Identifying and Correcting Publication Selection Bias in the Efficiency-Wage Literature: Heckman Meta-Regression

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1 Faculty of Busness and Law SCHOOL OF ACCOUNTING, ECONOMICS AND FINANCE School Workng Paper - Economc Seres 27 SWP 27/11 Identfyng and Correctng Publcaton Selecton Bas n the Effcency-Wage Lterature: Heckman Meta-Regresson T.D. Stanley and Hrstos Doucoulagos The workng papers are a seres of manuscrpts n ther draft form. Please do not quote wthout obtanng the author s consent as these works are n ther draft form. The vews expressed n ths paper are those of the author and not necessarly endorsed by the School.

2 Identfyng and Correctng Publcaton Selecton Bas n the Effcency-Wage Lterature: Heckman Meta-Regresson T.D. Stanley* and Hrstos Doucoulagos** * Department of Economcs, Hendrx College, 16 Washngton St., Conway, AR, 7232, USA Emal: Stanley@hendrx.edu. Phone: ** School of Accountng, Economcs and Fnance, Deakn Unversty, 221 Burwood Hghway, Burwood, 3125, Vctora, Australa. Emal: douc@deakn.edu.au. Phone: Abstract Publcaton selecton bas represents one of the most serous challenges to the ntegrty of emprcal economcs. We develop Heckman regresson methods to solve ths potentally persstent problem and apply these meta-regresson methods to seventy fve emprcal estmates from the effcency-wage lterature. Although many researchers fnd mxed or ambguous support for the effcency wage hypothess (EWH), our meta-analyses gve unambguous confrmaton of the EWH. After correctng for publcaton selecton bas, we estmate the wage elastcty of output to be.32, much smaller than what the neoclasscal verson of the effcency wage hypothess demands. Ths wage elastcty also depends sgnfcantly upon whether the researchers model accounts for the smultanety of wages and productvty and whether ther emprcal model ncludes captal. In both cases, the correct specfcaton ncreases the magntude of the wage elastcty of producton, thereby further corroboratng the EWH. JEL Classfcaton Codes: J3, C2, E24.

3 I. Introducton (P)ublcaton bas s leadng to a new formulaton of Gresham s law lke bad money, bad research drves out good Bland (1988, p. 45). Publcaton selecton bas s wdely acknowledged as another fundamental threat to vald emprcal economc nference (DeLong and Lang, 1992; Card and Krueger, 1995; Ashenfelter et al., 1999; Roberts and Stanley, 25). Lke debasng the con of the realm, publcaton selecton bas can greatly nflate the estmated magntude of key emprcal economc parameters. Such nflaton can be as hgh as 1, 2 or many hundreds of percent (Stanley, 25). Hence, t becomes mperatve to nvestgate the severty of publcaton selecton and to correct emprcal fndngs for the dstortng effects of selectvty. Effcency wage theory has long captured the attenton as well as the magnaton of researchers and polcy analysts. No fewer than two Nobel przes were awarded, at least n part, for work on the effcency wage hypothess (EWH) (Akerlof, 22; Stgltz, 22). For decades and n hundreds of papers, the EWH has been employed to ratonalze nvoluntary unemployment (Solow, 1979; Akerlof, 22), to explan the seemngly nexplcably wde varaton of wages across ndustres and frms (Krueger and Summers, 1988), and to queston the conventonal neoclasscal characterzaton of employers (Akerlof, 1982; Chen and Edn, 22). The effcency wage hypothess argues that frms pay hgher than market-clearng wages n order to ncrease productvty. But does the emprcal research record confrm or deny ths labor market complcaton? And, s the emprcal evdence consstent wth the neoclasscal formulaton of the EWH? Although emprcal research generally supports the EWH, t also contans ambguous fndngs wth a large varaton among the reported elastctes. Furthermore, the neoclasscal formulaton of the EWH contans very specfc mplcatons about the magntude of the wage elastctes of effort and producton (often referred to as the elastcty of output or effort wth respect to wages). In partcular, the Solow condton requres that effort elastcty be unty (Akerlof and Yellen, 1986, p.14), and the wage 1

4 elastcty of producton must equal labor s share (see secton 2, below). Does the research record confrm these specfc magntudes for wage elastctes? We conduct the frst meta-regresson analyss of the effcency-wage lterature and fnd confrmaton of the exstence of an effcency-wage effect as well as publcaton selecton bas. Once publcaton bas s fltered from the estmates of the wage elastcty of producton, the evdence clearly rejects the neoclasscal verson of the effcency wage hypothess. The paper s set out as follows. Secton 2 provdes a bref dscusson of effcency-wage theory. The technques for detectng publcaton selecton and correctng estmates for the dstortng effects of publcaton selecton are developed n secton 3, whle the meta-analyss of the EWH s reported and dscussed n secton 4. Secton 5 concludes the paper. II. Theoretcal Consderatons The most referenced and emprcally tested verson of the effcency wage hypothess s the shrkng model (Shapro and Stglz, 1984). In compettve markets, frms offer workers wth smlar sklls the market wage rate. There s no cost to shrkng on the job because another job can easly be found wth a competng employer at the same market rate. Although employers can montor ther employees, montorng s costly and sometmes nfeasble. Thus, t mght be advantageous to pay ther workers an effcency wage ; that s, one that s hgher than the market-clearng wage rate and reduces shrkng. Such an effcency wage mples a connecton between the effort (and therefore productvty) of a gven worker and her wage. An effcency wage premum buys loyalty, hard work, and lessens turnover wth all ts assocated costs. In ths way, effort and productvty become dependent on the level of wages. There are several other justfcatons of effcency wages, ncludng the adverse selecton model and the gft-exchange model (Stgltz, 1976; Akerlof, 1982). However, regardless of whch model s used, the wage-productvty channel has been central to the effcency wage hypothess, and t 2

5 s the dmenson of the EWH most frequently estmated and tested by emprcal research. Thus, the wage-productvty connecton s the focus of our meta-regresson analyss. Conventonal neoclasscal proft maxmzaton requres that the wage elastcty of producton (or the output elastcty wth respect to wages) be equal to labor s share. Ths s essentally equvalent to the Solow condton that the wage-effort elastcty s unty, but n dfferent unts of measurement. To see ths, recall the conventonal formulaton of producton and proft functons. When effcency wages are modeled, the textbook producton functon ncludes labor enhanced wth an explct channel from wages to effort, e(w). Q = f(e(w)l,k) (1) where Q s, of course, some measure of producton or output and L and K denote labor and captal, respectvely. Otherwse, producton behaves n the neo-classcal manner, and proft ( π ) takes on the textbook form. π = PQ wl rk (2) For smplcty and generalty across varyng market structures, let Y be producton n monetary unts (.e., Y=PQ). Frst order condtons gve: Y/ w = L (3) If we multply by w and dvde by Y, then we fnd that proft maxmzaton forces the wage elastcty of producton to equal labor s share, L. Y/Y w/w = wl / Y = L (4) 3

6 Ths condton s qute ntutve. When wages ncrease by x percent, total costs wll go up by xl percent, at the margn; thus, producton n monetary unts must ncrease by xl percent to break even at equlbrum. In practce, ths wage elastcty of producton, the left sde of equaton (4), s the most frequently reported emprcal magntude n the effcency-wage lterature and therefore the most approprate emprcal magntude for our focus. Next, we consder how to best model the wde varaton n reported wage elastctes. III. Identfyng and Correctng Publcaton Bas A. Identfyng publcaton selecton Research s generated through a knowledge producton process. Researchers transform ther labor and knowledge (as well as the labor of others, data, expertse, economc theory, nformaton technology and past research) nto research output. In emprcal economcs research, the research output s typcally estmates of the drecton of an effect (whether t s postve, neutral or negatve), and the magntude of key elastctes, or some other standardzed measure of assocaton, such as elastctes and partal correlatons. Denote the true wage elastcty as η and the estmated elastcty reported n the th study asηˆ. 1 In general, ˆ η = η + (5) ω where ω s an error term assocated wth the complex research producton process. Ths error term s more complcated than the usual random samplng error because t s lkely to contan varous msspecfcaton and selecton bases that may be systematcally related to observable research characterstcs. When ths complexty s taken nto account, equaton (1) may be modeled broadly as: ˆ η = β + α SE + Zβ + ε (6) 1 Ths assumes that there s a sngle populaton parameter. It s a smple matter to ntroduce a dstrbuton of populaton parameters see equaton (6) below. 4

7 where ε s a random error vector,..d. Z s a vector of moderator varables that acknowledge key dmensons n the varaton of the true elastcty (heterogenety) or dentfy large-sample bases that arse from model msspecfcaton. SE contans the reported standard errors of the estmated elastctes. Lastly, β denotes the true elastcty, whch can be estmated by meta-regresson analyss (MRA). However, the most wdely used estmate of the true elastcty s a smple weghted mean: ˆ η ˆ η (7) = w N / N where N s sample sze assocated wth the th study. 2 Ths weghted elastcty wll be an unbased estmate of the populaton elastcty, as long as the studes ncluded n the calculaton of equaton (7) are a random sample from the populaton of all estmates, whether reported or not (see Hunter and Schmdt 24). In the presence of publcaton selecton effects, the pool of avalable studes from whch nferences can be drawn wll be truncated. When only a selected subset of estmates s used to calculate ηˆ w, then the lterature s sad to be contamnated by publcaton selecton bas. Publcaton bas means that a non-random sample of the estmates reman unpublshed and unreported; hence, the pool of research avalable s a based representaton of the populaton of estmates. Publcaton bas can arse from both sdes of the market for research. On the supply sde, t s possble that n the course of ther emprcal analyss, authors may fnd results that are opposte to ther expectatons (e.g., negatve wage elastctes of producton) and decde to fle these results away, leavng them unpublshed and unknown to other researchers. 3 Authors may modfy ther methodology, data, model specfcaton, and estmaton procedure untl the expected correct results emerge (e.g., 2 Alternatve weghts such as the nverse of the estmate s varance are also used, but the general results are the same. 3 Ths dea of flng away unwelcome fndngs s called the fle drawer problem and s wdely dscussed n the meta-analyss lterature (Rosenthal, 1979). 5

8 postve wage elastctes of producton). 4 Alternatvely, authors may smply be seekng statstcal sgnfcance because t s perceved to be a necessary condton for scentfc mportance. The ssue of publcaton selecton bas has recently receved heghtened scrutny among medcal researchers as a consequence of the Voxx and Paxl scandals. As a result of these purposeful suppressons of unfavorable clncal evdence, most leadng medcal journals now requre the pror regstraton of all clncal trals as a condton for the later publcaton of ther fndngs (Krakovsky, 24). On the demand sde, journals may not publsh results that are contrary to the edtors or referees expectatons or that are not statstcally sgnfcant. Or, edtors and referees may encourage researchers to modfy ther methodology untl statstcal sgnfcance or the expected result s found. To the extent that anyone n the complex process of research and publcaton has a preference for results of a gven type, then publcaton selecton bas wll be present n that lterature. The knowledge (or the mere belef) that edtors and revewers have a preference for statstcally sgnfcant results s suffcent to cause authors to engage n selecton n order to satsfy ther customers perceved demand. It would be a great endorsement of academc research f publcaton selecton bas were absent from academc research. Sadly, the large majorty of meta-analytc nvestgatons of publcaton selecton have, thus far, found the statstcal traces of bas. Examples nclude: Card and Krueger (1995) on mnmum wage effects; Ashenfelter et al. (1999) on returns to educaton; Görg and Strobl (21) on the productvty effects of multnatonals; Stanley (25) on the prce elastcty of water; Abreu et al. (25) on economc growth convergence; Njkamp and Poot (25) on the wage curve; Rose and Stanley (25) on the trade effects of common currency; Doucoulagos (25) on the growth effects of economc freedom; and Mookerjee (26) on the growth effects of exports. 5 4 It s also possble that some authors wll even falsfy results, although ths s hopefully a very rare event. 5 The lterature on the mpact of unons on productvty levels s a rare excepton to ths pattern (Doucoulagos and Laroche 23). 6

9 But then much of the publcaton selecton found n economcs may be the result of an understandable desre to report fndngs that are statstcally sgnfcant (hence noteworthy) and of the correct sgn (hence perceved to be vald ). For example, all economsts know that there s an nverse relatonshp between prce and quantty demanded the Law of demand. Thus, when estmatng the prce elastcty of water demand, some researchers wll suppress aberrant postve estmates n the belef that they are correctng some data or specfcaton error and are thereby reportng more accurate and relable elastcty estmates. Lkewse, a fndng that rasng wages causes workers to be less productve seems to be an obvous error; some mportant consderaton must have been omtted. Such publcaton selecton s clearly seen among estmates of water prce elastctes, and t causes a three or four-fold exaggeraton of the magntude of ths elastcty (Stanley, 25). Even when selecton s for the best of scentfc reasons, t can greatly dstort any summary or average (whether weghted or not) of a research lterature. B. Heckman regresson Because we do not have access to all of the orgnal fndngs, manuscrpts, ther revsons, and the assocated referee reports, we have no effectve way of drectly detectng or modelng the selecton process. Avalable estmates consttute the ncdental truncaton of the populaton of estmates (Wooldrdge, 26). The problem of publcaton selecton bas s much the same as sample selecton bas n general (Heckman, 1977), but wth a twst. Wdely used econometrc models that correct for sample selecton bases typcally employ Heckman s two-equaton system: ˆ η = Zβ + ε (8) * P = Kα + u (9) where P * represents the probablty that an estmated effect s publshed (or reported), hence observable, and K s a matrx of varables that nfluence the probablty of beng publshed or reported 7

10 (Davdson and MacKnnon, 24, pp ). 6 Typcally, equaton (9) s estmated by a probt on the observable P = {,1} usng the entre sample of selected and non-selected observatons. However, n the case of publcaton selecton, we do not generally have access to the non-selected (unreported) estmates. 7 Thus, step one n the conventonal Heckman two-step method cannot be estmated for the publcaton selecton of emprcal economc research. To reveal the consequences of ths lmtaton, consder the Heckman regresson, whch s the second step of Heckman s method: ˆ η = Z β + ρσi ( K ˆ α) + e (1) where I ( K ˆ α) s the nverse Mlls rato, ρ s the correlaton between ε and u, and σ s the standard error of ε (Davdson and MacKnnon, 24, pp ). Unfortunately, the usual Heckman method s unavalable to us because the frst-step probt s needed to estmateαˆ and hence to dentfy the nverse Mlls rato, I ( K ˆ α). Wthout ths frst step, I ( K ˆ α) becomes an omtted varable n the estmaton of β from ˆ η = Zβ. Fortunately, emprcal economc research s blessed wth heteroscedastcty. That s, the standard error ofηˆ, SE, s lkely to vary greatly from one study to the next. Lke the nonlnearty of the nverse Mlls rato that permts the dentfcaton of equaton (1) n the typcal economc applcaton (Davdson and MacKnnon, 24, p.489), t s ths heteroscedastcty of emprcal economc estmates that makes possble the dentfcaton and estmaton of Heckman regresson for publcaton selecton, equaton (1). Replacng the nverse Mlls rato term n Heckman regresson wthα gves our SE 6 Although K was used prevously for captal, followng conventonal practce, t s used here and n the remander of ths paper to represent any varable related to publcaton selecton. 7 Of course, we all know the values of all the estmates for our own research but not the unreported fndngs of others. Even unpublshed workng papers and dssertatons may contan only selected results n antcpaton of future peer revew. 8

11 prevously reported equaton (6). In ths equaton, α represents systematc selecton for statstcal sgnfcance. Studes wth smaller samples and hence larger standard errors ( SE ) wll need to run and re-run ther models more ntensely to acheve a statstcally sgnfcant result. More precse studes (.e., those wth a smaller 1/ SE ) wll requre less searchng and less selecton to obtan the desred sgnfcant result. Thus, the telltale sgnal of publcaton selecton s a systematc relaton of reported SE effects wth ther standard errors revealed by a meta-regresson analyss (Stanley 25). However, the meta-regresson equaton (6) wll tself contan obvous heteroscedastcty, because SE s the estmated standard devaton of ηˆ, whch wll vary from observaton to observaton. The obvous remedy s to dvde equaton (6) by SE to obtan the weghted least squares (WLS) verson of ths MRA. ˆ / SE = t = α + β / SE + Σβk Z k / SE ν (11) η + where t s the reported t-value of the th estmated elastcty. It s ths meta-regresson model and ts extensons that form the bass of our nvestgaton of the effcency-wage lterature. Wthout makng the connecton to Heckman regresson, medcal researchers have wdely used the smple MRA verson of model (11) as a test for publcaton bas: t = α + β / SE + ν (12) (Egger et al., 1997; Sutton et al., 2). Testng whetherα = becomes the test for the presence of publcaton bas. Ths method for dentfyng publcaton selecton has been called a funnel-asymmetry test (FAT) due to ts relaton to funnel graphs (Stanley, 25). A funnel graph plots precson, the nverse of the standard error (1/ SE ), aganst the reported estmated effects. In our case, we use the wage elastcty of producton. Fgure 1 plots these estmated 9

12 elastctes and ther precson. 8 As can be seen n Fgure 1, effcency-wage elastctes are almost entrely postve and skewed to the rght, ndcatng publcaton selecton for postve wage elastctes. Even clearer, and more mportantly, s the asymmetry of the effcency-wage funnel graph. Ths asymmetry becomes more apparent when compared to what the absence of publcaton s expected to look lke see Fgure 2. However, any such speculaton quckly reveals the underlyng weakness of usng graphs to dentfy publcaton bas. Vsual nspecton s nherently subjectve and must reman somewhat ambguous; hence, the motvaton for a formal statstcal test such as: FAT (H : α =). FIGURE 1 ABOUT HERE FIGURE 2 ABOUT HERE Smulatons show that the MRA estmate of β n equaton (12) also serves as a test for genune emprcal effect corrected for publcaton bas (Stanley, 28). Because 1/ SE s the precson of ths estmate of emprcal effect, ths test (H: β =) has been called the precson-effect test (PET), whch makes the meta-regresson model (12) a FAT-PET-MRA. These smulatons also reveal that PET s very powerful and robust to the ntensty of publcaton selecton (Stanley 28). However, there s also a downward bas n the FAT-PET-MRA estmate of β when there s an authentc emprcal effect (.e., β ). Although there are several reasons for such a bas, the fact that the nverse Mlls rato term n equaton (1) s a nonlnear functon of SE s lkely a contrbutng factor. 9 8 See secton IV for the detals about how ths data was collected. 9 Below, we wll ntroduce a nonlnear MRA that smulatons show greatly reduces ths bas. See the Appendx of ths paper for smulatons of the bas of ths PET estmate relatve to several other estmates, ncludng an mproved nonlnear verson of PET PEESE. Another source of the attenuaton n the estmate of β s the fact that s an estmate and, hence, has SE 1

13 Equatons (6) and (11) may be expanded to capture the dependence of reported effects on the omtted selecton varables, K. ˆ η β + α SE + Σβ Z + SE Σα K + ε (13) = k k j j t = α + β / SE + Σβ Z / SE + Σα K + ν k k j j (14) for = 1, 2,..., l; where l s the number of estmates reported n a lterature. The varables, Z, explan the heterogenety n the true estmated effects or large sample bas that arses from msspecfcaton, and the K-varables dentfy factors that nfluence the propensty that an estmate gets publshed or reported. Because the nverse Mlls rato s a nonlnear functon of K αˆ, hgher order terms of K mght be needed n the FAT-PET-MRA equaton (14). However, n our applcaton to the effcency-wage lterature as well as many other meta-regresson analyses, the K-varables wll be dummy varables, makng K perfectly collnear to K 2 or any other power of K. Thus, the potental nonlnearty of K s rrelevant for our applcaton. As smulatons ndcate, equatons (6) and (13) may also be nonlnear n SE. Ths ssue s addressed explctly n the next secton. C. Correctng publcaton selecton PEESE Thus far we have dscussed how publcaton selecton may be dentfed and how genune emprcal effect beyond publcaton selecton can be tested. However, emprcal researchers and polcy makers also need an accurate estmate of the magntude of the underlyng effect, corrected for possble publcaton bas. If the nverse Mlls rato were constant and ndependent of SE, then MRA model (12) would be correctly specfed and could provde consstent estmates of the true effect samplng error. Such errors n the ndependent varable (errors-n-varables) cause a well-known downward bas. See Stanley (25) for a possble remedy. 11

14 corrected for publcaton bas. Unfortunately, nether s true, whch explans the low power of usng H : α = as a test for publcaton bas and the downward bas n the estmate of β (Egger et al., 1997; Stanley, 28, and see the Appendx). The nverse Mlls rato depends on SE. 1 Thus, the relatonshp between the observed effect and ts standard error wll be nonlnear n the presence of publcaton selecton. Ths nonlnearty wth respect to SE becomes the bass for a new approach to estmaton, corrected for publcaton bas precson-effect estmate wth standard error (PEESE). Typcally, nonlnear relatons are estmated usng a power seres. Thus, PEESE begns wth the square of SE (.e., the varance of each estmated elastcty) as the second term of MRA model (6). ˆ η + (15) 2 = β + α SE ε However as before, ths model has obvous heteroscedastcty; thus the WLS verson would dvde equaton (15) by SE gvng: t = α / + SE + β SE ν (16) Note that there s now an addtonal ndependent varable SE, compared wth equaton (12), and that there s no ntercept. In ths framework, ˆβ s the estmated value of η corrected for 1 An excepton to ths rule s for the case when there s no genune effect, Te=. In general, t can be shown that the dervatve of nverse Mlls rato wth respect to SE s: [TeI(T e /SE - c) / SE 2 ][(Te/ SE - c)+ I(Te/ SE - c)]; where c s the crtcal value from the t-dstrbuton. See Davdson and MacKnnon (24) and Wooldrdge (26, p. 598). Because ths dervatve s generally nonzero when Te, the nverse Mlls rato s not constant wth respect to SE. But conversely, when Te =, ths dervatve s zero, and the relatonshp between the observed effect and ts standard error wll be lnear. 12

15 publcaton selecton or the precson-effect estmate wth standard error (PEESE). The Appendx reports the results of Monte Carlo smulatons of ths corrected estmate of η, and shows that PEESE greatly reduces the potental bas of publcaton selecton, and compares ts bas to several other overall estmates of elastcty. Our approach n ths paper s to explore the effcency-wage lterature for evdence of publcaton selecton and to correct the overall estmate of wage elastcty for publcaton selecton usng (16) and varants of equaton (14). In ths framework, ˆβ, from equaton (16), s the estmated value of η, corrected for publcaton selecton. From equaton (14), ˆα s the drect measure of publcaton bas, βˆk are the coeffcents on other varables that affect the magntude of η, and αˆ j are the coeffcents on other varables that affect the propensty for publcaton selecton. In the more complex multvarate meta-regresson analyss (14), publcaton selecton s modeled by the combnaton of ˆα and theαˆ jk j terms. Ths framework s flexble enough to facltate the testng of several hypotheses. Frst, t enables us to test whether η vares: (a) across country (UK or US), (b) by the level of aggregaton (Frm or Industry), (c) through the choce of wage and output measures (RelWage for relatve wages, ValAdded, for value added, and PerWorker for output per worker), (d) by allowng potental smultanety between wages and productvty (Smul), and (e) by ncludng other relevant varables (a measure of captal stock, Cap, unon membershp, Unon, and human captal, HumCap). IV. Meta-Analyss of the Effcency Wage Hypothess A. The Data A comprehensve search of the lterature was conducted of several databases, ncludng ECONLIT for: effcency wage and ( emprcal or test or estmate ). Ths search produced over 2 hts. After cullng those that had any chance of contanng an appled or emprcal fndng, we obtaned over 13

16 1 workng papers, books, dssertatons, and journal artcles. In all cases, we erred on the sde of ncluson and dd not exclude any paper because of language. 11 Ths lterature search netted 75 estmates of the wage elastcty of producton contaned n 14 studes. Gven the mmense nterest n effcency wage theory, t s surprsng to fnd only 14 emprcal studes that offer comparable estmates of the effcency-wage elastcty. A complete reference lst s avalable from the authors. Although the effcency-wage lterature contans several dfferent emprcal outcome measures, the most commonly reported measure of the effcency-wage effect s the wage elastcty of producton, The average reported wage elastcty s.63 (p<.1). Ths magntude s approxmately equal to what the frst-order condtons of neoclasscal theory requre. Recall that proft maxmzaton, equaton (4), predcts that the wage elastcty of producton should be equal to labor s share. Labor s share s one of the few macroeconomc stylzed facts thought to be stable. Tradtonally, labor s share s assumed to be 2/3, although t s recognzed to vary by ndustry and over tme. More recent estmates report values around.6 or much hgher (Krueger, 1999). Superfcally, therefore, ths smple average of reported elastcty estmates seems to confrm the neoclasscal verson of the effcency wage hypothess. However, economc research usually contans publcaton bas and consderable heterogenety, both of whch call nto queston the valdty of usng any average elastcty estmate. Frst, we turn to the ssue of whether there s publcaton bas and how best to correct t. After that, we wll use MRA to explan any excess heterogenety. ηˆ. B. Identfyng and correctng publcaton selecton Table 1 reports MRA models (12) and (16). Recall that FAT s a test for publcaton selecton, and t tests H : α = n equaton (12). The effcency-wage lterature shows clear evdence of 11 We were unable to obtan only two papers that mght have contaned estmates. Both were publshed n narrowly crculated journals one n the Netherlands and the other n Turkey. The search for studes was ended July 26. Any studes subsequent to that date have not been ncluded n our dataset. 14

17 publcaton selecton (equaton (12); t= 2.67; p<.1). In spte of ths selecton, equaton (12) also contans strong evdence of a genune effcency-wage effect (PET; t= 18.41; p<.1) recall the precson-effect test of Secton 2 and see Stanley (25) and Stanley (28). 12 These results agan confrm the relevance of the EWH. Thus far, we have clear evdence of both publcaton bas and a real effcency-wage effect. Gven that a genune effcency-wage effect exsts, how large s t? Is t consstent wth the neoclasscal verson of EWH? The coeffcent on precson (1/SE) n equaton (16) Table 1 s.32. Ths s the PEESE estmate of the effcency-wage elastcty, corrected for publcaton selecton bas. Note that ths correcton nearly halves the prevous smple average. Even wth modest publcaton selecton, 13 t can nduce a bas that has practcally mportant mplcatons (estmated bas= =.31). Our corrected estmate of the effcency-wage elastcty rejects of any labor s share of.4 or larger. Because PEESE s so much less than any credble estmate of the aggregate labor s share, our meta-analyss of the effcency wage lterature calls nto queston the neoclasscal verson of the EWH. TABLE 1 ABOUT HERE If the neoclasscal vew s not vald, what s? (I)n ts most general form, effcency wage theory s vrtually tautologous (Akerlof and Yellen, 1988, p.44). Takng ths rsk of turnng the EWH nto a tautology, one mght observe that turnover costs are the most obvous omsson from our proft functon and add turnover costs, T(w), explctly nto t. Frst order condtons would then mandate that: 12 We do not use the MRA coeffcent on 1/SE from equaton (12), as an estmate of wage elastcty, because t s known to be based downward when there s an effect. See the Appendx. Ths s the man reason that we prefer to use PEESE nstead. Nonetheless, testng whether ths coeffcent on 1/SE from equaton (12) s or s not zero, (.e., H: β = ) remans a vald and robust test for a genune emprcal effect beyond publcaton bas (Stanley, 28). 13 The magntude of, whch s.98 n equaton (12) Table 1, serves as a good proxy for the severty of publcaton selecton (Doucoulagos and Stanley, 27). ^ α 15

18 η = Y/Y w/w = L + (T/Y)Ε t (17) where Ε t s the wage elastcty of turnover costs. Yet ths amended formulaton of proft maxmzaton gves lttle support to neoclasscal theory. Now, ether turnover costs need to be a large proporton of output or turnover costs need to be hghly elastc, or both. The emprcal magntudes revealed by our meta-analyss requre that hgher wages reduce turnover costs by roughly an amount equal to the value of the ncreased productvty. At a mnmum, our results gve addtonal credence to the labor turnover model of the effcency wage hypothess (Stgltz 1974; Salop 1979). C. Z and K varables and Multvarate MRA Surely, more than publcaton selecton s responsble for the wde varaton we observe among reported wage elastctes? To explan ths varaton n observed elastctes, we coded a number of research characterstcs that were thought to be potentally related to ether the magntude of η or to ts publcaton selecton. Our frst set of explanatory varables contans country dummes (US and UK). These are ncluded as part of the Z-vector to capture any underlyng dfferences n the effcency-wage effect across countres. They are ncluded also n the K-vector to test whether the selected country nfluences the publcaton process. Recall the dstncton between Z and K- varables made n Secton III. In our meta-regresson analyss, we allow country composton to affect both the genune effect and publcaton selecton bas. We also explore whether ncludng other relevant ndependent varables (captal stock, Cap, unon membershp, Unon, and human captal, HumCap) affects reported elastctes. The last Z-varable s Smul, whch denotes whether the researcher s model or methods allowed for the potental smultanety between wages and productvty. An obvous crtcsm of the effcency wage hypothess s that hgher productvty leads to hgher wages through rent-sharng, rather 16

19 than the other way around (Blanchard and Summers, 1986; Wadhwan and Wall, 1991). Thus, t s mportant to account for whether researchers took precautons for the lkely smultanety between wages and productvty. All of these Z-varables are dvded by the standard error, SE, and denote genune effects on the underlyng wage elastcty recall equaton (14). The K-varables are those that mght be correlated wth publcaton selecton and some are conceptually lnked to the frst step probt of Heckman s selecton method. Along wth dummes for countres (UK and US), they nclude the level of aggregaton (Frm or Industry) and the choce of wage and output measures used (RelWage for relatve wages, ValAdded for value added, and PerWorker for output per worker). Perhaps edtors or referees have a professonal preference about whch level of aggregaton s most approprate or whch measures are the best? Table 2 presents the results of estmatng a more complex multvarate MRA model, by addng the K and Z-varables. The specfc moderator varables ncluded n Table 2 were selected usng the general to specfc (GTS) approach; that s, all the above varables were ncluded n the metaregresson model and the nsgnfcant ones were omtted, one at a tme. The results reported n Table 2 are qute consstent wth our smple MRA fndngs, equaton (12) Table 1. The effcency wage effect s now contaned n the Z-varables: Smul/SE, Cap/SE, and UK/SE. Note how the coeffcents of Smul/SE and Cap/SE sum approxmately to the same value as the smple MRA coeffcent for 1/SE (equaton (12) Table 1). Studes that nclude a measure of captal n ther estmated producton functons, whch all should, produce sgnfcantly hgher wage-productvty elastctes. Together, these two varables, Smul/SE and Cap/SE, represent the genune postve effcency-wage effect and are jontly statstcally sgnfcant (F (2,68) = 314.7; p<.1). The postve effect of controllng for smultanety,.18 (t=3.73; p<.1), s especally mportant. If the crtque that t s productvty changes that cause movements n wages were vald, rather than the other way around, then those studes that explctly allowed smultanety would have margnally lower estmates of the effcency-wage elastcty. Yet, we fnd the opposte. It s also mportant to use captal 17

20 stock n one s emprcal producton functon to avod obvous omtted-varable bas, dong so also produces hgher wage elastctes (Table 2). Therefore, better specfed estmaton models produce larger estmates of the effcency-wage elastcty. All of these MRA results confrm the valdty of a postve effcency-wage effect. Note also that the UK exhbts hgher wage elastctes, by approxmately.2 (t=2.39; p<.1). But even n the UK, MRA estmates ths wage elastcty (.5= ) to be lower than the accepted stylzed fact. TABLE 2 ABOUT HERE Table 2 also shows that publcaton selecton depends on several factors: whether producton s measured by value added (ValAdded), whether wages are measured relatvely (RelWages) and whether the study measures these wages and producton at the ndustry level (Industry). Publcaton selecton s assocated wth those studes that use ndustry-level data, value added output measures, and that do not employ relatve wages. The jont sgnfcance of these varables agan confrm the presence of publcaton selecton (F (3,68) = 6.3; p<.1). Together these moderator varables explan 91% of the varaton n the observed t-values and 82% of the estmate-to-estmate varaton n reported wage elastctes. Not only are the multvarate MRA results consstent wth the smple FAT-PET-MRA (equaton (12) Table 1), but they are also qute robust. Column 2 of Table 2 presents the estmated MRA coeffcents from a random-effects mult-level model that allows for dependence of the reported elastctes wthn a gven study. Wthn study dependence has long been recognzed as a potental problem for meta-regresson, and mult-level models can compensate for any observed wthn-study dependence (Bateman and Jones, 23). Column 3 Table 2 reports the robust regresson verson of ths same multvarate MRA model. Note that these MRA results are qute robust to dfferent methods and models. The mportant coeffcents, especally on Smul/SE and Cap/SE, reman the same. The only noteworthy dfferences are that the 18

21 magntude and sgnfcance of UK/SE falls n the robust regresson, and ValAdded s no longer sgnfcant. Even f we remove these two varables, all of our prevous nferences reman unchanged ncludng the sgn, sgnfcance and the sum of the remanng coeffcents. Lastly, as a further check for robustness, ths MRA model s run agan by omttng each study, one at a tme. In all cases, we obtaned the same general fndngs concernng the sgn and sgnfcance of the mportant MRA coeffcents. V. Summary and Conclusons Ths paper extends Heckman s econometrc methods for selecton correcton to a crtcal research doman publcaton selecton and ts effects on our research base. Econometrcally, ths s a dffcult problem because data from non-selected (hence unpublshed) studes are generally unavalable. However, serendptous heteroscedastcty among reported econometrc estmates allows selecton effects to be dentfed from a background of random varaton and heterogenety among genune emprcal effects. In addton to Heckman s methods that dentfy publcaton selecton and test for genune effects beyond publcaton selecton, we offer a new MRA model (PEESE, precson-effect estmate wth standard error), whch reduces the bas caused by publcaton selecton. The resultant meta-regresson models are appled to the emprcal lterature on the effcency wage hypothess (EWH). Our meta-analyss of the wage elastctes of producton fnds the average reported elastcty to be.63, whch s roughly equal to the labor s share as requred by neoclasscal theory. However, after correctng for publcaton bas, ths value goes down to.32, practcally and statstcally very dfferent than labor s share. Thus, these Heckman meta-regresson methods call nto queston the conventonal neoclasscal verson of the EWH. Publcaton selecton has a large practcal effect on the effcency wage lterature. 19

22 Consstent wth ths large observed publcaton bas s evdence of publcaton selecton from the funnel-asymmetry test (t=2.67; one-taled p<.1). Nonetheless, the evdence of a genune effcency-wage effect, regardless of publcaton selecton, remans very strong (t=18.4; p<.1). Lke all other meta-regresson analyses n economcs, we fnd that the magntude of the reported wage elastcty depends on other factors. Regon, UK, usng a measure of captal stock, and employng an econometrc method that can accommodate possble smultanety between productvty and wages all have szable effects on the reported wage elastcty. Those studes that allow for smultanety report sgnfcantly larger wage elastctes (t=3.73; p<.1), corroboratng the valdty of the effcency wage hypothess. Lkewse, ncludng a measure of the captal stock n the researcher s model ncreases the reported wage elastcty (t=2.77; p<.1). Together, these effects sum to practcally the same elastcty estmate,.31, as that gven by PEESE s corrected estmate. All of our MRA results are consstent, confrm the presence of an effcency-wage effect, and are robust to varatons n methods and the studes ncluded n the meta-analyss. Although publcaton selecton need not be unethcal or nefarous, ts bas often has practcal consequences for economc research and polcy. Therefore, the effects of publcaton selecton must be understood and fltered from any summary of research fndngs f we are to have a clear or accurate dea about what research s tellng us. 2

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26 Stgltz, J E., The Effcency Wage Hypothess, Surplus Labor and the Dstrbuton of Income n LDCs, Oxford Economc Papers 28 (1976), Stgltz, J.E., Informaton and the Change n the Paradgm n Economcs, Amercan Economc Revew. 92 (22), Sutton, A.J., Abrams, K.R., Jones, D. R., Sheldon, T. A. and Song, F., Methods for Meta-analyss n Medcal Research, (Chchester: John Wley and Sons, 2). Wadhwan, S. B., and Wall, M., A Drect Test of Effcency Wage Model Usng UK Mcro-Data," Oxford Economc Papers 43 (1991), Wooldrdge,J.M., Introductory Econometrcs: A Modern Approach, (South-Western, 26). 24

27 Appendx: Smulatons of Precson-Effect wth Standard Error (PEESE) 14 Our smulatons are premsed on a research lterature that tests a regresson coeffcent (.e., H :β 1 =). Ths smple regresson estmate s merely llustratve of any statstcal estmate. Smlar statstcal propertes for precson-effect estmate wth standard error (PEESE), equaton (16), are found for other types of statstcal estmates. 15 The basc structure of our meta-regresson smulatons may be sketched as: 1. Generate the regresson varables randomly. 2. Use OLS to estmate and test H : β 1=. Select sgnfcant test results. Each selected test of H : β 1= comprses one study s reported result n our hypothetcal emprcal lterature. 3. Smulate these meta-regresson tests by repeatng the prevous steps ether 2 or 8 tmes. At ths stage, meta-regresson model (16) s estmated to provde one PEESE estmate. 4. Repeat all of the above steps 1, tmes. Step 1 s the data-generatng process. X 1 s the ndependent varable and s made to be a random unform varable (1, 2). Then, the dependent varable s generated by: Y =1 + β X 1 + β 2 X 2 +1 e =1, 2, n (A.1) e ~NID(,1). β s ether or 1. When β =1, the average R 2 s approxmately 9%, and the correlaton coeffcent s approxmately.3. The role of β 2 X 2 s to nduce msspecfcaton bas, n general, and omtted-varable bas, n partcular. 14 See Stanley (28) for a more detaled descrpton and dscusson of these smulatons. 15 A team of medcal researchers have corroborated these smulatons n the context of estmatng the logarthm of the odds rato (Moreno et al., 27). The odds rato s the conventonal measure of the effect of a new medcal treatment or new drug where there s a bnary outcome. 25

28 Omttng a relevant varable from a regresson model causes the estmate of β to be based and nconsstent. Such bas remans n large samples and does not lessen wth the standard error. Therefore, t can be mstaken for a genune effect, potentally confusng any statstcal estmate that tres to dentfy genune effect. Small-sample bas causes lttle or no dffculty for MRA methods. Although only omtted-varable bases are smulated here, they serve as a paradgm of any type of large-sample, msspecfcaton bas. Because large-sample bas s the most problematc for these MRA methods, t s the only type of bas used n these smulatons. Cases where there are no msspecfcaton bases have also been smulated but are not reported here because they are even more favorable to PEESE. Random msspecfcaton bas s nduced by makng β n equaton (A.1) a random normal varable, N(,.25). Ths random msspecfcaton bas acts as heterogenety, whch has been recognzed as a key parameter by meta-analysts (Sutton et al., 2; Moreno et al., 27). The dfference s largely a matter of nterpretaton. Here, we assume that varaton n the expected estmated effects s caused by some weaknesses n our emprcal methods; whereas, heterogenety s usually vewed as varaton n the true effect. The most mportant magntude for the performance of meta-regresson methods s the typcal sze of the msspecfcaton bas, relatve to the samplng error. The magntude of publcaton bas ncreases wth the rato of the standard devaton of these msspecfcaton bases (σ bas ) to the standard devaton of the samplng errors (σ b ). The average observed varaton among reported effects found among prevous meta-analyses mples a standard devaton of ths msspecfcaton bas of approxmately.1. To be conservatve, ths value s multpled by two and half tmes, σ bas =.25, and ths value s agan doubled to explore the robustness of our MRA methods. 16 We further assume that there are ether 2 or 8 studes. Twenty s chosen because t s a rather small sample sze for any regresson estmate, whle eghty s selected because many metaanalyses n economcs contan eghty or more estmates. A MRA sample sze of eghty s both 16 See Stanley (28) for greater detals. There, the sze of these msspecfcaton bases s doubled yet agan. 26

29 practcally feasble and gves these MRA methods hgh power (Stanley, 28). Statstcal power depends n the expected way on the MRA sample sze, σ bas, the magntude of the effect, and σ b. Sample szes for the orgnal studes are {3, 5, 75, 1, 2}. Publcaton bas s smulated as selectng a statstcally sgnfcant postve b 1. That s, f the random estmate does not provde a sgnfcantly postve t-value, a new sample s taken and the orgnal regresson s run agan wth dfferent random errors and random bases untl a sgnfcant t-value s obtaned by chance. For example, the 5% publcaton selecton condton assumes that exactly half of the studes estmate and re-estmate ther regresson models untl a random, yet statstcally sgnfcant, estmate s found and reported. For the other half, the frst random estmate, sgnfcant or not, s reported and used. In practce, not all reported results that are publshed wll have been selected for statstcal sgnfcance. Among prevous economc meta-analyses, the proporton of statstcally sgnfcant results vares from 29% to 79%. 1% publcaton selecton s not smulated, because we have observed no such case n economcs. Economc research s too contentous to permt unanmous agreement. Therefore, t s assumed that the ncdence of publcaton selecton s ether: %, 25%, 5% or 75%. The conventonal estmates of an overall effect (the smple average, fxed-effects and randomeffects) have large bases when there s publcaton selecton, and n all cases PEESE greatly reduces ths bas. Recall that fxed-effects and random-effects estmates are weghted averages, where the weghts are the nverse of each reported estmate s varance. The fxed-effects estmates assumes that the estmates come from a common populaton mean; whereas the random-effects estmate does not (Sutton et al., 2). When there are genune effects and publcaton selecton, Appendx Table 1 shows that PEESE has the smallest bas. However, the smple MRA estmate of β from equaton (12) has the smallest bas when there s no underlyng emprcal effect (.e., β 1 =). The superorty of ˆβ when Te= s an 27