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FRM Part I · FRM Exam Part I · Measuring and Monitoring Volatility

When estimating GARCH parameters by maximum likelihood under normality, which statement is correct?

Maximum likelihood for GARCH under normality selects omega, alpha and beta to maximize the sum of negative log conditional variance minus squared return divided by conditional variance. This rewards variances that fit observed returns well without being unnecessarily large.

  1. AParameters are chosen to maximize the sum of -ln(sigma_i^2) - r_i^2/sigma_i^2 over observationsCorrect
  2. BParameters are chosen to minimize the sum of squared returns
  3. CParameters are chosen so that the sample mean of returns equals zero
  4. DParameters are chosen to maximize the sum of sigma_i^2 times r_i^2

Explanation

With normal errors and zero mean, the log-likelihood is proportional to the sum of -ln(v_i) - u_i^2/v_i, where v_i is the conditional variance. The optimizer searches over omega, alpha and beta to maximize it. The other options are not the likelihood objective.

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