Berck, Peter

Villas-Boas, Sofia B

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UC BerkeleyPostprints from Department of Agricultural and Resource Economics, UCBTitleA note on the triple difference in economic modelsPermalinkhttps://escholarship.org/uc/item/77g020ssJournalApplied Economics Letters, 23(4)ISSN1350-4851 1466-4291AuthorsBerck, PeterVillas-Boas, Sofia BPublication Date2015-08-03DOI10.1080/13504851.2015.1068912 Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaFull Terms & Conditions of access and use can be found athttp://www.tandfonline.com/action/journalInformation?journalCode=rael20Download by: [University of California, Berkeley] Date: 30 October 2015, At: 09:54Applied Economics LettersISSN: 1350-4851 (Print) 1466-4291 (Online) Journal homepage: http://www.tandfonline.com/loi/rael20A note on the triple difference in economic modelsPeter Berck & Sofia B. Villas-BoasTo cite this article: Peter Berck & Sofia B. Villas-Boas (2015): A note on the triple difference ineconomic models, Applied Economics Letters, DOI: 10.1080/13504851.2015.1068912To link to this article:  http://dx.doi.org/10.1080/13504851.2015.1068912Published online: 03 Aug 2015.Submit your article to this journal Article views: 8View related articles View Crossmark dataA note on the triple difference in economic modelsPeter Berck and Sofia B. Villas-BoasDepartment of Agricultural and Resource Economics, University of California, Berkeley, CA, USAABSTRACTThis article shows when a triple difference strategy using an imperfect control category improveson the double difference strategy for estimating an average treatment effect. For example, aproduct is treated in one place and not another leading to a double difference strategy. Whendoes comparison with an untreated product in triple difference strategy improve accuracy?KEYWORDSDifference in difference;triple difference; policyevaluation; averagetreatment effectJEL CLASSIFICATIONC21; C25; C51I. IntroductionDifference-in-differences estimation is one of themost important identification strategies in appliedeconomics (Meyer 1995; Angrist and Krueger 2005;Bertrand, Duflo, and Mullainathan 2004; Athey andImbens 2006). Exploiting a quasi-experimental paneldesign identification of the treatment effect uses adifference-in-differences econometric estimationstrategy to measure the effect of a policy change onan outcome variable of interest. The difference-in-difference model measures the effect of policy byremoving the effects of time and place. When theoutcome variable is determined by policy, time,place and yet another variable, a triple differencestrategy may reduce the bias in the estimate of theeffect of the policy change. Published examples of atriple difference strategy include (1) comparingmen’s hours of labour before and after Californiaextended its requirement for overtime after 8 hoursworked in a day to men. The comparison to otherstates provided the double difference and the furthercomparison to women, the triple difference(Hamermesh and Trejo, 2000) and (2) the effect ofdaylight savings time on energy use was found bycomparing energy use in Sydney when it expanded itsuse of daylight savings time for the Olympics withother states in Australia. The double difference isgiven by comparing time and place and the tripledifference comes from comparing to mid-day whereenergy use is not affected by daylight savings time(Kellogg and Wolf 2008).II. ModelWe start by presenting the commonly known lineardifference-in-differences model for a continuous anduncensored outcome. A policy change is observed inplace c = 1, from time t = 1 forwards. The policydoes not change in place c = 0. The policy in place 1is p = p0 and then from time 1 is p1. For example, atax is imposed on water bottles in Washington inJuly 2010 and it is not imposed in Oregon.Furthermore, the tax is imposed on good i = 1 andnot on good i = 0, and in the example, the tax isimposed on water but not on juice. The outcome ofinterest is the number of water bottles sold, becausethe water bottles are a major ingredient in litter. Fora second example, Sweden reduces its tax on restau-rants while Norway does not. Moreover, the tax ischanged on restaurant meals but not on hotels or onother retail. The outcome of interest is restaurantemployment because the tax change was meant toincrease employment.The intuition behind the difference in differencesis to measure the effect of the policy p on the out-come xi in the treated group relative to changes inthe outcome xi in the control group. If the panel datahave an untreated outcome xi=0 that shares somepossible common determinants with the outcomeCONTACT: Sofia B. Villas-Boas sberto@berkeley.eduAPPLIED ECONOMICS LETTERS, 2015http://dx.doi.org/10.1080/13504851.2015.1068912© 2015 Taylor & FrancisDownloaded by [University of California, Berkeley] at 09:54 30 October 2015 of interest xi=1, another control structure can be usedin a triple difference strategy to estimate the treat-ment effect of the policy change by comparing thedouble differences in xi=1 with the double differencein xi=0 and in so doing, controlling for more factorsthat could bias the average treatment effect.These facts give the raw material for a tripledifference: time t<>1, place c = 0, 1 and goodi = 0, 1. But when is the triple difference, usingvariation from time, place and good, better thanthe double difference, just using time and placevariation in the data to estimate the average treat-ment effect of the policy change?The economic relationship at issue is the changein outcome x due to the policy p. As usual, theestimation is set up so that x is a function of timet, place c and the policy p. We add a possiblyconfounding variable y. For instance, in the exam-ples, y would be income, and it varies by time andplace and affects the demand for x (restaurant mealsand water bottles). In this article, we maintain thehypothesis that the policy does not affect either theuntreated place, time or good.Let the true equation to be estimated be given byxict ¼ αc þ βit þ γiln pict þ δiln yict; (1)where the Greek letters are parameters to be esti-mated and in particular γ denotes the average effectof the policy on the outcome x. The model is writtenin logs because it is natural in the examples to thinkof growth rates, but the algebra will be much thesame in the linear case. The variable δi is the semi-income elasticity of good i and it is not assumed thatit is the same as the semi-elasticity of good j. Nor isit assumed that income growth was the same in bothregions. Let Δ be the difference operator. In the caseof time, the difference is taken from the initial value,where t = 0. Differencing out in time yieldsΔtxict ¼ βit þ γilnðpict=pict¼0Þ þ δilnðyict=yict¼0Þ(2)And additionally differencing out in space yieldsΔtcxict ¼ γilnðpi;c¼1;t=pi;c¼1;t¼0Þþ δi lnðyi;c¼1;t=yi;c¼1;t¼0Þ lnðyi;c¼0;t=yi;c¼0;t¼0Þ(3)LettingΔtc ln p ¼ lnðpi;c¼1;t=pi;c¼1;t¼0Þ and Δtc ln y¼ lnðyi;c¼1;t=yi;c¼1;t¼0Þ  lnðyi;c¼0;t=yi;c¼0;t¼0Þ (4)we can write this more compactly asΔtcxict ¼ γiΔtc ln pþ δiΔtc ln y (5)Equation 5 is the double difference specification. Itdepends upon the policy in region 0 being the samebefore and after t = 1, so that lnðpi;c¼0;t=pi;c¼0;t¼0Þ = 0.As set up, the double difference removes the effect of ywhenever both regions have the same growth in y,which is in the example the same growth in income. Ifthe data on y are exogenous, then the data can directlybe included in a regression and easily controlled for.However if there are no available data on y in one ofthe periods or if y is an endogenous variable, then itwould be desirable to eliminate y with a further dif-ferencing strategy. In our example of water bottles, theobservations could be on purchases in stores wherethe income of the customers is not known. For therestaurants, the policy change was meant to affectincome by increasing employment and income ismay be endogenous to this system. With either miss-ing data or endogeneity, a triple difference may bedesirable. The triple difference specification as givenby taking an additional difference in good i:Δtcixict ¼ γilnðpi¼1;c¼1;t=pi¼1;c¼1;t¼0Þþ δi¼1  δi¼0ð Þ lnðyi;c¼1;t=yi;c¼1;t¼0Þ lnðyi;c¼0;t¼1=yi;c¼0;t¼0Þ:(6)orΔtcixict ¼ γiΔtc ln pþ ðδ1  δ0ÞΔtc ln y (7)Equation 6 is dependent upon there being nochange in p for good 0 and growth in y beingindependent of the good. If y was the income termfrom a demand equation, that would be the case.III. Specification error and estimationThe estimated models omit the terms in y (becauseof endogeneity or data problems). That is, one esti-mates the equations: Δtcxict ¼ γiΔtc ln p or Δtcixict ¼γiΔtc ln p. The bias in the estimators for these equa-tions can be derived using Thiel’s specification errortheorem (Thiel, 1971, 549).2 P. BERCK AND S. B. VILLAS-BOASDownloaded by [University of California, Berkeley] at 09:54 30 October 2015 Eγtc1 ¼ðΔtc lnp0Δtc lnpÞ1Δtc lnp0xtc¼ðΔtc lnp0Δtc lnpÞ1Δtc lnp0½γ1Δtc lnpþδ1Δtc lny¼γ1þfðΔtc lnp0Δtc lnpÞ1Δtc lnp0Δtc lnygδ1(8)The first line is the formula for ordinary leastsquares (OLS) and E is the expectation operator.The second line substitutes the formula for the truemodel for x. The third line shows the differencebetween the OLS estimator and the true γ is thecoefficient from regressing the income term onprice times the true coefficient on the income term.The double difference estimator is unbiased if δ1 ¼ 0or if the regression of twice differenced ln y on twicedifferenced ln p has a zero coefficient. Another wayto look at the bias is that whenever the productðΔtc ln yÞδ1 is near zero, the double difference esti-mator will have little bias.The same algebraic steps give the bias for thetriple differenceEγtci1 ¼ðΔtc lnp0Δtc lnpÞ1Δtc lnp0xtc¼ðΔtc lnp0Δtc lnpÞ1Δtc lnp0½γ1Δtc lnpþðδ1δ0ÞΔtc lny¼γ1þfðΔtc lnp0Δtc lnpÞ1Δtc lnp0Δtc lnygðδ1δ0Þ(9)The bottom line is that the triple difference haslower bias than the double difference whenever y isimportant and the two goods have a similar responseto the omitted variable ln y. Specifically wheneverδ1  δ0j j < δ1j j (10)IV. DiscussionWhen should δ1  δ0j j < δ1j j? When δ1 and δ0 areincome elasticities for goods, one would expect themto be different by less than their magnitude. Forinstance, in Park et al. (1996), the income elasticitiesfor foods eaten at home and foods away from homefor poverty and nonpoverty groups are given. Foreach group for foods eaten at home, the incomeelasticities are within 0.2 and larger than 0.2, so atriple difference will lessen the bias. Blanciforti andGreen (1983) look at income elasticities betweengroups of items. The income elasticities vary from4.4 to .24, so a triple difference strategy absent priorknowledge of the income elasticities may not reducebias.In the context of a markup type model, let x bethe price of a good, p be the policy and y is the costof distribution. So long as the markup on the cost ofdistribution is close in the two regions, the tripledifference is more accurate than the doubledifference.What drives these examples is that the goodsi = 0 and i = 1 share a common ‘shift’ variable y,either income or cost of distribution. So eventhough they may not respond exactly the same tothat variable, and they may not have exactly thesame changes in that variable, the product of thedifferences is all that matters and small times smallis very small.Disclosure statementNo potential conflict of interest was reported by the authors.FundingWe thank the Giannini Foundation for support.ReferencesAngrist, J. D., and A. B. Krueger. 2005. “Empirical Strategiesin Labor Economics.” Handbook of Labor Economics3 (Part A): 1277–1366.Athey, S., and G. Imbens. 2006. “Identification and Inferencein Nonlinear Difference-in-Differences Models.”Econometrica 74 (2, March): 431–497. doi:10.1111/ecta.2006.74.issue-2.Bertrand, M., E. Duflo, and S. Mullainathan. 2004. “HowMuch Should We Trust Differences-In-DifferencesEstimates?” The Quarterly Journal of Economics 119 (1):249–275. doi:10.1162/003355304772839588.Blanciforti, L., and R. Green. 1983. “An Almost IdealDemand System Incorporating Habits: An Analysis ofExpenditures on Food and Aggregate CommodityGroups.” The Review of Economics and Statistics 65 (3):511–515.Hamermesh, D., and S. Trejo. 2000. “The Demand for Hoursof Labor: Direct Evidence from California.” The Review ofEconomics and Statistics 82 (1): 38–47. doi:10.1162/003465300558614.Kellogg, R., and H. Wolf. 2008. “Daylight Time and Energy:Evidence from an Australian Experiment.” Journal ofAPPLIED ECONOMICS LETTERS 3Downloaded by [University of California, Berkeley] at 09:54 30 October 2015 Environmental Economics and Management 56: 207–220.doi:10.1016/j.jeem.2008.02.003.Meyer, B. D. 1995. “Natural and Quasi-Experiments inEconomics.” Journal of Business & Economic Statistics13 (2) JBES Symposium on Program and PolicyEvaluation (April): 151–161.Park, J. L., R. B. Holcom, K. C. Raper, and O. Capps Jr. 1996.“A Demand Systems Analysis of Food Commodities byU.S. Households Segmented by Income.” The Review ofEconomics and Statistics 65 (3, August): 511–515.Thiel, H. 1971. Principles of Econometrics. New York: JohnWiley.4 P. BERCK AND S. B. VILLAS-BOASDownloaded by [University of California, Berkeley] at 09:54 30 October 2015 

A note on the triple difference in economic models

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