Distribution assumptions

GARCH Models in Python

Chelsea Yang

Data Science Instructor

Why make assumptions

Volatility is not directly observable
GARCH model use residuals as volatility shocks $$ r_t = \mu_t + \epsilon_t $$
Volatility is related to the residuals: $$ \epsilon_t = \sigma_t * \zeta (WhiteNoise)$$

Standardized residuals

Residual = predicted return - mean return $$ residuals = \epsilon_t = r_t - \mu_t $$
Standardized residual = residual / return volatility $$ std\,Resid = \frac{\epsilon_t}{\sigma_t} $$

Residuals in GARCH

gm_std_resid = gm_result.resid / gm_result.conditional_volatility

plt.hist(gm_std_resid, facecolor = 'orange',label = 'standardized residuals')

Histogram of standardized residuals

Fat tails

Higher probability to observe large (positive or negative) returns than under a normal distribution

Fat tail example

Skewness

Measure of asymmetry of a probability distribution

Example of skewness

Student's t-distribution

t-distribution example

$\nu$ parameter of a Student's t-distribution indicates its shape

GARCH with t-distribution

arch_model(my_data, p = 1, q = 1,
           mean = 'constant', vol = 'GARCH', 
           dist = 't')

                              Distribution                              
========================================================================
                 coef    std err          t      P>|t|  95.0% Conf. Int.
.-----------------------------------------------------------------------
nu             4.9249      0.507      9.709  2.768e-22 [  3.931,  5.919]
========================================================================

GARCH with skewed t-distribution

arch_model(my_data, p = 1, q = 1,
           mean = 'constant', vol = 'GARCH', 
           dist = 'skewt')

                                Distribution                               
===========================================================================
                 coef    std err          t      P>|t|     95.0% Conf. Int.
.--------------------------------------------------------------------------
nu             5.2437      0.575      9.118  7.681e-20    [  4.117,  6.371]
lambda        -0.0822  2.541e-02     -3.235  1.216e-03 [ -0.132,-3.241e-02]
===========================================================================

Let's practice!

GARCH Models in Python