Background
Michael John McAleer with coauthors has in multiple articles (2013, 2019a, 2019b and other) criticized the BEKK, DCC and VCC sorts of multivariate GARCH models on the grounds that there is no stochastic process that leads to the model / that the model is based on / that the model relies on (these are the different formulations used by the author in different papers). I am trying to grasp what exactly this means, what specifications of stochastic processes are permitted and why this is necessary for establishing the properties of the models and/or estimators.
Beef
Since BEKK, DCC and VCC are all rather complex models, I would like to examine this question in a simpler case of ARCH(1). In McAleer (2019b), the author cites Tsay (1987) who showed that the ARCH(1) model can be derived from a random coefficient autoregressive stochastic process of order 1, RCAR(1), which is specified as
$$
\varepsilon_t=\phi_t\varepsilon_{t-1}+\eta_t
$$
where
\begin{aligned}
\phi_t &\sim iid(0, \alpha), \\
\eta_t &\sim iid(0, \omega) \\
\end{aligned}
and $\eta_t = \varepsilon_t/\sqrt{h_t}$ is the standardized residual. From this it is possible to derive that
$$
h_t = \mathbb{E}(\varepsilon_t^2|I_{t−1}) = \omega + \alpha\varepsilon_{t-1}^2,
$$
which is the conditional variance equation of the ARCH(1) model. Recall that
the classical specification of ARCH(1) is
\begin{aligned}
\varepsilon_t &= \sqrt{h_t}\eta_t, \\
h_t &= \omega+\alpha\varepsilon_{t-1}^2, \\
\eta_t &\sim iid(0,1),
\end{aligned}
assuming away any additive elements in the conditional mean specification besides the error term itself.
McAleer (2019b) further states
It should be emphasized that the random coefficient autoregressive process is a sufficient condition to derive ARCH, but to date the ARCH specification has not been derived from any other known underlying stochastic process.
This appears to be implying that before the derivation of the conditional variance equation of ARCH(1) from the RCAR(1) model by Tsay (1987), even the simplest form of autoregressive conditional heteroskedasticity models, the vanilla ARCH(1), could have been criticized in the same vein as DCC, VCC and BEKK now are. This leaves me wondering what the problem with the classical definition of ARCH(1) is.
Question: Why/How is the derivation of the ARCH(1) conditional variance equation from the RCAR(1) process somehow more valid (in terms of model and/or estimator properties) than the classical specification of ARCH(1)?
References
- Caporin, M., and M. McAleer (2013). Ten Things You Should Know About DCC. Unpublished working paper. Padova, Italy: University of Padova and Rotterdam: Erasmus University.
- McAleer, M. (2019). What they did not tell you about algebraic (non-) existence, mathematical (ir-) regularity and (non-) asymptotic properties of the dynamic conditional correlation (DCC) model. Journal of Risk and Financial Management, 12(2), 61.
- McAleer, M. (2019). What they did not tell you about algebraic (non-) existence, mathematical (ir-) regularity and (non-) asymptotic properties of the full BEKK dynamic conditional covariance model. Journal of Risk and Financial Management, 12(2), 66.
- Tsay, R. S. (1987). Conditional heteroscedastic time series models. Journal of the American Statistical Association, 82(398), 590-604.
