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.

**Load the airline passenger data.**

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

load(fullfile(matlabroot,'examples','econ','Data_Airline.mat')) y = log(Data); T = length(y); figure plot(dates,y) xlim([1,T]) datetick('x','mmmyy') axis tight title('Log Airline Passengers') ylabel('(thousands)')

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

**Plot the seasonally integrated series.**

Calculate the differenced series, , where 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.

**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.

**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.

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

Mdl = 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:

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.

`arima`

| `autocorr`

| `filter`

| `LagOp`

- Estimate Multiplicative ARIMA Model
- Simulate Multiplicative ARIMA Models
- Forecast Multiplicative ARIMA Model
- Check Fit of Multiplicative ARIMA Model
- Model Seasonal Lag Effects Using Indicator Variables

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