FIGURE 2: ESTIMATING THE MARGINAL COST OF A POLICY
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- Millicent Carson
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1 The Observed Choce roblem n Estmatng the Cost of olces November 7, 1997 Erc Rasmusen Abstract A polcy wll be used more heavly when ts margnal cost s lower. In a regresson settng, ths can mean that the equaton to be estmated s actually y = æ èæ è. The analyst who treats tmes and places as dentcal wll underestmate the polcy's average cost. OLS s based towards small coeæcents, and nstrumental varables should be used. Indana Unversty, Kelley School of Busness,BU 456, 1309 E 10th Street, Bloomngton, Indana, Oæce: è81è Fa: Emal: Erasmuse@ndana.edu. Web: hp.ndana.eduèçerasmuse. Copes of ths paper can be found at hp.ndana.eduèçerasmuseè@artclesèunpublshedèmchoce.pdf.
2 November 7, It s common to estmate polcy eæects by lookng at data from varous locatons. Suppose Impact = æ æ olcy, or y = æ ; è1è and that the mpact s undesrable. In ths settng, = èæ è because polces are chosen n recognton of ther margnal mpacts n partcular locatons, and æ vares across locatons. Ths causes a predctable bas n OLS estmaton whch I call ë the observed choce problem". Ths problem has not been drectly dscussed n the econometrcs lterature. The closest I have found s Garen è1984è. In my own Rasmusen è1996è I develop the problem more fully and apply t to the slghtly more complcated case where the polcy mpact s desrable. The followng three-equaton model llustrates the bas. y = æ + æ èè æ = æ + v = æ 1 + æ æ + æ 3 z + u è3è è4è Assume that: èè æ 1 + æ æ + æ 3 z N é 0; èè æé0, èè z and æ are nonstochastc, èvè æ; u and v are ndependent stochastc dsturbances wth mean zero and ænte varance, èvè v has a symmetrc dstrbuton, èvè æ é 0. Assumptons èè and èè are just normalzatons, but èvè represents that y s an undesrable mpact of, so s used less when æ s greater. The OLS estmate of æ s bæ OLS = y ; è5è whch has the epectaton è! è! èæ + v + æ è E = E æ + E è! è! v æ + E : è6è The ærst and last terms of è6è equal æ and 0, and the mddle term equals 0 f Eè v è=0. If and v are ndependent, OLS s unbased.
3 November 7, 1997 Ths model, however, volates the OLS assumptons n two ways, each harmless by tself, but bad n combnaton: random parameters and stochastc regressors. The smpler system of just èè and è3è has random parameters, and the smpler system of just èè and è4è èso æ = æè has stochastc regressors, but n each of those two smple systems, OLS would be unbased. To see that the OLS estmate of æ s based n the full system, combne equatons è3è and è4è to get = æ 1 + æ æ + æ v + æ 3 z + u : è7è The crtcal mddle term n equaton è6è, whch for unbasedness must equal zero, can be wrtten usng è7è as èæ1 + æ æ + æ v + æ 3 z + u è v : è8è The summed quantty n the numerator has the epectaton æ ëæ 1 + æ æ + æ 3 z ëç v ; è9è snce Eèv 3 è=0by assumpton èvè, and u and v are ndependent. Epresson è9è has the same sgn as æ ëæ 1 + æ æ + æ 3 z ë. Summed across the n observatons, ths takes the same sgn as æ, snce the term n square brackets s postve by assumpton èè. Snce æ é 0, æ s underestmated. Ths s smlar to the folk wsdom that estmaton problems lead to coeæcents beng too small. Instrumental varables can be used to solve the observed-choce problem, as I show n Rasmusen è1996è, f the analyst can observe z. Fgure 1 llustrates the problem. It shows two localtes wth ther own relatonshps between polcy and mpact y depcted as rays through the orgn. Localtes 1 and have slopes æ 1 and æ,anaverage slope of æ = èæ 1+æ. olcymakers 1 and choose ponts on ther respectve rays. If they choose gnorng local condtons, 1 and have the same epected value, and the epected average of the two observatons s on the mddle ray. Ths
4 November 7, corresponds to OLS beng unbased. If, however, y s a cost of, and a steeper slope makes a polcymaker choose a lower level of, then Localty 1, wth a greater margnal cost, chooses a lower than Localty : 1 é. If the econometrcan draws a lne through the orgn to le between the two observatons and mnmze the squared devatons, that lne wll have a slope of less than æ. OLS underestmates the margnal cost. Cost (y) Iowa average.. Wsconsn olcy () FIGURE : ESTIMATING THE MARGINAL COST OF A OLICY REFERENCES Garen, John è1984è. The returns to schoolng: A selectvty bas approach wth a
5 November 7, contnuous choce varable. Econometrca 5 èseptemberè: Rasmusen, Erc è1996è ëobserved Choce and Optmsm n Estmatng the Eæects of Government olces," forthcomng, ublc Choce.
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e i is a random error
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