Standard economic models based on rational expectations and homogeneity have problems explaining the complex and volatile nature of financial markets. Recently, boundedly rational and heterogeneous agent models have been developed and simulated returns are found to exhibit various stylized facts, such as volatility clustering and fat tails. Here, we are interested in how well the proposed models can explain all the properties seen in real data, not just one or a few at a time. Hence, we do a proper estimation of some simple versions of such a model by the use of efficient method of moments and maximum likelihood and compare the results to real data and more traditional econometric models. We discover two main findings. First, the similarities with observed data found in earlier simulations rely crucially on a somewhat unrealistic modeling of the noise term. Second, when the stochastic is more properly introduced we find that the models are able to generate some stylized facts, but that the fit generally is quite poor.