Building Econometric Models


The Unconditional Variance under the GARCH Specification



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Ch9 slides

The Unconditional Variance under the GARCH Specification

  • ‘Introductory Econometrics for Finance’ © Chris Brooks 2013
  • The unconditional variance of ut is given by
  • when
  • is termed “non-stationarity” in variance
  • is termed intergrated GARCH
  • For non-stationarity in variance, the conditional variance forecasts will not converge on their unconditional value as the horizon increases.

Estimation of ARCH / GARCH Models

  • ‘Introductory Econometrics for Finance’ © Chris Brooks 2013
  • Since the model is no longer of the usual linear form, we cannot use OLS.
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  • We use another technique known as maximum likelihood.
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  • The method works by finding the most likely values of the parameters given the actual data.
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  • More specifically, we form a log-likelihood function and maximise it.
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Estimation of ARCH / GARCH Models (cont’d)

  • ‘Introductory Econometrics for Finance’ © Chris Brooks 2013
  • The steps involved in actually estimating an ARCH or GARCH model are as follows
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  • Specify the appropriate equations for the mean and the variance - e.g. an AR(1)- GARCH(1,1) model:
  • Specify the log-likelihood function to maximise:
  • 3. The computer will maximise the function and give parameter values and their standard errors

Parameter Estimation using Maximum Likelihood

  • ‘Introductory Econometrics for Finance’ © Chris Brooks 2013
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  • Consider the bivariate regression case with homoscedastic errors for simplicity:
  • Assuming that ut  N(0,2), then yt  N( , 2) so that the probability density function for a normally distributed random variable with this mean and variance is given by
  • (1)
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  • Successive values of yt would trace out the familiar bell-shaped curve.
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  • Assuming that ut are iid, then yt will also be iid.

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