# Documentation

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## Simulate Multiplicative ARIMA Models

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

### Load the Data and Estimate a Model.

Load the data set `Data_Airline`.

```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); EstMdl = estimate(Mdl,y);```
``` 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} -0.377162 0.0667944 -5.64661 SMA{12} -0.572378 0.0854395 -6.69923 Variance 0.00126337 0.00012395 10.1926 ```
`res = infer(EstMdl,y);`

### Simulate Airline Passenger Counts.

Use the fitted model to simulate 25 realizations of airline passenger counts over a 60-month (5-year) horizon. Use the observed series and inferred residuals as presample data.

```rng('default') Ysim = simulate(EstMdl,60,'NumPaths',25,'Y0',y,'E0',res); mn = mean(Ysim,2); figure plot(y,'k') hold on plot(T+1:T+60,Ysim,'Color',[.85,.85,.85]); h = plot(T+1:T+60,mn,'k--','LineWidth',2); xlim([0,T+60]) title('Simulated Airline Passenger Counts') legend(h,'Simulation Mean','Location','NorthWest') hold off```

The simulated forecasts show growth and seasonal periodicity similar to the observed series.

### Estimate the Probability of a Future Event.

Use simulations to estimate the probability that log airline passenger counts will meet or exceed the value 7 sometime during the next 5 years. Calculate the Monte Carlo error associated with the estimated probability.

```rng default Ysim = simulate(EstMdl,60,'NumPaths',1000,'Y0',y,'E0',res); g7 = sum(Ysim >= 7) > 0; phat = mean(g7)```
```phat = 0.3910 ```
`err = sqrt(phat*(1-phat)/1000)`
```err = 0.0154 ```

There is approximately a 39% chance that the (log) number of airline passengers will meet or exceed 7 in the next 5 years. The Monte Carlo standard error of the estimate is about 0.02.

### Plot the Distribution of Passengers at a Future Time.

Use the simulations to plot the distribution of (log) airline passenger counts 60 months into the future.

```figure histogram(Ysim(60,:),10) title('Distribution of Passenger Counts in 60 months')```