98
8 Time Series
indicates the band that describes the 95% confidence level in the sense discussed in
Sect. 7.5.
After considering the MA processes and how to interpret the magnitude of the
coefficients in autocorrelation plots, we need to investigate the auto-covariances and
autocorrelations of the AR( p) processes from (8.6). As before, we first consider
the simplest process with p = 1, which is given by y i = φ 1 y i−1 + ε i . The autocovariances are given by (8.8) and can be calculated by considering a system at rest
y j = 0 for j < 0 before the sequence of shocks starts to excite the system. After the
first iteration we have y 1 = ε 1 and after the second iteration we have y 2 = ε 2 + φ 1 ε 1
and after the third y 3 = ε 3 + φ 1 ε 2 + φ
2
1 ε 1 . Obviously, the value of y i depends on the
contemporary and all previous shocks ε i− j with i ≥ 0 where each shock is weighted
by φ
j
1 . Generalizing this we can write
y i = ε i + φ 1 ε i−1 + φ
2
1 ε i−2 + · · · =
∞
j=0
φ
j
1 ε i− j .
(8.13)
In this form we can calculate the auto-covariances to be
γ 0 = =y i y i = σ
2
(1 + φ
2
1 + φ
4
1 + · · · )
γ j = =y i y i− j = σ
2
φ
j
1 (1 + φ
2
1 + φ
4
1 + · · · )
(8.14)
and, provided that |φ 1 | < 1, we can sum the series and arrive at
γ 0 =
σ
2
1 − φ
2
1
and
γ j =
σ
2
φ
j
1
1 − φ
2
1
.
(8.15)
The autocorrelations ρ j then turn out to be
ρ j =
γ j
γ 0
= φ
j
1 ,
(8.16)
which implies that the autocorrelations ρ j exponentially decay from j = 0 towards
larger j, which is the signature of an AR(1) process.
Returning to the data shown in Fig. 8.4, we display the ACF for the CO 2 data after
removing trend and seasonality in Fig. 8.5. The red lines denote the 95% confidence
level around zero. We see that a large number of autocorrelations are highly relevant
and no cutoff is visible. On the other hand there appears to be an exponential decay
of the ρ j as a function of the shift j which suggests that the underlying process is of
the AR type.
The large number of non-negligible autocorrelations makes it difficult to identify
the relevant parameters φ j directly. To remedy this deficiency, we discuss a method
to determine the most important parameters in the following section.
8 Time Series
indicates the band that describes the 95% confidence level in the sense discussed in
Sect. 7.5.
After considering the MA processes and how to interpret the magnitude of the
coefficients in autocorrelation plots, we need to investigate the auto-covariances and
autocorrelations of the AR( p) processes from (8.6). As before, we first consider
the simplest process with p = 1, which is given by y i = φ 1 y i−1 + ε i . The autocovariances are given by (8.8) and can be calculated by considering a system at rest
y j = 0 for j < 0 before the sequence of shocks starts to excite the system. After the
first iteration we have y 1 = ε 1 and after the second iteration we have y 2 = ε 2 + φ 1 ε 1
and after the third y 3 = ε 3 + φ 1 ε 2 + φ
2
1 ε 1 . Obviously, the value of y i depends on the
contemporary and all previous shocks ε i− j with i ≥ 0 where each shock is weighted
by φ
j
1 . Generalizing this we can write
y i = ε i + φ 1 ε i−1 + φ
2
1 ε i−2 + · · · =
∞
j=0
φ
j
1 ε i− j .
(8.13)
In this form we can calculate the auto-covariances to be
γ 0 = =y i y i = σ
2
(1 + φ
2
1 + φ
4
1 + · · · )
γ j = =y i y i− j = σ
2
φ
j
1 (1 + φ
2
1 + φ
4
1 + · · · )
(8.14)
and, provided that |φ 1 | < 1, we can sum the series and arrive at
γ 0 =
σ
2
1 − φ
2
1
and
γ j =
σ
2
φ
j
1
1 − φ
2
1
.
(8.15)
The autocorrelations ρ j then turn out to be
ρ j =
γ j
γ 0
= φ
j
1 ,
(8.16)
which implies that the autocorrelations ρ j exponentially decay from j = 0 towards
larger j, which is the signature of an AR(1) process.
Returning to the data shown in Fig. 8.4, we display the ACF for the CO 2 data after
removing trend and seasonality in Fig. 8.5. The red lines denote the 95% confidence
level around zero. We see that a large number of autocorrelations are highly relevant
and no cutoff is visible. On the other hand there appears to be an exponential decay
of the ρ j as a function of the shift j which suggests that the underlying process is of
the AR type.
The large number of non-negligible autocorrelations makes it difficult to identify
the relevant parameters φ j directly. To remedy this deficiency, we discuss a method
to determine the most important parameters in the following section.