On the Properties of the Likelihood Function of Spanos' Conditional t Heteroskedastic Model

Abstract

This article investigates the properties of the likelihood function of Spanos’ conditional t heteroskedastic model (Spanos, 1994) On modeling heteroskedasticity: the student's t and elliptical linear regression models. It is shown that estimability of the degrees of freedom of t distribution and the block-diagonality of the information matrix of the joint likelihood function with respect to conditional mean parameters and remaining parameters hold for the model. The joint maximum likelihood estimator and its inference based on the t-statistic and χ²-statistic are examined in finite samples by simulation when the degrees of freedom is known and unknown.

Keywords:

• Conditional variance
• Degrees of freedom
• t distribution

Mathematics Subject Classification:

• Primary 62F12
• Secondary 62J05

Acknowledgment

The author thanks Kimio Morimune, Jeffrey Wooldridge, and the referees for many helpful comments.

Notes

The result described at p. 364 of Rao (1973) where the parameter is univariate can be easily extended to that with a multivariate parameter, which is the case in the conditional t heteroskedastic model.

This interpretation was provided by the referee.

We gratefully acknowledge the statistics literature about parameter orthogonality was suggested by the referee.

Actually we also tried three other degrees of freedom, i.e., ν = 3, 4, and 20 to see the effects of different degrees of freedom on the simulation results. Existence of the expected information matrix of the joint likelihood function is satisfied when ν > 4 under our setup where the condition ν > 4m − 4 for existence of the expected information matrix becomes ν > 4 with m = 2. The simulation results under 3 and 4 degrees of freedom turn out to be similar to those under 5 degrees of freedom and hence they are not presented here. Thus, as far as our limited simulation experiment is concerned, existence of the expected information matrix of the joint likelihood function does not appear to be a critical factor to obtain reasonable simulation performance although it is sufficient for the JMLE to have the standard asymptotic normality result. The simulation results under 20 degrees of freedom are worse in the point estimate by the JMLE and its inference based on the t − statistic and χ²-statistic for ν while better for other parameters, compared to the simulation results under 9 degrees of freedom. As the degrees of freedom increases, the performance of the point estimate by the JMLE and its inference based on the t − statistic and χ²-statistic for ν tends to deteriorate while the performance for other parameters tends to improve.

The t-statistic and χ²-statistic are defined respectively as and where T denotes the sample size, is the JMLE of λ _i where λ _i is a subset of λ, is the JMLE of λ ₍₁₎ where λ ₍₁₎ is a subvector of λ, is an asymptotic variance estimator of and is an asymptotic covariance matrix estimator of . When ν is known, we just call the χ²-statistic testing the parameters except ν χ²-statstic. When ν is unknown, we call the χ²-statistic testing all the parameters χ²-statistic 1 and the χ²-statistic testing the parameters except ν χ²-statistic 2.

The MME of the degrees of freedom ν is obtained using the relationship κ −3 = 6/(ν −4), where κ denotes the kurtosis of the univariate t distribution with ν degrees of freedom.