1. Identification
    1. Autocorrelation Check for White Noise
      1. Empirical Autocorrelations of a GARCH
      2. Portmanteau Tests
      3. Sample Partial Autocorrelations (SPAC) of a GARCH
      4. Numerical Illustrations
    2. Identifying the ARMA Orders
      1. Sample Autocorrelations of an ARMA-GARCH
      2. Case of a Non Symmetric Noise
      3. Identifying the Orders (P,Q)
    3. Identifying the GARCH Orders
      1. Corner Method in the GARCH Case
      2. Applications
    4. LM Test for Conditional Homoscedasticity
      1. General Form of the LM Test
      2. LM Test for Conditional Homoscedasticity
    5. Application to Real Serie
    6. Bibliographical Notes
    7. Exercises

Christian Francq and Jean-Michel Zakoïan

Keywords: ARCH Effect Test, ARMA Identification, Coefficient of determination, CLT (Central Limit Theorem) for Stationary Martingale Differences, Corner Method, GARCH(p,q) Identification, Generalized Bartlett's Formula, HAC (Heteroscedasticity and Autocorrelation Consistent), Lagrange multiplier test, Long Run Variance, Partial Autocorrelation, Portmanteau Test, Sample Autocorrelation, Score test.

Description: In this chapter, we consider the problem of selecting an appropriate GARCH or ARMA-GARCH model for given observations X1, . . . ,Xn of a centered stationary process. A large part of the Finance theory rests on the assumption that prices follow a random walk. The price variations process, X = (Xt), should thus constitute a martingale difference sequence, and should coincide with its innovation process ε = (εt). The first question addressed within this chapter, in Section 5.1, will then be the test of this property, at least a consequence of it: absence of correlation. The problem is far from being 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) to the data before using a GARCH(p,q) 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 tests) will be considered in Section 5.4. 5.1 Autocorrelation Check for White Noise Let the

R programs used to R Figure 5.1 (autocorrelations of exchange rates) , R Figures 5.7 (autocorrelations of the CAC and FTSE returns) and C.4 (autocorrelations of the SP 500 and DAX returns)

Data sets used for Figures 5.7 and C.4 données CAC 40 , données DAX ,data FTSE ,data SP500 (obtained from yahoo).

Fortran 90 program for the portmanteau tests (tables similar to Table 5.7) : F90 F90 code for Absoft's compiler (needs to be adapted for other Fortran 90 compiler) , Doc a file with the FTSE data of Table 5.7, Exe executable program.

Fortran 90 program for identifying ARMA-GARCH with the corner method (similar to Tables 5.10-5.11) : F90 F90 code for Absoft's compiler , Exe executable program.