ECONOMICS 452* -- Stata 9 Tutorial 6. Stata 9 Tutorial 6. TOPIC: Estimating and Interpreting Probit Models with Stata: Introduction

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1 ECONOMICS 45* -- Stata 9 utoral 6 Stata 9 utoral 6 OPIC: Estmatng and Interpretng Probt Models wth Stata: Introducton DAA: mroz.raw (a text-format (ASCII) data fle) ASKS: Stata 9 utoral 6 demonstrates how to use Stata to compute maxmum lkelhood (ML) estmates of probt models for bnary dependent varables. It ntroduces you to hypothess testng n probt models, n partcular to testng that the margnal ndex effects (and hence the margnal probablty effects) of ndvdual explanatory varables are zero. Fnally, t demonstrates how to compute the margnal probablty effects of both contnuous and bnary explanatory varables n probt models. he Stata commands that consttute the prmary subject of ths tutoral are: probt dprobt test lncom Used to compute ML estmates of probt coeffcents n probt models of bnary dependent varables. Used to compute ML estmates of the margnal probablty effects of explanatory varables n probt models. Used after probt estmaton to compute Wald tests of lnear coeffcent equalty restrctons on probt coeffcents. Used after probt estmaton to compute and test the margnal effects of ndvdual explanatory varables. he Stata statstcal functons used n ths tutoral are: normalden(z) Computes value of the standard normal densty functon (p.d.f.) for a gven value z of a standard normal random varable. normal(z) Computes value of the standard normal dstrbuton functon (c.d.f.) for a gven value z of a standard normal random varable. nvnormal(p) Computes the nverse of the standard normal dstrbuton functon (c.d.f.); f normal(z) = p, then nvnormal(p) = z. NOE: Stata commands are case senstve. All Stata command names must be typed n the Command wndow n lower case letters. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 1 of 39 pages

2 ECONOMICS 45* -- Stata 9 utoral 6 Preparng for Your Stata Sesson Before begnnng your Stata sesson, you wll need to do two thngs. 1. Download the text-format data fle mroz.raw and ts descrpton from the course web ste. If necessary, consult Stata 9 utoral 1 to refresh your memory on how to do ths.. Use Wndows Explorer to copy the text-format data fle mroz.raw to the Stata workng drectory on the C:-drve or D:-drve of the computer at whch you are workng. On the computers n Dunnng 350, the default Stata workng drectory s usually C:\DAA or C:\Stata9. On the computers n MC B109/B111, the default Stata workng drectory s usually D:\courses. Start Your Stata Sesson o start your Stata sesson, double-clck on the Intercooled Stata con on the Wndows desktop. After you double-clck the Intercooled Stata con, you wll see the famlar screen of four Stata wndows. Record Your Stata Sesson and Stata Commands -- log usng, cmdlog usng o record your Stata sesson, ncludng all the Stata commands you enter and the results (output) produced by these commands, make a text-format.log fle named 45tutor6.log. o open (begn) the log fle 45tutor6.log, enter n the Command wndow: log usng 45tutor6.log hs command opens a text-format (ASCII) fle called 45tutor6.log n the current Stata workng drectory. ECON 45* -- Fall 007: Stata 9 utoral 6 Page of 39 pages

3 ECONOMICS 45* -- Stata 9 utoral 6 Note: It s mportant to nclude the.log fle extenson when openng a log fle; f you do not, your log fle wll be n smcl format, a format that only Intercooled Stata can read. Once you have opened the 45tutor6.log fle, a copy of all the commands you enter durng your Stata sesson and of all the results they produce s recorded n that 45tutor6.log fle. o record only the Stata commands you type durng your Stata sesson, use the Stata cmdlog usng command. o start (open) the command log fle 45tutor6.txt, enter n the Command wndow: cmdlog usng 45tutor6 hs command opens a plan text-format (ASCII) fle called 45tutor6.txt n the current Stata workng drectory. All commands you enter durng your Stata sesson are recorded n ths fle. Load the ext-format Data Fle mroz.raw nto Stata -- nfle Before startng your Stata sesson, you downloaded the text-format (or ASCII-format) data fle mroz.raw from the ECON45 web ste and placed t n a drectory or folder on the C:-drve of your computer. hs secton demonstrates how to nput that data fle nto Stata usng the nfle command. o load the text-format data fle mroz.raw nto memory, type n the Command wndow ether of the followng two commands: or nfle nlf hours kdslt6 kdsge6 age educ wage repwage hushrs husage huseduc huswage famnc mtr motheduc fatheduc unem cty exper nwfenc lwage expersq usng mroz.raw nfle nlf hours kdslt6 kdsge6 age educ wage repwage hushrs husage huseduc huswage famnc mtr motheduc fatheduc unem cty exper nwfenc lwage expersq usng mroz he data fle mroz.raw contans 753 observatons on varables. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 3 of 39 pages

4 ECONOMICS 45* -- Stata 9 utoral 6 Usng the varable defntons n the descrpton fle mroz.des, you may wsh to assgn varable labels to the varables n the data set. Use the label varable command for ths purpose. o refresh your memory on the label varable command, consult Stata 9 utoral 1. o save some keystrokes n ths exercse, I have renamed some of the varables n the orgnal mroz.raw data fle. In partcular, I have used the followng new varable names: ed = educ; exp = exper; expsq = expersq. Use the rename command to assgn the new names ed, exp and expsq to the orgnal varable names educ, exper and expersq, respectvely. Enter the followng rename commands: rename educ ed rename exper exp rename expersq expsq o save the data n memory as a Stata-format dataset named mroz, enter the followng save command: save mroz hs command saves on dsk n the Stata workng drectory the Stata-format dataset mroz.dta, but leaves unchanged the orgnal text-format data fle mroz.raw. Famlarze yourself wth the current data set -- descrbe and summarze o summarze the contents of the current dataset, use the descrbe command. Recall from Stata 9 utoral 1 that the descrbe command dsplays a summary of the contents of the current dataset n memory, whch n ths case s the Stata-format data fle mroz.dta. Enter the command: descrbe o compute summary statstcs for the varables n the current dataset, use the summarze command. Recall from Stata 9 utoral 1 that the summarze command computes descrptve summary statstcs for all numerc varables n the current dataset n memory. Enter the command: summarze ECON 45* -- Fall 007: Stata 9 utoral 6 Page 4 of 39 pages

5 ECONOMICS 45* -- Stata 9 utoral 6 o dsplay summary statstcs only for the varables that are used n ths tutoral, enter the command: summarze nlf nwfenc ed exp expsq age kdslt6 kdsge6 Note that the varable nlf s a bnary varable that takes only the two values 0 and 1. It s the dependent varable n the probt models estmated n ths tutoral. o obtan more detaled summary statstcs on the varables used n ths tutoral, enter the followng summarze command wth the detal opton: summarze nlf nwfenc ed exp expsq age kdslt6 kdsge6, detal Maxmum Lkelhood Estmaton of Probt Models -- probt A Generc Probt Model * Y = x + u = + X + X + L + X + u (1) where: * Y = a contnuous real-valued ndex varable for observaton that s unobservable, or latent; k k x = = ( 1 X1 X L Xk ), a 1 K row vector of regressor values for observaton ; 0 1 L k ) (, a K 1 column vector of regresson coeffcents; u = an d N(0, σ ) random error term for observaton. he observable outcomes of the bnary choce problem are represented by a bnary * ndcator varable Y that s related to the unobserved dependent varable Y as follows: Y = 1 f * Y > 0 (.1) Y = 0 f * Y 0 (.) ECON 45* -- Fall 007: Stata 9 utoral 6 Page 5 of 39 pages

6 ECONOMICS 45* -- Stata 9 utoral 6 he random ndcator varable Y represents the observed realzatons of a bnomal process wth the followng probabltes: * > Pr( Y = 1) = Pr(Y > 0) = Pr(x + u 0) (3.1) * Pr( Y = 0) = Pr(Y 0) = Pr(x + u 0) (3.) * * he bnomal probabltes Pr( Y = 1) = Pr( Y 0) and Pr( Y 0) = Pr( Y 0) > = are represented n terms of the c.d.f. (cumulatve dstrbuton functon) for the random varable u as follows: * Pr( Y = 1) = Y > = Φ( ) * Pr( Y = 0) = Pr( Y 0) = Φ( x ) where Φ( ) Model 1 x (4.1) 1 (4.) x s the c.d.f. for the standard normal dstrbuton. he followng s a model of marred women's labour force partcpaton, where the observed bnary dependent varable nlf s defned as follows: nlf = 1 f the -th marred woman s n the employed labour force = 0 f the -th marred woman s not n the employed labour force he probt ndex functon, or regresson functon, for Model 1 s: x = 0 + 1nwfenc + ed + 3 exp + 4 exp + 5age + 6kdslt6 + 7kdsge6 where nwfenc ed exp age = non-wfe famly ncome of the -th woman (n thousands of dollars per year); = years of formal educaton of the -th woman (n years); = years of actual work experence of the -th woman (n years); = age of the -th woman (n years); ECON 45* -- Fall 007: Stata 9 utoral 6 Page 6 of 39 pages

7 ECONOMICS 45* -- Stata 9 utoral 6 kdslt6 kdsge6 = number of chldren less than 6 years of age for the -th woman; = number of chldren 6 years of age or older for the -th woman. hs bnary dependent varable model s used extensvely as an example n the Wooldrdge textbook. See Wooldrdge (003): Sec. 7.5, pp ; Example 8.8, pp ; and Example 17.1, pp he ML probt coeffcent estmates ˆ are the partal dervatves of the estmated probt ndex functon ˆ wth respect to the ndvdual regressors: x ˆ j = x ˆ X j he estmated probt ndex functon for Model 1 s wrtten n general as: x ˆ ˆ ˆ ˆ ˆ ˆ = 0 + 1nwfenc + ed + 3 exp + 4 exp + 5age + 6kdslt6 + 7kdsge6 ˆ ˆ ˆ he ML probt coeffcent estmates for Model 1 are: ˆ ˆ x 1 = ; nwfenc ˆ ˆ x 5 = ; age ˆ ˆ x = ; ed ˆ ˆ x 6 = ; kdslt6 ˆ ˆ x 3 = ; ˆ x 4 = exp exp ˆ ˆ x 7 =. kdsge6 ˆ Notce that there s somethng very odd about the above nterpretaton of the probt coeffcent estmates ˆ 3 and ˆ 4. What does t mean to vary exp whle holdng exp constant, and vce versa? he answer s that the margnal ndex effect of the explanatory varable exp n Model 1 s not a constant, and therefore s not estmated by a sngle slope coeffcent estmate. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 7 of 39 pages

8 ECONOMICS 45* -- Stata 9 utoral 6 Margnal Index Effects of Contnuous Explanatory Varables Model 1 he margnal ndex effects of the ndvdual explanatory varables on the latent * (unobserved) dependent varable Y --.e., on the probt ndex functon x -- equal ther respectve probt coeffcents only for contnuous varables that enter the ndex functon lnearly. In Model 1, all the explanatory varables except exp enter the probt ndex functon lnearly. For these explanatory varables, the margnal ndex effects equal ther respectve probt coeffcents: x 1. margnal ndex effect of nwfenc = = 1 nwfenc x. margnal ndex effect of ed = = ed x 3. margnal ndex effect of age = = 5 age x 4. margnal ndex effect of kdslt6 = = 6 kdslt6 x 5. margnal ndex effect of kdsge6 = = 7 kdsge6 But the explanatory varable exp enters the ndex functon x nonlnearly. Consequently, the margnal ndex effect of exp does not correspond to a sngle slope coeffcent. In Model 1, the margnal ndex effect of exp s a lnear functon of exp wth coeffcents 3 and 4 : x 6. margnal ndex effect of exp = = exp exp ECON 45* -- Fall 007: Stata 9 utoral 6 Page 8 of 39 pages

9 ECONOMICS 45* -- Stata 9 utoral 6 Margnal Probablty Effects of Contnuous Explanatory Varables Model 1 he margnal probablty effects of contnuous explanatory varables n probt models are the partal dervatves of the standard normal c.d.f. Φ( x ) wth respect to the ndvdual explanatory varables: Φ margnal probablty effect of X j ( x ) ( ) = Φ x X j = x x X j = φ ( x ) x X j where ( ) φ x = the value of the standard normal p.d.f. at x X j = the margnal ndex effect of X j In other words, the margnal probablty effect of a contnuous explanatory varable φ x, the value of the X j s smply the margnal ndex effect of X j multpled by ( ) x standard normal p.d.f. evaluated at. x he margnal probablty effects of the explanatory varables n Model 1 are: Φ 1. margnal probablty effect of nwfenc ( x ) = nwfenc Φ. margnal probablty effect of ed ( x ) = = φ ( x ) ed Φ 3. margnal probablty effect of age ( x ) = = φ ( x ) 5 age Φ 4. margnal probablty effect of kdslt6 ( x ) = kdslt6 Φ 5. margnal probablty effect of kdsge6 ( x ) = kdsge6 = φ ( x ) 1 = φ ( x ) 6 = φ ( x ) 7 ECON 45* -- Fall 007: Stata 9 utoral 6 Page 9 of 39 pages

10 ECONOMICS 45* -- Stata 9 utoral 6 Φ 6. margnal probablty effect of exp ( x ) = exp = φ ( x ) ( + exp ) 3 4 Note that the margnal probablty effect of a contnuous explanatory varable X j s proportonal to the margnal ndex effect of X j, where the factor of proportonalty s φ x evaluated at : the standard normal p.d.f. ( ) margnal probablty effect of X j = φ( ) x x margnal ndex effect of X j Note too that estmaton of the margnal probablty effects of a contnuous explanatory varable X j requres one to choose a specfc vector of regressor values x. Common choces for x are the sample mean and sample medan values of the regressors. Maxmum Lkelhood Estmaton of Probt Models -- probt o compute ML estmates of probt Model 1 and dsplay the full set of saved results, enter the followng commands: probt nlf nwfenc ed exp expsq age kdslt6 kdsge6 ereturn lst o save as a column vector the ML probt coeffcent estmates ˆ for Model 1 and dsplay the results, enter the followng matrx commands: matrx bhat = e(b)' matrx lst bhat Note that the vector of ML probt coeffcent estmates ˆ s dsplayed n Stata format wth the ntercept coeffcent estmate as the last, rather than the frst, element. For Model 1, ˆ s dsplayed n Stata format as ( ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) Τ ˆ = ECON 45* -- Fall 007: Stata 9 utoral 6 Page 10 of 39 pages

11 ECONOMICS 45* -- Stata 9 utoral 6 estng margnal effects of contnuous explanatory varables -- test and lncom Computng and estng Margnal Index Effects he test command can be used after a probt estmaton command to compute large sample Wald tests of the margnal ndex effects of ndvdual explanatory varables. he Wald statstc computed by the test command after a probt command has the followng form: W ML = 1 ( Rˆ r) ( RVˆ R ) ( Rˆ r) ML ML ML where ˆ ML = the ML estmator of the probt coeffcent vector ; Vˆ ML = Vˆ ML( ˆ ML X ) = the ML estmator of VML ( ˆ ML X ) covarance matrx of ˆ ML., the varance- he null dstrbuton of the W statstc s a ch-square dstrbuton wth degrees of freedom equal to q, the number of lnear restrctons specfed by H 0 : a WML ~ χ [q] under H 0, where q = number of restrctons specfed by H 0. he lncom command can be used after a probt estmaton command to compute the estmated values of the margnal ndex effects of ndvdual explanatory varables, and to perform two-tal asymptotc t-tests of the null hypothess that these margnal ndex effects equal zero. he asymptotc null dstrbuton of the t-statstc s N(0, 1), the standard normal dstrbuton (whch s what the t dstrbuton converges to as degrees of freedom, or sample sze, becomes ndefntely large): t a ~ N(0, 1) under H 0. In Stata, the sample value of the t-statstc produced by the lncom command (and by the probt estmaton commands) s labelled as z, but s computed n exactly the same way as any t-statstc. hs secton presents some examples of hypothess tests on the estmated probt ndex functon of Model 1. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 11 of 39 pages

12 ECONOMICS 45* -- Stata 9 utoral 6 est 1 - Model 1: Proposton: he non-wfe ncome of the famly has no effect on marred women's labour force partcpaton; the margnal ndex effect and margnal probablty effect of nwfenc equal zero. Margnal Index and Probablty Effects: x margnal ndex effect of nwfenc = = 1 nwfenc Φ margnal probablty effect of nwfenc ( x ) = nwfenc = φ ( x ) 1 A suffcent condton for both the margnal ndex and margnal probablty effects of nwfenc to equal zero for all marred women s 1 = 0. Null and Alternatve Hypotheses: H 0 : 1 = 0 H 1 : 1 0 o calculate a Wald test of ths hypothess and the correspondng p-value for the calculated W-statstc, enter the followng test and return lst commands: test nwfenc or test nwfenc = 0 return lst dsplay sqrt(r(ch)) o calculate a two-tal asymptotc t-test of H 0 aganst H 1, enter the followng lncom and return lst commands: lncom _b[nwfenc] or lncom nwfenc return lst Note that the lncom command uses ether _b[nwfenc] or nwfenc to refer to the ML probt coeffcent estmate ˆ 1 of the regressor nwfenc. Compare the results of ths two-tal t-test wth those of the prevous Wald test. You should be able to explan why the two tests are equvalent. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 1 of 39 pages

13 ECONOMICS 45* -- Stata 9 utoral 6 est - Model 1: Proposton: A marred woman's educaton has no effect on the probablty that she s a labour force partcpant; the margnal ndex and probablty effects of ed equal zero. Margnal Index and Probablty Effects: x margnal ndex effect of ed = = ed Φ margnal probablty effect of ed ( x ) = = φ ( x ) ed A suffcent condton for both the margnal ndex and margnal probablty effects of ed to equal zero for all marred women s = 0. Null and Alternatve Hypotheses: H 0 : = 0 H 1 : 0 o calculate a Wald test of ths hypothess and the correspondng p-value for the calculated W-statstc, enter ether of the followng test commands: test ed or test ed = 0 o calculate a two-tal asymptotc t-test of H 0 aganst H 1, enter the followng lncom command: lncom _b[ed] Note that ths lncom command smply replcates the sample value of the z-statstc and ts two-tal p-value n the output of the probt command for Model 1. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 13 of 39 pages

14 ECONOMICS 45* -- Stata 9 utoral 6 est 3 - Model 1: Proposton: Years of actual work experence have no effect on the probablty of labour force partcpaton for marred women. In other words, the margnal probablty (and ndex) effect of exp equals zero. Margnal Index and Probablty Effects: margnal ndex effect of exp = x exp = 3 + Φ margnal probablty effect of exp ( x ) = exp 4 exp. = φ ( x ) ( + exp ) A suffcent condton for both the margnal ndex and margnal probablty effects of exp to equal zero for all marred women s 3 = 0 and 4 = Null and Alternatve Hypotheses: H 0 : 3 = 0 and 4 = 0 H 1 : 3 0 and/or 4 0 o calculate a Wald test of ths hypothess and the correspondng p-value for the calculated W-statstc, enter the followng test commands: test exp = 0 test expsq = 0, accum est 4 - Model 1: Proposton: For marred women wth fve years of actual work experence, years of actual work experence have no effect on the probablty of labour force partcpaton. In other words, the margnal probablty (and ndex) effects of exp equals zero for marred women for whom exp = 5, where margnal ndex effect of exp for exp = 5 = + exp = ECON 45* -- Fall 007: Stata 9 utoral 6 Page 14 of 39 pages

15 ECONOMICS 45* -- Stata 9 utoral x ( margnal probablty effect of exp for exp = 5 = φ ( x ) ( + exp ) = φ ( ) 5) A mnmally suffcent condton for both the margnal ndex and margnal probablty effects of exp to equal zero for marred women wth fve years of actual work experence (for whom exp = 5) s + 5 = Null and Alternatve Hypotheses: H 0 : = = 0 H 1 : + 5 = o calculate a Wald t-test of ths hypothess and the correspondng two-tal p-value for the calculated t-statstc, enter the followng lncom command: lncom _b[exp] + *_b[expsq]*5 est 5 - Model 1: Proposton: For the medan marred women n the sample, years of actual work experence have no effect on the probablty of labour force partcpaton. In other words, the margnal probablty and ndex effects of exp equal zero for marred women for whom exp = exp50p, where exp50p s the sample medan value of exp, margnal ndex effect of exp for exp = exp50p s + 4 exp = x ( margnal probablty effect of exp for exp = exp50p s φ ( x ) ( + exp ) exp50p = φ ( ) exp50p) A mnmally suffcent condton for both the margnal ndex and margnal probablty effects of exp to equal zero for marred women wth medan years of actual work experence (for whom exp = exp50p) s exp50p = 0, meanng that the margnal ndex effect of exp equals zero when evaluated at exp50p, the sample medan value of exp. Frst, use the summarze command wth the detal opton to compute the sample medan value of exp. Enter the commands: ECON 45* -- Fall 007: Stata 9 utoral 6 Page 15 of 39 pages

16 ECONOMICS 45* -- Stata 9 utoral 6 summarze exp, detal return lst Note that the sample medan value of exp s saved n the scalar r(p50) and equals 9 years. Use a scalar command to save the sample medan value of exp. Enter the scalar command: scalar exp50p = r(p50) Null and Alternatve Hypotheses: H 0 : exp50p = = 0 H 1 : + exp50p = o calculate a Wald t-test of ths hypothess and the correspondng two-tal p-value for the calculated t-statstc, enter ether of the followng lncom commands: lncom _b[exp] + *_b[expsq]*exp50p lncom _b[exp] + *_b[expsq]*9 Note that these two lncom commands produce dentcal results. est 6 - Model 1: Proposton: he number and ages of dependent chldren n the famly have no effect on the probablty of marred women's labour force partcpaton. In other words, the margnal probablty (and ndex) effects of both kdslt6 and kdsge6 equal zero. Margnal Index and Probablty Effects: x margnal ndex effect of kdslt6 = = 6 kdslt6 Φ margnal probablty effect of kdslt6 ( x ) = kdslt6 = φ ( x ) 6 ECON 45* -- Fall 007: Stata 9 utoral 6 Page 16 of 39 pages

17 ECONOMICS 45* -- Stata 9 utoral 6 x margnal ndex effect of kdsge6 = = 7 kdsge6 Φ margnal probablty effect of kdsge6 ( x ) = kdsge6 = φ ( x ) 7 A suffcent condton for the margnal ndex and probablty effects of both kdslt6 and kdsge6 to equal zero for all marred women s 6 = 0 and 7 = 0. Null and Alternatve Hypotheses: H 0 : 6 = 0 and 7 = 0 H 1 : 6 0 and/or 7 0 o calculate a Wald test of ths hypothess and the correspondng p-value for the calculated W-statstc, enter the followng test commands: test kdslt6 = 0 test kdsge6 = 0, accum Computng margnal probablty effects of contnuous explanatory varables -- dprobt Computng Margnal Index Effects: Background Recall that the margnal probablty effect of a contnuous explanatory varable X j φ x, the value of the s smply the margnal ndex effect of X j multpled by ( ) x standard normal p.d.f. evaluated at : Φ margnal probablty effect of X j ( x ) ( ) = Φ x X j = φ( ) = x x X j = φ ( x ) x margnal ndex effect of X j x X j where ECON 45* -- Fall 007: Stata 9 utoral 6 Page 17 of 39 pages

18 ECONOMICS 45* -- Stata 9 utoral 6 ( ) φ x = the value of the standard normal p.d.f. at x X j = the margnal ndex effect of X j x In order to compute estmates of the margnal probablty effects of contnuous φ ˆ, where ˆ s the explanatory varables, t s necessary to compute the value of ( ) vector of ML probt coeffcent estmates and x x s a vector of regressor values. It s also necessary to choose a specfc vector of regressor values. Common x choces for are the sample mean and sample medan values of the regressors, although others may certanly be approprate n partcular crcumstances. Introducton to the dprobt command x he dprobt command s an alternatve to probt for computng ML estmates of probt models. However, dprobt dffers from probt n that t reports the margnal probablty effects of the ndvdual regressors (and ther standard errors) rather than the probt coeffcent estmates ˆ j (and ther standard errors). he dprobt command estmates the margnal probablty effects of the ndvdual regressors at some user-selected vector of regressor values. x hs means that when usng the dprobt command, you must select a vector of regressor values x at whch to evaluate the margnal probablty effects of the explanatory varables. hs s done usng the at(vecname) opton on the dprobt command, where vecname s the name of a user-defned vector that contans the values of the regressors at whch the estmates of the margnal probablty effects are evaluated. he default verson of the dprobt command, whch does not use the at(vecname) opton, estmates the margnal probablty effects at the sample mean values of the regressors;.e., the dprobt command wthout the at(vecname) opton sets x Τ = x Τ, where the vector Τ x contans the sample means of all the regressors. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 18 of 39 pages

19 ECONOMICS 45* -- Stata 9 utoral 6 he dprobt command wthout the at(vecname) opton o obtan ML estmates of the margnal probablty effects for Model 1, evaluated at the sample mean values of the regressors x = ( nwfenc ed exp expsq age kdslt6 kdsge6 1), enter the commands: dprobt nlf nwfenc ed exp expsq age kdslt6 kdsge6 ereturn lst Carefully nspect the followng results of these two commands (whch are reproduced on the followng page). ECON 45* -- Fall 007: Stata 9 utoral 6 Page 19 of 39 pages

20 ECONOMICS 45* -- Stata 9 utoral 6. dprobt nlf nwfenc ed exp expsq age kdslt6 kdsge6 Iteraton 0: log lkelhood = (output omtted) Iteraton 4: log lkelhood = Probt regresson, reportng margnal effects Number of obs = 753 LR ch(7) = 7.14 Prob > ch = Log lkelhood = Pseudo R = nlf df/dx Std. Err. z P> z x-bar [ 95% C.I. ] nwfenc ed exp expsq age kdslt kdsge obs. P pred. P (at x-bar) z and P> z correspond to the test of the underlyng coeffcent beng 0. ereturn lst scalars: macros: matrces: functons: e(n) = 753 e(ll_0) = e(ll) = e(df_m) = 7 e(ch) = e(r_p) = e(pbar) = e(xbar) = e(offbar) = 0 e(cmd) : "dprobt" e(dummy) : " " e(ttle) : "Probt regresson" e(depvar) : "nlf" e(crttype) : "log lkelhood" e(predct) : "probt_p" e(propertes) : "b V" e(estat_cmd) : "probt_estat" e(chtype) : "LR" e(b) : 1 x 8 e(v) : 8 x 8 e(se_dfdx) : 1 x 7 e(dfdx) : 1 x 7 e(sample) = x Τ ˆ ECON 45* -- Fall 007: Stata 9 utoral 6 Page 0 of 39 pages

21 ECONOMICS 45* -- Stata 9 utoral 6 One of the saved scalars from the above dprobt command s x Τˆ, the estmated Τ value of the probt ndex functon x evaluated at the sample means of the Τ regressors x and the ML probt coeffcent estmates ˆ. o dsplay the value of x Τˆ, enter the command: dsplay e(xbar) he estmated value of Φ( ˆ ) x, the standard normal c.d.f. Φ( ) x evaluated at x Τˆ, can be computed usng the Stata statstcal functon normal(z). Enter the command: dsplay normal(e(xbar)) Note that the value returned by ths dsplay command equals the value of pred. P n the output of the dprobt command. he value of φ( ˆ ) x, the estmated value of the standard normal p.d.f. φ( x ) evaluated at x Τˆ, can be computed usng the Stata statstcal functon normalden(z). Enter the command: dsplay normalden(e(xbar)) o compare the output of the above dprobt command wth that of the correspondng probt command, enter the command: probt Note that the values of the t-statstcs (labelled as z) and ther two-tal p-values (labelled P > z ) are dentcal n the output of the probt and dprobt commands. hat's because the reported t-statstcs are testng the same hypothess, namely the null hypothess that the probt coeffcent j = 0 aganst the two-sded alternatve hypothess that j 0. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 1 of 39 pages

22 ECONOMICS 45* -- Stata 9 utoral 6 he dprobt command wth the at(vecname) opton An alternatve way to obtan ML estmates of margnal probablty effects n probt models s to use the at(vecname) opton on the dprobt command, where vecname s the name of a user-defned vector that contans the values of the regressors at whch the estmates of the margnal probablty effects are evaluated. o llustrate how the at(vecname) opton works, we use t to compute estmates of the margnal probablty effects of the regressors n Model 1 at the vector of sample means,.e., at x Τ = x Τ. Frst, use summarze and scalar commands to compute and save the sample means of the regressors. Enter the commands: summarze nwfenc scalar nwncbar = r(mean) summarze ed scalar edbar = r(mean) summarze exp scalar expbar = r(mean) summarze expsq scalar expsqbar = r(mean) summarze age scalar agebar = r(mean) summarze kdslt6 scalar klt6bar = r(mean) summarze kdsge6 scalar kge6bar = r(mean) scalar lst nwncbar edbar expbar expsqbar agebar klt6bar kge6bar Next, use matrx commands to create and dsplay a row vector xbar that contans the sample mean values of the regressors and a 1 for the ntercept coeffcent. Enter the commands: matrx xbar = (nwncbar, edbar, expbar, expsqbar, agebar, klt6bar, kge6bar, 1) matrx lst xbar ECON 45* -- Fall 007: Stata 9 utoral 6 Page of 39 pages

23 ECONOMICS 45* -- Stata 9 utoral 6 Now use the dprobt command wth the at(xbar) opton to compute ML estmates of the margnal probablty effects for Model 1 and ther standard errors at the sample mean values of the regressors. Enter the command: dprobt nlf nwfenc ed exp expsq age kdslt6 kdsge6, at(xbar) he output of ths command s dentcal to that for the dprobt command wthout the at(xbar) opton; the default opton for dprobt s to compute margnal probablty effects at the sample mean values of the regressors. o see ths, enter the above dprobt command wthout the at(xbar) opton: dprobt nlf nwfenc ed exp expsq age kdslt6 kdsge6 Carefully compare the results of these two dprobt commands. hey are dentcal. Margnal probablty effects for explanatory varables that enter nonlnearly n the probt ndex functon he only explanatory varable for whch the above dprobt command does not produce an estmate of the true margnal probablty effect s exp, years of actual work experence. Recall that the correct margnal probablty effect of exp s: Φ margnal probablty effect of exp ( x ) = exp = ( x ) ( + exp ) φ. Evaluated at the sample mean values of the regressors n the vector 3 4 x = (nwfenc ed exp expsq age kdslt6 kdsge6 1), the margnal probablty effect of exp s: margnal probablty effect of exp at x = ( x ) ( 3 + 4exp) φ. he ML estmate of the margnal probablty effect of exp at x s: estmate of margnal probablty effect of exp at φ ˆ (ˆ ˆ. x = ( x ) exp) ECON 45* -- Fall 007: Stata 9 utoral 6 Page 3 of 39 pages

24 ECONOMICS 45* -- Stata 9 utoral 6 φ x ˆ (ˆ ˆ 3 4 for Model 1, φ ˆ (ˆ + ˆ exp = 0. he followng steps demonstrate how to compute ( ) + exp) and how to test the null hypothess ( ) ) x 3 4 Frst, re-estmate the probt coeffcent estmates of Model 1. Enter the dprobt command: dprobt nlf nwfenc ed exp expsq age kdslt6 kdsge6, at(xbar) ereturn lst Note from the above ereturn lst command that the temporarly-saved scalar e(xbar) contans the value of x Τˆ, the estmated value of the probt ndex functon Τ Τ x evaluated at the sample means of the regressors x and the ML probt coeffcent estmates ˆ. o dsplay the value of x Τˆ, enter the dsplay command: dsplay e(xbar) Now use the normalden(z) statstcal functon to compute the value of φ( x ˆ ) save t as a scalar. Enter the followng dsplay and scalar commands: dsplay normalden(e(xbar)) scalar phxbarb = normalden(e(xbar)) scalar lst phxbarb he scalar phxbarb now equals the value of φ( ˆ ) x for Model 1. Fnally, use the lncom command both to compute the value of φ ( ) + exp) and to perform a t-test of the null hypothess that ( ) + exp) ether of the followng lncom commands: and x ˆ (ˆ ˆ 3 4 φ ˆ (ˆ ˆ = 0. Enter x 3 4 lncom phxbarb*(_b[exp] + *_b[expsq]*expbar) lncom phxbarb*(exp + *expsq*expbar) Note that these two lncom commands produce exactly the same output. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 4 of 39 pages

25 ECONOMICS 45* -- Stata 9 utoral 6 Another example of the dprobt command wth the at(vecname) opton he dprobt command wth the at(vecname) opton can be used to compute ML estmates of the margnal probablty effects for Model 1 at any user-selected values of the regressors. hs secton demonstrates how to do ths. Frst, use a matrx command to compute and save the selected values of the regressors. he selected regressor values are nwfenc = 0, ed = 14, exp = 10, expsq = 100, age = 40, kdslt6 = 0, and kdsge6 = 1. In other words, we are gong to estmate the margnal probablty effects n Model 1 for marred women whose nonwfe famly ncome s $0,000 per year, who have 14 years of formal educaton and 10 years of actual work experence, who are 40 years of age, who have no dependent chldren under 6 years of age, and who have one dependent chld 6 years of age or older. Enter the commands: matrx xvec = (0, 14, 10, 100, 40, 0, 1, 1) matrx lst xvec Note that the value 1 s ncluded as the last element of the regressor vector named xvec to allow for the probt ntercept coeffcent 0. Now use the dprobt command wth the at(xvec) opton to compute ML estmates of the margnal probablty effects n Model 1 for marred women wth the observed characterstcs descrbed by the regressor values n the vector xvec. Enter the commands: dprobt nlf nwfenc ed exp expsq age kdslt6 kdsge6, at(xvec) ereturn lst dsplay e(at) Compare the output of ths dprobt command wth that of the earler dprobt command. he estmates of the margnal probablty effects are dfferent because they are evaluated at dfferent values of the regressor vector. Note that the temporarly-saved scalar e(at) contans the value of Φ( ˆ ) x, the estmated value of the standard normal dstrbuton functon (or c.d.f.) evaluated at the ML probt coeffcent estmates ˆ and the specfed vector xvec, where xvec s x ECON 45* -- Fall 007: Stata 9 utoral 6 Page 5 of 39 pages

26 ECONOMICS 45* -- Stata 9 utoral 6 x = ( ). Compute an estmate of the margnal probablty effect of exp for marred women wth the observed characterstcs specfed by the regressor vector xvec, and test the null hypothess that ths margnal probablty effect equals zero. Frst, we have to compute the estmated value of the standard normal p.d.f. φ( x ) evaluated at ˆ and the specfed vector xvec; ths s done usng a scalar command together wth the Stata statstcal functons nvnormal(p) and normalden(z). Enter the commands: dsplay nvnormal(e(at)) scalar phxvec = normalden(nvnormal(e(at))) scalar lst phxvec lncom phxvec*(_b[exp] + *_b[expsq]*10) Snce the temporarly-saved scalar e(at) contans the value of Φ( ˆ ) statstcal functon nvnormal(e(at)) returns the value of phxvec thus equals the value of φ x ˆ. ( ) x x, the Stata ˆ. he saved scalar Compute estmates of the margnal probablty effects of nwfenc, ed, age, kdslt6, and kdsge6 for marred women wth the observed characterstcs specfed by the regressor vector xvec, and test the null hypothess that each of these margnal probablty effects equals zero. Enter the lncom commands: lncom phxvec*_b[nwfenc] lncom phxvec*_b[ed] lncom phxvec*_b[age] lncom phxvec*_b[kdslt6] lncom phxvec*_b[kdsge6] Compare the estmated margnal probablty effects of nwfenc, ed, age, kdslt6, and kdsge6 produced by these lncom commands wth those produced by the prevous dprobt command. hey should be dentcal because each of the explanatory varables nwfenc, ed, age, kdslt6, and kdsge6 enter lnearly the probt ndex functon for Model 1. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 6 of 39 pages

27 ECONOMICS 45* -- Stata 9 utoral 6 Margnal probablty effects of bnary explanatory varables Model Model All the explanatory varables n Model 1 are contnuous, or at least take more than two dscrete values. But the use of dummy varable regressors -- bnary varables that take only the two values 0 and 1 -- s as common n bnary dependent varables models as t s n conventonal lnear regresson models. hs secton explans how Stata computes estmates of the margnal probablty effects of dummy varable regressors n probt models. For ths purpose, we replace the explanatory varables kdslt6 and kdsge6 n Model 1 wth a sngle dummy varable dkdslt6 defned as follows: dkdslt6 = 1 f kdslt6 > 0 for the -th marred woman = 0 f kdslt6 = 0 for the -th marred woman he probt ndex functon, or regresson functon, for Model s: x = 0 + 1nwfenc + ed + 3 exp + 4 exp + 5age + 6dkdslt6 where all other explanatory varables apart from dkdslt6 are as prevously defned for Model 1; that s, all other explanatory varables n Model are contnuous varables. hs secton addresses the queston of how to compute the margnal probablty effect of the dummy explanatory varable dkdslt6. Background Margnal probablty effects of contnuous explanatory varables As we have prevously seen, the margnal probablty effects of contnuous explanatory varables n probt models are the partal dervatves of the standard Φ x wth respect to the ndvdual explanatory varables: normal c.d.f. ( ) Φ margnal probablty effect of X j ( x ) ( ) = Φ x X j = x x X j = φ ( x ) x X j ECON 45* -- Fall 007: Stata 9 utoral 6 Page 7 of 39 pages

28 ECONOMICS 45* -- Stata 9 utoral 6 where ( ) φ x = the value of the standard normal p.d.f. at x X j = the margnal ndex effect of X j, where X j s a contnuous varable. x he margnal probablty effects of the contnuous explanatory varables n Model are therefore: Φ 1. margnal probablty effect of nwfenc ( x ) = = φ ( x ) 1 nwfenc Φ. margnal probablty effect of ed ( x ) = = φ ( x ) ed Φ 3. margnal probablty effect of exp ( x ) = = φ ( x ) ( + exp ) exp Φ 4. margnal probablty effect of age ( x ) = = φ ( x ) 5 age 3 4 Margnal probablty effects of bnary (or dummy) explanatory varables he margnal probablty effect of a bnary, or dummy, explanatory varable D j n probt models s computed by evaluatng the standard normal c.d.f. Φ( x ) at the two dfferent values of the dummy varable D j, and then takng the dfference. Defne two dfferent vectors of regressor values: 1 0 x = any vector of regressor values wth D j = 1; x = the same vector of regressor values but wth D j = 0. he margnal probablty effect of the dummy varable D j s defned as: margnal probablty effect of D j = Φ( ) Φ( ) x. 1 x 0 ECON 45* -- Fall 007: Stata 9 utoral 6 Page 8 of 39 pages

29 ECONOMICS 45* -- Stata 9 utoral 6 Example: Compute the margnal probablty effect of the dummy varable dkdslt6 n Model. Frst, defne x 1 wth the contnuous varables set to any arbtrary values and the dummy varable dkdslt6 = 1: x = 1 ( 1 nwfenc ed exp exp age 1) Second, defne x 0 wth the contnuous varables set to the same arbtrary values but wth the dummy varable dkdslt6 = 0: x = 0 ( 1 nwfenc ed exp exp age 0 ) he condtonal probablty that nlf = 1 when dkdslt6 = 1 s: Pr ( nlf = 1 dkdslt6 = 1) = Φ( x 1 ) = Φ ( + nwfenc + ed + exp + exp + + ) age he condtonal probablty that nlf = 1 when dkdslt6 = 0 s: Pr ( nlf = 1 dkdslt6 = 0) = Φ( x 0 ) = Φ ( + nwfenc + ed + exp + exp + ) age he margnal probablty effect of the dummy varable dkdslt6 n Model s the dfference between the condtonal probablty that nlf = 1 when dkdslt6 = 1 and the condtonal probablty that nlf = 1 when dkdslt6 = 0: Pr ( nlf = 1 dkdslt6 1) Pr( nlf = 1 dkdslt6 0) = = Φ( x ) Φ( ) 1 x 0 = = Φ ( 0 + 1nwfenc + ed + 3 exp + 4 exp + 5age + 6 ) Φ ( + nwfenc + ed + exp + exp + ) age 6 ECON 45* -- Fall 007: Stata 9 utoral 6 Page 9 of 39 pages

30 ECONOMICS 45* -- Stata 9 utoral 6 Margnal probablty effects of bnary explanatory varable dkdslt6 -- dprobt Before estmatng Model, t s necessary to create the new dummy varable dkdslt6. Enter the commands: generate dkdslt6 = kdslt6 > 0 summarze dkdslt6 kdslt6 tab dkdslt6 kdslt6 Margnal Probablty Effects for Model -- dprobt Use the dprobt command wthout the at(vecname) opton to compute the margnal probablty effects n Model at the sample mean values of the regressors,.e., at Τ x = x. Enter the followng command: dprobt nlf nwfenc ed exp expsq age dkdslt6 he next seres of Stata commands wll demonstrate how to use the dprobt command wth the at(vecname) opton to compute the margnal probablty effect of the dummy varable dkdslt6 n Model for marred women whose non-wfe famly ncome s $0,000 per year (nwfenc = 0), who have 14 years of formal educaton (ed = 14) and 10 years of actual work experence (exp = 10, expsq = 100), and who are 40 years of age (age = 40): Pr ( nlf = 1 dkdslt6 1) Pr( nlf = 1 dkdslt6 0) = = Φ( x ) Φ( ) 1 x 0 = = Φ ( 0 + 1nwfenc + ed + 3 exp + 4 exp + 5age + 6 ) Φ ( + nwfenc + ed + exp + exp + ) age = Φ ( ) Φ ( ) he procedure for ths computaton conssts of three steps. Step 1: Compute an estmate of the probablty of labour force partcpaton for marred women wth the specfed characterstcs who have one or more dependent chldren under 6 years of age, for whom dkdslt6 = 1:.e., compute an estmate of ECON 45* -- Fall 007: Stata 9 utoral 6 Page 30 of 39 pages

31 ECONOMICS 45* -- Stata 9 utoral 6 ( x ) = Φ ( + nwfenc + ed + exp + exp + + ) Φ age = ( ) Φ. Step : Compute an estmate of the probablty of labour force partcpaton for marred women wth the specfed characterstcs who have no dependent chldren under 6 years of age, for whom dkdslt6 = 0:.e., compute an estmate of ( x ) = Φ ( + nwfenc + ed + exp + exp + ) Φ age = ( ) Φ. Step 3: Compute an estmate of the dfference Φ( x ) Φ( ) 1 x0, whch s the margnal probablty effect of havng one or more dependent chldren under 6 years of age for marred women who have the specfed characterstcs. 6 Step 1: Use the dprobt command wth the at(vecname) opton to compute the margnal probablty effects n Model for marred women whose non-wfe famly ncome s $0,000 per year (nwfenc = 0), who have 14 years of formal educaton (ed = 14) and 10 years of actual work experence (exp = 10, expsq = 100), who are 40 years of age (age = 40), and who have one or more dependent chldren under 6 years of age (dkdslt6 = 1). You wll frst have to create a vector contanng the specfed values of the regressors for Model, snce the dprobt command does not permt number lsts n the at( ) opton. Note that n Stata format, the vector x 1 wth the dummy varable dkdslt6 = 1 s wrtten as: x ( nwfenc ed exp exp age 1 1) ( ) 1 = = Enter the followng commands: matrx x1vec = (0, 14, 10, 100, 40, 1, 1) matrx lst x1vec dprobt nlf nwfenc ed exp expsq age dkdslt6, at(x1vec) ereturn lst. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 31 of 39 pages

32 ECONOMICS 45* -- Stata 9 utoral 6 Dsplay and save the value of Φ( x ˆ ) where Φ( ˆ ) Pr x 1 generated by the above dprobt command, nlf = 1 dkdslt6 1. Enter the commands: 1 s an estmate of ( = ) dsplay e(at) scalar PHIx1vec = e(at) scalar lst PHIx1vec Step : Now use the dprobt command wth the at(vecname) opton to compute the margnal probablty effects n Model for marred women whose non-wfe famly ncome s $0,000 per year (nwfenc = 0), who have 14 years of formal educaton (ed = 14) and 10 years of actual work experence (exp = 10, expsq = 100), who are 40 years of age (age = 40), and who have no dependent chldren under 6 years of age (dkdslt6 = 0). Frst, you wll have to create a vector contanng the specfed values of the regressors for Model ; the dprobt command does not permt number lsts n the at( ) opton. Note that n Stata format, the vector x 0 wth the dummy varable dkdslt6 = 0 s wrtten as: x ( nwfenc ed exp exp age 0 1) ( ) 0 = = Enter the followng commands: matrx x0vec = (0, 14, 10, 100, 40, 0, 1) matrx lst x0vec dprobt nlf nwfenc ed exp expsq age dkdslt6, at(x0vec) ereturn lst Dsplay and save the value of Φ( x ˆ ) where Φ( ˆ ) Pr x 0 generated by the above dprobt command, nlf = 1 dkdslt6 0. Enter the commands: 0 s an estmate of ( = ) dsplay e(at) scalar PHIx0vec = e(at) scalar lst PHIx0vec ECON 45* -- Fall 007: Stata 9 utoral 6 Page 3 of 39 pages

33 ECONOMICS 45* -- Stata 9 utoral 6 Step 3: Fnally, compute the estmate of the dfference Φ( x ) Φ( ) 1 x0, whch s the margnal probablty effect havng one or more dependent chldren under 6 years of age for marred women who have the specfed characterstcs. Enter the commands: scalar dffphi = PHIx1vec - PHIx0vec scalar lst PHIx1vec PHIx0vec dffphi Carefully compare the results of ths three-step procedure wth the output of the two dprobt commands you have estmated. Enter the followng commands: * Model at x0vec: dprobt dprobt nlf nwfenc ed exp expsq age dkdslt6, at (x0vec) * Model at x1vec: dprobt dprobt nlf nwfenc ed exp expsq age dkdslt6, at (x1vec) he Stata output lstng produced by these commands s reproduced on the followng page. Note n partcular the hghlghted results n the output lstng for these two dprobt commands. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 33 of 39 pages

34 ECONOMICS 45* -- Stata 9 utoral 6. * Model at x0vec: dprobt. dprobt nlf nwfenc ed exp expsq age dkdslt6, at (x0vec) Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton : log lkelhood = Iteraton 3: log lkelhood = Probt regresson, reportng margnal effects Number of obs = 753 LR ch(6) = Prob > ch = Log lkelhood = Pseudo R = nlf df/dx Std. Err. z P> z x [ 95% C.I. ] nwfenc ed exp expsq age dkdslt6* obs. P pred. P (at x-bar) pred. P (at x) (*) 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. * Model at x1vec: dprobt. dprobt nlf nwfenc ed exp expsq age dkdslt6, at (x1vec) Iteraton 0: log lkelhood = Iteraton 1: log lkelhood = Iteraton : log lkelhood = Iteraton 3: log lkelhood = Probt regresson, reportng margnal effects Number of obs = 753 LR ch(6) = Prob > ch = Log lkelhood = Pseudo R = nlf df/dx Std. Err. z P> z x [ 95% C.I. ] nwfenc ed exp expsq age dkdslt6* obs. P pred. P (at x-bar) pred. P (at x) (*) 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 ECON 45* -- Fall 007: Stata 9 utoral 6 Page 34 of 39 pages

35 ECONOMICS 45* -- Stata 9 utoral 6 Margnal probablty effects of contnuous explanatory varables n Model -- dprobt For any gven set of values of the contnuous explanatory varables nwfenc, ed, exp and age n Model, there are two dstnct sets of margnal probablty effects to compute: (1) those for marred women wth the specfed values of nwfenc, ed, exp and age who have no dependent chldren under 6 years of age (for whom dkdslt6 = 0); and () those for marred women wth the specfed values of nwfenc, ed, exp and age who have one or more dependent chldren under 6 years of age (for whom dkdslt6 = 1). Margnal Probablty Effects for Women for Whom dkdslt6 = 0 In ths secton, we use the dprobt command wth the at(vecname) opton to compute the margnal probablty effects of the contnuous explanatory varables n Model for marred women whose non-wfe famly ncome s $0,000 per year (nwfenc = 0), who have 14 years of formal educaton (ed = 14) and 10 years of actual work experence (exp = 10, expsq = 100), who are 40 years of age (age = 40), and who have no dependent chldren under 6 years of age (for whom dkdslt6 = 0). hese are gven by the followng expressons: φ x 1. margnal probablty effect of nwfenc = ( 0 ) 1 φ x. margnal probablty effect of ed = ( 0 ) 3. margnal probablty effect of exp = φ ( x ) ( + exp ) 4. margnal probablty effect of age = ( 0 ) x 0 ( = φ ( ) 10) φ x Note that n all these four expressons, the standard normal p.d.f. φ( ) x 0 x s evaluated at the vector wth the dummy varable dkdslt6 = 0, where n Stata format s x ( nwfenc ed exp exp age 0 1) ( ) 0 = = You have prevously created the vector x 0 and named t x0vec. x 0. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 35 of 39 pages

36 ECONOMICS 45* -- Stata 9 utoral 6 Frst, use the dprobt command wth the at(vecname) opton to re-estmate Model and evaluate the margnal probablty effects at. Enter the commands: dprobt nlf nwfenc ed exp expsq age dkdslt6, at(x0vec) ereturn lst dsplay e(at) Note that the temporarly saved scalar e(at) created by ths dprobt command Φ ˆ Pr nlf = 1 dkdslt6 0. equals the value of ( ) x x 0 0, whch s an estmate of ( = ) Second, use the Stata nvnormal(p) statstcal functon to save the value of x ˆ 0 as the scalar x0vecb. Enter the scalar commands: scalar x0vecb = nvnormal(e(at)) scalar lst x0vecb hrd, use the Stata normalden(z) statstcal functon to save the value of φ( ˆ ) the scalar phx0vec. Enter the scalar commands: scalar phx0vec = normalden(x0vecb) scalar lst x0vecb phx0vec x 0 as Fnally, use lncom commands to compute the values of the margnal probablty effects of the four contnuous explanatory varables n Model for women wth the specfed characterstcs who have no pre-school aged chldren, and to test the null hypothess that each margnal probablty effect equals zero. Enter the lncom commands: lncom phx0vec*_b[nwfenc] lncom phx0vec*_b[ed] lncom phx0vec*(_b[exp] + *_b[expsq]*10) lncom phx0vec*_b[age] Compare the estmated margnal probablty effects of the explanatory varables nwfenc, ed and age produced by these lncom commands wth those computed by the precedng dprobt command wth the at(x0vec) opton; they are dentcal because the varables nwfenc, ed and age enter the probt ndex functon for Model lnearly. ECON 45* -- Fall 007: Stata 9 utoral 6 Page 36 of 39 pages