Statistics and Algorithms / GARCH Models / GARCH Models (Univariate) /

UV GARCH I-GARCH

We can rearrange the GARCH(1,1) model as

\begin{equation} h_t = c_0 + a_1 \left( {u_{t - 1}^2 - h_{t - 1} } \right) + (a_1 + b_1 )h_{t - 1} \end{equation}

where (if the model is correct) the \(a_1\) term now has expected value zero. So \(a_1 + b_1\) is the proper measure of the persistence of volatility. In the IGARCH model (Nelson (1990)), this is constrained to be one. In general, for a GARCH(p,q) model the restriction is

\begin{equation} a_1 + a_2 + \ldots + a_q + b_1 + \ldots + b_p = 1 \label{eq:garch_igarchgeneral} \end{equation}

To estimate an IGARCH model, include the I=NODRIFT or I=DRIFT option on GARCH. I=NODRIFT imposes the constraint \eqref{eq:garch_igarchgeneral} and zeros out the constant \(c_0\) in the variance equation. I=DRIFT also imposes \eqref{eq:garch_igarchgeneral} but leaves the constant free. This can be added as an option to ARCH, GARCH and EGARCH models (for EGARCH, it constrains only the sum of the \(b\)’s).

To estimate an IGARCH(1,1) model (without variance drift), use

garch(p=1,q=1,i=nodrift) / dlogdm

Output

IGARCH Model - Estimation by BFGS

Convergence in 17 Iterations. Final criterion was 0.0000000 <= 0.0000100

Dependent Variable DLOGDM

Usable Observations 1866

Log Likelihood -2088.7331

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Mean(DLOGDM) -0.018534067 0.015549141 -1.19197 0.23327407

2. C 0.000000000 0.000000000 0.00000 0.00000000

3. A 0.082104140 0.008813022 9.31623 0.00000000

4. B 0.917895860 0.008813022 104.15223 0.00000000

The C is included in the output even though it is forced to be zero by the option. A and B have identical standard errors because they are constrained to sum to 1.