What are the conditions on standard GARCH coefficients

Question

it is hard to find the full list of restrictions on GARCH(p,q) coefficients. Let me clarify. First, define GARCH(p,q) for a zero-mean returns time series as:

\begin{equation} \label{eq:garch_pq} \sigma_t^2 = \omega + \sum_{i=1}^q\alpha_i r^2_{t-i} + \sum_{i=1}^p \beta_i \sigma_{t-i}^2 \end{equation} where $r_t = \sigma_t \epsilon_t$ and $\epsilon_t \sim i.i.d(0,1)$.

Now, I can't find what the full list of restrictions are on the $\alpha_i$'s and the $\beta_i$'s.

I've collected the following restrictions from various sources. Here is a list:

• $ \left(\sum_{i=1}^{q} \alpha_i + \sum_{i=1}^{p} \beta_i \right) < 1 $
• $\alpha_i \geq 0$ for any $i$
• $\beta_i \geq 0$ for any $i$

Are the above correct? What about $0 < \left(\sum_{i=1}^{q} \alpha_i + \sum_{i=1}^{p} \beta_i \right)$? This is implied if the last two above are correct.

Answer 1

The first one ensures that you have a covariance stationary process. The last two conditions in combination with $\omega >0$ are sufficient to ensure that $\sigma_t^2$ is non-negative. So if $\alpha_i \geq 0$ and $\beta_i \geq 0$, then $\sum_{i=1}^p\alpha_i +\sum_{i=1}^q\beta_i \geq 0$. However, since $\alpha_i$ and $\beta_i$ cannot be negative, $\sum_{i=1}^p\alpha_i +\sum_{i=1}^q\beta_i = 0$ implies that $\alpha_i=\beta_i=0$, which implies that $\sigma_t^2= \omega$. This means that your conditional variance is constant, i.e. you don't have any (G)ARCH effects in your time series. In real world applications it is common to test whether there is conditional heteroskedasticity in the time series (for instance, Engle LM-test) and then fit a GARCH-model. Following this approach, the case that all $\alpha_i$ and $\beta_i$ are equal to zero, will almost never occur (why you should fit a GARCH model when there is no conditional heteroskedasticity?). Furthermore, when dealing with financial time series, you often observe that the sum $\sum_{i=1}^p\alpha_i+\sum_{i=1}^q\beta_i$ is very close to one. So the discussion whether $ \sum_{i=1}^p\alpha_i+\sum_{i=1}^q\beta_i=1$, so called integrated GARCH (IGARCH), is more important.