Lecture 1: Empirical Research and Sampling

Transcription

1 Lectre Notes o Advaced coometrcs Lectre : mprcal Research ad Samplg Taash Yamao Fall Semester 5 Wh do we std coometrcs? For almost all of s ths class, we std ecoometrcs becase we eed to codct sold emprcal aalses, stead of advacg ecoometrc theor. More precsel for o, o std ecoometrcs becase o eed t as a tool to wrte a thess or dssertato to complete or program. Some research papers do ot clde ecoometrc aalses bt se coceptal framewors ad sbject-matter owledge to spport ther argmets. The ma add descrptve data wth some basc tables ad fgres. I most cases academc stdes, however, coceptal ad descrptve data aalses ma ot be eogh. Recet ears, data are becomg avalable more tha before ad powerfl ecoometrc pacages are easer to se tha ever. Ths, t s mch easer for researchers to test ther hpotheses emprcall wth good data ad sophstcated ecoometrcs methods. Ths, there are fewer ecses for ot codctg ecoometrc-aalses. Althogh ecoometrcs ca be a powerfl tool to test hpotheses, t s qte eas to msse t. To covce readers that o have terestg ad relable reslts, o frst eed to covce them that o have arrved robst coclsos based o sod data ad well-thoght-ot methods. I hope ths corse wll help o o ths pot. Ths corse ses two tetboos: Itrodctor coometrcs d edto b Jeffre Wooldrge ad coometrcs Aalss 5 th edto b Wllam H. Greee. The frst boo s wrtte for emprcal researchers who wat to lear how to codct ecoometrc aalses. As the sb-ttle of the frst boo -- A Moder Approach -- sggests, ths boo also covers recet developmets appled ecoometrcs. The feld of Appled coometrcs s developg rapdl de to mprovemets persoal compters ad epadg avalablt of data. B sg ecoometrc-pacages that are crretl avalable the maret, t does ot tae mch tme to estmate complcated models. Large data are avalable throgh Iteret or are sold CD-ROMs. Ths meas that a wde varet of methods are avalable to o. Prevosl, those advaced methods are ol avalable to people who had advaced owledge o ecoometrcs. Ths there was lttle eed for trodctor tetboos to epla advaced methods. However, becase those advaced methods are crretl avalable from software pacages, sch as STATA, there s a eed for a trodctor

2 tetboo to epla advaced methods for people who do ot have advaced owledge o ecoometrcs. Wooldrdge realzes ths ad has wrtte the tetboo. However, becase ths tetboo covers ma methods a sgle volme, t does ot epla theor detals. Ths, I se the secod tetboo Greee s coometrc Aalss to std ecoometrc theor detals. specall, the secod boo eplas ecoometrc theor matr. coometrc theor ca be eplaed clearl wth matr as o wll fd the lectres. I wll provde o a basc trag matr calclato ths corse. I start m lectre o how to codct emprcal projects becase the ma prpose of ths corse s to provde o a tool to wrte emprcal papers. I th t s mportat for o to have a clear dea o how o are gog to se ecoometrcs or research. Carrg ot a mprcal Project Posg a qesto There are some was to pose a research qesto: New Theor New Isses New Data New Methods Chaged or Dfferet vromets New Theor: Yo ma pose ths tpe of qestos whe o come p wth a ew theor to loo at a sse. Yo ma provde a formal theor to a old sse. For stace, people ew that edcated people gaed some retrs from edcato eve before Chcago ecoomsts, sch as T.W. Shltz ad Gar Becer. However, Chcago ecoomsts created a ew theor of hma captal b coceptalzg edcato as vestmet hma captal. New Isses: Yo ma pose ths tpe of qestos whe o recogze emergg sses. These sses ma have ested before, bt o ma arge that these sses have ga mportace. Iteratoal trade ad face were ot mportat whe t was dffclt to trasfer goods or moe across cotres. Bt toda, t s ver dffclt to fd a place wthot teratoall-traded goods the world. As a reslt, teratoal trade ad face sses have gaed ther mportace ecoomc theor. New Data: Althogh ma data sets have become avalable recetl, ths s a relatvel ew pheomeo. There are ma emprcal sses that have ot bee eamed emprcall becase approprate data were ot avalable before. Good data are hard to come b. So whe o come across a ewl avalable data wth hgh qalt, o shold wrte a paper b sag Ths paper ses a ewl avalable data set whch eables s to overcome lmtatos prevos stdes

3 New Methods: o ca advace the owledge amog researchers b applg a ew ad more approprate method o old sses. B stdg ecoometrcs, o wll be able to appl advaced methods o old sses. Chaged or Dfferet vromets: hs s the most commo practce. Ths s to appl a estg research qesto ad method a chaged over tme evromet or dfferet regos or cotres. Yo ma codct a old research whe o sspect that the evromet has chaged over tme ad that reslts wold be dfferet from prevos oes. Or o ma codct a estg research a dfferet place or cotr. Bt how do o ow f or dea s reall ew? The aswer: Dg the lteratre. Lteratre Revew The most commo place to loo for research qestos s papers jorals we call them lteratre. It trs ot that o are ot the ol oe who eeds to codct ecoomc aalses. Ma people have doe or are dog eactl what o are dog, ad o ca fd other people s efforts ther pblshed papers jorals, boos, ad o Iteret. Althogh o ma wat to come p wth a brad-ew-qesto that obod has thoght abot t before, there are few qestos that people have ot thoght abot. Chaces are that some ecoomsts have thoght abot or seemgl-brad-ew-deas ma ears ago, f ot ma decades ago, ad have doe fe aalses. As o revew lteratre, tr to arrow dow or focs ad fd a le of research that s o the same research qesto. For stace, o ma wat to std edcato developg cotres ad start readg abot t. As o go throgh prevos stdes o wll fd stdes o a the effects of edcato o farm prodctvt, b the effects of mothers edcato o chld health, c the ecoomc retrs of edcato labor marets, d the effects of school qalt o test-scorg, ad so o. ach oe of them cold be called a le of research. At the ed, alog the le of research, o shold loo for a gap the lteratre. Ad or paper, o shold be able to sa: Prevos stdes have fod these reslts before, bt we stll do ot ow the aswer to ths qesto. The prpose of ths paper, therefore, s to fd a aswer to ths qesto. I or aalss, o do ot eed to cover all of the prevos stdes o the sse o the le of research that o are terested. If o are wrtg a joral paper, or readers are sall eperts o the sbject. Probabl the ow abot prevos stdes better tha o do. Ths, o jst eed to meto relevat stdes to detf a gap the lteratre that o are gog to fll. 3

4 Data Collecto Hgh qalt data ca lead researchers to hgh qalt emprcal research. coometrcs ca help o to prodce hgh qalt emprcal research from hgh qalt data. Bt ecoometrcs ca ot help o to prodce hgh qalt emprcal research from low qalt data. The qalt of the data draws the pper lmt o the qalt of research. A emprcal aalss sg low qalt data wth ver advaced ecoometrc techqes s ot sefl practcall, eve f the techqe tself s mpressve. O the other had, a smple aalss wth hgh qalt data ca provde sefl formato. I am a strog belever of the latter method. Large data sets are sefl bt ot ecessarl better tha small data becase ofte large data sets lac detaled formato, althogh large observatos ca help arrowg stadard errors. More ad more data are becomg avalable from Iteret. Ths s great. Bt whe o se sch data, o frst eed to derstad what o get. It s dageros to se data wthot a fll owledge o how the data were collected ad arraged. Whe o collect data orself from felds or from secodar data sorces, o shold th ver carefll f o wll be able to aswer or research qestos wth the data o collect. If ot, o eed to search for dfferet data or chage or research qestos. Otherwse after ma moths or ears of efforts, o wold be told b or advsers or revewers of or paper that Well, t seems o ca t aswer or qestos wth or data coometrc Aalss The prpose of or research s to fd aswers to or research qestos ad covce people readers that o have relable reslts. To acheve or goal, o ma ot eed the most advaced ecoometrc methods. For the most cases, o probabl do ot eed the most advaced ecoometrc methods. Smple methods are sall sffcet ad powerfl. Yo shold, however, std advaced methods for followg reasos. Frst, o shold be able to choose the most approprate method amog all methods. Secod, o shold have a fll derstadg o the method of or choce. Thrd, o shold recogze the lmtatos of the method. Wrtg a mprcal Paper After or aalses, o eed to smmarze or fdgs a cocse maer. A short paper s desrable. Based o m lmted epereces, I wold sggest: a b Start wth a short paper wth a specfc qesto Wor wth people who have pblcatos 4

5 c d Mmc other papers Keep o wrtg; f o have bee stc oe secto of or paper, o shold start wrtg other sectos Wrtg a emprcal paper s ver dffclt. Bt o are ot aloe. Do t be dscoraged! Referece Brorse, B. Wade. 987 Observatos o the Joral Pblcato Process, North Cetral Joral of Agrcltral coomcs, Vol. 9, No.. O glsh Wrtg, I sggest: Str ad Whte The lemets of Stle, ew edto, Massachsetts: Ala ad Baco. [Ths s the classc.] O Coer, P.T. 996 Woe Is I: The grammarphobe s gde to better glsh pla glsh, New Yor: Pttam s Sos Pblshers. [Ths s f to read.] Data Ch. Wooldrdge; Ch. Lohr, 999 Oce o have research qestos, o eed to collect data. Yo ma collect data from secodar data sorces sch as offcal statstcs from govermets or other stttos, or o ma collect data orself wth or colleages. There are several tpes of data: Tme Seres Data Cross-Sectoal Data Pooled Cross Sectos Pael or Logtdal Data Whe o cosder sg a data set, o eed to derstad the followg: Observato Ut: a object o whch a measremet s tae Target poplato: the complete collecto of observatos we wat to std Sample: a sbset of poplato Sampled poplato: the poplato from whch the sample was tae Samplg t: the t of actal sample Samplg frame: the lst of samplg ts Yo eed to fd the data that wll eable o to fd aswers to or research qestos. For stace, f o are terested dvdal behavor, o wat to have dvdallevel data. Yo ma th ths s obvos bt ma cases o ma ot fd dvdallevel data avalable to o. Whe o caot fd dvdal-level data to std dvdal behavor, o ma be forced to se aggregated data. Sppose, for stace, that o are terested wrtg a research paper o the mpacts of a droght o farm prodcto bt ol have access to aggregated data. To carr ot a aalss, o eed to ow how the data were collected from farmers. Whe was the srve codcted? Before or after the droght? Who were the target poplato? 5

6 Cotr, some regos, or some dstrcts? Were hard-ht areas clded the sampled poplato? If ot, there s o pot of sg the data. Were hard-ht areas combed wth ot-so-hard-ht areas? Farmers ot-so-hard-ht areas mght have ejoed hgh otpt prces, created b the droght. Wthot a good derstadg o data, ecoometrc aalses caot provde good aalses. Some Importat Cocepts Casalt ad Ceters Parbs I scetfc research, researchers create epermetal evromet labs. I sch evromets, researchers ca carefll cotrol factors, sch as temperatre or hmdt. Uder carefll cotrolled evromets, t s relatvel eas to chage oe varable, holdg other varables costat, ad eame a reacto a object. Becase all of the other varables are held costat, ceters parbs, t s eas ad relable to fd a casal effect of the varable, whch was chaged, o the object. I socal scece, sch as ecoomcs, t s mpossble to cotrol the socal evromet. What we ca do s to observe chages people s behavor der smltaeos chages ma factors. Whe ma factors chage at the same tme, we ma fd two varables chagg the same drecto. Ths cold be jst a cocdece a assocato. Or ths cold be a casal effect of oe factor to aother a casal effect. For stace, we fd that the demad for ar-codtoers was hgh drg the past smmer ad that the hosehold come was also hgh at the same tme. Ths, o fd a assocato betwee the demad for ar-codtoers ad the hosehold come. Bt o also fd that the temperatre was ver hgh the past smmer. Ths, t s possble that the demad for ar-codtoer was hgh ot becase of the hgh come bt becase of the hgh temperatre. Ths, the hosehold come dd ot have a casal effect o the demad for the ar-codtoers bt the hgh temperatre had a casal effect. Dstgshg a assocato from a casal effect s oe of the most dffclt sses emprcal research. Omtted Varables Wth ecoometrcs, what we hope to do s to solate a effect of oe varable from other effects of other varables. B cldg all of the varables that affect a otcome, o are hopg to cotrol for other factors. A major problem s that o ofte do ot have formato o some mportat varables. We call those varables observed varables. Some of observed varables are observable bt ot observed, whle other varables cold smpl be observable. People s ablt ad taste are eamples of observable varables. 6

7 Whe there are observed varables, those varables wll be omtted from ecoometrcs aalses. Whe mportat varables are omtted from ecoometrc models, omtted varables ma case bases estmated coeffcets the omtted varables problem. For stace, sppose that o wat to measre a casal effect of agrcltral credt o farm prodctvt. Yo have formato o farm prodctvt ad agrcltral credt, bt o do ot have formato o the edcato of farmers. Sppose o fd a postve assocato betwee farm prodctvt ad agrcltral credt: the hgher the amot of agrcltral credt, the hgher the farm prodctvt. Bt t s possble that the credt s gve to edcated farmers whose farm prodctvt s hgh eve wthot the credt. I ths case, we wold fd a postve assocato betwee credt ad farm prodctvt eve whe credt does ot have a postve mpacts o farm prodctvt. I ths case, what we wold le to do s to measre the mpacts of credt o farm prodctvt amog farmers wth the same level of edcato. Bt we wll ot be able to do ths wthot the edcato formato of farmers. ve whe edcato formato s avalable, we do ot have the formato o farmers ablt whch ca ot be measred fll b formal edcato geeral. The omtted varable problem s oe of the most commo ad seros problems crosssecto aalses. We wll dscss ths problem etesvel ths corse. Ad later ths corse, we wll lear methods to overcome ths problem. Smltaet The drecto of the casal effect ma ot be oe-wa. Two factors ma flece each other. I ths sese the drecto of casalt s ot clear. For stace, the prevos eample of farm prodctvt ad agrcltral credt ca be cosdered as a smltaeos problem. Oe cold arge that agrcltral credt creases farm prodctvt, whle other wold arge that agrcltral credt s gve to hghl prodctve farmers. I ma cases, we ca cosder a smltaeos problem as a omtted varables problem. For stace, the above smltaeos problem ca be cosdered as a omtted varables problem: f we have the formato o all of the characterstcs that represet farmers ablt to repa credt, the we ca compare two farmers who have the same ablt to repa bt oe s gve credt whle the other s ot. If we fd that the farmer who s gve credt has hgher farm prodctvt tha the other farmer, wthot credt, who otherwse has the same repamet ablt, the we ma coclde that there s a casal effect of credt o farm prodcto. The problem, however, s that t s dffclt to observe all of the characterstcs that represet farmers ablt as data. Ths the smltaeos problem betwee agrcltral credt ad farm prodctvt rema as a omtted varables problem of farmers ablt. For ow, I wold jst wat o to be aware of dffereces betwee a assocato ad a 7

8 casal effect. What we sall wat to ow s a casal effect of a varable, whch s ofte a mportat polc varable, ot a assocato. Ubasedess ad Precseess I coometrcs ad Statstcs, t s mportat to clearl dstgsh basedess ad precseess. A estmate cold be Based ad Imprecse, Based bt Precse, Ubased bt Imprecse, ad Ubased ad Precse Accrate. See Fgre. Lohr 999. I coometrcs ad Statstcs, we sppose that there s oe tre vale, b. Ths cold be a average mber e.g., the average come, epedtre, age or a sze of a mpact of oe varable o aother e.g., the mpact of smog oe pac of tobacco o the probablt of havg the lg cacer. B sg data ad ecoometrc methods, we estmate the average mber or the sze of the mpact. Let s deote the estmated mber as. Whe the estmated mber estmator,, s ver dfferet from the tre mber, b, for some sstematc reasos, we call the estmator based. For stace, f we sample people from a telephoe boo, we mss people who are ot regstered the telephoe boo. If the people who are ot o the telephoe boo are sstematcall dfferet from the people o the telephoe boo, the a estmator b sg data from the telephoe-boo sample wold be based from the tre vale of the etre poplato, cldg both people who are o the telephoe boo ad who are ot. The estmator from the telephoe-boo sample, however, cold be precse. For stace, as the mber of sample from the telephoe boo creases, the estmator becomes more precsel estmated. Bt becase the sample s ot represetg the etre poplato, the precsel estmated estmator s stll based. We wll dscss more o ths sse throghot ths corse. Fgre smmarzes the dscsso o basess ad precseess. A: based bt ot precse B: precse bt based C: based ad precse Fgre. Basess ad Precseess Sorce: Fgre., page 9, Lohr 999 8

9 Taash Yamao Fall Semester 5 Lectre : The Smple Regresso Model I ths lectre we revew the smple bvarate lear regresso model. We focs o statstcal assmptos to obta based estmators. A good derstadg o the assmptos of the sgle lear regresso model wll help o to derstad the mportace of each assmpto of the mltvarate regresso model. Defe the smple lear regresso SLR model: $ $ where s a depedet varable, s a depedet varable or a covarate, ad s the error term or dstrbace. We wat to obta ow two parameters $ ad $. These two parameters are called tre vales that ca ot be observed. What we ca do s to estmate these two parameters. We deote the estmated parameters as ad. We ma call them as estmated coeffcets or estmators. There are several was to estmate the two parameters. The most commo method s the least sqared method. Least Sqared Resdals Method: Defe $ ad $ : ow parameters or coeffcets ad : Least Sqared estmates b ad b : coeffcets that are sed to obta ad b ad b are two varables that cold tae a vales. We se these two varables whle we are searchg for ad. Whe the sm of sqared resdals s mmzed, b ad b wold be eqal to ad, respectvel. Defe b b s the resdal. Mmzg Problem: Fd a par of b ad b that mmzes the sm of sqared resdals: 9

10 m b b Frst order codtos F.O.C. wth respect to w.r.t b ad b are w.r.t. b w.r.t. b Notce that b ad b are replaced b ad becase the frst order dervatves are set to be zero ad. From, we have From, B sg, we have b

11 We ca orgaze ths to fd, Ths, the least sqared estmates are 3 4 Ubasedess of SLR I ths sb-secto, we eame the baasedess of the smple ler regresso SLR estmates. To show the baasedess, we eed for assmptos, whch are specfed below. We eed to derstad wh we eed each assmpto ad how the SLR estmates wll be based whe oe of the for assmptos s volated. Assmptos: SLR Lear parameters: $ $ SLR Radom samplg ad are radom SLR 3 Zero codtoal mea SLR 4 Sample varato - > From 4, we have

12 Here, we have sed SLR. 5 At ths pot, we ca see that: f the epected vale of the secod term s zero, the the epected vale of wll be eqal to, the ow tre vale. B tag the epectato of the both sdes of 5, we have 6 Net, we have mportat steps: becase s jst a costat. B the defto, we ow that Cov. Uder SLR3, we assme Cov, ths we have.

13 Ths, we have Therefore der assmptos SLR-4, the least sqared estmate s based. Whe s based, s also based. However, SLR 3 s a ver strog assmpto especall a sgle lear regresso model becase the dstrbace,, cldes so ma mportat omtted varables. Whe SLR 3 s volated, we have a omtted varable problem. We wll dscss abot the omtted varables problem the et lectre. For ow, we smpl assme that the for assmptos are satsfed. Varaces of OLS stmators For the OLS estmates to be the most effcet estmates amog ma other estmates, we eed to add oe more assmpto: SLR 5 Homosedastct: Var F The homosedastct assmpto dcates that the sze of the varace of s costat or does ot deped o. Whe ths assmpto s volated, the we sa the error term ehbts heterosedastct. Note that eve der the heterosedastct, the least sqared estmates are ot based as we have show prevosl. For ow, we assme the homosedastct ad obta the varaces of estmates. B sg 4, we have Var From SLR5, Var F Var Var [ ] σ 3

14 σ σ 7 Note: the larger the F, the larger the Var, ad the larger the varace, the smaller the Var. Chec the tetboo page 56 for the varace of the estmated costat term, Var. The Stadard rror of We ca get the stadard devato of b tag the postve sqared root of the varace: sd Var σ However, F s ow. Ths, we have to estmate F. The based estmator of F s SSR σ. Ths s adjsted b the - degrees of freedom. The degree of freedom s - becase we have observatos wth two estmators, whch clde a tercept.. Ths the stadard error of s se σ SSR / The stadard error for ca be obtaed b replacg the tet. σ wth σ eqato.58 4

15 R-sqared The R-sqared s the fracto of the sample varato that s eplaed b. R SS / SST SSR / SST SS: the eplaed sm of sqares; SSR: the resdal sm of sqares; SST: the total sm of sqares. SST SS SSR Regresso Throgh the Org Sppose we have a model sch as $. Least Sqared Approach: m b Frst order codtos F.O.C. wth respect to w.r.t b ad b are ~ w.r.t. b ~ 8 Method of Momets Approach To tae the method of momet approach, we start wth two assmptos. The frst assmpto s ot strog. Bt the secod assmpto s ver strog. For the least sqared estmates to be based, however, we stll eed to assme the same secod assmpto. 5

16 Ths, these assmptos are also eeded for least sqared estmates to be based. Both approaches reach the same estmates. Two assmptos are: 9 Ths we have two restrctos ad two ow parameters $ ad $. Ths we are able to solve these two eqatos. These are eactl the same as ad. Ths we obta the same estmates: 3 ad 4. 6

17 Lectre 3: Smple Omtted Varables Problem I ths lectre, we epad the smple lear regresso model to mltvarate ler model. The ma motvato to clde more tha oe varable to a regresso model s that t s ver dffclt to detf the effect of oe depedet varable,, o the depedet varable,, wthot cosderg the other factors. For stace, we ca measre the effect of schoolg more precsel f we compare people who have the eactl the same characterstcs, ecept the schoolg. There have bee ma stdes o tws. If we fd a dfferece wage rates betwee the two detcal people ecept the schoolg, the we ca coclde that the dfferece s de to the dfferece the schoolg. Ufortatel, socal scece, t s mpossble to fd eve two persos eactl the same ecept oe factor. ve tws are dfferet ma was. The problem of omttg mportat varables regresso aalses s called the omtted varables problem, ad ths s the core of ma problems ecoometrcs. I start ths lectre wth the smplest case of the omtted varables problem. Omtted Varables Problem: a smple case Sppose that the tre mode shold be: $ $ $. 3- Bt sppose that we have formato ol o bt ot so that we ca ol estmate a smple bvarate model wth bt ot. Accordg to the least sqared approach from the prevos lectre, we ow that the least sqared estmators are However, the tre model s 3-. Ths, we ca replace 3-3 b sg the tre model 3-, 7

18 8 As we dd before, we ca smplf the rght had sde of the eqato: Ths, b tag the epectato of both sdes of the eqato, the last term becomes zero der assmptos SLR -4, whle the secod term remas: ~ 3-4 Ths dcates that the least sqared estmate of, wll be based f the secod term of 3-4 s ot zero,.e., ad are ot correlated. Note, frther, that the last part of the secod term s smpl a least sqared estmator of a smple bvarate regresso model betwee ad : * * e ~ δ Ths, 3-4 ca be wrtte as ~ δ Therefore the bas s ~ δ. Ths, the sgs of ~ δ ad determe the drecto of the bas pward or dowward. If the two parameters, ~ δ ad, have the same sgs both

19 postve or both egatve, the the drecto of bas s postve. Bt f the parameters have the opposte sgs, the the drecto of bas s egatve. For stace, let s sa that we wat to estmate the effect of agrcltral credt o farm prodctvt bt do ot have formato o the edcato of farmers. Ths, the error term of a bvarate model cldes edcato as a omtted varable: farm_prod $ $ credt edc Becase edc s correlated wth farm_prod ad credt postvel, postve. The drecto of a potetal bas s postve. ~ adδ are both ~ The magtde of the bas depeds o the absolte vales of ad δ. If the correlatos betwee ad ad betwee ad are strog, the the sze of the bas becomes bgger. O the other had, f oe of the two correlatos s wea the sze of the bas becomes small. ample 3-: Idvdal No-farm Icome Ugada I rral areas of Ugada, o-farm come provdes mch eeded cash to farm hoseholds. Most edcated me ad wome rral areas have opporttes to hold reglar jobs, mag a costat mothl wage throghot a ear. Other people who are ot fortate eogh to hold reglar jobs are lel to ear o-farm come from small self-emploed bsesses sch as mag basets or tradg goods. Sppose that we are terested the geder dffereces o-farm come ad wat to test a hpothess that wome mae less o-farm come tha me. Bt a smple comparso of o-farm come betwee me ad wome does ot provde a relable test for ths hpothess f characterstcs of me ad wome are ot smlar. Oe major factor o-farm come s edcato. If me are better edcated tha wome ad me mae more o-farm come tha wome, we ca ot be sre f the hgh o-farm come s a reslt of geder or edcato. For eample, let s se the data from Ugada. The data are collected b FASID collaborato wth Maerere Uverst Ugada 3. The data come from 94 hoseholds. Amog them, we fd 648 people who eared some come from o-farm actvtes. Table dcates average o-farm come US$, average schoolg ears, age, ad observatos for me ad wome. Table. Idvdal No-farm Icome Ugada 9

20 table female, cmea ocus mea ed mea age age f%8.f row; female meaocus meaed meaage Nage Total As o ca see Table, me are slghtl better edcated ad mae more o-farm come. To compare o-farm come betwee me ad wome whle holdg edcato levels costat, we have created categores for edcato: o edcato categor, -4 ears of schoolg, 5-7 ears of schoolg, 8- ears of schoolg 3, ad more tha ears of schoolg 4. The we have calclated average o-farm come for each categor for me ad wome separatel. Althogh wome mae less o-farm come tha me geeral, t-tests dcate that the dfferece s statstcall sgfcat at the 5 percet level ol for Categor 3. Table. No-farm Icome b Geder ad dcato. table edcat female, cmea ocus f%8.f row; female edcat Dfferece p-vale Total Now, let s cosder whch drecto the bas wold be f we estmate the o-farm come eqato wth a female dmm bt ot edcato. dcato the depedet varable s postvel correlated wth edcato the omtted varable, ad edcato the omtted varable s egatvel correlated wth the female dmm the depedet varable. Ths, accordg to the theor above, the drecto of the bas shold be egatve. lo-farm come $ $ female edc. reg loc female Sorce SS df MS Nmber of obs F, Model Prob > F. Resdal R-sqared Adj R-sqared.84 Total Root MS

21 loc Coef. Std. rr. t P> t [95% Cof. Iterval] female _cos reg loc female ed Sorce SS df MS Nmber of obs F, Model Prob > F. Resdal R-sqared Adj R-sqared.894 Total Root MS loc Coef. Std. rr. t P> t [95% Cof. Iterval] female ed _cos The reslts the frst model dcate that wome mae abot 63 percet less o-farm come tha me, bt the reslts the secod model dcate that the dfferece s abot 55 percet. Ths, the reslts dcate that the estmated coeffcet was based dowward whe ed was ot clded. Of corse, there are probabl ma other varables that shold be clded ths model to estmate the geder dfferece more precsel. We wll come bac to ths model later ths corse. d of ample Mltvarate Lear Regresso Model: Illstrato It s clear that we eed to clde more tha oe varable regresso models to tae to accot ma factors. How ma varables shold be clded? We wll std how we shold select varables based o derlg ecoomc theor ad emprcal aalses later. For ow, let s cosder a geeral case wth depedet varables pls a tercept. A mltvarate ler regresso model wth -depedet varables s $ $ $. $. for,,. where,, ad are observato s, ad. As the bvarate model, let s start wth the least sqared resdals approach, whch s called the Ordar Least Sqared OLS regresso. Least Sqared Resdals Approach

22 As the bvarate regresso, we wat to fd a set of estmators, $ j, that mmze the sm of sqared resdals: m b b b b F.O.C. w.r.t. b w.r.t. b 3- w.r.t. b Whe there s ol oe depedet varable ths, we cold solve the frst order codtos easl. Bt whe there are depedet varables, solvg the frst order codtos for varables wold be complcated. Matr maes t easer.

23 Lectre 4: Mltvarate Regresso Model Matr Form I ths lectre, we rewrte the mltple regresso model matr form. A geeral mltple-regresso model ca be wrtte as $ $ $. $. for,,. I matr form, we ca rewrte ths model as X % We wat to estmate %. Least Sqared Resdal Approach Matr Form Please see Lectre Note A for detals The strateg the least sqared resdal approach s the same as the bvarate lear regresso model. Frst, we calclate the sm of sqared resdals ad, secod, fd a set of estmators that mmze the sm. Ths, the mmzg problem of the sm of the sqared resdals matr form s m t X bt - Xb Notce here that t s a scalar or mber sch as, becase ts a matr ad s a matr ad the prodct of these two matrces s a matr ths a scalar. The, we ca tae the frst dervatve of ths object fcto matr form. Frst, we smplf the matrces: t t btxt - Xb t btxt - txb btxtxb t btxt - btxt btxtxb A-: btxt txb 3

24 4 t btxt btxtxb The, b tag the frst dervatve wth respect to b, we have: t/ b Xt XtX b So the frst order codto F.O.C. s Xt XtX Β 4A- ad 4A-3 The, XtX Β Xt 4- Notce that I have replaced b wth Β becase Β satsf the F.O.C, b defto. Mltpl the verse matr of XtX o the both sdes, ad we have: Β XtX - Xt 4- Ths s the least sqared estmator for the mltvarate regresso lear model matr form. We call t as the Ordar Least Sqared OLS estmator. Note that the frst order codtos 4- ca be wrtte matr form as Xt - X Β Ths s the same as the frst order codtos, codtos, we derved the prevos lectre ote: b

25 5 b b ample 4- : A bvarate lear regresso matr form As a eample, let s cosder a bvarate model matr form. A bvarate model s $ $. for,,. I matr form, ths s X % From 4-, we have Β XtX - Xt 4-3 Let s cosder each compoet 4-3. XtX Ths s a sqare matr. Ths, the verse matr of XtX s, XtX -

26 6 The secod term s Xt Therefore we have all the compoets of the rght-had sde of 3-4. Ths the OLS estmators are: Β XtX - Xt

27 d of ample 4- Ubasedess of OLS Wooldrdge Apped I ths sb-secto, we show the basedess of OLS der the followg assmptos. Assmptos: Lear parameters: X % Zero codtoal mea: X 3 No perfect colleart: X has ra. From 4-3, we ow the OLS estmators are Β XtX - Xt We ca replace wth the poplato model, Β XtX - Xt X % XtX - XtX % XtX - Xt % XtX - Xt 4-4 Tae the epectato o the both sdes of the eqato: Β % XtX - Xt X From MLR 3, we have X. Ths, Β % The OLS estmators are based. The Varace of OLS stmators Net, we 7

28 8 Assmpto: 4 Homosedastct: Var XF ad Cov, j, ths Var X F I Becase of ths assmpto, we have [ ] I σ σ σ σ Therefore, VarΒ Var [% XtX - Xt] Var [XtX - Xt ] [XtX - Xt t X XtX - ] XtX - Xt [ t] X XtX - XtX - Xt F I X XtX - 4: Homosedastct F XtX - XtX XtX - VarΒ F XtX GAUSS-MARKOV Theorem: Uder assmptos 4, Β s the Best Lear Ubased stmator BLU.

29 ample 4-: Step b Step Regresso stmato b STATA I ths sb-secto, I wold le to show o how the matr calclatos we have stded are sed ecoometrcs pacages. Of corse, practces o do ot create matr programs: ecoometrcs pacages alread have blt- programs. The followg are matr calclatos wth STATA sg data called, NFIcomeUgada.dta. Here we wat to estmate the followg model: l_fcome $ $ female $ edc $ edcsq. All the varables are defed ample 3-. Descrptve formato abot the varables are here:. s; Varable Obs Mea Std. Dev. M Ma female ed edsq l_fcome Frst, we eed to defe matrces. I STATA, o ca load specfc varables data to matrces. The commad s called mmat. Here we create a matr, called, cotag the depedet varable, l_fcome, ad a set of depedet varables, called, cotag female, edc, edcsq.. mmat l_fcome, matr. mmat female edc edcsq, matr The, we create some compoets: XtX, XtX -, ad Xt:. matr '*;. mat lst ; smmetrc [4,4] female ed edsq cost female 44 ed edsq cost matr smv;. mat lst ; smmetrc [4,4] female ed edsq cost female.944 ed edsq e-6 cost

30 Here s Xt:. matr '*;. mat lst ; [4,] l_fcome female ed edsq cost Therefore the OLS estmators are XtX - Xt:. ** stmatg b hat;. matr bhat*;. mat lst bhat; bhat[4,] l_fcome female ed.4488 edsq cost ** stmatg stadard error for b hat;. matr e-*bhat;. matr sse'*e/648--3;. matr vecdag;. mat lst ss; smmetrc ss[,] l_fcome l_fcome mat lst ; [,4] female ed edsq cost r e Let s verf what we have fod.. reg l_fcome female ed edsq; Sorce SS df MS Nmber of obs F 3, Model Prob > F. Resdal R-sqared Adj R-sqared.93 Total Root MS l_fcome Coef. Std. rr. t P> t [95% Cof. Iterval] female ed edsq _cos ed of do-fle 3

31 Lectre 5: OLS Iferece der Fte-Sample Propertes I ths lectre, we std OLS ferece der fte-sample propertes where estmators are eact ad dstrbtos are precsel defed eve wth small sample szes. We start wth the basedess of OLS der the followg assmptos. Ubasedess of OLS Wooldrdge Apped Assmptos: Lear parameters: X % Zero codtoal mea: X 3 No perfect colleart: X has ra. Aga, der fte-sample propertes, these assmptos hold for a sample szes. From 4-3, we ow the OLS estmators are Β XtX - Xt We ca replace wth the poplato model, Β XtX - Xt X % XtX - XtX % XtX - Xt % XtX - Xt 5- Tae the epectato o the both sdes of the eqato: Β % XtX - Xt X From, we have X. Ths, Β % The OLS estmators are based. The Varace of OLS stmators 3

32 3 Net, we add oe more assmpto abot the varace of the error term ad cosder the varace the OLS estmators. Assmpto: 4 Homosedastct: Var XF ad Cov, j, ths Var X F I Becase of ths assmpto, we have [ ] I σ σ σ σ Therefore, VarΒ ] [ X ΒΒ Β Β ] [ X X X X X X X X X X I X X X σ 4: Homosedastct F XtX - XtX XtX - VarΒ F XtX - 5- GAUSS-MARKOV Theorem: Uder assmptos 4, Β s the Best Lear Ubased stmator BLU. See Wooldrdge, pp 79.

33 So far, we have obtaed OLS estmatos for Β ad Var Β. Bt we eed to ow the shape of the fll samplg dstrbto of Β order to codct statstcal tests, sch as t-tests or F-tests. The dstrbto of OLS estmator Β depeds o the derlg dstrbto of the errors. Ths, we mae the followg assmpto aga, der ftesample propertes. Assmpto 5 Normalt of rrors: ~ N, F I Note that N, F I dcates a mltvarate ormal dstrbto of wth mea ad the varace-covarace matr F I. Remember aga that ol assmptos -3 are ecessar to have based OLS estmators. I addto, assmpto 4 s eeded to show that the OLS estmators are the best lear based estmator BLU, the Gass-Marov theorem. We eed assmpto 5 to codct statstcal tests. Assmptos -5 are collectvel called as the Classcal Lear Model CLM assmptos. The model wth all assmptos -5 s called the classcal lear model. The OLS estmators wth the CLM assmptos are the mmm varace based estmators. Ths dcates that the OLS estmators are the most effcet estmators amog all models ot ol amog lear models. Normalt of Β Uder the CLM assmptos -5, Β codtoal o X s dstrbted as mltvarate ormal wth mea Β ad varace-covarace matr F XtX -. Β ~ N[ Β, σ X X Ths s a mltvarate ormal dstrbto, whch meas each elemet of Β s ormall dstrbted: ~ N[, σ X X ] XtX - s the -th dagoal elemet of XtX -. Let s deote the -th dagoal elemet of XtX - as S. Ths, ] 33

34 σ X X S. σ. S σ S σ S... S.... σ S.. Ths s the varace-covarace matr of the OLS estmator. O the dagoal, there are varaces of the OLS estmators. Off-the dagoal, there are covarace betwee estmators. The, becase each OLS estmator s assmed to be ormall dstrbted, we ca obta a stadard ormal dstrbto of a OSL estmator b sbtractg the mea ad dvdg t b the stadard devato: z. σ S However, F s ow. Ths we se a estmator of F stead. A based estmator of F s s s the sm of sqared errors. Remember s a prodct of a matr ad a matr, whch gves a sgle mber. Therefore b replacg F wth s, we have t s S / σ S. [ s / σ ]/ Ths rato has a t-dstrbto wth -- degree of freedom. It has a t-dstrbto becase t s a rato of a varable that has a stadard ormal dstrbto the omator the parethess ad a varable that has a ch-sqared dstrbto dvded b --. The stadard error of, se, s s S. Testg a Hpothess o I most cases we wat to test the ll hpothess H : wth the t-statstcs t-test: - / se ~ t --. Whe we test the ll hpothess, the t-statstcs s jst a rato of a OLS estmator over ts stadard error. 34

35 We ma test the ll hpothess agast the oe-sded alteratve or two-sded alteratves. Oe-sded Alteratve H : > We reject the ll hpothess H whe the t-statstcs s greater tha the crtcal vale for a reasoable crtcal vale, sch as 5 percet: t > c Whe the oe-sded alteratve s the opposte: H : < We reject the ll hpothess H whe the t-statstcs s smaller tha the crtcal vale for a reasoable crtcal vale, sch as 5 percet: t < -c. Two-sded Alteratves H : We reject the ll hpothess H whe the absolte vale of t-statstcs s greater tha the crtcal vale for a reasoable crtcal vale, sch as 5 percet: t > c For a two-taled test, c s chose for the half of the crtcal level. For the 5 percet level, the crtcal vale for.5 percet wth -- degree of freedom shold be chose. Testg agast a o-zero costat Yo ca test a hpothess test le H : agast H : b sg a t-test t-test: - / se ~ t --. Testg a Jot Hpotheses Test o ' s Sppose we have a mltvarate model: $ $ $ $ 3 $ 4 $ 5. Sometmes we wat to test to see whether a grop of varables jotl has effects of. Sppose we wat to ow whether depedet varables 3, 4, ad 5 jotl have effects o. Ths the ll hpothess s 35

36 H : The ll hpothess, therefore, poses a qesto whether these three varables ca be eclded from the model. Ths the hpothess s also called eclso restrctos. A model wth the eclso s called the restrcted model: $ $ $. O the other had, the model wthot the eclso s called the restrcted model: $ $ $ $ 3 $ 4 $ 5. We ca geeralze ths problem b chagg the mber of restrctos from three to q. The jot sgfcace of q varables s measred b how mch the sm of sqared resdals SSR creases whe the q-varables are eclded. Let deote the SSR of the restrcted ad restrcted models as SSR r ad SSR r, respectvel. Of corse the SSR r s smaller tha the SSR r becase the restrcted model has more varables tha the restrcted model. Bt the qesto s how mch compared wth the orgal sze of SSR. The F-statstcs s defed as F-test: F SSR SSR r r SSRr / q. / The merator measres the chage SSR, movg from restrcted model to restrcted model, per oe restrcto. Le percetage, the chage SSR s dvded b the sze of SSR at the startg pot, the SSR r stadardzed b the degree of freedom. The above defto s based o how mch the models caot epla, SSR s. Istead, we ca measre the cotrbto of a set of varables b asg how mch of the eplaator power s lost b ecldg a set of q varables. The F-statstcs ca be re-defed as F-test: F R R r r Rr / q. / Aga, becase the restrcted model has more varables, t has a larger R-sqared tha the restrcted model. Ths the merator s alwas postve. The merator measres the loss the eplaator power, per oe restrcto, whe movg from the restrcted model to the restrcted model. Ths chage s dvded b the eplaed varato b the restrcted model, stadardzed b the degree of freedom. If the decrease eplaator power s relatvel large, the the set of q-varables s cosdered a jotl sgfcat the model. Ths these q-varables shold sta the model. 36

37 Lectre 6: OLS Asmptotc Propertes Revewg: Fte-Sample Propertes of Least Sqares I the prevos secto, we have stded fte-sample propertes of OLS estmators. Uder the fte-sample propertes, the OLS estmators are based ad the error terms are ormall dstrbted eve whe sample szes are small. Here s a qc revew: Assmptos: Lear parameters: X % Zero codtoal mea: X 3 No perfect colleart: X has ra. Uder these assmptos, the OLS estmators are based becase: Ths, Β XtX - Xt becase of 3 XtX - ests Β XtX - Xt X % from % XtX - Xt Β % XtX - Xt X from Β % based To fd the varace of the OLS estmators, we added oe more assmpto. 4 Homosedastct: Var XF ad Cov, j, ths Var X F I Therefore, VarΒ [ Β Β ΒΒ X ] X X X [ X ] X X X X X X σ I X X X from 4 F XtX - Fall, for s to be able to codct statstcal tests, we eed to ow the dstrbto of the OLS estmators, the we added a assmpto abot the dstrbto of the error term: 5 Normalt of rrors: ~ N, F I Uder the assmptos -5, each of followg estmators has a eact dstrbto. Β ~ N[ Β, σ X X ] Normal dstrbto 37

38 Ths holds for t t dstrbto s S F SSR SSR r r SSRr / q / F dstrbto However, these are strog assmptos ad ca be relaed easl b sg asmptotc theor. Therefore, ths lectre, we std the asmptotc propertes or large sample propertes of the OLS estmators. Uder the asmptotc propertes, the propertes of the OLS estmators deped o the sample sze. I short, we ca show that the OLS estmators cold be based wth a small sample sze bt cosstet wth a sffcetl large sample sze. Cosstec stead of basedess Frst, we eed to defe cosstec. Sppose W s a estmator of θ o a sample of Y, Y,, Y of sze. The, W s a cosstet estmator of θ f for ever e >, P W - θ > e as. Ths sas that the probablt that the absolte dfferece betwee W ad θ beg larger tha e goes to zero as gets bgger. Whch meas that ths probablt cold be o-zero whle s ot large. For stace, let s sa that we are terested fdg the average come of Amerca people ad tae small samples radoml. Let s assme that the small samples clde Bll Gates b chace. The sample mea come s wa over the poplato average. Ths, whe sample szes are small, the probablt that the dfferece betwee the sample ad poplato averages s larger tha e, whch s a postve mber, ca be o-zero. However, the dfferece betwee the sample ad poplato averages wold be smaller as the sample sze gets bgger as log as the samplg s properl doe. As a reslt, as the sample sze goes to ft, the probablt that the dfferece betwee the two averages s bgger tha e o matter how small e s becomes zero. I other words, we sa that θ s the probablt lmt of W : plm W θ. Uder the fte-sample propertes, we sa that W s based, W θ. Uder the asmptotc propertes, we sa that W s cosstet becase W coverges to θ as gets larger. The OLS estmators From prevos lectres, we ow the OLS estmators ca be wrtte as Β XtX - Xt 38

39 39 Β B XtX - Xt I the matr form, we ca eame the probablt lmt of OLS Β Β X p X X p lm lm Here, we assme that Q X X p lm ad assme that Q - ests. From, we have lm X p. Ths, lm Β Β Q p Ths, we have show that the OLS estmator s cosstet. Nest, we focs o the asmmetrc ferece of the OLS estmator. The asmptotc dstrbto of the OLS estmator s derved b wrtg Β Β X X X ΒΒ X X X The probablt lmt of Β Β goes to zero becase the cosstec of Β. The varace of Β Β s ΒΒ Β Β X X X X X X X X X X X X From 4, the probablt lmt of goes to F I. Ad also we assmed plm of X X s Q. Ths, Q X X Q σ QQ Q σ

40 4 Q σ Therefore, the asmptotc dstrbto of the OLS estmator s ] [, ~ Β Β Q N a σ. From ths, we ca treat the OLS estmator, Β, as f t s appromatel ormall dstrbted wth mea Β ad varace-covarace matr Q / σ. ample 6-: Cosstec of OLS stmators Bvarate Lear stmato A bvarate model: $ $ ad To eame the basedess of the OLS estmator, we tae the epectato Uder the assmpto of zero codtoal mea SLR 3:, we ca separate the epectato of ad :. Ths we eed the SLR 3 to show the OLS estmator s based. Now, sppose we have a volato of SLR 3 ad ca ot show the basedess of the OLS estmator. We cosder a cosstec of the OLS estmator. lm lm lm p p p lm lm lm p p p

41 cov, p lm var p lm f cov, Ths, as log as the covarace betwee ad s zero, the OLS estmator of a bvarate model cosstet. d of ample 6-4

42 Lectre 7: OLS Frther Isses I ths lectre, we wll dscss some practcal sses related to OLS estmatos, sch as fctoal forms ad terpretatos of several tpes of varables. For detals, please read Wooldrdge chapter 6 ad 7. Measremet rror the Depedet Varable Let * deote the varable that we wold le to epla: * $ $ $ $. However, we ca ol observe whch s a measred varable of * wth measremet errors. e - * B replacg * wth ad e, we get $ $ $ $ e. Ths, f e satsf the OLS assmptos sch as e X, the OLS estmators are based or cosstet. Bt the varace of the dstrbace s larger b Vare wth the measremet error e tha wthot. Note, however, that the measremet error the depedet varable cold be correlated wth depedet varables [Cov e ]. I that case, the estmators wll be based. Measremet rror a Idepedet Varable Let * deote a depedet varable, whch cold be observed wth the measremet error, e, where e. e - * 8- * $ $ $ $ - e. Assmpto : Cov, e Uder ths assmpto, the error term - $ e has zero mea ad correlated wth the depedet varables. Ths the estmators are based cosstet. The error varace, however, s bgger b $ e. Assmpto : Cov *, e 4

43 43 Ths assmpto s called the Classc rrors--varables CV assmpto. Becase e - *, ad e mst be correlated der the assmpto : Cov, e e * e e σ e Ths, we have the omtted varables problem, whch gves cosstet estmators of all depedet varables. The Atteato Bas For a bvarate regresso model t s eas to show the eact bas cased b the CV. Now, s measred wth the measremet errors, stead of. I a bvarate regresso model, the least sqare estmator ca be wrtte as e Ths, the probablt lmt of s, cov lm σ σ σ σ σ σ σ < e e e e e Var Var e p Ths, the plm s alwas closer to zero or based toward zero tha. Ths s called the famos atteato bas OLS de to classcal errors--varables. For a mltvarate regresso model, the probablt lmt of s

44 σ r p lm < σ σ r e where r α α s the poplato error the eqato α r Aga the mplcato s the same as before. The estmated coeffcet of the varable wth measremet errors s based toward zero or less lel to reject the ll hpothess. Data Scalg Ma varables we se have ts, sch as moetar ts ad qatt ts. The bottom le s that data scalg does ot chage mch. Scalg p/dow a depedet varable: α. α α α α If o scale p/dow the depedet varable b ", the OLS estmators ad stadard errors wll be also scaled p/dow b ", bt ot t-statstcs. Ths the sgfcace level remas the same as before scalg. Scalg p/dow a depedet varable: / α α. If o scale p/dow oe depedet varable, the estmated coeffcet of the depedet varable wll be scale dow/p b the same scale. Aga the t-statstcs or sgfcace level does ot chage. Logarthmc Forms For a small chage, a chage log tmes, log, s appromatel close to a percetage chage, /. Therefore, we ca terpret the followg cases sg percetage chages: log-log: / log log / Oe percet chage chages b percet. s a elastct. 44

45 / log-level: log Oe t chage chages b percet. 3 level-log: log / Oe percet chage chages b. 4 level-level: Oe t chage chages b. Whe a chage log s ot small, the appromato betwee a chage log ad a chage ma ot be accrate. For stace, the log-level model gves s log log. 8- If the chage log s small, the there s o problem of terpretg ths as oe t of chages b percet, becase log log /. Bt whe a chage log s ot small, the appromato ma ot be appromate. Ths we eed to trasform 8- as: log log log / / ep / ep / ep % [ep ] Ths oe t chages b [ep ] percetage. Qadratc Form / Iterpretato: > ad < a crease creases wth a dmshg rate < ad > a crease decreases wth a dmshg rate Trg Pot: At the trg pot, the frst dervatve of wth respect to s zero: 45

46 46 / Ths the vale of at the trg ths pot s / * Iteracto Terms 3 The mpact of o s 3 / A Dmm Varable or where A grop of observatos wth s called a base, bechmar, or referece grop. The estmated coeffcet of measres the dfferece averages ŷ amog observatos for ad the base grop, holdg other varables costat.,,,,,, The grop B, wth, has a lower or hgher tha the base grop, wth. Iteracto Terms wth Dmmes 3 or where Whe s a cotos varable, 3 measres a dfferece the effect of o betwee a grop wth ad a grop wth, or a dfferece slopes of : / / 3 whe whe

47 Mltcolleart From the prevos lectre, we ow that the varace of OLS estmators s Var Β σ X X. The varace of a estmator,, s σ S. Ths ca be wrtte as Var σ R See Wooldrdge pp94 or Greee pp57 R s the R-sqared the regresso of agast all other varables. I other words, R s the proporto of the total varato that ca be eplaed b the other depedet varables. If two varables are perfectl correlated, the R wll be oe for both of the perfectl correlated varables, ad the varace of those two varables wll ot be measred. Obvosl, o eed to drop oe of the two perfectl correlated varables. I STATA, STATA drops oe of perfectl correlated varables atomatcall. So f o see a dropped varable STATA otpts, o shold sspect that o have clded perfectl correlated varables wthot realzg. ve f two or more varables are ot perfectl correlated, f the are hghl correlated hgh R, the varace of estmators wll be large. Ths problem s called mltcolleart. The problem of mltcolleart ca be avoded to some etet b collectg more data ad crease varace depedet varables. For stace, let s th abot a sample of for dvdals. All of for cold be male ad marred. I ths case, a geder varable ad a martal stats varable wll be perfectl correlated. Sppose three of them have collage edcato. The a edcato varable wll be hghl correlated wth the geder ad martal-stats varables. Of corse, correlatos betwee these varables wll dsappear to some etet as the sze of sample ad the varato varables crease. Whe dobt, codct a F-test o varables that o sspect casg mltcolleart. A tpcal smptom of mltcolleart s A hgh jot sgfcace ad low dvdal sgfcace a hgh F-statstcs bt low t-statstcs A smple solto s to eep them the model. If or ma focs s o a varable whch s ot the part of mltcolleart, the t s ot a seros problem to have mltcolleart or model. Yo cold drop oe of hghl correlated varables, bt b dog so ma create a omtted varable problem. Remember that a omtted varable problem ca case bases o all of estmators. Ths cold be a more seros problem tha mltcolleart. 47