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It could be much more simple: if you use the method of moments (MM) then you estimate the mean and the variance and for example the kurtosis of your sample. Then you fit the parameters to these statistics. Alternatively you use maximum-likelihood (MLE). For MM: from wikipedia you get the mean and the variance. In your notation you can fit $b = \bar{r}$ so ...


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I think the following two questions and related answers should help in answering the question: Why use swap-rates in a yield curve? and Is there an Australian Interbank Rate? Essentially to derive funding curves you gotta use what is left with the constraint that the source instrument has to be liquid enough and closely enough reflect true market ...


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It really depends on how/where do you plan to use final values. I would not use extrapolation since it will ignore market realities. Forward rates across long end tend to be increasing while dumb extrapolation might give you the opposite result. In case of treasuries one can use treasury and swap spread and while you do not have 50 Y treasuyy one can find ...



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