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')

The time series has a clear upward trend.

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

$$\Delta y_t = (1-L)y_t = y_t - y_{t-1}.$$

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')

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

Take a second difference of the series,

$$\Delta^2 y_t = (1-L)^2y_t = y_t - 2 y_{t-1} + y_{t-2}.$$

D2 = D1*D1;
ddY = filter(D2,Y);

figure
plot(3:N,ddY)
xlim([0,N])
title('Second Differenced GDP Series')

The second-differenced series appears more stationary.

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