Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.


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1 Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement. Ths note derves an exact analytcal relatonshp between the accountng perod and nequalty as measured by the n ndex. The relatonshp s smlar to the decomposton of the coeffcent of varaton. The methodology s llustrated wth panel data on urban wages from Mexco. It s found that the effect of the accountng perod on nequalty s senstve to the propertes of the n correlatons between the perodcal ncomes. JEL categores: C, J6, O5 Keywords: nequalty, tme, Decomposton, n correlaton Correspondng author: Shlomo Ytzha Department of Economcs Hebrew Unversty Jerusalem, 995 Israel emal: fax
2 . Introducton bson, Huang and Rozelle () demonstrate that nequalty n urban Chna s low relatve to other countres n large part because the accountng perod s based on a full year of ncome, whle n many other countres, surveys record household ncome for a month or less. They estmate that f the accountng perod n Chna would be one month, then nequalty would be between 7 and 69 percent hgher. Smlar fndngs have been observed for the Unted States and ermany by Burhauser and Poupore (997). In ths paper, usng nformaton on the n correlaton of ncome between dfferent tme perods, we derve the exact analytcal relatonshp between nequalty and the accountng perod for the n ndex of nequalty. The methodology s llustrated wth panel data on urban wages from Mexco.. Methodology Ths secton derves the relatonshp between the n ndex of nequalty for the weghted average of several tme perods and the n nequalty ndces for each perod taen separately. To smplfy notaton, we restrct the proof to two perods. The extenson to many perods s mmedate. Let (Y,Y ) be drawn from a bvarate contnuous dstrbuton, where Y s the ncome dstrbuton n perod. Let Y = b Y + b Y, where b > (=,) s a constant. The value of the constant determnes whether we are dealng wth the sum of the ncome, or the average ncome, possbly weghted. For example, f b=.5 then Y represents the straght average ncome measured over two perods. Denotng by F(Y ) the cumulatve dstrbuton and µ the expected ncome, the n coeffcent (Lerman and Ytzha, 984) s: = cov(y,f(y ))/ µ. () Denote by Γ = COV(Y, F(Y )),, =,,, the n correlaton between ncomes from COV(Y,F(Y )) perods Y and Y, or between ncome from one perod and average ncome. As dscussed n Shchechtman and Ytzha (987, 999), the propertes of the n correlatons are a mxture of Pearson s and Spearman s correlaton coeffcents. In partcular, Γ s bounded by mnus one and Other statstcs may also be senstve to the accountng perod. For example, Behrman and Taubman (989) fnd that the estmated ntergeneratonal correlaton of parental ncome and offsprngs s.58 when ten years of earnngs are used, compared to.37 for a sngle year. See Bowles and nts () for addtonal evdence and Creedy (979,99) for surveys on the effect of usng a longer accountng perod.
3 one, but Γ s not necessarly equal to Γ. Defne also D = Γ  Γ, for =, (here, the n correlatons are taen between the ncome n each of the two perods and the average ncome over tme), and a = b (µ /µ ), where µ >. Proposton: (a) The followng dentty holds: + [a D D ] = a a (Γ Γ ). () (b) Provded that D =, for =,, and Γ = Γ = Γ, then: = a a Γ. (3) Proof: The proof s n the appendx. The extenson of equatons () and (3) to perods s trval. Let Y = = a Y, then = = = a D = a a Γ. (4) If D =, for =,, and Γ = Γ, then: = a + a a = = < Γ. (5) Equaton (5) s dentcal n ts structure to the decomposton of the varance and the coeffcent of varaton. For t to hold, the n correlatons between each par of varables Y, Y, and Y must be equal. Schechtman and Ytzha (987) show that a suffcent condton for Γ = Γ s that the varables are exchangeable up to a lnear transformaton. Examples of such dstrbutons are the multnormal and the multvarate lognormal, provded that σ = σ, where σ s the logarthmc standard devaton. If the n correlatons between pars of varables are not equal, we need to use equaton (4), where each volaton of the equalty of the n correlatons s captured by an addtonal term n the decomposton (hence, we can treat each volaton 3
4 separately). Lower values for Γ and Γ wll yeld a larger decrease n the n ndex of nequalty over several perods of tme, as compared to the average level of nequalty for the varous perods taen separately. 3. Illustraton We use data from Mexco s 996 Natonal Urban Employment Survey, a panel of ndvduals lvng n 6 dfferent metropoltan areas. The ndvduals are ntervewed on a quarterly bass for fve quarters. Table gves the varous statstcal results. The frst part of the table gves the n ndces of wage nequalty for each quarter among males aged 5 to 65 (we have 66 observatons n the sample). Wages for the fve quarters are converted nto real terms usng the Consumer Prce Index. The quarterly n ndces vary from.567 to.653. vng equal weght to the fve perods, the ncome shares per quarter vary from 9.3 percent to.99 percent. The second part of the table gves the matrx of parwse correlatons Γ between the varous perods. As expected, the farther the perods, the lower the correlatons, wth the excepton that for most quarters, the correlaton wth the quarter four perods later (or earler) ncreases a bt, whch suggests mld seasonalty n earnngs. Another nterestng property s that for most perods, there are dfferences between Γ and Γ, whch means that the margnal dstrbutons are not exchangeable. The thrd and fourth parts of the table provdes the parwse correlatons Γ and Γ. The correlatons wth aggregate ncome tends to be hgher than the correlatons wth another perod, whch s reasonable snce the aggregate ncome ncludes as a component the perodcal ncome. The larger the number of quarters ncluded n average ncome, the lower the correlatons are, snce each perod of tme represents a smaller share of overall ncome. The last part of the table provdes the decomposton from equaton (4) appled to the varous accountng perods, from two to fve quarters. Fgure shows how the n ndex of nequalty s reduced by tang more perods n consderaton. Specfcally, the n ndex for average ncome between quarters one and two s For the frst three quarters, t s.57. For the frst four perods, t s.555. For all fve quarters, t s Wth fve quarters, a total of 48 correlatons must be computed n order to mplement the decomposton n equaton (4). In the varancele decomposton of equaton (5), the number of correlatons requred s reduced to for fve quarters. How good an approxmaton s equaton 4
5 (5)? Table can be used to show that usng equaton (5) would have yelded a n for the fve perods of.5388, whch accounts for 98.4 percent of the actual n obtaned wth equaton (4). Equaton (5) accounts for an even larger share of the actual n for fewer perods. Hence, t s a useful frst order approxmaton for evaluatng the mpact of the accountng perod on the n. 4. Concluson The n ndex of nequalty for a sum of random varables can be decomposed n a way that resembles the decomposton of the varance, plus an addtonal term, whch reflects the devaton of the underlyng dstrbutons from exchangeablty up to a lnear transformaton. To be able to mae quanttatve nferences on the effect of the accountng perod on the n coeffcent, we should evaluate the n correlaton for dfferent types of varables (households or ndvdual, monthly or quarterly etc.). If we fnd the magntude of those correlatons, and f they are relatvely stable over tme (and possbly over countres), we may be able to predct the mpact of the accountng perod on nequalty n qute general settngs. 5
6 Appendx: Proof of equatons () and (3). For smplcty, we defne a = b (µ /µ ), and provde the decomposton for = COV[a Y + a Y, F(Y )]. Ths normalzaton enables us to wor wth varables wth unt means, but t does not affect the generalty of the proof. Usng the propertes of the covarance we can wrte: (A.) = COV[a Y Y, F(Y )] = a COV[Y, F(Y )] COV[Y, F(Y ))] Defne the dentty: (A.) = a Γ Γ. Γ = Γ + D for =,, where D s the dfference between the two n correlatons defned between Y and Y. Usng (A.) and (A.), we get: = a (Γ + D ) (Γ + D ). Rearrangng terms:  a D  a D = a Γ Γ. Usng the propertes of the covarance: Γ cov(y,f(y )) = = cov(y,f(y )) a Γ = cov(y,f(y Wrtng Γ n a smlar manner, we get equaton (): [a D = a D a ]. (Γ = a (a + Γ ). {acov(y,f(y )) )) Γ ) (a Γ cov(y,f(y ))} = ) Assumng equalty of the n correlaton coeffcents between Y and Y sets D =. A smlar assumpton for Y and Y sets D =. The assumpton Γ=Γ =Γ completes the proof of (3). 6
7 References Behrman, J. R. and P. Taubman (989). Is Schoolng Mostly n the enes?, Journal of Poltcal Economy 97: Burhauser, R. V. and J.. Poupore (997), A CrossNatonal Comparson of Permanent Inequalty n the Unted States and ermany, Revew of Economcs and Statstcs 79: 7 Bowles, S. and H. nts (). Intergeneratonal Inequalty, Journal of Economc Perspectves, forthcomng. Lerman, R. and S. Ytzha (984). A Note on the Calculaton and Interpretaton of the n Index, Economcs Letters 5: Creedy, J. (979). The Inequalty of Earnng and the Accountng Perod, Scottsh Journal of Poltcal Economy 6: Creedy, J. (99). Lfetme Earnng and Inequalty, Economc Record 67: bson, J., Huang, J. and Rozelle, S. (). Why s Income Inequalty so Low n Chna Compared to Other Countres? The Effect of Household Survey Method, Economcs Letters 7: Schechtman, E. and S. Ytzha (987). A Measure of Assocaton Based on n's Mean Dfference, Communcatons n Statstcs: Theory and Methods, A6: 73. Schechtman, E. and S. Ytzha (999). On The Proper Bounds of The n Correlaton, Economcs Letters 63:
8 Table : n ndex of nequalty and accountng perod, Mexco 996 n ndces of nequalty ( ) and ncome shares (a ) st quarter nd quarter 3 rd quarter 4 th quarter 5 th quarter n ndex Income share (fve quarters) n correlatons matrx (Γ ) st quarter nd quarter 3 rd quarter 4 th quarter 5 th quarter st quarter nd quarter rd quarter th quarter th quarter n correlatons wth aggregate ncome (Γ ) One quarter Two quarters Three quarters Four quarters Fve quarters st quarter nd quarter rd quarter th quarter th quarter n correlatons wth aggregate ncome((γ ) st quarter nd quarter 3 rd quarter 4 th quarter 5 th quarter One quarter Two quarters Three quarters Four quarters Fve quarters a D = = a = a a Γ One quarter Two quarters Three quarters For quarters All quarters Source: Authors estmaton from ENEU data. 8
9 .64 Fgure : Accountng perod and wage nequalty n Mexco, 996 n ndex of nequalty (quarter ).5855 (quarters ).57 (quarters 3).555 (quarters 4).5475 All quarters Accountng perod (number of quarters) Source: Authors estmaton from ENEU data. 9
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