# 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!

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.
• Thank you so much for your informative response! This is exactly what I was looking for. Using a realized GJR-GARCH model will allow my project to move forward (as opposed to me being forced to abandon this interesting project). May I ask a followup question? Is there a way to justify assuming that Beta is zero? This assumption would make things a lot easier for me. Thank you! Commented May 15, 2020 at 20:28
• @BillB, thank you for your feedback! I have little experience with realized GARCH models, but out of the top of my head I do not think such an assumption is sensible. See e.g. the slides of Peter. R. Hansen. He notes $\alpha$ may be zero, but does not say so about $\beta$. Commented May 15, 2020 at 20:53
• Thank you so much for your kind help! This is very helpful. Commented May 16, 2020 at 6:00
• @BillB, if you look at the slides I linked above starting with the slide Completing the GARCH-X, you will see what I mean. Commented May 16, 2020 at 8:42
• @BillB, I am mostly off Cross Validated this July, so I cannot promise you anything much. But thanks for pinging me; at least I will be aware of the questions when I find the time. Commented Jul 18, 2020 at 13:39