Deterministic time-aggregation of principal component factors. Is it wrong? I have estimated the first factor/score using PCA on a set of 190 monthly timeseries. For my analysis I also need the quarterly factor. Two choices come to mind:

*

*Take the 3-month average of the constructed monthly factor,

*Take the 3-month average of the 190 timeseries and then estimate the quarterly factor via PCA.

Can you please give any reasons why to do one over the other? Thank you.
 A: I did a little research on the matter, and I am providing my insights to the community.
The time-aggregation literature distinguishes between flow and stock variables. An example from economics is that Investment (I) is a flow, whereas Capital (K; which is the cumulated investment) is a stock time-series. The distinction is important because if you want to convert a monthly variable to quarterly, then in the case of investment (i.e. flow) you would add the 3 monthly values of each quarter [I(t)+I(t-1)+I(t-2)], whereas in the case of capital (i.e. stock) you would only need to take the monthly figure that corresponds to the LAST month of the quarter [K(t)] (-in other words, only the figures for March, June, September, December are kept, and values from all other months are scrapped).
Now, assume a 3rd case where we have the investments timeseries (i.e. flow), in monthly frequency, and we calculate its (m-o-m) growth rate (g). If we want to time-aggregate this monthly growth rate into a quarterly growth rate series (and hence, q-o-q), Mariano-Murasawa (2003) showed that this can be approximated by: g(t)+2g(t-1)+3g(t-2)+2g(t-3)+g(t-4). Of course, if you wanted the quarterly growth series, you could alternatively, first time-aggregate the monthly series in LEVELS (into quarterly), and then calculate the growth rate on the resulted quarterly LEVELS series. Both ways would be equivalent.
Going to the PCA, it is customary (at least when one deals with economic time-series), that any non-stationary series are turned into stationary, I(0), before the principal components are extracted. However, let me add that I have come across published studies that do not remove unit roots before applying PCA. If you do stationarize your monthly series before PCA, then the resulted monthly factors would be in the form of growth rates (i.e. they are considered to be I(0)). As a result, and given all the above, if you want to time-aggregate monthly factors into quarterly, you will need to do so by applying the time-aggregation function f(t)+2f(t-1)+3f(t-2)+2f(t-3)+f(t-4) to your monthly factor time-series.
If you would like a published academic paper where they time-aggregate their PCA factors using the abovementioned approximation, you can see Hepenstrick-Marcellino (2019) p.86. They do it in the second iterated forecasting approach, where they mention that they 'aggregate the monthly factor to the quarterly frequency'. They also provide the proof for the MM (2003) approximation.
References:
Hepenstrick, C., & Marcellino, M. (2019). Forecasting gross domestic product growth with large unbalanced data sets: the mixed frequency three‐pass regression filter. Journal of the Royal Statistical Society: Series A (Statistics in Society), 182(1), 69-99.
Mariano, Roberto S., and Yasutomo Murasawa. "A new coincident index of business cycles based on monthly and quarterly series." Journal of applied Econometrics 18.4 (2003): 427-443.
