Unformatted text preview: hood estimators, • From (6), (9)
24 Parameter Estimation using Maximum Likelihood (cont’d)
• From (7), (10) • From (8),
25 Parameter Estimation using Maximum Likelihood (cont’d)
(11) • How do these formulae compare with the OLS estimators?
(9) & (10) are identical to OLS
(11) is different. The OLS estimator was • Therefore the ML estimator of the variance of the disturbances is biased,
although it is consistent.
But how does this help us in estimating heteroscedastic models? • 26 Estimation of GARCH Models Using
• Now we have yt = μ + φyt-1 + ut , ut ∼ N(0, ) • Unfortunately, the LLF for a model with time-varying variances cannot be
maximised analytically, except in the simplest of cases. So a numerical
procedure is used to maximise the log-likelihood function. A potential
problem: local optima or multimodalities in the likelihood surface.
The way we do the optimisation is:
1. Set up LLF.
2. Use regression to get initial guesses for the mean parameters.
3. Choose some initial guesses for the conditional variance parameters.
4. Specify a convergence criterion - either by criterion or by value. • 27 Non-Normality and Maximum Likelihood • Recall that the conditional normality assumption for ut is essential.
• We can test for normality using the following representation
vt ∼ N(0,1)
u t = vt σ t • The sample counterpart is
• Are the
are still leptokurtic, although less so than
. Is this a problem? Not really, as we can use the ML with a robust
variance/covariance estimator. ML with robust standard errors is called QuasiMaximum Likelihood or QML.
28 Extensions to the Basic GARCH Model • Since the GARCH model was developed, a huge number of extensions
and variants have been proposed. Three of the most important
examples are EGARCH, GJR, and GARCH-M models. • Problems with GARCH(p,q) Models:
- Non-negativity constraints may still be violated
- GARCH models cannot account for leverage effects • Possible solutions: the exponential GARCH (EGARCH) model or the
GJR model, which are asymmetric GARCH models. 29 The EGARCH Model • Suggested by Nelson (1991). The variance equation is given by • Advantages of the model
- Since we model the log(σt2), then even if the parameters are negative, σt2
will be positive.
- We can account for the leverage effect: if the relationship between
volatility and returns is negative, γ, will be negative.
30 The GJR Model • Due to Glosten, Jaganathan and Runkle where It-1 = 1 if ut-1 < 0
= 0 otherwise
• For a leverage effect, we would see γ > 0. • We require α1 + γ ≥ 0 and α1 ≥ 0 for non-negativity. 31 An Example of the use of a GJR Model • Using monthly S&P 500 returns, December 1979- June 1998 • Estimating a GJR model, we obtain the following results. 32 News Impact Curves The news impact curve plots the next period volatility (ht) that would arise from various
positive and negative values of ut-1, given an estima...
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