This paper generalizes the popular stochastic volatility in mean model of Koopman and Hol Uspensky (2002) to allow for time-varying parameters in the conditional mean. The estimation of this extension is nontrival since the volatility appears in both the conditional mean and the conditional variance, and its coefficient in the former is time-varying. We develop an efficient Markov chain Monte Carlo algorithm based on band and sparse matrix algorithms instead of the Kalman filter to estimate this more general variant. We illustrate the methodology with an application that involves US, UK and Germany inflation. The estimation results show substantial time-variation in the coefficient associated with the volatility, high-lighting the empirical relevance of the proposed extension. Moreover, in a pseudo out-of-sample forecasting exercise, the proposed variant also forecasts better than various standard benchmarks.