# Model comparison via AIC or BIC for different likelihood maximization procedures

Maximum likelihood estimation of different models (which all model the same variable and assume the same likelihood function) is done by a different method for each model. Simple numerical maximization of the likelihood is not possible due to the "curse of dimensionality" and consequently methods like

• composite likelihood estimation
• estimating a subset of parameters by a moment estimator (and the rest by ML)
• iteratively estimating subsets of parameters

are employed. Assuming all estimation procedures yield consistent parameter estimates but some are more efficient than others, is it sensible to compare the models via AIC or BIC? What if some procedures are inconsistent?

• Are your models actually different? Or are only the estimation procedures different but the model is one and the same? – Richard Hardy Jun 17 '16 at 16:59
• Yes. The models have different autoregressive specifications for the scale matrix of Wishart distribution. F. ex. a stationary MGARCH (which lends itself to covariance targeting) or a DCC-GARCH. More complicated models can only exploit that submatrices of a Wishart distributed random matrix are also Wishart distributed, so many low dimensional models are estimated and a parameter estimates is obtained taking the average of its low dimensional estimates (this is actually consistent and called composite likelihood estimation). link is a reference this. – stollenm Jun 18 '16 at 17:15
• So what do you mean by "[different models] assume the same likelihood function"? – Richard Hardy Jun 18 '16 at 20:33
• I'm sorry for the misleading statement. The models are different specifications for the same parameter of the Wishart distribution (the scale matrix). Hence they all have the same wishart log-likelihood function. – stollenm Jun 19 '16 at 18:42
• You probably mean the errors are Wishart-distributed or something like that, not that the likelihood functions are the same. Because the same likelihood functions mean the same models (except probably for some very special cases). – Richard Hardy Jun 19 '16 at 19:06