Systems of Equations (SUR, GMM, and 3SLS)

Transcription

1 Lecture otes on Advanced Econometrcs Takash Yamano Fall Semester 4 Lecture 4: Sstems of Equatons (SUR, MM, and 3SLS) Seemngl Unrelated Regresson (SUR) Model Consder a set of lnear equatons: $ + ɛ $ + ɛ () $ + ɛ where g s a -b-k g matrx (k g ma dffer across equatons but often not n applcatons). (g,,, ) g, $, ɛ are all -b- vectors. g contans onl exogenous varables. The sstem () s called Seemngl Uncorrelated because each equaton seems unrelated wth other equatons. However, t s often assumed that error terms are correlated across equatons. For example, man applcatons estmated a set of demand functons for famles. n ths case, error terms are most lkel to be correlated across equatons. The sstem () can be wrtten as + x x K K x x () Where K s a sum of k g for g,,. The error term has the followng propertes: ɛ [ɛ', ɛ',, ɛ' ]' E[ɛ,,, ] E[ɛɛ',,, ] ɛ s a -b- vector, and E[ɛɛ'] V s a -b- matrx. We further assume that

2 E[ɛ t ɛ js,,, ] j f t s and otherwse. For nstance, ths ndcates that the error term for the demand for pork (-th equaton) at tme t, ɛ t, s correlated wth the error term for the demand for chcken (j-th equaton) at tme t, ɛ jt. But the error terms are uncorrelated across tme. For one equaton, the varance-covarance matrx of ɛ s E[ɛ ɛ ',,, ] The varance-covarance matrx of ɛ and ɛ j s E[ɛ ɛ j ',,, ] j Thus, for all the observatons (3) Ω E s ],,, [ (What s the order of ths square matrx?) From (), we have (4) + + Thus, the error term of the SUR model n (4) has a varance-covarance of (3). Thus, a LS estmaton can be appled to the SUR model. Defne Σ so, n (3), Ω and Σ Ω Σ

3 j Denotng the j th element of Σ b, thus The LS estmator s Σ ˆ LS SUR [ Ω ] Ω [ ( Σ ) ] ( Σ ). See reene pp34 for more detals. Some mportant results on the SUR Model:. SUR SUR SUR3 f the equatons are actuall unrelated, then there s no paoff to LS estmaton. f the same set of ndependent varables are used for each equaton,, then LS and OLS are dentcal. f the ndependent varables on one block of equatons are a subset of those n another, then LS brngs on effcenc gan over OLS n estmaton of the smaller set of equatons. We wll examne these results n examples. Testng SUR Defne (,, ), (,,, ), u ( u, u,, u ), and dag(, Then, we can wrte (-4) as,,, + u Then the LS estmator s ˆ LS ) ( Ω ) Ω ( ( Σ ) ) ( Σ (5) Suppose all equatons have the same set of ndependent varables,. Then, we can wrte and. Thus, ˆ LS [( )( Σ )( )] ( )( Σ ) [( Σ )( )] ( Σ [( Σ ) ( )] ( Σ ) ). 3

4 [( Σ ( ) ]( Σ ) [( ( ) ] ( ) ( ) ( ) ( ) ( ) ˆ, OLS ( ) ˆ, OLS ( ) ˆ, OLS (We have used a rule on Kronecker Product that ( A B)( C D) ( AC BD) when AC and BD can be defned. See reene A.5.5. n Appendx A.) Thus, the LS estmators are dentcal to the OLS estmators. Example 4-: Wage and Frnge Benefts (FRE.dta) n ths example, we estmate a two-equaton sstem for hourl wage and hourl benefts. There are 66 workers n the data set. The FLS results for a SUR model are presented below. FL Estmaton for a SUR Model. sureg (hrearn educ exper exper unon) ( hrbens educ exper exper unon), small dfk Seemngl unrelated regresson Equaton Obs Parms RMSE "R-sq" F-Stat P hrearn hrbens Coef. Std. Err. t P> t [95% Conf. nterval] hrearn educ exper exper unon _cons hrbens educ exper exper unon _cons

5 Here, we examne a correlaton between resduals from two equatons. The test ndcates that the resduals from the two equatons are correlated wth a correlaton of The Breush-Pagan test suggests that the two are correlated sgnfcantl. t s not surprsng to fnd a strong correlaton because the same unobserved varables, such as ablt, across two equatons should be correlated. Correlaton matrx of resduals: hrearn hrbens hrearn. hrbens Breusch-Pagan test of ndependence: ch() 7.59, Pr. ow, we estmate the two equatons separatel b OLS. The results below are exactl the same as n the FLS-SUR model above. Ths confrms SUR, LS and OLS are dentcal f the same set of ndependent varables s used n a SUR model. The model above s not useless because we could at least test f the resduals from equatons are correlated. OLS on Hourl Wage:. reg hrearn educ exper exper unon Source SS df MS umber of obs F( 4, 6) 8.6 Model Prob > F. Resdual R-squared Adj R-squared.496 Total Root MSE hrearn Coef. Std. Err. t P> t [95% Conf. nterval] educ exper exper unon _cons OLS on Hourl Benefts:. reg hrbens educ exper exper unon Source SS df MS umber of obs F( 4, 6) Model Prob > F. Resdual R-squared Adj R-squared.95 Total Root MSE hrbens Coef. Std. Err. t P> t [95% Conf. nterval] educ exper exper unon _cons To test SUR 3, we add two ndependent varables, marred and male, to the frst equaton but not n the second. Thus, the ndependent varables n the second equaton are a subset 5

6 of the set of ndependent varables n the frst equaton. The results below confrms SUR3, the results below on the second equaton are dentcal to the OLS results.. sureg (hrearn educ exper exper unon marred male) ( hrbens educ exper exper unon), small dfk corr Seemngl unrelated regresson Equaton Obs Parms RMSE "R-sq" F-Stat P hrearn hrbens Coef. Std. Err. t P> t [95% Conf. nterval] hrearn educ exper exper unon marred male _cons hrbens educ exper exper unon _cons End of Example 4- eneralzed Method of Moments (MM) Estmaton Suppose we have a set of equatons and for observaton, we have $ + ɛ where s a -b- vector (-b- f there are two equatons), s a -b-k matrx, and ɛ s a -b- error vector. Assumpton : E(Z' u ), where Z s a -b-l matrx of observable nstrumental varables. Assumpton : rank (Z' ) K. stands for sstem nstrumental varables. We also assume E(u ). The order condton, L K, should be satsfed when the rank condton ( ) s met. We have a set of equatons wth potentall endogenous varables: $ + ɛ $ + ɛ (6) $ + ɛ 6

7 where we can also wrte Y $ + ɛ Where Y s a -b- vector, s a -b-k matrx, and ɛ s a -b- vector. Under Assumptons and, ˆ s the unque K-b- vector solvng the lnear set populaton moment condtons: E[Z' (Y - ˆ )] (7) Because sample averages are consstent estmators of populaton moments, the analog appled to condton (7) suggests choosng to solve ˆ Z ( Y ˆ ) (8) Z s a -b-l matrx. When L K, ˆ (Z') - Z'Y Ths s called the sstem V () estmator. When L > K (so we have more columns n the V matrx Z than we need for dentfcaton), choosng s more complcated. The equaton (3-8) does not have a ˆ soluton. nstead, we choose ˆ to make the vector n equaton (3-8) as small as possble n the sample. One dea s to mnmze ths: Z ( Y ˆ ) Z ( Y ˆ ) But, a more general class of estmators s obtaned b usng a weghtng matrx n the quadratc form. Let Wˆ be an L-b-L smmetrc, postve semdefnte matrx. The generalzed method of moments (MM) estmator that solves the problem mn or mn Z ( Y ˆ ) Wˆ Z ( Y ˆ [ ˆ Z ( Y )] ˆ [ ( ˆ W Z Y )] [ Y Z ˆ Z ][ WZ ˆ Y WZ ˆ ˆ ] mn mn Y ZWZ ˆ Y ˆ ZWZ ˆ Y Y ZWZ ˆ ˆ + ˆ ZWZ ˆ ˆ F.O.C. s ZWZ ˆ Y Y ZWZ ˆ + ZWZ ˆ ˆ ) 7

8 ZWZ ˆ ˆ ZWZ ˆ Y ˆ ZWZ ˆ ) ( ZWZ ˆ Y (9) MM Assumpton 3: W ˆ p W as, where W s nonrandom, smmetrc, L-b-L postve defnte matrx. Theorem (Consstenc of MM): Under assumptons -3, the MM estmators are consstent and asmptotcall normall dstrbuted. The Sstem SLS Estmator Suppose we choose ˆ W ( Z Z / ) whch s a consstent estmator of [E( Z Z )] -. Assumpton 3 smpl requres that E( Z Z ) exst and be nonsngular. When we plug ths nto equaton (9), we get ˆ S SLS ( Z( Z Z) Z ) Z( Z Z Z Y ) () Ths s called the sstem SLS estmator. The Optmal Weghtng Matrx The Procedure. Let ~ n be an ntal consstent estmator of. n most cases ths s the sstem SLS estmator.. Obtan the -b- resdual vectors ~ ~ for,,, n 3. A generall consstent estmator of Λ s Λ ˆ Z ~ ~ Z 4. Choose W ˆ ˆ ~ ~ Λ Z Z and use ths matrx to obtan the asmptotcall optmal MM estmator. The Three-Stage Least Squares (3SLS) Estmator The 3SLS estmator s a MM estmaton that uses a partcular weghtng matrx. Let u~ ~ S SLS be the resdual from an ntal estmaton (usuall a sstem SLS), then defne 8

9 Ω ~ ~ u u Usng the same arguments as n the FLS case, ˆ P Ω Ω E( u ~ ~ u ). The weghtng matrx used b 3SLS s Wˆ ˆ Z Ω [ ( Ωˆ Z Z ) Z / ] Pluggng ths nto (9), we get ˆ 3 SLS [ Z{ Z ( Ωˆ ) Z} Z ] Z{ Z ( Ωˆ ) Z} Z Y () Ths s dfferent from the tradtonal 3SLS estmator n most textbooks, such as reene (pp. 46 n 5e). The tradtonal 3SLS estmator s ˆ ˆ ˆ T SLS ˆ [ ( ) ] ˆ ( ˆ 3 Ω Ω ) Y where ˆ Z Πˆ and Π ˆ ( Z Z) Z. (ote that n reene notatons are dfferent.) n short, the MM defnton of 3SLS s more generall vald, and t reduces to the standard defnton n the tradtonal smulaton equatons settng. 9