@sadia: linear filtering typically means regressing on a constant and a trend. The constant will take out the mean. If you do not use a constant, you also need to demean.
Note that the Stock-Watson argument applies to DSGE models as well. Their solution takes the form of a VARMA-process and the likelihood minimizes the forecast error.
Two papers using the one-sided HP-filter in the context of DSGE model estimation are
- Mingming Jiang (2016): “By force of demand: Explaining cyclical fluctuations of international trade and government spending”, JEDC, Volume 69,p. 249–267
- Born/Pfeifer (2014): “Policy risk and the business cycle”, JME, 68,p. 68-85
@JoseDNino: yes, there are papers using the regular HP-filter. Christensen/Dib (2008): The financial accelerator in an estimated New Keynesian model, RED is one example. But nowadays this is really frowned upon.