Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function
Umberto Triacca
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universit`adell’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 unit of measurement of the random variables. Thus it is very difficult to evaluate the dependence of random variables of a stochastic process by using autocovariances.
A more useful measure of dependence, since it is dimensionless, is the 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 γx (k) ρx (k) = ∀k ∈ Z γx (0) 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 autocovariance function normalized so that ρx (0) = 1.
We note that ρx (k) measures the correlation between xt and xt−k
In fact, we have
γx (k) γx (k) Cov(xt , xt−k ) ρx (k) = = p p = p p γx (0) γx (0) γx (0) 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 autocovariance function is given by
T 1 X γˆ(k) = (x − x¯)(x − x¯) for k = 0, 1,..., T − 1. T t t−k t=k+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 autocovariance γ(k), by γ(k) ρ(k) = ∀k ∈ γ(0) Z a natural estimate of ρ(k) is
γˆ(k) ρˆ(k) = γˆ(0)
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function Estimation of Autocorrelation Function
When we calculate the sample autocorrelation,ρ ˆ(k), of any given series with a fixed sample size T , we cannot put too much confidence in the values ofρ ˆ(k) for large k’s, since fewer pairs of (xt−k , xt ) will be available for computingρ ˆ(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 evaluateρ ˆ(k) for k > T /3.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function The correlogram
The plot ofρ ˆ(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 ˆ(k), can be computed for any given time series and are not restricted to observations from a stationary 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 following figure.
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 autocorrelation function,ρ ˆ(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 (like seasonality), the sample autocorrelation function,ρ ˆk , exhibits a behavior with the same periodicity.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function The correlogram
Thusρ ˆ(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
−1 φk = Rk ρk with ρ(0) ρ(1) ··· ρ(k − 1) ρ(1) ρ(0) ··· ρ(k − 2) R = k . . .. . . . . . ρ(k − 1) ρ(k − 2) ··· ρ(0) 0 and ρk = (ρ(1), ..., ρ(k)) is called partial autocorrelation function of 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)
and xt−k − E(xt−k |xt−1, ..., xt−k+1)
Thus πx (k) measures the linear link between xt and xt−k once the influence 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 a stationary processes xt is given by the sequence
πˆx (0) = 1
πˆx (k) k ≥ 1
whereπ ˆx (k), is the last component of ˆ ˆ−1 φk = Γk γˆk
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function Estimation of Partial Autocorrelation Function
The plot ofπ ˆx (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 this example 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 this example are
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function Correlogram and Partial Correlogram
As we will see the correlogram and the partial correlogram will be used in the model identification stage for Box-Jenkins autoregressive moving average time series models.
Umberto Triacca Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function