Computes the sample partial autocorrelation function of a stationary time series.

The type double function is imsls_d_partial_autocorrelation.

Pointer to an array of length lagmax containing the partial autocorrelations of the time series x.

Function imsls_f_partial_autocorrelation estimates the partial autocorrelations of a stationary time series given the K = lagmax sample autocorrelations

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

where φkj 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 nonstationarity boundary. A possible alternative would be to estimate {φkk} 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.

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. Function imsls_f_partial_autocorrelation is used to compute the estimated partial autocorrelations.