Is it possible to fit GJR-GARCH using BOTH daily variances and daily returns? I have to fit a GJR-GARCH to daily variances (computed using the 5-minute stock price data for that day) in addition to using the daily returns (computed using the daily stock price data) as the input. Are there any software packages that could do that?
Thank you!
 A: Here is the GJR-GARCH model by Glosten et al. (1993):
\begin{aligned}
x_t &= \mu_t+\varepsilon_t, \\
\varepsilon_t &= \sigma_t z_t, \\
\sigma^2_t &= \omega + (\alpha+\gamma \mathbb{I}_{t-1})\varepsilon^2_{t-1} + \beta\sigma^2_{t-1}, \\
z_t &\sim D(0,1),
\end{aligned}
where $\mu_t$ is the conditional mean of $x_t$, $D$ is some distribution with mean $0$ and variance $1$ and $\mathbb{I}_{t-1}$ is the indicator function:
$\mathbb{I}_{t-1}(\varepsilon_{t-1})=\varepsilon_{t-1}$ for $\varepsilon_{t-1}>0$ and 
$\mathbb{I}_{t-1}(\varepsilon_{t-1})=0$ otherwise.
Estimation of the model parameters is based on a single time series $x_t$ as an input. 
Now you have an additional time series $\tilde\sigma_t^2$ corresponding to the fitted values of conditional variance based on 5-minute data. You can add it as a regressor to the conditional variance equation:
$$
\sigma^2_t = \omega + (\alpha+\gamma \mathbb{I}_{t-1})\varepsilon^2_{t-1} + \beta\sigma^2_{t-1} \color{blue}{+ \delta\tilde\sigma_{t-1}^2}.
$$
This is almost a realized GJR-GARCH model. (For the seminal paper on realized GARCH, see Hansen et al. (2012).) You would need to specify an equation for $\tilde\sigma_t^2$ to complete the model, but in your case this might not be necessary (depends on what exactly you want to do with the model). 
In R, you can use the rugarch package with the main functions ugarchspec and ugarchfit to specify and fit the model. $\tilde\sigma_t^2$ would come in via the argument external.regressors within the argument variance.model of the ugarchspec function. Do not forget to lag the series appropriately. The package has a nice vignette that you might find helpful.
References


*

*Glosten, L. R., R. Jagannathan, D. E. Runkle. (1993). On The Relation between The Expected Value and The Volatility of Nominal Excess Return on stocks. Journal of Finance 48: 1779-1801.

*Hansen, P. R., Huang, Z., & Shek, H. H. (2012). Realized GARCH: A Joint Model for Returns and Realized Measures of Volatility. Journal of Applied Econometrics, 27(6), 877-906.

