Parameter Constraints for GJR-GARCH Models with Exogenous Inputs

# How to deal with negative intercept terms on GJR-GARCH(1,1) model?


# How to deal with negative intercept terms on GJR-GARCH(1,1) model?












Recently, I have been studying the relationship between COVID-19 and stock returns using a GJR form of threshold ARCH model. However, I got some unusual estimation results I can't figure out whether they are fine or not. The model I have estimated is written below:

In the equation, D1 is 0 before the pandemic and 1 during the pandemic. I used the following command in STATA for obtaining the result:

```
arch djones covid19, arch(1/1) tarch(1/1) garch(1/1) het(covid19)
```

The estimation results of the above models show negative intercept terms, α in the volatility equation (-2.97). However, the estimated intercept is significant at 1%. Full results are shown below:

Furthermore, in the case of S&P500, the sum of alpha1 and beta1 also would be negative but significant.

However, when I remove the dummy term and focus on GARCH(1,1) model in the equation written above, the problem gets solved mostly. But I won't be able to study the impacts of COVID-19 on stock returns. I also checked AIC and BIC and found that still, the GJR-GARCH model has lower BIC than that of the standard GARCH model.

My question is how to deal with this type of problem? Can the negative intercept term still valid if it is significant? or

- Do I ignore the intercept of the volatility equation?

- Should I conduct two GARCH models separately: one for before the pandemic and another during the pandemic?

I would be thankful if you could provide some insights on it.

Thank you.

## Answer by Pleb (score 1)

https://quant.stackexchange.com/a/69339

My answer is based on the fact that the above equations corresponds to the provided STATA code and output. Thus, I disregard my own questions in the comments below the post.

## Consider bounding your GJR-GARCH parameters:

You need to bound your parameters in order to ensure positivity (positive estimates) and covariance stationarity. I will disregard the mean-dynamics and focus on the GJR-GARCH type model.

Let us vaguely define the GJR-GARCH-X(1,1) model which is a simplified version of the model seen in equation (3) (the "X" implies that the model contains exogenous input(s), aka. $\lambda D_1$): \begin{align*} r_t \vert \mathcal{F}_{t-1} &= \mu + \varepsilon_t\\ \varepsilon_t &= \sigma_t \cdot z_t\\ \sigma^2_t &= \alpha_0 + \alpha_1 \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 + \gamma I_{t-1} \varepsilon_{t-1}^2 + \lambda D_1, \end{align*} where $z_t \overset{iid}{\sim} D(0,1)$ (which in your case is the Gaussian distribution),

$$I_{t-1} =\begin{cases} 1 & \text{if } \varepsilon_{t-1} < 0 \\ 0 & \text{if } \varepsilon_{t-1} \geq 0 \end{cases}$$

and $D_1$ is an indicator function specifying the time of the covid pandemic (this is vaguely specified in your question).

When we have a negative intercept term, $\alpha_0 <0$, we need $\alpha_1 + \beta + \gamma + \lambda > -\alpha_0$ to ensure positivity, which in your scenario $0.2489 + 0.8655 - 0.3069 + 1.875\not> 2.97$ is clearly not satisfied. If we derive the unconditional variance of the GJR-GARCH-X type model:

$$ \mathbb{V}ar(r_t) := \sigma^2 = \frac{\alpha_0 + \lambda \cdot \rho}{1 - (\alpha_1 + \beta + \gamma \kappa)} $$

where $\rho = \mathbb{E}\left[D_1\right] = \text{"}\mathbb{P}(\text{pandemic})\text{"}$ and $\kappa = \mathbb{E}\left[I_{t-1} z_{t-1}^2\right] = \mathbb{P}(z_{t-1}<0)$ (is 0.5 for symmetric distributions), then for both $\kappa$ and $\rho \in [0,1]$, your estimates yield negative values for the unconditional variance when $\alpha_0 = -2.97$ and $\lambda = 1.875$.

In conclusion your above estimates also violate the unconditional variance. Therefore you need to bound your parameters as is also detailed in one of my earlier answers found here. You need to let $\alpha_0 + \lambda\cdot \rho > 0$ and $0<\alpha_1 + \beta + \gamma \kappa<1$ in order to ensure covariance stationarity.

Further imposing $\beta, \alpha_1, \alpha_0>0$, $\alpha_0 + \lambda > 0$ and $\alpha_1 + \gamma >0 $ allows $\lambda$ and $\gamma$ to become negative (ie. vary more freely). This is a broader statement than eg. imposing $\alpha_1, \gamma > 0$ and positivity constraints on both estimators in the numerator, $\alpha_0, \lambda > 0$, therefore giving your parameter estimates more leeway, which might help the with the model fit.