Characterizing Time Series: The Autocorrelation Function
The autocorrelation function is very useful because it provides a partial
description of the process for modeling purposes.
The autocorrelation function tells us how much correlation there is and
how much interdependency there is between neighboring data points in the
series Yt .
Define the autocorrelation with lag K as
E [(Yt  Y )(Yt K  Y )]
K 

E [(Yt  Y )2 ]E [(Yt K  Y )2 ]
cov(Yt ,Yt K )
 t t K
For a stationary series, the variance at t is the same as the variance at time
t  K , thus the denominator is just the variance of Yt ,
K 
E [(Yt  Y )(Yt K  Y )]
 Y2
The numerator is the covariance between Yt and Yt  K , or
K ,
So we have
K 
K
0
Notice that by definition: 0  1 for any stochastic process.
Example:
Yt   t
Where  t is an independently distributed random variable with zero mean, then
the autocorrelation function for this process is given by
0  1, and K  0 for K  0.
This process is called white noise, and there is no model that can provide a
forecast any better than Y t  l  0, for all l . Thus, if the autocorrelation function is
zero, or close to zero for all K  0, there is little or no value in using a model
to forecast the series.
Estimation: How to Calculate An Estimate of the Autocorrelation
Function
The sample autocorrelation function of Yt is
T K
K 

t 1
(Yt  Y ) * (Yt K  Y )
T
 (Yt
t 1
 Y )2
The autocorrelation functions are symmetrical, i.e., that the correlation for a
positive displacement is the same as that for a negative displacement, so that
K  K
When plotting an autocorrelation function (plotting  K for different values of K ),
one need consider only positive values of K .
Spreadsheet Example.
How to Test a Particular Value of PK Equal to Zero
If Yt has been generated by a white noise process, the sample
autocorrelation coefficients (K  0) are approximately distributed according to a
1
normal distribution with mean zero and standard deviation
, where T is the
T
1
number of observations in the series. For example: T  100 ,
 0.1, which is
T
a standard error to each autocorrelation coefficient. For example, if a particular
coefficient was in magnitude greater than 0.2, we could be 95 percent sure that
the time autocorrelation coefficient is not zero.
How to Test the Joint Hypothesis That All the Autocorrelation
Coefficients Are Zero
Box and Pierce’s Q test
M
Q  T  K2
K 1
is approximately distributed as  2 with M degree of freedom. Thus, if the
calculated value of Q is greater than the critical 5 percent level, we can be 95
percent sure that the autocorrelation coefficients 1, 2 ,....M are not all zero.
SAS Examples.
PROC ARIMA;
IDENTIFY statement.
SAS Simulated Series.
The Partial Autocorrecation Function



The partial autocorrelation function can be used to determine the order of
AR processes.
The reason why it is called partial autocorrelation function is because it
describes the correlation between Yt and Yt  K minus the part explained
linearly by the intervening lags.
The idea here is to use Yule-Walker equations to solve for successive
values of p , which is the order of the AR process.
Example
Suppose we start from the assumption that the autoregressive order is
one, i.e. p  1 . Then we have 1 = 1, or sample autocorrelation ˆ1  ˆ1. If
the calculated value ˆ , is significantly different from zero,  the
1
autoregressive order is at least one (using  1 to denote ˆ1.)
Now, consider p  2 . Solving Yule-Walker equations for p  2 ,
 ˆ and ˆ . If ˆ is significantly different from zero,  the process is
1
2
2
at least order 2 (denote  2  ˆ2 ) . If ˆ2 is approximately zero,  the order
is one.
Repeating the process and get 1,...  . We call 1,...  the partial
autocorrelation function and we can determine the order of AR processes
from its behavior.
In particular, if the true order is p , then  j  0 for  j  P .

To test  j and see if it is zero, we use the fact that it is
approximately normally distributed with mean zero and standard
1
2
error
. So 5 percent level is to see whether it exceeds
in
T
T
magnitude.