# Exercise 8. Panel Data (Answers) (the values of any variable are correlated over time for the same individuals)

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1 ercse 8. Panel Daa Answers. Gven m a,.,. I follows ha Varm m s Varm Varm s - Covm, m s Wh anel daa s lkel ha Covm, m s > he vales of an varable are correlaed over me for he same ndvdals Wh ooled daa hen he cross-secon ns are ndeenden over me and so Covm, m s Hence Varm m s anel < Varm m s ooled And so anel daa esmaon s lkel o gve more effcen esmaes of changes assmng no measremen error B ooled daa s lkel o gve more effcen esmaes of sms or averages snce Varm m s Varm Varm s Covm, m s oe ha he error erms from ooled esmaon are osvel correlaed Cove e Cova Cova Vara > so ha esmaed OLS sandard errors n a ooled regresson are neffcen. b b b b 3 a,.,. and d, a Show ha s dfferencng wll nrodce negave aocorrelaon no he dfferenced error erm wh correlaon coeffcen ρ.5 Assmng as n he qeson ha he orgnal resdals are homoskedasc and ncorrelaed over me, hen s dfferencng gves b b,.,. whch elmnaes he fed effec and gves conssen esmaes as long as,, lm lm k k k for all and all k hs s sasfed b assmon he observed vales are ncorrelaed wh he resdal as eher or more lkel n a cal anel [oe a smlar argmen ales o he conssenc of he whn-gro esmaor whch reles on

2 lm Y However n he s dfference model Var Var - Var Var - - Cov, - for all Assmng ha hese resdals are no correlaed over me hen Cov, - and Var consan, homoskedasc varance Smlarl Cov, -, - snce, - [ ], - [ - ] [ -, - ] - [, - ] - snce no aocorrelaon he nd erm s he onl non-zero erm. So he form of aocorrelaon s AR. hs can be sefl o know when rng o f he sandard errors I follows ha /,, Var Var Cov corr So s dfferencng nrodces aocorrelaon no he resdals whch leads o neffcen sandard errors Fnd he degree of aocorrelaon ndced b whn-gros esmaon of hs model Whn-gros esmaon mles esmae for,. and,. Assmng no aocorrelaon or heeroskedasc n he error erm hen, Consder he covarance beween an whn-gro resdals s s s s Snce

3 3 assmng orgnal levels resdals are no correlaed e s all s no eqal Smlarl So s whch s also he eqaon for he covarance snce s Hence s s s Var Var Cov corr Wh Var Hence corr s corr s so whn-gro resdals are also serall correlaed, b as hen ρ noe ha wh 3 he degree of aocorrelaon s he same as ha wh s dfferencng. When >3 hen aocorrelaon n he whn-gro model s smaller han n he s dfferencng model

4 3. Gven b b D b 3 b 4 D,,.. hen a me Y b f Y b b 3 f A me Y b b b 3 b 4 If & D Y b b If & D So he change n he vale of he deenden varable for hose gven he reamen s Y - Y reaed b b b 3 b 4 b b 3 b b 4 and for he conrol gro Y - Y conrol b b b b so he dfference n dfference esmaor s gven b Y - Y reaed - Y - Y conrol b 4 whch s he coeffcen on he neracon erm D n he model above If > hen here are effecvel several dfference n dfferences n esence Y - Y - reaed - Y - Y - conrol for,3.. he roblem hen becomes wheher he average of he dfference-n-dfferences s sgnfcanl dfferen n he erod afer he nervenon ook lace comared wh he average n he erod before Sose he nervenon occrs a I follows ha b b Y b 3 b 4 Y D,,.. wll do hs snce he d--d before he nervenon ook lace s Y - Y reaed - Y - Y conrol [ b b b 3 b b b 3 ] - [ b b b ] b n he erod afer he nervenon Y 3 - Y reaed - Y 3 - Y conrol [ b b 3 b 3 b 4 b b b 3 ] - [ b b 3 b b ] b 4 and hs holds for an > So he coeffcen b 4 measres he average 4

5 5 4. Snce conssenc of he whn-gro esmaor reles on lm Y where.. hs effecvel reqres ha he resdal a me s ncorrelaed wh he vales n ALL me erods hs s called srong eogene and so an model wh lagged vales of wll be endogenos f esmaed sng whn-gros esmaon 5. Gven he beween gro esmaor b b a,. hen OLS on and conssenc of hs esmaor gven b lm lm lm v a lm lm lm lm lm lm v a lm a where a a lm and lm

6 Assmng lm v Snce hese are groed resdals se resl from ercse 4 Heeroskedasc ha Var and so lm a So n general he beween-gro esmaor s nconssen f he nobserved comonens are correlaed wh he observed varable means and he nconssenc n he beween gros esmaor wll ncrease wh he nmber of me erods wh he drecon of he nconssenc deendng on he covarance beween a and he gro means 6. o show ha ooled OLS s a weghed average of he whn and beween gro esmaors -Assmng here are no nobserved ndvdal effecs hen he ooled OLS model s b Smmng over all observaons on ndvdal I and dvdng b he nmber of me seres observaons gves he beween gro means model b and gves he whn-gro esmaor b where We know see lecre noes ha he marces of sms and cross rodcs can be wren as Smlarl he whn-gro eqvalens are e he devaons of each ndvdal observaon from s gro mean 6

7 and he beween gro sms of sqares are gven b he devaon of each gro mean from he overall samle mean across ndvdals n an one of me erods where I follows ha he oal varaon n across ndvdals and me erods can be wren as e oal Varaon Whn-Gro Varaon Beween-Gro Varaon S S whn S beween a smlar eresson holds for Y or an connos varable and also ha S S whn S beween I follows ha he Pooled OLS esmaor of b n n mean devaon form can be wren as OLS 4 whch sng he above means ha whn beween whn beween OLS S S S S 5 Smlarl he beween gro esmaor n can be wren as beween beween beween S S 6 and he whn gro esmaor from 3 whn whn whn S S 7 so from 6 S whn whn whn S 7

8 S beween beween S beween and from 7 sb. hs no 5 gves OLS whn beween whn beween S S S whn S beween S whn beween S whn beween whn whn beween beween S S S S so ha he ooled OLS esmaor s a weghed average of he whn and beween gro esmaes where he wegh reflecs he conrbon of each comonen o he oal varaon n. he larger smaller he varaon n beween gros he closer frher awa he ooled esmaor s o he beween gro esmae and herefore more lke a cross-secon esmae. 7. One assmon needed for nbased OLS esmaes n an model s ha whch n rn reqres ha for each row of he mar [ - ] hs s called a momen resrcon and he samle eqvalen o hs s so ha on average he samle correlaon beween he observed vales and he resdals s zero oe ha he vale of whch mnmses hs vale s eacl ha whch mnmses he sm of sqares n he normal eqaons sed o defne he OLS esmaor [F.O.C. mnmm whch gves k normal eqaons ] For hs reason OLS s also known as a mehod of momens esmaor For he whn-gro model hs s based on he dea ha for all and all where he whn gro resdal Y Y he momen condon eqvalen o mnmsng he sm of sqares can herefore be wren as 8

9 9 Y Y hs s called he samle momen resrcon snce s nqe o hs arclar samle when hen and can be wren as Y Y whch s eacl he s order condon sed o mnmse he sm of sqares n he s dfferenced model Y - Y - oe ha hs eqvalence does no hold when > snce

10 Y Y 3, 3 8. Show ha he fed effecs esmaor s conssen n an nbalanced anel In an nbalanced anel some observaons for some b no necessarl all ndvdals are mssng n ceran me erods Gven he anel daa model b he sal whn-gro fed effecs esmaor can be wren as F Le d d, d,.d be a b vecor of dmm varables somemes called selecon ndcaors akng he vale f ndvdal I s observed n me erod, eqal oherwse. hen for an nbalanced anel he fed effec esmaor can be wren as s s fe s s Sb. n for s s F So conssenc deends on s s Snce hs nvolves he whn-gro mean of hs reles on a src eogene condon he assmon ha he resdal n erod s ncorrelaed wh he vales from an erod js as he conssenc reqremen for a balanced anel. he nrodcon of he ndcaor erm s does no change hs reqremen and so he roof of conssenc s he same noe ha he convergence o a fne vale of he frs erm n n he lm rles o an me-nvaran varables, snce he wold be collnear and so an nverse of hs mar wll no es 9. Consder a smle varable anel daa model

11 b f,.,. where now he varable s measred wh error e so ha he observed model s b f be b f v Consder he conssenc of he ooled OLS esmaor OLS on gves Consder e v f b lm lm lm f b snce all oher correlaons vansh a he lm Smlarl e e lm lm lm e So lm e e b e f b e f b OLS 3 Hence n he resence of measremen error n anel daa he ooled OLS esmaor s nconssen de o a he sal aenaon effec from measremen error he hrd erm n 3 b an addonal bas de o he resence of he fed effec f noe ha he fed effecs esmaor wll be nconssen onl becase of an measremen error. A beween-gro esmaor n he resence of measremen error can be shown o be conssen wh he ndvdal secfc dmmes acng as nsrmens for he ms-measred varable

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