The empirical properties of financial market data, in particular the leptokurtosis and clustering of volatilities, have been investigated systematically by Mandelbrot (1963) and Fama (1965). ARCH Models were introduced by Engle (1982) and generalized to GARCH by Bollerslev (1986). For an excellent overview of ARCH models we refer to Gouriéroux (1997). An extensive discussion of the estimation of the tail exponent and the Hill estimator can be found in Embrechts et al. (1997). A comparison of the efficiency of the Yule Walker estimator with ML estimators of ARCH and GARCH models is given in Maercker (1997).
The theory of QML estimation was developed by White (1982) and Gouriéroux et al. (1984). Weiss (1986) applied the theory to ARCH models, Lee and Hansen (1994) and Lumsdaine (1996) to GARCH models. Bollerslev and Wooldridge (1992) considered QML estimation for general conditionally heteroscedastic models. The idea of the smooth transition specification stems from Teräsvirta (1994) who used it for AR models. It was applied to GARCH models by Lubrano (1998). Hafner and Herwartz (2000) discussed various methods to test for significance of the AR parameter under conditional heteroscedasticity. Moreover, they compare empirical results of AR models versus ARCH-M models, applied to several German stock returns.
In the multivariate case, Jeantheau (1998) has shown the consistency of the QML estimator, and Comte and Lieberman (2003) derived asymptotic normality. Analytical expressions for the score and Hessian matrices are provided by Hafner and Herwatz (2003).