PhD/MA Econometrics Examination. January, 2019

Transcription

1 Economercs Comprehensve Exam January 2019 Toal Tme: 8 hours MA sudens are requred o answer from A and B. PhD/MA Economercs Examnaon January, 2019 PhD sudens are requred o answer from A, B, and C. The answers should be presened n erms of equaons, sascal deals, and wh necessary proofs and sascal deducon. Verbal and bref descrpve dscussons wll no be suffcen. PART A (Answer any TWO from Par A) Q1. Use he able below. HWSEI s a consruced varable ha assgns a Hauser and Warren Socoeconomc Index (SEI) score o each occupaon usng he modfed verson of he 1990 occupaonal classfcaon scheme avalable n he OCC1990 varable. The HWSEI varable s a measure of occupaonal saus based upon he earnngs and educaonal aanmen assocaed wh each caegory n he 1990 occupaonal scheme. Source SS Df MS Number of obs = F(???,???) = Model Prob > F = 0 Resdual R-squared =??? Adj R- squared = Toal Roo MSE = HWSEI Coef. SE. P> [95% Conf. Inerval] Age ??? 20.55????????? Age^ Female ????????? Nursery school o grade Grade 5, 6, 7, or Grade Grade Grade Grade year of college years of college years of college years of college Consan Omed groups: "male" and "no school or N/A" 1 of 11

2 Economercs Comprehensve Exam January 2019 a) Inerpre he coeffcen (and oher relevan nformaon) on Grade 11. Grade 11 s a dummy varable for people who have compleed hrough 11 h grade,.e. one year shor of fnshng hgh school. b) Now, lookng a he coeffcen on he varable Grade 12, s here an mporan lesson ha we can learn? c) Calculae RR 2. Explan your answer. Wha does he number mean? d) Fnd he degrees of freedom for he F. e) Wha s he F value? Wha does ell you? Can you calculae from he nformaon gven above? If you can calculae F, nerpre he value. f) Calculae he mssng values n he row labeled Age. g) Calculae he margnal effec of age from he regresson resuls. h) Calculae he mssng values n he row labeled Female. ) Wha s he Dummy Varable Trap? *** You should have flled n 10 blanks (???) n he able. **** 2 of 11

3 Economercs Comprehensve Exam January 2019 Q2. Fundamenals of OLS a. Wre ou he OLS equaon n marx form. Also, wre ou he marces and sae her dmensons. b. Sae he OLS assumpons n mahemacal saemens and n senences (words). c. Show ha he OLS esmaor s BLUE and defne BLUE. Show all pars: B, L, U, and E. d. Wha are he properes (hn: here are sx) of he OLS esmaor? Sae hem n mahemacs and words. Also, sae any requremens whch are necessary for hese properes o hold. e. Gven he properes n par d, wha can you nfer abou he dsurbances from he resduals? f. Wre ou a smple OLS model. Defne your varables and descrbe how your model mgh mee or no mee all he assumpons you saed above. 3 of 11

4 Economercs Comprehensve Exam January 2019 Q3. Probably Theory, Dsrbuons, and More a. Sae Bayes Theorem b. In words, descrbe wha Bayes Theorem means c. Wha dsrbuon s below: d. Fnd he frs and second momens of hs dsrbuon drecly from he dsrbuon self. e. Fnd he frs and second momens usng he momen generang funcon. Also, dscuss f hese momens are he same or dfferen from he momens you found n par d and why or why no. f. Ths dsrbuon has a propery called memoryless. Prove ha hs dsrbuon s memoryless. g. Name a praccal applcaon of hs dsrbuon,.e., where does naurally occur? h. Name he hrd and fourh cenral momens name hem do no calculae hem.. Defne and draw he PDF and CDF of he dsrbuon. 4 of 11

5 Economercs Comprehensve Exam January 2019 Par B: Answer any wo of he followng hree quesons [Shor verbal descrpve answer whou mahemacal proofs, seps, and necessary dervaon wll no earn you full cred.] Q4. You have been commssoned o nvesgae he relaonshp beween he brh weghs of newborn females and he number of prenaal vss o a physcan or mdwfe ha her mohers made durng pregnancy. The dependen varable s bwgh, he brh wegh of he h newborn female measured n grams. The explanaory varable s pnvss, he number of prenaal vss of he h newborn s moher durng pregnancy, measured n number of vss. The model you propose o esmae s gven by he populaon regresson equaon bwgh β + pnvss + u. = 0 β1 Your research asssan has used 857 sample observaons on bwgh and pnvss o esmae he followng OLS sample regresson equaon, where he fgures n parenheses below he coeffcen esmaes are he esmaed sandard errors of he coeffcen esmaes: bwgh = pnvss + uˆ = 1,..., N, N=857 ( ) ( ) a. Perform a es of he null hypohess H 0 : β1 = 0 agans he alernave hypohess: H : β 0 A 1 a he 1% sgnfcance level (.e., for sgnfcance level α = 0. 01). Noe, he crcal value for he -dsrbuon a he one percen level s Show how you calculaed he es sasc. Sae he decson rule you use, and he nference you would draw from he es. Wha would you conclude form he resuls of hs es? b. Compue he wo-sded 95% confdence nerval for he nercep coeffcen β 0. Use hs wosded 95% confdence nerval for β 0 o es he hypohess ha he mean brh wegh of newborn females whose mohers made no prenaal vss o a physcan or mdwfe equals 3,000 grams. Sae he null hypohess H 0 and he alernave hypohess H A. Sae he decson rule you use, and he nference you would draw from he es. Noe, a he 95% confdence level, he crcal value for he -dsrbuon s c. Wha s he nerpreaon of he coeffcen on pnvss? d. Wha assumpon s necessary n order o nerpre he coeffcen on Wha s he nerpreaon of he coeffcen on pnvss as causal? Do you hnk ha assumpon s sasfed? Why or why no? 5 of 11

6 Economercs Comprehensve Exam January 2019 Q5. Suppose y 1, y2,..., yn be d wh a posson dsrbuon such ha f ( y λ y e λ λ) = y! a. Wha s he lkelhood of observng your daa (.e., wha s he lkelhood funcon for your sample)? b. Derve he log lkelhood and score funcons for esmang he parameer λ. c. Derve he Maxmum Lkelhood Esmaor for λ. Derve he asympoc varance for λˆ MLE usng he nformaon marx mehod (Hn: Remember ha EE[θθ] = θθ MMMMMM because MLE gves a conssen esmae of he parameer. Also use he relaonshp derved n par c o subsue ou all random varables from he asympoc varance). 6 of 11

7 Economercs Comprehensve Exam January 2019 Q6. Consder he model where y = Xβ + ε 2 var( ε ) = Σ σ I a. Whch OLS assumpon fals? Wha are he mplcaons of ha falure for he OLS esmaor? b. Derve he properes for he OLS esmaor n hs scenaro (.e., wha s he mean and varance of he OLS esmaor for β ). 2 1 c. Assumng Σ = σ ε Ω = ( P' P), derve he GLS esmaor for β. Show ha βˆ GLS s 2 unbased and he varance for he GLS predced resdual s σ I d. Suppose he resdual ε akes he followng form ε ρε + u = 1 2 where u ~ N(0, σ ) Wha s he saonary assumpon? Derve he properes of ε assumng he saonary assumpon holds (.e. derve he mean and varance of ε ). e. Derve he correlaon beweenε and s u ε, where s 1 n he scenaro presened n c. 2 f. Wha do he marces for Σ and Ω look lke n hs scenaro, where Σ = σ Ω. ε 7 of 11

8 Economercs Comprehensve Exam January 2019 PART C: Answer any Two [Shor verbal descrpve answer whou mahemacal proofs, seps, and necessary dervaon wll no earn you full cred.] Q7. The Nepal Sudy Cener s plannng o conduc a sudy o help a clnc n a rural vllage n Nepal s Gulm Dsrc o mplemen a mcro healh nsurance program. I plans o use a dchoomous choce expermen desgn o carry ou he sudy. The plan s o sample 420 households randomly from he hree communes ha lay around he clnc --s cachmen area. Each communy has nne wards. The samplng wll be performed by usng he proporonal samplng desgn represenng all he wards from each of he clnc. The households are presened wh opons o enroll n one of hree mcro healh nsurance plans: Basc (clnc vss), General (clnc + plus pharmacy), Comprehensve (clnc vss, pharmacy + mnor surgery). The hree alernaves are presened below: c=(comprehensve, General, Basc) We would expec a person s uly relaed o each of he hree alernaves o be a funcon of boh personal characerscs (such as ncome, age ec..) and characerscs of he healh care plan (such as s prce/premum). We colleced daa would look lke he able below: person s age (dvded by 10), he person s household ncome (n Rs10,00 / monh), and he prce of a plan (n Rs100 / 6 monhs). The frs hree cases from he daa are shown below. I s n he long form. HHd MH_Al ch Choce hhnc age Premum 1 1 Comprehensve General Basc Comprehensve General Basc Comprehensve General Basc Addonally, we wll also collec nformaon on he followng varables: Receve Remance (yes/no), No Of Chldren, No of Clnc Vss Per Sx Monh, and Dsance o Clnc (mnues of walkng dsance). These varables are no shown n he able o save space. Takng he frs case (d==1), we see ha he case-specfc varables hhnc, age, Remance, NoChldren, and Dsance are consan across alernaves, whereas he alernave-specfc varable prce vares over alernaves. Addonally, we also colleced nformaon on he followng varables: Receve Remance, No Of Chldren, No of Clnc Vss Per Sx Monh, and Dsance o Clnc 8 of 11

9 Economercs Comprehensve Exam January 2019 The varable MHal (mcro healh nsurance alernaves) labels he alernaves, and he bnary varable choce ndcaes he chosen alernave ( s coded 1 for he chosen plan, and 0 oherwse). Q7.1. For smplcy, consder only hree varables for model se up (age, ncome, and prce/premum) and a bnary decson Basc versus Oherwse opons. Y()* = a+ b*age() + c*ncome()+ u() where f y*() >0 => y() = 1 (Non-basc opon chosen) else 0 f opon Basc chosen Se up a RUM framework wh lnear ndrec ules o analyze hs model. Show all he seps and ules ec Derve he log-lkelhood funcon usng he logsc assumpon. Q7.2 Now, add he premum varable o hs model and consder all hree opons: basc, general, and comprehensve. Y()* = a+ b*age() + c*ncome()+ d*premum() +u() Se up a RUM framework wh lnear ndrec ules o analyze hs model. Show all he seps n deals. (You may assume ha he age and prce have he same mpac on he preference funcons, bu allow ncome o have dfferen mpac.) Presen he correspondng daa able n he wde form as a se up for a long-hand mle scrp Usually, we use some sor of cluserng adjusmen for he sandard errors vce (cluser d). In hs case, whch cluserng d would you use ndvdual d, communy d, or ward d and why? 9 of 11

10 Economercs Comprehensve Exam January 2019 Q8. Dscuss he followng wh proper noaons (whenever approprae) and examples (NOT JUST WORDS and DEFINITIONAL SENTENCES): 8.1 Mulnomal Log versus condonal log. 8.2 Exreme value dsrbuon and logsc dsrbuon. 8.3 Mulnomal log versus Ordered log. 8.4 Mulnomal log and s lmaon, and he Nesed Log and s benef. 8.5 Posson, Negave Bnomal, Zero-Inflaed, and Hurdle 10 of 11

11 Economercs Comprehensve Exam January 2019 Q9. Consder he followng 2-equaon model for a wo-counry arff realaon: TTTTTTTTTTTT_UUUU() = CC1 + φφ11 TTTTTTTTTTTT_UUUU( 1) + φφ12 TTTTTTTTTTTT_CCh( 1) + uu1() TTTTTTTTTTTT_CCh() = CC2 + φφ21 TTTTTTTTTTTT_UUUU( 1) + φφ22 TTTTTTTTTTTT_CCh( 1) + uu2() a. Derve he var-cov marx of he error vecor of hs 2-equaon model. Show seps n deal. b. Dscuss he sep-by-sep GLS mehod of esmaon. c. Why s called a seemngly unrelaed regresson (SUR), and no a smulaneous model? d. Show ha he OLS and GLS esmaors become dencal when 1) he cross-error covarances beween u1 and u2 are zero OR 2) he rgh hand varables are dencal. Se up he FIML log-lkelhood for hs model. 11 of 11