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 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); res = infer(EstMdl,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

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.

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) err = sqrt(phat*(1-phat)/1000)

phat = 0.3910 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.

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')
```

`arima`

| `estimate`

| `infer`

| `simulate`

- Specify Multiplicative ARIMA Model
- Estimate Multiplicative ARIMA Model
- Forecast Multiplicative ARIMA Model
- Check Fit of Multiplicative ARIMA Model

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