100
8 Time Series
ρ 1 = φ 1 + φ 2 ρ 1 + · · · + φ p ρ p−1
ρ 2 = φ 1 ρ 1 + φ 2 + · · · + φ p ρ p−2
. . .
(8.20)
ρ p = φ 1 ρ p−1 + φ 2 ρ p−2 + · · · + φ p .
This set of equations can be cast into a matrix-valued equation with the result
⎛
⎜
⎜
⎜
⎜
⎜
⎝
ρ 1
ρ 2
. . .
ρ p−1
ρ p
⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1
ρ 1 . . . ρ p−1
ρ 1
1 . . . ρ p−2
. . .
. . .
. . .
. . .
ρ p−2 ρ p−3 . . . ρ 1
ρ p−1 ρ p−2 . . . 1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎝
φ 1
φ 2
. . .
φ p−1
φ p
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(8.21)
and, if the matrix is non-degenerate, we can determine the coefficients φ j by inverting
the matrix. We find
⎛
⎜
⎜
⎜
⎜
⎜
⎝
φ 1
φ 2
. . .
φ p−1
φ p
⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1
ρ 1 . . . ρ p−1
ρ 1
1 . . . ρ p−2
. . .
. . .
. . .
. . .
ρ p−2 ρ p−3 . . . ρ 1
ρ p−1 ρ p−2 . . . 1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
−1 ⎛
⎜
⎜
⎜
⎜
⎜
⎝
ρ 1
ρ 2
. . .
ρ p−1
ρ p
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,
(8.22)
provided that we know the order p of the AR( p)-process. But we do not know this
a-priori. However, after initially calculating the estimates of the autocorrelations, we
can set up (8.22) for increasing p = 1, 2, . . . , which results in a sequence of linear
systems of order p = 1, 2, . . . . At each step, we record the coefficient of the highest
order p, which we call φ pp . This coefficient is small if all information is already
accounted for by the lower-order coefficients. Conversely, if it is large, it carries new
information. By redoing the fit with progressively increasing order p and always
saving the highest-order coefficient φ pp , we progressively check whether the new
highest coefficient carries new information. Moreover, if the order of the process that
generated the time-series from which the autocorrelations are derived, is ˆ
p, all φ pp
with p > ˆ
p should be zero or at least small. The coefficients φ pp are called partial
auto correlation function (PACF). Since they are small or zero beyond the value of
ˆ
p, characteristic for the underlying process, they provide a cutoff that helps us to
identify the largest p needed to adequately describe the process dynamics.
In Fig. 8.6 we display the PACF derived from the autocorrelations depicted in
Fig. 8.5 for our CO 2 data with 95% confidence level around zero indicated as red
lines. Here we find that the coefficients for p = 1 and 12 exceed the 95% confidence
level significantly and those at p = 7, 9, and 13 only slightly. In the spirit of building a
parsimonious model we dare to neglect the latter, because the lowest order model with
p = 1 contains most of the dynamics and the p = 12 component is a residual seasonal
8 Time Series
ρ 1 = φ 1 + φ 2 ρ 1 + · · · + φ p ρ p−1
ρ 2 = φ 1 ρ 1 + φ 2 + · · · + φ p ρ p−2
. . .
(8.20)
ρ p = φ 1 ρ p−1 + φ 2 ρ p−2 + · · · + φ p .
This set of equations can be cast into a matrix-valued equation with the result
⎛
⎜
⎜
⎜
⎜
⎜
⎝
ρ 1
ρ 2
. . .
ρ p−1
ρ p
⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1
ρ 1 . . . ρ p−1
ρ 1
1 . . . ρ p−2
. . .
. . .
. . .
. . .
ρ p−2 ρ p−3 . . . ρ 1
ρ p−1 ρ p−2 . . . 1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎝
φ 1
φ 2
. . .
φ p−1
φ p
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(8.21)
and, if the matrix is non-degenerate, we can determine the coefficients φ j by inverting
the matrix. We find
⎛
⎜
⎜
⎜
⎜
⎜
⎝
φ 1
φ 2
. . .
φ p−1
φ p
⎞
⎟
⎟
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1
ρ 1 . . . ρ p−1
ρ 1
1 . . . ρ p−2
. . .
. . .
. . .
. . .
ρ p−2 ρ p−3 . . . ρ 1
ρ p−1 ρ p−2 . . . 1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
−1 ⎛
⎜
⎜
⎜
⎜
⎜
⎝
ρ 1
ρ 2
. . .
ρ p−1
ρ p
⎞
⎟
⎟
⎟
⎟
⎟
⎠
,
(8.22)
provided that we know the order p of the AR( p)-process. But we do not know this
a-priori. However, after initially calculating the estimates of the autocorrelations, we
can set up (8.22) for increasing p = 1, 2, . . . , which results in a sequence of linear
systems of order p = 1, 2, . . . . At each step, we record the coefficient of the highest
order p, which we call φ pp . This coefficient is small if all information is already
accounted for by the lower-order coefficients. Conversely, if it is large, it carries new
information. By redoing the fit with progressively increasing order p and always
saving the highest-order coefficient φ pp , we progressively check whether the new
highest coefficient carries new information. Moreover, if the order of the process that
generated the time-series from which the autocorrelations are derived, is ˆ
p, all φ pp
with p > ˆ
p should be zero or at least small. The coefficients φ pp are called partial
auto correlation function (PACF). Since they are small or zero beyond the value of
ˆ
p, characteristic for the underlying process, they provide a cutoff that helps us to
identify the largest p needed to adequately describe the process dynamics.
In Fig. 8.6 we display the PACF derived from the autocorrelations depicted in
Fig. 8.5 for our CO 2 data with 95% confidence level around zero indicated as red
lines. Here we find that the coefficients for p = 1 and 12 exceed the 95% confidence
level significantly and those at p = 7, 9, and 13 only slightly. In the spirit of building a
parsimonious model we dare to neglect the latter, because the lowest order model with
p = 1 contains most of the dynamics and the p = 12 component is a residual seasonal