In the paper on Double Heston model (2009) from Peter Christoffersen, they say: "Our focus is on explaining why a two-factor model works better than a one-factor model for the purpose of option pricing". However, they calibrate the model using market data from every Wednesday in a year period. So in the same calibration they include option (market) prices from 52 different dates. It seems strange to me that they are including historical data (1 year of data) to test a model for option pricing. If we have 52 sets of market data, I would expect 52 calibrations using market data only as of one date, similar to Bakshi's paper (1997), where they do daily calibrations for a period of 3 years. Any thoughts on this?
I read a similar empirical analysis on the Heston-Nandi model where the model parameters were calibrated based on historical option data over a couple of years. For the same period, they also did a rolling estimation of the parameters (using historical log-returns). I believe the intention was to compare the parameter level and variance of the two methods over time. E.g. under risk-neutral distribution, they find that the leverage effect is stronger (consistent with the skewness premium).
I don't have professional experience with these types of models but my understanding is that you only use the most recent data when calibrating.