Building Econometric Models

Heteroscedasticity Revisited

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• An example of a structural model is
• with ut  N(0, ).
•  
• The assumption that the variance of the errors is constant is known as
• homoscedasticity, i.e. Var (ut) = .
•  
• What if the variance of the errors is not constant?
• - heteroscedasticity
• - would imply that standard error estimates could be wrong.
•  
• Is the variance of the errors likely to be constant over time? Not for financial data.

Autoregressive Conditionally Heteroscedastic (ARCH) Models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• So use a model which does not assume that the variance is constant.
• Recall the definition of the variance of ut:
• = Var(ut ut-1, ut-2,...) = E[(ut-E(ut))2 ut-1, ut-2,...]
• We usually assume that E(ut) = 0
• so = Var(ut  ut-1, ut-2,...) = E[ut2 ut-1, ut-2,...].
•  What could the current value of the variance of the errors plausibly depend upon?
• Previous squared error terms.
• This leads to the autoregressive conditionally heteroscedastic model for the variance of the errors:
• = 0 + 1
• This is known as an ARCH(1) model
• The ARCH model due to Engle (1982) has proved very useful in finance.

Autoregressive Conditionally Heteroscedastic (ARCH) Models (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• The full model would be
• yt = 1 + 2x2t + ... + kxkt + ut , ut  N(0, )
• where = 0 + 1
• We can easily extend this to the general case where the error variance depends on q lags of squared errors:
• = 0 + 1 +2 +...+q
• This is an ARCH(q) model.
•  
• Instead of calling the variance , in the literature it is usually called ht, so the model is
• yt = 1 + 2x2t + ... + kxkt + ut , ut  N(0,ht)
• where ht = 0 + 1 +2 +...+q