8.3 Auto-Covariance and Autocorrelation
97
where we use (8.1) to evaluate the average over the shocks. For the other autocovariances γ j we find
γ 1 = =(ε i − θ 1 ε i−1 )(ε i−1 − θ 1 ε i−2 ) = −θ 1 σ
2
γ 2 = =(ε i − θ 1 ε i−1 )(ε i−2 − θ 1 ε i−3 ) = 0
(8.10)
and all higher γ j with j > 1 are zero as well. This is noteworthy, because it will help
us to estimate and characterize the order of the process when calculating estimates
of the covariances from a given time series.
The autocorrelations (ACF) ρ j are given by the auto-covariances γ j , divided by
γ 0
ρ j =
γ j
γ 0
,
(8.11)
which makes the autocorrelations dimensionless. They neither depend on the physical
units of the process nor the numerical magnitude of the time series values y i .
Generalizing to higher order MA(q)-processes we find that γ 0 and the γ j of
higher-order can be calculated in the same way as before. They are given by
γ 0 = =(ε i − θ 1 ε i−1 − · · · − θ q ε i−q )
2
= σ
2
(1 + θ
2
1 + · · · + θ
2
q )
(8.12)
γ j = σ
2
(−θ j + θ 1 θ j+1 + · · · + θ q− j θ q ) for 1 < j < q
γ j = 0
f o r j > q.
Again, we see that all coefficients for j > q vanish. The ACF ρ j are defined in the
same way as for the q = 1 case by normalizing the γ j by γ 0 as in (8.11). In practice we
calculate estimates for the γ j from the time series data y i such as from Fig. 8.4, plot
them and inspect whether some distinct peaks show up and whether the covariances
or correlations vanish beyond a cutoff, which we can interpret as the order q of the
process.
Let us briefly discuss what “vanish” or “small” means in this context with a
very heuristic and qualitative discussion. The auto-covariances are calculated from
the sum of n products y i y i− j and we can intuitively understand that an increasing
number of data points n will improve the precision to which the auto-covariances can
be calculated. Since the random component of the y i are uncorrelated, we can assume
that the product of two such parameters are uncorrelated. The random component of
coefficient γ j will therefore grow like
√
n while the magnitude of γ j and also γ 0 will
grow linearly with n. From these two dependencies on the length of the time series
n we can intuitively understand that the random component of the autocorrelations
ρ j , will decrease as 1/
√
n, where this indicates the standard error of the random
component. See Quenouille [4] for a more detailed discussion of this dependency.
But we can use this information to specify what we mean by “small,” namely that the
coefficient ρ j is small compared to two times the standard error or ±2/
√
n, which
97
where we use (8.1) to evaluate the average over the shocks. For the other autocovariances γ j we find
γ 1 = =(ε i − θ 1 ε i−1 )(ε i−1 − θ 1 ε i−2 ) = −θ 1 σ
2
γ 2 = =(ε i − θ 1 ε i−1 )(ε i−2 − θ 1 ε i−3 ) = 0
(8.10)
and all higher γ j with j > 1 are zero as well. This is noteworthy, because it will help
us to estimate and characterize the order of the process when calculating estimates
of the covariances from a given time series.
The autocorrelations (ACF) ρ j are given by the auto-covariances γ j , divided by
γ 0
ρ j =
γ j
γ 0
,
(8.11)
which makes the autocorrelations dimensionless. They neither depend on the physical
units of the process nor the numerical magnitude of the time series values y i .
Generalizing to higher order MA(q)-processes we find that γ 0 and the γ j of
higher-order can be calculated in the same way as before. They are given by
γ 0 = =(ε i − θ 1 ε i−1 − · · · − θ q ε i−q )
2
= σ
2
(1 + θ
2
1 + · · · + θ
2
q )
(8.12)
γ j = σ
2
(−θ j + θ 1 θ j+1 + · · · + θ q− j θ q ) for 1 < j < q
γ j = 0
f o r j > q.
Again, we see that all coefficients for j > q vanish. The ACF ρ j are defined in the
same way as for the q = 1 case by normalizing the γ j by γ 0 as in (8.11). In practice we
calculate estimates for the γ j from the time series data y i such as from Fig. 8.4, plot
them and inspect whether some distinct peaks show up and whether the covariances
or correlations vanish beyond a cutoff, which we can interpret as the order q of the
process.
Let us briefly discuss what “vanish” or “small” means in this context with a
very heuristic and qualitative discussion. The auto-covariances are calculated from
the sum of n products y i y i− j and we can intuitively understand that an increasing
number of data points n will improve the precision to which the auto-covariances can
be calculated. Since the random component of the y i are uncorrelated, we can assume
that the product of two such parameters are uncorrelated. The random component of
coefficient γ j will therefore grow like
√
n while the magnitude of γ j and also γ 0 will
grow linearly with n. From these two dependencies on the length of the time series
n we can intuitively understand that the random component of the autocorrelations
ρ j , will decrease as 1/
√
n, where this indicates the standard error of the random
component. See Quenouille [4] for a more detailed discussion of this dependency.
But we can use this information to specify what we mean by “small,” namely that the
coefficient ρ j is small compared to two times the standard error or ±2/
√
n, which