Outline. 9. Heteroskedasticity Cross Sectional Analysis. Homoskedastic Case

Transcription

1 Outlne 9. Heteroskedastcty Cross Sectonal Analyss Read Wooldrdge (013), Chapter 8 I. Consequences of Heteroskedastcty II. Testng for Heteroskedastcty III. Heteroskedastcty Robust Inference IV. Weghted Least Square Estmaton I. Consequences of Heteroskedastcty Homoskedastc Case Motvaton: Consder the model sav = 0 nc + u y = sav =savng x = nc =ncome Constant varances (MLR. 5) Var(u nc ) =, whch mples that Var(sav nc ) = f(y x). y. E(y x) = 0 x =5,000 x =100,000 I. Consequences of Heteroskedastcty 3 I. Consequences of Heteroskedastcty 4

2 Heteroskedastcty Example of Heteroskedastcty Volaton of homoskedastcty: What f the varablty of savngs of the rch s less than that of the lower ncome group? Here we say that the varance of savngs y (or unobserved factors u) ncreases wth ncome VAR(sav nc = 5,000) = 5 (see ) VAR(sav nc = 100,000) = 100 (see x ) f(y x).. E(y x) = 0 x When varances are unequal, ths problem s called heteroskedastcty.. (See Graph) =5,000 x =100,000 x I. Consequences of Heteroskedastcty 5 I. Consequences of Heteroskedastcty 6 MLR.5 volated: Heteroskedastcty Propertes nvald under heteroskedastcty: unequal varances Consder a model y = 0 + x + + k x k + u Heteroskedastcty VAR(u,,x k ) = Propertes unaffected by heteroskedastcty: 1) OLS estmators are stll unbased and consstent. ) The nterpretaton s the same for goodness of ft measures, R and R bar. 1) The estmators of the varances, Var( ), are based. ) t, F and LM statstcs no longer have t, F and LM dstrbutons. 3) OLS s no longer best lnear unbased estmator (BLUE). I. Consequences of Heteroskedastcty 7 I. Consequences of Heteroskedastcty 8

3 II. Testng for Heteroskedastcty Breusch Pagan Test There are many tests for heteroskedastcy, but we wll learn two modern tests: 1) Breusch Pagan Test for Heteroskedastcty ) Whte Test use no cross terms. use cross terms. use ftted values of the LHS varable Gven MLR.1 MLR.4, consder the Model y = 0 + x u Want to test H 0 : whether MLR.5 s true H 0 : VAR(u,, ) =E(u ) = u = 0 + x + + k x k + v These modern tests assume that the varance of the error depends or does not depend upon the explanatory varables. To test whether u s related to x s H 0 : 1 = = = k = 0 II. Testng for Heteroskedastcty 9 II. Testng for Heteroskedastcty 10 Use resduals for u Snce we don t observe u, but we have estmates of resduals. = 0 + x + + k x k + v Use F test or LM test to test the overall sgnfcance H 0 : 1 = = = k = 0 R F (1 R uˆ ) uˆ / k / n k LM = n* ~ k F 1 k,( n k 1) Example: Consder cgarette demand functon ncome: annual ncome n dollars cgprc: state cgarette prce, cents per pack educ: years of schoolng age: age measured n years restaurn =1 f a state has restaurant smokng restrctons =0 f a state has no restaurant smokng restrctons cgs = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u BP Test for heteroskedastcty Step 1: Estmate the above equaton. Step : obtan resduals from the cgs equaton or In Evews, obtan resdual seres resd01 Step 3: Regress on all x s. resd01^ = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u Step 4: Use F and LM Tests for heteroskedastcty. Compute F and LM statstcs and compare to crtcal values of the F k,n-k-1 and k dstrbutons II. Testng for Heteroskedastcty 11 II. Testng for Heteroskedastcty 1

4 LM verson of the Bruesch Pagan Test Step 4: Use F test and LM test 1) F statstc = 5.55 p value = ( ) ) LM statstc = Obs*R squared = 807* = 3.6 Ch square dstrbuton wth 6 DFs c = 1.59 (5% sgnfcance level) c = 16.81(1% sgnfcance level) What can we say about heteroskedastcty? II. Testng for Heteroskedastcty 13 Evews: Step 1: Estmate the cgarette demand equaton Dependent Varable: CIGS Sample: Included observatons: 807 C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat.0185 Prob(F-statstc) 0 Proc/Make Resdual Seres. Step : name for resdual seres: resd01 II. Testng for Heteroskedastcty 14 Step 3: Regress resd01^ ( ) on all x s Whte Test of Heteroskedastcty Dependent Varable: RESID01^ Sample: C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd 1.06E+08 Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc) Consder the three varable model y = 0 + x u H 0 : Var(u, x, ) = Weaker assumpton by Whte (1980) u s uncorrelated wth (, x, ), (, x, ), ( x,, x ) = 0 + x x x x + v H 0 : 1 = = = 9 = 0 Use F and LM Test What are the rejecton rules? II. Testng for Heteroskedastcty 15 II. Testng for Heteroskedastcty 16

5 Intractable: more regressors Easer way to mplement Whte Test Consder the model wth 6 regressors cgs = 0 log(ncome) + (cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u = regressors + v LOG(INCOME), (LOG(INCOME))^, (LOG(INCOME))*(LOG(CIGPRIC)), (LOG(INCOME))*EDUC, (LOG(INCOME))*AGE, (LOG(INCOME))*(AGE^), (LOG(INCOME))*RESTAURN, LOG(CIGPRIC), (LOG(CIGPRIC))^, (LOG(CIGPRIC))*EDUC, (LOG(CIGPRIC))*AGE, (LOG(CIGPRIC))*(AGE^), (LOG(CIGPRIC))*RESTAURN EDUC, EDUC^, EDUC*AGE, EDUC*(AGE^), EDUC*RESTAURN AGE, AGE^, AGE*(AGE^), AGE*RESTAURN, (AGE^)^, (AGE^)*RESTAURN RESTAURN H 0 : 1 = = = 5 = 0 k = 5 n k 1 = n 6 What are the rejecton rules? Idea : use OLS ftted values n a test for heteroskedastcty = + + x + + x k When we square, we get a partcular functon of all the squares and cross products Smpler form of Whte Test = v What are the null hypothess and rejecton rules? II. Testng for Heteroskedastcty 17 II. Testng for Heteroskedastcty 18 Model: cgs = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u Evews: Step 1: Estmate the cgarette demand equaton A Specal case of the Whte Test for heteroskedastcty Step 1: Estmate the above equaton. Step : obtan ftted value from the cgarette equaton. In Evews, obtan resdual seres resd01 (or ) Note that y = + or cgs = + Generatng seres for called cgshat cgshat = cgs resd01 Step 3: Regress on and resd01^ = 0 cgshat + cgshat + v Step 4: Use F and LM Tests for heteroskedastcty. Compute F and LM statstcs and compare to crtcal values of the F,n-3 and dstrbutons Dependent Varable: CIGS Sample: Included observatons: 807 C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat.0185 Prob(F-statstc) 0 Step : Generatng seres for cgs ^ called cgshat II. Testng for Heteroskedastcty 19 II. Testng for Heteroskedastcty 0

6 Step 3: Regress resd01^ on and ^ Step 4. LM and F test for Whte Test Dependent Varable: RESID01^ Method: Least Squares Sample: Included observatons: 807 C CIGSHAT CIGSHAT^ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd 1.06E+08 Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc) resd01^ = 0 cgshat + cgshat + v H 0 : 1 = = 0 F test F = p value = Could you fnd crtcal values to verfy ths? LM test LM = obs*r squared = 807*.0398 = 6.57 Ch square dstrbuton wth DFs c = 5.99 (5% sgnfcance level) c = 9.1 (1% sgnfcance level) II. Testng for Heteroskedastcty 1 II. Testng for Heteroskedastcty Model: cgs = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u We could follow smlar steps to test for heteroskedastcty usng Whte Tests. 3) Whte Test wth no cross terms = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + 7 [log(ncome)] + 8 [log(cgprc)] + 9 educ 0 [age ] + v 4) Whte Test wth cross terms. = regressors + v Evews: It has commands to fnd F and LM statstcs usng Whte Tests (methods 3 4). In the equaton output wndow, (3) Choose Vew/Resdual Tests/Whte heteroskedastcty (wth no cross terms) (4) Choose Vew/Resdual Tests/Whte heteroskedastct (wth cross terms) for (4) In the Equaton wndow, run the followng regresson. Dependent Varable: CIGS Sample: Included observatons: 807 C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat.0185 Prob(F-statstc) 0 II. Testng for Heteroskedastcty 3 II. Testng for Heteroskedastcty 4

7 Vew/Resdual Tests/Whte heteroskedastcty (no cross terms) for (3) Whte Heteroskedastcty Test: F-statstc Probablty Obs*R-squared Probablty Dependent Varable: RESID^ Included observatons: 807 C LOG(INCOME) (LOG(INCOME))^ LOG(CIGPRIC) (LOG(CIGPRIC))^ EDUC EDUC^ AGE AGE^ (AGE^)^ 1.9E-05 1.E RESTAURN R-squared Mean dependent var Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc) II. Testng for Heteroskedastcty Vew/Resdual Tests/Whte heteroskedastcty (wth cross terms) for (4) Whte Heteroskedastcty Test: F-statstc Probablty Obs*R-squared Probablty Dependent Varable: RESID^ Included observatons: 807 Varable (5 regressors) Coeffcent Std. Error t-statstc Prob. C LOG(INCOME) (LOG(INCOME))^ (LOG(INCOME))*(LOG(CIGPRIC)) RESTAURN R-squared Mean dependent var Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc) II. Testng for Heteroskedastcty III. Heteroskedastcty Robust Inference after OLS Estmaton In the presence of heteroskedastcty, Evews can adjust standard errors, t, F, and LM, statstcs so that they are vald. Ths method s called the heteroskedastc robust procedure. Techncally, ths procedure s vald, at least n large samples, whether or not the errors have constant varances. Sketch the procedure Consder the smple regresson model y = 0 x + u Var(u x ) = Steps to fnd robust standard errors: 1) Fnd estmator of 1 ) Under the assumptons MLR.1 MLR.4, the varance can be found. 3) Whte(1980) suggests usng n place of. Thus, we can fnd the estmator of VAR( ) 4) A heteroskedastc robust standard error can be found. Sketch for the general case? III. Heteroskedastcty Robust Inference after OLS Estmaton 7 III. Heteroskedastcty Robust Inference after OLS Estmaton 8

8 Varance wth Heteroskedastcty Varance wth Heteroskedastcty For the smple case, ˆ x x u, so x x 1 1 x x 1 SSTx Var ˆ, where SST x x x For the general multple regresson model, a vald j j estmator of Var ˆ wth heteroskedastcty s ˆˆ ˆ ru Varˆ j, SSR th j A vald estmator for ths when s x x ˆ x u, SST where uˆ are are the OLS resduals where rˆ s the resdual from regressng x on j all other ndependent varables, and SSR j s the sum of squared resduals from ths regresson j III. Heteroskedastcty Robust Inference after OLS Estmaton 9 III. Heteroskedastcty Robust Inference after OLS Estmaton 30 Example: cgs equaton Steps n Evews: Consder the model cgs = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u Fnd the cgs equaton usng heteroskedastc robust procedure. III. Heteroskedastcty Robust Inference after OLS Estmaton 31 Step 1: Estmate the log equaton n usual OLS method. cgs = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u Step : Fnd the log equaton wth heteroskedastcty-robust standard errors. In the Equaton wndow, Choose Estmate. In the Equaton Estmaton box, clck opton button. Then, clck heteroskedastcty consstent coeffcent covarance. clck Whte III. Heteroskedastcty Robust Inference after OLS Estmaton 3

9 Step 1: Estmate the cgs equaton n usual way Step : Whte Heteroskedastcty-Consstent s.e. s Dependent Varable: CIGS Included observatons: 807 C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat.0185 Prob(F-statstc) 0 Dependent Varable: CIGS Included observatons: 807 Whte Heteroskedastcty-Consstent Standard Errors & Covarance C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat.0185 Prob(F-statstc) 0 III. Heteroskedastcty Robust Inference after OLS Estmaton 33 III. Heteroskedastcty Robust Inference after OLS Estmaton 34 OLS and Robust estmates: compared Dependent Varable: CIGS Interpretaton: cgs demand equaton cgs = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u Varable Coeffcent s.e s.e (robust) Prob. Prob. (robust) C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ ) In ths applcaton, any varable that s statstcally sgnfcant at 1% level usng the usual t test s stll sgnfcant under the heteroskedastcty robust procedure. varables: educ, age, age and restaurn. ) The robust standard errors are ether larger or smaller than the usual standard error 3) The robust s.e. on log(ncome) becomes smaller, but that on log(cgprce) s larger. 4) How to nterpret the coeffcents of varous varables n the model? RESTAURN III. Heteroskedastcty Robust Inference after OLS Estmaton 35 III. Heteroskedastcty Robust Inference after OLS Estmaton 36

10 Robust F test and Wald Test cgs = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u Want to test the null hypothess: H 0 : 1 = =0 In the equaton wth robust standard errors, we can easly obtan heteroskedastc robust F statstc also called heteroskedastc robust Wald statstc. Step 1. Whte Heteroskedastcty Consstent s.e. s Step. Vew/Coeffcent Tests/Wald Coeffcent Restrctons Step 3. Type n c()=0, c(3)=0 The F statstc s F = ; p value = (ncorrect) robust F statstc = 1.099; p value=.3338 (correct) Snce p value >, we do not reject H 0 and conclude that ncome and prce together do not have an effect on cgarette demand. Step 1: Whte Heteroskedastcty-Consstent s.e. s Dependent Varable: CIGS Included observatons: 807 Whte Heteroskedastcty-Consstent Standard Errors & Covarance C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat.0185 Prob(F-statstc) 0 III. Heteroskedastcty Robust Inference after OLS Estmaton 37 III. Heteroskedastcty Robust Inference after OLS Estmaton 38 Vew/Coeffcent Tests/Redundant Varables Redundant Varables: LOG(INCOME) LOG(CIGPRIC) F-statstc Probablty Log lkelhood rato Probablty Assume that MLR.1 MLR.4 hold, but MLR.5 does not. Vew/Coeffcent Tests/Wald Coeffcent Restrctons Wald Test: Let xdenote, x,., x k VAR(u x) = (heteroskedastcty) Equaton: Unttled Test Statstc Value df Probablty F-statstc (, 800) Ch-square Let = h(x) VAR(u x) = h(x) h(x) > 0 (snce VAR > 0) Null Hypothess Summary: Normalzed Restrcton (= 0) Value Std. Err. C() Heteroskedastty can be corrected under two cases: 1) = h(x). h(x) s known up to a multplcatve constant ) h(x) has to be estmated feasble GLS C(3) Restrctons are lnear n coeffcents. III. Heteroskedastcty Robust Inference after OLS Estmaton 39 40

11 Case 1: h(x) s known up to a multplcatve constant Weghted Least Squares Consder the model Let y = sav ; x = nc ; h = x y = 0 x + u var(u x ) = h Trck : Dvde the equaton by sqr(h ) y 1 x u 0 1 h h h h (OLS) (heteroskedastc) (IV.1) Show that the errors n (IV.1) are homoskedastc! Weghted least squares obtan the values of j* that makes the weghted SSR as small as possble: n u h 1 1 ( y x... x ) / h * * * k k where each squared resdual s weghted by 1/h. Brng 1/h nsde the squared resdual: GLS s an effcent procedure. n u y 1 x x n n * * 1 * k k 1 h 1 h h h h 41 4 Example: Savng equaton OLS & WLS compared OLS : sav = 0 nc + u VAR(u nc ) = = nc s not constant. and are not BLUE. OLS WLS MLR.1-MLR-4 yes yes GLS : transformed equaton sav /(nc) 1/ = 0 /(nc) 1/ nc /(nc) 1/ + u /(nc) 1/ VAR[u /nc ] = s constant. 0* and 1* are BLUE MLR.5 heteroskedastcty homoskedastcty error varance not constant constant BLUE no yes The GLS estmators after correctng the error for heteroskedastcty s called weghted least squares (WLS) estmators. t and F Dst nvald vald R-squared meanngful not meanngful 43 44

12 OLS and WLS Results: savng equaton Step 1: OLS Regress sav on nc (wth ntercept) OLS: WLS: sav = nc prob {.8493} {0.014} n=100, R =.061 R bar=.056 sav/nc^.5 = 15.0[1/nc^.5] nc^.5 prob {.7955} {.003} n=100, R =.05 R bar=.015 Dependent Varable: SAV Method: Least Squares Sample: Included observatons: 100 C INC Evew Trck: WLS Step 1: Obtan Orgnal output Step : In the [Equaton:..] wndow, Choose Estmate and then optons. clck Weghted LS/TSLS Type n the weght, 1/nc^.5 or 1/sqr(nc) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd 1.00E+09 Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc) WLS: Regress sav/(nc^.5) on 1/nc^.5 and nc^.5 (wth no ntercept) Dependent Varable: SAV Included observatons: 100 Step : Compare to the regresson of sav /(nc )1/ = 0 /(nc ) 1/ nc /(nc ) 1/ + u /(nc ) 1/ Weghtng seres: 1/INC^.5 Dependent Varable: SAV/(INC^.5) Method: Least Squares Sample: Included observatons: 100 C INC Weghted Statstcs R-squared Mean dependent var /(INC^.5) INC^ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd 6.93E+08 Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc) Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood Durbn-Watson stat Unweghted Statstcs R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Sum squared resd 1.00E+09 Durbn-Watson stat

13 WLS: In practce, we rarely know h(x). Consder the model: sav = 0 nc + sze + 3 educ + 4 age + 0 black +u Does the varance depend on age or educaton? Average data at the frm level: contrb 0 earn 1 u Indvdual level data vs. Averages of data There s a case where weghts needed arse naturally. Example: the effect of earnngs on the contrbuton to Prvate Provdent Fund. Indvdual data: contrb,e = 0 earn,e + u,e Assume MLR.1 MLR.4 and Var(u,e ) = : denote a partcular frm e: an employee wthn the frm contrb,e : annual contrbuton earn,e : annual earnngs VAR(u bar) = /m (Heteroskedastc) h =1/m weght = m = m 1/ (n the transformed equaton and Evews) Show the error n the transformed equaton s homoskedastc! WLS: to use wth per capta data A smlar weghtng arses when we use data at the cty, provnce, or country level. In summary, WLS gves us an effcent way to treat averages of data Case : h(x) must be estmated Feasble GLS Estmator, Suppose the heteroskedastcty functon s unknown.e., VAR(u,, x k ) = h(x) where h(x) s unknown and must be estmated;.e., fnd. Assume that VAR(u x) = exp ( k x k ) u = exp( k x k )v log(u ) = k x k + e Usng n GLS transformaton yelds an estmator, called an FGLS estmator. FGLS s no longer unbased but consstent n large samples. FGLS s no longer BLUE but asymptotcally more effcent than OLS. Estmate log( ) = k x k + e = ftted value of log( ) The estmates of h are smply = exp( ) 51 5

14 Propertes: OLS, GLS and FGLS Example: FGLS and cgarette equaton cgs = 0 log(ncome) + log(cgprc) + 3 educ + 4 age + 5 age + 6 restaurn + u y = 0 + x x x x 6 + u Unbasedness yes yes OLS known h : FGLS BLUE no yes t and F dst. no exact t and F dstrbutons no longer unbased but consstent no longer BLUE but asymptotcally more effcent approxmately t and F dstrbutons Usng Breusch Pagan Test or Whte Tests, we found that the varances are nonconstant. Steps n runnng FGLS equaton Step 1: Run the regresson of y on, x,..., x 6 and obtan resduals (called, resd01) n Evews. Step : Run the regresson of log( ) or log(resd01^) on,, x k and obtan resduals (called, lresd0 n Evews) Step 3: From step, obtan and and the ftted values of log( ) (called ) = log(resd01^) lresd0 Snce h ^ = exp(g ^), then n Evew h01 = exp(log(resd01^) lresd0) Step 4: Run the FGLS equaton usng 1/h01 as weghts. (1/ 01 are weghts n Evews) Evews: Step 1: Estmate the cgarette demand equaton Dependent Varable: CIGS Sample: Included observatons: 807 C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat.0185 Prob(F-statstc) 0 Proc/Make Resdual Seres. Then, name for resdual seres: resd01 55 Step : Run the regresson of log( ) or log(resd01^) on,, x k Dependent Varable: LOG(RESID01^) Included observatons: 807 C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc) 0 Proc/Make Resdual Seres. Then, name for resdual seres: lresd0 Step 3: Generatng Seres: h01 = exp(log(resd01^)-lresd0) 56

15 Step 4: FGLS equaton wth weghts, 1/sqr(h01) Evew Trck: WLS; In the Equaton wndow, Choose Estmate and then optons. Clck Weghted LS/TSLS. Type n the weght, 1/sqr(h01). Dependent Varable: CIGS Included observatons: 807 Weghtng seres: 1/SQR(H01) C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN Weghted Statstcs R-squared Mean dependent var Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc) 0 Unweghted Statstcs R-squared Mean dependent var Alternatvely, Step 4: Regress cgs/sqr(h01) on 1/sqr(h01), log(ncome)/sqr(h01), wth no ntercept Dependent Varable: CIGS/SQR(H01) Included observatons: 807 1/SQR(H01) LOG(INCOME)/SQR(H01) LOG(CIGPRIC)/SQR(H01) EDUC/SQR(H01) AGE/SQR(H01) AGE^/SQR(H01) RESTAURN/SQR(H01) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood Durbn-Watson stat OLS & FGLS Results Compared Dep. Var = cgs OLS FGLS OLS FGLS Varable Coeffcent Coeffcent Prob. Prob. C LOG(INCOME) LOG(CIGPRIC) EDUC AGE AGE^ RESTAURN Interpretaton: 1. Income effect s now statstcally sgnfcant and larger n magntude.. Prce effect s stll statstcally nsgnfcant. 3. Cgarette smokng s negatvely related to schoolng. Recap of Heteroskedastcy Consequences of Heteroskedastcty Testng for Heteroskedastcty Heteroskedastcty Robust Inference Weghted Least Square Estmaton 4. Age has a dmnshng margnal effect on smokng. Smokng ncreases wth age up untl 4.8 years old and then smokng decreases wth age. 5. Cgarette smokng s negatvely affected by restaurant smokng restrctons