 34, 66, 86–87
Box-Cox transformations 116
Box-Jenkins ARIMA models
about 79–82
seasonal adjustments and 116–118, 123–127
unobserved components models and 132, 134
C
Census Bureau X11 method
See X11 method
confidence limits
applying for forecasts 42–43
calculating for forecasts 42
Danish fertility example 43–46
continuous time series 4
CONVERT statement, EXPAND procedure
METHOD= option 23
TRANSFORMOUT= option 25
CYCLE statement, UCM procedure 145, 176–177
cycles, trend and business 101–103, 124–125
D
damped-trend method 49–51
Danish fertility example
component estimation 137–139
confidence limits for forecasts 43–46
estimating smoothing parameter with ESM procedure 47–49
forecast errors 38–40
forecasting by exponential smoothing 33–38
forecasting using damped-trend method 50–51
forecasting with fitted ARMA models 82–85
model extensions 142–148
outlier detection 139–142
UCM procedure and 137–148
Danish hotel overnight stays by US citizens example
exponential smoothing and 57–61
seasonal adjustment and 121–128
data sets, datetime variables and 16–17
date formats in procedure calls 10–11
DATEPART function 11, 14
DATESTYLE system option 13, 16
datetime variables
about 9
calculating time span between dates 17
extracting data from date variable example 15
handling 14–16
importing 12–14
output formats 9–12
datetime variables (continued)
plotting quarterly data by year example 14–15
time series data sets and 16–17
DATETIME20. format 11
DATETIME20.3 format 12
deleting outliers 185
DEPLAG statement, UCM procedure
about 177
ARMA models and 172
LAG= option 146–147
DIF function 16–17, 20
DIF= option, MODEL statement (VARMAX) 84
DIF12 function 16–17
discrete time series 4
dollar sign ($) 9
double exponential smoothing
about 30–33
Danish fertility example 36, 45
forecasting variance for 42
E
e-commerce example
ESM procedure supporting 63–74
estimation of components 149–155
model fit 158–160
regression components 155–158
seasonal adjustment in 103–108, 116–120
UCM procedure and 149–160
Winters method 63–68
X11 method and 103–108
X12 procedure and 116–120
ESM procedure
See also FORECAST statement, ESM procedure
about 5
BACK= option 34, 66, 86–87
damped-trend method 50–51
Danish fertility example 33–36, 43–53
Danish hotel overnight stays by US citizens example 59–63
e-commerce example 63–74
estimating smoothing parameter 47–49
ID statement 64
LEAD= option 65
OUTFOR= option 33–34
PLOT= option 44
PRINT= option 33–34, 47, 51, 66, 87
Swiss Business Indicator example 86–90
transformation facilities in 70–74
UCM procedure and 147, 160
Winters method 63–68
X12 procedure and 116, 128
ESTIMATE statement, UCM procedure
about 146, 158, 164
EXTRADIFFUSE= option 170
PLOT= option 139, 141, 146, 158, 161, 167
EURDFDD10. format 11
EURDFMY5. format 10
EURDFMY7. format 10
EXPAND procedure
about 5, 23
Arctic ice coverage example 181–182, 185
CONVERT statement 23, 25
time series aggregation example 22
TO= option 23
exponential smoothing
about 29–30, 53, 55–56
additive smoothing 61–63
applying confidence limits for forecasts 42–43
basic formula 40–41
calculating confidence limits for forecasts 42
damped-trend method 49–53
Danish fertility example 33–40, 44
Danish hotel overnight stays by US citizens example 57–61
determining smoothing constant 46–47
double 30–33, 36, 42
ESM procedure supporting 61–63, 67–74
estimating smoothing parameter in ESM procedure 47–49
Holt method 49–50
moving averages and 40–41
nonseasonal 29–53
parameterized models versus 75–95
relative importance of e-commerce example
67–74
seasonal 55–74
US retail e-commerce example 63–67
Winters method 55–56, 63–68
EXTRADIFFUSE= option, ESTIMATE statement (UCM) 170
F
fertility example
See Danish fertility example
FIRSTOBS= option, PRINT procedure 12
FORCE= option, X11 statement (X12) 123
forecast errors
See prediction errors
FORECAST statement, ESM procedure
ALPHA= option 45
MEDIAN option 72
METHOD= option 50–51, 59, 61, 64, 87
MODEL= option 35, 44
TRANSFORM= option 74
FORECAST statement, UCM procedure
about 138, 163–164
ALPHA= option 138, 163
forecasts
applying confidence limits for 42–43
calculating confidence limits for 42
confidence limits for Danish fertility example
43–46
exponential smoothing of nonseasonal series
29–53
exponential smoothing of seasonal series 55–74
exponential smoothing versus parameterized models 75–95
X12 procedure and 115–116
formats, including in procedure calls 10–12
G
Gaussian distribution
Danish fertility example 39
unobserved components models 135
H
Henderson moving average 124
Holt exponential smoothing method 49–50
I
ID statement
ESM procedure 64
TIMESERIES procedure 168, 179
UCM procedure 137, 161, 168, 181
VARMAX procedure 82
importing datetime variables 12–14
INTCK function 17
interpolation of time series 23–26
INTERVAL= option
ID statement (ESM) 64
ID statement (TIMESERIES) 168, 179
ID statement (UCM) 137, 168, 181
ID statement (VARMAX) 82
IRREGULAR statement, UCM procedure 172, 182
K
Kalman filter 134–135
KEEPH= option, SEASON statement (UCM) 175
L
LAG= option, DEPLAG statement (UCM) 146–147
LAG12 function 16–17
LEAD= option
ESM procedure 65
OUTPUT statement (VARMAX) 91
LEVEL statement, UCM procedure
about 149, 157
PLOT= option 138, 142
PRINT= option 142
likelihood function 81, 135
likelihood ratio testing 81
Ljung-Box test 118
M
maximum likelihood estimation 81
MEDIAN option, FORECAST statement (ESM) 72
METHOD= option
CONVERT statement (EXPAND) 23
FORECAST statement (ESM) 50–51, 59, 61, 64, 87
MODEL statement (VARMAX) 91
MINIC= option, MODEL statement (VARMAX) 91
missing values
autoregressive models and 76
EXPAND procedure and 23
TIMESERIES procedure and 21–22, 181–182
UCM procedure and 185
MMDDYY6. format 10
MMDDYY10. format 10
MODE= option, X11 statement (X12) 121
model fitting
ARMA models 82–85
autoregressive models 77–78
X12 procedure and 115–116
MODEL= option, FORECAST statement (ESM) 35, 44
MODEL statement, UCM procedure 137, 156
MODEL statement, VARMAX procedure
about 82
DIF= option 84
METHOD= option 91
MINIC= option 91
NOINT option 84
NSEASON= option 94
PRINT= option 94
TREND= option 94
monthly averages, aggregation to 168–173
MONYY5. format 9–10
MONYY7. format 9–10
moving averages 40–41, 124
N
NLDATE12. format 11
NOEST option
SEASON statement (UCM) 168
SLOPE statement (UCM) 143, 151–153, 162
NOINT option, MODEL statement (VARMAX) 84
nonseasonal exponential smoothing
about 29–30, 53
applying confidence limits for forecasts 42–43
basic formula 40–41
calculating confidence limits for forecasts 42
damped-trend method 49–53
Danish fertility example 33–40, 44
determining smoothing constant 46–47
double 30–33, 36, 42
estimating smoothing parameter in ESM procedure 47–49
Holt method 49–50
moving averages and 40–41
NOPRINT option, OUTPUT statement (VARMAX) 91
normal distribution, in exponential smoothing 42
NSEASON= option, MODEL statement (VARMAX) 94
O
OBS= option, PRINT procedure 12
ODS Graphics System 91, 104
ODS OUTPUT statement 120
OECD (Organization for Economic Co-operation and Development) 86
Organization for Economic Co-operation and Development (OECD) 86
OUT= option, OUTPUT statement (VARMAX) 91
OUTFOR= option, ESM procedure 33–34
OUTLIER statement
UCM procedure 137, 139
X12 procedure 121–122
outliers
additive 115, 117, 121–122, 139–140
Arctic ice coverage example 185–186
deleting 185
exponential smoothing of 43, 57, 61
UCM procedure and 139–142
X11 method and 103–104, 111, 113, 115
X12 procedure and 115–118, 121–122
output formats for datetime variables 9–12
OUTPUT statement
VARMAX procedure 82, 91
X12 procedure 104–108, 110–113, 118–120,
124–128
P
parameterized models versus exponential smoothing 75–95
percent sign (%) 9
period (.) 9
PLOT= option
ESM procedure 44
ESTIMATE statement (UCM) 139, 141, 146, 158, 161, 167
LEVEL statement (UCM) 138, 142
RANDOMREG statement (UCM) 157
SEASON statement (UCM) 154
SLOPE statement (UCM) 142
VARMAX procedure 83
PLOTS= option, UCM procedure 169, 182
prediction errors
Danish fertility example 34, 39–40, 52
Danish hotel overnight stays by US citizens example 61
e-commerce example 66, 69, 72, 74
exponential smoothing and 38–39
Swiss Business Indicator 89–90
PRINT= option
ESM procedure 33–34, 47, 51, 66, 87
LEVEL statement (UCM) 142
MODEL statement (VARMAX) 94
SEASON statement (UCM) 169
PRINT procedure
FIRSTOBS= option 12
OBS= option 12
procedure calls, including formats in 10–12
R
RANDOMREG statement, UCM procedure
about 156–157
PLOT= option 157
regression coefficient 155–158
REGRESSION statement, X12 procedure 116, 122
S
SAS/ETS software 5
SBC (Schwarz’s Bayesian Criterion) 77, 91
Schwarz’s Bayesian Criterion (SBC) 77, 91
SEASON statement, UCM procedure
about 149, 157, 174, 177
KEEPH= option 175
NOEST option 168
PLOT= option 154
PRINT= option 169
VAR= option 168
seasonal adjustments
unobserved components models and 133
X11 method for 99–113
X12 procedure and 115–128
seasonal exponential smoothing
about 55–56
additive smoothing 61–63
Danish hotel overnight stays by US citizens example 57–61
e-commerce examples 63–74
ESM procedure supporting 61–63, 67–74
Winters method 55–56, 63–68
seasonality 99–101
SET statement 120
SETMISSING= option, TIMESERIES procedure 21
SGPLOT procedure
Danish fertility example 33–34, 39–40
Danish hotel overnight stays by US citizens example 57–58, 124–128
e-commerce example 104–105, 118–120
time series aggregation example 20–21
time series interpolation example 24
SLOPE statement, UCM procedure
about 149, 157
NOEST option 143, 151–153, 162
PLOT= option 142
VAR= option 143, 151–153, 162
smoothing constant 46–47
smoothing parameter, ESM procedure 47–49
SORT procedure 16–17
state space models 135
STATESPACE procedure 135
Statistical Graphics facility 43–44
STEPwise AutoRegression (STEPAR) method 78, 90
Student’s t-distribution 39, 42
Swiss Business Indicator example
ESM procedure and 86–90
VARMAX procedure and 90–95
T
time formats in procedure calls 12
time series
additional information 6
aggregation of 19–22, 161–186
datetime variables 9–18
examples of 3–4
interpolation of 23–26
SAS procedures supporting 5–6
trends in 101–103, 124–125
types of 4–5
TIMESERIES procedure
about 5, 161
aggregation example 19–22, 161–186
ID statement 168, 179
SETMISSING= option 21
WHERE statement 161
TO= option, EXPAND procedure 23
TOD12. format 11
TRANSFORM= option, FORECAST statement (ESM) 74
TRANSFORM statement, X12 procedure 116
TRANSFORMOUT= option, CONVERT statement (EXPAND) 25
TREND= option, MODEL statement (VARMAX) 94
TRENDMA= option, X11 statement (X12) 124
trends in time series 101–103, 124–125
TYPE= option, OUTLIER statement (X12) 121
U
UCM procedure
See also ESTIMATE statement, UCM procedure
See also SEASON statement, UCM procedure
See also SLOPE statement, UCM procedure
about 5, 25
Arctic ice coverage example 161–185
AUTOREG statement 164
CYCLE statement 145, 176–177
Danish fertility example 137–148
DEPLAG statement 146–147, 172, 177
e-commerce example 108, 149–160
FORECAST statement 138, 163–164
ID statement 137, 161, 168, 181
IRREGULAR statement 172, 182
LEVEL statement 138, 142, 149, 157
MODEL statement 137, 156
OUTLIER statement 137, 139
PLOTS= option 169, 182
RANDOMREG statement 156–157
state space models and 135
TIMESERIES procedure and 161
unobserved components models and 131
unemployment in UK example 108–113
UNIVARIATE procedure 137
unobserved components models
about 131–132
Arctic ice coverage example 161–186
ARIMA representation 132
Danish fertility example 137–148
e-commerce example 149–160
estimation of 134–135
extensions of 132–134
state space models and 135
V
VAR= option
SEASON statement (UCM) 168
SLOPE statement (UCM) 143, 151–153, 162
VARMAX procedure
about 5–6
ARIMA (Box-Jenkins) models and 81–82
ARMA models and 172
Danish fertility example 82–85
fitted ARMA models and 82–85
ID statement 82
MODEL statement 82, 84, 91, 94
OUTPUT statement 82, 91
PLOT= option 83
Swiss Business Indicator example 90–95
X12 procedure and 116
W
weekly averages, aggregation to 173–179
WHERE= option, SET statement 120
WHERE statement, TIMESERIES procedure 161
Winters method
Danish hotel overnight stays by US citizens example 62–63
e-commerce examples 63–68
for seasonal forecasting 56–57
transformation and 70
X
X11-ARIMA method, X12 procedure 101, 116–117
X11 method
about 101–103
e-commerce example 103–108
“forgetting” algorithm 113
outliers and 115
unemployment in UK example 108–113
X11 procedure 101, 104
X11 statement, X12 procedure 123
MODE= option 121
TRENDMA= option 124
X12 procedure
about 5, 115–116
ARIMA (Box-Jenkins) models and 116
ARIMA statement 118
AUTOMDL statement 117–118, 123
X12 procedure (continued)
Danish hotel overnight stays by US citizens example 121–128
e-commerce example 104, 108, 116–120
model fitting and forecasting 115–116
OUTLIER statement 121–122
OUTPUT statement 104–108, 110–113, 118–120, 124–128
REGRESSION statement 116, 122
seasonal adjustment and 116–128
TRANSFORM statement 116
UCM procedure and 160
unemployment in UK example 110–113
X11-ARIMA method 101, 116–117
X11 statement 121, 123–124
Y
yearly averages, aggregation to 161–168
Symbols
. (period) 9
% (percent sign) 9
$ (dollar sign) 9