Binary Dependent Variables. In some cases the outcome of interest rather than one of the right hand side variables is discrete rather than continuous

Transcription

1 Bnary Dependent Varables In some cases the outcome of nterest rather than one of the rght hand sde varables s dscrete rather than contnuous The smplest example of ths s when the Y varable s bnary so that t can take only 1 or 2 possble values (eg Pass/Fal, Proft/Loss, Wn/Lose) Representatng ths s easy just use a dummy varable on the left hand sde of your model, so that Becomes Y = β 0 + β 1 X 1 + β 2 X 2 + u (1) D = β 0 + β 1 X 1 + β 2 X 2 + u (2) Where D = 1 f the ndvdual (or frm or country etc) has the characterstc (eg Wn) = 0 f not (eg Lose) Equaton (2) can be estmated by OLS. When do ths t s called a lnear probablty model and can nterpret the coeffcents n the same way as wth other OLS models So dd /dx 1 = β 1 now gves the mpact of a unt change n the value of X 1 on the chances of belongng to the category coded D=1 (eg of wnnng) - hence the name lnear probablty model Example: Usng the dataset marks.dta can work out the chances of gettng a frst depend on class attendance and gender usng a lnear probablty model gen frst=mark>=70 /* frst set up a dummy varable that wll become the dependent varable */ tab frst frst Freq. Percent Cum Total So 39 students got a frst out of 118. If we summarse ths varable then the mean value of ths (or any) bnary varable s the proporton of the sample wth that characterstc c frst Varable Obs Mean Std. Err. [95% Conf. Interval] frst So n ths case 33% of the course got a frst class mark (.33 33%) To see what determnes ths look at the OLS regresson output 1

2 reg frst num_sems female Source SS df MS Number of obs = F( 2, 115) = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE =.4333 frst Coef. Std. Err. t P> t [95% Conf. Interval] num_sems female _cons Ths says that the chances of gettng a frst rse by 3.4 percentage ponts for every extra class attended and that, on average, women are 21 percentage ponts more lkely to get a frst, even after allowng for the number of classes attended. However, can show that OLS estmates when the dependent varable s bnary 1. wll suffer from heteroskedastcty, so that the t-statstcs are based 2. may not constran the predcted values to le between 0 and 1 (whch need f gong to predct behavour accurately) Usng the example above predct phat (opton xb assumed; ftted values). su phat Varable Obs Mean Std. Dev. Mn Max phat Can see that there are some negatve predctons whch s odd for a varable that s bnary. Note that not many of the predcted values are zero (and none are 1). Ths s not unusual snce the model s actually predctng the probablty of belongng to one of the two categores. Because of ths however t s better to use a model that explctly rather than mplctly models ths probablty and does not suffer from heteroskedastcty So that model Probablty (D =1) as a functon of the rght hand sde varables There are 2 alternatves 1. Logt Model Assumes that the Probablty model s gven by 1 Prob (D =1) = ( β 0+ β1x1+ β2x 2 ) 1+ e 2

3 2. The Probt Model Assumes that the Probablty model s gven by Prob (D =1) = ( β0 + β1x1+ β2x 2 ) 2 e 2π The dea s to fnd the values of the coeffcents that make ths probablty as close to the values n the dataset as possble. The technque s called maxmum lkelhood estmaton Example: Usng the marks.dta data set above the logt and probt equvalents of the OLS lnear probablty estmates above are, respectvely logt frst num_sems female Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Logstc regresson Number of obs = 118 LR ch2(2) = Prob > ch2 = Log lkelhood = Pseudo R2 = frst Coef. Std. Err. z P> z [95% Conf. Interval] num_sems female _cons probt frst num_sems female Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Probt regresson Number of obs = 118 LR ch2(2) = Prob > ch2 = Log lkelhood = Pseudo R2 = frst Coef. Std. Err. z P> z [95% Conf. Interval] num_sems female _cons The standard errors and t values on these varables should be free of the bas nherent n OLS though they could stll be subject to other types of heteroskedastcty so t s a good dea to use the, robust adjustment even wth logt and probt estmators 3

4 logt frst num_sems female, robust Iteraton 0: log pseudolkelhood = Iteraton 1: log pseudolkelhood = Iteraton 2: log pseudolkelhood = Iteraton 3: log pseudolkelhood = Iteraton 4: log pseudolkelhood = Logstc regresson Number of obs = 118 Wald ch2(2) = Prob > ch2 = Log pseudolkelhood = Pseudo R2 = Robust frst Coef. Std. Err. z P> z [95% Conf. Interval] num_sems female _cons (n ths example t make lttle dfference) However n both cases the estmated coeffcents look very dfferent from those of the OLS lnear probablty estmates. Ths s because they do not have the same nterpretaton as wth OLS (they are smply values that maxmse the lkelhood functon). To obtan coeffcents whch can be nterpreted n a smlar way to OLS, need margnal effects In the case of probt model ths s gven by δ Pr ob( T δx = 1) = β φ ( Z γ ) and for the logt model ths s gven by δ Pr ob( T δx = 1) = β exp( Z γ ) ( 1+ exp( Z γ )) = β Pr ( = 1) * (1 Pr ( = 1)) 2 ob T ob T In both cases the nterpretaton of these margnal effects s the mpact that a unt change n the varable X has on the probablty of belongng to the treatment group (just lke OLS coeffcents) 4

5 To obtan margnal effects n Stata run ether the logt or probt command then smply type logt frst num_sems female mfx Margnal effects after logt y = Pr(frst) (predct) = varable dy/dx Std. Err. z P> z [ 95% C.I. ] X num_sems female* (*) dy/dx s for dscrete change of dummy varable from 0 to 1 probt frst num_sems female mfx Margnal effects after probt y = Pr(frst) (predct) = varable dy/dx Std. Err. z P> z [ 95% C.I. ] X num_sems female* (*) dy/dx s for dscrete change of dummy varable from 0 to 1 (or wth probt you can also type dprobt frst num_sems female Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton 2: log lkelhood = Iteraton 3: log lkelhood = Iteraton 4: log lkelhood = Probt regresson, reportng margnal effects Number of obs = 118 LR ch2(2) = Prob > ch2 = Log lkelhood = Pseudo R2 = frst df/dx Std. Err. z P> z x-bar [ 95% C.I. ] num_sems female* obs. P pred. P (at x-bar) (*) df/dx s for dscrete change of dummy varable from 0 to 1 z and P> z correspond to the test of the underlyng coeffcent beng 0 These estmates are smlar to those of OLS (as they should be snce OLS, logt and probt are unbased) 5

6 Note also that the predcted values from the logt and probt regressons wll le between 0 and 1 predct phat_probt (opton p assumed; Pr(frst)) su phat_probt phat_logt Varable Obs Mean Std. Dev. Mn Max phat_probt phat_logt whle the means are the same the predctons are not dentcal for the two estmaton technques Goodness of Ft Tests The maxmum lkelhood equvalent to the F test of goodness of ft of the model as a whole s gven by the Lkelhood Rato (LR) test whch compares the value of the (log) lkelhood when maxmsed wth the lkelhood functon wll all coeffcents set to zero Can show that 2 [Log L max Log L 0 ] ~ χ 2 (k-1) where k s the number of rght hand sde varables ncludng the constant If the estmate ch-squared value exceeds the crtcal value then reject the null that the model as no explanatory power Ths value s gven n the top rght hand corner of the logt/probt output n Stata Can also use the LR test to test restrctons on subsets of the coeffcents n a smlar way. A maxmum lkelhood equvalent of the R 2 s the pseudo-r 2 = 1 (Log L max /Log L 0 ) Ths value les between 0 and 1 and the closer to one the better the ft of the model (agan ths value s gven n the top rght hand corner of the logt/probt output n Stata) 6