# 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 Data and Estimate Model

y = log(DataTimeTable.PSSG);
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) (Gaussian Distribution):

Value      StandardError    TStatistic      PValue
_________    _____________    __________    __________

Constant            0              0           NaN             NaN
MA{1}        -0.37716       0.066794       -5.6466      1.6364e-08
SMA{12}      -0.57238       0.085439       -6.6992      2.0952e-11
Variance    0.0012634     0.00012395        10.193      2.1406e-24
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);
fh = DataTimeTable.Time(end) + calmonths(1:60);

figure
plot(DataTimeTable.Time,y,'k')
hold on
plot(fh,Ysim,'Color',[.85,.85,.85]);
h = plot(fh,mn,'k--','LineWidth',2);
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 Probability of 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.

Ysim = simulate(EstMdl,60,'NumPaths',1000,'Y0',y,'E0',res);

g7 = sum(Ysim >= 7) > 0;
phat = mean(g7)
phat = 0.3820
err = sqrt(phat*(1-phat)/1000)
err = 0.0154

The probability that the (log) number of airline passengers will meet or exceed 7 in the next 5 years is approximately a 0.38. 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')