Using the Fixed Variance process

The FixedVariance volatility process can be used to implement zig-zag model estimation where two steps are repeated until convergence. This can be used to estimate models which may not be easy to estimate as a single process due to numerical issues or a high-dimensional parameter space.

This setup code is required to run in an IPython notebook

[1]:
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn

seaborn.set_style("darkgrid")
plt.rc("figure", figsize=(16, 6))
plt.rc("savefig", dpi=90)
plt.rc("font", family="sans-serif")
plt.rc("font", size=14)

Setup

Imports used in this example.

[2]:
import datetime as dt

import numpy as np

Data

The VIX index will be used to illustrate the use of the FixedVariance process. The data is from FRED and is provided by the arch package.

[3]:
import arch.data.vix

vix_data = arch.data.vix.load()
vix = vix_data.vix.dropna()
vix.name = "VIX Index"
ax = vix.plot(title="VIX Index")
../_images/univariate_univariate_using_fixed_variance_6_0.png

Initial Mean Model Estimation

The first step is to estimate the mean to filter the residuals using a constant variance.

[4]:
from arch.univariate.mean import HARX, ZeroMean
from arch.univariate.volatility import GARCH, FixedVariance

mod = HARX(vix, lags=[1, 5, 22])
res = mod.fit()
print(res.summary())
                    HAR - Constant Variance Model Results
==============================================================================
Dep. Variable:              VIX Index   R-squared:                       0.876
Mean Model:                       HAR   Adj. R-squared:                  0.876
Vol Model:          Constant Variance   Log-Likelihood:               -2267.95
Distribution:                  Normal   AIC:                           4545.90
Method:            Maximum Likelihood   BIC:                           4571.50
                                        No. Observations:                 1237
Date:                Tue, Mar 09 2021   Df Residuals:                     1233
Time:                        18:11:02   Df Model:                            4
                                   Mean Model
================================================================================
                      coef    std err          t      P>|t|     95.0% Conf. Int.
--------------------------------------------------------------------------------
Const               0.6335      0.189      3.359  7.831e-04    [  0.264,  1.003]
VIX Index[0:1]      0.9287  6.589e-02     14.095  4.056e-45    [  0.800,  1.058]
VIX Index[0:5]     -0.0318  6.449e-02     -0.492      0.622  [ -0.158,9.463e-02]
VIX Index[0:22]     0.0612  3.180e-02      1.926  5.409e-02 [-1.076e-03,  0.124]
                            Volatility Model
========================================================================
                 coef    std err          t      P>|t|  95.0% Conf. Int.
------------------------------------------------------------------------
sigma2         2.2910      0.396      5.782  7.361e-09 [  1.514,  3.068]
========================================================================

Covariance estimator: White's Heteroskedasticity Consistent Estimator

Initial Volatility Model Estimation

Using the previously estimated residuals, a volatility model can be estimated using a ZeroMean. In this example, a GJR-GARCH process is used for the variance.

[5]:
vol_mod = ZeroMean(res.resid.dropna(), volatility=GARCH(p=1, o=1, q=1))
vol_res = vol_mod.fit(disp="off")
print(vol_res.summary())
                     Zero Mean - GJR-GARCH Model Results
==============================================================================
Dep. Variable:                  resid   R-squared:                       0.000
Mean Model:                 Zero Mean   Adj. R-squared:                  0.001
Vol Model:                  GJR-GARCH   Log-Likelihood:               -1936.93
Distribution:                  Normal   AIC:                           3881.86
Method:            Maximum Likelihood   BIC:                           3902.35
                                        No. Observations:                 1237
Date:                Tue, Mar 09 2021   Df Residuals:                     1237
Time:                        18:11:02   Df Model:                            0
                              Volatility Model
===========================================================================
                 coef    std err          t      P>|t|     95.0% Conf. Int.
---------------------------------------------------------------------------
omega          0.2355  9.134e-02      2.578  9.932e-03  [5.647e-02,  0.415]
alpha[1]       0.7217      0.374      1.931  5.353e-02 [-1.098e-02,  1.454]
gamma[1]      -0.7217      0.252     -2.859  4.255e-03    [ -1.217, -0.227]
beta[1]        0.5789      0.184      3.140  1.692e-03    [  0.218,  0.940]
===========================================================================

Covariance estimator: robust
[6]:
ax = vol_res.plot("D")
../_images/univariate_univariate_using_fixed_variance_11_0.png

Re-estimating the mean with a FixedVariance

The FixedVariance requires that the variance is provided when initializing the object. The variance provided should have the same shape as the original data. Since the variance estimated from the GJR-GARCH model is missing the first 22 observations due to the HAR lags, we simply fill these with 1. These values will not be used to estimate the model, and so the value is not important.

The summary shows that there is a single parameter, scale, which is close to 1. The mean parameters have changed which reflects the GLS-like weighting that this re-estimation imposes.

[7]:
variance = np.empty_like(vix)
variance.fill(1.0)
variance[22:] = vol_res.conditional_volatility ** 2.0
fv = FixedVariance(variance)
mod = HARX(vix, lags=[1, 5, 22], volatility=fv)
res = mod.fit()
print(res.summary())
Iteration:      1,   Func. Count:      7,   Neg. LLF: 255807017422.13824
Iteration:      2,   Func. Count:     19,   Neg. LLF: 930336.6903539689
Iteration:      3,   Func. Count:     28,   Neg. LLF: 3486.6609351352117
Iteration:      4,   Func. Count:     36,   Neg. LLF: 2885.718175566
Iteration:      5,   Func. Count:     44,   Neg. LLF: 65535540.6667968
Iteration:      6,   Func. Count:     53,   Neg. LLF: 1935.9527544129544
Iteration:      7,   Func. Count:     59,   Neg. LLF: 1935.9470521802314
Iteration:      8,   Func. Count:     65,   Neg. LLF: 1935.9470515696541
Optimization terminated successfully    (Exit mode 0)
            Current function value: 1935.9470515696541
            Iterations: 8
            Function evaluations: 65
            Gradient evaluations: 8
                      HAR - Fixed Variance Model Results
==============================================================================
Dep. Variable:              VIX Index   R-squared:                       0.876
Mean Model:                       HAR   Adj. R-squared:                  0.876
Vol Model:             Fixed Variance   Log-Likelihood:               -1935.95
Distribution:                  Normal   AIC:                           3881.89
Method:            Maximum Likelihood   BIC:                           3907.50
                                        No. Observations:                 1237
Date:                Tue, Mar 09 2021   Df Residuals:                     1233
Time:                        18:11:02   Df Model:                            4
                                    Mean Model
==================================================================================
                      coef    std err          t      P>|t|       95.0% Conf. Int.
----------------------------------------------------------------------------------
Const               0.5584      0.153      3.661  2.507e-04      [  0.260,  0.857]
VIX Index[0:1]      0.9376  3.625e-02     25.866 1.608e-147      [  0.867,  1.009]
VIX Index[0:5]     -0.0249  3.782e-02     -0.657      0.511 [-9.899e-02,4.926e-02]
VIX Index[0:22]     0.0493  2.102e-02      2.344  1.909e-02  [8.064e-03,9.044e-02]
                            Volatility Model
========================================================================
                 coef    std err          t      P>|t|  95.0% Conf. Int.
------------------------------------------------------------------------
scale          0.9986  8.081e-02     12.358  4.420e-35 [  0.840,  1.157]
========================================================================

Covariance estimator: robust

Zig-Zag estimation

A small repetitions of the previous two steps can be used to implement a so-called zig-zag estimation strategy.

[8]:
for i in range(5):
    print(i)
    vol_mod = ZeroMean(res.resid.dropna(), volatility=GARCH(p=1, o=1, q=1))
    vol_res = vol_mod.fit(disp="off")
    variance[22:] = vol_res.conditional_volatility ** 2.0
    fv = FixedVariance(variance, unit_scale=True)
    mod = HARX(vix, lags=[1, 5, 22], volatility=fv)
    res = mod.fit(disp="off")
print(res.summary())
0
1
2
3
4
                    HAR - Fixed Variance (Unit Scale) Model Results
=======================================================================================
Dep. Variable:                       VIX Index   R-squared:                       0.876
Mean Model:                                HAR   Adj. R-squared:                  0.876
Vol Model:         Fixed Variance (Unit Scale)   Log-Likelihood:               -1935.74
Distribution:                           Normal   AIC:                           3879.48
Method:                     Maximum Likelihood   BIC:                           3899.96
                                                 No. Observations:                 1237
Date:                         Tue, Mar 09 2021   Df Residuals:                     1233
Time:                                 18:11:02   Df Model:                            4
                                    Mean Model
=================================================================================
                      coef    std err          t      P>|t|      95.0% Conf. Int.
---------------------------------------------------------------------------------
Const               0.5602      0.152      3.681  2.324e-04     [  0.262,  0.858]
VIX Index[0:1]      0.9381  3.616e-02     25.939 2.389e-148     [  0.867,  1.009]
VIX Index[0:5]     -0.0262  3.774e-02     -0.693      0.488   [ -0.100,4.781e-02]
VIX Index[0:22]     0.0499  2.099e-02      2.380  1.733e-02 [8.809e-03,9.109e-02]
=================================================================================

Covariance estimator: robust

Direct Estimation

This model can be directly estimated. The results are provided for comparison to the previous FixedVariance estimates of the mean parameters.

[9]:
mod = HARX(vix, lags=[1, 5, 22], volatility=GARCH(1, 1, 1))
res = mod.fit(disp="off")
print(res.summary())
                        HAR - GJR-GARCH Model Results
==============================================================================
Dep. Variable:              VIX Index   R-squared:                       0.876
Mean Model:                       HAR   Adj. R-squared:                  0.875
Vol Model:                  GJR-GARCH   Log-Likelihood:               -1932.61
Distribution:                  Normal   AIC:                           3881.23
Method:            Maximum Likelihood   BIC:                           3922.19
                                        No. Observations:                 1237
Date:                Tue, Mar 09 2021   Df Residuals:                     1233
Time:                        18:11:03   Df Model:                            4
                                   Mean Model
================================================================================
                      coef    std err          t      P>|t|     95.0% Conf. Int.
--------------------------------------------------------------------------------
Const               0.7796      1.190      0.655      0.513    [ -1.554,  3.113]
VIX Index[0:1]      0.9180      0.291      3.156  1.597e-03    [  0.348,  1.488]
VIX Index[0:5]     -0.0393      0.296     -0.133      0.894    [ -0.620,  0.541]
VIX Index[0:22]     0.0632  6.353e-02      0.994      0.320 [-6.136e-02,  0.188]
                            Volatility Model
========================================================================
                 coef    std err          t      P>|t|  95.0% Conf. Int.
------------------------------------------------------------------------
omega          0.2357      0.250      0.944      0.345 [ -0.254,  0.725]
alpha[1]       0.7091      1.069      0.664      0.507 [ -1.386,  2.804]
gamma[1]      -0.7091      0.519     -1.367      0.172 [ -1.726,  0.308]
beta[1]        0.5579      0.855      0.653      0.514 [ -1.117,  2.233]
========================================================================

Covariance estimator: robust