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Non stationary time series
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Time Series Analysis - ARIMA Models - Non stationary time series

[Home] [Up] [Basics] [AR(1) process] [AR(2) process] [AR(p) process] [MA(1) process] [MA(2) process] [MA(q) process] [ARMA(1,1) process] [ARMA(p,q) process] [Wold's decomp.] [Differencing] [Behavior] [Inverse Autocorr.] [Unit Root Tests] [Non stationarity]

k. Non stationary time series

Most economic (and also many other) time series do not satisfy the stationarity conditions stated earlier for which ARMA models have been derived. Then these times series are called non stationary and should be re-expressed such that they become stationary with respect to the variance and the mean.

It is not suggested that the description of the following re-expression tools is exhaustive! They rather form a set of tools which have shown to be useful in practice. It is quite evident that many extensions are possible with respect to re-expression tools: these are discussed in literature such as in JENKINS (1976 and 1978), MILLS (1990), MCLEOD (1983), etc...

Transformation of time series

If we write a time series as the sum of a deterministic mean and a disturbance term

Time Series Analysis - ARIMA Models - Non stationary time series



where h is an arbitrary function.


This can be used to obtain the variance of the transformed series


which implies that the variance can be stabilized by imposing


Accordingly, if the standard deviation of the series is proportional to the mean level




from which it follows that


In case the variance of the series is proportional to the mean level, then


from which it follows that


With the use of a Standard Deviation / Mean Procedure (SMP) we are able to detect heteroskedasticity in the time series. Above that, with the help of the SMP, it is quite often possible to find an appropriate transformation which will ensure the time series to be homoskedastic. In fact, it is assumed that there exists a relationship between the mean level of the time series and the variance or standard deviation as in


which is an explicitly assumed relationship, in contrast to (V.I.1-194).

The SMP is generated by a three step process:


the time series is spilt into equal (chronological) segments;


for each segment the arithmetic mean and standard deviation is computed;


the mean and S.D. of each segment is plotted or regressed against each other.

By selecting the length of the segments equal to the seasonal period one ensures that the S.D. and mean is independent from the seasonal period.

In practice one of the following patterns will be recognized (as summarized in the graph). Note that the lambda parameter should take a value of zero when a linearly proportional association between S.D. and the mean is recognized.

The value of lambda is in fact the transformation parameter which implies the following:


(Figure V.I.1-10)

Differencing of time series

With the use of the Autocorrelation Function (ACF) (with autocorrelations on the y axis and the different time lags on the x axis) it is possible to detect unstationarity of the time series with respect to the mean level.

(figure V.I.1-11)

When the ACF of the time series is slowly decreasing, this is an indication that the mean is not stationary. An example of such an ACF is given in figure (V.I.1-11).

The differencing operator (nabla) is used to make the time series stationary.

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AR(1) process
AR(2) process
AR(p) process
MA(1) process
MA(2) process
MA(q) process
ARMA(1,1) process
ARMA(p,q) process
Wold's decomp.
Inverse Autocorr.
Unit Root Tests
Non stationarity
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