Solved – Dealing with shape parameter of standardized skew normal distribution in DCC-GARCH

garchnormal distributionskew-normal-distributionvolatility-forecasting

Say we have a GARCH(1,1) equation such as

$h_t = \omega + \alpha\varepsilon^2_{t-1} + \beta h_{t-1}$

and $\varepsilon_{t-1}$ follows a standardized skew normal distribution. Using MLE we get the parameter estimates of $\omega$, $\alpha$ and $\beta$. We also get the shape parameter $\gamma$ of the skew normal distribution. If this $\gamma$ equals one we get the symmetric normal distribution.

Say, we want to estimate a DCC-GARCH model using this skew normal distributional assumption, where does this shape parameter fit? Should the GARCH equation be adjusted, for example, or should one use this parameter in the DCC equation (i.e. $Q_t = (1-a-b) + au_{t-1}u_{t-1}' +b\bar{Q}$).

The equations I stated are the well-known equations used when $\varepsilon_{t-1}$ has the normal distribution.

Basically, my question is, what should I do with this $\gamma$ shape parameter of the skew normal distribution in the context of GARCH(1,1) or DCC(1,1)-GARCH(1,1)?

Best Answer

The distributional assumption for a DCC-GARCH model considers standardized model residuals (a multivariate time series). Standardization is done by scaling the raw residuals (from the conditional mean model, if any) by premultiplying them by the square root of inverted conditional variance matrix, $\hat\Sigma^{-1/2}$, for each time period, so as to make the residuals roughly uncorrelated with approximately unit variances.

In theory, each univariate series could have a different marginal distribution, and different copulas could be used to obtain different joint distributions. In practice, software implementations usually have a more limited choice of distributions; for example, you may choose between multivariate normal, multivariate skew normal and multivariate $t$-distribution. The main thing is to achieve an empirical distribution that is not too far from the assumed one. However, perhaps the idea of quasi maximum likelihood estimation could be used to defend a mismatch if the assumed distribution is multivariate normal.

The DCC-GARCH model is estimated in two stages; first, the univariate GARCH models, and second, the DCC part. In the first stage, the relevant distributional assumptions will be those of the marginal distributions. So if the assumed multivariate distribution is multivariate skew normal, I guess the marginal distributions will be univariate skew normal, and the skew parameter(s) will be important in the first stage. I am not sure how one builds a multivariate skew normal distribution from univariate skew normal distributions; if the skew parameter(s) have a role there, then they will be important also in the second stage of the estimation (the DCC part). I do not have full control of the details here, but the idea should be clear.

Related Solutions

Solved – Quasi Maximum Likelihood estimator for GARCH with jump (Compoud poisson process with normal distribution)

We can show that the density of $Y_t$ knowing $F_{t-1}$ is given by:

$$ \sum_{k=0}^{\infty} \frac{\lambda^k e^{-\lambda}}{k!} \frac{1}{\sqrt{2 \pi(1+k\sigma^2)h_t^{*}}} e^{\frac{(y-(r+m_t^{*}+\sqrt{h_t^{*}}k\mu))^2}{2(1+k\sigma^2)h_t^{*}}} $$

with:

$$ m_t^{*} = m - \frac{\sqrt{h_t^{*}}\lambda\mu}{\sqrt{1+\lambda(\mu^2+\sigma^2)}} $$

$$ h_t^{*} = \frac{h_t}{1 + \lambda(\mu^2 + \sigma^2)}$$

So the following code will compute the log-likelihood of the model and can be optimized to fit the parameters to a dataset.

density_condi<-function(y,r,m,mu,lambda,h,sigma){
res=0
for(k in 0:5){
res=res+(exp(-lambda)*(lambda^k)/factorial(k))*(1/(sqrt(2*pi*(1+k*sigma^2)*h)))*exp(-((y-
(r+m+sqrt(h)*k*mu))^2)/(2*(1+k*sigma^2)*h))
}
return(res)
}
garch_gjr_jumps_loglik<-function(para,Y,r){
omega=para[1]
alpha=para[2]
beta=para[3]
theta=para[4]
mu=para[5]
sigma=para[6]
lambda=para[7]
m=para[8]
loglik=0
h=var(Y)
chi=(1+lambda*(mu^2+sigma^2))
h_star=h/chi
for (i in 2:length(Y)){
h=omega+alpha*(Y[i-1]-r-m)^2+beta*h+theta*(max(0,Y[i-1]-r-m))^2
h_star=h/chi
m_star=m-sqrt(h)*lambda*mu/(sqrt(chi))
loglik=loglik+log(density_condi(Y[i],r,m_star,mu,lambda,h_star,sigma))
}
print(-loglik)
return(-loglik)
}

FYI: on the SP500 this model give an alpha of 0.1579and a theta of -0.1635, which show the importance to modelize the leverage effect (GJR model), the use of jump is justified by the non-gaussianity of residuals in the simple GJR model.

GARCH – Evaluating Persistence in GJR-GARCH (Asymmetric GARCH) Models

I found an answer in the "vignette" to the "rugarch" package in R. Here is a quote from pages 7-8 (emphasis is mine):

Because of the presence of the indicator function, the persistence of the model now crucially depends on the asymmetry of the conditional distribution used. The persistence of the model $\hat P$ is,

$$ \hat P = \sum_{j=1}^q \alpha_j + \sum_{j=1}^p \beta_j + \sum_{j=1}^q \gamma_j\kappa $$

where $\kappa$ is the expected value of the standardized residuals $z_t$ below zero (effectively the probability of being below zero),

$$ \kappa = \mathbb{E}(\mathbb{I}_{t-j} z_{t-j}^2) = \int_{-\infty}^0 f(z,1,0,\dotsc) dz $$

where $f$ is the standardized conditional density with any additional skew and shape parameters $(\dotsc)$. In the case of symmetric distributions the value of $\kappa$ is simply equal to 0.5.

Best Answer

Related Solutions

Solved – Quasi Maximum Likelihood estimator for GARCH with jump (Compoud poisson process with normal distribution)

GARCH – Evaluating Persistence in GJR-GARCH (Asymmetric GARCH) Models

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