Lets say I have two time series $X_t$ and $Y_{t,q}$. As an examples, lets say $X_t$ is a series that measures year over year changes in the level of output of a good (say number of widgets). So $X_t = \frac{Widgets_t}{Widgets_{t-1}} - 1$. I have another series $Y_{t,q}$ that is quarterly and measures changes in number of workers for the company (and would like to use this series because I think that the changes in workers $w_{t,q}$ would be indicative of the change in number of widgets sold. The quarterly series would hopefully provide a good indicator
How would I actually best compare the two series?
I could take the mean of the quarters of a given year for $Y_t$ so then I would get $Y^{mean}_t = \frac{1}{4}\sum \limits_{i=1}^4Y_{t,i}-1=\frac{1}{4} \left( \frac{w_{t,1}}{w_{t-1,4}}+\frac{w_{t,2}}{w_{t,1}}+\frac{w_{t,3}}{w_{t,2}}+\frac{w_{t,4}}{w_{t,3}} \right) - 1 $
Or alternatively, I could take a geometric mean. $Y^{geomean}_t = \left( \prod \limits_{i=1}^4Y_{t,i} \right)^{1/4}=\left( \frac{w_{t,4}}{w_{t-1,4}}\right)^{\frac{1}{4}} - 1 $
Both dont seem like the most ideal way since the mean method measures more of an interyear change and the geomean measures last quarter of the year changes.