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We provide three new results concerning quasi-maximum likelihood (QML) estimators in generalized autoregressive conditional heteroskedastic in mean (GARCH-M) models. We first show that, depending on the functional form that we impose in the mean equation, the properties of the model may change and the conditional variance parameter space may be restricted, in contrast to the theory of traditional GARCH processes. Second, we also present a new test for GARCH effects in the GARCH-M context which is simpler to implement than alternative procedures such as in Beg et al. . We propose a new way of dealing with parameters that are not identified by creating composites of parameters that are identified. Third, the finite sample properties of QML estimators are explored in a restricted ARCH-M model and bias and variance approximations are found which show that the larger the volatility of the process the …