The behavior of non stationary time series
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.
time series that is generated by
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
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
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
ARIMA(p,d,q) model can be expanded by introducing deterministic
d-order polynomial trends.
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
same properties can be achieved by writing (V.I.1-215) as an
invertible ARMA process
c is a parameter-constant. This is because
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!
ARIMA model can be generally written as a difference equation. For instance, the ARIMA(1,1,1) can be
illustrates the postulated fact. This form of the ARIMA model is
used for recursive forecasting purposes.
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
if j is the maximum of (p + d - 1, q)
follows that the y-weights
implies that large-lagged
are composed of polynomials, exponentials (damped), and sinusoids
(damped) with respect to index j.
form of the ARIMA model (c.q. eq. (V.I.1-219)) is used to compute
the forecast confidence intervals.
third way of writing an ARIMA model is the truncated random shock
parameter k may be interpreted as the time origin of the observable
data. First, we observe that if Yt' is a particular
(V.I.1-213), thus if
it follows from (V.I.1-213), and
the general solution of (V.I.1-213)
is the sum of
(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
that the general solution of the homogeneous difference equation with
respect to time origin k < t is
general complementary function for
Di described in
(V.I.1-231) it can be concluded that the complementary function
involves a mixture of:
of the random shock model form) satisfying the ARIMA model structure
(where B operates on t, not on k)
can be easily proved on noting that
if t - k > q eq. (V.I.1-233) is the particular integral of
in an extreme case k = -¥
the nontruncated random shock
form of the ARIMA model.
this result with (V.I.1-237)).
remark that it is evident that
implies that when using the complementary function for forecasting
purposes, it is advisable to update the forecast as new observations