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Ex
P
{}
t
,
t
= 1, 2, …
Var(
xEx
P
)
{(
) }
2
,
t
= 1, 2, …
N
t
t
0
Cov(
xx
,
)
E x
{(
P
)(
x
PN
)}
,
t
t
d
t
t
d
d
with
t
= 1, 2, …,
d
= ..., -2, -1, 0, 1, 2, ..., and where P,
0
N and
N are some
finite-value constants.
In statistical terms, a time series is stationary when the underlying stochastic
process is in a particular state of statistical equilibrium,
i.e.
when the joint
distributions of
X
(
t
) and
X
(
t-
W) depend only on W but not on
t
. Consequently, the
stationary model
of a time series can be easily built if the process (or the dynamics
generating the time series) remains in the equilibrium state for all times around a
constant mean level.
It is difficult to verify whether a given time series meets the three stationarity
conditions formulated above simultaneously. In earlier practice, the stationarity of
a time series was roughly checked by inspection of the time series pattern. A given
time series was recognized as stationary when it is represented by a flat-looking
pattern, with no trend or seasonality, and with time-invariant variance and
autocorrelation structure. When the time series model is available, the stationarity
of the process generating the time series observation values can be easily checked.
For instance, for the first-order autoregressive process
x
T
H
t
t
1
t
the stationarity condition requires that the condition
Var(
x
)
Var(
x
)
t
t
1
or the equality
Ex
{[
T
H
t
] }
2
Ex
{[
T
H
(
t
1)] }
2
t
1
t
2
holds. Therefore, because of mutual independence of
H and
x
, the equality
2
Var(
x
)
T
Var(
x
)
Var(
H
)
t
t
1
t
follows, and finally the equality
2
2
NTNV
, _T_ < < 1
0
0
where
N does not depend on time
t
.
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