Is the Hamilton regression filter really superior to Hodrick-Prescott detrending?

An article published in 2018 by J.D. Hamilton gained significant attention due to its provocative title, "Why you should never use the Hodrick-Prescott filter." Additionally, an alternative method for detrending, the Hamilton regression filter (HRF), was introduced. His work was frequently interpreted as a proposal to substitute the Hodrick-Prescott (HP) filter with HRF, therefore utilizing and understanding it similarly as HP detrending. This research disputes this perspective, particularly in relation to quarterly business cycle data on aggregate output. Focusing on economic fluctuations in the United States, this study generates a large amount of artificial data that follow a known pattern and include both a trend and cyclical component. The objective is to assess the effectiveness of a certain detrending approach in accurately identifying the real decomposition of the data. In addition to the standard HP smoothing parameter of $\lambda = 1600$ , the study also examines values of $\lambda <^>{\star }$ from earlier research that are seven to twelve times greater. Based on three unique statistical measures of the discrepancy between the estimated and real trends, it is evident that both versions of HP significantly surpass those of HRF. Additionally, HP with $\lambda <^>{\star }$ consistently outperforms HP-1600.