Estimate Multiplicative ARIMA Model

This example shows how to estimate a multiplicative seasonal ARIMA model using estimate. The time series is monthly international airline passenger numbers from 1949 to 1960.

Load Data and Specify Model.

Load the airline data set.

load Data_Airline
y = log(Data);
T = length(y);

Mdl = arima('Constant',0,'D',1,'Seasonality',12,...
    'MALags',1,'SMALags',12);

Estimate Model.

Use the first 13 observations as presample data, and the remaining 131 observations for estimation.

y0 = y(1:13);
[EstMdl,EstParamCov] = estimate(Mdl,y(14:end),'Y0',y0)

 
    ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution):
 
                  Value      StandardError    TStatistic      PValue  
                _________    _____________    __________    __________

    Constant            0              0           NaN             NaN
    MA{1}        -0.37716       0.073426       -5.1366      2.7972e-07
    SMA{12}      -0.57238       0.093933       -6.0935      1.1047e-09
    Variance    0.0013887     0.00015242        9.1115      8.1249e-20

EstMdl = 
  arima with properties:

     Description: "ARIMA(0,1,1) Model Seasonally Integrated with Seasonal MA(12) (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
               P: 13
               D: 1
               Q: 13
        Constant: 0
              AR: {}
             SAR: {}
              MA: {-0.377161} at lag [1]
             SMA: {-0.572379} at lag [12]
     Seasonality: 12
            Beta: [1×0]
        Variance: 0.00138874

EstParamCov = 4×4

         0         0         0         0
         0    0.0054   -0.0015   -0.0000
         0   -0.0015    0.0088    0.0000
         0   -0.0000    0.0000    0.0000

The fitted model is

$$\Delta {\Delta }_{12}{y}_t=(1-0.38L)(1-0.57L^{12}){\epsilon }_t,$$

with innovation variance 0.0014.

Notice that the model constant is not estimated, but remains fixed at zero. There is no corresponding standard error or t statistic for the constant term. The row (and column) in the variance-covariance matrix corresponding to the constant term has all zeros.

Infer Residuals.

Infer the residuals from the fitted model.

res = infer(EstMdl,y(14:end),'Y0',y0);

figure
plot(14:T,res)
xlim([0,T])
title('Residuals')
axis tight

When you use the first 13 observations as presample data, residuals are available from time 14 onward.

References:

Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.