Time Series Analysis - ARIMA Models - Differencing Operators

l. Differencing (Nabla and B operator)

We have already used the back shift operator in previous sections. As we know the back shift operator (B-operator) transforms an observation of a time series to the previous one

(V.I.1-205)

Also, it can easily be shown that

(V.I.1-206)

Note that if we back shift k times seasonally we get

(V.I.1-207)

The back shift operator can be useful in defining the nabla operator which is

(V.I.1-208)

or in general

(V.I.1-209)

which is sometimes also called the differencing operator.

As stated before, a time series which is not stationary with respect to the mean can be made stationary by differencing. How can this be interpreted ?

(figure V.I.1-12)

In figure (V.I.1-12) a function is displayed with two points on a graph (a, f(a)), and (b, f(b)).

Assume that a time series is generated by the function f(x). Then the derivative of the function gives the slope of a line tangent with respect to the graph in every point of the function's domain.

The derivative of a function is defined as

(V.I.1-210)

If we compute the slope of the cord in (figure V.I.1-12), this is in fact the same as the derivative of f(x) between a and b with a discrete step in stead of an infinitesimal small step.

This results in computing

(V.I.1-211)

Although we have assumed the time series to be generated by f(x), in practice we only observe sample values at discrete time intervals. Therefore the best approximation of f(x) between two known points (a, f(a)) and (b, f(b)) is a straight line with slope given by (V.I.1-211).

If this approximation is to be optimal, the distance between a and b should be as small as possible. Since the smallest difference between observations of equally spaced time series is the time lag itself, the smallest value of h in eq. (V.I.1-211) is in fact equal to 1.

Therefore (V.I.1-211) reduces to

(V.I.1-212)

which is nothing else but the differencing operator.

We conclude that by differencing a time series we 'derive' the function by which it is generated, and therefore reduce the function's power by 1. If e.g. we would have a time series generated by a quadratic function, we could make it stationary by differencing the series twice.

Furthermore it should be noted that if a time series is non stationary, and must therefore be 'derived' to induce stationarity, the series is often called to be generated by an integrated process. Now the ARMA models which have been described before, can be elaborated to the class of ARIMA (c.q. Autoregressive Integrated Moving Average) models.