Section 11 Simultaneous Equations

Transcription

1 Sectin 11 Simultaneus Equatins The mst crucial f ur OLS assumptins (which carry er t mst f the ther estimatrs that we hae studied) is that the regressrs be exgenus uncrrelated with the errr term This assumptin is ilated if we hae reerse causality in which e y x System estimatin s single-equatin The first essential questin t ask in a situatin where the regressr may be endgenus is What is the mdel that determines the endgenus regressr? This questin, which must be answered at least partially t use any f the techniques in this sectin, suggests that ur single ecnmetric equatin shuld be thught f as part f a system f simultaneus equatins that jintly determine bth ur y and ur endgenus x ariables Fr example, ne f the mst cmmn applicatins in ecnmics is attempting t estimate a demand cure: quantity is a functin f price Hweer, shcks t demand (e) affect price, s price cannt generally be taken as exgenus The demand cure is part f a system f simultaneus equatins alng with the supply cure that jintly determine quantity and price Thinking f the jint determinatin f y and (at least sme) x fcuses ur attentin n a crucial set f ariables: the exgenus ariables that are in the ther equatin that determines x but that are nt in the equatin as separate determinants f y Whether we end up mdeling the secnd equatin explicitly r nt, these ariables are crucial t identifying the effects f x n y The tw main appraches t endgeneity rele arund ur degree f interest in the determinatin f the endgenus regressrs: System estimatin inles estimating a full set f equatins with tw r mre dependent ariables that are n the left-hand side f ne equatin and the righthand side f thers (Example: bth the supply and demand equatins) Single-equatin estimatin inles estimating nly the ne equatin f interest, but we still need t cnsider the ariables that are in the ther equatin(s) (Example: estimate nly the demand equatin, but the exgenus ariables in the supply equatin are used as instruments) Simultaneus equatins and the identificatin prblem In the simple case abe, we had ne endgenus ariable n the right-hand side and ne exgenus ariable aailable t act as an instrument ~ 102 ~

2 In the mre general case, there may be multiple endgenus ariables and multiple instruments This frces us t think abut the prblem f whether there is sufficient exgenus ariatin t identify the cefficients we want t estimate: the identificatin prblem We will examine an extended example f a set f supply and demand cures t explre the identificatin prblem Mdel I: Demand cure: Q 0 P P u Supply cure: Q 0 P P Sling fr the reduced frm: 0 PP 0 PP u P PP 0 0u 0 0 u P P 0 P, P P P P 0 0 u Q 0 P u P P P P 0 P P P 0 0 P P P Q P P P P u Q u u P 0 P 0 P P Q0 Q P P P P The equatins P P,0 P Q Q,0 Q are called the reduced-frm equatins We hae sled the system f simultaneus linear equatins fr separate linear equatins each f which has an endgenus ariable n the left and nne n the right The cefficients are the reduced-frm cefficients: they are nnlinear cmbinatins f the structural cefficients and We can estimate the reduced-frm cefficients by OLS because there are n endgenus ariables n the right-hand side In this case, there are n ariables at all n the RHS! We can estimate P,0 and Q,0 as the means f P and Q Des this gie us enugh infrmatin t identify the and parameters? N There are fur structural cefficients (tw and tw ) and nly tw reduced-frm cefficients () There is n way t ~ 103 ~

3 cnstruct a unique estimatr f and f the r cefficients frm the estimate f Thus, in Mdel I neither f the equatins is identified Shw graph: all ariatin in P and Q are due t unbsered errr terms Mdel II: Demand cure: Q0 PPMM u, where M is incme and is exgenus Supply cure: Q 0 P P Sling fr the reduced frm: 0 PP 0 PP MM u 0 0 M u P M P 0 PM M P, P P P P P P u u 0 0 M Q 0 P M P P P P P P P 0 P 0 M P P P Q M Q0 QMM Q P P P P P P Suppse we estimate the fur reduced-frm cefficients P0, PM, Q0, QM by OLS Can we identify the fie structural cefficients? Obiusly nt: can t identify fie cefficients uniquely frm fur Hweer, we can identify sme f them: MP QM P P P PM M P P P 0 P Q0 P P0 P 0 P P P P This is called indirect least squares and is an antiquated methd fr estimating these mdels The presence f the incme term in the demand equatin identifies the slpe and intercept f the supply equatin Changes in incme affect demand but nt supply, s we can use these changes t trace ut the slpe f the supply cure Hw much des an increase incme affect P and hw much des it affect Q? The supply equatin is just identified because there is nly ne way f extracting the structural parameters frm the reduced-frm parameters 2SLS f the supply equatin using incme as an instrument gies us the same estimatr as ILS in the justidentified case ~ 104 ~

4 The demand equatin is nt identified: the nly ariatin in the supply cure is the unbsered randm shck What wuld happen if incme als affected supply? Mdel III: Demand cure: Q0 PP MM u Supply cure: Q 0 PP MM Sling fr the reduced frm: 0 PP MM 0 PP MM u 0 0 M M u P M M P P P P P P P 0 PM P, u u 0 0 M M Q 0 P M P P P P P P P 0 P 0 M P M P P P Q M Q0 QMM Q P P P P P P It s n lnger pssible t identify either equatin Nne f the six structural cefficients can be identified frm estimates f the fur reduced-frm cefficients We can n lnger use changes in M t trace ut either cure because it affects bth cures Nte that nthing in the data has changed: we hae merely changed ur assumptin (lens analgy) abut hw the data were generated If the assumptin in Mdel II that incme des nt affect supply is incrrect, ur estimates f the supply cure wuld be nnsense Mdel IV: Demand cure: Q0 PP MM u Supply cure: Q 0 PP RR, where R is rainfall (exgenus) Sling fr the reduced frm: 0 PP RR 0 PP MM u 0 0 M R u P M R M R P P P P P P P P P 0 PM PR P, u 0 0 M R Q 0 P M R R R P P P P P P P P u P 0 P 0 M P R P P P Q M R Q0 QMM QRRQ P P P P P P P P ~ 105 ~

5 There are nw six estimable cefficients and six structural cefficients we wuld like t estimate Just identificatin f all cefficients is pssible based n the numbers In fact, as befre, MP QM P P P PM M P P P 0 P Q0 P P0 P 0 P P P P Nw, we can d the same thing with the rainfall cefficients: P R QR P P P PR R P P P0 P P 0 0 PM P P M RM P P R Bth equatins are just identified: Rainfall identifies the demand equatin because it is exgenus, affects the endgenus ariable price, and is nt in the demand equatin n its wn Incme identifies the supply equatin because it is exgenus, affects the endgenus ariable price, and is nt in the supply equatin n its wn Again, 2SLS gies us the same estimatrs as ILS in the just-identified case: iregress 2sls q m (p = r) t estimate the demand equatin iregress 2sls q r (p = m) t estimate the supply equatin Mdel V: Demand cure: Q0 PP MM u Supply cure: Q 0 PP RR WW, where W is wages (exgenus) We nw hae tw exgenus ariables in the supply equatin that are nt in the demand equatin Tw alternatie ways f identifying the demand cure ~ 106 ~

6 Sling fr the reduced frm: P R W P M u 0 P R W 0 P M u 0 0 M R W P M R P P P P P P P P P P P0 PM PR PW P, P M R W 0 0 M R W u Q 0 P M R W RRWW P P P P P P P P P P P 0 P 0 MP RP WP PuP Q M R W P P P P P P P P P P Q0 QM QR QW Q Q M R W There are nw eight estimable reduced-frm cefficients and seen structural cefficients All six f the equatins that we used in Mdel IV t get the thse six cefficients still wrk QR QW Nw we can estimate P either as P r as P The demand equatin is nw eridentified because there are tw exgenus ariables that affect the single endgenus ariable P that are nt separately in the demand equatin Will they be the same? Will QR PR PR Generally they wn t be identical een if the mdel is crrect because f sampling errr Is there mre inequality than wuld be expected randmly? We can test this nnlinear null hypthesis If the mdel is alid, we shuld nt be able t reject this null hypthesis Rejecting these eridentifying restrictins suggests that the mdel is nt alid There are tw different ILS estimates fr the cefficients f the demand equatin 2SLS will be a cmbinatin f them Estimate demand equatin by iregress 2sls q m (p = r w) The instrument used is the predictin f Q based n R and W Nte seeral prperties f identificatin Identificatin is usually by equatin/cefficient, nt necessarily f the whle system It s pssible t hae ne equatin that is identified with thers nt QW PW? PW ~ 107 ~

7 If there are multiple endgenus regressrs it is pssible t hae ne identified and thers nt Identificatin depends crucially n three assumptins: That the instrument is exgenus That the instrument des nt itself appear in the equatin That the instrument des appear in anther equatin that influences the endgenus regressr If any f these assumptins is ilated, then the 2SLS estimatr is biased and incnsistent In general, there needs t be ne mitted exgenus ariable fr each included endgenus ariable (Order cnditin fr identificatin) Hweer, if yu hae tw instruments that are crrelated with ne endgenus ariable but neither is crrelated with the ther, then identificatin f the secnd endgenus regressr fails Order is nt enugh; the rank cnditin applies as well Matrix ntatin f the 2SLS estimatr Cnsider ur 2-equatin system f Mdel V Let y = [Q P] be an N 2 matrix f the tw endgenus ariables Let Z = [1 M R W ] be an N 4 matrix f the fur exgenus ariables (which are instruments fr ne equatin and included exgenus ariables fr the ther) Let e = [u ] be an N 2 matrix f errr terms, which are prbably crrelated within a single bseratin Let be the 2 2 matrix f cefficients applied t the endgenus ariables (which will ften be 1 r 1 by nrmalizatin) Let B be the 4 2 matrix f cefficients applied t the exgenus ariables, which must hae sme elements that are knwn t be zer in rder fr identificatin t be achieed The tw equatins f the mdel can be written as Y Z e, where Q1 P1 1 M1 R1 W1 Q2 P M2 R2 W2 Y,, Z, P P Qn Pn 1 Mn Rn Wm 0 0 u1 1 M 0 u, 2 2 e 0 R 0 W un n The reduced-frm equatins are btained by pst-multiplying the equatin by the inerse f (which must exist fr the mdel t be ~ 108 ~

8 1 1 slable): Y Z e 1 1 1, r Y Z e Z, where 1 and Q1 P1 1 Q2 P2 e Qn Pn If the system is identified, then there are enugh restrictins n the and B matrices (fie in the mdel abe tw 1s and three 0s) t assure that the remaining elements can be btained uniquely frm the matrix Estimatin f systems f equatins (nt in text) The methd f instrumental ariables ffers us a means t estimate a single equatin frm a larger system f simultaneus equatins Smetimes we want r need t estimate the entire system Estimates are generally mre efficient if all equatins are estimated tgether Taking accunt f the crrelatin between the errr terms is beneficial Suppse that we knw that the errr terms are psitiely crrelated and that equatin 2 seems t hae a large psitie errr fr bseratin i Jint estimatin allws us t take accunt f the likely psitie errr in equatin 1 and nt attempt t fit the utlying bseratin t clsely Adds infrmatin and thus impres efficiency We may want t impse and/r test cefficient restrictins acrss the equatins f a system Demand equatins deried frm a cmmn utility functin (r factr demands frm a cmmn cst functin) hae crss-equatin symmetry restrictins (The Slutsky cnditin fr demand says that the incmecmpensated crss-price elasticity f demand fr x with respect t the price f y equals the elasticity f demand fr y with respect t the price f x) Might want t test whether the incme elasticity f demand fr apples exceeds that f bananas Might want t test whether all f the cefficients f the demand fr apples are the same as thse f bananas s that we can aggregate them tgether Tw kinds f jint-system estimatin Seemingly unrelated regressins (SUR) (als called Zellner-efficient regressin) System f equatins with n endgenus ariables n right-hand side Efficiency gains frm taking accunt f crrelatin f errr Pssibility f testing/impsing crss-equatin cefficient restrictins Three-state least squares (3SLS) ~ 109 ~

9 System f equatins with endgenus regressrs Example wuld be estimating bth demand and supply equatins tgether Adds efficiency gains (r crss-equatin tests) t 2SLS/IV cnsistent estimatr f equatin(s) with endgenus regressrs Estimatin by seemingly unrelated regressins Here we hae a set f equatins that hae n endgenus regressrs, but we want t estimate the equatins jintly We can d this by stacking the regressins: Suppse that there are 3 equatins t be jintly estimated with dependent ariables y 1, y 2, and y 3, sets f regressrs (which might erlap) X 1, X 2, and X 3, and errr terms e 1, e 2, and e 3 Separately, the equatins can be written y1 X11 e1, y2 X22 e2, y3 X33 e3 Let y be the 3N 1 element clumn ectr that stacks the 3 y ectrs: y1 y y2 y 3 X1 0 0 Let X be the 3N (K 1 + K 2 + K 3 ) matrix X 0 X X3 e1 Let e be the 3N 1 element clumn ectr e e2 e3 1 Let be the (K 1 + K 2 + K 3 ) 1 element ectr 2 3 We can then write the cmbined system f equatins as y X e Can we estimate this system by OLS? Yes, except this nt efficient because f prbably crrelatin between the ith bseratin s errr term acrss equatins Specificatin f errr term If bseratins are IID, then, c(e mi, e lj ) = 0 if i j (First subscript is equatin; secnd is the bseratin) Hweer, it is likely that within each bseratin, c(e mi, e li ) = ml 0 ~ 110 ~

10 Heterskedasticity is als almst certain since we hae different dependent ariables fr each third f the stacked regressin Let u Assume that there is n crrelatin acrss bseratins either within any f the equatins r between them Then the cariance matrix f the stacked errr term is the 3n 3n 11IN 12IN 13IN matrix IN 12IN 22IN 23IN 13IN 23IN 33I N Gien the nn-scalar cariance matrix, this is anther ptential applicatin f ˆ 1 1 GLS X X X y generalized least square: 1 (This general frmula specializes t weighted least-squares when there is heterskedasticity but n autcrrelatin We will als see a GLS applicatin fr serial crrelatin f the errr) Of curse, we dn t knw ml, s we must estimate it We can d s based n OLS residuals because OLS is cnsistent (if nt efficient) SUR is a tw-step prcedure First estimate the three regressins by OLS and calculate the residual ectrs uˆ1, uˆ ˆ 2,and u 3 Next estimate these estimatrs int n 1 ˆ uˆ uˆ, m = 1, 2, 3; l = 1, 2, 3, and assemble ml mi li n i 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ u ˆ ˆ ˆ and ˆ ˆ u I n Finally, use feasible GLS t estimate as 1 ˆ ˆ 1 ˆ 1 FGLS X X X Y This prcedure can be iterated: Because ˆ FGLS is a mre efficient estimatr than the OLS estimatr, we shuld get better residuals be calculating them based n ˆ FGLS rather than n OLS Iterated seemingly unrelated regressins (ISUR) repeatedly re-estimates ml based n the FGLS cefficient estimatr, then recalculates ˆ and reestimates by FGLS This can be repeated er and er until the elements f ˆ d nt change frm iteratin t iteratin ~ 111 ~

11 SUR is mre efficient than separate OLS except in tw situatins (in which they are identical): First, there is n crrelatin between errr terms acrss equatins In ther wrds, ml = 0 fr all m l Secnd, the same regressrs appear in all equatins: X m = X l fr all m, l In Stata, we use sureg (dar1 indars1) (dar2 indars2) (dar3 indars3) The ptin isure iterates t cnergence The ptin cnstraints ([dar1]indar1j = [dar2]indar2j) impses the cnstraint that the indar1j cefficient in the equatin fr dar1 equals the indar2j cefficient in the equatin fr dar2 If cnstraints are cmplex, can als use cnstraint 1 [dar1]indar1j = [dar2]indar2j cnstraint 2 [dar2]indar2j = [dar3]indar3j sureg (dar1 indars1) (dar2 indars2) (dar3 indars3), cnstraint(1 2) Estimatin by three-stage least squares If endgenus ariables appear n the RHS f equatins, then we must cmbine the system estimatin f SUR with the instrumental ariables methd f 2SLS The resulting estimatr is 3SLS: Estimate the reduced-frm equatins by OLS Dn t need SUR because all exgenus ariables in the system appear in each equatin, s the X m matrices are identical and OLS is equialent t SUR Calculate fitted alues f the endgenus ariables based in the reduced-frm regressins n the exgenus ariables as in 2SLS Estimate the indiidual equatins by 2SLS, using their fitted alues in place f the endgenus regressrs Calculate the residuals f each equatin frm the 2SLS regressins Calculate estimates f ml and assemble them int ˆ Estimate the system f equatins jintly by FGSL using the estimated ˆ As with SUR, this can be iterated 3SLS has the same adantages relatie t 2SLS that SUR has relatie t OLS: Efficiency gain by taking accunt f crss-equatin crrelatin f errr (if it exists) Pssibility f impsing r testing crss-equatin cefficient restrictins Stata will d 3SLS using the reg3 cmmand, which cmbines the frms f the iregress (with endregrs = instars in the ariable list f each regressin) and multiple equatins enclsed in parentheses Maximum-likelihd estimatrs fr simultaneus equatins Unlike OLS, 2SLS is nt an MLE, nr is 3SLS ~ 112 ~

12 There are MLEs that apply t these mdels under (usually) the nrmal distributin Limited-infrmatin maximum likelihd is a single-equatin MLE that is analgus t 2SLS Full-infrmatin maximum likelihd is the multiple-equatin MLE analg f 3SLS Stata will d LIML (and GMM) by changing the 2SLS ptin in the iregress cmmand t liml r gmm Each f these has different ptins that will need t be set I can t find any FIML prcedure in Stata, but it may be pssible t prgram it in the general-purpse ml cmmand ~ 113 ~