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## Specify Multiplicative ARIMA Model

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

Step 1. Load the airline passenger data.

Copy the data file Data_Airline.mat from the folder \help\toolbox\econ\examples in your MATLAB® root, and paste it into your current working folder (or set \help\toolbox\econ\examples as your current working folder).

Load the data, and then plot the natural log of the monthly passenger totals.

```load Data_Airline
Y = log(Dataset.PSSG);
N = length(Y);
mos = get(Dataset,'ObsNames');

figure
plot(Y)
xlim([1,N])
set(gca,'XTick',[1:18:N])
set(gca,'XTickLabel',mos([1:18:N]))
title('Log Airline Passengers')
ylabel('(thousands)')```

The data look nonstationary, with a linear trend and seasonal periodicity.

Step 2. Plot the seasonally integrated series.

Calculate the differenced series, , where yt is the original log-transformed data. Plot the differenced series.

```A1 = LagOp({1,-1},'Lags',[0,1]);
A12 = LagOp({1,-1},'Lags',[0,12]);
dY = filter(A1*A12,Y);

figure
plot(dY)
title('Differenced Log Airline Passengers')```

The differenced series appears stationary.

Step 3. Plot the sample autocorrelation function (ACF).

```figure
autocorr(dY,50)```

The sample ACF of the differenced series shows significant autocorrelation at lags that are multiples of 12. There is also potentially significant autocorrelation at smaller lags.

Step 4. Specify a seasonal ARIMA model.

Box, Jenkins, and Reinsel suggest the multiplicative seasonal model,

for this data set (Box et al., 1994).

Specify this model.

```model = arima('Constant',0,'D',1,'Seasonality',12,...
'MALags',1,'SMALags',12)```
```model =

ARIMA(0,1,1) Model Seasonally Integrated with ...
Seasonal MA(12):
--------------------------------------------- ...
------------------
Distribution: Name = 'Gaussian'
P: 13
D: 1
Q: 13
Constant: 0
AR: {}
SAR: {}
MA: {NaN} at Lags [1]
SMA: {NaN} at Lags [12]
Seasonality: 12
Variance: NaN```

The property P is equal to 13, corresponding to the sum of the nonseasonal and seasonal differencing degrees (1 + 12). The property Q is also equal to 13, corresponding to the sum of the degrees of the nonseasonal and seasonal MA polynomials (1 + 12). Parameters that need to be estimated have value NaN.

## References

[1] 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.