Estimating m-regimes STAR-GARCH model using QMLE with parameter transformation

It is well known in the literature that obtaining the parameter estimates for the Smooth Transition Autoregressive-Generalized Autoregressive Conditional Heteroskedasticity (STAR-GARCH) can be problematic due to computational difficulties. Conventional optimization algorithms do not seem to perform well in locating the global optimum of the associated likelihood function. This makes Quasi-Maximum Likelihood Estimator (QMLE) difficult to obtain for STAR-GARCH models in practice. Curiously, there has been very little research investigating the cause of the numerical difficulties in obtaining the parameter estimates for STAR-GARCH using QMLE. The aim of the paper is to investigate the nature of the numerical difficulties using Monte Carlo Simulation. By examining the surface of the log-likelihood function based on simulated data, the results provide several insights into the difficulties in obtaining QMLE for STAR-GARCH models. Based on the findings, the paper also proposes a simple transformation on the parameters to alleviate these difficulties. Monte Carlo simulation results show promising signs for the proposed transform. The asymptotic and robust variance–covariance matrices of the original parameter estimates are derived as a function of the transformed parameter estimates, which greatly facilitates inferences on the original parameters

Nonlinear volatility models in economics: smooth transition and neural network augmented GARCH, APGARCH, FIGARCH and FIAPGARCH models

Recently, Donaldson and Kamstra (1997) proposed a class of NN-GARCH models which are extended to a class of NN-GARCH family by Bildirici and Ersin (2009). The study aims to analyze the nonlinear behavior and leptokurtic distribution in petrol prices by utilizing a newly developed family of econometric models that deal with these concepts by benefiting from both LSTAR type and ANN based nonlinearity. With this purpose, the study proposed several LSTAR-GARCH-NN family models. It is noted that the multilayer perceptron (MLP) neural network and LSTAR models have significant architectural similarities. Accordingly, linear GARCH, fractionally integrated FI-GARCH, asymmetric power APGARCH and fractionally integrated asymmetric power APGARCH models are augmented with a family of Neural Network models. The study has following contributions: i. STAR-GARCH and LSTAR-GARCH are extended to their fractionally integrated asymmetric power versions and STAR-ST-FIGARCH and STAR-ST-APGARCH, STAR-ST-FIAPGARCH models are developed and evaluated. ii. By extending these models with neural networks, LSTAR-LST-GARCH-MLP family models are developed and investigated. These models benefit from LSTAR type nonlinearity and NN based nonlinear NN-GARCH models to capture time varying volatility and nonlinearity in petrol prices. ANN augmented versions of LSTAR-LST-GARCH models are as follows: LSTAR-LST-GARCH-MLP, LSTAR-LST-FIGARCH-MLP, LSTAR-LST-APGARCH-MLP and LSTAR-LST-FIAPGARCH-MLP. Empirical findings are collected as follows. i. To model petrol prices, fractionally integrated and asymmetric power versions provided improvements among the GARCH family models in terms of forecasting. ii. LSTAR-LST-GARCH model family is promising and show significant gains in out-of-sample forecasting. iii. MLP-GARCH family provided similar results with the LSTAR-LST-GARCH family models, except for the MLP-FIGARCH and MLP-FIAPGARCH models. iv. Volatility clustering, asymmetry and nonlinearity characteristics of petrol prices are captured most efficiently with the LSTAR-LST-GARCH-MLP models benefiting from forecasting capabilities of neural network techniques, whereas, among the newly developed models, LSTAR-LST-APGARCH-MLP model provided the best performance overall

Nonlinear volatility models in economics: smooth transition and neural network augmented GARCH, APGARCH, FIGARCH and FIAPGARCH models

Recently, Donaldson and Kamstra (1997) proposed a class of NN-GARCH models which are extended to a class of NN-GARCH family by Bildirici and Ersin (2009). The study aims to analyze the nonlinear behavior and leptokurtic distribution in petrol prices by utilizing a newly developed family of econometric models that deal with these concepts by benefiting from both LSTAR type and ANN based nonlinearity. With this purpose, the study proposed several LSTAR-GARCH-NN family models. It is noted that the multilayer perceptron (MLP) neural network and LSTAR models have significant architectural similarities. Accordingly, linear GARCH, fractionally integrated FI-GARCH, asymmetric power APGARCH and fractionally integrated asymmetric power APGARCH models are augmented with a family of Neural Network models. The study has following contributions: i. STAR-GARCH and LSTAR-GARCH are extended to their fractionally integrated asymmetric power versions and STAR-ST-FIGARCH and STAR-ST-APGARCH, STAR-ST-FIAPGARCH models are developed and evaluated. ii. By extending these models with neural networks, LSTAR-LST-GARCH-MLP family models are developed and investigated. These models benefit from LSTAR type nonlinearity and NN based nonlinear NN-GARCH models to capture time varying volatility and nonlinearity in petrol prices. ANN augmented versions of LSTAR-LST-GARCH models are as follows: LSTAR-LST-GARCH-MLP, LSTAR-LST-FIGARCH-MLP, LSTAR-LST-APGARCH-MLP and LSTAR-LST-FIAPGARCH-MLP. Empirical findings are collected as follows. i. To model petrol prices, fractionally integrated and asymmetric power versions provided improvements among the GARCH family models in terms of forecasting. ii. LSTAR-LST-GARCH model family is promising and show significant gains in out-of-sample forecasting. iii. MLP-GARCH family provided similar results with the LSTAR-LST-GARCH family models, except for the MLP-FIGARCH and MLP-FIAPGARCH models. iv. Volatility clustering, asymmetry and nonlinearity characteristics of petrol prices are captured most efficiently with the LSTAR-LST-GARCH-MLP models benefiting from forecasting capabilities of neural network techniques, whereas, among the newly developed models, LSTAR-LST-APGARCH-MLP model provided the best performance overall