I have been tasked with a long-run analysis of the Japanese and Brazilian economies, and as part of this, I have decomposed GDP growth into the contributions from capital, labour and TFP (following the standard Solow model and using Penn World Table's data). My method is fairly standard: the capital contribution to GDP growth is the growth in the capital stock growth times 'alpha', and the labour contribution to GDP growth is employment growth to the power of 'beta', with 'beta' being the labour share of income and 'alpha' being the capital share of income'. The TFP contribution to GDP growth is then the residual, GDP growth minus the capital contribution minus the labour contribution.
However, this approach means that all the short-run business cycles and other noise from real GDP is included in the TFP contribution to the GDP growth. I feel that this is unhelpful for long-run analysis and that, by using the HP Filter to smooth real GDP across the business cycle, this would give a better picture of the long run. For instance, the first chart below shows my results for Brazil without smoothing GDP using the HP Filter, and the second chart shows the results for Brazil after smoothing GDP with the HP Filter.
Thank you in advance!