Abstract
This study evaluates the empirical performance of four power transformation families: extended Tukey, Modulus, Exponential, and Yeo--Johnson, in modeling the return in the context of GARCH(1,1) models with two error distributions: Gaussian (normal) and Student-t. We employ an Adaptive Random Walk Metropolis method in Markov Chain Monte Carlo scheme to draw parameters. Using 19 international stock indices from the Oxford-Man Institute and basing on the log likelihood, Akaike Information Criterion, Bayesian Information Criterion, and Deviance Information Criterion, the use of power transformation families to the return series clearly improves the fit of the normal GARCH(1,1) model. In particular, the Modulus transformation family provides the best fit. Under Student's t-error distribution assumption, the GARCH(1,1) models under power transformed returns perform better in few cases.
Highlights
Normality is assumed in various statistical methods such as hypothesis testing, linear regression analysis, econometric modeling, and quality control problems
This study investigated the performance of four power transformation families, i.e. the extended Tukey, exponential, Modulus, and Yeo–Johnson, in transforming the returns of Generalized ARCH (GARCH)(1,1) models
We performed the Bayesian inference of the proposed model by the Adaptive Random Walk Metropolis (ARWM) sampler in the Markov Chain Monte Carlo (MCMC) method
Summary
Normality is assumed in various statistical methods such as hypothesis testing, linear regression analysis, econometric modeling, and quality control problems. When the simple version of extended Box-Cox was applied—which is none but the extended Tukey transformation (Tukey 1957), i.e. T (x, λ) = sign(x)|x|λ for λ > 0—the parameter estimation is significant and the extended model outperforms the basic model in terms of likelihood ratio test. This result confirms the findings of Nugroho, Susanto, Prasetia, and Rorimpandey (2019). Nugroho (2019) used the Yeo–Johnson function to transform the returns data that is applied to estimate a normal GARCH(1,1) model.
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