GARCH-M(1,1) where ARMA(0,0) is "removed" in R Which of the following is the correct code for fitting a GARCH-M(1,1) model where the ARMA(0,0) is "removed"? Or what is the correct code?


*

*modm11 = ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1,1)), 
                     mean.model=list(archm=T, archpow=2, include.mean=F))

*modm11 = ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1,1)), 
                     mean.model=list(armaOrder=c(0,0), archm=T, archpow=2, 
                     include.mean=F))

*modm11 = ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1,1)), 
                     mean.model=list(armaOrder=c(0,0), archm=T, archpow=2))
 A: The ARFIMAX(p, d, q)-GARCHX-in-mean(l, m) process as encoded in the rugarch package consists of 
the conditional mean specification and the conditional variance specification.
Conditional mean specification
The process is specified as
$$
\boldsymbol{\phi}(L)\left(1-L\right)^d\left(Y_t - \mu_t\right) = \boldsymbol{\theta}(L)\varepsilon_t 
$$
where $d\in(0, 1)$ is the order of fractional integration, $\boldsymbol{\phi}(L)$, 
 and $\boldsymbol{\theta}(L)$ (controlled by armaOrder) are the autoregressive and moving averages lag polynomials 
 of order $p$ and $q$ respectively, and the conditional mean is specified as
 $$
 \begin{align}
\mu_t &= \mu + \sum_{k = 1}^{K-K_1}\gamma_k X_{kt} +\sum_{k = K_1 + 1}^{K}\gamma_k X_{kt}\sigma_t + \xi\sigma_t^r  
\end{align}
$$
where 


*

*$r \in \{1, 2\}$ is the ARCH-in-mean parameter (archpow),

*inclusion of the unconditional mean $\mu$ is controlled by include.mean,

*
and the conditional mean specification is
general enough to allow the last $K-K_1$ covariates to be scaled by the conditional variance $\sigma_t$
which is to be defined shortly. 


Conditional variance specification
The conditional variance of the process is specified as 
$$
\begin{align}
\sigma_t^2 &= \left(\omega + \sum_{k = 1}^K \upsilon_k X_{kt}\right) + \sum_{s = 1}^l\alpha_s\varepsilon_{t-s}^2 + \sum_{u = 1}^m\beta_u\sigma_{t-u}^2
\end{align}
$$
where again, the conditional variance is allowed to depend on covariates, and the parameters $l$ and $m$ are controlled by garchOrder.
Given that you are saying that you have pre-demeaned your process using an ARMA(0, 0) filter (by which 
presumably you mean that you have removed an unconditional mean), and that you believe that there are no
further mean dynamics (you do not want an ARMA process to model your conditional mean), the appropriate specification
is your 2, that is 
modm11 = ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1,1)), 
                     mean.model=list(armaOrder=c(0,0), archm=T, archpow=2, 
                     include.mean=F))

