How do economists and climatologists reconstruct their time series from the available data? I know in particular that economists have built models to extimate the GDP per capita between the year 0 and today from the last 100 years data.
Also, given that differintegral (as defined in fractional calulus) has a memory effect, can we use it for the reconstruction of time series? I know there are models such as fractional autoregressive integrated moving average that accounts for long memory effects, is it possible to define something like FARIMA models that reconstruct the past?