# Documentation

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## 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 the Data and Specify the model.

```load(fullfile(matlabroot,'examples','econ','Data_Airline.mat')) y = log(Data); T = length(y); Mdl = arima('Constant',0,'D',1,'Seasonality',12,... 'MALags',1,'SMALags',12);```

### Estimate the Model Using Presample Data.

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): --------------------------------------------------------------- Conditional Probability Distribution: Gaussian Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 0 Fixed Fixed MA{1} -0.377161 0.0734258 -5.13663 SMA{12} -0.572379 0.0939327 -6.0935 Variance 0.00138874 0.000152417 9.1115 ```
```EstMdl = 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: {-0.377161} at Lags [1] SMA: {-0.572379} at Lags [12] Seasonality: 12 Variance: 0.00138874 ```
```EstParamCov = 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

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 the 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.