Building Econometric Models

Problems with ARCH(q) Models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• How do we decide on q?
• The required value of q might be very large
• Non-negativity constraints might be violated.
• When we estimate an ARCH model, we require i >0  i=1,2,...,q (since variance cannot be negative)
•  
• A natural extension of an ARCH(q) model which gets around some of these problems is a GARCH model.

Generalised ARCH (GARCH) Models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Due to Bollerslev (1986). Allow the conditional variance to be dependent upon previous own lags
• The variance equation is now
• (1)
• This is a GARCH(1,1) model, which is like an ARMA(1,1) model for the variance equation.
• We could also write
• Substituting into (1) for t-12 :

Generalised ARCH (GARCH) Models (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Now substituting into (2) for t-22
•  
• An infinite number of successive substitutions would yield
•  
• So the GARCH(1,1) model can be written as an infinite order ARCH model.
•  
• We can again extend the GARCH(1,1) model to a GARCH(p,q):
•  

Generalised ARCH (GARCH) Models (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• But in general a GARCH(1,1) model will be sufficient to capture the volatility clustering in the data.
•  
• Why is GARCH Better than ARCH?
• - more parsimonious - avoids overfitting
• - less likely to breech non-negativity constraints