How could one go about estimating a long term aggregate production function, when there is a sharp decline in both factor inputs (i.e. investment and hours worked) and output for a few consecutive periods? I have seen some research, where the authors either adapt a regime-switching errors technique Bartkus, A., 2016 (although this technique was used in forecasting GDP as a random-walk process, and not estimating a production function), and in another - the authors omitted the "undesireable", "anomalous" data (the three or four consecutive quarters that the sharp fall in GDP was observed in the GFC) from the estimation alltogether.
What other, robust, methods are used when estimating the aggregate production function from such heteroscedastic(?) time-series data? Without going too much into business cycle theory teritory.