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Time Series Analysis - ARIMA models - The behavior of 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.] [Non stationarity] [Differencing] [Inverse Autocorr.] [Unit Root Tests] [Behavior]

m. The behavior of non stationary time series

In the previous subsections, non stationarity has been discussed at a rather intuitive level. Now we will discuss some more fundamental properties of the behavior of non stationary time series.

A time series that is generated by

Time Series Analysis - ARIMA models - The behavior of non stationary time series


with g(B) an AR operator which is not stationary: g(B) has d roots equal to 1; all other roots lie outside the unit circle. Thus eq. (V.I.1-213) can be written by factoring out the unit roots


where f(B) is stationary.

In general a univariate stochastic process as (V.I.1-214) is denoted an ARIMA(p,d,q) model where p is the autoregressive order, d is the number of non-seasonal differences, and q is the order of the moving average components.

Quite evidently, time series exhibiting non stationarity in both variance and mean, are first to be transformed in order to induce a stable variance, and then to be differenced enabling stationarity with respect to the mean level. The reason for this is that power, and logarithmic transformations are not always defined for negative (real) numbers.

The ARIMA(p,d,q) model can be expanded by introducing deterministic d-order polynomial trends.

This is simply achieved by adding a parameter - constant to (V.I.1-214), expressed in terms of a (non-seasonal) non-stationary time series Zt


The same properties can be achieved by writing (V.I.1-215) as an invertible ARMA process


where c is a parameter-constant. This is because


Also remark that the p AR parameters must not add to unity, since this would, according to (V.I.1-217), imply (in the limit) an infinite mean level, an obvious nonsense!

An ARIMA model can be generally written as a difference equation. For instance, the ARIMA(1,1,1) can be formulated as


which illustrates the postulated fact. This form of the ARIMA model is used for recursive forecasting purposes.

The ARIMA model can also be generally written as a random shock model (c.q. a model in terms of the y-weights, and the white noise error components) since


it follows that


Hence, if j is the maximum of (p + d - 1, q)


it follows that the y-weights satisfy


which implies that large-lagged y-weights are composed of polynomials, exponentials (damped), and sinusoids (damped) with respect to index j.

This form of the ARIMA model (c.q. eq. (V.I.1-219)) is used to compute the forecast confidence intervals.

A third way of writing an ARIMA model is the truncated random shock model form.

The parameter k may be interpreted as the time origin of the observable data. First, we observe that if Yt' is a particular solution of (V.I.1-213), thus if


then it follows from (V.I.1-213), and (V.I.1-223) that


Hence, the general solution of (V.I.1-213) is the sum of

Yt'' (c.q. a complementary function which is the solution of (V.I.1-224)), and Yt' (c.q. a particular integral which is a particular solution of (V.I.1-213)).


and that the general solution of the homogeneous difference equation with respect to time origin k < t is given by






see also (V.I.1-227).

The general complementary function for




with Di described in


From (V.I.1-231) it can be concluded that the complementary function involves a mixture of:


(with y-weights of the random shock model form) satisfying the ARIMA model structure (where B operates on t, not on k)


which can be easily proved on noting that


such that


Hence, if t - k > q eq. (V.I.1-233) is the particular integral of (V.I.1-234).

If in an extreme case k = - then


called the nontruncated random shock form of the ARIMA model.


(compare this result with (V.I.1-237)).

Also remark that it is evident that


This implies that when using the complementary function for forecasting purposes, it is advisable to update the forecast as new observations become available.

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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.
Non stationarity
Inverse Autocorr.
Unit Root Tests
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