1 i. δ ε. so all the estimators will be biased; in fact, the less correlated w. (i.e., more correlated to ε. is perfectly correlated to x 4 i.

Transcription

1 Specal Topcs I. Use Instrmental Varable to F Specfcaton Problem (e.g., omtted varable 3 3 Assme we don't have data on If s correlated to,, or s mssng from the regresson Tradtonal Solton - pro varable: Plg that nto orgnal model: To be more general, assme 3, then cold have regressors correlated wth error term f w δ δ ε of 8 w w δ ε δ 33 δ (.e., constant: δ ε δ ε 33 w δ δ δ Ths s the actal eqaton we rn the regresson on Problem - w s correlated wth ε so all the estmators wll be based; n fact, the less correlated w s to (.e., more correlated to ε, the worse the problem s; f w s perfectl correlated to ths problem doesn't occr Fng It - easest wa s to test H : wth standard t-test b sng the t-rato for /δ wthot... f we can't reject H, then we can remove the pro and rn the model Use Instrmental Varable - add another pro If ε s not correlated to ε, then we can se II. Generated Regressors λ w λ ε w as nstrment for 3 3, where and 3 are estmates for and 3 (.e., η and 3 3 η3 Eample - have to estmate epected prce n model to forecast GDP Ke - as sample sze gets larger, η and η 3 "go awa" (.e. and 3 are better estmates for and 3 Problem - OLS estmates ( are OK, bt statstcal tests aren't vald becase the standard error compted b software packages s ncorrect; the correct verson s ver complcated (.e., "beond the scope of ths corse" Ecepton - F-test and Wald Test for are OK (becase f o fal to reject, 3 o can drop the varables and not worr abot generatng them! Ecepton - techncall we se generated regressors n SLS, bt the problem above (.e., standard error not beng correct doesn't appl becase we know how the nstrments are generated w

2 III. Generated Instrments wth E( If 3 3 E (, E ( 3 ; w nstrment for 3 ŵ s consstent estmate of w t can be sed and there's no effect on SLS (see ecepton n prevos secton IV. Testng Nonlneart (Fnctonal Form 3 3 Assme we want to know f model s msspecfed (e.g., shold there be a hgher order term or an nteracton term? Rn OLS and get and generate resdals: 3 3 If model s msspecfed û wll be correlated wth hgher order (or nteracton terms Method - rn b b b b b b b etc.; cold nclde cbed terms (or hgher... sall end p wth too man terms to check, bt ths s techncall better than the Ramse Test Method (Ramse Test - ( Regress û on,, 3... ths s restrcted regresson; techncall t shold be nsgnfcant becase û s orthogonal to the regressors (b desgn ( Generate ftted vales: 33 3 (3 Rn δ δ δ δ δ δ 5 δ ε... ths s nrestrcted regresson (note also that we're sng generated regressors ( Use F-test to test δ δ δ... f fal to reject, then there probabl aren't an hgher 5 6 order terms (5 Optonal - can look at δ, δ 5, δ 6 ndvdall to get hnt on correct fnctonal form; techncall can't se t-test, bt f t-rato s ver small (or ver bg, we're probabl safe to sa t's not (or s sgnfcant... have to se some jdgment there V. Dfference n Dfference Cross secton over tme where we observe grops at tmes Treatment Grop - grop A; receves treatment (.e., polc change, tranng, reorganzaton, etc. Control Grop - grop B; don't receve treatment Observatons - s grop 's average at tme j j Grop A Grop B t t Bred Treatment Effects - two of them: ( A A - ncldes treatment effect and other factors (e.g., tastes change over tme; doesn't accont for control ( A B - ncldes treatment effect, bt becase t ncldes dfferent grops there are other factors (e.g., dfferent tastes between grops Assmpton - n order to solate the actal treatment effect we have to assme that f behavor (e.g., tastes or preferences changes over tme, t changes the same wa for both grops A B A B of 8

3 Actal Treatment Effect - ( A A B B (second term acconts for change n behavor CAUTION - ths s onl vald for a controlled eperment; otherwse we need more complcated technqes (wll cover n second ear corse VI. Seemngl Unrelated Regresson Estmate (SURE Model Pooled Cross-Secton Tme-Seres Data - collect dfferent cross-secton data over tme; ma not necessarl be from same sorce (.e., mght not be same people n sample from ear to ear Panel Data - random selecton of ndvdals bt all ftre observatons come form same ndvdals... we'll std ths later Pooled Regresson - combnes more than regresson model and ; cold have dfferent or same regressors ( and or regressands ( and ; eamples: (a demand eqaton for ear and ear ('s same; 's same (b demand for pork and demand for beef ('s dfferent; 's same [some of them anwa] (c demand for pork vs. prce of pork and demand for beef vs. prce of beef ('s dfferent; 's dfferent Wh Use Pooled Regresson - ( mght be able to get better estmate b ncreasng sample sze ( mght be able to get a "better" standard error (3 to test cross-eqaton restrctons (e.g., Solvng - can solve each regresson separatel b smple OLS or pool them together and appl OLS of "stacked data" n n n or or Y X n n n Theoretcal Mathematcs - ths "stackng" s done for mathematcal reasons; there's no economc theor at work here... n fact, the new "dependent varable" cold be gbbersh from an economc perspectve Parameter Estmates - ( X' X X' Y... wll be the same as rnnng the regressons OLS separatel Problems - pooled regresson estmates var(, bt sngle regressons estmate var( and var( and these are not the same (f and standard error and t-ratos wll not be the same Solvng Correlaton Problem - se GLS estmaton are correlated... that means 3 of 8

4 Defne varance-covarance matr E Use Cholesk Decomposton to break t down nto a tranglar matr ' (apparentl, the eact technqe for ths sn't mportant... statstcal package wll do t "Crop" data - par data so we set n mn( n, n Rewrte model: Y X Whch can be wrtten: or Y X Note: X s not block dagonal lke X s Now have var( var( and cov(, New Estmates - ( X' X X Y ' GLS Estmatng. Rn each regresson ndvdall. Save resdals: and n 3. n. Use statstcal package to do the decomposton: ' Wasted Effort? - and wll be (asmptotcall the same n cases: OLS GLS ( and are not correlated (.e., cov(, or s a dagonal matr ( (.e., same vales for regressors If ether of these s not the case, wll be more effcent GLS Problem - procedre doesn t handle heteroskedastct of seral correlaton... f ether of those ests, o have to do t b hand (knd of Create new varable to rn clstered regresson (see Stata notes VII. SURE wth Endogenos Regressors ( Endogenet - at least one regressor s correlated wth the error term; those regressors that are correlated at called endogenos regressors Specfcaton - f a model has one or more endogenos regressors, t s nder specfed (there are more nknowns that eqatons so the model can't be solved; f there s one IV for each endogenos regressor, the model s eactl specfed; f there are more IVs than endogenos regressors, the model s over specfed Model - E ( E( and E ( E( E ( and E ( z s an nstrment for ( z E( z... no problem there (.e. E and E ( z,,, n,,, n,,, n,,, n n d n of 8

5 SLS - can estmate coeffcents b applng SLS to each eqaton ndvdall:. Regress on,, z and predct. Rn and Pooled Regresson ( - combnes SLS wth SURE model; wold do t for same reasons covered n prevos secton (better estmates, "better" standard error, cross-eqaton restrctons. Regress on,, z and predct and. Rn to get a. coeffcent estmates:,,,,, b. resdals: and 3rd Stage - Theor - n n n n n n or n n Problem - pooled regresson estmates var(, bt sngle regressons estmate var( and var( and these are not the same (f and are correlated... that means standard error and t-ratos wll not be the same σ ρ Varance-Covarance Matr - E Iσ ρ σ Cholesk Decomposton - same as before; break matr down nto a tranglar matr ' "Crop" data - f had dfferent amont of data for each eqaton, we have to par data and set n mn( n, n... drop etra data ponts jst lke we covered n the prevos secton Rewrte Model - E I (.e., var( var( and cov(, New Estmates - ( X' X X Y ' 5 of 8

6 Practce - note that there was onl endogenos varable above (the same one n both eqatons; to make ths more general, we'll now look at two endogenos varables, one n each eqaton; snce we have two of them, we need at least to IVs: Model - (same as before 3 ( E( 3 E ( E( E ( and E ( (two new varables E and... no problem there z and z are nstrments for and Stage - se IVs to get ftted vales for endogenos varables a. Regress on,, z, z and predct b. Regress on, 3, z, z and predct Stage - plg ftted vales nto orgnal eqatons to get resdals a. Rn to get b. Rn 3 to get 3 Stage 3 - n a. Estmate n b. Use statstcal package to do the decomposton: ' c. Rn regresson wth transformed data to get Wasted Effort? - SLS and secton: ( and are not correlated (.e., cov(, ( same vales for regressors wll be nmercall the same n cases (same as prevos ρ ; s a dagonal matr Better - If above cases don t hold (and below case doesn t hold, then are asmptotcall the same, bt SLS and wll be better estmate (.e., have smaller varance f the model s over specfed... that's good to know for the Hasman test whch reqres estmates [ better than the other] Idea - etra nformaton from data n second model cold help get better predctons Worse - s less robst; f we're nterested n the frst eqaton, bt the second eqaton s msspecfed (e.g., (a thnk there are no endogenos varables, bt there are; (b fnctonal form s wrong; (c mssng varables, etc., then wll be based Iteraton - after stage 3, we can se to estmate new resdals, then repeat stage 3 (.e., reestmate and ; eventall wll converge to the mamm lkelhood estmate Fll Informaton Mamm Lkelhood Estmaton (FIML - specf everthng (e.g.,, jont normal; s the same as FIML (bt easer; SLS s same as LIML ( (lmted nformaton 6 of 8

7 When s Sstem Identfed - Lnear Sstem - D p and S α α p ε D S Q so Q and p are jontl determned... p s endogenos (correlated to error term Need nstrments for p If we add ncome to demand eqaton, we can se I as an IV for prce n sppl eqaton, bt we stll have D nder specfed becase we don't have an nstrment for prce n the demand eqaton; f we add wage to the sppl eqaton, we can se t as the IV n the demand eqaton D p I and S α α p α w ε Nonlnear Endogenos Varable - D p I and S α α ln p ε Second eqaton s OK becase we can se ncome from the frst one as an IV for ln n the second one Fndng IV for frst eqaton s trck... looks lke we don't have one becase there are no etra varables n the second eqaton; here's the trck: D p I S α α ln p ε... we don t have to solve for p to know t won't be a lnear fncton of ncome se I or ln I as IV for prce n frst eqaton Seral Correlaton - D p I and S α α p ε wth ε ρε γ As n prevos case, second eqaton s OK becase we can se ncome from the frst one as an IV for p n the second one D S p I α p ρε Solve for p : p α I α α α γ ρε can se ε as IV for p n frst eqaton γ α Lessons -. For lnear model, need one IV for each endogenos varable; f a separate varable s not avalable, the IV cold be one of the varables form the other eqaton. For model wth nonlnear endogenos regressor wll alwas be dentfed (eactl or over specfed becase there wll be plent of IVs to se [jst se fnctons of the eogenos varables] 3. Dnamc sstem (ncldes seral correlaton or an lagged regressor or regressand wll alwas be dentfed p 7 of 8

8 Need Rght Nmber of IVs - wth E ( Assme z s IV for Wrong Wa - se SLS. Regress on, z and get. Rn ( Ths s not consstent (.e., wll be based Rght Wa - need IVs... snce z s correlated to, we know a. Regress on, z, b. Regress on, z, SLS z and get z and get. Rn (ths s not the same as z s correlated to ( 8 of 8