PhD/MA Econometrics Examination. August PART A (Answer any TWO from Part A)

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1 Economercs Comprehensve Exam Augus 08 Toal Tme: 8 hours MA sudens are requred o answer from A and B. PhD/MA Economercs Examnaon Augus 08 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) Q. Fundamenals of Economercs a. Defne. Economercs. A sasc. An esmae v. An esmaor b. Wre ou an OLS model.. Defne your varables (nclude a leas 4 ndependen varables) and descrbe your model.. Wre ou he hypohess for each coeffcen (null and alernave n mahemacal noaon) and use economc heory o defend your expeced resul (senences). c. Defne mulcollneary and descrbe how you would es for. If mulcollneary exss n heory bu no n pracce, wha should you do? d. In words and mahemacal noaon descrbe how o deal wh measuremen error n your daa:. When he error s n your dependen varable. When he error s n one of your ndependen varables Q. Ordnary Leas Squares (OLS) a. Wre ou he OLS equaon n marx form. b. Wre ou he marces and sae her dmensons. c. Sae he classcal assumpons n words and equaons. d. Derve he normal equaons. e. Demonsrae ha he OLS esmaor s BLUE. f. From XX ee = 0, we can derve several properes. Sae hese properes. Hn: here are 6 and 5 of hem requre ha he OLS regresson ncludes a consan. Q3. The Varance of Leas Squares We know vvvvvv(bb) = σσ (XX XX), bu σσ s an unknown parameer. Therefore, o fnd vvvvvv (bb), we need o fnd a good esmaor for σσ. a. Derve ha esmaor. b. Wre down he sandard error of bb. c. Wha s he sandard of error of bb used for? Or explan why dd we derve? Page of 9

2 Economercs Comprehensve Exam Augus 08 PART B: Answer any Two [Shor verbal descrpve answer whou mahemacal proofs, seps, and necessary dervaon wll no earn you full cred.] Q4. Consder he model y = Xβ + ε where 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 ). c. Assumng Σ = σ ε Ω = ( P' P), derve he GLS esmaor for β. Show ha βˆ GLS s unbased and he varance for he GLS predced resdual s σ I d. Suppose he resdual ε akes he followng form ε = ρε + u where u ~ (0, σ u ) 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 ε, where s n he scenaro presened n c. f. Wha do he marces for Σ and Ω look lke n hs scenaro, where Σ = σ Ω. s ε Page of 9

3 Economercs Comprehensve Exam Augus 08 Q5. Le ~ y be some unobserved laen varable such ha y~ = xβ + ε where εε~(0, σσ ) You observe y y and x, f y~ > 0 = 0 oherwse =,...,, such ha Defne φ (θ ) as he pdf for a sandard normal and Φ (θ ) as he cdf for he sandard normal. oe: Φ( z) θ z = φ( z) θ a.) Wha s θ, he denfable parameer of neres n hs problem? b.) Derve he probables ha y = and y = 0 for ndvdual. c.) Derve he conrbuon of each ndvdual n your sample o he overall lkelhood funcon (.e., derve L (θ )) and he ndvdual log-lkelhood funcon. d.) Derve he score funcon needed o denfy θˆ MLE. e.) Explan wha s mpled by he smplfed form of he Score funcon (.e., wha s he mpled orhogonaly condon). Page 3 of 9

4 Economercs Comprehensve Exam Augus 08 Q6. A researcher usng daa for a sample of 340 female employees 5 years of age and over o nvesgae he relaonshp beween employees hourly wage raes Y (measured n dollars per hour) and her age X (measured n years). The populaon regresson equaon akes he form of equaon () Y = β 0 + βx + ε. Prelmnary analyss of he sample daa produces he followng sample nformaon: V = 340 ( y y) = ( x x) y e y ( x x)( y y) = x = = = x = = x y = = g. Use he above nformaon o compue OLS esmaes of he nercep coeffcen β 0 and he slope coeffcen β. (Hn: Remember he formula for bea based on he normal equaons) h. Inerpre he slope coeffcen esmae you calculaed n par a.e., explan n words wha he numerc value you calculaed for ˆβ means.. Calculae an esmae of s, he esmaed error varance. (Remember ha you echncally have explanaory varables snce here s a column of ones mulpled by he nercep, meanng k=) j. Calculae he esmae of var( ˆ β ). R, he coeffcen of deermnaon for he esmaed OLS k. Compue he value of sample regresson. Brefly explan wha he value ha you have calculaed for R means. l. Calculae he sample value of he -sasc for esng he null hypohess H : β = 0 agans he alernave hypohess H : β 0. (oe: You are no requred o oban or sae he nference of hs es. Jus calculae he es-sasc self). o Page 4 of 9

5 Economercs Comprehensve Exam Augus 08 PART C: Answer any Two [Shor verbal descrpve answer whou mahemacal proofs, seps, and necessary dervaon wll no earn you full cred.] Q7. Gven he followng -equaon sysem, CCh = αα 0 + αα MMMMMMheeeeeeeeeeee + αα CCh + αα 3 FFFFFFheeeeeeeeeeeeeeeeeeeeeeee + uu FFFFFFheeeeeeeeeeeeeeeeeeeeeeee = ββ 0 + ββ PPPPPPPPPPPPPP + ββ FFFFFFheeeeAAAAAA + uu FaherRemance varable capures he absence of fahers workng overseas sendng money back home o he famly. a. How would you esmae he chld labor equaon usng a -sls mehod? Brefly dscuss he wo seps. b. Lkewse, how would you esmae he Remance equaon? Explan he necessary seps. c. Se up hs sysem n a grand marx noaon YY = ZZ ββ + UU () d. Derve he Var-Cov(U). You can use he generc noaon, where u and u are sacked whn U. e. Dscuss he erave 3-SLS esmaon mehod o oban he esmae of ββ and s varance-covarance marx; show all he necessary seps and dervaons. f. Wha s he benef of usng a 3-SLS mehod over he -SLS? g. BOUS exercse: how would you se up a model lke hs as a FIML? Page 5 of 9

6 Economercs Comprehensve Exam Augus 08 Q8. epal Sudy Cener s plannng o conduc a sudy o help a clnc n a rural vllage n epal 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 40 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 0), he person s household ncome (n Rs0,00 / monh), and he prce of a plan (n Rs00 / 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 Comprehensve General Basc Comprehensve General Basc Comprehensve General Basc Addonally, we wll also collec nformaon on he followng varables: Receve Remance (yes/no), o Of Chldren, o 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==), we see ha he case-specfc varables hhnc, age, Remance, ochldren, 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, o Of Chldren, o of Clnc Vss Per Sx Monh, and Dsance o Clnc The varable MHal (mcro healh nsurance alernaves) labels he alernaves, and he bnary varable choce ndcaes he chosen alernave ( s coded for he chosen plan, and 0 oherwse). For smplcy, consder only hree varables for model se up (age, ncome, and prce). Page 6 of 9

7 Economercs Comprehensve Exam Augus 08 A) Se up a Random Uly Model (RUM). Show all he seps. B) Presen he log lkelhood funcon. Show all he seps. (You may assume ha he ncome and age have he same mpac on he uly funcons.) C) Show an example of he daa able for hs problem. D) Wha are dfferences beween he smple log, he mul-nomal log, and he condonal log? Page 7 of 9

8 Economercs Comprehensve Exam Augus 08 Q9. Consder he followng concepual framework around he concep of boom and bus (collapse-and-regeneraon) whn a socal-ecologcal sysems: Creang nsuons o mee he challenge of susanably s arguably he mos mporan ask confronng socey; s also daunngly complex. Ecologcal, economc, and socal elemens all play a role, bu despe ongong effors, researchers have ye o succeed n negrang he varous dscplnes n a way ha gves adequae represenaon o he nsghs of each. Panarchy, a erm devsed o descrbe evolvng herarchcal sysems wh mulple nerrelaed elemens, offers an mporan new framework for undersandng and resolvng hs dlemma. Panarchy s he srucure n whch sysems, ncludng hose of naure (e.g., foress) and of humans (e.g., capalsm), as well as combned human-naural sysems (e.g., nsuons ha govern naural resource use such as he Fores Servce), are nerlnked n connual adapve cycles of growh, accumulaon, resrucurng, and renewal. These ransformaonal cycles ake place a scales rangng from a drop of waer o he bosphere, over perods from days o geologc epochs. By undersandng hese cycles and her scales, researchers can denfy he pons a whch a sysem s capable of accepng posve change, and can use hose leverage pons o foser reslence and susanably whn he sysem. Dagrammacally, he collapse funcon s represened by he green funcon and he regenerave phase s represened by he grey funcon: Conex: In he far norh-wes regon of Musang Valley near he border of epal-tbe, wo vllages began facng severe waer crss as a resul of he exend drough perod, hus causng hese vllages (and vllagers) o relocae (Swss Sudy). The green cycle represens he collapsng of he socal-ecologcal sysems. The followng non-lnear model s posulaed o represen he collapse phase: PP = αα [ + exp {µ ( xx ββ)}] + uu Where, x = annual drough ndex. ββ = delay parameer around whch me he collapse begns o ranson from bad o worse; P = Agrculure producon ndex (mosly from grazng, apple farmng, and hgh alude herbal plans, all seem o have been affeced by he drough.). a. Do you know wha αα represens? Page 8 of 9

9 Economercs Comprehensve Exam Augus 08 b. If you were o smulae hs collapse scenaro, wha sgn (posve or negave) would you assgn o he parameer: µ? c. Presen he sep-by-sep mehod of esmang hs funcon usng he non-lnear leas squares (use he algorhm of your choce, e.g.: Gauss-ewon). OR Se up a maxmum lkelhood funcon and dscuss he opmzaon algorhm (dervaon of ewon-raphson). Page 9 of 9