Lecture 8: Instrumental Variables Estimation

Transcription

1 Lecture Notes on Advanced Econometrcs Lecture 8: Instrumental Varables Estmaton Endogenous Varables Consder a populaton model: Takash Yamano Fall Semester 5 y = $ y y + $ + $ x + + $ k- x k- + u (7-) We call y an endogenous varable when y s correlated wth u. As we have studed earler, y would be correlated wth u f (a) there are omtted varables that are correlated wth y and y, (b) y s measured wth errors, and (c) y and y are smultaneously determned (we wll cover ths ssue n the next lecture note). All of these problems, we can dentfy the source of the problems as the correlaton between the error term and one or some of the ndependent varables. For all of these problems, we can apply nstrumental varables (IV) estmatons because nstrumental varables are used to cut correlatons between the error term and ndependent varables. To conduct IV estmatons, we need to have nstrumental varables (or nstruments n short) that are (R) uncorrelated wth u but (R) partally and suffcently strongly correlated wth y once the other ndependent varables are controlled for. It turns out that fndng proper nstruments s very dffcult! In practce, we can test the second requrement (b), but we can not test the frst requrement (a) because u s unobservable. To test the second requrement (b), we need to express a reduced form equaton of y wth all of exogenous varables. Exogenous varables nclude all of ndependent varables that are not correlated wth the error term and the nstrumental varable, z. The reduced form equaton for y s y = δ Z z + δ + δ x + + δ k x k- + u For the nstrumental varable to satsfy the second requrement (R), the estmated coeffcent of z must be sgnfcant. In ths case, we have one endogenous varable and one nstrumental varable. When we have the same number of endogenous and nstrumental varables, we say the endogenous varables are just dentfed. When we have more nstrumental varables than endogenous varables, we say the endogenous varables are over-dentfed. In ths case,

2 we need to use two stage least squares (SLS) estmaton. We wll come back to SLS later. Defne x = (y,, x,, x k- ) as a -by-k vector, z = (z,, x,, x k- ) a -by-k vector of all exogenous varables, X as a n-by-k matrx that ncludes one endogenous varable and k- ndependent varables, and Z as a n-by-k matrx that nclude one nstrumental varable and (k-) ndependent varables: y x x3 x k z x x3 x k = y x x3 xk = z x x3 xk X, Z. yn xn xn3 xnk zn xn xn3 xnk The nstrumental varables (IV) estmator s ˆ β = ( Z X ) Z Y (7-) IV Notce that we can take the nverse of Z'X because both Z and X are n-by-k matrces and Z'X s a k-by-k matrx whch has full rank, k. Ths ndcates that there s no perfect co lnearty n Z. The condton that Z'X has full rank of k s called the rank condton. The consstency of the IV estmators can be shown by usng the two requrements for IVs: ˆ β = ( Z X ) Z ( Xβ u) IV + = β + ( Z X ) Z u = β + ( Z X / n) Z u / n From the frst requrement (R), p lm Z u / n. From the second requrement (R), p lm Z X / n A, where A E( z x). Therefore, the IV estmator s consstent when IVs satsfy the two requrements. A Bvarate IV model Let s consder a smple bvarate model: y = $ y y + $ + u We suspect that y s an endogenous varable, cov(y, u). Now, consder a varable, z, whch s correlated x but not correlated wth u: cov(z, y ) but cov(z, u) =. And consder cov(z, y ): cov(z, y ) = cov(z, $ y y + $ + u) = $ y cov(z, y ) + cov(z, u)

3 Because cov(z, u) =, $ y = cov(z, y ) / cov(z, y ) = n = n = ( z z)( y ( z z)( y y ) y ) The problem n practce s the frst requrement, cov(z, u) =. We can not emprcally confrm ths requrement because u cannot be observed. Thus, the valdty of ths assumpton s left to economc theory or economsts common sense. Recent studes show that even the frst requrement can be problematc when the correlaton between the endogenous and nstrumental varables s weak. Here s a bvarate case. Weak Correlaton between the IVs and the Endogenous Varables In a bvarate model, we wrte plm ˆ β IV = β + cov( z, u) cov( z, ) y because Corr(z,u)=cov(z,u)/[sd(z)sd(u)] (see Wooldrdge pp74) plm plm ˆ β IV = β + corr( z, u) /( sd( z) sd( u)) corr( z, y ) /( sd( z) sd( )) y corr( z, u) sd( y ) corr( z, y ) sd( ) ˆ IV = β + u β Thus f z s only weakly correlated wth the endogenous varable, y, --corr(z, y ) s very small--, the IV estmator could be severely based. Example 7-: Card (995), CARD.dta. A dummy varable grew up near a 4 year collage as an IV on educ. OLS. reg lwage educ Source SS df MS Number of obs = F(, 38) = Model Prob > F =. Resdual R-squared =.987 3

4 Adj R-squared =.984 Total Root MSE =.439 lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ _cons Correlaton between nearc4 (an IV) and educ. reg educ nearc4 Source SS df MS Number of obs = F(, 38) = 63.9 Model Prob > F =. Resdual R-squared = Adj R-squared =.5 Total Root MSE =.6494 educ Coef. Std. Err. t P> t [95% Conf. Interval] nearc _cons Thus, nearc4 satsfes the one of the two requrements to be a good canddate as an IV. IV Estmaton:. vreg lwage (educ=nearc4) Instrumental varables (SLS) regresson Source SS df MS Number of obs = F(, 38) = 5.7 Model Prob > F =. Resdual R-squared = Adj R-squared =. Total Root MSE = lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ _cons Instrumented: educ Instruments: nearc4 Note that you can obtan the same coeffcent by estmatng OLS on lwage wth the predcted educ (predcted by nearc4). However, the standard error would be ncorrect. In the above IV Estmaton, the standard error s already corrected. End of Example 7-4

5 The Two-Stage Least Squares Estmaton Agan, let s consder a populaton model: y = $ y y + $ + $ x + + $ k- x k- + u (7-3) where y s an endogenous varable. Suppose that there are m nstrumental varables. Instruments, z = (, x,, x k-, z,, z m ), are correlated wth y. From the reduced form equaton of y wth all exogenous varables (exogenous ndependent varables plus nstruments), we have y = δ + δ x + + δ k- x k- + δ k z + + δ k+m- z m + r y (7-4) = ŷ + r y ŷ s a lnear projecton of y wth all exogenous varables. Because ŷ s projected wth all exogenous varables that are not correlated wth the error term, u, n (7-3), ŷ s not correlated wth u, whle r y s correlated wth u. Thus, we can say that by estmatng y wth all exogenous varables, we have dvded nto two parts: one s correlated wth u and the other s not. The projecton of y wth Z can be wrtten as ˆ y ˆ = Zδ = Z( Z Z) Z y When we use the two-step procedure (as we dscuss later), we use ths ŷ n the place of y. But now, we treat y as a varable n X and project X tself wth Z: Xˆ = ZΠ= ˆ Z( Z Z) Z X = P X Πˆ s a (k+m-)-by-k matrx wth coeffcents, whch should look lke: δ δ Π= ˆ δ k. + m Z Thus, y n X should be expressed as a lnear projecton, and other ndependent varables n X should be expressed by tself. P Z = Z( Z Z) Z s a n-by-n symmetrc matrx and dempotent (.e., P Z PZ = PZ ). We use Xˆ as nstruments for X and apply the IV estmaton as n ˆ β = ( Xˆ X ) Xˆ Y = ( X P X ) X P Y = ( X Z( Z Z) Z X ) X Z( Z Z Z Y (7-5) SLS Z Z ) 5

6 Ths can be also wrtten as ˆ SLS = Xˆ Xˆ ) β ( Xˆ Y (7-6) Ths s the SLS estmator. It s called as two-stage because t looks lke we tale two steps by creatng projected X to estmate SLS estmator n (-6). As matter of a fact, we do not need to take two steps, as you can see n (-6), we can just estmate SLS estmators n one step by usng X and Z. (Ths s what econometrcs packages do.) The Two-Step procedure It s stll a good dea to know how to estmate the SLS estmators by a two-step procedure: Step : Obtan ŷ by estmatng an OLS aganst all of exogenous varables, ncludng all of nstruments (the frst-stage regresson) Step : Use ŷ n the place of y to estmate y aganst ŷ and all of exogenous ndependent varables, not nstruments (the second stage regresson) The estmated coeffcents from the two-step procedure should exactly the same as SLS from (-5) or (-6). However, you must be aware that the standard errors from the two-step procedure are ncorrect, usually smaller than the correct ones. Thus, n practce, t s always safe not to use the two-step procedure. Avod usng predcted varables as much as you can! Econometrc packages wll provde you SLS results based on (-5) or (-6). So you do not need to use the two-step procedure. We use the frst step procedure to test the second requrement for IVs. In the frst stage regresson, we should conduct a F-test on all nstruments to see f nstruments are jontly sgnfcant n the endogenous varable, y. As we dscuss later, nstruments should be strongly correlated wth y to have relable SLS estmators. Consstency of SLS Assumpton SLS.: For some -by-(k+m-) vector z, E(z'u)=, where z = (, x,, x k-, z,, z m ). Assumpton SLS.: (a) rank E(z'z) = k+m-; (b) rank E(z'x) = k. (b) s the rank condton for dentfcaton that z s suffcently lnearly related to x so that rank E(z'x) has ful column rank. The order condton s k-+m k-+h, where h s the number of endogenous varable. Thus, the order condton ndcates that we must have at least as many nstruments as endogenous varables. 6

7 Under assumpton SLS and SLS, the SLS estmators n (-5) are consstent. Under homoskedastcty, ˆ σ ( Xˆ X n ˆ ) = ( n k) uˆ ( Xˆ Xˆ = ) s a vald estmator of the asymptotc varance of ˆ β. SLS Under heteroskedastcty, the heteroskedastcty-robust (Whte) standard errors s ( X ˆ ˆ ) ˆ ˆ ˆ ( ˆ ˆ ) X XΣX X X Here are some tests for IV estmaton. See Wooldrdge (Introductory Econometrcs) for detals. Testng for Endogenety () () () (v) Estmate the reduced form model usng the endogenous varable as the dependent varable: q aganst z s and x s. Obtan the resdual, vhat Estmate y = $ + $ q + $ x + + $ k- x k- + $ v vhat + u If $ v s sgnfcant, then q s endogenous. Testng the Over-Identfcaton () () () Estmate $ IV and obtan vhat. Regress vhat on z s and x s. Get R and get NR, whch s ch-squared. Example 7-: Card (995), card.dta agan. OLS. reg lwage educ exper expersq black smsa south Source SS df MS Number of obs = F( 6, 33) = 4.93 Model Prob > F =. Resdual R-squared = Adj R-squared =.89 Total Root MSE =.3749 lwage Coef. Std. Err. t P> t [95% Conf. Interval]

8 educ exper expersq black smsa south _cons IV Estmaton: nearc nearc4 as IVs. vreg lwage (educ= nearc nearc4) exper expersq black smsa south Instrumental varables (SLS) regresson Source SS df MS Number of obs = F( 6, 33) =.3 Model Prob > F =. Resdual R-squared = Adj R-squared =.438 Total Root MSE =.465 lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper expersq black smsa south _cons Instrumented: educ Instruments: nearc nearc4 + exper expersq... south. vreg lwage (educ= nearc nearc4 fatheduc motheduc) exper expersq black sms > a south Instrumental varables (SLS) regresson Source SS df MS Number of obs = F( 6, 3) = Model Prob > F =. Resdual R-squared = Adj R-squared =.57 Total Root MSE =.386 lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper expersq black smsa south _cons Instrumented: educ Instruments: nearc nearc4 fatheduc motheduc + exper expersq... south 8

9 IV Estmaton: fatheduc motheduc as IVs. vreg lwage (educ= fatheduc motheduc) exper expersq black smsa south Instrumental varables (SLS) regresson Source SS df MS Number of obs = F( 6, 3) = Model Prob > F =. Resdual R-squared = Adj R-squared =.58 Total Root MSE =.3857 lwage Coef. Std. Err. t P> t [95% Conf. Interval] educ exper expersq black smsa south _cons Instrumented: educ Instruments: fatheduc motheduc + exper expersq... south Whch ones are better? End of Example 7-9

10 Lecture 9: Heteroskedastcty and Robust Estmators In ths lecture, we study heteroskedastcty and how to deal wth t. Remember that we dd not need the assumpton of Homoskedastcty to show that OLS estmators are unbased under the fnte sample propertes and consstency under the asymptotc propertes. What matters s how to correct OLS standard errors. Heteroskedastcty In ths secton, we consder heteroskedastcty, whle mantanng the assumpton of noautocorrelaton. The varance of dsturbance, u, s not constant across observatons but not correlated wth u j : =Σ = = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( n n n n n n n u u E u u E u u E u u E u u E u u E u u E u u E u u E uu E σ σ σ or Ω = = / / / ) ( σ σ σ σ σ σ σ σ n uu E Notce that under homoskedastcty, I = Ω. Under heteroskedastcty, the sample varance of OLS estmator (under fnte sample propertes) s Var(Βˆ ) = Var [% + (XtX) - Xtu] = E [(XtX) - Xtu utx (XtX) - ] = (XtX) - Xt E [u ut] X (XtX) - = (XtX) - XtF S X (XtX) - = (XtX) - XtG X (XtX) - (7-) (See Theorem. n Greene (3))

11 Unless you specfy, however, econometrc packages automatcally assume homoskedastcty and wll calculate the sample varance of OLS estmator based on the homoskedastcty assumpton: Var(Βˆ ) = F (XtX) - Thus, n the presence of heteroskedastcty, the statstcal nference based on s (XtX) - would be based, and t-statstcs and F-statstcs are napproprate. Instead, we should use (7-) to calculate standard errors and other statstcs. Fnte Sample Propertes of OLS Estmators The OLS estmators are unbased and have the samplng varance specfed n (6-). If u s normally dstrbuted, then the OLS estmators are also normally dstrbuted: ˆ Β X ~ N[ B, σ ( X X ) ( XΩ X )( X X ) ] Asymptotc Propertes of OLS Estmators If Q = plm(xtx) and plm(xtωx/ n) are both fnte postve defnte matrces, then Βˆ s consstent for B. Under the assumed condtons, plmβˆ = B. (See Greene (3), page 93-94, for detals.) Robust Standard Errors If G s known, we can obtan effcent least square estmators and approprate statstcs by usng formulas dentfed above. However, as n many other problems, G s unknown. One common way to solve ths problem s to estmate G emprcally: Frst, estmate an OLS model, second, obtan resduals, and thrd, estmate G: Σˆ = uˆ uˆ uˆ n (We may multply ths by (n/(n-k-)) as a degree-of-freedom correcton. But when the number of observatons, n, s large, ths adjustment does not make any dfference.) Thus by usng the estmated G, we have

12 = = Σ = ˆ ˆ ˆ ˆ ˆ k kn k n n x x x x x x x x x x x u X u u u X X X. Therefore, we can estmate the varances of OLS estmators (and standard errors) by usng ˆ : Var(Βˆ ) = (XtX) - XtΣˆ X (XtX) - (7-) Standard errors based on ths procedure are called (heteroskedastcty) robust standard errors or Whte-Huber standard errors. Or t s also known as the sandwch estmator of varance (because of how the calculaton formula looks lke). Ths procedure s relable but entrely emprcal. We do not mpose any assumptons on the structure of heteroskedastcty. Sometmes, we may mpose assumptons on the structure of the heteroskedastcty. For nstance, f we suspect that the varance s homoskedastc wthn a group but not across groups, then we obtan resduals for all observatons and calculate average resduals for each group. Then, we haveσˆ whch has a constant ˆ j u for group j. (In STATA, you can specfy groups by usng cluster.) In practce, we usually do not know the structure of heteroskedastcty. Thus, t s safe to use the robust standard errors (especally when you have a large sample sze.) Even f there s no heteroskedastcty, the robust standard errors wll become just conventonal OLS standard errors. Thus, the robust standard errors are approprate even under homoskedastcty. A heteroskedastcty-robust t statstc can be obtaned by dvdng an OSL estmator by ts robust standard error (for zero null hypotheses). The usual F-statstc, however, s nvald. Instead, we need to use the heteroskedastcty-robust Wald statstc. Suppose the hypotheses can be wrtten as H : RB = r Where R s a q x (k+) matrx (q < (k+)) and r s a q x vector wth zeros for ths case. Thus, [ ] = = = +, : ) ( r I R q k q q.

13 The heteroskedastcty-robust Wald statstcs for testng the null hypothess s W= ( R ˆ β r) ( RVˆ R ) ( R ˆ β r) where Vˆ s gven n (7-). The heteroskedastcty-robust Wald statstcs s asymptotcally dstrbuted ch-squared wth q degree of freedom. The Wald statstcs can be turned nto an approprate F-statstcs (q, q-k- ) by dvdng t by q. Tests for Heteroskedastcty When should we use robust standard errors? My personal answer to ths queston s almost always. As you wll see n Example 7-, t s very easy to estmate robust standard errors wth STATA or other packages. Thus, at least I suggest that you estmate robust standard errors and see f there are any sgnfcant dfferences between conventonal standard errors and robust standard errors. If results are robust,.e., when you do not fnd any sgnfcant dfferences between two sets of standard errors, then you could be confdent n your results based on homoskedastcty. Statstcally, you can use followng two heteroskedastcty tests to decde f you have to use robust standard errors or not. The Breusch-Pagan Test for Heteroskedastcty If the homoskedastcty assumpton s true, then the varance of error terms should be constant. We can make ths assumpton as a null hypothess: H : E(u X) = F To test ths null hypothess, we estmate û = * + * x + * x + + * k x k + e. (7-3) Under the null hypothess, ndependent varables should not be jontly sgnfcant. The F- statstcs that test a jont sgnfcance of all ndependent varables s F k, n k The LM test statstcs s R / k = ( R ) /( n k) LM = n R ~ P k The Whte Test for Heteroskedastcty 3

14 Whte proposed to add the squares and cross products of all ndependent varables to (9- ): uˆ δ + δ x + δ x δ x + λ x + λ x λ x + φ x x + φ x x φ x = k k k k 3 k k + Because ŷ ncludes all ndependent varables, ths test s equvalent of conductng the followng test: u ˆ = δ + δ yˆ + δ yˆ + v We can use F-test or LM-test on H: δ = and δ. = x v Example7-: Step-by-Step Estmaton for Robust Standard Errors In the followng do-fle, I frst estmate a wage model: logwage = $ + $ female + $ educ + $ exper + $ expersq + u by usng WAGE.dta. Then, by usng resduals from ths conventonal OLS, I estmate Σˆ and obtan robust standard errors by step-by-step wth matrx. Fnally, I verfy what I get wth robust standard errors provded by STATA. Of course, you do not need to use matrx to obtan robust standard errors. You just need to use STATA command, robust, to get robust standard errors (e.g., reg y x x x3 x4, robust). But at least you know how robust standard errors are calculated by STATA.. *** on WAGE.dta. *** Ths do-fle estmates Whte-Huber robust standard errors. set matsze 8. clear. use c:\docs\fasd\econometrcs\homework\wage.dta.. * Varable constructon. gen logwage=ln(wage). gen expsq=exper*exper. gen x=.. * Obtan conventonal OLS resduals. reg logwage female educ exper expsq Source SS df MS Number of obs = F( 4, 5) = Model Prob > F =. Resdual R-squared = Adj R-squared =.395 Total Root MSE =.4345 logwage Coef. Std. Err. t P> t [95% Conf. Interval]

15 female educ exper expsq _cons predct e, resdual. gen esq=e*e.. * Create varables wth squared resduals. gen efemale=esq*female. gen eeduc=esq*educ. gen eexper=esq*exper. gen eexpsq=esq*expsq. gen ex=esq*x.. * Matrx constructon. mkmat logwage, matrx(y). mkmat efemale eeduc eexper eexpsq ex, matrx(ex).. * Whte-Huber robust standard errors. matrx xx=symnv(x'*x). matrx sgma=ex'*x. matrx b_robust=(56/(56-5))*xx*sgma*xx. * Here s the Whte-Huber robust var(b^). mat lst b_robust b_robust[5,5] female educ exper expsq x female e-7.647e educ e-6.57e exper -.77e-7-3.8e e expsq.647e-9.57e-7-4.5e-7.9e e-7 x e * Take square root of the dagonal elements>> sd.error. * Thus sd.er. for female s (.3)^.5=.36, educ s (.59)^.5=.768,. * and so on. 5

16 But, you do not need to go through ths calculaton yourself. STATA has a command called robust.. * Verfy wth STATA verson of robust standard errors. reg logwage female educ exper expsq, robust Regresson wth robust standard errors Number of obs = 56 F( 4, 5) = 8.97 Prob > F =. R-squared =.3996 Root MSE =.4345 Robust logwage Coef. Std. Err. t P> t [95% Conf. Interval] female educ exper expsq _cons end of do-fle Example 7-: the Breusch-Pagan test. *** on WAGE.dta. *** Ths do-fle conducts the Breusch-Pagan heteroskedastcty test. set matsze 8. clear. use c:\docs\fasd\econometrcs\homework\wage.dta. * Varable constructon. gen logwage=ln(wage). gen expsq=exper*exper. * Obtan resduals from the level model. reg wage female educ exper expsq (Output s omtted). predct u, resdual. gen uu=u*u End of Example 7-. * the BP test. reg uu female educ exper expsq Source SS df MS Number of obs = F( 4, 5) =.79 Model Prob > F =. Resdual R-squared = Adj R-squared =.84 Total Root MSE =.94 uu Coef. Std. Err. t P> t [95% Conf. Interval] female educ exper expsq _cons test female educ exper expsq 6

17 ( ) female =. ( ) educ =. ( 3) exper =. ( 4) expsq =. F( 4, 5) =.79 Prob > F =. The LM test statstcs s 56 x.894 = 47.. Ths s sgnfcant at percent level because the crtcal level s 3.8 for a ch-square dstrbuton of four degree of freedom. End of Example 7- Example 7-3: the Whte test. * Obtan resduals from the log model. reg logwage female educ exper expsq (Output omtted). predct yhat (opton xb assumed; ftted values). predct v, resdual. gen yhatsq=yhat*yhat. gen vsq=v*v. * the Whte test. reg vsq yhat yhatsq Source SS df MS Number of obs = F(, 53) = 3.96 Model Prob > F =.97 Resdual R-squared = Adj R-squared =. Total Root MSE =.7656 vsq Coef. Std. Err. t P> t [95% Conf. Interval] yhat yhatsq _cons test yhat yhatsq ( ) yhat =. ( ) yhatsq =. F(, 53) = 3.96 Prob > F =.97 LM stat s 56 x.49 = 7.84, whch s sgnfcant at 5 percent level but not at percent level. End of Example 7-3 7

18 Lecture : GLS, WLS, and FGLS Generalzed Least Square (GLS) So far, we have been dealng wth heteroskedastcty under OLS framework. But f we knew the varance-covarance matrx of the error term, then we can make a heteroskedastc model nto a homoskedastc model. As we defned before E ( uu ) = σ Ω=Σ. Defne further that Ω = P P P s a n x n matrx Pre-multply P on a regresson model or Py = PXB + Pu y * = X * B + u * (9-) In ths model, the varance of u * s E(u * ut * ) = E(PuutPt) = PE(uut)Pt= P σ Ω Pt= σ P Ω Pt= σ I. Note that PSPt= I, because defne PSPt=A, then PtPSPt= PtA, by the defnton of P S - SPt= PtA, thus Pt= PtA. Therefore, A must be I. Because E(u * ut * ) = σ I, the model (9-3) satsfes the assumpton of homoskedastcty. Thus, we can estmate the model (9-3) by the conventonal OLS estmaton. Hence, ˆ Β = ( X * X *) X * y* = ( X P PX ) X P Py = ( X Ω X ) X Ω y s the effcent estmator of Β. Ths s called the Generalzed Least Square (GLS) estmator. Note that the GLS estmators are unbased when E( u * X * ) =. The varance of GLS estmator s var( Βˆ ) = Ω σ ( X * X *) = σ ( X X ). Note that, under homoskedastcty,.e., Ω =I, GLS becomes OLS. 8

19 The problem s, as usual, that we don t know σ Ω or Σ. Thus we have to ether assume Σ or estmate Σ emprcally. An example of the former s Weghted Least Squares Estmaton and an example of the later s Feasble GLS (FGLS). Weghted Least Squares Estmaton (WLS) Consder a general case of heteroskedastcty. Var(u ) = σ = σ ω. Then, ω ω ω = = Ω Ω = ω E( uu ) σ σ, thus. ω n ω n Because of Ω = P P, P s a n x n matrx whose -th dagonal element s / ω. By pre-multplyng P on y and X, we get y / y / y = Py= yn / ω ω ω n / / = PX = / k * and X *. ω ω ω n x x x n / / / ω... x n k ω... x ω... x nk / ω / ω / ω n The OLS on y * and X * s called the Weghted Least Squares (WLS) because each varable s weghted by ω. The queston s: where can we fnd ω? Feasble GLS (FGLS) Instead of assumng the structure of heteroskedastcty, we may estmate the structure of heteroskedastcty from OLS. Ths method s called Feasble GLS (FGLS). Frst, we estmate Ωˆ from OLS, and, second, we use Ω ˆ nstead of Ω. ˆ β FGLS = ( X Ωˆ X ) X Ω ˆ y There are many ways to estmate FGLS. But one flexble approach (dscussed n Wooldrdge page 77) s to assume that var( u X ) = u = σ exp( δ + δ x + δ x δ k x k ) 9

20 By takng log of the both sdes and usng û nstead of log( uˆ ) = α + δx+ δ x δ k xk + The predcted value from ths model s exponental nto ˆ ω = exp( ˆ / ˆ or / uˆ ω. g u, we can estmate e. ˆ = log( uˆ ). We then convert t by takng the g ) = exp(log( uˆ )) = uˆ. We now use WLS wth weghts Example 9-. * Estmate the log-wage model by usng WAGE.dta wth WLS. * Weght s educ. * Generate weghted varables. gen w=/(educ)^.5. gen wlogwage=logwage*w. gen wfemale=female*w. gen weduc=educ*w. gen wexper=exper*w. gen wexpsq=expsq*w. * Estmate weghted least squares (WLS) model. reg wlogwage weduc wfemale wexper wexpsq w, noc Source SS df MS Number of obs = F( 5, 59) = 66.6 Model Prob > F =. Resdual R-squared = Adj R-squared =.946 Total Root MSE =.75 wlogwage Coef. Std. Err. t P> t [95% Conf. Interval] weduc wfemale wexper wexpsq w End of Example 9- Example 9-. * Estmate reg. reg logwage educ female exper expsq (Output omtted). predct e, resdual. gen logesq=ln(e*e)

21 . reg logesq educ female exper expsq (output omtted). predct esqhat (opton xb assumed; ftted values). gen omega=exp(esqhat). * Generate weghted varables. gen w=/(omega)^.5. gen wlogwage=logwage*w. gen wfemale=female*w. gen weduc=educ*w. gen wexper=exper*w. gen wexpsq=expsq*w. * Estmate Feasble GLS (FGLS) model. reg wlogwage weduc wfemale wexper wexpsq w, noc Source SS df MS Number of obs = F( 5, 59) = Model Prob > F =. Resdual R-squared = Adj R-squared =.9374 Total Root MSE =.997 wlogwage Coef. Std. Err. t P> t [95% Conf. Interval] weduc wfemale wexper wexpsq w End of Example 9-

22 Lecture : Unobserved Effects and Panel Analyss Panel Data There are two types of panel data sets: a pooled cross secton data set and a longtudnal data set. A pooled cross secton data set s a set of cross-sectonal data across tme from the same populaton but ndependently sampled observatons each tme. A longtudnal data set follows the same ndvduals, households, frms, ctes, regons, or countres over tme. Many governments conduct natonwde cross sectonal surveys every year or every once n a whle. Census s an example of such surveys. We can create a pooled cross secton data set f we combne these cross sectonal surveys over tme. Because these surveys are often readly avalable from governments (well ths s not true for most of the tmes because of many government offcals do not see any benefts of makng publcly funded surveys publc!), t s relatvely easer to obtan pooled cross secton data. Longtudnal data, however, can provde much more detaled nformaton as we see n ths lecture. Because longtudnal data follow the same samples over tme, we can analyze behavoral changes over tme of the samples. Nonetheless, pooled cross secton data can provde nformaton that a sngle cross secton data cannot. Pooled Cross Secton Data Wth pooled cross secton data, we can examne changes n coeffcents over tme. For nstance, y t = $ + $ T t + $ x t + $ 3 x t + $ 4 x t3 + u t () t =, =,,, N, N +, N +,, N +N where T t = f t = and d t = f t =. The coeffcent of the tme dummy T t measures a change n the constant term over tme. If we are nterested n a change n a potental effect of one of the varables, then we can use an nteracton term between the tme dummy and one of the varables: y t = $ + $ d t + $ x t + * (T t x t ) + $ 3 x t + $ 4 x t3 + u t () * measures a change n the coeffcent of x over tme. How about f there are changes n all of the coeffcents over tme? To examne f there s a structural change, we can use the Chow test. To conduct the Chow test, consder the followng model:

23 y t = $ + $ x t + $ x t + $ 3 x t3 + $ 4 x t4 + u t (3) for t =,. We consder ths model as a restrcted model because we are mposng restrctons that all the coeffcents reman the same over tme. There are k+ restrctons (n ths case 5 restrctons). Unrestrcted models are y t = * + * x t + * x t + * 3 x t3 + * 4 x t4 + v t for t = (4) and y t = ( + ( x t + ( x t + ( 3 x t3 + ( 4 x t4 + e t for t = (5) The coeffcents of the frst model (t = ) are not restrcted to be the same as n the second model (t = ). If all of the coeffcents reman the same over tme,.e., $ j = * j = ( j, then the sum of squared resduals from the restrcted model (SSR r ) should be equal to the sum of the sums of squared resduals from the two unrestrcted models (SSR ur + SSR ur ). On the other hand, f there s a structural change,.e., changes n the coeffcents over tme, then the sum of SSR ur and SSR ur should be smaller than SSR r, because unrestrcted coeffcents n unrestrcted models should match the data more precsely than the restrcted model. Then we take the dfference between SSR r and (SSR ur + SSR ur ) and examne f there s statstcally sgnfcant dfference between the two: F = [ SSRr ( SSRur+ SSRur )]/ ( k+ ) ( SSR + SSR ) /[ N + N ( k+ )] ur ur Ths s called the Chow test. Alternatvely, we can create nteracton terms on all of ndependent varables (ncludng the constant term) and conduct a F-test on the coeffcents of the k+ nteracton terms. Ths s just the same the Chow test. Dfference-n-Dfferences Estmator In many economc analyses, we are nterested n some of polcy varables or poltcally nterestng varables and how these varables affect people s lves. However, evaluatng 3

24 the mpacts of varous polces s dffcult because most of polces are not done under expermental desgns. For nstance, suppose a government of a low-ncome country decded to nvest n health facltes to mprove chld health. Suppose ths partcular government decded to start wth the most-needy communtes. After some years, the government wanted to evaluate the mpacts of the nvestment n health facltes. The government conducts a crosssectonal survey. However, the government fnds negatve correlaton between newlybuld health facltes and chld health. What happened? Chld health Communtes wthout E(H T : z = ) E(H T : z = ) * Communtes wth nvestments Tme The problem s that the government bult health facltes n communtes wth poor chld health. In the fgure above, the chld health n poor communty wth the governmentnvestments (z) has mproved over tme, but ts absolute level s stll not as good as the chld health n rch communtes wthout the government nvestments. Thus, an OLS model wth a dummy varable for the government nvestments n health facltes wll fnd a coeffcent of z: H t = $ + $ z t + u t (6) for =,, N communtes. When we fnd a negatve coeffcent or an opposte effect of what expected, we call t the reverse causalty. From the fgure, t s obvous that we need to measure a dfference between the two groups for each tme perod and measure a net change n the dfferences over tme: * = [E(H :z =) E(H :z =)] [E(H : z =) E(H : z =)]. (7) 4

25 Although both dfferences are negatve, the dfference between the two groups n the second perod s much smaller than the dfference n the frst perod. Thus, the net change s postve, whch measures the net mpact of z on H. We call the * n (-7) the dfference-n-dfferences (DID) estmator. The dfference-n-dfferences estmator can be estmated by estmatng the followng model: H t = $ + $ T + $ z + * (T z ) + u t (8) We can thnk ths example as a knd of the omtted varables problem. We can rewrte (9-6) as H = $ + $ z + " + u (9) where " s an mportant unobserved varable (or an unobserved fxed effect) whch s correlated wth both the government nvestments and the chld health. Let s say that " measures the lack of basc nfrastructure n communty : the larger the ", the poorer the basc nfrastructure. Because the government targets the poor communtes for the nvestments, " and z are correlated postvely. But " and H are correlated negatvely because " measures the lack of basc nfrastructure. Therefore, the estmated coeffcent of z wll be based downward, whch produces a reverse causalty. The Frst Dfferenced Estmaton Let s go back to the DID estmator and rearrange t so that the frst term measure a dfference n H t of communty over tme: * = [E(H T :z =) E(H T :z =)] [E(H C : z =) E(H C : z =)]. () Here the frst term measures a change over tme for the treatment group (T) and the second term measures for the comparson group (C). In a regresson form, we can also rearrange (8). Let s wrte the equaton (8) wth an unobserved fxed effect: H t = $ + $ T + $ z + * (T z ) + " + u t () Now, the problem s that z could be correlated wth ", whch may be also correlated wth H t. For the frst perod (thus T = ), the equaton () s H t = $ + $ z + " + u t, and for the second perod (T = ): 5

26 H t+ = ($ + $ ) + ($ + *) z + " + u t+. Then by takng the frst-dfference, we have H t+ - H t = $ + * z + v t+ () Notce that the unobserved fxed effect, ", has been excluded from ths model because the unobserved fxed effect s fxed over tme. In the frst-dfferenced equaton (), z wll not be correlated wth the error term. Quas-expermental and Expermental Desgns From ths pont of vew, t s obvous that under a nonrandom assgnment of z (or a quasexpermental desgn), * n (8) could be based because z (a program ndcator) could be correlated wth unobserved factors whch may be also correlated wth H (a dependent varable). In contrast, under a random assgnment of z (or an expermental desgn), z wll not be correlated wth any of unobserved factors. Thus the dfference-n-dfferences estmator wll provde relable estmators of the mpacts of programs on outcomes. Under an expermental desgn, a group of ndvduals or observatons that receve benefts from a gve polcy s called a treatment group or an expermental group. And a group of non-benefcares s called a control group. Under a quas-expermental desgn, a group of non-benefcares s called a comparson group, and reserve the name the control group for expermental desgns. In socal scence, t s dffcult to conduct expermental desgned programs because of ethcs and poltcal dffcultes. But expermental desgned programs can provde very useful nformaton about the effectveness of publc (or prvate) polces. Recent Example of an expermental desgned project: PROGRESA n Mexco, desgned and research by IFPRI. See For a summary of US experence, see Grossman (994) Evaluatng socal polces: prncples and U.S. experence, World Bank Research Observer, vol9: Thus, we have dealt wth an omtted varable problem by takng a dfference over tme. Next, we study the omtted fxed effect problem n general. The Omtted Varables Problem Revsted Suppose that a correctly specfed regresson model would be 6

27 y Xβ + u = X β + X + u = β X and X have k and k columns, respectvely. But, suppose we regress y on X wthout ncludng X (X represent omtted varables). The OLS estmator s ˆ β = ( X X ) X Y = ( X X ) X ( X β+ X β + u) = β + ( X ) X X β + ( X X X u X ) By takng the expectaton on both sdes, we have E ˆ β = + = + ( ) β ( X X ) X X β β δβ Note, however, that the second term ndcates the column of slopes ( ˆ δ ) n least squares regresson of the correspondng column of X on the columns of X. Thus, unless ether ˆ δ = orβ =, ˆβ s based. To overcome the omtted varables problem, we can take two dfferent methods. Frst method s to use panel data. As you see later, by usng panel (longtudnal) data, we can elmnate unobserved varables that are specfc to each sample and fxed (or tmenvarant or tme-constant) over tme. Second method s to use nstrumental varables that are correlated wth ndependent varables that are consdered to be correlated wth unobserved varables but uncorrelated wth the dependent varable. We wll dscuss nstrumental varables n the next lecture note. Before we dscuss about estmaton methods that use panel data, let us start wth types of panel data. ˆ Lnear Unobserved Effects What are unobserved varables? It s mpossble to collect all varables n surveys that affect people s economc actvtes. Thus, t s nevtable to have unobserved varables n our estmaton models. What, then, we should do? Frst, we should start wth characterzng possble unobserved varables. The most common type of unobserved varables s a fxed effect. A fxed effect s a tme nvarant characterstc of an ndvdual or a group (or cluster). For nstance, a may represent a fxed characterstc of group. Ths could be a regonal fxed effect or a cluster fxed effect. Another example s a j whch represents a fxed characterstc of a group (cluster) j. Suppose, we want to estmate the followng model wth a group fxed effect, 7

28 y = $ + $ x + + $ k x k + a j + u (3) In ths case, as long as unobserved varables (that are correlated wth ndvdual varables and the dependent varable) are fxed characterstcs of groups, then we can elmnate the omtted varables problem by explctly ncludng group dummes: y = $ + $ x + + $ k x k + "' d j + u (4) where "' s a -by-j vector, and d j s a j-by- vector. For nstance, t s common practce to nclude dstrct or vllage dummes n cross-sectonal data. However, n a crosssectonal data set, t s mpossble to nclude ndvdual dummes for all samples because we have only one observaton per sample. We wll have n- dummes for n observatons. If we have multple observatons for each sample (thus we need longtudnal data not a pooled cross-sectonal data over tme), then t s possble to have n- dummes for s x n observatons. (s s the number of observatons per sample.) Thus, we estmate y t = $ + $ x t + + $ k x t k + "' d + u t (5) Ths s called the Dummy Varable Regresson model. In ths model, we have elmnated the unobserved fxed effects by explctly ncludng ndvdual dummy varables. A dfferent way of elmnatng the fxed effects s to use the frst dfference model, as we have seen earler. Here let us reconsder the frst dfference model n a general treatment. Suppose, agan, that we have the followng model for tme t= and t=: and y = $ + $ x + + $ k x k + a j + u y = $ + $ x + + $ k x k + a j + u By subtractng the model for t= from the model for t=, we have or y - y = $ (x - x ) + + $ k (x k - x k ) + u - u y = β x β x k k + u Ths s called the frst dfference model. Notce that the ndvdual fxed effect, a j, has been elmnated. Thus as long as the new error term s uncorrelated wth the new ndependent varables, then the estmators should be unbased. Some notes: Frst, a frst dfferenced ndependent varable, x k, must have some varaton across. For nstance, a gender dummy varable does not change over tme, the frst-dfferenced gender dummy s zero for all. Thus, you can not estmate coeffcent on tme-nvarant ndependent varables n frst dfference models. Second, dfferenced ndependent varables loose varaton. Thus, estmators often have large standard errors. Large sample sze helps to estmate parameters precsely. 8

29 Fxed Effect Estmaton In the prevous lecture, we studed the frst dfferenced model, concernng the correlaton between a polcy varable and an unobserved fxed effect. In ths lecture, we generalze the model. Consder the followng model wth T perod and k varables y t = $ + $ x t + $ x t + + $ k x t k + " + u t (6) The omtted unobserved fxed effect could be correlated wth any of k ndependent varables. To take the fxed effect away, one can subtract the mean of each varable: y y = β + + ( x x ) + v (7) t ( xt x )... β tk k t As you can see, the unobserved fxed effect has been excluded from the model. Ths model s called the fxed effect estmaton. To estmate the fxed effect model, you need to transform each varable by takng the mean out and estmate the OLS wth the transformed data (the tme-demeaned data). In STATA, you don t need to transfer the data yourself. Instead you just need to use a command xtreg y x x x k, fe (d). See the manuals under xtreg. One drawback of the fxed effect estmaton s that some of tme-nvarant varables wll be also excluded from the model. For nstance, consder a typcal wage model, where the dependent varable s log(wage). Some of ndvdual characterstcs, such as educaton and gender are tme-nvarant (or fxed over tme). Thus f you are nterested n the effects of tme-nvarant varables you cannot estmate the coeffcents of such varables. However, what you can do s to estmate the changes n the effects of such tme-nvarant varables. Instead of takng the fxed effects out, we can explctly nclude them as dummes: y t = $ + $ x t + $ x t + + $ k x t k + * " + +* n " n + u t (8) Ths s called the least squares dummy varable (LSDV) model. The LSDV model provdes the exactly the same results as Fxed Effect model. When T=, the frst dfferenced (FD) model, the Fxed Effect (FE) model, and LSDV model all provde the same results. 9

30 Example -: OLS, Fxed Effect, Frst-Dfferenced, and LSDV models. use c:\docs\fasd\econometrcs\homework\jtrain.dta;. keep f year==988 year==989; (57 observatons deleted). replace sales=sales/; (54 real changes made). ** OLS;. reg hrsemp grant employ sales unon d89; Source SS df MS Number of obs = F( 5, 4) =.7 Model Prob > F =.73 Resdual.39e R-squared = Adj R-squared =.39 Total.3934e Root MSE = 49. hrsemp Coef. Std. Err. t P> t [95% Conf. Interval] grant employ sales unon d _cons ** Fxed Effect Model;. xtreg hrsemp grant employ sales unon d89, fe (fcode); Fxed-effects (wthn) regresson Number of obs = Group varable () : fcode Number of groups = 4 R-sq: wthn =.3 Obs per group: mn = between =.6 avg =.9 overall =. max = F(4,) =.85 corr(u_, Xb) = -.44 Prob > F =.497 hrsemp Coef. Std. Err. t P> t [95% Conf. Interval] grant employ sales unon (dropped) d _cons sgma_u sgma_e rho.3349 (fracton of varance due to u_) F test that all u_=: F(3,) =.87 Prob > F =.776 3

31 . ** LSDV model;. x: reg hrsemp grant employ sales unon d89.fcode;.fcode Ifcod-57 (Ifcod for fcode==43 omtted) Source SS df MS Number of obs = F(7, ) =.9 Model Prob > F =.676 Resdual R-squared = Adj R-squared = Total.3934e Root MSE = hrsemp Coef. Std. Err. t P> t [95% Conf. Interval] grant employ sales unon d Ifcod Ifcod Output omtted. ** Frst Dfferenced Model;. reg dhrsemp dgrant demploy dsales; Source SS df MS Number of obs = F( 3, ) =.8 Model Prob > F =.498 Resdual.356e R-squared = Adj R-squared = -.55 Total.3884e Root MSE = dhrsemp Coef. Std. Err. t P> t [95% Conf. Interval] dgrant demploy dsales _cons End of Example - Example -: Transferrng The Panel Data In general, the panel data are stacked vertcally. For nstance, n JTRAIN.dta, two observatons (actually there are three years of observatons, but I dropped one year) for each frm s stacked vertcally:. lst fcode year d89 employ frm code year d89 # of employees

32 To construct dfferenced varables, you need to lnearze the vertcal data. Here s an example:. do "C:\WINDOWS\TEMP\STDc.tmp". #delmt; delmter now ;. clear;. set more off;. set matsze 8;. set memory m; (4k). *** Obtan data from 988;. clear;. use c:\docs\fasd\econometrcs\homework\jtrain.dta;. keep year fcode employ;. keep f year==988; (34 observatons deleted). rename employ employ88;. drop year;. sort fcode;. save c:\docs\tmp\jtran88.dta, replace; fle c:\docs\tmp\jtran88.dta saved. *** Obtan data from 989;. clear;. use c:\docs\fasd\econometrcs\homework\jtrain.dta;. keep fcode year employ;. keep f year==989; (34 observatons deleted). rename employ employ89;. drop year;. sort fcode;. save c:\docs\tmp\jtran89.dta, replace; fle c:\docs\tmp\jtran89.dta saved. ** Combne the two years;. clear;. use c:\docs\tmp\jtran89.dta;. sort fcode;. merge fcode usng c:\docs\tmp\jtran88.dta;. gen demploy=employ89-employ88; ( mssng values generated). lst fcode employ89 employ88 demploy n /3; fcode employ89 employ88 demploy End of Example - 3

33 Lecture : Random Effect Models and the Hausman Test Random Effect Estmaton Let s go back to a general longtudnal model wth T perods: y t = $ + $ x t + $ x t + + $ k x t k + " + u t () The purpose of the fxed effect estmaton and the frst dfferenced estmaton s to elmnate " because we suspect " to be correlated wth some of ndependent varables. However, what f we are wrong (that there s no correlaton between " I and any of ndependent varables)? If ths s case, then the FE and FD estmatons wll be neffcent because we lose n- degree of freedom (thnk about havng n- dummes). Even when there s no correlaton between " and ndependent varables, the OLS estmaton on (-4) wll have a problem because of heteroskedastcty. In (-4) the new error term s v t = " + u t () Let s make some assumptons on " and u t : E(" ) =, E(u ) = Var(" ) = F ", Var(u ) = F u E(" " j ) = E(u u j ) = Then for the same observaton unt, Var(v t ) = F "+ F u and cov(v t v s ) = F " So for the same observaton unt for T perod, the varance-covarance matrx s Σ T T σ + σ α σ u u σ u σα + σ u = E( vv ) = σ u σ u σ u σ u σ + σ α u Because there are N observaton unts for T perods, the varance-covarance matrx for all observaton NT: 33