Model Seasonal Lag Effects Using Indicator Variables

This example shows how to estimate a seasonal ARIMA model:

• Model the seasonal effects using a multiplicative seasonal model.

• Use indicator variables as a regression component for the seasonal effects, called seasonal dummies.

Subsequently, their forecasts show that the methods produce similar results. The time series is monthly international airline passenger numbers from 1949 to 1960.

Load the data set Data_Airline, and plot the natural log of the monthly passenger totals counts.

dat = log(Data); % Transform to logarithmic scale
T = size(dat,1);
y = dat(1:103);  % Estimation sample

y is the part of dat used for estimation, and the rest of dat is the holdout sample to compare the two models' forecasts.

Step 2. Define and fit the model specifying seasonal lags.

Create an ARIMA$\left(0,1,1\right)\left(0,1,1{\right)}_{12}$ model

$\left(1-L\right)\left(1-{L}^{12}\right){y}_{t}=\left(1+{\theta }_{1}L\right)\left(1+{\Theta }_{12}{L}^{12}\right){\epsilon }_{t},$

where ${\epsilon }_{t}$ is an independent and identically distributed normally distributed series with mean 0 and variance ${\sigma }^{2}$. Use estimate to fit Mdl1 to y.

Mdl1 = arima('Constant',0,'MALags',1,'D',1,...
'SMALags',12,'Seasonality',12);
EstMdl1 = estimate(Mdl1,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}       -0.35732       0.088031        -4.059      4.9286e-05
SMA{12}     -0.61469       0.096249       -6.3864      1.6985e-10
Variance    0.001305      0.0001527        8.5467      1.2671e-17

The fitted model is

$\left(1-L\right)\left(1-{L}^{12}\right){y}_{t}=\left(1-0.357L\right)\left(1-0.615{L}^{12}\right){\epsilon }_{t},$

where ${\epsilon }_{t}$ is an iid normally distributed series with mean 0 and variance 0.0013.

Step 3. Define and fit the model using seasonal dummies.

Create an ARIMAX(0,1,1) model with period 12 seasonal differencing and a regression component,

$\left(1-L\right)\left(1-{L}^{12}\right){y}_{t}=\left(1-0.357L\right)\left(1-0.615{L}^{12}\right){\epsilon }_{t},$

$\left\{{x}_{t};t=1,...,T\right\}$ is a series of T column vectors having length 12 that indicate in which month observation $t$ was measured. A 1 in row i of ${x}_{t}$ indicates that the observation was measured in month i, the rest of the elements are 0s.

Note that if you include an additive constant in the model, then the T rows of the design matrix X are composed of the row vectors $\left[\begin{array}{cc}1& {x}_{t}^{\prime }\end{array}\right]$. Therefore, X is rank deficient, and one regression coefficient is not identifiable. A constant is left out of this example to avoid distraction from the main purpose. Format the in-sample X matrix

X  = dummyvar(repmat((1:12)',12,1));

% Format the presample X matrix
X0 = [zeros(1,11) 1; dummyvar((1:12)')];
Mdl2 = arima('Constant',0,'MALags',1,'D',1,...
'Seasonality',12);
EstMdl2   = estimate(Mdl2,y,'X',[X0; X]);

ARIMAX(0,1,1) Model Seasonally Integrated (Gaussian Distribution):

Value       StandardError    TStatistic      PValue
__________    _____________    __________    __________

Constant             0              0            NaN            NaN
MA{1}         -0.40711       0.084387        -4.8242     1.4053e-06
Beta(1)      -0.002577       0.025168       -0.10239        0.91845
Beta(2)     -0.0057769       0.031885       -0.18118        0.85623
Beta(3)     -0.0022034       0.030527      -0.072179        0.94246
Beta(4)     0.00094737       0.019867       0.047686        0.96197
Beta(5)     -0.0012146       0.017981      -0.067551        0.94614
Beta(6)        0.00487       0.018374        0.26505        0.79097
Beta(7)     -0.0087944       0.015285       -0.57535        0.56505
Beta(8)      0.0048346       0.012484        0.38728        0.69855
Beta(9)       0.001437       0.018245       0.078758        0.93722
Beta(10)      0.009274       0.014751        0.62869        0.52955
Beta(11)     0.0073665         0.0105        0.70158        0.48294
Beta(12)    0.00098841       0.014295       0.069146        0.94487
Variance     0.0017715     0.00024657         7.1848     6.7329e-13

The fitted model is

$\left(1-L\right)\left(1-{L}^{12}\right){y}_{t}={X}_{t}\underset{}{\overset{ˆ}{\beta }}+\left(1-0.407L\right){\epsilon }_{t},$

where ${\epsilon }_{t}$ is an iid normally distributed series with mean 0 and variance 0.0017 and $\underset{}{\overset{ˆ}{\beta }}$ is a column vector with the values Beta1 - Beta12. Note that the estimates MA{1} and Variance between Mdl1 and Mdl2 are not equal.

Step 4. Forecast using both models.

Use forecast to forecast both models 41 periods into the future from July 1957. Plot the holdout sample using these forecasts.

yF1 = forecast(EstMdl1,41,y);
yF2 = forecast(EstMdl2,41,y,'X0',X(1:103,:),'XF',X(104:end,:));
l1 = plot(100:T,dat(100:end),'k','LineWidth',3);
hold on
l2 = plot(104:144,yF1,'-r','LineWidth',2);
l3 = plot(104:144,yF2,'-b','LineWidth',2);
hold off
title('Passenger Data: Actual vs. Forecasts')
xlabel('Month')
ylabel('Logarithm of Monthly Passenger Data')
legend({'Observations','Polynomial Forecast',...
'Regression Forecast'},'Location','NorthWest') Though they overpredict the holdout observations, the forecasts of both models are almost equivalent. One main difference between the models is that EstMdl1 is more parsimonious than EstMdl2.

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.