UNIVERSITY OF CAMBRIDGE FACULTY OF ECONOMICS. 1. Consider the textbook example of IV regression

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1 UNIVERSITY OF CAMBRIDGE FACULTY OF ECONOMICS MP E MP E R S M300 E M S E S 2 Cosder the textbook examle of IV regresso l = β 0 + β l Q cgarettes P cgarettes + u wth strumet Z = SalesT ax Here l Q cgarettes s the logarthm of the umber of acks of cgarettes sold er cata state, l P cgarettes s the logarthm of the average real rce er ack of cgarettes state cludg all taxes, SalesT ax s the orto of the tax o cgarettes arsg from the geeral sales tax state a The results of the frst ad the secod stage regressos of the 2SLS method of estmatg β are: l l P cgarettes Q cgarettes = SalesTax = l P cgarettes where the stadard errors of the estmates the frst stage regresso 2SLS are gve below the coeff cet estmates What s the value of ˆβ? 2SLS Suggested soluto to a ˆβ = 08 b Check whether we have a weak strumet roblem the 2SLS regressos from a Suggested soluto to b F-statstc from the frst stage regresso equals the square of t-statstc because there s oly oe strumet Therefore, F = = 3844 > 0 It does ot seem that there s a weak strumet roblem c Gve the 2SLS results from a, what s the value of the J statstc for testg the strumet s exogeety? Suggested soluto to c The value of J statstc s zero because we have a just-detfed case here,

2 d Dscuss why SalesTax s a good or bad strumet for l P cgarettes from the ot of vew of the two requremets for the strumet valdty Suggested aswer to d Aswers here ca vary Studets should dscuss both strumet relevace ad exogeety The relevace s more or less obvous Sales taxes vary from state to state ad assed o to cosumers, so that they rereset a art of the rce The exogeety s somewhat more trcky I ths artcular examle, come s a omtted varable, ad t may well be correlated wth sales taxes volatg exogeety 2 Ths roblem s the emrcal exercse E22 from the Stock ad Watso s textbook We reroduce t here for coveece How does fertlty affect labour suly? That s, how much does a woma s labour suly fall whe she has a addtoal chld? I ths exercse you wll estmate ths effect usg data for marred wome from the 980 US Cesus The data are avalable o the textbook Web ste wwwearsohgheredcom/stock_watso the fle Fertlty ad descrbed the fle Fertlty_Descrto from the age wth the above address, you wll eed to go to the Comao Webste for the 3-rd edto of the textbook From there, you ll go to Studet Resources/Data for Emrcal Exercses The data set cotas formato o marred wome aged 2-35 wth two or more chldre a Regress weeksworked o the dcator varable morekds usg OLS O average, do wome wth more tha two chldre work less tha wome wth two chldre? How much less? Suggested solutos to a As ca be see from Table, the coeff cet s 5387, whch dcates that wome wth more tha 2 chldre work 5387 fewer weeks er year tha wome wth 2 or fewer chldre b Exla why the OLS regresso estmated a s arorate for estmatg the causal effect of fertlty morekds o labour suly weeksworked Suggested soluto to b Both fertlty ad weeks worked are choce varables A woma wth a ostve labor suly regresso error a Estmato method Regressor OLS IV IV Morekds Addtoal regressors cost cost cost, agem, black, hsa, othrace Frst stage F Table : Results usg full dataset 2

3 woma who works more tha average may also be a woma who s less lkely to have a addtoal chld Ths would mly that Morekds s egatvely correlated wth the regresso error, so that the OLS estmator of β Morekds s egatvely based c The data set cotas the varable samesex, whch s equal to f the frst two chldre are of the same sex boy-boy or grl-grl ad equal to 0 otherwse Are coules whose frst two chldre are of the same sex more lkely to have a thrd chld? Is the effect large? Is t statstcally sgfcat? Suggested soluto to c The lear regresso of morekds o samesex a lear robablty model yelds Morekds = samesex so that coules wth samesex= are 66% more lkely to have a addtoal chld that coules wth samesex=0 The effect s hghly sgfcat t-statstc=352 d Exla why samesex s a vald strumet for the strumetal varable regresso of weeksworked o morekds Suggested soluto to d Samesex s radom ad s urelated to ay of the other varables the model cludg the error term the labor suly equato Thus, the strumet s exogeous From c, the frst stage F-statstc s large F=238 so the strumet s relevat Together, these mly that samesex s a vald strumet e Is samesex a weak strumet? Suggested soluto to e No, see the aswer to d f Estmate the regresso of weeksworked o morekds usg samesex as a strumet How large s the fertlty effect o labour suly? Suggested soluto to f See colum 2 of Table The estmated value of β Morekds s 633 Ths s somewhat surrsg because we exected a egatve bas of OLS, ad IV s suosed to correct for ths g Do the results chage whe you clude the varables agem, black, hsa, ad othrace the labour suly regresso treatg these varables as exogeous? Exla why or why ot Suggested soluto to g See colum 3 of Table The results do ot chage a mortat way The reaso s that samesex s urelated to agem, black, hsa, othrace, so that there s o omtted varable bas IV regresso colum 2 3 Cosder a regresso y = β 0 + β x + u 3

4 wth oe edogeous regressor x Let z be a vald strumet for x Assume that y, x, z form a d sequece Deote the ftted value ˆπ 0 + ˆπ z from the frst stage regresso as ˆx x = π 0 + π z + e a Deote the vector, ˆx as V, ad let β = β 0, β Show that ˆβ 2SLS = β + V V V u Suggested soluto to a Sce ˆβ 2SLS s a OLS estmator from the secod stage regresso, we have ˆβ 2SLS = V V V y O the other had, y = β 0 + β x + u = V β + u + β x ˆx Usg ths the above formula, we get ˆβ 2SLS = β + V V V u + β x ˆx It remas to rove that V x ˆx = 0 But ths s true because x ˆx s the resdual from the frst stage regresso, ad V s the vector of regressors from the frst stage regresso Resduals are always orthogoal to regressors, hece, V x ˆx = 0 b Let Ṽ =, π 0 + π z Show that V V = ṼṼ + 0 ˆπ 0 π 0 + ˆπ π z ˆπ 0 π 0 + ˆπ π z ˆπ 0 + ˆπ z 2 π 0 + π z 2 Usg the Slutsky theorem, rove that V V E Ṽ Ṽ Suggested soluto to b Note that 0 V = Ṽ + ˆπ 0 π 0 + ˆπ π z 4

5 Therefore, V V = ṼṼ + Ṽ 0, ˆπ 0 π 0 + ˆπ π z 0 + Ṽ ˆπ 0 π 0 + ˆπ π z 0 + 0, ˆπ ˆπ 0 π 0 + ˆπ π z 0 π 0 + ˆπ π z Oeg u the brackets, we get V V = ṼṼ 0 ˆπ 0 π 0 + ˆπ π z + ˆπ 0 π 0 + ˆπ π z ˆπ 0 + ˆπ z 2 π 0 + π z 2 Now, ˆπ 0 π 0 + ˆπ π z = ˆπ 0 π 0 + ˆπ π sce ˆπ 0 π0, ˆπ π, ad z Ez, we have, by Slutsky s theorem ˆπ 0 π 0 + ˆπ π z 0 3 Further, [ ˆπ 0 + ˆπ z 2 π 0 + π z 2] = ˆπ ˆπ 0ˆπ π 2 0 2π 0 π z 2 z + ˆπ 2 z π 2 z 2 = ˆπ 2 0 π ˆπ0ˆπ 2π 0 π + ˆπ 2 π 2 z 2 By Slutsky s theorem, ˆπ 2 0 π 2 0 0, 2ˆπ 0ˆπ 2π 0 π 0, ad ˆπ 2 π 2 0 Therefore, usg Slutsky oce aga, we coclude that [ ˆπ 0 + ˆπ z 2 π 0 + π z 2] 0 4 z 2 z Fally, by LLN, Ṽ Ṽ E Ṽ Ṽ 5 5

6 Combg 2, 3, 4, ad 5, we obta c Show that V u = Ṽu + V V E Usg the Slutsky theorem, rove that Ṽ Ṽ 0 ˆπ 0 π 0 u + ˆπ π z u d V u N 0, E u 2 ṼṼ Suggested soluto to c Equalty V u = Ṽu 0 + ˆπ 0 π 0 u + ˆπ π z u follows from Next, = ˆπ 0 π 0 ˆπ 0 π 0 u + ˆπ π z u u + ˆπ π z u O the other had, ˆπ 0 π 0 0, ˆπ π 0, whereas, by CLT, u d N 0, V ar u ad z d u N 0, V ar z u By the Slutsky theorem, ˆπ 0 π 0 u + ˆπ π z u 0, ad V u coverges dstrbuto to the same lmt as Ṽu Secfcally, d V u N 0, E u 2 ṼṼ d Usg your results a, b, ad c, show that [ ] d ˆβ2SLS β N 0, E Ṽ Ṽ E u 2 ṼṼ [ E ] Ṽ Ṽ 6

7 Suggested soluto to d From a, From b, From c, ˆβ2SLS β = V V V V E d V u N 0, E Ṽ Ṽ u 2 ṼṼ V u Combg ths usg the Slutsky theorem ad the cotuous mag theorem, we get [ ] d [ ] ˆβ2SLS β N 0, E Ṽ Ṽ E u 2 ṼṼ E Ṽ Ṽ 4 Cosder a geeral IV regresso model Y = β 0 + β X + + β k X k + β k+ W + + β k+r W r + u wth strumets Z,, Z m a Let ˆX,, ˆX k be ftted values from the frst stage regressos, let ũ be the resduals from the secod stage regresso, ad let û = Y ˆβ 2SLS 0 ˆβ 2SLS Show that û = ũ X ˆβ 2SLS k ˆβ 2SLS X ˆX Suggested soluto to a By defto, X k ˆβ 2SLS k+ W ˆβ 2SLS k+r W r ˆβ 2SLS k X k ˆX k ũ = Y ˆβ 2SLS 0 ˆβ 2SLS ˆX ˆβ 2SLS k ˆX k ˆβ 2SLS k+ W ˆβ 2SLS k+r W r It follows that û = ũ ˆβ 2SLS X ˆX ˆβ 2SLS k X k ˆX k b Show that, for ay j =,, k, the OLS estmates of all the coeff cets the regresso of X ˆX o costat ad Z,, Z m, W,, W r equal zero 7

8 Suggested soluto to b X ˆX are resduals from the frst stage regresso Z,, Z m, W,, W r are exlaatory varables the frst stage regresso OLS resduals are always orthogoal to the corresodg regressors c Show that the OLS estmates of all the coeff cets the regresso of ũ o costat ad ˆX,, ˆX k, W,, W r equal zero Suggested soluto to c ũ s the OLS resdual from a regresso of Y o costat ad ˆX,, ˆX k, W,, W r Sce OLS resduals are orthogoal to regressors, all the coeff cets the regresso of ũ o costat ad ˆX,, ˆX k, W,, W r equal zero d Let Z be a m matrx wth -th row Z,, Z m ad let X be a k matrx wth -th row X,, X k Assume that the umber of strumets m equals the umber of the edogeous regressors k, ad that both Z Z ad Z X are vertble matrces Argue that the ftted value from the regresso of ũ o costat ad ˆX,, ˆX k, W,, W r must be the same as the ftted value from the regresso of ũ o costat ad Z,, Z m, W,, W r Usg ths ad the result from c, show that the OLS estmates of all the coeff - cets the regresso of ũ o costat ad Z,, Z m, W,, W r equal zero Suggested soluto to d Let W be the r matrx wth -th row W,, W r, ˆX the k matrx wth -th row ˆX,, ˆX k, ũ = ũ,, ũ, ad =,, a -dmesoal vector of oes Further, let ˆα, ˆβ, ad ˆγ be the vectors of the OLS coeff cet estmates the regresso of ũ o costat ad Z,, Z m, W,, W r, ad let α, β, ad γ be the OLS coeff cet estmates the regresso of ũ o costat ad ˆX,, ˆX k, W,, W r That s, { ˆα, ˆβ, } ˆγ { = arg m ũ α Zβ W γ ũ α Zβ W γ }, α,β,γ 6 ad { α, β, } { γ = arg m ũ α ˆXβ W γ ũ α ˆXβ W γ } α,β,γ I artcular, the ftted values from the regresso of ũ o costat ad Z,, Z m, W,, W r equal ˆα + Z ˆβ + W ˆγ, ad that from the regresso of ũ o costat ad ˆX,, ˆX k, W,, W r equal α + ˆX β + W γ 8

9 Now, by defto, ˆX = Z Z Z Z X Sce Z X s assumed to be vertble, we ca wrte Z = ˆX Z X Z Z = ˆXA, where A = Z X Z Z s a vertble k-dmesoal matrx Relacg Z 6 by ˆXA, we get { ˆα, ˆβ, } { ˆγ = arg m ũ α ˆXAβ W γ ũ α ˆXAβ W γ } α,β,γ Ths mles that { } ˆα, Aˆβ, ˆγ = arg m α,δ,γ ad hece { ũ α ˆXδ W γ ũ α ˆXδ W γ }, α = ˆα, β = Aˆβ, ad γ = ˆγ Ths, ad the equalty Z = ˆXA mly that ˆα + Z ˆβ + W ˆγ = α + ˆX β + W γ Furthermore, from c, ˆα = 0, ˆβ = 0, ad ˆγ = 0 Therefore, α = 0, β = 0, ad γ = 0 too e Combg the results from a, b, ad d, rove that the value of J statstc the just-detfed case m = k must be zero Suggested soluto to e From a, b ad d, the OLS estmates of the coeff cets the regresso of û o costat ad Z,, Z m, W,, W r must be zero Therefore, the F statstc for testg the ull that the coeff cets o Z are zero must be equal to zero Ths mles that J = 0 5 Ths roblem s the emrcal exercse E23 from the Stock ad Watso textbook We reroduce t here for coveece O the textbook Web ste wwwearsohgheredcom/stock_watso you wll fd the data set WeakIstrumet that cotas 200 observatos o Y, X, Z for the strumetal varable regresso Y = β 0 + β X + u a Costruct for β ˆβ 2SLS, ts stadard error, ad the 95% cofdece terval Suggested soluto to a The 2SLS estmate of beta s 577 Assumg homoskedastcty, the stadard error of the 2SLS estmate of beta equals 04269, ad the 95 ercet cofdece terval s [0320, 9945] b Comute the F -statstc for the regresso of X o Z Is there evdece of a "weak strumet" roblem? 9

10 Fgure : Values of Aderso-Rub statstc The horzotal le at the crtcal level 384 Crcles o the horzotal axs rereset the o-rejected ots, whch sa the cofdece set Suggested soluto to b Assumg homoskedastcty, F = 4566 < 0 Ths suggests that there may be a weak strumet roblem c Comute a 95% cofdece terval for β usg the Aderso-Rub rocedure To mlemet the rocedure, assume that 5 β 5 Suggested soluto to c Assumg homoskedastcty, the cofdece terval s [ 5, 6667] Fgure shows a lot of the values of the AR-statstc for varous 5 β 5 The o-rejected ots are deoted as crcles o the horzotal axs The o-rejecto haes whe the value of the F-statstc s above the horzotal le wth ordate 384 d Commet o the dffereces the cofdece tervals a ad c Whch s more relable? Suggested soluto to d The cofdece terval a s ot relable because of the weak strumet roblem The cofdece terval c s relable eve whe strumets are weak Studets were welcome to do the roblem usg a rogram of ther choce, be t Excel, stata, E-vews, or aythg else I have doe the roblem Matlab Here s the corresodg code Matlab code for roblem 5 %frst stage regresso %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =200;%umber of observatos c=oes200,; %create costat Z=[c,z]; %create matrx of exlaatory varables hat=vz *Z*Z *x; %OLS estmates of the coeff cets xhat=z*hat; %get ftted values 0

11 ehat=x-xhat;% resduals from the frst stage sgehat=ehat *ehat/-2;%estmate of the varace of the %error the frst stage regresso var=sgehat*vz *Z; %covarace matrx of the %frst stage estmates tstat=hat2,/sqrtvar2,2;%t-statstc from the frst stage Fstat=tstat^2;%F-statstc from the frst stage ds F-statstc from the frst stage regresso equals dsfstat %secod stage regresso %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Xhat=[c,xhat]; %create matrx of exlaatory varables beta2sls=vxhat *Xhat*Xhat *y; %obta 2SLS estmates beta2sls=beta2sls2,; %2SLS estmate of beta ds the 2SLS estmate of beta s dsbeta2sls; %asymtotc covarace matrx %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X=[c,x]; %create matrx of exlaatory varables uhat=y-x*beta2sls; %comute 2SLS resduals %assumg homoscedastcty sgmahat=uhat *uhat/-2; %estmate of the varace of error Omegahat=sgmahat*vXhat *Xhat; %estmate of the asymtotc % varace %stadard error %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% staerr=sqrtomegahat2,2;%stadard error estmate ds the stadard error of the 2SLS estmate of beta equals dsstaerr %cofdece terval %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ds a 95 ercet cofdece terval s ds[beta2sls-96*staerr,beta2sls+96*staerr] %Ivertg Aderso Rub test %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% betagrd=lsace-5,5,00;%creat a grd of equally saced %ot betwee -5 ad 5 kee=zeros00,;% create a dcator varable, whose values % wll be chaged to oe for those ots o the % grd ot rejected by AR test for :00 %start a loo

12 yew=y-x*betagrd,;%trasform left-had varable betaar=vz *Z*Z *yew; %ru Aderso-Rub regresso resd=yew-z*betaar;%get resduals from A-R regresso sgar=resd *resd/-2;%estmate of the varace of the error term OmegaAR=sgAR*vZ *Z;%covarace matrx of estmates tar=betaar2,/sqrtomegaar2,2;%t-statstc Fstatstc,=tAR^2;% F-statstc f Fstatstc,>384 %do othg else %do ot reject kee,=;%remember that you ve ot rejected ths value ed ed lotbetagrd,fstatstc%lot the values of AR statstc hold o%use ths to kee revous lot le[-5 5],[384,384]%%crtcal value for AR statstc lotbetagrdkee==,zerossumkee,, o % lot o for o-rejected 6 Cosder a regresso y = x β + u, =,,, where x = x,, x k ad E u x 0 Suose you have m strumets z = z,, z m, ad m > k Assume that y, x, z s a d sequece Deote the m matrx wth -th row z as Z, the k matrx wth -th row x as X, ad the vector wth -th elemet y as y a State the assumtos that z,, z m must satsfy to be vald strumets Suose these assumtos hold Let Ŵ = Z Z Z Z X Prove that ˆβ 2SLS = Ŵ X Ŵ y β Suggested soluto to a The weakest IV assumtos mlyg cosstecy are Z Z C > 0, Z X D, whch s full colum rak relevace Z u 0 exogeety 2

13 The followg shows cosstecy ˆβ 2SLS β = X Z Z Z Z X X Z Z Z Z u X Z Z Z Z X X Z Z Z Z u = D C D D C 0 = 0 b Suose that heteroskedastcty s reset, wth E u 2 z = σ 2 Prove that the varace of the asymtotc dstrbuto of ˆβ2SLS β s gve by D C D D C BC D D C D, where C = E z z, D = E z x, ad B = E u 2 z z Suggested soluto to b X ˆβ2SLS β Z Z Z Z X X Z = Z Z Z u O the other had, X Z Z Z Z X X Z Z Z D C D D C ad Z u d N 0, B Therefore, by Slutsky s theorem ˆβ2SLS β d N 0, D C D D C BC D D C D c Take the resduals û = y X ˆβ 2SLS ad form the m matrx ˆQ = [z û,, z û ] The form ˆR = ˆQ ˆQ Cosder ˆβ { = arg m y Xβ Z ˆR } Z y Xβ Fd the frst order codtos Usg these codtos, or otherwse, show that ˆβ = X Z ˆR Z X X Z ˆR Z y Prove that ˆβ s cosstet 3

14 Suggested soluto to c The frst order codtos are 2X Z ˆR Z y X ˆβ = 0 Therefore, ˆβ = X Z ˆR Z X X Z ˆR Z y Sce y = Xβ + u, we have ˆβ β = = X Z ˆR Z X X Z ˆR Z u X Z ˆR Z X X Z ˆR Z u D B D D B 0 = 0 d Show that the varace of the asymtotc dstrbuto of ˆβ β equals DB D Suggested soluto to d X ˆβ Z β = O the other had, ˆR Z X X Z ˆR Z u X Z ˆR Z X X Z ˆR D B D D B to show the covergece of ˆR to B oe eeds to aly Slutsky theorem smlarly to how ths s doe roblem 3 ad Z u d N 0, B By the Slutsky theorem, we have ˆβ β d N 0, D B D D B BB D D B D Performg the cacellatos, we obta ˆβ β d N 0, D B D e Bous questo Prove that, uder heteroskedastcty, ˆβ s asymtotcally more eff cet tha ˆβ 2SLS 4

15 Suggested soluto to e Note that D C D D C BC D D C D D B D ca be rereseted the form D C D D C B /2 I H H H H B /2 C D D C D, where H = B /2 D O the other had, I H H H H 0 because ths s a demotet matrx ts square s equal to tself Ideed I H H H H I H H H H = I 2H H H H +H H H H H H H H = I 2H H H H +H H H H = I H H H H Ay matrx of the form M M 2 M, where M 2 0 s a ostve semdefte matrx I artcular, D C D D C BC D D C D D B D 0, whch mles that D C D D C BC D D C D D B D 5

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