Lesson 7: Estimation of Autocorrelation and Partial Autocorrela

Transcription

1 Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it

2 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:

3 Estimation of Autocorrelation Function Let {x t ; t Z} be a weakly stationary process. The function ρ x : Z R defined by ρ x (k) = γ x(k) γ x (0) k Z is called autocorrelation function of the weakly stationary process {x t ; t Z}.

4 Estimation of Autocorrelation Function We see that the autocorrelation function is the autocovariance function normalized so that ρ x (0) = 1. We note that ρ x (k) measures the correlation between x t and x t k In fact, we have ρ x (k) = γ x(k) γ x (0) = and hence 1 ρ x (k) 1. γ x (k) γx (0) γ x (0) = Cov(x t, x t k ) Var(xt ) Var(x t k )

5 Estimation of Autocorrelation Function We have seen that a consistent estimator for the autocovariance function is given by ˆγ(k) = 1 T T t=k+1 (x t x)(x t k x) for k = 0, 1,..., T 1.

6 Estimation of Autocorrelation Function Since the autocorrelation ρ(k) is related to the autocovariance γ(k), by ρ(k) = γ(k) γ(0) k Z a natural estimate of ρ(k) is ˆρ(k) = ˆγ(k) ˆγ(0)

7 Estimation of Autocorrelation Function When we calculate the sample autocorrelation, ˆρ(k), of any given series with a fixed sample size T, we cannot put too much confidence in the values of ˆρ(k) for large k s, since fewer pairs of (x t k, x t ) will be available for computing ˆρ(k) when k is large. Only one if k = T 1.

8 Estimation of Autocorrelation Function One rule of thumb is not to evaluate ˆρ(k) for k > T /3.

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

10 The correlogram Consider the time series represented in the following figure.

11 The correlogram The correlogram for the data of this example is

12 The correlogram The sample autocorrelation function, ˆρ(k), can be computed for any given time series and are not restricted to observations from a stationary stochastic process.

13 The correlogram As an example consider the time series plotted in the following figure. The correlogram for the data of this example is

14 The correlogram Let us consider the time series plotted in the following figure. The correlogram for the data of this example is

15 The correlogram For the time series containing a trend, the sample autocorrelation function, ˆρ(k), exhibits slow decay as k increases.

16 The correlogram For the time series containing a periodic component (like seasonality), the sample autocorrelation function, ˆρ k, exhibits a behavior with the same periodicity.

17 The correlogram Thus ˆρ(k) can be useful as an indicator of nonstationarity.

18 The correlogram Consider again the number of airline passengers The correlogram for the data of this series is

19 The correlogram Now, the first difference of the log tranformation The correlogram for this series is

20 The correlogram Finally, we consider the correlogramm of the series Figure : (1 L)(1 L 12 ) Log international airline passengers

21 The Partial Autocorrelation Function Definition. The function π : Z R defined by the equations π x (0) = 1 π x (k) = φ kk k 1 where φ kk, is the last component of with φ k = R 1 k ρ k ρ(0) ρ(1) ρ(k 1) ρ(1) ρ(0) ρ(k 2) R k = ρ(k 1) ρ(k 2) ρ(0) and ρ k = (ρ(1),..., ρ(k)) is called partial autocorrelation function of the stationary process {x t ; t Z}.

22 Estimation of Partial Autocorrelation Function It is possible to show that π x (k) = φ kk k 1 is equal to the coefficient of correlation between x t E(x t x t 1,..., x t k+1 ) and x t k E(x t k x t 1,..., x t k+1 ) Thus π x (k) measures the linear link between x t and x t k once the influence of x t 1,..., x t k+1 has been removed.

23 Estimation of Partial Autocorrelation Function The sample partial autocorrelation function (SPACF) of a stationary processes x t is given by the sequence ˆπ x (0) = 1 ˆπ x (k) k 1 where ˆπ x (k), is the last component of ˆφ k = 1 ˆΓ k ˆγ k

24 Estimation of Partial Autocorrelation Function The plot of ˆπ x (k) vs. k is called the partial correlogram.

25 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

26 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

27 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.