Simultaneous estimation of ARMA and GARCH components

Question

I am not allowed to comment on older questions relevant to this question, therefore I ask this myself. The question was dealt with here, here and here with a clear result: the ARMA part should be estimated simulataneously. However, I wonder how to show or prove clearcut that the ARMA coefficients are inconsistent and the ACF bounds are invalid? Further, why is it not possible to assume the regular null distribution under GARCH-type errors when conducting the Ljung-Box test? Does someone have relevant resources related to this issues?

Answer 1

[W]hy is it not possible to assume the regular null distribution under GARCH-type errors when conducting the Ljung-Box test?

Wooldridge "On the application of robust, regression-based diagnostics to models of conditional means and conditional variances" (1991) is a very relevant source. It says (emphasis in bold is mine):

One popular LM-based procedure begins with an estimated model of the conditional mean, usually under the assumption of homoskedasticity. The residuals are used to compute regression-based tests for serial correlation, nonlinearities, nonnested alternatives, and various other types of misspecification. Once the model for the conditional mean is deemed satisfactory on the basis of such tests, models for the conditional variance - such as ARCH models [Engle (1982a)] - are estimated and tested. At a minimum the careful researcher checks for evidence of heteroskedasticity or, more generally, for violation of whatever conditional variance assumption that has been maintained in the computation of the conditional mean tests. <...>

While this ‘bottom-up’ approach follows a natural progression, it suffers from logical inconsistency. The problem is that at any given stage the validity of standard specification tests relies on auxiliary assumptions that are tested only at a future stage. This has been observed by several authors, including White (1981, 1989), Bera and Jarque (1982), Pagan and Hall (1983), and Godfrey (1987). As an example, consider the usual LM test for serial correlation in a dynamic model. The LM test is invalid in the presence ot ARCH or other forms of conditional heteroskedasticity. <...>

I recommend skimming through the whole article. Other relevant references recommended to me by Jean-Michel Zakoian are as follows:

• Francq et al. "Diagnostic checking in ARMA models with uncorrelated errors" (2005);
• Duchesne & Francq "On Diagnostic Checking Time Series Models with Portmanteau Test Statistics Based on Generalized Inverses and {2}-Inverses" (2008);
• Section 8.4 of Francq & Zakoian "GARCH models: structure, statistical inference and financial applications" (2011).