GARCH Models

Appendix B

Autocorrelation and Partial Autocorrelation

B.1 Partial Autocorrelation

Definition

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

and

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

(B.1)

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

(B.2)

We have

(B.3)

(B.4)

and

(B.5)

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