Model reduction by moment matching does not preserve, in a systematic way, the transient response of the system to be reduced, thus limiting the use of this model reduction technique in control problems. With the final goal of designing reduced-order models which can effectively be used (not just for analysis but also) for control purposes, we determine, using a data-driven approach, an estimate of the moments and of the transient response of an unknown system. We compute the unique, up to a change of coordinates, reduced-order model which possesses the estimated transient and, simultaneously, achieves moment matching at the prescribed interpolation points. The error between the output of the system and the output of the reduced-order model is minimized and we show that the resulting system is a constrained optimal (in a sense to be specified) reduced-order model. The results of the paper are illustrated by means of a simple numerical example.