lesson 7: estimation of autocorrelation and partial ... · lesson 7: estimation of autocorrelation...

Lesson 7: Estimation of Autocorrelation andPartial Autocorrelation Function

Umberto Triacca

Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversita dell’Aquila,

[email protected]

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:



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}.



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.



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.



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)



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.



One rule of thumb is not to evaluate ρ(k) for k > T/3.


The correlogram

The plot of ρ(k) vs. k , is called the correlogram.


The correlogram

Consider the time series represented in the following figure.


The correlogram

The correlogram for the data of this example is


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.


The correlogram

As an example consider the time series plotted in the followingfigure.



The correlogram

Let us consider the time series plotted in the following figure.



The correlogram

For the time series containing a trend, the sample autocorrelationfunction, ρ(k), exhibits slow decay as k increases.


The correlogram

For the time series containing a periodic component (likeseasonality), the sample autocorrelation function, ρk , exhibits abehavior with the same periodicity.


The correlogram

Thus ρ(k) can be useful as an indicator of nonstationarity.


The correlogram

Consider again the number of airline passengers

The correlogram for the data of this series is


The correlogram

Now, the first difference of the log tranformation

The correlogram for this series is


The correlogram

Finally, we consider the correlogramm of the series

Figure : (1− L)(1− L12) Log international airline passengers


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}.


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.



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



The plot of πx(k) vs. k is called the partial correlogram.


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


Correlogram and Partial Correlogram

As we will see the correlogram and the partial correlogram will beused in the model identification stage for Box-Jenkinsautoregressive moving average time series models.


lesson 7: estimation of autocorrelation and partial ... · lesson 7: estimation of autocorrelation...

Documents