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

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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

(V.I.1-213)

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

(V.I.1-214)

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

(V.I.1-215)

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

(V.I.1-216)

where c is a parameter-constant. This is because

(V.I.1-217)

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

(V.I.1-218)

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

(V.I.1-219)

it follows that

(V.I.1-220)

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

(V.I.1-221)

it follows that the y-weights satisfy

(V.I.1-222)

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

(V.I.1-223)

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

(V.I.1-224)

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)).

(V.I.1-225)

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

(V.I.1-226)

(V.I.1-227)

(V.I.1-228)

since

(V.I.1-229)

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

The general complementary function for

(V.I.1-230)

is

(V.I.1-231)

with Di described in

(V.I.1-232)

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

(V.I.1-233)

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

(V.I.1-234)

which can be easily proved on noting that

(V.I.1-235)

such that

(V.I.1-236)

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

(V.I.1-237)

called the nontruncated random shock form of the ARIMA model.

(V.I.1-238)

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

Also remark that it is evident that

(V.I.1-239)

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