In this paper we study the problem of minimum variance prediction for linear time-varying systems. We consider the standard time-varying autoregression moving average (ARMA) model and develop a predictor which guarantees minimum variance prediction for a large class of linear time-varying systems. The predictor is developed based on a pseudocommutation technique for dealing with noncommutativity of linear time-varying operators in a transfer operator framework. We also show connections between this input-output predictor and the Kalman predictor via an example.