garch_11

sim.garch_11(n, sigma_sq_0, mu, alpha, beta, omega)

Generate GARCH(1, 1) log-returns of size n. This function is accelerated via JIT with Numba.

Parameters

nint

The length of the wished time series.

sigma_sq_0float > 0

The variance starting value.

mufloat:

The drift of log-returns.

alphafloat >= 0

The volatility shock parameter. A higher value will lead to larger spikes in volatility. A.k.a short-term persistence.

betafloat >= 0

The volatility persistence parameter. A larger value will result in stronger persistence. A.k.a long-term persistence.

omegafloat > 0

The variance constant. A higher value results in a higher mean variance.

Returns

rnumpy.ndarray

The GARCH log-returns time series.

sigma_sqnumpy.ndarray

The resulting variance time series with which each log-return was generated.

Notes

In general, the conditional variance of a GARCH(p,q) model is given by

\[\sigma_{t}^{2}=\omega+\sum_{i=1}^{q} \alpha_{i} \varepsilon_{t-i}^{2}+\sum_{j=1}^{p} \beta_{j} \sigma_{t-j}^{2}.\]

The unconditional variance is given by

\[\sigma^{2}=\frac{\omega}{1-\sum_{i=1}^{q} \alpha_{i}-\sum_{j=1}^{p} \beta_{j}}.\]

Here, \(p=q=1\), and \(\epsilon_{t} \sim \mathcal{N}\left(0, 1\right)\)