MODULE TWO, PART TWO: ENDOGENEITY, INSTRUMENTAL VARIABLES AND TWO-STAGE LEAST SQUARES IN ECONOMIC EDUCATION RESEARCH USING LIMDEP

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1 MODULE TWO, PART TWO: ENDOGENEITY, INSTRUMENTAL VARIABLES AND TWO-STAGE LEAST SQUARES IN ECONOMIC EDUCATION RESEARCH USING LIMDEP Part Two of Module Two provdes a cookbook-type demonstraton of the steps requred to use LIMDEP to address problems of endogenety usng a two-stage least squares, nstrumental varable estmator. The Durbn, Hausman and Wu specfcaton test for endogenety s also demonstrated. Users of ths model need to have completed Module One, Parts One and Two, and Module Two, Part One. That s, from Module One, users are assumed to know how to get data nto LIMDEP, recode and create varables wthn LIMDEP, and run and nterpret regresson results. From Module Two, Part One, they are expected to have an understandng of the problem of and source of endogenety and the basc dea behnd an nstrumental varable approach and the two-stage least squares method. The Becker and Johnston (1999) data set s used throughout ths module for demonstraton purposes only. Module Two, Parts Three and Four demonstrate n STATA and SAS what s done here n LIMDEP. THE CASE As descrbed n Module Two, Part One, Becker and Johnston (1999) called attenton to classroom effects that mght nfluence multple-choce and essay type test takng sklls n economcs n dfferent ways. For example, f the student s n a classroom that emphaszes sklls assocated wth multple choce testng (e.g., rsk-takng behavor, queston analyzng sklls, memorzaton, and keen sense of judgng between close alternatves), then the student can be expected to do better on multple-choce questons. By the same token, f placed n a classroom that emphaszes the sklls of essay test queston answerng (e.g., organzaton, good sentence and paragraph constructon, obfuscaton when uncertan, logcal argument, and good penmanshp), then the student can be expected to do better on the essay component. Thus, Becker and Johnston attempted to control for the type of class of whch the student s a member. Ther measure of teachng to the multple-choce questons s the mean score or mark on the multple-choce questons for the school n whch the th student took the 12 th grade economcs course. Smlarly, the mean school mark or score on the essay questons s ther measure of the th student s exposure to essay queston wrtng sklls. In equaton form, the two equatons that summarze the nfluence of the varous covarates on multple-choce and essay test questons are wrtten as the follow structural equatons: _ J j j = 4 M = ρ + ρ W + ρ M + ρ X + U _ J j j = 4 W = ρ + ρ M + ρ W + ρ X + V j j *. *. W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 1

2 M and W are the th student s respectve scores on the multple-choce test and essay test. M and W are the mean multple-choce and essay test scores at the school where the th student took the 12 th grade economcs course. The varables are the other exogenous varables (such X j as gender, age, Englsh a second language, etc.) used to explan the th student s multple-choce and essay marks, where the ρs are parameters to be estmated. The ncluson of the mean multple-choce and mean essay test scores n ther respectve structural equatons, and ther excluson from the other equaton, enables both of the structural equatons to be dentfed wthn the system. As shown n Module Two, Part One, the least squares estmaton of the ρs nvolves bas * * because the error term U s related to W, n the frst equaton, and V s related to M, n second equaton. Instruments for regressors W and M are needed. Because the reduced form equatons express W and M solely n terms of exogenous varables, they can be used to generate the respectve nstruments: J j j = 4 M = Γ + Γ W + Γ M + Γ X + U J j j = 4 W = Γ + Γ M + Γ W + Γ X + V j j ** **. The reduced form parameters (Γs) are functons of the ρs, and the reduced form error terms U ** and V ** are functons of U * and V *, whch are not related to any of the regressors n the reduced form equatons.. We could estmate the reduced form equatons and get Mˆ andw ˆ. We could then substtute Mˆ andwˆ nto the structural equatons as proxy regressors (nstruments) for M and W. The least squares regresson of M on W, M and the Xs and a least squares regresson of W on M ˆ, ˆ W and the Xs would yeld consstent estmates of the respectve ρs, but the standard errors would be ncorrect. LIMDEP automatcally does all the requred estmatons wth the two-stage, least squares command: 2SLS; LHS= ; RHS= ; INST= $ TWO-STAGE, LEAST SQUARES IN LIMDEP The Becker and Johnston (1999) data are n the fle named Bll.CSV. Before readng these data nto LIMDEP, however, the Project Settngs must be ncreased from cells (222 W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 2

3 rows and 900 columns) to accommodate the 4,178 observatons. Ths can be done wth a project settng of cells (4444 rows and 900 columns), followng the procedures descrbed n Module One, Part Two. After ncreasng the project settng, fle Bll.CSV can be read nto LIMDEP wth the followng read command (the fle may be located anywhere on your hard drve but here t s located on the e drve): READ; NREC=4178; NVAR=44; FILE=e:\bll.csv; Names= student,school,sze,other,brthday,sex,eslflag,adultst, mc1,mc2,mc3,mc4,mc5,mc6,mc7,mc8,mc9,mc10,mc11,mc12,mc13, mc14,mc15,mc16,mc17,mc18,mc19,mc20,totalmc,avgmc, essay1,essay2,essay3,essay4,totessay,avgessay, totscore,avgscore,ma081,ma082,ec011,ec012,ma083,en093$ Usng these recode and create commands, yelds the followng relevant varable defntons: recode; sze; 0/9=1; 10/19=2; 20/29=3; 30/39=4; 40/49=5; 50/100=6$ create; smallest=sze=1; smaller=sze=2; small=sze=3; large=sze=4; larger=sze=5; largest=sze=6$ TOTALMC: Student s score on 12 th grade economcs multple-choce exam ( M ). AVGMC: Mean multple-choce score for students at school ( M ). TOTESSAY: Student s score on 12 th grade economcs essay exam ( W ). AVGESSAY: Mean essay score for students at school ( W ). ADULTST = 1, f a returnng adult student, and 0 otherwse. SEX = GENDER = 1 f student s female and 0 s male. ESLFLAG = 1 f Englsh s not student s frst language and 0 f t s. EC011 = 1 f student enrolled n frst semester 11 grade economcs course, 0 f not. EN093 = 1 f student was enrolled n ESL Englsh course, 0 f not MA081 = 1 f student enrolled n the frst semester 11 grade math course, 0 f not. MA082 = 1 f student was enrolled n the second semester 11 grade math course, 0 f not. MA083 = 1 f student was enrolled n the frst semester 12 grade math course, 0 f not. SMALLER = 1 f student from a school wth 10 to 19 test takers, 0 f not. SMALL = 1 f student from a school wth 20 to 29 test takers, 0 f not. LARGE = 1 f student from a school wth 30 to 39 test takers, 0 f not. LARGER = 1 f student from a school wth 40 to 49 test takers, 0 f not. In all of the regressons, the effect of beng at a school wth more than 49 test takers s captured n the constant term, aganst whch the other dummy varables are compared. The smallest schools need to be rejected to treat the mean scores as exogenous and unaffected by any ndvdual student s test performance, whch s accomplshed wth the followng command: Reject; smallest = 1$ W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 3

4 The descrptve statstcs on the relevant varables are then obtaned wth the followng command, yeldng the LIMDEP output table shown: Dstat;RHS=TOTALMC,AVGMC,TOTESSAY,AVGESSAY,ADULTST,SEX,ESLFLAG, EC01,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE,LARGER$ Descrptve Statstcs All results based on nonmssng observatons. =============================================================================== Varable Mean Std.Dev. Mnmum Maxmum Cases =============================================================================== All observatons n current sample TOTALMC AVGMC TOTESSAY AVGESSAY ADULTST E E SEX ESLFLAG E EC EN E MA MA MA SMALLER SMALL LARGE LARGER E For comparson wth the two-stage least squares results, we start wth the least squares regressons shown after ths paragraph. The least squares estmatons are typcal of those found n multple-choce and essay score correlaton studes, wth correlaton coeffcents of 0.77 and The essay mark or score, W, s the most sgnfcant varable n the multple-choce score regresson (frst of the two tables) and the multple-choce mark, M, s the most sgnfcant varable n the essay regresson (second of the two tables). Results lke these have led researchers to conclude that the essay and multple-choce marks are good predctors of each other. Notce also that both the mean multple-choce and mean essay marks are sgnfcant n ther respectve equatons, suggestng that somethng n the classroom envronment or group experence nfluences ndvdual test scores. Fnally, beng female has a sgnfcant negatve effect on the multple choce-test score, but a sgnfcant postve effect on the essay score, as expected from the least squares results reported by others. We wll see how these results hold up n the two-stage least squares regressons. Regress;LHS=TOTALMC;RHS=TOTESSAY,ONE,ADULTST,SEX,AVGMC, ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE,LARGER$ Ordnary least squares regresson Weghtng varable = none Dep. var. = TOTALMC Mean= , S.D.= Model sze: Observatons = 3710, Parameters = 15, Deg.Fr.= 3695 Resduals: Sum of squares= , Std.Dev.= Ft: R-squared= , Adjusted R-squared = Model test: F[ 14, 3695] = , Prob value = Dagnostc: Log-L = , Restrcted(b=0) Log-L = LogAmemyaPrCrt.= 1.868, Akake Info. Crt.= Autocorrel: Durbn-Watson Statstc = , Rho = W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 4

5 Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X TOTESSAY E Constant ADULTST E-02 SEX E AVGMC E ESLFLAG E-01 EC E E EN E-01 MA MA MA SMALLER E SMALL E LARGE E LARGER E E-01 (Note: E+nn or E-nn means multply by 10 to + or -nn power.) Regress;LHS=TOTESSAY;RHS=TOTALMC,ONE, ADULTST,SEX,AVGESSAY, ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE,LARGER$ Ordnary least squares regresson Weghtng varable = none Dep. var. = TOTESSAY Mean= , S.D.= Model sze: Observatons = 3710, Parameters = 15, Deg.Fr.= 3695 Resduals: Sum of squares= , Std.Dev.= Ft: R-squared= , Adjusted R-squared = Model test: F[ 14, 3695] = , Prob value = Dagnostc: Log-L = , Restrcted(b=0) Log-L = LogAmemyaPrCrt.= 3.509, Akake Info. Crt.= Autocorrel: Durbn-Watson Statstc = , Rho = Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X TOTALMC E Constant ADULTST E-02 SEX AVGESSAY E ESLFLAG E-01 EC EN E-01 MA MA MA SMALLER SMALL LARGE LARGER E E-01 (Note: E+nn or E-nn means multply by 10 to + or -nn power.) Theoretcal consderatons dscussed n Module Two, Part One, suggest that these least squares estmates nvolve a smultaneous equaton bas that s brought about by an apparent reverse causalty between the two forms of testng. Consstent estmaton of the parameters n ths smultaneous equaton system s possble wth two-stage least squares, where our nstrument ( M ˆ ) for M s obtaned by a least squares regresson of M on SEX, ADULTST, AVGMC, AVGESSAY, ESLFLAG,SMALLER,SMALL, LARGE, LARGER, EC011, EN093, MA081, MA082, and MA083. Our nstrument ( Wˆ ) for W s obtaned by a least squares regresson of W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 5

6 W on SEX, ADULTST, AVGMC, AVGESSAY, ESLFLAG, SMALLER, SMALL, LARGE, LARGER, EC011, EN093, MA081, MA082, and MA083. LIMDEP wll do these regressons and the subsequent regressons for M and W employng these nstruments va the followng commands, whch yeld the subsequent output: 2SLS; LHS = TOTALMC; RHS = TOTESSAY,ONE, ADULTST,SEX,AVGMC, ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE, LARGER; INST = ONE,SEX, ADULTST,AVGMC,AVGESSAY,ESLFLAG, SMALLER,SMALL,LARGE,LARGER,EC011,EN093,MA081,MA082,MA083$ Two stage least squares regresson Weghtng varable = none Dep. var. = TOTALMC Mean= , S.D.= Model sze: Observatons = 3710, Parameters = 15, Deg.Fr.= 3695 Resduals: Sum of squares= , Std.Dev.= Ft: R-squared= , Adjusted R-squared = (Note: Not usng OLS. R-squared s not bounded n [0,1] Model test: F[ 14, 3695] = 67.63, Prob value = Dagnostc: Log-L = , Restrcted(b=0) Log-L = LogAmemyaPrCrt.= 2.529, Akake Info. Crt.= Autocorrel: Durbn-Watson Statstc = , Rho = Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X TOTESSAY E E Constant ADULTST E-02 SEX E AVGMC E ESLFLAG E-01 EC EN E-01 MA MA MA SMALLER SMALL E LARGE LARGER E E-01 2SLS; LHS = TOTESSAY; RHS = TOTALMC,ONE, ADULTST,SEX,AVGE SSAY,ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL, LARGE,LARGER; INST = ONE,SEX, ADULTST,AVGMC,AVGESSAY,ESLFLAG, SMALLER,SMALL,LARGE,LARGER,EC011,EN093,MA081,MA082,MA083$ Two stage least squares regresson Weghtng varable = none Dep. var. = TOTESSAY Mean= , S.D.= Model sze: Observatons = 3710, Parameters = 15, Deg.Fr.= 3695 Resduals: Sum of squares= , Std.Dev.= Ft: R-squared= , Adjusted R-squared = (Note: Not usng OLS. R-squared s not bounded n [0,1] Model test: F[ 14, 3695] = , Prob value = Dagnostc: Log-L = , Restrcted(b=0) Log-L = LogAmemyaPrCrt.= 4.005, Akake Info. Crt.= Autocorrel: Durbn-Watson Statstc = , Rho = Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X TOTALMC E Constant ADULTST E-02 W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 6

7 SEX AVGESSAY E ESLFLAG E-01 EC EN E-01 MA MA MA SMALLER SMALL LARGE LARGER E E-01 The 2SLS results dffer from the least squares results n many ways. The essay mark or score, W, s no longer a sgnfcant varable n the multple-choce regresson and the multplechoce mark, M, s lkewse nsgnfcant n the essay regresson. Each score appears to be measurng somethng dfferent when the regressor and error-term-nduced bas s elmnated by our nstrumental varable estmators. Both the mean multple-choce and mean essay scores contnue to be sgnfcant n ther respectve equatons. But now beng female s nsgnfcant n explanng the multple-choce test score. Beng female contnues to have a sgnfcant postve effect on the essay score. DURBIN, HAUSMAN AND WU TEST FOR ENDOGENEITY The theoretcal argument s strong for treatng multple-choce and essay scores as endogenous when employed as regressors n the explanaton of the other. Nevertheless, ths endogenety can be tested wth the Durbn, Hausman and Wu specfcaton test, whch s a two-step procedure n LIMDEP versons pror to Ether a Wald statstc, n a Ch-square ( χ ) test wth K* degrees of freedom, or an F statstc wth K* and n (K + K*) degrees of freedom, s used to test the jont sgnfcance of the contrbuton of the predcted values ( Xˆ *) of a regresson of the K* endogenous regressors, n matrx X*, on the exogenous varables (and column of ones for the constant term) n matrx Z: y=xβ+ X*γ ˆ + ε*, where X* = Zλ +u, X*=Zλ ˆ ˆ, and λˆ s a least squares estmator of λ. H o : γ = 0, the varables n Z are exogenous H : γ 0, at least one of the varables n Z s endogenous A In our case, K* = 1 when the essay score s to be tested as an endogenous regressor n the multple-choce equaton and when the multple-choce regressor s to be tested as endogenous n the essay equaton. ˆX* s an n 1vector of predcted essay scores from a regresson of essay scores on all the exogenous varables (for subsequent use n the multple-choce equaton) or an n 1vector of predcted multple-choce scores from a regresson of multple-choce scores on all the exogenous varables (for subsequent use n the essay equaton). Because K* = 1, the relevant W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 7

8 test statstc s ether the t, wth n (K + K*) degrees of freedom for small n or the standard normal, for large n. In LIMDEP, the predcted essay score s obtaned by the followng command, where the specfcaton ;keep=essayhat tells LIMDEP to predct the essay scores and keep them as a varable called Essayhat : Regres; lhs= TOTESSAY; RHS= ONE,ADULTST,SEX, AVGESSAY,AVGMC, ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE,LARGER ;keep=essayhat$ The predcted essay scores are then added as a regressor n the orgnal multple-choce regresson: Regress;LHS=TOTALMC;RHS=TOTESSAY,ONE,ADULTST,SEX,AVGMC, ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE, LARGER, Essayhat$ T he test statstc for the Essayhat coeffcent s then used n the test of endogenety. In the below LIMDEP output, we see that the calculated standard normal test statstc z value s , whch far exceeds the absolute value of the 0.05 percent Type I error crtcal 1.96 standard normal value. Thus, the null hypothess of an exogenous essay score as an explanatory varable for the multple-choce score s rejected. As theorzed, the essay score s endogenous n an explanaton of the multple-choce score. --> Regres; lhs= TOTESSAY; RHS= ONE,ADULTST,SEX, AVGESSAY,AVGMC, ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE,LARGER;keep=Esayhat$ Ordnary least squares regresson Weghtng varable = none Dep. var. = TOTESSAY Mean= , S.D.= Model sze: Observatons = 3710, Parameters = 15, Deg.Fr.= 3695 Resduals: Sum of squares= , Std.Dev.= Ft: R-squared= , Adjusted R-squared = Model test: F[ 14, 3695] = , Prob value = Dagnostc: Log-L = , Restrcted(b=0) Log-L = LogAmemyaPrCrt.= 4.025, Akake Info. Crt.= Autocorrel: Durbn-Watson Statstc = , Rho = Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X Constant ADULTST E-02 SEX AVGESSAY E AVGMC E ESLFLAG E-01 EC EN E-01 MA MA MA SMALLER SMALL W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 8

9 LARGE LARGER E E-01 (Note: E+nn or E-nn means multply by 10 to + or -nn power.) --> Regress;LHS=TOTALMC;RHS=TOTESSAY,ONE, ADULTST,SEX,AVGMC, ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE,LARGER, Essayhat$ Ordnary least squares regresson Weghtng varable = none Dep. var. = TOTALMC Mean= , S.D.= Model sze: Observatons = 3710, Parameters = 16, Deg.Fr.= 3694 Resduals: Sum of squares= , Std.Dev.= Ft: R-squared= , Adjusted R-squared = Model test: F[ 15, 3694] = , Prob value = Dagnostc: Log-L = , Restrcted(b=0) Log-L = LogAmemyaPrCrt.= 1.825, Akake Info. Crt.= Autocorrel: Durbn-Watson Statstc = , Rho = Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X TOTESSAY E Constant ADULTST E-02 SEX E E AVGMC E ESLFLAG E-01 EC E EN E-01 MA MA MA SMALLER SMALL E LARGE LARGER E E-01 ESSAYHAT E (Note: E+nn or E-nn means multply by 10 to + or -nn power.) The smlar estmaton routne to test for the endogenety of the multple-choce test score n the essay equaton yelds a calculated z test statstc of , whch far exceeds the absolute value of ts 1.96 crtcal value. Thus, the null hypothess of an exogenous multple-choce score as an explanatory varable for the essay score s rejected. As theorzed, the multple-choce score s endogenous n an explanaton of the essay score. --> Regress;LHS=TOTALMC; RHS=ONE, ADULTST,SEX, AVGMC,AVGESSAY, ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE,LARGER; keep=mchat$ Ordnary least squares regresson Weghtng varable = none Dep. var. = TOTALMC Mean= , S.D.= Model sze: Observatons = 3710, Parameters = 15, Deg.Fr.= 3695 Resduals: Sum of squares= , Std.Dev.= Ft: R-squared= , Adjusted R-squared = Model test: F[ 14, 3695] = , Prob value = Dagnostc: Log-L = , Restrcted(b=0) Log-L = LogAmemyaPrCrt.= 2.376, Akake Info. Crt.= Autocorrel: Durbn-Watson Statstc = , Rho = Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 9

10 Constant ADULTST E-02 SEX AVGMC E AVGESSAY E E ESLFLAG E-01 EC EN E-01 MA MA MA SMALLER SMALL E LARGE LARGER E E-01 --> Regress;LHS=TOTESSAY;RHS=TOTALMC,ONE, ADULTST, SEX,AVGESSAY, ESLFLAG,EC011,EN093,MA081,MA082,MA083,SMALLER,SMALL,LARGE,LARGER, MChat$ Ordnary least squares regresson Weghtng varable = none Dep. var. = TOTESSAY Mean= , S.D.= Model sze: Observatons = 3710, Parameters = 16, Deg.Fr.= 3694 Resduals: Sum of squares= , Std.Dev.= Ft: R-squared= , Adjusted R-squared = Model test: F[ 15, 3694] = , Prob value = Dagnostc: Log-L = , Restrcted(b=0) Log-L = LogAmemyaPrCrt.= 3.473, Akake Info. Crt.= Autocorrel: Durbn-Watson Statstc = , Rho = Varable Coeffcent Standard Error b/st.er. P[ Z >z] Mean of X TOTALMC E Constant ADULTST E-02 SEX AVGESSAY E ESLFLAG E-01 EC EN E-01 MA MA MA SMALLER SMALL LARGE LARGER E E-01 MCHAT (Note: E+nn or E-nn means multply by 10 to + or -nn power.) CONCLUDING COMMENTS Ths cookbook-type ntroducton to the use of nstrumental varables and two-stage least squares regresson and testng for endogenety has just scratched the surface of ths controversal problem n statstcal estmaton and nference. It was ntended to enable researchers to begn usng nstrumental varables n ther work and to enable readers of that work to have an dea of what s beng done. To learn more about these methods there s no substtute for a graduate level textbook treatment such as that found n Wllam Greene s Econometrc Analyss. W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 10

11 REFERENCES Becker, Wllam E. and Carol Johnston (1999). The Relatonshp Between Multple Choce and Essay Response Questons n Assessng Economcs Understandng, Economc Record (Economc Socety of Australa), Vol. 75 (December): Greene, Wllam (2003). Econometrc Analyss. 5 th Edton, New Jersey: Prentce Hall. ENDNOTES In the default mode, relatvely large samples are requred for 2SLS n LIMDEP because a routne amed at provdng consstent estmators s employed; thus, for example, no degrees of freedom adjustment s made for varances;.e., 2 th 2 ˆ (1 / ) ( ) σ = n predcton error As Wllam Greene states, ths s consstent wth most publshed sources, but (curously enough) nconsstent wth most other commercally avalable computer programs. The degrees of freedom correcton for small samples s obtanable by addng the followng specfcaton to the 2SLS command: ;DFC In Lmdep verson 9.0.4, the followng command wll automatcally test x3 for endogenety: Regress; lhs=y; rhs=one,x2,x3; nst=one,x2,x4; Wu test$ Because x3 s not an nstrument, LIMDEP knows the test for endogenety s on ths varable. W. E. Becker Module Two, Part Two: LIMDEP IV Hands On Secton Sept. 15, 2008 p. 11

MODULE TWO, PART THREE: ENDOGENEITY, INSTRUMENTAL VARIABLES AND TWO-STAGE LEAST SQUARES IN ECONOMIC EDUCATION RESEARCH USING STATA

MODULE TWO, PART THREE: ENDOGENEITY, INSTRUMENTAL VARIABLES AND TWO-STAGE LEAST SQUARES IN ECONOMIC EDUCATION RESEARCH USING STATA Part Three of Module Two demonstrates how to address problems of endogenety