&& 
&   
&
Many economic and business variables are affected by seasonal factors.
For example, power usage is highest in the months when temperatures are
most extreme. The most common type of seasonality is variation due to
the time of year, but other types of seasonality are also found in time
series data.
Seasonal models are often multiplicative rather than
additive. A multiplicative model includes the product of one or more
nonseasonal parameters with one or more seasonal parameters. For
example, a multiplicative model with both autoregressive and moving
average terms (an ARMA model) and with yearly seasonality for a time
series, yt, can be written as:
&

& 

&& 
&   

& where
& 

&& 
&   

&&  is the intercept parameter.
&   

&& 
&   

&&  is the nonseasonal first-order autoregressive parameter.
&   

&& 
&   

&&  is the seasonal autoregressive parameter.
&   

&& 
&   

&&  is the nonseasonal first-order moving average parameter.
&   

&& 
&   

&&  is the seasonal moving average parameter.
&   

& 

To identify a seasonal model, you need to examine the autocorrelation
function (ACF) and the inverse autocorrelation function (IACF) plots.
For multiplicative MA processes, there are small spikes in the ACF plot
q lags before and after the seasonal lag, where q
is the number of nonseasonal MA parameters necessary to model the data.
These small spikes are usually in the opposite direction of the
seasonal spike. For example, a multiplicative MA(1, 12) process
typically has small spikes at lags 11 and 13 on either side of, and in
the opposite direction of, a large spike at lag 12.

& 

&    An additive MA process typically has small spikes q lags before the seasonal lag, where
&    q is the number of nonseasonal MA parameters necessary to model the data. For example,
&    an additive MA(1, 12) process typically has a small spike at lag 11 and a larger spike at lag 12.
&

& 

To identify an AR process, look for the patterns described previously
in the IACF plot rather than in the ACF plot. If a process contains
both AR and MA components, the patterns may appear in both the ACF and
IACF plots.

& 

&    This example develops an ARMA model for steel shipments from U.S. steel mills.
&

& 

&    Analysis
&

The identification and estimation of Autoregressive Integrated Moving
Average (ARIMA) models is more of an art than a science. Generally, the
most parsimonious model fitting the data is considered the best. This
example uses steel shipments data taken from Metal Statistics 1993. The
values represent monthly totals of steel products shipped from U.S.
steel mills, in thousands of net tons, for the period from January 1984
to December 1991. The following statements create the data set STEEL.

&

   data steel;& input date:monyy5. steelshp @@;& format date monyy5.;& title 'U.S. Steel Shipments Data';& title2 '(thousands of net tons)';& datalines;   JAN84 5980 FEB84 6150 MAR84 7240 APR84 6472 MAY84 6948 JUN84 6686    JUL84 5820 AUG84 6033 SEP84 5454 OCT84 6087 NOV84 5317 DEC84 4867    ... more data lines ...   ;

& 
The analysis performed by the ARIMA procedure
is divided into three stages, corresponding to the stages described by
Box and Jenkins (1976). The IDENTIFY, ESTIMATE, and FORECAST statements
perform these three stages. In the identification stage, you use the
IDENTIFY statement to specify the response series and identify
candidate ARIMA models for it. The IDENTIFY statement reads time series
that are to be used in later statements, possibly differencing them,
and computes autocorrelations, inverse autocorrelations, partial
autocorrelations, and cross correlations. The analysis of this output
usually suggests one or more ARIMA models that could be fit. The VAR=
option specifies the variable to be identified.

   proc arima data=steel;& i var=steelshp;   run;

& 
&& 
&
&&
U.S. Steel Shipments Data
 

(thousands of net tons)
 

&&

&&

&&

&&
&&
The ARIMA Procedure
 

&&

&&

&&

&&
&&
Autocorrelations
 

Lag
 
Covariance
 
Correlation
 
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
 
Std Error
 

0
 
406442
 
1.00000
 
|&&|********************|
 
0
 

1
 
262630
 
0.64617
 
|&&.   |*************&  |
 
0.102062
 

2
 
261597
 
0.64363
 
|&   .&|*************&  |
 
0.138258
 

3
 
235909
 
0.58042
 
|&  .& |************&   |
 
0.166570
 

4
 
168515
 
0.41461
 
|&  .& |********& |
 
0.186451
 

5
 
201896
 
0.49674
 
|& .&  |**********& |
 
0.195820
 

6
 
129000
 
0.31739
 
|& .&  |****** .& |
 
0.208533
 

7
 
152701
 
0.37570
 
|&.&   |********.&|
 
0.213506
 

8
 
113117
 
0.27831
 
|&.&   |******  .&|
 
0.220285
 

9
 
127532
 
0.31378
 
|&.&   |******  .&|
 
0.223918
 

10
 
137000
 
0.33707
 
|&.&   |******* .&|
 
0.228452
 

11
 
130723
 
0.32163
 
|&.&   |******  .&|
 
0.233575
 

12
 
200408
 
0.49308
 
|& .&|**********& |
 
0.238144
 

13
 
112496
 
0.27678
 
|& .&|******   .& |
 
0.248551
 

14
 
135119
 
0.33244
 
|& .&|*******  .& |
 
0.251741
 

15
 
103295
 
0.25414
 
|& .&|*****    .& |
 
0.256273
 

16
 
62982.090
 
0.15496
 
|& .&|***& .& |
 
0.258885
 

17
 
108381
 
0.26666
 
|& .&|*****    .& |
 
0.259850
 

18
 
42836.479
 
0.10539
 
|&.& |**&   .&|
 
0.262685
 

19
 
65840.039
 
0.16199
 
|&.& |***&  .&|
 
0.263125
 

20
 
37765.859
 
0.09292
 
|&.& |**&   .&|
 
0.264162
 

21
 
27790.106
 
0.06837
 
|&.& |*&.&|
 
0.264502
 

22
 
40303.846
 
0.09916
 
|&.& |**&   .&|
 
0.264686
 

23
 
46097.710
 
0.11342
 
|&.& |**&   .&|
 
0.265073
 

24
 
76317.464
 
0.18777
 
|&.& |****& .&|
 
0.265578
 

&&

&&

&&

&&
&&
"." marks two standard errors
 

&&

&&

&&

&&  

The large spike at lag 12 in the ACF plot provides evidence that the
steel shipments time series has a seasonal autoregressive component.
The lack of a large spike at lag 24 indicates that the series is
stationary at the seasonal level.

&

& 
&& 
&
&&
U.S. Steel Shipments Data
 

(thousands of net tons)
 

&&

&&

&&

&&
&&
The ARIMA Procedure
 

&&

&&

&&

&&
&&
Inverse Autocorrelations
 

Lag
 
Correlation
 
-1 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 1
 

1
 
-0.37291
 
|&  *******|   .&&|
 

2
 
0.08136
 
|&&.   |** .&&|
 

3
 
-0.31032
 
|&   ******|   .&&|
 

4
 
0.16197
 
|&&.   |***.&&|
 

5
 
-0.20750
 
|&&****|   .&&|
 

6
 
0.16115
 
|&&.   |***.&&|
 

7
 
-0.02341
 
|&&.   |   .&&|
 

8
 
0.06910
 
|&&.   |*  .&&|
 

9
 
0.00628
 
|&&.   |   .&&|
 

10
 
0.02046
 
|&&.   |   .&&|
 

11
 
0.02875
 
|&&.   |*  .&&|
 

12
 
-0.23279
 
|&    *****|   .&&|
 

13
 
0.03755
 
|&&.   |*  .&&|
 

14
 
0.04050
 
|&&.   |*  .&&|
 

15
 
0.03498
 
|&&.   |*  .&&|
 

16
 
0.09969
 
|&&.   |** .&&|
 

17
 
-0.10703
 
|&&. **|   .&&|
 

18
 
0.04901
 
|&&.   |*  .&&|
 

19
 
-0.08634
 
|&&. **|   .&&|
 

20
 
0.02281
 
|&&.   |   .&&|
 

21
 
0.00844
 
|&&.   |   .&&|
 

22
 
0.10510
 
|&&.   |** .&&|
 

23
 
-0.10923
 
|&&. **|   .&&|
 

24
 
0.02676
 
|&&.   |*  .&&|
 

&&

&&  

The spikes at lags 1 and 3 in the IACF plot indicate that other
components are necessary to fit an adequate model. The null hypothesis
of white noise residuals is resoundingly rejected.

&

& 
&& 
&
&&
U.S. Steel Shipments Data
 

(thousands of net tons)
 

&&

&&

&&

&&
&&
The ARIMA Procedure
 

&&

&&

&&

&&
&&
Autocorrelation Check for White Noise
 

To Lag
 
Chi-Square
 
DF
 
Pr > ChiSq
 
Autocorrelations
 

6
 
170.51
 
6
 
<.0001
 
0.646
 
0.644
 
0.580
 
0.415
 
0.497
 
0.317
 

12
 
255.47
 
12
 
<.0001
 
0.376
 
0.278
 
0.314
 
0.337
 
0.322
 
0.493
 

18
 
296.96
 
18
 
<.0001
 
0.277
 
0.332
 
0.254
 
0.155
 
0.267
 
0.105
 

24
 
309.34
 
24
 
<.0001
 
0.162
 
0.093
 
0.068
 
0.099
 
0.113
 
0.188
 

&&

&&  

& In the estimation and diagnostic checking
stage, you use the ESTIMATE statement to specify the ARIMA model to fit
to the variable specified in the previous IDENTIFY statement and to
estimate the parameters of that model. The ESTIMATE statement also
produces diagnostic statistics to help you judge the adequacy of the
model.
Significance tests for parameter estimates indicate
whether some terms in the model may be unnecessary. Goodness-of-fit
statistics aid in comparing this model to others. Tests for white noise
residuals indicate whether the residual series contains additional
information that might be used by a more complex model. If the
diagnostic tests indicate problems with the model, you try another
model, then repeat the estimation and diagnostic checking stage.

& 
The following statement fits a
seasonal ARMA model to the time series. In the syntax of the ESTIMATE
statement, the two multiplicative AR terms, denoted by the P= option,
are enclosed in separate parentheses. The two additive MA terms,
denoted by the Q= option, are separated by a space within a single set
of parentheses.

& e p=(2)(12) q=(1 3);   run;

& 
&& 
&
&&
U.S. Steel Shipments Data
 

(thousands of net tons)
 

&&

&&

&&

&&
&&
The ARIMA Procedure
 

&&

&&

&&

&&
&&
Autocorrelation Check of Residuals
 

To Lag
 
Chi-Square
 
DF
 
Pr > ChiSq
 
Autocorrelations
 

6
 
2.42
 
2
 
0.2979
 
-0.009
 
-0.051
 
0.071
 
0.070
 
0.104
 
0.018
 

12
 
3.63
 
8
 
0.8891
 
-0.084
 
0.032
 
-0.024
 
0.013
 
-0.033
 
-0.035
 

18
 
11.86
 
14
 
0.6176
 
-0.082
 
0.168
 
0.014
 
-0.137
 
0.107
 
0.073
 

24
 
16.16
 
20
 
0.7066
 
0.023
 
0.019
 
-0.010
 
-0.047
 
0.174
 
-0.000
 

&&

&& 
&
&&
Model for variable steelshp
 

Estimated Mean
 
6057.122
 

&&

&& 
&
&&
Autoregressive Factors
 

Factor 1:
        
 
1 - 0.54234 B**(2)
 

Factor 2:
        
 
1 - 0.64802 B**(12)
 

&&

&& 
&
&&
Moving Average Factors
 

Factor 1:
        
 
1 + 0.55505 B**(1) + 0.43689 B**(3)
 

&&

&&  

The Autocorrelation Check of Residuals shows that none of the
Q-statistics are statistically significant. This indicates that the
model provides an adequate fit to the data.

&

& 
&& 
&
&&
U.S. Steel Shipments Data
 

(thousands of net tons)
 

&&

&&

&&

&&
&&
The ARIMA Procedure
 

&&

&&

&&

&&
&&
Conditional Least Squares Estimation
 

Parameter
 
Estimate
 
Standard Error
 
t Value
 
Approx

Pr > |t|
 
Lag
 

MU
 
6057.1
 
232.96713
 
26.00
 
<.0001
 
0
 

MA1,1
 
-0.55505
 
0.08021
 
-6.92
 
<.0001
 
1
 

MA1,2
 
-0.43689
 
0.07936
 
-5.51
 
<.0001
 
3
 

AR1,1
 
0.54234
 
0.09903
 
5.48
 
<.0001
 
2
 

AR2,1
 
0.64802
 
0.09392
 
6.90
 
<.0001
 
12
 

&&

&& 
&
&&
Constant Estimate
 
975.7391
 

Variance Estimate
 
126334.1
 

Std Error Estimate
 
355.4351
 

AIC
 
1404.983
 

SBC
 
1417.805
 

Number of Residuals
 
96
 

&&

&&  

& All of the estimated parameters have relatively large t-statistics, which indicates that
& these parameters cannot be omitted from the model.
& 

In the forecasting stage, you use the FORECAST statement to forecast
future values of the time series and to generate confidence intervals
for these forecasts from the ARIMA model produced by the preceding
ESTIMATE statement.

& 
The following statements produce
forecasts and upper and lower 95% confidence limits for 12 future
periods and creates the output data set STEEL2.

& f lead=12&   out=steel2&   id=date&   interval=month&   noprint;   run;

& 
To prepare the output data set for plotting,
change the values for the forecasts and confidence limits to missing
for all dates prior to the future forecast periods.

   data steel3;& set steel2;& if date lt '01jan92'd then do;&    forecast=.;&    l95=.;&    u95=.;& end;   run;

& 

&    Use the GPLOT procedure to plot the data.
&

   proc gplot data=steel3;& format date year4.;& plot steelshp*date=1&  forecast*date=2&  l95*date=3&  u95*date=3 / overlay cframe=ligrnbsp;  haxis=axis1 vaxis=axis2nbsp;  vminor=1 href='01jan92'd;& title 'U.S. Steel Shipments Data';& title2 '(thousands of net tons)';& axis1 offset=(1 cm)& label=('Year') minor=none& order=('01jan84'd to '01jan93'd by year);& axis2 label=(angle=90 'Steel Shipments')& order=(4500 to 8500 by 1000);& symbol1 c=blue  i=join l=1 v=star;& symbol2 c=red   i=join l=1 v=F;& symbol3 c=green i=join l=20;   run;   quit;

& 
&&
    
&&  

The values of the original steel shipments time series are plotted with
the star symbol. The forecasts are plotted with the F symbol, and the
upper and lower 95% confidence limits for the forecasts are plotted
with dashed lines.
Because the model fit to the steel shipments data
includes a seasonal component, the forecasts do not follow a simple
linear trend. Instead, the forecasts show variability due to the season
(month of the year).

[此贴子已经被angelboy于2008-6-5 10:32:01编辑过]