
ARMAName:
Autoregressive models are defined by:
where X_{t} is the series and \(\bar{X}\) is the mean of the series, A_{t} represent normally distributed random errors, and the \(\phi_1, ..., \phi_p\) are the parameters of the model. Autoregressive models are simply a linear regression of the current value of the series against one or more prior values of the series. The value of p is called the order of the model. Moving average models are defined by:
where X_{t} is the series and \(\bar{X}\) is the mean of the series, A_{ti} represent random shocks of one or more prior points of the series, and the \(\theta_1, ..., \theta_q\) are the parameters of the model. The random shocks are assumed to come from a common (typically normal) distribution with common location and scale. The primary idea behind the moving average model is that the random shocks are propogated to future values of the series. Fitting moving average models require iterative, nonlinear fitting techniques. The power of ARMA models is that they can incorporate both autoregressive terms and moving average terms. The use of ARMA models was popularized by Box and Jenkins. Although both AR and MA models were previously known and used, Box and Jenkins provided a systematic approach for modeling both AR and MA terms in the model. ARMA models are also commonly known as BoxJenkins models or ARIMA models. ARMA models assume that the data are stationary, i.e. the data have constant location and scale. Trend can often be removed from a nonstationary series to achieve stationarity. Differencing is a common approach for removing trend. The first difference is defined as X_{t}  X_{t1}. In most cases, a single differencing is sufficient. However, more than one differencing can be applied if necessary. You can also fit a linear or nonlinear model to remove trend. ARMA models can also incorporate seasonal terms (and seasonal differencing). See Box and Jenkins for the complete mathematical description of this model. ARMA models typically require fairly long series (at least 50 points is recommended by some authors). Also, if the series is dominated by trend and seasonal components, a trend/seasonality/residual decomposition method may be preferred. Dataplot supports a SEASONAL LOWESS command for this type of decomposition. The typical components for fitting ARMA models is:
The ARMA command addresses (2), model fitting. Model identification for ARMA models can be difficult and require a fair amount of experience. See the various time series texts that describe ARMA modeling for more guidance on ARMA model identification. Model validation is similar to the nonlinear fitting case (i.e., various residual plots). Again, see texts that describe nonlinear model fitting.
<SUBSET/EXCEPT/FOR qualification> where <y> is the response variable; <ar> is the order of the autoregressive terms; <diff> is the number of differences to apply (typically 0, 1, or 2); <ma> is the order ot the moving average terms; <sar> is the order of seasonal autoregressive terms; <sdiff> is the number of seasonal differences to apply (typically 0, 1, or 2); <sma> is the order of seasonal moving average terms; <speriod> is the period for seasonal terms (defaults to 12); and where the <SUBSET/EXCEPT/FOR qualification> is optional. If there is no seasonal component, the <sar>, <sdiff>, <sma>, and <speriod> terms can be omitted.
ARMA Y 2 0 1 1 0 1 12
In addition, you can define the variable ARFIXED to fix certain parameters to their start values. That is, you define ARPAR to specify the start values. If the corresponding element of ARFIXED is zero, the parameter is estimated as usual. If ARFIXED is one, then the parameter is fixed at the start value. The most common use of this is to set certain parameters to zero. For example, if you fit an AR(2) model and you want the AR(1) term to be zero, you could enter the following:
LET ARFIXED = DATA 1 0
Chatfield (1989), "The Analysis of Time Series: An Introduction," Fourth Edition, Chapman & Hall.
skip 0 set read format 3e16.4 read negiz4.dat junk1 junk2 y set read format delete junk1 junk2 let arpar = data 0.1 0.1 0.1 arma y 3 0 0This program generates the following output. ************************************ ** let arpar = data 0.1 0.1 0.1 ** ************************************ THE NUMBER OF VALUES GENERATED FOR THE VARIABLE ARPAR = 3 THE FIRST COMPUTED VALUE OF ARPAR = 0.1000000E+00 (ROW 1) THE LAST ( 3TH) COMPUTED VALUE OF ARPAR = 0.1000000E+00 (ROW 3) THE CURRENT COLUMN FOR THE VARIABLE ARPAR = 2 THE CURRENT LENGTH OF THE VARIABLE ARPAR = 3 ******************** ** arma y 3 0 0 ** ******************** ############################################################# # NONLINEAR LEAST SQUARES ESTIMATION FOR THE PARAMETERS OF # # AN ARIMA MODEL USING BACKFORECASTS # ############################################################# SUMMARY OF INITIAL CONDITIONS  MODEL SPECIFICATION FACTOR (P D Q) S 1 3 0 0 1 DEFAULT SCALING USED FOR ALL PARAMETERS. ##STEP SIZE FOR ######PARAMETER ##APPROXIMATING #################PARAMETER DESCRIPTION STARTING VALUES #####DERIVATIVE INDEX #########TYPE ##ORDER ##FIXED ##########(PAR) ##########(STP) 1 AR (FACTOR 1) 1 NO 0.10000000E+00 0.22896898E05 2 AR (FACTOR 1) 2 NO 0.10000000E+00 0.22688602E05 3 AR (FACTOR 1) 3 NO 0.10000000E+00 0.22438846E05 4 MU ### NO 0.10000000E+01 0.25174593E05 NUMBER OF OBSERVATIONS (N) 559 MAXIMUM NUMBER OF ITERATIONS ALLOWED (MIT) 500 MAXIMUM NUMBER OF MODEL SUBROUTINE CALLS ALLOWED 1000 CONVERGENCE CRITERION FOR TEST BASED ON THE FORECASTED RELATIVE CHANGE IN RESIDUAL SUM OF SQUARES (STOPSS) 0.1000E09 MAXIMUM SCALED RELATIVE CHANGE IN THE PARAMETERS (STOPP) 0.1489E07 MAXIMUM CHANGE ALLOWED IN THE PARAMETERS AT FIRST ITERATION (DELTA) 100.0 RESIDUAL SUM OF SQUARES FOR INPUT PARAMETER VALUES 0.3537E+07 (BACKFORECASTS INCLUDED) RESIDUAL STANDARD DEVIATION FOR INPUT PARAMETER VALUES (RSD) 79.84 BASED ON DEGREES OF FREEDOM 559  0  4 = 555 NONDEFAULT VALUES.... AFCTOL.... V(31) = 0.2225074307 ##### RESIDUAL SUM OF SQUARES CONVERGENCE ##### ESTIMATES FROM LEAST SQUARES FIT (* FOR FIXED PARAMETER) ######################################################## PARAMETER STD DEV OF ###PAR/ ##################APPROXIMATE ESTIMATES ####PARAMETER ####(SD 95 PERCENT CONFIDENCE LIMITS TYPE ORD ###(OF PAR) ####ESTIMATES ##(PAR) #######LOWER ######UPPER FACTOR 1 AR 1 0.58969407E+00 0.41925732E01 14.07 0.52061708E+00 0.65877107E+00 AR 2 0.23795137E+00 0.47746327E01 4.98 0.15928434E+00 0.31661840E+00 AR 3 0.15884704E+00 0.41922036E01 3.79 0.89776135E01 0.22791795E+00 MU ## 0.11472145E+03 0.78615948E+00 145.93 0.11342617E+03 0.11601673E+03 NUMBER OF OBSERVATIONS (N) 559 RESIDUAL SUM OF SQUARES 108.9505 (BACKFORECASTS INCLUDED) RESIDUAL STANDARD DEVIATION 0.4430657 BASED ON DEGREES OF FREEDOM 559  0  4 = 555 APPROXIMATE CONDITION NUMBER 89.28687 PARAMETERS, SD(PARAMETERS), 1/SD(PAR), LOWER AND UPPER 95% CONFIDENCE INTERVAL WRITTEN OUT TO FILE DPST1F.DAT ORDER IS: 1. AUTO_REGRESSIVE TERMS 2. SEASONAL AUTO_REGRESSIVE TERMS 3. MU (MEAN TERM) 4. MOVING AVERAGE TERMS 5. SEASONAL MOVING AVERAGE TERMS FOLLOWING WRITTEN OUT TO FILE DPST2F.DAT 1. ROW NUMBER 2. PREDICTED VALUES 3. STANDARD DEVIATION OF PREDICTED VALUES 4. RESIDUALS 5. STANDARDIZED RESIDUALS RESULTS OF ITERATIONS WRITTEN OUT TO FILE DPST3F.DAT PARAMETER VARIANCECOVARIANCE MATRIX WRITTEN OUT TO FILE DPST4F.DAT FORECAST, STANDARD DEVIATION OF FORECASTS, AND 95% CONFIDENCE INTERVAL FOR FORECAST WRITTEN TO FILE DPST5F.DAT  
Date created: 06/05/2001 Last updated: 12/04/2023 Please email comments on this WWW page to alan.heckert@nist.gov. 