CHATER 5: DUMMY DEENDENT VARIABLES AND NON-LINEAR REGRESSION. Th roblm of Dummy Dpndnt Varabls You alrady larnd about dumms as ndpndnt varabls. But what do you do f th dpndnt varabl s a dummy? On answr s: Logstc rgrsson Of cours, you could also run OLS, whch, howvr, has obvous lmtatons. Fgur : OLS n Dummy Dpndnt Estmaton 0 X 70
roblms wth OLS whn th dpndnt varabl s a bnomnal dummy ar: Th rror trm s obvously not normally dstrbutd. Th rror trm s htroskdastc. R-squard bcoms a uslss masur Most mportantly, th modl s problmatc for forcastng purposs. On would lk to forcast th probablty of a crtan st of ndpndnt varabls to crat a crtan bnomnal outcom. OLS could crat probablts of gratr than on or smallr than zro. Logstc rgrsson s a non-lnar stmaton tchnqu, whch solvs th problm of unbounddnss of OLS.. Th Logt - Modl Th Logt-Modl s dfnd as: LN ( = β 0 β X β X ε () It s basd on th cumulatv logstc dstrbuton 7
7 () an b rarrangd to: [ ε ] β β β = 0 X X () How? Dfn LN = Tak Antlog = Tms (-) = Dvd by ` = lus = Tak nvrs = Multply by = Expand rght sd by = Thus, ε β β β = 0 X X LN ( = ] [ ε β β β = 0 X X
If 0 0 and f Thrfor, logstc rgrsson solvs th unbounddnss problm of OLS. Fgur : Logstc Rgrsson 0 X You wll also ncountr robt modls. Thr da s smlar to th logt. Th only dffrnc s that th probt stmats ar drvd out of th cumulatv normal dstrbuton. In practc, thr mthod ylds prtty dntcal rsults. 73
3. Intrprtaton A postv coffcnt n stmatng LN ( = β 0 β X β X ε tlls you that as X ncrass, th lklhood that th DV taks th valu ncrass. Most statstcal softwar packags actually calculat th probablty Statstcal softwar packags also rport th so-calld odds rato. If t s postv, > 0.5; f t s ngatv <0.5. An odds rato of : (), for xampl, tlls you that = 0.66; and an odds rato of : (0.5), that = 0.33. 74
4. Exampl: War Rsk Upload th datast War.xls, whch classfs countrs as a war country f thy had at last on yar of armd conflct btwn 960 and 005. Th othr varabls ar pr capta ncom n (000 USD, ln), olty (a masur of dmocracy), ncom nqualty, manufacturng xport shar (% of GD), and Muslm Chrstan olarzaton (th lklhood of obtanng a Muslm and a Chrstan n a random drawng from th populaton). Th datast also contans nghborhood ffcts for th rgons DvMENA (dvrsfd conoms of th Mddl East), OlMENA (ol conoms of th Mddl East), Sub Saharan Afrca (SSA), Latn Amrca and th Carbban (LAC), South Asa (SA), East Asa and th acfc (EA), East Asan Tgrs (EAT), North Amrca (NAM), Wstrn Europ (WE), and Eastrn and Cntral Europ (ECE). Th nghborhood ffcts ar populaton wghtd rgonal polty and rgonal ol (ful xports as a prcntag of GD). It also contans th numbr of rfugs pr 00,000 (RgRf). 75
A look at som scattr plots s usful to s why OLS s problmatc wth dummy dpndnt varabls. Th dpndnt varabl s always War Country whl th ndpndnt varabls ar pr capta ncom (ln), polty, nqualty, and manufacturng xport shars rspctvly. War Country vs. Incom War Country vs. olty War Country vs. Inqualty War Country vs. Manufacturng In all cass th prdctd valu ar dffcult to ntrprt, spcally n th cas of War Country vs. Manufacturng, whch llustrats th problm of unbounddnss. 76
Logstc rgrsson s mor manngful. War Country vs. Incom War Country vs. olty War Country vs. Inqualty War Country vs. Manufacturng Th abov scattr plots ar gnratd as follows. Go to Modl Nonlnar modls Logt Bnary. Th dpndnt varabl s War Country. Th ndpndnt varabl s, for xampl, pr capta ncom. Run th modl. Sav th fttd valus. Crat scattr plot of Fttd valu vs. Incom. 77
Rportng logstc rgrsson rsults. Modl Ln WarCtry NoWarCtry Rgrsson rsults (Exampl) Const LnYCA OLITY INEQ MANU = β βinc β olty β Inq β Manu ε 0 I II III IV V 5.35 0.6-3.5.00.9 (<0.0) (<0.0) (<0.0) (<0.0) (0.9) -0.68-0.53 (<0.0) (<0.0) -0.09 (<0.0) 0.09 (<0.0) 3-0.09 (<0.0) 4 0.05 (0.09) N 8 60 5 76 45 % Class. 70.9% 70.6% 69.7% 67.0% 73.% Thr s obvously a multcollnarty problm btwn nqualty, polty, and manufacturng. olcy smulaton: Assum Lbanon has today a pr capta ncom of $5,000 and a Gn coffcnt of 60. What s Lbanon s war country lklhood? By how much would Lbanon s lvl of nqualty b rducd n ordr to drv th war country lklhood blow 50%? 78
5. Truncatd Data and Logstc Rgrsson Logstc rgrsson, whch undrls th logt modl, can also b appld to data whch s somhow cut off. Such cut off data s calld truncatd or cnsord. Truncatd data causs a truncaton bas, whch maks th trnd ln flattr than t would b f th data was normally dstrbutd. DV Non-truncatd data DV Truncatd data IV IV A strkng mprcal rgularty s that th maxmum Dvdng th OLS stmats by th proporton of nonlmt obsrvatons n th sampl s a common practtonr s soluton to corrct for ths bas (Grn (003): Economtrc Analyss, p. 768). Howvr, an vn bttr modl would b agan a non-lnar ft, smlar to th logt modl. 79
An xampl for truncatd data s th olty datast whch taks valus btwn -0 (Autocracy) and 0 (Dmocracy). Runnng from th War.xls datast olty on pr capta ncom (LnYCA) ylds th followng scattr plot. Scattr lot of olty (truncatd) on r Capta Incom 80
Runnng from th War.xls datast olty on pr capta ncom (LnYCA) ylds th followng scattr plot. A closr look at th data rvals that out of th 53 obsrvatons, 0 obsrvatons ar lmt obsrvatons (rproduc ths rsult n xcl), whch gvs a prcntag of non-lmt obsrvatons of (33/53=87%). Th truncaton-adjustd coffcnt would thrfor b./0.87=.55. Instad of runnng a lnar rgrsson, truncatd data s always a natural canddat for logstc rgrsson. 8
Whn runnng a logstc rgrsson on truncatd data, t s ncssary to transform th data frst such that that all valus of th dpndnt varabl ar postv. It s rcommndd to add to th dpndnt varabl th mnmum plus on, whch s lvn n th cas of olty. In runnng a logstc rgrsson wth truncatd data, grtl also wll ask you to spcfy th asymptotc maxmum, whch n th cas of olty s now. For computatonal rasons, th asymptotc must b slghtly abov th maxmum valu, for xampl,.00 In grtl you opn th logstc rgrsson modul n Modl Nonlnar modls Logstc Th rgrsson rsults ar summarzd blow. A comparson of th adjustd R shows that th logstc rgrsson s a much bttr ft, ncrasng th R by almost 7 prcntag ponts. 8
Logstc rgrsson rsults Scattr plot Actual vs. rdctd usng logstc rgrsson 83
6. Furthr Radngs Dummy dpndnt stmaton tchnqus ar ncly xpland n th followng txts: Camron, S. (005), Economtrcs, McGraw Hll, Boston, Chaptr 8. Schmdt, S., Economtrcs, McGraw Hll, Boston, 005, chaptr 9. Studnmund, A., Usng Economtrcs, A ractcal Gud, 4 th Edton, arson Educaton, 00, Chaptr 3. 84