Building Econometric Models


Problems with ARCH(q) Models



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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

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