3/3/2014. CDS M Phil Econometrics. Vijayamohanan Pillai N. CDS Mphil Econometrics Vijayamohan. 3-Mar-14. CDS M Phil Econometrics.

Transcription

1 Dummy varable Models an Plla N Dummy X-varables Dummy Y-varables Dummy X-varables Dummy X-varables Dummy varable: varable assumng values 0 and to ndcate some attrbutes To classfy data nto mutually exclusve categores Also called: ndcator varable, bnary varable, dchotomous varable, categorcal varable, qualtatve varable Y = α + βd + u Dummy X-varables Y = Wage rate of an agrcultural labourer D =, f male worker 0, otherwse. Mean wage of a male agr. worker? E(Y D = ) = α + β 5 6

2 Dummy X-varables Dummy X-varables Y = α + βd + u Y = Wage rate of an agrcultural labourer D =, f male worker 0, otherwse. Mean wage of a female agr. worker? E(Y D = 0) = α Y = α + βd + u Y = Wage rate of an agrcultural labourer D =, f male worker 0, otherwse. H 0 : no sex dscrmnaton H 0 : β= = Dummy X-varables Dummy X-varables Y = α + βd + u Y = Wage rate of an agrcultural labourer D =, f male worker 0, otherwse. Analyss of Varance (ANOVA) Model: Mean dfference test In eco applcatons, control for other soco-eco factors: caste, nature of work, experence, Both quanttatve and qualtatve varables: Analyss of Covarance (ANCOVA) Model 9 0 Dummy X-varables Dummy X-varables Dfferental ntercept: α Y = α 0 + α D +βx + u D =, f male worker = 0, otherwse. Mean wage of a female agr. worker? E(Y D = 0) = α 0 + βx Mean wage of a male agr. worker? Agrcultural wage rate (α 0 + α ) α α 0 E(Y D = ) = (α 0 + α ) + βx X

3 Dummy X-varables Dummy X-varables Agrcultural wage rate (α 0 + α ) α α 0 β β Agrcultural wage rate α 0 α (α 0 +α ) f m X X 3 4 Y = α 0 + α D +β X +β D X + u Interacton term D =, f male worker = 0, otherwse. Dummy X-varables Mean wage of a female agr. worker? E(Y D = 0) = α 0 + β X Mean wage of a male agr. worker? E(Y D = ) = (α 0 +α ) + (β +β )X Agrcultural wage rate α 0 α (α 0 +α ) β β + β X 3 March 04 CDS 5 6 Dummy Y-varables Dummy Y-varable Dscrete Choce Models Many stuatons n whch the dependent varable s not a contnuous varable. Dscrete or, qualtatve 3

4 Dummy Y-varable Bnary Choce Model In general types:.dependent varables whch take one of two values (bnary/ dchotomous choce), and Indvdual. dependent varables whch can take more than two values but are fnte (polychotomous; multple choce). Choose A To be Don t Choose A Not to be 9 Bnary Choce Model Multnomal Choce Model Indvdual Indvdual Choose A Don t Choose A By car Not Alternatves walk cycle 3 car bus tran J Bnary Choce Model Examples Bnary Choce Model Examples Labour Force Partcpaton: - occupatonal choce (multple choce) - employed or unemployed (bnary choce) - to be employed full-tme, part-tme tme or unemployed (multple choce) Votng Behavour: - to vote or not to vote (bnary choce) - to vote Congress, BJP, Communsts, or abstan (multple choce) 4

5 Bnary Choce Model Examples Bnary Choce Model Censored Data: Lmted dependent varable: Housng expendture some may not have purchased: so zero Two questons () Can we stll use OLS to estmate such outcomes? () If not, how do we model such outcomes? Bnary Choce () Lnear Probablty Model (LPM) () Logt/ Probt Model Bnary Choce Model Censored/ Lmted Dependent Varable Regresson Model Lnear Probablty Model We focus on sngle equaton bnary outcomes: y { 0,} A fundamental dfference between a quanttatve response model and a qualtatve response model: The latter s a probablty model. () Tobt Model 8 In general: Lnear Probablty Model Prob(event j occurs) = P(Y = j) = f (relevant varables; parameters) = f(x, β) where x = [ x,..., x ] k are the varables and β s a vector of parameters. Gven Lnear Probablty Model y { 0,} P(y = x ) = P = f(x, β); P(y = 0x ) = P = f(x, β); y follows Bernoull probablty dstrbuton 9 5

6 Lnear Probablty Model How do we specfy f(x,β)? An obvous choce s the famlar least squares procedure: f(x, β) = x β y = x β + u Ths leads to the lnear probablty model (LPM). Lnear Probablty Model Assumng E(u) = 0, t follows that E(y x ) = 0 P(y = 0) + P(y = ) E(y x ) = P(y = x ) = xβ Condtonal expectaton = condtonal probablty The regresson eqn descrbes the probablty that y = gven nformaton on x 3 Lnear Probablty Model y, P y = β +β X β +β X k β The case of a sngle explanatory varable: 0 The probablty of the event occurrng, p, s assumed to be a lnear functon of the varable X. Now an example. X k X Changed nto bnary: 0 = none; = F 35 M 86 LPM Ray Far Model Kurtoss N Skewness Mn Max Mean Std. Std. Std. Statstc Statstc Statstc Dev Statstc Error Statstc Error Have extramartal affars Sex Age group Marred years group Have chldren Relgocty group Educaton group Occupaton group Marrage ratng group Interpretaton? 36 6

7 LPM Ray Far Model LPM Ray Far Model LPM Ray Far Model LPM Ray Far Model Predcted Probabltes Chap 3-39 LPM Ray Far Model The Lnear Probablty Model Predcted Probabltes sex Age years marred chldren relgou s Unstandardzed Standardzed t Sg. B Std. Error Beta (Constant) Sex Age Marred years Have chldren Relgocty Educaton Occupaton Marrage ratng Dependent Varable: Extramartal affars Interpretaton? years relgou Educatoaton ratng Prob Occup- marrag Pred sex Age marred chldren s Educaton Occupaton marrag ratng Pred Prob Negatve probablty! Some serous shortcomngs. () The dstrbuton of the dsturbance s non-normal. As y can take only one of two values, the error term also has a dscrete (non-normal) normal) dstrbuton. The probablty dstrbuton of u s: y = u = xβ y = 0 u = xβ wth P u wth ( P ) In effect, u follows a Bernoull dstrbuton y x β = 7

8 LPM Ray Far Model The Lnear Probablty Model Some serous shortcomngs. () the error term s heteroskedastc Var(u) = (xβ)( xβ) Whch clearly vares wth the value of x Normal PP Plot of Regresson Standardzed Resduals Hstogram 44 LPM Ray Far Model The Lnear Probablty Model these problems not nsurmountable: Problem of non-normalty normalty can be crcumvented provded we have a large sample sze (nvoke the central lmt theorem) Problem of heteroskedastctycan can be removed by usng Whte s heteroskedastc standard errors Chap The Lnear Probablty Model The Lnear Probablty Model () The man problem s the Non-fulflment of 0 E(Y ) There s no guarantee that the predcted values of Y wll all le between 0 and. sex Age years marred chldren relgou Educatoaton Occup- marrag s ratng Pred Prob Negatve probablty! 47 What we requre therefore s a way of constranng the LPM so that the predcted probabltes do le n the [0,] range. In general we use alternatve estmaton models to do ths. 8

9 The Soluton The Soluton The usual way of avodng ths problem s to hypothesze that the probablty s a sgmod (S-shaped) functon of, F(), where s a functon of the explanatory varables Several mathematcal functons are sgmod n character. Alternatves Alternatves to The Lnear Probablty Model The dstrbuton Normal: PROBIT, natural for behavor Logstc: LOGIT, allows thcker tals Gompertz: asymmetrc, underles the basc logt model for multple choce 5 Underlyng Probablty Dstrbutons for Bnary Choce The Logt Model Several mathematcal functons are sgmod n character. One s the logstc functon. F( ) e F() = = + e + e where = x β 3 March 04 CDS 54 9

10 F ( ) The Logt Model e F() = = + e + e where = x β Normal dstrbuton vs. Logstc dstrbuton As,, e - 0 and p (but cannot exceed ). As,, e - and p 0 (but cannot be below 0). 3 March 04 CDS Logstc dstrbuton Logstc dstrbuton The Logstc dstrbuton has densty functon: (z a)/b (/b)e f(z) = (z a)/b ( + e ) where a s the mean of the dstrbuton b s the scale parameter e s the base of the natural logarthm, Euler's e (.7...) Wth a = 0 and b =, the Logstc dstrbuton has densty functon: z e f(z) = < z < z ( + e ) Integratng the pdf gves the dstrbuton functon: F(z) = < z < z + e Here a = 0; b =,, and 3 Here a = 0; b =,, and The Logt Model: Odds Rato The Logt Model: Odds Rato.00 F ( ) P = ; P = ; z z + e + e Odds rato P = E(y = X ) = F() e = = + e + e z P + e = P + e where = e z = x β.00 F ( ) Takng log of the odds rato, L L s called Logt. Hence the Logt model Odds rato z P + e z = = e P + e where = x β P = ln = = xβ P Now an example. 3 March 04 CDS 60 0

11 Changed nto bnary: 0 = none; = F 35 M 86 Logt Ray Far Model Regresson varables estmates B S.E. Wald df Sg. Exp(B) Sex Age Marred years Have chldren Relgocty Educaton Occupaton Marrage ratng Constant Logt Ray Far Model Regresson varables estmates Logt Ray Far Model Odds rato estmates Logt Ray Far Model Regresson varables estmates B S.E. Wald Sg. Exp(B) Sex Age Marred years Have chldren Relgocty Educaton Occupaton Marrage ratng Constant Wald = (B/SE) = t Only 4 varables sgnfcantly dfferent from zero at α = 0.05

12 Exp(B ) = odds rato = Logt Ray Far Model P z = e P Factor by whch the odds change when the th ndependent varable by one unt. B S.E. Wald Sg. Exp(B) Sex Age Marred years Have chldren Relgocty Educaton Occupaton Marrage ratng Constant If B > 0, OR > ; If B < 0, OR <. If B = 0, odds unchanged e.g., When No. of years marred by unt, log of odds for affars by.099 or 9.9%, ceters parbus. Exp(B ) = odds rato = P = z e P e 0. 8 =.34 Logt Ray Far Model B S.E. Wald Sg. Exp(B) Sex Age Marred years Have chldren Relgocty Educaton Occupaton Marrage ratng Constant e.g., Relgoctysgnfcantly reduces ncdence of extramartal affars! Marred years sgnfcantly contrbutes to extramartal affars! 68 Logt Ray Far Model Logt Ray Far Model Exp(B ) = odds rato = P = z e P B S.E. Wald Sg. Exp(B) Sex Age Marred years Have chldren Relgocty Educaton Occupaton Marrage ratng Constant Those who are marred longer are about. tmes more lkely to have extramartal affars than those recently marred! 69 * ŷ Model Dscrmnaton: Goodness of Ft 0f ŷ 0.5 = f ŷ > 0.5 Compares the observed and predcted group membershps. Cases wth a cut value of 0.5 or greater are classfed as havng extramartal affars. Logt Ray Far Model Predcted 0 Observed 0 Goodness of Ft. The lkelhood (L): probablty of the observed results, gven the parameter estmates. 5/50 435/45 5/4 435/560 As L s a small number, <, we use tmes the log of L ( LL) A good model s one wth a hgh L of the observed results small value for LL. (If a model fts perfectly, L = ; LL = 0.) Model Dscrmnaton Usually compare wth the LL of a model wth only the constant. Model Dscrmnaton

13 Logt Ray Far Model Goodness of Ft The lkelhood (L): Usually compare wth the LL of a model wth only the constant. Model s good, f LL(wth all varables) < LL(wth only constant), Model Dscrmnaton Logt Ray Far Model Goodness of Ft Pseudo R : smlar to R. to quantfy the proporton of explaned varaton n the logstc regresson model.. Cox & Snell R and Nagelkerke R : (n SPSS) (a) Cox & Snell R : /N L(0) R CS = L(Max) 73 Model Dscrmnaton 74 (a) Cox & Snell R : Logt Ray Far Model Goodness of Ft /N L(0) R CS = L(Max) Logt Ray Far Model Goodness of Ft Model Dscrmnaton L(0) = L(wth only the constant): Constraned L(Max) = L(wth all varables): Unconstraned N = sample sze. Cannot acheve a maxmum value of. Model Dscrmnaton (b)nagelkerke R : where R R CS N = R max R CS = Cox & Snell R R max = [L(0)] /N 75 Logt Ray Far Model Logt Ray Far Model: Goodness of Ft Goodness of Ft Model Dscrmnaton Model Dscrmnaton No. of correct predctons = = 460 N (No. of observatons) = 60 Count R = (No. of correct predctons)/n 460/60 =

14 Logt Ray Far Model B Sex 0.80 Predcted Probabltes Age Marred years e Have chldren P( y = ) = + e or Relgous Educaton 0.0 P = ; Occupaton 0.03 z + e Marrage ratng Constant.377 Marredy Educat- Occup- marrag sex Age ears chldren relgous on aton ratng P PROBIT MODEL Another commonly used dstrbuton: the probt. Here the sgmod functon s the cumulatve standardzed normal dstrbuton. F() = Φ() = e π f() = φ() = e π < z < COMPARISON OF LOGIT AND PROBIT How do logt and probt models compare? Results qute smlar although the logstc dstrbuton has slghtly fatter tals Varance of the probt s (standard normal dstrbuton). For the logt t s π 3 COMPARISON OF LOGIT AND PROBIT How do logt and probt models compare? Amemya (98) : the relatonshp between probt and logt models : β probt = 0.65β logt β logt =.6β probt and 8 8 ESTIMATION OF BINARY PROBIT AND LOGIT MODELS The logt and probt are non-lnear. The parameters enter the regresson model n a non-lnear fashon. We can no longer use OLS. Hence the method of Maxmum Lkelhood. 4