This paper proposes a unified framework for the effective rate analysis over arbitrary correlated and not necessarily identical multiple inputs single output (MISO) fading channels, which uses moment generating function (MGF) based approach and H transform representation. The proposed framework has the potential to simplify the cumbersome analysis procedure compared to the probability density function (PDF) 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.