IDL Analyst Reference Guide: Time Series and Forecasting

IMSL_PARTIAL_AC

The IMSL_PARTIAL_AC function computes the sample partial autocorrelation function of a stationary time series.

Syntax

Result = IMSL_PARTIAL_AC(cf [, /DOUBLE])

Return Value

One-dimensional array containing the partial auto-correlations of the time series x.

Arguments

cf

One-dimensional array containing the auto-correlations of the time series x.

Keywords

DOUBLE

If present and nonzero, double precision is used.

Discussion

IMSL_PARTIAL_AC estimates the partial auto-correlations of a stationary time series given the K = (N_ELEMENTS(cf) – 1) sample auto-correlations:

for k = 0, 1, ..., K. Consider the AR(k) process defined by:

where f_kj denotes the j-th coefficient in the process. The set of estimates:

for k = 1, ..., K is the sample partial autocorrelation function. The autoregressive parameters:

for j = 1, ..., k are approximated by Yule-Walker estimates for successive AR(k) models where k = 1, ..., K. Based on the sample Yule-Walker equations:

a recursive relationship for k = 1, ..., K was developed by Durbin (1960). The equations are given by:

and:

This procedure is sensitive to rounding error and should not be used if the parameters are near the non-stationary boundary. A possible alternative would be to estimate {f_kk} for successive AR(k) models using least or maximum likelihood. Based on the hypothesis that the true process is AR(p), Box and Jenkins (1976, page 65) note:

See Box and Jenkins (1976, pages 82–84) for more information concerning the partial autocorrelation function.

Example

Consider the Wolfer Sunspot Data (Anderson 1971, page 660) consisting of the number of sunspots observed each year from 1749 through 1924. The data set for this example consists of the number of sunspots observed from 1770 through 1869. Routine IMSL_PARTIAL_AC is used to compute the estimated partial auto-correlations.

data  =  IMSL_STATDATA(2)  
x  =  data(21:120,1)  
result  =  IMSL_AUTOCORRELATION(x, 20)  
partial  =  IMSL_PARTIAL_AC(result)  
PRINT, 'LAG      PACF'  
FOR i  =  0, 19 DO PM, i + 1, partial(i), FORMAT = '(I2, F11.3)'  
  
LAG      PACF  
 1      0.806  
 2     -0.635  
 3      0.078  
 4     -0.059  
 5     -0.001  
 6      0.172  
 7      0.109  
 8      0.110  
 9      0.079  
10      0.079  
11      0.069  
12     -0.038  
13      0.081  
14      0.033  
15     -0.035  
16     -0.131  
17     -0.155  
18     -0.119  
19     -0.016  
20     -0.004