SLS Estmates ECON 3033 Bll Evas Fall 05 Two-Stage Least Squares (SLS Cosder a stadard lear bvarate regresso model y 0 x. I ths case, beg wth the assumto that E[ x] 0 whch meas that OLS estmates of wll roduced based ad cosstet estmates. Suose however that there s a varable z that drectly macts x but t s ucorrelated wth. Ths ca be exressed as E[ z] 0. The varable z s kow as a strumet the model. The mlct assumto s that z macts x ad x macts y. Therefore, f we were to somehow shock x wth z, we ca fer the relatosh betwee x ad y. Ths s doe by cosderg the relatosh betwee x ad z as a smle lear model. Ths frst-stage relatosh betwee x ad z s gve by the equato x z u 0 Where we assume z ad u are ucorrelated. Note that x has two comoets. Oe s a determstc oe that s a fucto of z, x 0 z. Because z s ucorrelated wth ε the ths redctve comoet of x s also ucorrelated wth ε. Sce x s correlated wth ε ad x s ot, t must be that what s drvg the correlato betwee x ad ε s the u. The suggesto the s stead of usg x the regresso y 0 x -- whch we kow would roduce based estmates -- why ot use x stead. The x comoet of x cotas formato about x, but t s NOT correlated wth ε so t should roduce estmates wth ce roertes. That s a great dea excet that we do ot kow 0 or ad therefore, we caot use x a regresso. We ca however use the ext best thg: a ubased estmate of x whch s roduced from a frst-stage regresso of x o z. Ths two-stage least squares rocedure s very straght forward. Ru a regresso of the form x z u. It should be o surrse that the OLS estmates for the arameters 0 ad are 0 x z ad 0 x. ( z z ( x x. Wth these estmates we ca costruct a redcted value for ( z z 0 z Istead of usg x the estmates for -- whch roduces ols ( x x( y y ( x x
use the redcted value of x, x 0 z, stead. The redcted value s a lear fucto of z whch s ucorrelated wth so the SLS estmate for should have ce roertes. The SLS estmate for s therefore sls x( y y. x Workg wth the SLS estmate Gve that x 0 z t s clear that x 0 zad hece x ( x z z. Substtute ths to the sls equato for ad we obta x( y y ( z z ( y y ( z z ( y y ( z z ( y y sls x ( z z ( z z ( z z Substtute the defto of to ths equato ad we obta ( z z ( y y ( z z ( y y ( z z ( y y sls ( z z ( x x ( z z ( z z ( z z ( z z ( x x Dvde the umerator ad deomator by (- ad we ote the SLS estmte s therefore the rato of the covarate betwee y ad z dvded by the covarace betwee x ad z. ( z z ( y y ( z z ( y y / ( ( z z ( x x ( z z ( x x / ( sls yz sls Is a ubased estmate?
Aytme we have a ew radom varable, we always ask what s the exected value ad the varace? Ths s a dffcult questo. I the ed, we are uable to show that sls s a ubased estmate ths cotext? Why? I order to demostrate the roertes of the estmate, we always substtute the truth back to the model. Start wth the estmate for sls ad ote that we ca dro oe of the meas both the umeatror ad deomator. ( z z ( y y ( z z y sls ( z z ( x x ( z z x At ths ot, we usually substtute the defto of y to the umerator. However, x s also a radom varable that s a fucto of the error v so we would eed to substtute that defto as well. ( z z y ( z z ( x 0 sls ( z z x ( z z ( z u 0 The roblem s that we ow have radom varables the umerator ad the deomator. If a ad b are radom varables E[a/b] E[a]/E[b]. Wthout kowg the jot dstrbuto of the umerator ad deomator, we caot fgure out the exected value of the estmate. Our oly oto s the to cosder whether the estmate s cosstet. The cosstecy of SLS estmates ( z z ( y y To show whether the SLS estmate s cosstet, start wth the defto sls dro ( x x( z z y ad substtute the defto of y. Ths roduces sls ( z z ( 0 x. Exad the ( x x( z z umerator, ad we get ( z z ( z z x ( z z. The frst ( x x( z z ( x x( z z ( x x( z z 0 sls 3
term equals zero because sums of devato from meas equal zero. The secod term equals ad the ( z z whole term reduces to sls. Dvde the umerator ad deomator of the rght ( x x( z z ( z z / ( had term by (- ad you get ( x x( z z / ( sls ze. Now we ask what haes to ths estmate as the samle szes creases to fty. Note that f a ad b are estmates, the lm(a/b=lm(a/lm(b. Note also that the lm( ad lm(, ad therefore, t s z z o surrse that SLS ze lm(. I ths case, the estmate SLS wll oly be cosstet f lm( 0. To roduce cosstet estmates, t must be the case that z oly roduces a chage z z y through a tal chage z. If z has a drect effect o y the lm( 0 ad the model wll z z roduce cosstet estmates. A alteratve terretato of SLS estmates The structural equato of terest s y 0 x. We atcate that x s correlated wth ε so we caot estmate ths model by OLS. We do however have a varable z that we beleve s redctve of x but t s ot drectly correlated wth ε. The frst-stage equato that relatos x to z s gve by the equato x z u. Substtute ths equato to the equato for y ad oe obtas the equato 0 y ( z u ( z u 0 0 0 0 I ths model, defe ( 0 0 0 let ad let u v. Ths allows us to wrte the equato for y as y 0 z v. Ths equato s referred to as the reduced-form ad t reresets the y correlato betwee z ad y. The coeffcet s. Note that the coeffcet has a terestg z terretato. The model we are suggestg s that y s a fucto of x ad x s a fucto of z. Therefore y=f(x(z. If we take the dervatve of y wth resect to z, because we assume z oly chages y through a 4
y y x chage x, we obta z x z. Note that yx =, whch s exactly. Therefore, f we xz yx take the reduced form estmate for whch equals xz = ad dvde t by the st stage coeffcet we get. It s also the case that sls. ( z z ( y y ( z z ( z z ( y y sls ( z z ( x x ( z z ( x x ( z z whch s exactly the same estmate we got above. To reca, we call the tal equato the structural equato of terest : y 0 x To estmate ths va SLS or drect least squares, we eed a frst-stage relatosh x 0 z u The drect relatosh betwee the strumet ad the outcome of terest s called the reduced-form. y z v 0 sls The varace of Note that f we were to estmate the regresso y 0 x by OLS, the varace for would be ols Var( ols ( x x. The varace s roortoal to how much varace x s used to roduce the estmate. I the case of SLS, we are ot use x but stead, x. Therefore Recall that a regresso, SST=SSM+SSE ad ths case result, by costructo, ols sls ( x ( x x x Var( sls ( x ( x x x v x. As a whch meas also that by costructo Var( Var(. The beeft of SLS s that we are oly usg a art of the varato x that art. 5
that s ucorrelated wth -- so our estmates are cosstet. However, because we are usg smaller varato x that the OLS case, the cost of SLS s a reducto recso the value sls Var( creases cosderably. What wll roduce smaller values for ( sls Var? Start wth the defto Var( sls z z remember that x x ( z z. Substtute ths ad we get sls Var( ( ( z z ( x x ad substtute ths to the varace equato ad you get ( z z x ad. Note that Var( ( z z sls ( z z ( x x ( ( z z x x ( z z ( z z The deomator ths term ( z z ( x x s othg more that (- xz whle the umerator ca be thought of as ( z z (. Therefore, z Var( ( z z sls ( z z ( ( xz xz ( z z ( x x ad we wll reduce the ( sls Var s f we ca fd a varable z that s strogly correlated wth x. At the other ed of the sectrum, suose xz aroaches zero ths meas z does ot exla much of x. What s the cost? I ths case, f z does ot exla much of x, the we caot lear x s mact o y ad the sls Var( wll exlode. 6