An aggregation-consistent implementation of the Hamilton filter

I propose a modified implementation of the popular Hamilton filter, to make the cyclical component extracted from an aggregate variable consistent with the aggregation of the cyclical components extracted from its underlying variables. This procedure is helpful in many circumstances, for instance when dealing with a variable that comes from a definition or when the empirical relationship is based on an equilibrium condition of a growth model. The procedure consists of the following steps: 1) build the aggregate variable, 2) run the Hamilton filter regression on the aggregate variable and store the related OLS estimates, 3) use these estimated parameters to predict the trends of all the underlying variables, 4) rescale the constant terms to obtain mean-zero cyclical components that are aggregation-consistent. I consider two applications, exploiting U.S. and Canadian data. The former is based on the GDP expenditure components, while the latter on the GDP of its Provinces and Territories. I find sizable differences between the cyclical components of aggregate GDP computed with and without the adjustment, making it a valuable procedure for both assessing the output gap and validating empirically DSGE models.