This paper proposes a semiparametric approach by introducing a smooth
scale function into the standard generalized autoregressive conditional
heteroskedastic (GARCH) model so that conditional heteroskedasticity
(CH) and scale change in financial returns can be modeled
simultaneously. An estimation procedure combining kernel estimation of
the scale function and maximum likelihood estimation of the GARCH
parameters is proposed. Asymptotic properties of the estimators are
investigated in detail. It is shown that asymptotically normal,
-consistent parameter
estimation is available. A data-driven algorithm is developed for
practical implementation. Finite sample performance of the proposal is
studied through simulation. The proposal is applied to model CH and
scale change in the daily S&P 500 and DAX 100 returns. It is shown
that both series have simultaneously significant scale change and
CH.We are very grateful to the co-editor
and two referees for their helpful comments and suggestions, which led to a
substantial improvement of this paper. The paper was finished under the
advice of Professor Jan Beran, Department of Mathematics and Statistics,
University of Konstanz, Germany, and was financially supported by the
Center of Finance and Econometrics (CoFE), University of Konstanz. We thank
colleagues in CoFE, especially Professor Winfried Pohlmeier, for their
interesting questions at a talk of the author. It was these questions that
motivated the author to write this paper. Our special thanks go to Dr. Erik
Lüders, Department of Finance and Insurance, Laval University, and
Stern School of Business, New York University, for his helpful
suggestions.