论文部分内容阅读
Abstract： The need to capture the heterogeneous and volatility nature of bothfinancial and economic time series theory and modeling their behavior in practical work have stimulated interest in the empirical modeling of variances which forms the basis for this study. In the study we augmented GARCH models with smooth transition model by dropping the assumption of autoregression of the model； necessary theoretical frame work was derived and properties of the new model established and illustrated with foreign exchange rate data from Federal Republic of Nigeria （Naira）， Great Britain （Pound）， Botswana （Pula） and Japanese （Yen） against United States of America （Dollar）. The smooth transition GARCH model is better than the classical GARCH model as there were reduction in the variances of the augmented model； this claim is confirmed by the empirical illustration with foreign exchange data. Within the group of smooth transition GARCH model， Logistic Smooth Transition is adjudged the best as it produced the least variance.
Key words： GARCH models； ST-GARCH models； ET-GARCH； ESTGARCH； LST-GARCH； Foreign exchange data
1. INTRODUCTION
Financial time series experts have discovered evidence of occasional sudden breaks in many economic time series. For example， currency exchange rates often move suddenly as governments devalue due to speculative pressure and deteriorating economic conditions. Commodity prices， such as oil， change in response to shocks from exogenous geopolitical events or supply disruptions due to weather related catastrophes like hurricane Katrina， andfinancial markets can shift abruptly in response tofinancial crises [1]. Perhaps the best example of this is economic growth， where the rate of growth of the economy alternates between periods of high growth（economic expansion） and periods of declining or negative growth （recession）. It is an established fact that Generalized autoregressive model cannot capture nonlinearity in economic andfinancial series adequately； [2] utilized GARCH to model stock market indices and concluded that the model fails to capture the statistical structure of the market returns series for all the countries economies investigated.[3] employed the Hinich portmanteau bicorrelation test to determine the adequacy of GARCH model for eight Asian stock markets and equally concluded that GARCH is not suitable. Due to the inadequacy of GARCH model there is need to augment it with non-linear models， in this paper we combine GARCH model with Smooth transition model， the various transition models used are Exponential transition （ET）， Exponential smooth transition （EST） and Logistic smooth transition （LST） models. The smooth transition is an extension of the regime switching model that allows intermediate states or regimes. The idea of smooth transition was proposed to allow a more gradual change for the parameter of transition. Thus， the ST-GARCH allows enriching the class of GARCH models， through asymmetry， or the leverage effect， that is the difference in the volatility response to positive and negative return shocks [4]. This model emphasizes the nonlinearities in the conditional volatility equation. Although a variety of models were already presented in the literature in order to explore different forms of nonlinearity， Hagerud （1997） [5] affirms that the ST-GARCH model presents new characteristics， very advantageous for the modeling of the volatility. So also this model provides moreflexibility in the transition mechanism of the conditional volatility. Contrary to the traditional threshold models， that allow only two volatility regimes （a low volatility regime and a high volatility regime）， the ST-GARCH model offers the possibility of intermediate regimes and allows the introduction of a smoother transition mechanism in the GARCH specification [6]. In this paper， the Smooth Transition Generalized Autoregressive Conditional Heteroscedastic （ST-GARCH） model is considered， where the possibility of intermediate regimes is modeled with the introduction of a smooth transition mechanism in a Generalized Autoregressive Conditional Heteroscedastic（GARCH） specification. The transition functions utilized are logistic （the Logistic Smooth Transition GARCH （LST-GARCH） model） Exponential （the Exponential Transition GARCH （ET-GARCH） and the Exponential Smooth Transition GARCH（EST-GARCH） model）. An important characteristic of the LST-GARCH model is that it highlights the asymmetric effect of unanticipated shocks on the conditional volatility. On the other side， the ET-GARCH and EST-GARCH model allow the dynamics of the conditional variance to be independent of the sign of past news. Indeed， this model allows us to highlight the size effect of the shocks， so that small and big shocks have separate effects. The remaining part of this paper is organized as follows： section 2-5 covers the Specification of transition models with GARCH， estimation of the parameters of STGARCH models， efficiency of ST-GARCH models with respect to GARCH model， section 6， Empirical analysis， estimation ofγand c estimation of ET-GARCH， ESTGARCH and LST-GARCH for all series section 7 Empirical comparisons of models section 8 Conclusion.
6. EMPIRICAL RESULTS/DATA ANALYSIS WITH EXCHANGE RATE DATA
This section examines the empirical results obtained for Smooth transition models with GARCH （ST-GARCH） for four sets of exchange rates data namely British（Pounds）， Japanese （Yen）， Nigerian （Naira） and Batswana （Pula） against American （Dollar）.Here the Parameters of Exponential transition GARCH models（ET-GARCH）， Exponential smooth transition GARCH （EST-GARCH） and Logistic transition GARCH （LST-GARCH） models were obtained using the derived equations for all the series.
In the tables 3 and 4 above， all the asterisk values are selected because they have minimum values and are subsequently used in equations （8）， （13） and （18）.
We can now illustrate the empirical implication of these theories as follows：
Using equations （8）， （13） and （18） the variances of all the series for Smooth transition GARCH models （ET-GARCH， EST-GARCH and LT-GARCH） with classical GARCH model are computed and shown/displayed in Table 5.
Table 5 shows the variances of GARCH model and those of ST-GARCH models（ET-GARCH， EST-GARCH and LST-GARCH）. The superiority of Logistic Smooth transition model within the group of smooth transition was asserted here as this model gave us the minimum variances for all the series under study， this is followed by EST-GARCH and ET-GARCH in that order as seen in Table 5 below：
The variances of all ST-GARCH models with GARCH as displayed in Table 5 shows that all ST-GARCH outperformed the classical GARCH model， however， the LST-GARCH performed best， followed by the performances of LST-GARCH and ET-GARCH in that order. The implication is that the use of LSTCGARCH produces the best result； however EST-GARCH and ET-GARCH may be utilized in some occasions. But LST-GARCH produces optimal result.
REFERENCES
[1] Brooks， C.， & Hinich， M. （1998）. Episodic nonstationarity in exchange rates. Applied Economics Letters， 5， 719-722.
[2] Bonilla， C. A.， & Sepulveda， J. （2011）. Stock returns in emerging markets and the use of GARCH models. Journal of Applied Economics Letters， 18（14）， 1321-1325. [3] Lim， K. P.， Hinich， M. J.， & Liew， V. （2005）.Statistical inadequacy of GARCH models for Asian stock markets： evidence and implications.International Journal of Emerging Market Finance， 4， 263-79.
[4] Bollerslev， T. （1986）. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics， 31， 307-327.
[5] Hagerud， G. E. （1997）. A new non linear GARCH model （Doctoral Dissertation）. Stochlom School of Economics.
[6] Bonilla， C.， Romero-Meza， R.， & Hinich， M. J. （2006）. Episodic nonlinearities in the Latin American stock market indices. Applied Economics Letters， 13， 195-9.
[7] Ter¨asvirta， T. （1994）. Specification， estimation， and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association， 89， 208-218.
[9] Ter¨asvirta， T.， & Anderson， H. （1992）. Characterizing nonlinearities in business cycles using smooth transition autoregressive models. Journal of Applied Econometrics， 7， S119-S136.
Key words： GARCH models； ST-GARCH models； ET-GARCH； ESTGARCH； LST-GARCH； Foreign exchange data
1. INTRODUCTION
Financial time series experts have discovered evidence of occasional sudden breaks in many economic time series. For example， currency exchange rates often move suddenly as governments devalue due to speculative pressure and deteriorating economic conditions. Commodity prices， such as oil， change in response to shocks from exogenous geopolitical events or supply disruptions due to weather related catastrophes like hurricane Katrina， andfinancial markets can shift abruptly in response tofinancial crises [1]. Perhaps the best example of this is economic growth， where the rate of growth of the economy alternates between periods of high growth（economic expansion） and periods of declining or negative growth （recession）. It is an established fact that Generalized autoregressive model cannot capture nonlinearity in economic andfinancial series adequately； [2] utilized GARCH to model stock market indices and concluded that the model fails to capture the statistical structure of the market returns series for all the countries economies investigated.[3] employed the Hinich portmanteau bicorrelation test to determine the adequacy of GARCH model for eight Asian stock markets and equally concluded that GARCH is not suitable. Due to the inadequacy of GARCH model there is need to augment it with non-linear models， in this paper we combine GARCH model with Smooth transition model， the various transition models used are Exponential transition （ET）， Exponential smooth transition （EST） and Logistic smooth transition （LST） models. The smooth transition is an extension of the regime switching model that allows intermediate states or regimes. The idea of smooth transition was proposed to allow a more gradual change for the parameter of transition. Thus， the ST-GARCH allows enriching the class of GARCH models， through asymmetry， or the leverage effect， that is the difference in the volatility response to positive and negative return shocks [4]. This model emphasizes the nonlinearities in the conditional volatility equation. Although a variety of models were already presented in the literature in order to explore different forms of nonlinearity， Hagerud （1997） [5] affirms that the ST-GARCH model presents new characteristics， very advantageous for the modeling of the volatility. So also this model provides moreflexibility in the transition mechanism of the conditional volatility. Contrary to the traditional threshold models， that allow only two volatility regimes （a low volatility regime and a high volatility regime）， the ST-GARCH model offers the possibility of intermediate regimes and allows the introduction of a smoother transition mechanism in the GARCH specification [6]. In this paper， the Smooth Transition Generalized Autoregressive Conditional Heteroscedastic （ST-GARCH） model is considered， where the possibility of intermediate regimes is modeled with the introduction of a smooth transition mechanism in a Generalized Autoregressive Conditional Heteroscedastic（GARCH） specification. The transition functions utilized are logistic （the Logistic Smooth Transition GARCH （LST-GARCH） model） Exponential （the Exponential Transition GARCH （ET-GARCH） and the Exponential Smooth Transition GARCH（EST-GARCH） model）. An important characteristic of the LST-GARCH model is that it highlights the asymmetric effect of unanticipated shocks on the conditional volatility. On the other side， the ET-GARCH and EST-GARCH model allow the dynamics of the conditional variance to be independent of the sign of past news. Indeed， this model allows us to highlight the size effect of the shocks， so that small and big shocks have separate effects. The remaining part of this paper is organized as follows： section 2-5 covers the Specification of transition models with GARCH， estimation of the parameters of STGARCH models， efficiency of ST-GARCH models with respect to GARCH model， section 6， Empirical analysis， estimation ofγand c estimation of ET-GARCH， ESTGARCH and LST-GARCH for all series section 7 Empirical comparisons of models section 8 Conclusion.
6. EMPIRICAL RESULTS/DATA ANALYSIS WITH EXCHANGE RATE DATA
This section examines the empirical results obtained for Smooth transition models with GARCH （ST-GARCH） for four sets of exchange rates data namely British（Pounds）， Japanese （Yen）， Nigerian （Naira） and Batswana （Pula） against American （Dollar）.Here the Parameters of Exponential transition GARCH models（ET-GARCH）， Exponential smooth transition GARCH （EST-GARCH） and Logistic transition GARCH （LST-GARCH） models were obtained using the derived equations for all the series.
In the tables 3 and 4 above， all the asterisk values are selected because they have minimum values and are subsequently used in equations （8）， （13） and （18）.
We can now illustrate the empirical implication of these theories as follows：
Using equations （8）， （13） and （18） the variances of all the series for Smooth transition GARCH models （ET-GARCH， EST-GARCH and LT-GARCH） with classical GARCH model are computed and shown/displayed in Table 5.
Table 5 shows the variances of GARCH model and those of ST-GARCH models（ET-GARCH， EST-GARCH and LST-GARCH）. The superiority of Logistic Smooth transition model within the group of smooth transition was asserted here as this model gave us the minimum variances for all the series under study， this is followed by EST-GARCH and ET-GARCH in that order as seen in Table 5 below：
The variances of all ST-GARCH models with GARCH as displayed in Table 5 shows that all ST-GARCH outperformed the classical GARCH model， however， the LST-GARCH performed best， followed by the performances of LST-GARCH and ET-GARCH in that order. The implication is that the use of LSTCGARCH produces the best result； however EST-GARCH and ET-GARCH may be utilized in some occasions. But LST-GARCH produces optimal result.
REFERENCES
[1] Brooks， C.， & Hinich， M. （1998）. Episodic nonstationarity in exchange rates. Applied Economics Letters， 5， 719-722.
[2] Bonilla， C. A.， & Sepulveda， J. （2011）. Stock returns in emerging markets and the use of GARCH models. Journal of Applied Economics Letters， 18（14）， 1321-1325. [3] Lim， K. P.， Hinich， M. J.， & Liew， V. （2005）.Statistical inadequacy of GARCH models for Asian stock markets： evidence and implications.International Journal of Emerging Market Finance， 4， 263-79.
[4] Bollerslev， T. （1986）. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics， 31， 307-327.
[5] Hagerud， G. E. （1997）. A new non linear GARCH model （Doctoral Dissertation）. Stochlom School of Economics.
[6] Bonilla， C.， Romero-Meza， R.， & Hinich， M. J. （2006）. Episodic nonlinearities in the Latin American stock market indices. Applied Economics Letters， 13， 195-9.
[7] Ter¨asvirta， T. （1994）. Specification， estimation， and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association， 89， 208-218.
[9] Ter¨asvirta， T.， & Anderson， H. （1992）. Characterizing nonlinearities in business cycles using smooth transition autoregressive models. Journal of Applied Econometrics， 7， S119-S136.