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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?

  1. 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 $

  2. 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.

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You need to think in terms of autocorrelations and volatility to make your choice:

  • in your example you have the change in the number of workers $Y_{t,q}$
  • what is the meaning of the average change per quarter compared to the yearly production ?
  • probably you should sum your quarterly changes to have a yearly one : I would recommend $\sum_q Y_{t,q}$.
  • if you believe the agitation in the number of workers has an influence on the production, you can add another time series made of the average of changes $\frac{1}{Q} \sum_q |Y_{t,q}|$.
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Why don't you construct the annual value of $Y_t$ from the data, so in your example it would be $Y_{t,annual} = \sum_{i = 1}^{4}Y_{i, quart}$. This is of course only relevant if levels are important, and the time series is in absolute values. If it is a percentage, the geo-mean would be the correct (see https://en.wikipedia.org/wiki/Geometric_mean#Proportional_growth).

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the geometric mean is appropriate.

rule of thumb:

  • geometric mean for percentage numbers
  • arithmetic mean for absolute numbers and continuous rates
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An alternative approach would be to use a procedure similar to that described in

Chow, G. C. and Lin, A.-l. (1971). Best linear unbiased interpolation, distribution, and extrapolation of time series by related series, The Review of Economics and Statistics 53(4): 372 – 75.

This procedure is used to produce, for example, quarterly national accounts aggregates where only annual data are available but some indicator variables are available. A linear relationship with an auto-correlated disturbance is assumed between the unobserved quarterly variable and the indicators. Chow-Lin estimates this relationship using some "tricks". The estimation procedure can be replace by a maximum likelihood estimator. The Chow-Lin procedure is implemented in various software packages but I would recommend that it be programmed using maximum likelihood when you can adapt the procedure to better meet your needs.

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