Building Econometric Models


Heteroscedasticity Revisited



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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

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