In statistics, the order of integration, denoted I(d), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series (i.e., a time series whose mean and autocovariance remain constant over time).
The order of integration is a key concept in time series analysis, particularly when dealing with non-stationary data that exhibits trends or other forms of non-stationarity.
Integration of order d
A time series is integrated of order d if
-
(
1
−
L
)
d
X
t
{\displaystyle (1-L)^{d}X_{t}\ }
is a stationary process, where
L
{\displaystyle L}
-
(
1
−
L
)
X
t
=
X
t
−
X
t
−
1
=
Δ
X
.
{\displaystyle (1-L)X_{t}=X_{t}-X_{t-1}=\Delta X.}
In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process.
In particular, if a series is integrated of order 0, then
(
1
−
L
)
0
X
t
=
X
t
{\displaystyle (1-L)^{0}X_{t}=X_{t}}
Constructing an integrated series
An I(d) process can be constructed by summing an I(d − 1) process:
- Suppose
X
t
{\displaystyle X_{t}}
is I(d − 1)
- Now construct a series
Z
t
=
∑
k
=
0
t
X
k
{\displaystyle Z_{t}=\sum _{k=0}^{t}X_{k}}
- Show that Z is I(d) by observing its first-differences are I(d − 1):
-
Δ
Z
t
=
X
t
,
{\displaystyle \Delta Z_{t}=X_{t},}
-
Δ
Z
t
=
X
t
,
{\displaystyle \Delta Z_{t}=X_{t},}
- where
-
X
t
∼
I
(
d
−
1
)
.
{\displaystyle X_{t}\sim I(d-1).\,}
-
X
t
∼
I
(
d
−
1
)
.
{\displaystyle X_{t}\sim I(d-1).\,}
See also
References
- Hamilton, James D. (1994) Time Series Analysis. Princeton University Press. p. 437. ISBN 0-691-04289-6.