A three-sector model with a suitably chosen distribution of price stickiness can closely approximate the response to aggregate shocks of New Keynesian models with a much larger number of sectors, allowing for their estimation at much reduced computational cost.
Carvalho, Carlos, and Fernanda Nechio. 2017. “Approximating Multisector New Keynesian Models,” Federal Reserve Bank of San Francisco Working Paper 2017-12. Available at https://doi.org/10.24148/wp2017-12