  Range-measured return contains more information than the traditional
scalar-valued return. In this paper, we propose to model the [low, high] price
range as a random interval and suggest an interval-valued GARCH (Int-GARCH)
model for the corresponding range-measured return process. Under the general
framework of random sets, the model properties are investigated. Parameters are
estimated by the maximum likelihood method, and the asymptotic properties are
established. Empirical application to stocks and financial indices data sets
suggests that our Int-GARCH model overall outperforms the traditional GARCH for
both in-sample estimation and out-of-sample prediction of volatility.
