archette

GARCH is a minimal implementation of the GARCH(1,1) model with zero mean and Gaussian noise.

Usage

Installation

Install latest from the GitHub repository:

$ pip install git+https://github.com/xiaochuany/archette.git

Documentation

Documentation can be found hosted on this GitHub repository’s pages.

How to use

import numpy as np
from archette import GARCHETTE

y = np.random.randn(20)
mod = GARCHETTE()
mod.fit(y)

<archette.core.GARCHETTE>

One can get inspect the fit params: a triplet of omega, alpha and beta as in

$$ \begin{align*} Y_t &= \sigma_t W_t \ \sigma_t^2 &= \omega + \alpha Y_{t-1}^2 + \beta \sigma^2_{t-1} \end{align*} $$

mod.params

array([1.46162489e+00, 0.00000000e+00, 4.34165554e-16])

The conditional variance of the fit model is computed by the recursion

$$ \hat{\sigma}t^2 = \hat{\omega} + \hat{\alpha} + \hat{\beta} \hat{\sigma}{t-1}^2 $$

which depends entirely on the unobserved $\sigma_0^2$. The model sets a sensible default value for it.

mod.vs

array([1.6637509 , 1.46162489, 1.46162489, 1.46162489, 1.46162489,
       1.46162489, 1.46162489, 1.46162489, 1.46162489, 1.46162489,
       1.46162489, 1.46162489, 1.46162489, 1.46162489, 1.46162489,
       1.46162489, 1.46162489, 1.46162489, 1.46162489, 1.46162489])

The standardised residual is deduced from the conditional variance

$$ r_t = \frac{y_t}{\hat{\sigma}_t} $$

mod.std_resids

array([ 1.83606806,  0.40202783, -0.05913285, -0.47174874, -0.59056739,
       -2.20575562,  0.71528908,  0.26541164, -0.20129438,  1.33214741,
       -0.50849529,  0.2951227 , -0.84969512,  1.53542218,  0.44267321,
       -1.95023795, -1.68947028, -0.23377285, -0.18693467,  0.7959041 ])

Finally, one can forecast the condtional variance and simulate the process with the fit parameters with a given horizon

mod.forecast_vs(horizon=5)

array([1.46162489, 1.46162489, 1.46162489, 1.46162489, 1.46162489])

mod.simulate(horizon=5, method="bootstrap", n_rep=2, seed=1)

array([[-2.66670736,  0.35679649, -1.02726169, -0.24336023,  1.61053531],
       [ 0.35679649, -2.66670736, -2.35779243,  2.21976368, -2.04253036]])