8.4 Partial Autocorrelation Function
99
Fig. 8.5 The
autocorrelations ρ j versus j
of the CO 2 concentration
with trend removed and
differenced with lag 12 to
remove seasonality. The red
lines denote the 95%
confidence levels
8.4 Partial Autocorrelation Function
First, we determine relations between the experimentally determinable autocorrelations ρ j and the sought-after parameters φ k . Such a relation for AR( p) processes can
be found by inserting the definition of the AR-process from (8.6) into the definition
of the auto-covariances. First we consider γ 0
γ 0 = =y i y i = φ 1 γ 1 + · · · + φ p γ p + σ
2
for j = 0 ,
(8.17)
where the term σ
2 comes from ε i y i = =ε i
ε i +
p
j=1 φ j y i− j
= σ
2
. For γ j with
j > 0 we calculate
γ j = =(φ 1 y i−1 + · · · + φ p y i− p + ε i )y i− j
= φ 1 y i−1 y i− j + · · · + φ p y i− p y i− j + +ε i y i− j
= φ 1 γ j−1 + · · · + φ p γ j− p
for j > 0,
(8.18)
where ε i y i− j = 0, because the shocks at later time i are uncorrelated to the time
series value y i− j that precede it.
Dividing the equations for γ j by γ 0 , we obtain the relation for the autocorrelations ρ j
ρ j = φ 1 ρ j−1 + φ 2 ρ j−2 + · · · + φ p ρ j− p .
(8.19)
with ρ 0 = 1 and ρ j = ρ − j . These equations are called the Yule-Walker equations.
They show that the autocorrelations ρ j obey the same recursion relation as the
original time series, but without the random shocks ε k .
If we write (8.19) for consecutive j, we obtain a system of equations for the AR( p)
coefficients φ j