Chapter 12 The FORECAST Procedure

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1 Chapter 12 The FORECAST Procedure Chapter Table of Contents OVERVIEW GETTING STARTED Introduction to Forecasting Methods SYNTAX FunctionalSummary PROCFORECASTStatement BYStatement IDStatement VARStatement DETAILS MissingValues DataPeriodicityandTimeIntervals Forecasting Methods Specifying Seasonality DataRequirements OUT=DataSet OUTEST= Data Set EXAMPLES Example 12.1 Forecasting Auto Sales Example 12.2 Forecasting Retail Sales Example 12.3 Forecasting Petroleum Sales REFERENCES

2 Part 2. General Information 578

3 Chapter 12 The FORECAST Procedure Overview The FORECAST procedure provides a quick and automatic way to generate forecasts for many time series in one step. The procedure can forecast hundreds of series at a time, with the series organized into separate variables or across BY groups. PROC FORECAST uses extrapolative forecasting methods where the forecasts for a series are functions only of time and past values of the series, not of other variables. You can use the following forecasting methods. For each of these methods, you can specify linear, quadratic, or no trend. The stepwise autoregressive method is used by default. This method combines time trend regression with an autoregressive model and uses a stepwise method to select the lags to use for the autoregressive process. The exponential smoothing method produces a time trend forecast, but in fitting the trend, the parameters are allowed to change gradually over time, and earlier observations are given exponentially declining weights. Single, double, and triple exponential smoothing are supported, depending on whether no trend, linear trend, or quadratic trend is specified. Holt two-parameter linear exponential smoothing is supported as a special case of the Holt-Winters method without seasons. The Winters method (also called Holt-Winters) combines a time trend with multiplicative seasonal factors to account for regular seasonal fluctuations in a series. Like the exponential smoothing method, the Winters method allows the parameters to change gradually over time, with earlier observations given exponentially declining weights. You can also specify the additive version of the Winters method, which uses additive instead of multiplicative seasonal factors. When seasonal factors are omitted, the Winters method reduces to the Holt two-parameter version of double exponential smoothing. The FORECAST procedure writes the forecasts and confidence limits to an output data set, and can write parameter estimates and fit statistics to an output data set. The FORECAST procedure does not produce printed output. PROC FORECAST is an extrapolation procedure useful for producing practical results efficiently. However, in the interest of speed, PROC FORECAST uses some shortcuts that cause some statistical results (such as confidence limits) to be only approximate. For many time series, the FORECAST procedure, with appropriately chosen methods and weights, can yield satisfactory results. Other SAS/ETS procedures can produce better forecasts but at greater computational expense. 579

4 Part 2. General Information You can perform the stepwise autoregressive forecasting method with the AUTOREG procedure. You can perform exponential smoothing with statistically optimal weights as an ARIMA model using the ARIMA procedure. Seasonal ARIMA models can be used for forecasting seasonal series for which the Winters and additive Winters methods might be used. Additionally, the Time Series Forecasting System can be used to develop forecasting models, estimate the model parameters, evaluate the models ability to forecast and display the results graphically. See Chapter 23, Getting Started with Time Series Forecasting, for more details. 580

5 Chapter 12. Getting Started Getting Started To use PROC FORECAST, specify the input and output data sets and the number of periods to forecast in the PROC FORECAST statement, then list the variables to forecast in a VAR statement. For example, suppose you have monthly data on the sales of some product, in a data set, named PAST, as shown in Figure 12.1, and you want to forecast sales for the next 10 months. Obs date sales 1 JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL Figure Example Data Set PAST The following statements forecast 10 observations for the variable SALES using the default STEPAR method and write the results to the output data set PRED: proc forecast data=past lead=10 out=pred; var sales; run; The following statements use the PRINT procedure to print the data set PRED: proc print data=pred; run; The PROC PRINT listing of the forecast data set PRED is shown in Figure

6 Part 2. General Information Obs _TYPE LEAD_ sales 1 FORECAST FORECAST FORECAST FORECAST FORECAST FORECAST FORECAST FORECAST FORECAST FORECAST Figure Forecast Data Set PRED Giving Dates to Forecast Values Normally, your input data set has an ID variable that gives dates to the observations, and you want the forecast observations to have dates also. Usually, the ID variable has SAS date values. (See Chapter 2, Working with Time Series Data, for information on using SAS date values.) The ID statement specifies the identifying variable. If the ID variable contains SAS date values, the INTERVAL= option should be used on the PROC FORECAST statement to specify the time interval between observations. (See Chapter 3, Date Intervals, Formats, and Functions, for more information on time intervals.) The FORECAST procedure uses the INTERVAL= option to generate correct dates for forecast observations. The data set PAST, shown in Figure 12.1, has monthly observations and contains an ID variable DATE with SAS date values identifying each observation. The following statements produce the same forecast as the preceding example and also include the ID variable DATE in the output data set. Monthly SAS date values are extrapolated for the forecast observations. proc forecast data=past interval=month lead=10 out=pred; var sales; id date; run; Computing Confidence Limits Depending on the output options specified, multiple observations are written to the OUT= data set for each time period. The different parts of the results are contained in the VAR statement variables in observations identified by the character variable TYPE and by the ID variable. For example, the following statements use the OUTLIMIT option to write forecasts and 95% confidence limits for the variable SALES to the output data set PRED. This data set is printed with the PRINT procedure. proc forecast data=past interval=month lead=10 out=pred outlimit; var sales; id date; run; 582

7 Chapter 12. Getting Started proc print data=pred; run; The output data set PRED is shown in Figure Obs date _TYPE LEAD_ sales 1 AUG91 FORECAST AUG91 L AUG91 U SEP91 FORECAST SEP91 L SEP91 U OCT91 FORECAST OCT91 L OCT91 U NOV91 FORECAST NOV91 L NOV91 U DEC91 FORECAST DEC91 L DEC91 U JAN92 FORECAST JAN92 L JAN92 U FEB92 FORECAST FEB92 L FEB92 U MAR92 FORECAST MAR92 L MAR92 U APR92 FORECAST APR92 L APR92 U MAY92 FORECAST MAY92 L MAY92 U Figure Output Data Set Form of the OUT= Data Set The OUT= data set PRED, shown in Figure 12.3, contains three observations for each of the 10 forecast periods. Each of these three observations has the same value of the ID variable DATE, the SAS date value for the month and year of the forecast. The three observations for each forecast period have different values of the variable TYPE. For the TYPE =FORECAST observation, the value of the variable SALES is the forecast value for the period indicated by the DATE value. For the TYPE =L95 observation, the value of the variable SALES is the lower limit of the 95% confidence interval for the forecast. For the TYPE =U95 observation, the value of the variable SALES is the upper limit of the 95% confidence interval. You can control the types of observations written to the OUT= data set with the PROC FORECAST statement options OUTLIMIT, OUTRESID, OUTACTUAL, OUT1STEP, OUTSTD, OUTFULL, and OUTALL. For example, the OUTFULL option outputs the confidence limit values, the one-step-ahead predictions, and the actual data, in addition to the forecast values. See the sections "Syntax" and "OUT= Data Set" later in this chapter for more information. 583

8 Part 2. General Information Plotting Forecasts The forecasts, confidence limits, and actual values can be plotted on the same graph with the GPLOT procedure. Use the appropriate output control options on the PROC FORECAST statement to include in the OUT= data set the series you want to plot. Use the TYPE variable in the GPLOT procedure PLOT statement to separate the observations for the different plots. In this example, the OUTFULL option is used, and the resulting output data set contains the actual and predicted values, as well as the upper and lower 95 proc forecast data=past interval=month lead=10 out=pred outfull; id date; var sales; run; proc gplot data=pred; plot sales * date = _type_ / haxis= 1jan90 d to 1jan93 d by qtr href= 15jul91 d; symbol1 i=none v=star; /* for _type_=actual */ symbol2 i=spline v=circle; /* for _type_=forecast */ symbol3 i=spline l=3; /* for _type_=l95 */ symbol4 i=spline l=3; /* for _type_=u95 */ where date >= 1jan90 d; run; The TYPE variable is used in the GPLOT procedure s PLOT statement to make separate plots over time for each type of value. A reference line marks the start of the forecast period. (Refer to SAS/GRAPH Software: Reference, Volume 2, Version 7, First Edition for more information on using PROC GPLOT.) The WHERE statement restricts the range of the actual data shown in the plot. In this example, the variable SALES has monthly data from July 1989 through July 1991, but only the data for 1990 and 1991 are shown in the plot. The plot is shown in Figure

9 Chapter 12. Getting Started Figure Plot of Forecast with Confidence Limits Plotting Residuals You can plot the residuals from the forecasting model using PROC GPLOT and a WHERE statement. 1. Use the OUTRESID option or the OUTALL option in the PROC FORECAST statement to include the residuals in the output data set. 2. Use a WHERE statement to specify the observation type of RESIDUAL in the PROC GPLOT code. The following example adds the OUTRESID option to the preceding example and plots the residuals: proc forecast data=past interval=month lead=10 out=pred outfull outresid; id date; var sales; run; proc gplot data=pred; where _type_= RESIDUAL ; plot sales * date / haxis= 1jan89 d to 1oct91 d by qtr; symbol1 i=circle; run; The plot of residuals is shown in Figure

10 Part 2. General Information Figure Plot of Residuals Model Parameters and Goodness-of-Fit Statistics You can write the parameters of the forecasting models used, as well as statistics measuring how well the forecasting models fit the data, to an output SAS data set using the OUTEST= option. The options OUTFITSTATS, OUTESTTHEIL, and OUT- ESTALL control what goodness-of-fit statistics are added to the OUTEST= data set. For example, the following statements add the OUTEST= and OUTFITSTATS options to the previous example to create the output statistics data set EST for the results of the default stepwise autoregressive forecasting method: proc forecast data=past interval=month lead=10 out=pred outfull outresid outest=est outfitstats; id date; var sales; run; proc print data=est; run; The PRINT procedure prints the OUTEST= data set, as shown in Figure

11 Chapter 12. Getting Started Obs _TYPE_ date sales 1 N JUL NRESID JUL DF JUL SIGMA JUL CONSTANT JUL LINEAR JUL AR1 JUL AR2 JUL91. 9 AR3 JUL AR4 JUL AR5 JUL AR6 JUL AR7 JUL AR8 JUL SST JUL SSE JUL MSE JUL RMSE JUL MAPE JUL MPE JUL MAE JUL ME JUL MAXE JUL MINE JUL MAXPE JUL MINPE JUL RSQUARE JUL ADJRSQ JUL RW_RSQ JUL ARSQ JUL APC JUL AIC JUL SBC JUL CORR JUL Figure The OUTEST= Data Set for STEPAR Method In the OUTEST= data set, the DATE variable contains the ID value of the last observation in the data set used to fit the forecasting model. The variable SALES contains the statistic indicated by the value of the TYPE variable. The TYPE =N, NRESID, and DF observations contain, respectively, the number of observations read from the data set, the number of nonmissing residuals used to compute the goodnessof-fit statistics, and the number of nonmissing observations minus the number of parameters used in the forecasting model. The observation having TYPE =SIGMA contains the estimate of the standard deviation of the one-step prediction error computed from the residuals. The TYPE =CONSTANT and TYPE =LINEAR contain the coefficients of the time trend regression. The TYPE =AR1, AR2,..., AR8 observations contain the estimated autoregressive parameters. A missing autoregressive parameter indicates that the autoregressive term at that lag was not included in the model by the stepwise model selection method. (See the section "STEPAR Method" later in this chapter for more information.) The other observations in the OUTEST= data set contain various goodness-of-fit statistics that measure how well the forecasting model used fits the given data. See "OUTEST= Data Set" later in this chapter for details. 587

12 Part 2. General Information Controlling the Forecasting Method The METHOD= option controls which forecasting method is used. The TREND= option controls the degree of the time trend model used. For example, the following statements produce forecasts of SALES as in the preceding example but use the double exponential smoothing method instead of the default STEPAR method: proc forecast data=past interval=month lead=10 method=expo trend=2 out=pred outfull outresid outest=est outfitstats; var sales; id date; run; proc print data=est; run; The PRINT procedure prints the OUTEST= data set for the EXPO method, as shown in Figure Obs _TYPE_ date sales 1 N JUL NRESID JUL DF JUL WEIGHT JUL S1 JUL S2 JUL SIGMA JUL CONSTANT JUL LINEAR JUL SST JUL SSE JUL MSE JUL RMSE JUL MAPE JUL MPE JUL MAE JUL ME JUL MAXE JUL MINE JUL MAXPE JUL MINPE JUL RSQUARE JUL ADJRSQ JUL RW_RSQ JUL ARSQ JUL APC JUL AIC JUL SBC JUL CORR JUL Figure The OUTEST= Data Set for METHOD=EXPO See the "Syntax" section later in this chapter for other options that control the forecasting method. See "Introduction to Forecasting Methods" and "Forecasting Methods" later in this chapter for an explanation of the different forecasting methods. 588

13 Chapter 12. Getting Started Introduction to Forecasting Methods This section briefly introduces the forecasting methods used by the FORECAST procedure. Refer to textbooks on forecasting and see "Forecasting Methods" later in this chapter for more detailed discussions of forecasting methods. The FORECAST procedure combines three basic models to fit time series: time trend models for long-term, deterministic change autoregressive models for short-term fluctuations seasonal models for regular seasonal fluctuations Two approaches to time series modeling and forecasting are time trend models and time series methods. Time Trend Models Time trend models assume that there is some permanent deterministic pattern across time. These models are best suited to data that are not dominated by random fluctuations. Examining a graphical plot of the time series you want to forecast is often very useful in choosing an appropriate model. The simplest case of a time trend model is one in which you assume the series is a constant plus purely random fluctuations that are independent from one time period to the next. Figure 12.8 shows how such a time series might look. Figure Time Series without Trend 589

14 Part 2. General Information The x t values are generated according to the equation x t = b 0 + t where t is an independent, zero-mean, random error, and b 0 is the true series mean. Suppose that the series exhibits growth over time, as shown in Figure Figure Time Series with Linear Trend A linear model is appropriate for this data. For the linear model, assume the x t values are generated according to the equation x t = b 0 + b 1 t + t The linear model has two parameters. The predicted values for the future are the points on the estimated line. The extension of the polynomial model to three parameters is the quadratic (which forms a parabola). This allows for a constantly changing slope, where the x t values are generated according to the equation x t = b 0 + b 1 t + b 2 t 2 + t PROC FORECAST can fit three types of time trend models: constant, linear, and quadratic. For other kinds of trend models, other SAS procedures can be used. Exponential smoothing fits a time trend model using a smoothing scheme in which the weights decline geometrically as you go backward in time. The forecasts from 590

15 Chapter 12. Getting Started exponential smoothing are a time trend, but the trend is based mostly on the recent observations instead of on all the observations equally. How well exponential smoothing works as a forecasting method depends on choosing a good smoothing weight for the series. To specify the exponential smoothing method, use the METHOD=EXPO option. Single exponential smoothing produces forecasts with a constant trend (that is, no trend). Double exponential smoothing produces forecasts with a linear trend, and triple exponential smoothing produces a quadratic trend. Use the TREND= option with the METHOD=EXPO option to select single, double, or triple exponential smoothing. The time trend model can be modified to account for regular seasonal fluctuations of the series about the trend. To capture seasonality, the trend model includes a seasonal parameter for each season. Seasonal models can be additive or multiplicative. x t = b 0 + b 1 t + s(t)+ t (Additive) x t =(b 0 + b 1 t)s(t) + t (Multiplicative) where s(t) is the seasonal parameter for the season corresponding to time t. The Winters method is similar to exponential smoothing, but includes seasonal factors. The Winters method can use either additive or multiplicative seasonal factors. Like exponential smoothing, good results with the Winters method depend on choosing good smoothing weights for the series to be forecast. To specify the multiplicative or additive versions of the Winters method, use the METHOD=WINTERS or METHOD=ADDWINTERS options, respectively. To specify seasonal factors to include in the model, use the SEASONS= option. Many observed time series do not behave like constant, linear, or quadratic time trends. However, you can partially compensate for the inadequacies of the trend models by fitting time series models to the departures from the time trend, as described in the following sections. Time Series Methods Time series models assume the future value of a variable to be a linear function of past values. If the model is a function of past values for a finite number of periods, it is an autoregressive model and is written as follows: x t = a 0 + a 1 x t,1 + a 2 x t,2 + :::+ a p x t,p + t The coefficients a i are autoregressive parameters. One of the simplest cases of this model is the random walk, where the series dances around in purely random jumps. This is illustrated in Figure

16 Part 2. General Information Figure Random Walk Series The x t values are generated by the equation x t = x t,1 + t In this type of model, the best forecast of a future value is the present value. However, with other autoregressive models, the best forecast is a weighted sum of recent values. Pure autoregressive forecasts always damp down to a constant (assuming the process is stationary). Autoregressive time series models can also be used to predict seasonal fluctuations. Combining Time Trend with Autoregressive Models Trend models are suitable for capturing long-term behavior, whereas autoregressive models are more appropriate for capturing short-term fluctuations. One approach to forecasting is to combine a deterministic time trend model with an autoregressive model. The stepwise autoregressive method (STEPAR method) combines a time-trend regression with an autoregressive model for departures from trend. The combined time-trend and autoregressive model is written as follows: x t = b 0 + b 1 t + b 2 t 2 + u t u t = a 1 u t,1 + a 2 u t,2 + :::+ a p u t,p + t The autoregressive parameters included in the model for each series are selected by a stepwise regression procedure, so that autoregressive parameters are only included at those lags at which they are statistically significant. 592

17 Chapter 12. Getting Started The stepwise autoregressive method is fully automatic and, unlike the exponential smoothing and Winters methods, does not depend on choosing smoothing weights. However, the STEPAR method assumes that the long-term trend is stable; that is, the time trend regression is fit to the whole series with equal weights for the observations. The stepwise autoregressive model is used when you specify the METHOD=STEPAR option or do not specify any METHOD= option. To select a constant, linear, or quadratic trend for the time-trend part of the model, use the TREND= option. 593

18 Part 2. General Information Syntax The following statements are used with PROC FORECAST: PROC FORECAST options; BY variables; ID variables; VAR variables; Functional Summary The statements and options controlling the FORECAST procedure are summarized in the following table: Description Statement Option Statements specify BY-group processing identify observations specify the variables to forecast BY ID VAR Input Data Set Options specify the input SAS data set PROC FORECAST DATA= specify frequency of the input time series PROC FORECAST INTERVAL= specify increment between observations PROC FORECAST INTPER= specify seasonality PROC FORECAST SEASONS= specify number of periods in a season PROC FORECAST SINTPER= treat zeros at beginning of series as missing PROC FORECAST ZEROMISS Output Data Set Options specify the number of periods ahead to PROC FORECAST LEAD= forecast name output data set containing the forecasts PROC FORECAST OUT= write actual values to the OUT= data set PROC FORECAST OUTACTUAL write confidence limits to the OUT= data set PROC FORECAST OUTLIMIT write residuals to the OUT= data set PROC FORECAST OUTRESID write standard errors of the forecasts to the PROC FORECAST OUTSTD OUT= data set write one-step-ahead predicted values to the PROC FORECAST OUT1STEP OUT= data set write predicted, actual, and confidence limit PROC FORECAST OUTFULL values to the OUT= data set write all available results to the OUT= data set PROC FORECAST OUTALL specify significance level for confidence limits PROC FORECAST ALPHA= control the alignment of SAS Date values PROC FORECAST ALIGN= 594

19 Chapter 12. Syntax Description Statement Option Parameters and Statistics Output Data Set Options write parameter estimates and goodness-of-fit PROC FORECAST OUTEST= statistics to an output data set write additional statistics to OUTEST= data PROC FORECAST OUTESTALL set write Theil statistics to OUTEST= data set PROC FORECAST OUTESTTHEIL write forecast accuracy statistics to OUTEST= data set PROC FORECAST OUTFITSTATS Forecasting Method Options specify the forecasting method PROC FORECAST METHOD= specify degree of the time trend model PROC FORECAST TREND= specify smoothing weights PROC FORECAST WEIGHT= specify order of the autoregressive model PROC FORECAST AR= specify significance level for adding AR lags PROC FORECAST SLENTRY= specify significance level for keeping AR lags PROC FORECAST SLSTAY= start forecasting before the end of data PROC FORECAST START= specify criterion for judging singularity PROC FORECAST SINGULAR= Initializing Smoothed Values specify number of beginning values to use in PROC FORECAST NSTART= calculating starting values specify number of beginning values to use in PROC FORECAST NSSTART= calculating initial seasonal parameters specify starting values for constant term PROC FORECAST ASTART= specify starting values for linear trend PROC FORECAST BSTART= specify starting values for the quadratic trend PROC FORECAST CSTART= PROC FORECAST Statement PROC FORECAST options; The following options can be specified in the PROC FORECAST statement: ALIGN= option controls the alignment of SAS dates used to identify output observations. The ALIGN= option allows the following values: BEGINNING BEG B, MID- DLE MID M, and ENDING END E. BEGINNING is the default. 595

20 Part 2. General Information ALPHA= value specifies the significance level to use in computing the confidence limits of the forecast. The value of the ALPHA= option must be between.01 and.99. You should use only two digits for the ALPHA= option because PROC FORECAST rounds the value to the nearest percent (ALPHA=.101 is the same as ALPHA=.10). The default is ALPHA=.05, which produces 95% confidence limits. AR= n NLAGS= n specifies the maximum order of the autoregressive model. The AR= option is only valid for METHOD=STEPAR. The default value of n depends on the INTERVAL= option and on the number of observations in the DATA= data set. See "STEPAR Method" later in this chapter for details. ASTART= value ASTART= ( value... ) specifies starting values for the constant term for the exponential smoothing, Winters, and additive Winters methods. This option is ignored if METHOD=STEPAR. See "Starting Values for EXPO, WINTERS, and ADDWINTERS Methods" later in this chapter for details. BSTART= value BSTART= ( value... ) specifies starting values for the linear trend for the exponential smoothing, Winters, and additive Winters methods. This option is ignored if METHOD=STEPAR or TREND=1. See "Starting Values for EXPO, WINTERS, and ADDWINTERS Methods" later in this chapter for details. CSTART= value CSTART= ( value... ) specifies starting values for the quadratic trend for the exponential smoothing, Winters, and additive Winters methods. This option is ignored if METHOD=STEPAR or TREND=1 or 2. See "Starting Values for EXPO, WINTERS, and ADDWINTERS Methods" later in this chapter for details. DATA= SAS-data-set names the SAS data set containing the input time series for the procedure to forecast. If the DATA= option is not specified, the most recently created SAS data set is used. INTERVAL= interval specifies the frequency of the input time series. For example, if the input data set consists of quarterly observations, then INTERVAL=QTR should be used. See Chapter 3, Date Intervals, Formats, and Functions, for more details on the intervals available. INTPER= n when the INTERVAL= option is not used, INTPER= specifies an increment (other than 1) to use in generating the values of the ID variable for the forecast observations in the output data set. 596

21 Chapter 12. Syntax LEAD= n specifies the number of periods ahead to forecast. The default is LEAD=12. The LEAD= value is relative to the last observation in the input data set and not to the end of a particular series. Thus, if a series has missing values at the end, the actual number of forecasts computed for that series will be greater than the LEAD= value. METHOD= method-name specifies the method to use to model the series and generate the forecasts. METHOD=STEPAR specifies the stepwise autoregressive method. METHOD=EXPO specifies the exponential smoothing method. METHOD=WINTERS specifies the Holt-Winters exponentially smoothed trendseasonal method. METHOD=ADDWINTERS specifies the additive seasonal factors variant of the Winters method. For more information, see the section "Forecasting Methods" later in this chapter. The default is METHOD=STEPAR. NSTART= n NSTART= MAX specifies the number of beginning values of the series to use in calculating starting values for the trend parameters in the exponential smoothing, Winters, and additive Winters methods. This option is ignored if METHOD=STEPAR. For METHOD=EXPO, n beginning values of the series are used in forming the exponentially smoothed values S1, S2, and S3, where n is the value of the NSTART= option. The parameters are initialized by fitting a time trend regression to the first n nonmissing values of the series. For METHOD=WINTERS or METHOD=ADDWINTERS, n beginning complete seasonal cycles are used to compute starting values for the trend parameters. For example, for monthly data the seasonal cycle is one year, and NSTART=2 specifies that the first 24 observations at the beginning of each series are used for the time trend regression used to calculate starting values. When NSTART=MAX is specified, all the observations are used. The default for METHOD=EXPO is NSTART=8; the default for METHOD=WINTERS or METHOD=ADDWINTERS is NSTART=2. See "Starting Values for EXPO, WIN- TERS, and ADDWINTERS Methods" later in this chapter for details. NSSTART= n NSSTART= MAX specifies the number of beginning values of the series to use in calculating starting values for seasonal parameters for METHOD=WINTERS or METHOD=ADDWINTERS. The seasonal parameters are initialized by averaging over the first n values of the series for each season, where n is the value of the NSSTART= option. When NSSTART=MAX is specified, all the observations are used. 597

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