Averaging quarterly level data to the annual level

Question

From the Quarterly Workforce Indicators, I have obtained a measure of data measured at the quarterly level for earnings and employment.

so in each quarter, they take the total earnings, and divide by the total employment for that quarter. I want to create an 'implied' annual average earnings from this. A few questions:

1. would it be invalid to just take the average of the average quarterly earnings, and multiply that by 4?

2. alternatively, since I can obtain the denominator for each quarter, I wished to do the following: for each quarter q, multiply average quarterly earnings by the total employment in that quarter. this should yield the total quarterly earnings. Do this for each quarter, and then add each to get the total annual earnings. divide by the total annual employment.

is 2. equivalent to a 'weighted' mean? or can it be directly interpreted as an annual average earnings? my concern is that since each measurement is calculated independently each quarter, the denominator and the numerator could be double counting many jobs.

Answer 1

Answer to Q1: While imperfect, that approach is preferable out of the two you've presented here. Your 'implied' annual average will generally be higher than the actual annual average, with the extent of this error rising with employment churn. Consider calling the result the "quarterly earnings per employment, averaged for the year" or similar, so that your figure isn't misconstrued as annual earnings per employment. Or, if you have the time and inclination, you could report something more sophisticated than a ratio; consider an Autocorrelated Error Model, e.g. Chapter 16 of Griffiths et al, below.

Answer to Q2: This is often done when the data points are independent; the established techniques of “ratio estimation” are covered in, for example, Section 7.4 of John Rice’s textbook (below). However, your data aren't independent. You have four highly dependent readings of a two-dimensional signal (one dimension is earnings, one is employment). You’re quite right to be concerned about what you call double counting, for this particular problem.

Good luck!

References:

• The 3rd edition of John Rice’s Mathematical Statistics & Data Analysis (2007)
• Learning and Practicing Econometrics by W. Griffiths et al (1993)