96
8 Time Series
back of the output to the input. Again the nomenclature varies a little bit and we write
y i = φ 1 y i−1 + · · · + φ p y i− p + ε i =
p
j=1
φ j y i− j + ε i
(8.6)
to denote an AR( p) process.
It is not surprising that the combined effect of a moving average process and an
autoregressive process is called an ARMA( p, q) process and its functional description is the sum of the effect of a MA and an AR process
y i = φ 1 y i−1 + · · · + φ p y i− p + ε i − θ 1 ε i−1 − · · · − θ q ε i−q ,
(8.7)
where we use the conventionally used variables θ j and φ j to denote the coefficients
describing this ARMA( p, q) process. Here we only consider stationary processes in
the sense that the coefficients θ j and φ k are constant for the entire duration of the
time series that we record.
Let us now return to the CO 2 time series y i , shown in Fig. 8.4, and investigate the
suitability of the processes discussed in this section to characterize that time series.
We therefore need to to determine the coefficients θ j and φ j that generate the y i from
shocks ε i , as defined in (8.1). But first we need to find out which orders of processes
p and q are suitable. This is facilitated by the autocorrelation function (ACF) and
partial autocorrelation (PACF) function.
8.3 Auto-Covariance and Autocorrelation
We first consider the MA(q) process from (8.3) and calculate the auto-covariances
γ j , which are the averaged sums of the time series y i , weighted by the time series
shifted by j samples y i− j , where we assume that undefined samples are zero
γ j = =y i y i− j ,
(8.8)
where the angle brackets denote an average over different realizations of the noise
process ε i . Note that the auto-covariances γ j are symmetric in the sense that γ j = γ − j .
In order to explore how these quantities can be useful, let us calculate the autocorrelation γ j for an MA(q) process given by (8.3) with q = 1, thus y i = ε i − θ 1 ε i−1 .
For γ 0 we then obtain
γ 0 = =(ε i − θ 1 ε i−1 )
2
= =ε
2
i − 2θ 1 ε i ε i−1 + θ
2
1 ε
2
i−1
(8.9)
= σ
2
(1 + θ
2
1 ) ,
8 Time Series
back of the output to the input. Again the nomenclature varies a little bit and we write
y i = φ 1 y i−1 + · · · + φ p y i− p + ε i =
p
j=1
φ j y i− j + ε i
(8.6)
to denote an AR( p) process.
It is not surprising that the combined effect of a moving average process and an
autoregressive process is called an ARMA( p, q) process and its functional description is the sum of the effect of a MA and an AR process
y i = φ 1 y i−1 + · · · + φ p y i− p + ε i − θ 1 ε i−1 − · · · − θ q ε i−q ,
(8.7)
where we use the conventionally used variables θ j and φ j to denote the coefficients
describing this ARMA( p, q) process. Here we only consider stationary processes in
the sense that the coefficients θ j and φ k are constant for the entire duration of the
time series that we record.
Let us now return to the CO 2 time series y i , shown in Fig. 8.4, and investigate the
suitability of the processes discussed in this section to characterize that time series.
We therefore need to to determine the coefficients θ j and φ j that generate the y i from
shocks ε i , as defined in (8.1). But first we need to find out which orders of processes
p and q are suitable. This is facilitated by the autocorrelation function (ACF) and
partial autocorrelation (PACF) function.
8.3 Auto-Covariance and Autocorrelation
We first consider the MA(q) process from (8.3) and calculate the auto-covariances
γ j , which are the averaged sums of the time series y i , weighted by the time series
shifted by j samples y i− j , where we assume that undefined samples are zero
γ j = =y i y i− j ,
(8.8)
where the angle brackets denote an average over different realizations of the noise
process ε i . Note that the auto-covariances γ j are symmetric in the sense that γ j = γ − j .
In order to explore how these quantities can be useful, let us calculate the autocorrelation γ j for an MA(q) process given by (8.3) with q = 1, thus y i = ε i − θ 1 ε i−1 .
For γ 0 we then obtain
γ 0 = =(ε i − θ 1 ε i−1 )
2
= =ε
2
i − 2θ 1 ε i ε i−1 + θ
2
1 ε
2
i−1
(8.9)
= σ
2
(1 + θ
2
1 ) ,