Nonseasonal Differencing

This example shows how to take a nonseasonal difference of a time series. The time series is quarterly U.S. GDP measured from 1947 to 2005.

Load the GDP data set included with the toolbox.

load Data_GDP
Y = Data;
N = length(Y);

figure
plot(Y)
xlim([0,N])
title('U.S. GDP')

Figure contains an axes object. The axes object with title U.S. GDP contains an object of type line.

The time series has a clear upward trend.

Take a first difference of the series to remove the trend,

$Δ y_{t} = (1 - L) y_{t} = y_{t} - y_{t - 1} .$

First create a differencing lag operator polynomial object, and then use it to filter the observed series.

D1 = LagOp({1,-1},'Lags',[0,1]);
dY = filter(D1,Y);

figure
plot(2:N,dY)
xlim([0,N])
title('First Differenced GDP Series')

Figure contains an axes object. The axes object with title First Differenced GDP Series contains an object of type line.

The series still has some remaining upward trend after taking first differences.

Take a second difference of the series,

$Δ^{2} y_{t} = (1 - L)^{2} y_{t} = y_{t} - 2 y_{t - 1} + y_{t - 2} .$

D2 = D1*D1;
ddY = filter(D2,Y);
  
figure
plot(3:N,ddY)
xlim([0,N])
title('Second Differenced GDP Series')

Figure contains an axes object. The axes object with title Second Differenced GDP Series contains an object of type line.

The second-differenced series appears more stationary.

See Also