Maximum Likelihood Estimation for ARCH/GARCH

Procedure

Estimating ARCH/GARCH parameters using conditional maximum likelihood under normality assumptions.

Setup

Consider an ARCH(m) model:

$$a_t={\sigma }_t{ϵ}_t,{\sigma }_t^2={\alpha }_0+\sum _{i=1}^m{\alpha }_i{a}_{t-i}^2$$

Assume ${ϵ}_t\sim \text{i.i.d. }N(0,1)$.

Conditional Distribution

Given the information set $F_{t-1}$:

$$a_t|F_{t-1}\sim N(0,{\sigma }_t^2)$$

The conditional PDF is:

$$f(a_t|F_{t-1})=\frac{1}{\sqrt{2\pi {\sigma }_t^2}}\exp \left(-\frac{a_t^2}{2{\sigma }_t^2}\right)$$

Conditional Likelihood Function

For observations $a_1,...,a_T$, dropping the complicated joint density of initial values:

$$L(\mathbf{\alpha }|a_1,...,a_T)=\prod _{t=m+1}^T\frac{1}{\sqrt{2\pi {\sigma }_t^2}}\exp \left(-\frac{a_t^2}{2{\sigma }_t^2}\right)$$

where $\mathbf{\alpha }=({\alpha }_0,{\alpha }_1,...,{\alpha }_m{)}^{\prime }$.

Log-Likelihood Function

Taking the logarithm:

$$\ell (\mathbf{\alpha })=\sum _{t=m+1}^T\left[-\frac{1}{2}\ln (2\pi )-\frac{1}{2}\ln ({\sigma }_t^2)-\frac{1}{2}\frac{a_t^2}{{\sigma }_t^2}\right]$$

Simplified form (dropping constants):

$$\boxed{\ell (\mathbf{\alpha })=-\frac{1}{2}\sum _{t=m+1}^T\left[\ln ({\sigma }_t^2)+\frac{a_t^2}{{\sigma }_t^2}\right]}$$

For GARCH Models

The log-likelihood has the same form, but ${\sigma }_t^2$ follows the GARCH specification:

$${\sigma }_t^2={\alpha }_0+\sum _{i=1}^m{\alpha }_i{a}_{t-i}^2+\sum _{j=1}^s{\beta }_j{\sigma }_{t-j}^2$$

Parameters: $\mathbf{\theta }=({\alpha }_0,{\alpha }_1,...,{\alpha }_m,{\beta }_1,...,{\beta }_s{)}^{\prime }$

Estimation

Objective:

$$\hat{\mathbf{\theta }}=\arg \underset{\mathbf{\theta }}{\max }\ell (\mathbf{\theta })$$

Subject to constraints:

• ${\alpha }_0>0$
• ${\alpha }_i\ge 0$, ${\beta }_j\ge 0$
• Stationarity condition: $\sum ({\alpha }_i+{\beta }_i)<1$

Methods:

• Numerical optimization (Newton-Raphson, BFGS)
• Most statistical software uses quasi-Newton methods

Alternative Distributional Assumptions

Distribution	Purpose	Heavy Tails
Normal (default)	Standard case	No
Student-t	Capture heavy tails	Yes
Skew-Student-t	Capture skewness & heavy tails	Yes
Generalized Error Distribution (GED)	Flexible	Adjustable

Related

• ARCH(m) Model
• GARCH(m,s) Model
• Building a Volatility Model