Tincho Almuzara, Gabriele Fiorentini, and Enrique Sentana have a fascinating new paper, "Aggregate Output Measurements: A Common Trend Approach," https://www.cemfi.es/ftp/wp/2101.pdf.
Totally reasonable to explore the cointegration route, and quite striking how much it reduces the extraction error variance for long-run objects. At the same time, it’s remarkable how little difference it makes for short-run objects when the signal-to-noise ratio is high.
That is, the extractions in the left and right panels of their Fig 5 appear almost identical. I would like to see a plot of their divergence (surely close to 0, although it would be interesting to see if/when the divergence is large -- e.g., are there business cycle effects?), along with a confidence interval (surely it would include 0?).
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