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.

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 $\frac{1}{Q} \sum_q |Y_{t,q}|$.

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).

the geometric mean is appropriate.

rule of thumb:

  • geometric mean for percentage numbers
  • arithmetic mean for absolute numbers and continuous rates

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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