What does it mean 'to difference out'?
A Google books search for the phrase "to difference out" yields more than a dozen matches, including this one from Harvey Poniachek, International Corporate Finance: Markets, Transactions and Financial Management (2012):
Finally, Professor Stone and Wood came up with a day-of-month and day-of-week forecasting system called a payment cycle forecasting system that works very well for disbursements. Used in a vacuum, each of these methodologies has its own type of errors, which are rather perverse because they do not reflect each of the other methodology's strengths. For example, a Stone and Wood model used alone would not reflect seasonality, cycle, or trend. It is possible to use all four of these methodologies to forecast all four of these kinds of factors simultaneously. If you examine a curve for disbursements on a given week, it is possible to difference out, or "layer down," that curve. Remove the trend, remove the cycle, remove the seasonality, and take out the day of month and day of week; each of those factors sits on top of the one below it.
As used by Poniachek, "difference out" seems to mean something like "subtract or remove specific computational elements from." This sounds suspiciously similar to the simpler (and more widely understood) phrase "factor out."
A similar meaning is at work in Ben Agger, Public Sociology: From Social Facts to Literary Acts:
We use the difference-of-logs measure here for theoretical reasons, to difference out Z (equation 5).
In this case, the differencing out consists in removing the variable Z from a complex computation by using a method that enables the user to avoid having to account directly for that variable.
Yet another instance involving a complex equation, from E. Kathleen Adams & Li-Nien Chien, "Racial Disparities in Breast and Cervical Cancer: Can Legislative Action Work," in Cancer Disparities: Causes and Evidence-Based Solutions (2012) uses the expression as follows:
The effect of BCCPTA was derived by using difference-in-difference analysis. That is, we estimated the effect of BCCPTA by using our control cancers to "difference" out the effects of other factors that changed pre- and post-BCCPTA that could affect the probability that any woman with cancer, not just those affected by BCCPTA, enrolled in Medicaid.
The difference-in-differences statistical technique, mentioned by Adams & Chien above, appears again in Teresa Gibson, Three Essays on Prescription Drugs, Prices and Chronic Illness (2003) [combined snippets]:
Each deductible increase is small and similar in size so our difference-in-differences modeling approach is likely to 'difference-out' the effects, if any, of the deductible increase.
The crucial notion in all of these instances is that "differencing out" removes problematic variables or effects in a scientifically rigorous way. The earliest instance of "to difference out" in the Google Books search results I obtained was from 1956, but the vast majority were from the past twenty years.
'Differencing out' through the years
The term "differencing out" is far more common than "to difference out" in Google Books search results. A search for the term yields more than a hundred unique matches. Again, most of the matches are from the past twenty years, but one in particular is surprisingly early. From Robert Sanderson, "Sermon II" (April 24, 1621), reprinted in The Works of Robert Sanderson, D. D. Sometime Bishop of Lincoln (1854):
Things not simply evil may accidentally become such, as by sundry other means, so especially by one of these three, Conscience, Scandal and Comparison. First, conscience, in regard of the agent. Though the thing be good, yet if the agent do it with a condemning, or but a doubting conscience, the action becometh evil. .. Secondly, scandal, in regard of other men. Though the thing be good, yet if a brother stumble, or be offended, or be made weak by it, the action becometh evil. ... Thirdly, comparison, in regard of other actions. Though the thing be good, yet, if we prefer it before better things, and neglect or omit them for it, the action becometh evil. ...
The stuff thus prepared, by differencing out those things which, undistinguished, might breed confusion, our next business must be, to lay the rule, and to apply it to the several kinds of evil, as they have been differenced. I foresaw we should not have time to go through all that was intended; and therefore we will content ourselves for this time with the consideration of this rule, applied to things simply evil.
By "differencing out," Sanderson seems to mean "distinguishing" or "identifying the differences." If so, he is not using it in the modern way. Far more common is the sense of the term that appears in Austin Fyfe, Notes on Summation and Interpolation: Abstract of Discussion Thereon (1908), where "diifferencing out" amounts to "subtracting out" or "simplifying":
The method generally followed to get the series subdivided, is to find the “subdivided differences,” by forming a series of equations expressing ux in terms of u0, and its differences Δu0, Δ2u0 etc., then by differencing out the series of equations so obtained, the subdivided differences are ready at hand.
Even more common is the usage in Patrick Bayer, Why Do Some Families Choose Poor Quality Schools for Their Children (1999), where "differencing out" seems roughly equivalent to "removing from consideration through scientifically acceptable techniques":
Much of the education production function literature attempts to control for this problem by differencing out unobserved fixed effects. By looking, for example, at the increment to a child's test score when the child is in a small class versus a larger class, it is possible to estimate the effect of class size on achievement.
Especially if the child is essentially a widget. And again in Tomas Philipson, Observational Agency and Supply Side Econometrics (1997):
Now compensation for errorless supply takes place only for those who have the condition. This allows for differencing out the ones who do not have the condition.
And again, finally, in John Taylor & Michael Woodford, Handbook of Macroeconomics (1999):
As I stressed several times during the chapter, the price one pays in dealing with Euler equations is not negligible: one loses the ability of saying anything about the level of consumption. The empirical tractability of the Euler equation is obtained [by] differencing out the marginal utility of wealth, and therefore one of the main determinants of consumption levels. The challenge for future research is to construct a consumption function that incorporates the insights of the Euler equation and yet allows us to say something about the levels of consumption and about how consumption reacts to changes in the economic environment. Such a consumption function is necessary to make predictions about future consumption, about saving behaviour, and about the effects of alternative policy measures.
The phrase "differencing out" has appeared in print since at least 1621, and in a strictly mathematical or statistical sense since the early 1900s. Nevertheless, its current popularity does not go back much farther than the 1980s, when discussions of economic and social science models latched onto it as a way of saying one or another of a complex of related things: "factor out," "subtract," "simplify," "remove from consideration."
From the many instances of its use in technical journals, textbooks, and other specialized works, it appears that the phrase is firmly established in modern statistical and social science settings. But we can wish at least that people who do choose to use it will make an effort to define precisely that they mean when they use it, since distinguishing which particular meaning a user has in mind can be difficult—even when one has access to multiple similar instances for reference.