In many application domains there is growing interest in the fitting of stochastic differential equations (SDEs, diffusions) to data. The data may come from experimental observations or from large scale computer simulations. In many case the data has a multiscale character which is incompatible with a (or the desired) diffusion process at small scales. However it may be compatible at intermediate scales. In order to understand this situation I will study the fitting of SDEs to data generated by multiscale diffusion processes, in situations where averaging and homogenization apply. The parametric model will be the averaged or homogenized equation; the data, however, will be chosen from the multiscale model. Understanding this mismatch between data and model will shed light on the original problem of incompatibility between model and data at small scales.