GARCH Models, 2nd Edition

5 Identification

In this chapter, we consider the problem of selecting an appropriate GARCH or ARMA‐GARCH model for given observations X ₁,…, X _n of a centred stationary process. A large part of the theory of finance rests on the assumption that prices follow a random walk. The price variation process, X = (X _t ), should thus constitute a martingale difference sequence, and should coincide with its innovation process, ε = (ε _t ). The first question addressed in this chapter, in Section 5.1, will be the test of this property, at least a consequence of it: absence of correlation. The problem is far from trivial because standard tests for non‐correlation are actually valid under an independence assumption. Such an assumption is too strong for GARCH processes which are dependent though uncorrelated.

If significant sample autocorrelations are detected in the price variations – in other words, if the random walk assumption cannot be sustained – the practitioner will try to fit an ARMA(P, Q) model to the data before using a GARCH(p, q) model for the residuals. Identification of the orders (P, Q) will be treated in Section 5.2, identification of the orders (p, q) in Section 5.3. Tests of the ARCH effect (and, more generally, Lagrange multiplier (LM) tests) will be considered in Section 5.4.

5.1 Autocorrelation Check for White Noise

Consider the GARCH(p, q) model

5.1

with (η _t ) a sequence of iid centred variables with unit variance, ω > 0, α _i  ≥ 0 (i = 1,…, ...