The consistency of certain identification methods for Linear Parameter Varying systems is considered. More precisely, methods for the identification of SISO input-output models are analysed. In order to perform a consistency analysis the application of ergodicity is required, which is not obviously applicable with these types of time-varying systems. It is therefore shown that, when the scheduling parameter satisfies certain conditions, ergodicity type results can be applied to the methods considered. An analysis is then carried out for two cases: that of noise-free measurements of the scheduling parameter, and then the more realistic case of noisy scheduling parameter measurements. The latter is shown to be an errors-in-variables type problem. In both cases the least squares technique is shown to typically give biased estimates and the instrumental variables method is proposed as a way of resolving this. The analysis carried out is reinforced by results in simulation.