1. Why does the autocorrelation matter when making inferences?
Consider estimation of the mean of a stationary series
E[X(t)]=m for all t
If X(1),…,X(n) are iid what is the sampling distribution of the estimator for m, namely the sample mean?
Spring 2005
K. Ensor, STAT 421

2. Why does the autocorrelation matter?
What if X(t) has the following structure (autoregressive model of order 1 AR(1) )
X(t)-m = a (X(t-1)-m ) + e (t)
Then Corr(X(t),X(t+h))= a|h| for all h
And Var(X)= (1+ a)/(1- a) Var(X)/n
Spring 2005
K. Ensor, STAT 421

3. Comparing the samples size?
Let m denote the number of iid obs.
Let n denote the number of correlated obs.
Setting the variances equal and solving for m as a function of n yields
m=n(1- a)/(1+a)
Let n=100, a=.9 then m=5 iid obs.
If n=100 and a=-.9 then the equivalent number of iid observations is 1900.
For positive and negative a (correlation of lag 1) the equivalent sample sizes are 33 and 300.
Spring 2005
K. Ensor, STAT 421

4. Why?
Why does the autocorrelation make such a big difference in our ability to estimate the mean?
The same arguments hold for other mean functions of the process or other functions of the process we want to estimate.
Spring 2005
K. Ensor, STAT 421

5. Summary
Times series is a sequentially observed series exhibiting correlation between the observations.
The autocorrelation, partial autocorrelation and cross-correlations are measures of the this correlation.
This dependence structure along with proper assumptions allows us to forecast the future of the process.
Correct inference requires incorporating knowledge of the dependence structure.
Spring 2005
K. Ensor, STAT 421