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ARIMA

The ARIMA command provides features for the Box-Jenkins approach (see Box and Jenkins [1976]) to the analysis of AutoRegressive Integrated Moving Average models of univariate time series. SHAZAM uses a modified version of programs written by Charles Nelson and described in Nelson [1973].

There are 3 forms of the command: IDENTIFICATION, ESTIMATION, and FORECASTING. The specified options determine which form of the ARIMA command is in effect.

The identification stage reports the sample autocorrelation function and the sample partial autocorrelation function that can be inspected to determine a specification for an ARIMA model. The estimation stage estimates the parameters of an ARIMA model and gives diagnostic tests for checking the model adequacy. The Box-Jenkins method is to repeat the identification and estimation stage until a suitable model is found. The forecasting stage provides point forecasts and confidence intervals.


IDENTIFICATION PHASE

In general, the format is:

  ARIMA vars / options

where vars is a list of variables. The available options are:

OPTION DESCRIPTION
ACF= Saves the sample AutoCorrelation Function in the variable specified.
ALL Computes the time series properties for ALL orders of differencing up to the values specified with NDIFF= and NSDIFF=.
BEG= Specifies the BEGinning observation to be used in estimation. This option overrides the SAMPLE command and defaults to the sample range in effect.
END= Specifies the ENDing observation to be used in estimation. This option overrides the SAMPLE command and defaults to the sample range in effect.
IAC Computes inverse autocorrelations. The number of lags is as specified with the NLAG= option (the maximum is 20).
GRAPHAC Plots the sample autocorrelation function using the gnuplot program. The number of lags is specified with the NLAG= option. An approximate 95% confidence interval for the autocorrelations is calculated based on +/-2 standard errors.
GRAPHDATA Plots the data using the gnuplot program. If the LOG, NDIFF= or NSDIFF= options are used then the plot is of the transformed data.
GRAPHPAC Plots the sample partial autocorrelation function using the gnuplot program. The number of lags is specified with the NLAGP= option. An approximate 95% confidence interval for the partial autocorrelations is calculated based on +/- 2 standard errors.
LOG Takes logs of the data.
NDIFF= Specifies the order of differencing to transform the data.
NLAG= Specifies the number of lags to consider in the calculation of the autocorrelations. The default is 24.
NLAGP= Specifies the number of lags to consider in the calculation of the partial autocorrelations. The default is 12. (The value for NLAGP= must not exceed that for NLAG=).
NSDIFF= Specifies the order of seasonal differencing. If this is specified then NSPAN= must be set.
NSPAN= Specifies the number of periods in the seasonal cycle. For example, set NSPAN=4 for quarterly data and NSPAN=12 for monthly data.
PLOTAC Plots the sample autocorrelation function with text characters on the SHAZAM output. Also see the GRAPHAC option.
PLOTDATA Plots the data with text characters on the SHAZAM output. Also see the GRAPHDATA option.
PLOTPAC Plots the sample partial autocorrelation function with text characters on the SHAZAM output. Also see the GRAPHPAC option.
PACF= Saves the sample Partial AutoCorrelation Function in the variable specified.
TESTSTAT= Saves the Ljung-Box-Pierce statistics (computed at every lag during IDENTIFICATION) in the variable specified.
WIDE/NOWIDE Uses a wider page width for the display of results. For printing, the page set-up should be set to landscape orientation.

The available temporary variables are:
$N - the number of observations used in the identification phase.


ESTIMATION PHASE

In general, the format is:

  ARIMA var / NAR= NMA= options

where var is a variable. The available options as used for the IDENTIFICATION phase are:
LOG, WIDE, BEG=, END=, NDIFF=, NSDIFF=, NSPAN=.

The available options as used for the OLS command are:
ANOVA, PCOR, PCOV, COV=, STDERR=, TRATIO=.

Other options are:

OPTION DESCRIPTION
DN Computes the estimated variance of the regression by dividing SIGMA**2 by N instead of (N-K).
GRAPHRES Prepare GNUPLOT plots for the PLOTRES option.
NOCONSTANT No intercept in model.
PITER Print every iteration.
PLOTRES Plot the residuals.
RESTRICT Use zero starting values as zero restrictions.
START Starting values for coefficients follow the ARIMA command.
ACF= Saves the AutoCorrelation Function of the estimated residuals in the variable specified.
COEF= Save Coefficients in variable specified.
ITER= Maximum number of iterations. The default is 50.
NAR= Order of the AR process. (REQUIRED)
NMA= Order of the MA process (REQUIRED).
NSAR= Order of the Seasonal AR process. If this is specified then NSPAN= must be set.
NSMA= Order of the Seasonal MA process. If this is specified then NSPAN= must be set.
PREDICT= Save Predicted values in variable specified.
RESID= Save Residuals in variable specified.
START=  A vector of starting values.
TESTSTAT=  Saves the Ljung-Box-Pierce statistics (computed at every lag -up to 60 lags) in the variable specified.

Following model estimation the available temporary variables as for the OLS command are:
$ERR, $K, $N, $R2, $SIG2, $SSE, $SSR, $SST.


FORECASTING PHASE

In general, the format is:

  ARIMA var / COEF= NAR= NMA= FBEG= FEND= options

where var is a variable. The available options as used for the IDENTIFICATION and ESTIMATION phase are:
LOG, NOCONSTANT, BEG=, END=, NAR=, NDIFF=, NMA=, NSAR=,
NSDIFF=, NSMA=, NSPAN=, PREDICT=, RESID=.

Other options are:

OPTION DESCRIPTION
GRAPHFORC Prepare GNUPLOT plots for the PLOTFORC option.
PLOTFORC Plot the forecast with error bounds.
COEF= Input coefficient variable. If this is not specified then coefficients must be entered on the line following the ARIMA command.
FBEG=  The origin date of the forecast (required).
FEND=  Last observation to forecast (required). (The maximum number of forecasts is 200.)
FCSE=  Saves the forecast standard errors in the variable.
SIGMA=  Used in calculating the forecast standard errors.

By default the data for the current sample period will be used to estimate SIGMA. A recommended approach is following estimation enter:

GEN1 S=SQRT($SIG2).

Then use the option SIGMA=S for the ARIMA forecasting.