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GAUSS - GAUSS Applications
CML- Constrained Maximum Likelihood NEW in CML 2.0!
CML solves the general maximum likelihood problem subject to general constraints on the parameters - linear or nonlinear, equality or inequality. CML uses the Sequential Quadratic Programming method in combination with several descent methods selectable by the user - Newton-Raphson, quasi-Newton (i.e, DFP and BFGS), scaled quasi-Newton, and BHHH. There are also several selectable line search methods. A Trust Region method is also available which prevents saddle point solutions. Gradients can be user-provided or numerically calculated. CML provides for statistical inference for constrained statistical models. Confidence limits may be computed from selected methods, bootstrap, Bayesian (using a weighted likelihood bootstrap), or inversion of three types of statistics, the Wald, the likelihood ratio, or the Lagrange Multiplier. Confidence limits from the inversion of the likelihood ratio statistic are also called profile likelihood confidence limits. The bootstrap and Bayesian procedures generate simulated parameter sets from the bootstrap and posterior distributions respectively. Procedures may be applied to these parameter sets to either produce confidence limits, expected values, or kernel density plots of the distributions Example
In the first example, a TGARCH(2,2) model is estimated where the residuals are assumed to have a Student's t distribution in order to measure the "fatness" or platykurtosis of the tails of the observed distribution of a well-known stock index measured monthly. The "NU" parameter, the "degrees of freedom" parameter in the t distribution, must be greater than 2, but the extent to which it is greater than 2 indicates the amount of platykurtosis. In this case, the index is clearly lplatykurtotic. The "delta2" parameter is on the constraint floor. A Lagrange multiplier is available for testing the constraint, which in this case is the same as the gradient and is equal to .0011. This, plus the fact that the lower confidence limits of the "alpha" parameters are on the constraint boundary, suggest that a TGARCH(1,1) model might be a better model. The following are the estimates for the TGARCH(1,1) model:
The likelihood ratio test of the TGARCH(2,2) model over the TGARCH(1,1) model is .4478 (=265*(2.91808-2.91639)) which is not statistically significant. The likelihood ratio of the TGARCH(1,1) over the GARCH(1,1) model, in which the errors are assumed to have a Normal distribution, is 9.9665 with 1 degree of freedom. We thus accept the TGARCH(1,1) model under the rule of parsimony over both the TGARCH(2,2) and GARCH(1,1) models. The likelihood ratio statistic for the GARCH(1,1) model over an ordinary least squares model is 75.2043 with 4 degrees of freedom which is highly significant and is strong evidence for the GARCH specification of the stock index.
CML requires GAUSS/GAUSS Light version 3.6.23 or greater |
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