LECTURE 13 SIMULTANEOUS EQUATIONS

Transcription

1 NOVEMBER 5, 26 Demad-upply ytem LETURE 3 SIMULTNEOUS EQUTIONS I thi lecture, we dicu edogeeity problem that arie due to imultaeity, i.e. the left-had ide variable ad ome of the right-had ide variable are determied imultaeouly. major example i the demad-upply ytem of equatio Q d i = i + U i Q i = 2 i + U 2i where Q d i ad Q i are quatitie demaded ad upplied repectively, ad i i the price (we ca aume that < ad 2 > ). The ytem alo iclude the followig idetity or equilibrium coditio Q d i = Q i = Q i Naturally, ecoometricia doe ot oberve Q d i ad Q i, but oly Q i determied i the equilibrium together with i a reult, a imple regreio of Q i agait i i meaigle, ice Q i come from both equatio. Further, we ca how that i i correlated with both U i ad U 2i. Firt, we olve the ytem i term of U i ad U 2i Subtract the demad equatio from the upply ad ue the equilibrium coditio to obtai U 2i i = U i ad Therefore, aumig that E (U i U 2i ) = Q i = 2U i U 2i EU 2 i E ( i U i ) = 6= Similarly, we ca how that other three covariace E ( i U 2i ) E (Q i U i ) E (Q i U 2i ) ad, therefore, both Q i ad i are edogeou, which violate oe of the critical aumptio of the regreio aalyi. a reult, it i impoible to etimate coitetly ad 2 Next, aume that the demad equatio iclude aother variable, ay, icome (I i ). Further, aume that I i i excluded from the upply equatio ad predetermied, i.e. i ot a ected by Q i ad i I fact, we aume that I i i exogeou E (I i U i ) = E (I i U 2i ) = Now, the ytem i give by Q d i = i + I i + U i () Q i = 2 i + U 2i (2) Q d i = Q i = Q i (3) gai, we ca olve the ytem i term of the predetermied variable I i ad the hock U i ad U 2i i = Q i = I i + U i U 2i I i + 2U i U 2i

2 or i = I i + V i (4) Q i = 2 I i + V 2i (5) where = V i = U i U 2i V 2i = 2U i U 2i The ytem of equatio ()-(3) i called tructural it parameter are referred a the tructural coe ciet. Equatio (4) ad (5) are called the reduced form equatio, ad, 2 - reduced form coe ciet. Note that the reduced form error V i ad V 2i are correlated eve if the demad ad upply hock U i ad U 2i are idepedet. Sice I i i exogeou, oe ca coitetly etimate reduced form equatio by the uual LS etimatio. However, the ecoomit are uually itereted i the tructural equatio. The tructural equatio i called ideti ed if it coe ciet ca be recovered from the reduced form parameter. I the above example, we have that (6) Note that we aume that 6= or 6= Thu, the upply equatio i ideti ed while the demad i ot. Ideti catio of the upply equatio i poible becaue variatio i I i itroduce exogeou hift of the demad equatio, which allow u to "ee" the poit o the upply lie. Oe ca coitetly etimate 2 by Idirect LS (ILS). Let b ad b 2 be the LS etimator of the reduced coe ciet ad 2 repectively. The ILS etimator of 2 i give by b ILS b 2 b The etimator b ILS 2 i coitet if b ad b 2 are coitet a follow from (6). Further, it i eay to how that the ILS etimator i, i fact, idetical to the IV etimator. We have that b = b I i i I2 i I iq i I2 i Therefore, b ILS I iq i I i i = b IV 2 The aymptotic ditributio of the IV etimator ha bee dicued i Lecture. Next, coider the followig ytem Q d i = i + I i + U i Q i = 2 i + 2 I i + U 2i (7) Q d i = Q i = Q i 2

3 The reduced form equatio are the ame a i (4)-(5), however, ow we have = which caot be olved for either of the tructural coe ciet. The ytem i ot ideti ed becaue chage i I i hift both equatio. Thu, i order for the upply to be ideti ed, I i mut be excluded from the upply equatio ( ). Next, uppoe that the demad equatio cotai two exogeou variable excluded from the upply equatio Q d i = i + I i + 2 W i + U i Q i = 2 i + U 2i Q d i = Q i = Q i where W i i exogeou. The reduced form i with i = I i + 2 W i + V i Q i = 2 I i + 22 W i + V 2i = Now, there are two olutio for 2 2 ad 22 2 a reult, the ILS will produce two di eret etimate of 2 I thi cae, we ay that the model i overideti ed, ad a better approach i the 2SLS. De e i = The 2SLS etimator of 2 i give by b 2SLS ii ( ii ) iq i ii ( ii ) i i If U 2i are heterokedatic (coditioal o i ), oe ca ue the two tep procedure to obtai the e ciet GMM etimator, a dicued i Lecture 2 Ii W i where b U 2i = Q i b 2SLS 2 i b GMM ii b U2i 2 ii iq i ii b U2i 2 ii i i 3

4 Ideti catio ad etimatio We will ue the followig otatio to decribe the ytem of m imultaeou equatio. Let y i y mi be the m radom edogeou variable, ad i = (z i z li ) be the radom l-vector of exogeou variable. The variable y ji appear o the left-had ide of equatio j, j = m Let Y ji to deote the m j -vector of the right-had ide edogeou variable icluded i the j-th equatio. Similarly, the radom l j -vector j deote the right-had ide exogeou variable icluded i equatio j Let u ji be the radom hock to equatio j Thu, we ca write the j-th equatio a y ji = Y ji j + ji j + u ji where E ( i u ji ) = j 2 R mj ad j 2 R lj Thi equatio decribe a IV regreio model. De e further, Yji X ji = ji j j = The, the above equatio ca be writte a j y ji = X ji j + u ji where we kow that m j out of m j + l j regreor are edogeou. We have total l itrumetal variable i i available to u. The momet coditio for equatio j i give by = E i y ji X ji j = E i y ji i X ji j (8) The GMM etimatio require that the l (m j + l j ) matrix E i X ji ha the full colum rak m j + l j (the rak coditio). Thu, the (eceary) order coditio for ideti catio of equatio j i l m j + l j, or l l j m j The order coditio ay, that for equatio j to be ideti ed, the umber of exogeou regreor excluded from that equatio mut be at leat a large a the umber of icluded edogeou regreor. If the order ad rak coditio are ati ed, oe ca etimate j by GMM a e GMM j = X X ji i! X X j j i Xji X ji i X j j i y ji The e ciet igle equatio GMM etimator i uch that I the cae of homokedatic error, i.e. whe j j! p Eu 2 ji i i E u 2 jij i = jj for all i (9) we eed that We ca et j j! p (E i i) X j j = i i! 4

5 ad the e ciet GMM reduce to the 2SLS etimator! 2SLS X X X e j = X ji i i i i Xji X X ji i! X X i i i y ji The 2SLS procedure ha bee dicued i Lecture 2. The 2SLS etimator for the ideti ed equatio are coitet ad joitly aymptotically ormal. The aymptotic variace of the j-th 2SLS etimator i give by jj EX ji i (E i i) E i X ji Let aume further that, i additio to (9), the error atify E (u ri u i j i ) = r for all i () The 2SLS etimator are aymptotically correlated acro equatio if r 6= umig that equatio r ad are both ideti ed, the aymptotic covariace of e 2SLS r ad e 2SLS i give by the followig reult e 2SLS =2 r r e 2SLS! d N rr Q r Q r r Q r Q r Q r Q Q Q!! r Q Q where Q r = E i X ri Q = E i X i = E i i The aymptotic covariace betwee equatio are required if oe wat to tet retrictio o parameter acro equatio, ad r 6= Whe the error are ucorrelated acro equatio (give i ), the idividual 2SLS etimator are aymptotically ucorrelated ad, therefore, aymptotically idepedet, ice the aymptotic ditributio i ormal. Sytem etimatio If the error correlated acro equatio, the ytem GMM etimatio ca be viewed a the GLS procedure for imultaeou equatio. I thi cae, the ytem etimatio i e ciet, while 2SLS etimatio of idividual equatio i ot. Suppoe that all m equatio are ideti ed. The momet coditio for the ytem are give by i (y i Xi B ) = E. i (y mi Xmi m) i y i i Xi BB B = E.. i y mi i Xmi m BB = E i y i. i y mi ix i i X mi B. m () 5

6 Let j be a l l matrix of full rak that give the weight aiged to the momet coditio aociated with equatio j omparig (8) ad (), we deduce that the ytem GMM etimator i give by = B GMM b. GMM b m X ii m ixi X mii m m m ixmi X ii m B iy i X. mii m m m iy mi rovided that homokedaticity coditio (9) ad () hold, the optimal weight matrice have to atify m m m m! p 2 E i i 2 me i i 2 me i i 2 mme i i Let aume that the error are ucorrelated acro the equatio coditioal o i, i.e. r = for all r = m ad r 6= The, the above coditio for optimal weight matrice become m m m m! p 2 E i i 2 me i i I thi cae, oe ca et X j j = i i! ad r = for r 6= to obtai that the e ciet ytem GMM etimator ad 2SLS etimator for idividual equatio are idetical. 6