# 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.

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)"
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

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

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.