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Lesson 7: Estimation of Autocorrelation andPartial Autocorrelation Function
Umberto Triacca
Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversita dell’Aquila,
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Autocorrelation Function
We observe that the autocovariance function depends on the unitof measurement of the random variables. Thus it is very difficult toevaluate the dependence of random variables of a stochasticprocess by using autocovariances.
A more useful measure of dependence, since it is dimensionless, isthe autocorrelation function, defined as follows:
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Autocorrelation Function
Let {xt ; t ∈ Z} be a weakly stationary process.The function
ρx : Z→ R
defined by
ρx(k) =γx(k)
γx(0)∀k ∈ Z
is called autocorrelation function of the weakly stationary process{xt ; t ∈ Z}.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Autocorrelation Function
We see that the autocorrelation function is the autocovariancefunction normalized so that ρx(0) = 1.
We note that ρx(k) measures the correlation between xt and xt−k
In fact, we have
ρx(k) =γx(k)
γx(0)=
γx(k)√γx(0)
√γx(0)
=Cov(xt , xt−k)√
Var(xt)√
Var(xt−k)
and hence −1 ≤ ρx(k) ≤ 1.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Autocorrelation Function
We have seen that a consistent estimator for the autocovariancefunction is given by
γ(k) =1
T
T∑t=k+1
(xt − x)(xt−k − x) for k = 0, 1, . . . ,T − 1.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Autocorrelation Function
Since the autocorrelation ρ(k) is related to the autocovarianceγ(k), by
ρ(k) =γ(k)
γ(0)∀k ∈ Z
a natural estimate of ρ(k) is
ρ(k) =γ(k)
γ(0)
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Autocorrelation Function
When we calculate the sample autocorrelation, ρ(k), of any givenseries with a fixed sample size T , we cannot put too muchconfidence in the values of ρ(k) for large k ’s, since fewer pairs of(xt−k , xt) will be available for computing ρ(k) when k is large.Only one if k = T − 1.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Autocorrelation Function
One rule of thumb is not to evaluate ρ(k) for k > T/3.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
The plot of ρ(k) vs. k , is called the correlogram.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
Consider the time series represented in the following figure.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
The correlogram for the data of this example is
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
The sample autocorrelation function, ρ(k), can be computed forany given time series and are not restricted to observations from astationary stochastic process.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
As an example consider the time series plotted in the followingfigure.
The correlogram for the data of this example is
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
Let us consider the time series plotted in the following figure.
The correlogram for the data of this example is
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
For the time series containing a trend, the sample autocorrelationfunction, ρ(k), exhibits slow decay as k increases.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
For the time series containing a periodic component (likeseasonality), the sample autocorrelation function, ρk , exhibits abehavior with the same periodicity.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
Thus ρ(k) can be useful as an indicator of nonstationarity.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
Consider again the number of airline passengers
The correlogram for the data of this series is
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
Now, the first difference of the log tranformation
The correlogram for this series is
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The correlogram
Finally, we consider the correlogramm of the series
Figure : (1− L)(1− L12) Log international airline passengers
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
The Partial Autocorrelation Function
Definition. The function π : Z→ R defined by the equations
πx(0) = 1
πx(k) = φkk k ≥ 1
where φkk , is the last component of
φφφk = R−1k ρρρk
with
Rk =
ρ(0) ρ(1) · · · ρ(k − 1)ρ(1) ρ(0) · · · ρ(k − 2)...
.... . .
...ρ(k − 1) ρ(k − 2) · · · ρ(0)
and ρρρk = (ρ(1), ..., ρ(k))′ is called partial autocorrelation functionof the stationary process {xt ; t ∈ Z}.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Partial Autocorrelation Function
It is possible to show that
πx(k) = φkk k ≥ 1
is equal to the coefficient of correlation between
xt − E (xt |xt−1, ..., xt−k+1)
andxt−k − E (xt−k |xt−1, ..., xt−k+1)
Thus πx(k) measures the linear link between xt and xt−k once theinfluence of xt−1, ..., xt−k+1 has been removed.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Partial Autocorrelation Function
The sample partial autocorrelation function (SPACF) of astationary processes xt is given by the sequence
πx(0) = 1
πx(k) k ≥ 1
where πx(k), is the last component of
φφφk = Γ−1k γγγk
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Estimation of Partial Autocorrelation Function
The plot of πx(k) vs. k is called the partial correlogram.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Correlogram and Partial Correlogram
Let us consider the time series plotted in the following figure.
The correlogram and the partial correlogram for the data of thisexample are
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Correlogram and Partial Correlogram
Let us consider the time series plotted in the following figure.
The correlogram and the partial correlogram for the data of thisexample are
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function