[R] pacf, round three

Martyn Plummer plummer at iarc.fr
Mon Jun 26 16:36:37 CEST 2000

I am the author of the offending help page.  I am sorry that you find
it ambiguous, but sending three long messages on a rather fine point of
interpretation risks exhausting the interest in the topic.

If I may reiterate Brian's response to your original message, solving
the Yule-Walker equations is a standard method for fitting autoregressive
models, and is the default method for the ar() function.  It is not clear,
to me, why you thought that OLS was used instead, unless you object to
the use of the word "fit" in this context.

If it will help, perhaps the help page could be changed to

     The partial correlation coefficient is estimated by fitting
     autoregressive models of successively higher orders up to
     `lag.max', by the Yule-Walker method (see ar).


On 26-Jun-00 Christian Kleiber wrote:
> 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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