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GARCH Models
book

GARCH Models

by Christian Francq, Jean-Michel Zakoian
August 2010
Intermediate to advanced content levelIntermediate to advanced
504 pages
12h 59m
English
Wiley
Content preview from GARCH Models

Appendix B

Autocorrelation and Partial Autocorrelation

B.1 Partial Autocorrelation

Definition

The (theoretical) partial autocorrelation at lag h > 0, rX(h), of a second-order stationary process X = (Xt) with nondegenerate linear innovations,1 is the correlation between

bapp02ue001_fmt

and

bapp02ue002_fmt

where EL(Y|Y1,…, Yk) denotes the linear regression of a square integrable variable Y on variables Y1,…,Yk. Let

(B.1) bapp02e001_fmt

The number rX(h) can be interpreted as the residual correlation between Xt and Xth, after the linear influence of the intermediate variables Xt−1, Xt−2,…, Xth+1 has been subtracted. Assume that (Xt) is centered, and consider the linear regression of Xt on Xt−1,…, Xth:

(B.2) bapp02e002_fmt

We have

(B.3) bapp02e003_fmt

(B.4) bapp02e004_fmt

and

(B.5) bapp02e005_fmt

Proof of (B.3) and (B.4). We obtain (B.3) from (B.2), using the linearity of EL(·|Xt−1

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Publisher Resources

ISBN: 9780470683910