Introductory Econometrics ECON 4720
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Date Created: 09/21/15
Qualitative Choice and Limited Dependent Variables September 2006 1 Binary Discrete Choice Models Consider the problem of whether to enter the labor force or whether to marry or whether to commit a crime etc In all of these cases7 the dependent variable is binary Because it is discrete7 we need alternative methods of estimation because the linear model must be mispeci ed De ne Xi 6239 N iidFla where is the latent value of doing it and a is a vector of parameters associated with F l 1 De ne the set of parameters to be 9 134 We observe yi1ygt039 Then Prlyi1lXil Frill gt llXil PrXiBaigt0 Prlai gt Xi l 1 i F Xi Prlyi 0 Xi Nixie Suppose we have a sample yiXi1 The log likelihood function is n L log 1 7 F em6 1 7 y logF iXiB 1 11 If we maximize L over 6 we get maximum likelihood estimates MLEs They are consistent7 ef cient7 and asymptotically normal 11 Example Logit expizl Fz 17 1expz39 Then equation 1 becomes 7 n V eXPi Xig V exp7XvB L 7 gm 10g 1 7 1 exp Xi 1 7 yz og 1 exp iXi n 1exp7Xi8 exp Xig yi log 7 Z 19XP i Xi5 1exp7Xi8 V exPi XiB 1 1 10g 7 n v 4 V expXz 7 gm 10g 1 expi XiB 1 7 yz 10g 1 exp7Xv8 exp Xi5 1 exp 71 Z 11239 10g i1 Zini 10g1 9X13 Xi5 11 r 1 1 1 7 yi og 1 expXiB If we take FOCs we get where 8L exp Xia W i 7 n V expXz 7 y17 1expXi X2 ZM PiXi0 i1 PiPriyi1iXil Thus we can think of the FCC as an orthogonality condition residuals should be orthogonal to Xi Note that for the linear models we considered we could interpret 3E 11239 i Xi 8X For this model 3mg Xi 7 3Pryi 1 X 3X 7 3X 3 expm m 8X 7 exp Xi 1 1 exp 1 exp Pi1ipiB7E Also note that eXPX13l 1 PF 112 7 1 l Xil 7 1 1exPIX ngr0 X 0g 1 2 Z 1expm 10g exp Xi XiB Some people use this fact usually incorrectly to estimate 12 Example Probit a N iidN 003 Fzl0395ltlgti Us Then equation 1 becomes 7 V ext6 V 7Xi6 L 7 gyzlogliltlgt lyzlogltlgt UH 0395 W 1 fl Note that While o39g is identi ed neither is identi ed separately Thus With out loss of generality we can set 0395 l L Zyilog 1 Xi5 1 yillog 1 i 1 Xz lll If we take FOCs we get 8L 0M WM a Zyi ltxmXi ltl yigtTm 139 at 1 7 mm 7 lt1 7 m Xi5 Xi 7 dgtXi5 Z W 1 7 ltIgt Xi Xi where ltIgtX Prlyi 1 Xi Thus again we can think of the F00 as an orthogonality condition residuals should be orthogonal to 15095 V X l1 I Xz ll Z Note that for this model ammxi 7 aPriyillXi 3X 7 3X 8ltIgtXz 3X Xz 2 Tobit Consider a model for the number of hours worked in a particular week Let yi be the observed number of hours worked by person i Assume that Xi l ui 11239 max 0 11 u N iidN0a Then the log likelihood contribution for observation i is L1ygt01og1 1y01ogqgt fi 039u We can maximize Li over 9 dam to get MLEs of 9 3 Polychotomous Discrete Choice Models In the binary discrete Choice problem the dependent variable had two possible values or ow we consider a set of problems where the dependent variable can take on more than two possible values We consider three types of models multinomial models ordered models and count models 3 1 Mult inomial Models Let 11 Xij zi7j5ijv j17277J 127 1yfj2yfchk a N iidF Give examples occupation Choice car brand Choice transportation to work Choice of college where to live We observe MX Zi L1 and we want to estimate 9 7 First note that we can not identify all of the elements of 7 because we get to only see a measure of values relative to each other So we set at least one of the elements 0137 0 Then PF 112739 1 Xivzil Pr Z yngk l X2321 Pr X2775 Z j 62739 2 XiicB Z k Eika Pr V27 3 5 7 aika where Vi Vij Vij X2775 ZN j W XikXij5Zi7k7j Note again that only the relative 7 s matter Continuing PF 112739 1 thZil Pr S 62739 Sika 2 Pr V27 3 5 7 5ka 57 dFJ 57 32 Multinomial Logit If we assume that 5 iidEV 3 F a exp7e E 3 f a exp76 7575 139 lzX Aid 4 t t 5565de 4 szfz Z smu Zz lX f g ZZ ZX I Mm WZ WXdequ 4 a U2 9 axe d eleqm oz 5 quozde 4 IZXIWZ qLZZquXdX9 013 4 X Z WZ WXHdXWX 99 79 oz 5 WZ 90150 393301 4 W2 gsz 4253 g em sousums 9100s sq pm 91 5 A mzz goszdX933014UZ gsz 5923 oz 5 911Adx933014 fFAWZ 5 5 5923 4 Mil23 oz 5 5 MA 1er 301 ME 4 IMMZ 2 91 301m3 zj 3 2 I0 Hounqumoo Pooqnwm 301 eqi 1250 zmwoumnw panes s epom sgqi 39918IIIEA9 01 29 s 0ng Wm qz A was 01 911AdXSZ 301 91 WA dxe Z 3014 71 ZZ ZX I M 1d 01 seggdulgs 5 110112an 119111 which we can think of as an orthogonality condition Similarly 3L39 3 Z Zyij X2775 Z j1OgZeXPXik5 ZWQ 3 37139 J 7c 7 ynzv 7 ZieXPHXiJWZW Z 2 2k eXP XikB Z170 yijZi PijZi yij Pij Zi Note that log PF 112739 1 l Xi Zil 10g eXP Vii Zk eXP iVikl Pr 1 l Xi Zil 9X13 6X13 eXP V27 1 V 7 V 0g exp 2 2m which does not depend upon characteristics of any choice other than j and m Discuss the Red BusBlue Bus problem HA Also discuss correlations of errors associated with the location problem 3 3 Mult inomial Probit If instead we assume that 6239 N iidN 0 9 the probability in equation 2 becomes a J 7 1 dimensional integral On the one hand this problem no longer suffers from the HA problem On the other equation 2 is very dif cult to evaluate analytically or numerically Instead we can use simulation methods to evaluate it These methods are not discussed here 34 Ordered Discrete Choice Models Consider problems where the dependent variable is discrete and ordinal but not cardinal For example health is usually measured as an ordered discrete vari able Sometimes income continuous is bracketed and turned into an ordered discrete variable We might think of the number of children one has as an ordered discrete variable in a sense we will discuss later et 11 Xi 5i yik liHTkSyfltTk1y k07177K 6139 N iidF The 739 variables are called threshholds We observe yiXi1 and we want to estimate 9 n Without loss of generality7 we can set 7390 700 7391 07 TK1 00 Then Prlyik1lXil Prl c 11 ltT7c1 l Xil Prl c SXi Ei ltT7c1l Pr Tic Xi S 5239 lt 739Ic1 XiBl F m1 7 X278 7 F m 7 XM lf we assume that Q iidN 003 then Pryik1 Xi M q I 05 05 Note that US is not identi ed separately from 738 Thus7 we can set 0395 l7 and Prlyik 1 l Xi 7k1 Xi q7397c Xi The log likelihood contribution for i is K Li 10g 1 7k1 X275 7 k Xz ICZO This is called ordered probit If7 instead we assume that 6239 N iidLogistic then 6XPTk1 Xi expifk Xi P 39 l X39 7 r gm l 2 1 exp Tk1 7 X278 1 exp Th 7 X278 K eXP Tk1 X275 exp Tic X275 L39 39 lo 7 Z gym glexp739k1 iXiB lexp7 C iXiB and this is called ordered logit What happens if the threshholds are data eg7 think about the bracketed income example Think about a nonparametric speci cation of the income equation 1 9Xi65i 35 Count Data In count models7 the dependent variable is a nonnegative integer Examples include the number of patents led by a rm in a year7 the number of trips made by an individual in a month7 or the number of hurricanes in a year In these cases7 it is useful to take into account the integer nature of the dependent varia e Let the dependent variable be git N indPoisson A11 v f gm 1m Elli Vann Ait If we model 10 An X2157 then we can allow observed explanatory to affect the statistical properties of git The log likelihood contribution of observation i 7 exp iAit Vt Li 7 Zlog yit Z t Z Ait yit 10 An 710gynl t Z 7 exp X yitX B 7 log 1mm t with score statistic BLZ39 3 W 217 expXnB yan5 710gynll t 21an exp XitB yitXitl t 2 1m 9X13 Xit l l Xit t which we can think of as an orthogonality condition because Eyn l Xn 6XPXit5 Discuss problems with residuals exhibiting heteroskedasticity and serial cor relation We can deal with these problems by amending equation 3 to logit Xit ui ui N iidGamma66
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