GARCH Model

What is GARCH Model?

GARCH is short for Generalised Autoregressive Conditional Heteroscedasticity. The GARCH model, developed by Robert Engel and Tim Bollerslev, can be regarded as an extension of EWMA. In GARCH (1,1), we also give some weight to a long-run average variance rate. The updated formula for the variance rate is:

α²_n = αr²_n-1 + βr²_{n – 1} + γV_L

Here, VL is the long-run average variance rate. The parameters, , , and are the weights given to the most recent squared return, the previous variance rate estimate, and the long-run average variance rate (respectively).

Because the weights must sum to one:

α + β ≤ 1,and
γ = (1 – α – β)

where, α and are positive, and the unconditional variance has been normalised to one.

Example

As an example of GARCH (1,1) calculations, suppose

ω=0.000003,     α= 0.12,      β=0.87

So that

α²_n= 0.000003 + 0.12r²_n-1 + 0.87r²_n-1

In this case
γ = (1 – α – β) = 0.01 and V_L= (0.000003/0.01) = 0.0003
The long-run average variance rate is 0.0003.

Why is it important?

The GARCH model is more suitable for illustrating the time series data than other forecast models adopted generally. Financial organisations commonly use this model to estimate the volatility of stock, bond, and market indices returns.