This paper proposes a unified framework for the effective rate analysis over arbitrary correlated and not necessarily identical multiple-input single-output (MISO) fading channels, which uses the moment generating function (MGF) based approach and H transform representation. The proposed framework has the potential to simplify the cumbersome analysis procedure compared with the probability density function-based approach. Moreover, the effective rates over two specific fading scenarios are investigated, namely, independent but not necessarily identical distributed (i.n.i.d.) MISO hyper Fox's H fading channels and arbitrary correlated generalized K fading channels. The exact analytical representations for these two scenarios are also presented. By substituting corresponding parameters, the effective rates in various practical fading scenarios, such as Rayleigh, Nakagami-m, Weibull/Gamma, and generalized K fading channels, are readily available. In addition, asymptotic approximations are provided for the proposed H transform and MGF-based approach as well as for the effective rate over i.n.i.d. MISO hyper Fox's H fading channels. Simulations under various fading scenarios are also presented, which support the validity of the proposed method.