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

Load the data set Data_Airline.

load Data_Airline
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')