[R] pacf, round three

[Christian Kleiber]

Dear Professor Ripley,

> I don't see how the theoretical PACF can be defined by methods of
> estimation.  Partial correlation coefficients have nothing to do with OLS.

I should have written 'regression' instead of 'OLS'. I agree that, in general,
partial correlation coefficients and regression coefficients are not the same thing.
Yet in the case of univariate stationary processes the two concepts happen to
coincide, in the sense that the coefficient on x_1 in the regression of x_k+1 on x_k,
..., x_1 is also the k-th partial autocorrelation coefficient (this is implicit in
the proof of Proposition 5.2.1 of Brockwell and Davis, 1991, it is explicitly stated
in e.g. Hannan, 1970, pp. 21-22). Without this coincidence it would in fact not be
possible to extract the PACF directly from the Durbin-Levinson recursion. After all,
the latter is an algorithm for computing regression coefficients and the Yule-Walker
equations are estimating equations. To put it differently: if the OLS estimate of the
PACF really was invalid, the YW estimate would be invalid too.

BTW, apart from Box and Jenkins, Priestley (1981, p. 371) also mentions both
estimators.

> > In the S-PLUS Help on acf() one finds
> >
> > "... For the partial autocorrelation function, the Levinson-Durbin recursion is
> > used to fit AR(p) models to x successively for p = 1, ..., lag.max, and from
> > the AR-coefficients the partial autocorrelation function values are derived."
> >
> > This is unambiguous. I feel R users would be better off if the algorithm
>
> No, it isn't. To what is the Levinson-Durbin recursion applied?
> There are lots of the correlation matrix estimates to start it.

I agree. I wrote "..." for brevity, in full "..." reads

"The autocovariance function is estimated by summing the lagged products and dividing
by the length of the series. For the autocorrelation function, all covariances are
further divided by the geometric mean of the corresponding variances."

Regards,
Christian Kleiber


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