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1 Bnary Regresson Models ( 二分类变量模型.1 Bnary Outcomes bnary deendent varables have two values, tycally coded as 0 for a negatve outcome (.e., the event dd not occur and 1 as a ostve outcome (.e., the event dd occur. Bnary Regresson Models: exlore how each ndeendent varable affects the robablty of the event occurrng bnary logt model( 二分胜算对数模型 bnary robt model( 二分概率单元模型.3 The Statstcal Model.3.1 A Latent Varable Model ( 隐性变量模型 :assume a latent or unobserved varable y* rangng from - to - τ 0 y* y For examle ( 女性参与就业市场 y s observed n two states: a woman s n the labor force she s not n the labor force The dea of a latent y* s that: There s an underlyng roensty that generates the observed event We cannot drectly observe y*, but at some ont a change n y* results n a change n what we observe As the number of young chldren n the famly ncreases, a woman s roensty to work would decrease. At some ont the roensty crosses a threshold that results n a decson to leave the labor force 9

2 Structural model y * x β + ε Measurement model: the lnk between the observed bnary y and the latent y* s made wth a smle measurement equaton: * 1f y y * 0 f y > 0 0 τ s called the threshold ( 临界点 or cutont( 分界点 the dstrbuton of y* s shown by the bell-shaed curves whch should be thought of as comng out of the fgure nto a thrd dmenson when y* s larger than τ, ndcated by the shaded area, we observe y1. Snce y* s contnuous, the model avods the roblems encountered wth the LPM. However, snce the deendent varable s unobserved, the model cannot be estmated wth OLS. Instead, we use ML estmaton, whch requres assumtons about the dstrbuton of the errors. In general, the robt model and the logt model rovde smlar results excet that the robt model assumes that error terms have a normal dstrbuton and the logt model assumes that error terms have a logstc dstrbuton. the sze of the coeffcents of the logt model s 1.7 to 1.8 larger than those of the robt model 10

3 .3. A Nonlnear Probablty Model( 非线性概率模型 : The BRM can also be derved wthout aealng to latent varable through secfyng a nonlnear model relatng the xs to the robablty of an event Ω ( x Pr Pr ( y 1x ( y 0 x Pr 1 Pr ( y 1x ( y 1x snce the robablty s transformed nto odds( 胜算, 或概率比, t ndcates how often somethng haens (y1 relatve to how often t does not haen (y0, and the values range from 0 when Pr ( y 1 x 0 to when Pr ( y 1 x 1. The log ( 自然对数 of the odds, or logt( 胜算对数, ranges from - to. Ths suggests a model that s lnear n the logt: ( y 1x ( y 1x Pr ln Ω( x ln 1 Pr xβ β + β x + β x β k x k.3..1 什么是胜算 (odds? 胜算是某事件发生的概率与该事件不发生的概率的比率 胜算 ( Ω 13 e. g., 1 39 概率 ( f N e.g., 13 5 胜算 4, 发生的概率为 0.8, 不发生的概率为 0.; 发生的概率为不发生的概率的 4 倍 中文 : 五五波, 六四开 胜算与概率相似处 : 11

4 都以 0 为最低值, 无负数 当胜算与概率的值增大时, 都表示某事件愈可能发生 ; 概率愈大, 胜算愈大 ; 概率愈小, 胜算愈小 胜算与概率不同处 : 胜算以该事件不发生的概率为分母, 概率以所有发生与不发生的事件为分母 概率最大值为 1, 胜算可达无穷大, 消除上界为 1 的问题 概率趋近于 1 时, 概率少许的变动会对胜算有很大的影响 ( 胜算 odds >1, 发生的概率大于不发生的概率 0< odds <1, 发生的概率小于不发生的概率 odds 1, 发生的概率等于不发生的概率 胜算与概率的关系 : Ω Ω Ω 1 Ω Ω Ω 1+ Ω ( 1 Ω + Ω ( 1+ Ω.3.. 胜算比 (odds rato: 一个胜算可与另一个胜算来比较 Ω θ Ω

5 如果胜算比大于 1, 表示第一个事件组的胜算大于第二个事件组的胜算 胜算比愈大, 第一个事件组的胜算愈大 如果胜算比小于 1, 第一个事件组的胜算小于 第二个事件组的胜算.3..3 什么是自然胜算对数 (logt, the natural log of odds? 自然胜算对数就是将胜算 ( Ω 1 取自然对数, 即 ln Ω ( ln ( 胜算 自然胜算 对数 从正无穷大到负无穷大, 消除概率上下限为 0 的问题 以概率为 0.5 为界, 成对称型 等量的概率变化会造成自然胜算对数不等量的改变 总之, 胜算大于 1, 自然胜算对数为正值 胜算等于 1, 自然胜算对数为 0 胜算大于 0 但小于 1, 自然胜算对数为负值 自然胜算对数以 0. 5为界, 成对称型 概率为 1 或 0, 相对的自然胜算对数不存在 13

6 .3..4 二分胜算对数模型 Pr ( y 1x ln Ω( x ln 1 Pr( y 1x xβ β + β x + β x β logt transformaton lnearzes the nonlnear relatons the model s lnear n logt, the same change n x have constant effects the logt mles a nonlnear relatons between x and the robabltes 上式可在两边取指数函数 (exonent, 左侧以胜算表示依变量, 右侧为自变量 等式右侧呈现相乘关系, 表示依变量与自变量是非线性关系 所以, 二分胜算对数模型将变量间的非线性关系, 以自然胜算对数将之线性化 (lnear n logt k x k Pr 1 Pr ( y 1x ( y 1x β0 + β1x1 + βx βk x ex( xβ e e β 0 e β x 1 1 e β x... e β x k k Examle: Labor force artcaton of women (fle name: bnlf The samle conssts of 753 whte, marred women between the ages of 30 and 60. The deendent varable lf equals 1 f a woman s emloyed and 0 otherwse. The hyothess s that the robablty of a marred woman n the ad labor force s affected by a seres of factors, ncludng the number of younger chldren (K5, the number of older chldren (K618, the wfe s age (age, the wfe s educaton (wc, the husband s educaton (hc, log of wfe s estmated wage rate (lwg and famly ncome excludng wfe s wages (nc. The model to be estmated s lf 1 F( β + β K5 + β K618 + β age + β wc Pr( 0 k 5 k β hc + β lwg + β nc hc lwg nc age wc Estmaton wth STATA (a do-fle 14

7 *logt command, suressng teraton log and storng the results logt lf k5 k618 age wc hc lwg nc, nolog (or statstcs/bnary outcomes/logstc regresson estmates store logt (or statstcs/general ost-estmaton/manage estmaton results/store estmaton results *robt command, suressng teraton hstory by the nolog oton and storng the results robt lf k5 k618 age wc hc lwg nc, nolog (or statstcs/bnary outcomes/robt regresson estmates store robt *creatng a table that combnes the results estmates table logt robt, b(%9.3f se label varwdth(30 (or statstcs/general ost-estmaton/manage estmaton results/table of estmaton results Outut age. logt lf k5 k618 age wc hc lwg nc, nolog Logt estmates Number of obs 753 LR ch( Prob > ch Log lkelhood Pseudo R lf Coef. Std. Err. z P> z [95% Conf. Interval] k k age wc hc lwg nc _cons estmates store logt.. *robt command, suressng teraton hstory by the nolog oton and storng > the results. robt lf k5 k618 age wc hc lwg nc, nolog Probt estmates Number of obs 753 LR ch( Prob > ch Log lkelhood Pseudo R

8 lf Coef. Std. Err. z P> z [95% Conf. Interval] k k age wc hc lwg nc _cons estmates store robt. *creatng a table that combnes the results. estmates table logt robt, b(%9.3f t label varwdth( Varable logt robt # kds < # kds Wfe's age n years Wfe College: 1yes 0no Husband College: 1yes 0no Log of wfe's estmated wages Famly ncome excludng wfe's Constant legend: b/se.3.4 Hyothess Testng wth test and lrtest If the assumtons of the model hold, the ML estmators are dstrbuted asymtotcally normally and thus can be tested wth the corresondng z- statstcs n the outut. The t-test and z-test are the same when the N s large. The z-test ncluded n the outut of estmaton command s a Wald test, whch can be comuted usng test. For examle, to test H β 0, : 0 k 5 *sgnfcant test of a sngle coeffcent test k5 (or statstcs/general ost-estmaton/tests/tests arameters 16

9 Outut age.test k5 ( 1 k5 0 ch( Prob > ch The effect of havng young chldren on the robablty of enterng the labor force s sgnfcant at the.01 level test can be used to test multle coeffcents. For examle, to test H : β β 0, 0 wc hc *sgnfcant test of multle coeffcents test hc wc Outut age.test hc wc ( 1 hc 0 ( wc 0 ch( Prob > ch The hyothess that the effects of the husband s and the wfe s educaton are smultaneously equal to zero can be rejected at the.01 level Comarng comettve (nested models usng LR test: the LR test assesses a hyothess by comarng the log lkelhood from the full model (M U (.e., the model that does not nclude the constrants mled by H 0 and a restrcted model (M C that moses the constrants. If the constrants sgnfcantly reduce the log lkelhood, the H 0 (that the mosed constrants are true s rejected. Consder the followng models: M1: Pr( y 1 x Λ( β 0 + β1x1 + β x + β 3x3 + β 4 x4 M: Pr( y 1 x Λ( β 0 + β1x1 + β x M3: Pr( y 1 x Λ( β 0 + β x + β 4 x4 17

10 The LR test can be defned as follows: 1. M c wth arameters β c s nested n M u wth β u.. H 0 s that the constrants mosed to create M c are true. 3. L(M u s the value of the lkelhood functon at the unconstraned estmates. 4. L(M c s the value at the constraned estmated. 5. The lkelhood rato statstc equals G ( M C M U ln L( M U ln L( M C 6. Under very general condtons, f H 0 s true, then G s asymtotcally dstrbuted as ch-square wth degrees of freedom equal to the number of ndeendent constrants Assume that a restrcted model (wthout k5 and k618, or H : β 0 k β 5 k s tested aganst a full model *comarng nested models usng an LR test logt lf k5 k618 age wc hc lwg nc, nolog estmates store fmodel logt lf age wc hc lwg nc, nolog estmates store rmodel lrtest fmodel rmodel (or statstcs/general ost-estmaton/tests/lkelhood-rato test Outut age. *comarng nested models usng an LR test. logt lf k5 k618 age wc hc lwg nc, nolog (outut omtted. estmates store fmodel.. logt lf age wc hc lwg nc, nolog (outut omtted. estmates store rmodel.. lrtest fmodel rmodel lkelhood-rato test LR ch( (Assumton: rmodel nested n fmodel Prob > ch

11 Snce the LR ch-square s statstcally sgnfcant, we reject the null hyothess that the coeffcents of k5 and k618 are smultaneously equal to zero (or, reject H : β 0 0 k β 5 k 618 When conductng LR test: 1 the two models must be nested (.e., a nested model s created by mosng constrants on the coeffcents n the ror model, and the two models must be ftted on exactly the same samle..4 Interretaton: In general, the estmated arameters from the BRM do not rovde drectly useful nformaton for understandng the relatonsh between the deendent and ndeendent varables. Substantvely meanngful nterretatons are thus based on redcted robabltes and functons of those robabltes (e.g., ratos, dfferences. There are four aroaches: - Predctons can be comuted for each observaton n the samle usng redct - Predcted values for substantvely meanngful rofles of the ndeendent varables can be comared usng rvalue or rtab - The margnal or dscrete change n the outcome can be comuted at reresentatve value of the ndeendent varables usng rchange - The nonlnear model can be transformed to a model that s lnear n some other outcome and lstcoef can be used.4.1 Interretaton wth Predcted Probablty usng redct: *logt command, suressng teraton log logt lf k5 k618 age wc hc lwg nc, nolog redct rlogt (or statstcs/general ost-estmaton/obtan redctons, resduals robt lf k5 k618 age wc hc lwg nc, nolog redct rrobt summarze r* *Correlatng rlogt and rrobt wcorr rlogt rrobt Outut age. *logt command, suressng teraton log. logt lf k5 k618 age wc hc lwg nc, nolog 19

12 Logt estmates Number of obs 753 LR ch( Prob > ch Log lkelhood Pseudo R lf Coef. Std. Err. z P> z [95% Conf. Interval] k k age wc hc lwg nc _cons redct rlogt (oton assumed; Pr(lf. robt lf k5 k618 age wc hc lwg nc, nolog Probt estmates Number of obs 753 LR ch( Prob > ch Log lkelhood Pseudo R lf Coef. Std. Err. z P> z [95% Conf. Interval] k k age wc hc lwg nc _cons redct rrobt (oton assumed; Pr(lf. summarze r* Varable Obs Mean Std. Dev. Mn Max rlogt rrobt

13 redct can be used to examne the range of redcted robabltes from the model. For nstance, after storng the redcted robabltes n the corresondng varables, rlogt and rrobt, summarze shows the range of the redcted robabltes are from.014 to.96 and.006 to.974 resectvely. redct can also be used to demonstrate that the redctons from logt and robt models are essentally dentcal. As the followng outut shows, Outut age. *Correlatng rlogt and rrobt. wcorr rlogt rrobt rlogt rrobt rlogt rrobt Indvdual Predcted Probabltes wth rvalue: a table of robabltes for deal tyes of eole (or countres, cows, etc. can quckly summarze the effects of key varables. E.g., n the examle of labor force artcaton Young, low ncome and low educaton famles wth young chldren Hghly educated, mddle aged coules wth no chldren An average famly defned as havng the mean on all varables *logt command, suressng teraton log logt lf k5 k618 age wc hc lwg nc, nolog *Indvdual Predcted Probabltes wth rvalue *Young, low ncome and low educaton famles wth young chldren rvalue, x(age35 k5 wc0 hc0 nc15 rest(mean (or statstcs/general ost-estmaton/table of adjusted means and roortons *Hghly educated, mddle aged coules wth no chldren rvalue, x(age50 k50 k6180 wc1 hc1 rest(mean *An average famly defned as havng the mean on all varables rvalue, rest(mean Outut age *Young, low ncome and low educaton famles wth young chldren. rvalue, x(age35 k5 wc0 hc0 nc15 rest(mean 1

14 logt: Predctons for lf Pr(ynLF x: % c: (0.073,0.8 Pr(yNotInLF x: % c: (0.7718,0.977 k5 k618 age wc hc lwg nc x *Hghly educated, mddle aged coules wth no chldren. rvalue, x(age50 k50 k6180 wc1 hc1 rest(mean logt: Predctons for lf Pr(ynLF x: % c: (0.666,0.791 Pr(yNotInLF x: % c: (0.079, k5 k618 age wc hc lwg nc x *An "average famly" defned as havng the mean on all varables. rvalue, rest(mean logt: Predctons for lf Pr(ynLF x: % c: (0.5388, Pr(yNotInLF x: % c: (0.3841,0.461 k5 k618 age wc hc lwg nc x Wth these redctons, we can summarze the results on factors affectng a wfe s labor force artcaton Table of Predcted Probabltes at Selected Values Ideal Tyes Probabltes of LFP Young, low ncome and low educaton famles.013 wth young chldren Hghly educated, mddle aged coules wth no chldren An average famly defned as havng the mean on all varables.7.58

15 .4.3 Indvdual Predcted Probabltes wth rtab: f the focus s one two or three categorcal ndeendent varables. Predctons for all combnatons of the categores of these varables could be resented as Table of Predcted Probabltes at Selected Values Number of Young Chldren The above table s created by rtab k5 wc, rest(mean Predcted Probablty Dd not Attend Attended College Dfference Outut age. *Indvdual Predcted Probabltes wth rtab. rtab k5 wc, rest (mean logt: Predcted robabltes of ostve outcome for lf Wfe College: # kds < 1yes 0no 6 NoCol College k5 k618 age wc hc lwg nc x Changes n Predcted Probabltes wth rchange: ths command comutes the margnal/dscrete change at the values of the ndeendent varables secfed wth x( or rest(. For nstance, *comutng margnal change wth all varables at ther mean rchange, rest (meanhel *the above command can be sulemented by fromto oton 3

16 rchange, fromto *(outut omtted Outut age rchange, rest (mean hel logt: Changes n Predcted Probabltes for lf mn->max 0->1 -+1/ -+sd/ MargEfct k k age wc hc lwg nc NotInLF nlf Pr(y x k5 k618 age wc hc lwg nc x sd(x Pr(y x: robablty of observng each y for secfed x values Avg Chg : average of absolute value of the change across categores Mn->Max: change n redcted robablty as x changes from ts mnmum to ts maxmum 0->1: change n redcted robablty as x changes from 0 to 1 -+1/: change n redcted robablty as x changes from 1/ unt below base value to 1/ unt above -+sd/: change n redcted robablty as x changes from 1/ standard dev below base to 1/ standard dev above MargEfct: the artal dervatve of the redcted robablty/rate wth resect to a gven ndeendent varable varyng age (age from ts mnmum of 30 to ts maxmum of 60 decreases the redcted robablty by.44 changng famly ncome (nc from ts mnmum to ts maxmum decreases the robablty of a woman beng n the labor force by.64 for a woman who s average on all characterstcs, an addtonal young chld decreases the robablty of emloyment by.348 f a woman attends college, her robablty of beng n the labor force s.1881 greater than a woman who does not attend college, holdng all other varables at ther mean..4.5 Interretaton usng Odds Ratos ( 胜算比 wth lstcoef: - For bnary outcomes, we tycally consder the odds of observng a ostve outcome versus a negatve one: 4

17 Ω Pr( y 1 Pr( y 0 Pr( y 1 1 Pr( y 1 The log of odds s logt and the model s lnear n the logt Pr( y 1 ln Ω ( x ln β + β x + β x β Pr( y 1 - Odds Rato Takng the exonental ( 指数 of the logt equaton: β0 + β1x1 + βx βk x Ω( x ex ln Ω( x ex( xβ e e β 0 e β1x1 e βx [ ]... e β k xk Let x change by 1: k x k Ω( x, x e β 0 e β1x1 + 1 ex( xβ e e βx e β... e β k xk β0 + β1x1 + β ( x βk x To comare the odds before and after the change of x, we take odds rato: β0 β Ω( x, x + 1 e e e β0 Ω( x, x e e β e... e x βk... e 1x1 βx βk xk β e β1x1 β xk e For a unt change n x k, the odds are exected to change by a factor of ex(β k, holdng all other varables constant For ex(β k >1, the odds are ex(β k tmes larger For ex(β k <1, the odds are ex(β k tmes smaller The effect does not deend on the level of x k or the level of any other varable. Factor change ( 倍数改变 n odds ratos for both a unt change and a standard devaton change of the ndeendent varables can be obtaned wth lstcoef: * obtanng factor change n odds ratos lstcoef, hel 5

18 Outut age * obtanng factor change n odds ratos lstcoef, hel logt (N753: Factor Change n Odds Odds of: nlf vs NotInLF lf b z P> z e^b e^bstdx SDofX k k age wc hc lwg nc b raw coeffcent z z-score for test of b0 P> z -value for z-test e^b ex(b factor change n odds for unt ncrease n X e^bstdx ex(b*sd of X change n odds for SD ncrease n X SDofX standard devaton of X For each addtonal young chld, the odds of beng emloyed decrease by a factor of 0.3, holdng all other varables constant ( 在其它变项不变的情形下, 每增加一个小孩, 已婚妇女参与就业市场的胜算 ( 或概率比 就会减少 0.3 倍. For a standard devaton ncrease n the log of the wfe s exected wages, the odds of beng emloyed are 1.43 tmes greater, holdng all other varables constant. The nterretaton of the odds rato assumes that the other varables are held constant, but t does not requre that they be held at any secfc values. It s mortant to know that a constant factor change n the odds does not corresond to a constant change or constant factor change n the robablty. For examle, f the odds are 1/1 (robablty.5 and double to /1 (robablty0.333, the robablty ncreases by Percent change n odds rato can also be obtaned wth lstcoef: { ex( δ 1} β k 100 whch s lsted by lsfcoef wth the ercent oton 6

19 Outut age lstcoef, ercent logt (N753: Percentage Change n Odds Odds of: nlf vs NotInLF lf b z P> z % %StdX SDofX k k age wc hc lwg nc For each addtonal young chld, the odds of beng emloyed decrease by 76.8%, holdng all other varables constant ( 在其它变项不变的情形下, 每增加一个小孩, 已婚妇女参与就业市场的胜算 ( 或概率比 就会减少 76.8% For a standard devaton ncrease n the log of the wfe s exected wages, the odds of beng emloyed ncreased 4.7%, holdng all other varables constant.5 Resduals and Influence Usng Predct.5.1 Detectng Resduals ( 残差值 Resduals are the dfference between a model s redcted and observed outcome for each observaton n the samle. When the dfferences are large, the observatons are consdered outlers( 离群值. For a bnary model, defne π The devaton E( y x Pr( y 1 x1 y π are heteroscedaststc, wth Var( y π x Var( y x π (1 π Whch suggests the Pearson resdual r y ˆ π (1 ˆ π π 7

20 Large values of r suggest a falure of the model to ft a gven observaton Pregbon roosed the standardzed Pearson resdual: r std r Var ( r An ndex lot s a useful way to examne resduals by smly lottng resduals aganst the observaton number. There s no hard-and-fast rule for what counts as a large resdual. One way to search for roblematc resduals s to sort the resduals by the value of a varable that you thnk may be a roblem for the model. If ths varable s rmarly resonsble for the lack of ft of some observatons, the lot would show a dsroortonate number of cases wth large resduals. The examle below sorts the data by ncome before lottng. *comute standardzed resduals and lottng them aganst ndex logt lf k5 k618 age wc hc lwg nc, nolog redct rstd, rs (or statstcs/general ost-estmaton/obtan redctons, resduals sort nc generate ndex_n grah twoway scatter rstd ndex, msymbol(none mlabel(ndex lst rstd ndex f rstd>.5 rstd<-.5 Outut age. //Resduals and nfluence usng redct. logt lf k5 k618 age wc hc lwg nc, nolog (outut omtted. redct rstd, rs. sort nc. generate ndex_n. grah twoway scatter rstd ndex, msymbol(none mlabel(ndex 8

21 standardzed Pearson resdual ndex The above cases can be lsted usng the command lst.. lst rstd ndex f rstd>.5 rstd< rstd ndex Detectng Influental Cases( 重要观察值 Influental cases are also called hgh-leverage onts, whch have a strong nfluence on the estmated arameters. They can be determned by examnng the change n the estmated βˆ that occurs when the th observaton s deleted. Ths s the counterart to Cook s dstance for the lnear regresson model, whch s the dbeta n Stata and s defned as C r h ( 1 h * Detectng nfluental cases redct cook, dbeta (or statstcs/general ost-estmaton/obtan redctons, resduals 9

22 generate ndex_n grah twoway scatter cook ndex, msymbol(none mlabel(ndex The commands roduce the followng lot, whch shows that cases 338, 534, and 635 mert further examnaton: Pregbon's dbeta ndex Scalar Measures of Ft ( 适合度 usng ftstat A scalar measure of ft can be used to summarze the overall ft of a model. A scalar measure of ft can be useful n comarng cometng models. Wthn a substantve area, measures of ft rovde a rough ndex of whether a model s adequate. However, there s no convncng evdence that selectng a model that maxmzes the value of a gven measure of ft results n a model that s otmal n any sense other than the model havng a larger value of that measure. Measures of fts thus only rovde artal nformaton that must be assessed wthn the context of the theory motvatng the analyss..6.1 Pseudo-R Based on R n the LRM Several seudo-r s for models wth CDVs have been defned by analogy to the formula for R n the LRM: McFadden s R, also know as the lkelhood-rato ndex, comares a model wth just the ntercet to a model wth all arameters. It s defned as 30

23 R McF 1 ln Lˆ( M ln Lˆ( M Full Intercet Maxmum Lkelhood R R ML L( M 1 L( M Intercet Full N Cragg & Uhler s R 1 { L( M L( M } Intercet Full R C& U N 1 L( M Intercet Efron s R N R Efron ( y ˆ π 1 ( y y.6. Informaton Measures AIC: Akake s nformaton crteron s defned as AIC { ln L ˆ( M k + P} N. Other thngs beng equal, the model wth smaller AIC s consdered the better-fttng model. BIC: The Bayesan nformaton crteron s defned as BICk D( M k df k ln N. The more negatve the BIC, the better the ft. When comarng two models, f BIC 1 - BIC <0, then the frst model s referred. If BIC 1 - BIC >0, then the second model s referred. Examle: Consder two models, M 1 has the orgnal secfcaton of ndeendent varables M adds a squared age term AGE and dros the varables K618, HC and LWG 31

24 * Scalar Measures of Ft usng ftstat gen ageage*age logt lf k5 k618 age wc hc lwg nc, nolog ftstat, save logt lf k5 age age wc nc, nolog ftstat, df Outut age. * Scalar Measures of Ft usng ftstat. quetly logt lf k5 k618 age wc hc lwg nc, nolog. estmates store model1. ftstat, save Measures of Ft for logt of lf Log-Lk Intercet Only: Log-Lk Full Model: D(745: LR(7: Prob > LR: McFadden's R: 0.11 McFadden's Adj R: Maxmum Lkelhood R: 0.15 Cragg & Uhler's R: 0.04 McKelvey and Zavona's R: 0.17 Efron's R: Varance of y*: 4.03 Varance of error: 3.90 Count R: Adj Count R: 0.89 AIC: 1.3 AIC*n: BIC: BIC': (Indces saved n matrx fs_0. gen ageage*age. quetly logt lf k5 age age wc nc, nolog. estmates store model. estmates table model1 model, b(%9.3f t Varable model1 model k k age wc hc lwg nc age _cons legend: b/t 3

25 . ftstat, df Measures of Ft for logt of lf Current Saved Dfference Model: logt logt N: Log-Lk Intercet Only: Log-Lk Full Model: D: ( ( ( LR: ( ( ( Prob > LR: McFadden's R: McFadden's Adj R: Maxmum Lkelhood R: Cragg & Uhler's R: McKelvey and Zavona's R: Efron's R: Varance of y*: Varance of error: Count R: Adj Count R: AIC: AIC*n: BIC: BIC': Dfference of n BIC' rovdes ostve suort for saved model. Note: -value for dfference n LR s only vald f models are nested. All seudo-r s are slghtly larger for M 1 Both AIC and BIC statstcs are smaller for M 1, whch rovdes suort for that model 33