Chapter 14 Advanced Panel Data Methods

Transcription

1 Chapter 14 Advanced Panel Data Methods 1. (From Wooldrdge 14.2) Wth a sngle explanatory varable, the equaton used to obtan the between estmator s y 0 1x a u where the overbar represents the average over tme. We can assume that E(a) = 0 because we have ncluded an ntercept n the equaton. Suppose that u s uncorrelated wth x, but Cov(x t,a ) = xa for all t (and because of random samplng n the cross secton). ~ () Lettng 1 be the between estmator, that s the OLS estmator usng the tme averages, show that ~ xa plm 1 1 where the plm s defned as N var( x ) (The between estmator s just the OLS estmator from the cross-sectonal regresson of y on x (ncludng an ntercept). Because we just have a sngle explanatory varable x and the error term s a + u, we have, from Secton 5.1, plm ( 1 ) = 1 + Cov( x,a + u )/Var( x ). But E(a + u ) = 0 so Cov( x,a + u ) = E( x (a + u )] = E( x a ) + E( x because E( x Now E( x a ) = u ) = Cov( x, u ) = 0 by assumpton. T T 1 E( xta ) = xa. Therefore, t 1 plm ( 1 ) = 1 + xa /Var( x ), whch s what we wanted to show. u ) = E( x a ) () () Assume further that the xt, for all t=1,2, T are uncorrelated wth constant varance 2 x. Show that plm ~ T xa 1 2 If {x t } s serally uncorrelated wth constant varance x then Var( x ) = 2 2 /T, and so plm 1 = 1 + xa /( x /T) = 1 + T( xa / x ). 1 If the explanatory varables are not very hghly correlated across tme, what does part () suggest about whether the nconsstency n the between estmator s smaller when there are more tme perods? 2 x 2 x As part () shows, when the x t are parwse uncorrelated the magntude of the nconsstency actually ncreases lnearly wth T. The sgn depends on the covarance between x t and a.

2 2. (From 14.6) Usng the cluster opton n STATA, the fully robust (robust to seral correlaton and heteroskedastcty) standard errors for the pooled OLS estmates n Table 14.2 are se( ˆ educ) =.011, se( ˆ black) =.051, se( ˆ hspan) =.039, se( ˆ exper) =.020, se( ˆ exper2) =.001, se( ˆ marred) =.026, se( ˆ unon) =.027 () How do these standard errors generally compare wth the nonrobust ones? Why? The fully robust standard errors are larger n each case, roughly double for the tme-constant regressors educ, black, and hspan. On the tmevaryng explanatory varables marred and unon, the fully robust standard errors are roughly 60 percent larger. The dfferences are smaller for exper and exper 2 but hardly trval. We expect ths f we thnk the composte error term, v t, contans an unobserved effect, a. Ths nduces postve seral correlaton and, as we saw n Secton 12.1 for tme seres, the usual OLS standard errors tend to understate the actual samplng varaton n the OLS estmates. The same holds true for pooled OLS wth panel data. () How do the robust standard errors for the pooled OLS compare wth the standard errors for random effects? Does t seem to matter whether the explanatory varable s tme-varyng or tme-constant? On the tme constant explanatory varables educ, black, and hspan, the RE standard errors and the robust standard errors for pooled OLS are roughly the same. (The coeffcent estmates are very smlar, too.) The man dfferences arse n the standard errors (and coeffcents) on the tme-varyng explanatory varables. For example, the RE standard errors on the marred and unon coeffcents are.017 and.018, respectvely, compared wth the robust standard errors for pooled OLS of.026 and.027. We expect ths to be true because, under the under the RE assumptons, RE s more effcent than pooled OLS.

3 3. A fellow classmate has wrtten ther own code to estmate wthn, between, and random effects models wth balanced panels. As a check on ther work, they estmate between, wthn and random effects models for the one-way effects models wth one covarate. (a) The equaton that descrbes the fxed and random effects models s Y t =X t β + u + ε t. The results are summarzed below. You look at the results and tell the student they have a programmng error. What tpped you off? Parameter Estmates and Standard Errors Varable Between Fxed Random Xt ( ) ( ) ( ) The random effects estmator can be thought of as a weghted average of the between and wthn (fxed effect) estmators. Here t s larger than both, so these results cannot be rght. (b) Your classmates fxes ther codng error and generates the followng results for a dfferent problem. These are correct. What s the Hausman test statstc for the null hypothess that u and X t are uncorrelated? Can you reject or not reject the null? Parameter Estmates and Standard Errors Varable Between Fxed Random Xt ( ) ( ) ( ) ˆ 2 2 ˆ / ( se ˆ ) ( se ˆ ) H = /.0112 =1.33 whch s FE RE FE RE dstrbuted as a t-statstc. Ths s clearly not sgnfcant at any conventonal level. Ths means that we cannot statstcally dstngush the two estmates, leadng us to not be able to reject the null. 4. Download the Stata dataset called panel_hw.dta. Ths dataset examnes voter turnout n 49 US states (Lousana s omtted because of an unusual electon n 1982) plus the Dstrct of Columba over 11 electons (contans data on 50 unts over 11 tme perods). Submt wrte-ups for these problems as well as the log fles. () Regress turnout as a percent of votng age populaton on the number of days before the general electon by whch an ndvdual needs to regster,

4 state per capta ncome, the dummy varable for mdterm electons, and the dummy varables for West North Central, the South, and the Border states.. regress vaprate gsp mdterm regdead WNCentral South Border Source SS df MS Number of obs = F( 6, 543) = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = vaprate Coef. Std. Err. t P> t [95% Conf. Interval] gsp mdterm regdead WNCentral South Border _cons test WNCentral South Border ( 1) WNCentral = 0 ( 2) South = 0 ( 3) Border = 0 F( 3, 543) = Prob > F = Whch coeffcents are sgnfcant? Are there any regonal effects of these regons? Use F-test to determne ths. All of the ndependent varables are sgnfcant. Yes regonal effects are sgnfcant both separately and jontly usng the F test. () Part () assumed that poolng the data was vald. Instead, estmate ths wth a fxed effects regresson. Whch varables are omtted from the estmaton? Why?. s stcode. ts year. xtreg vaprate mdterm gsp regdead WNCentral South Border, fe Fxed-effects (wthn) regresson Number of obs = 550 Group varable: stcode Number of groups = 50 R-sq: wthn = Obs per group: mn = 11 between = avg = 11.0 overall = max = 11 F(2,498) = corr(u_, Xb) = Prob > F =

5 vaprate Coef. Std. Err. t P> t [95% Conf. Interval] mdterm gsp regdead (dropped) WNCentral (dropped) South (dropped) Border (dropped) _cons sgma_u sgma_e rho (fracton of varance due to u_) F test that all u_=0: F(49, 498) = Prob > F = estmates store fe Regdead and the regonal dummes dropped because these do not vary wthn states over tme. Obvously, the regon of the country a state s n does not vary. The results ndcate that the number of days before the general electon by whch an ndvdual needs to regster also dd not change wthn states durng ths tme perod and therefore also dropped out.. () Test for whether there s evdence of unobserved heterogenety that s, whether a fxed effects model s more approprate than the pooled model. What s your testng hypothess? What s your test statstc? Is poolng approprate n lght of the results of your test? The testng hypothess s that the state dummes are equal to zero. The F-test s a test of ths. The fnal lne of the table s F test that all u_=0: F(49, 498) = Prob > F = We can reject the null that the dummes are all equal to zero wth a p-value of As a result, we conclude that poolng s not approprate the state fxed effects are sgnfcant. (v) Now estmate a random-effects model. Why do those results dffer from the fxed effects results? Is there evdence of unobserved heterogenety? Show your testng hypothess, and decde on t. Are some varables omtted from the estmaton? Why or why not?. xtreg vaprate mdterm gsp regdead WNCentral South Border, re Random-effects GLS regresson Number of obs = 550 Group varable: stcode Number of groups = 50 R-sq: wthn = Obs per group: mn = 11

6 between = avg = 11.0 overall = max = 11 Random effects u_ ~ Gaussan Wald ch2(6) = corr(u_, X) = 0 (assumed) Prob > ch2 = vaprate Coef. Std. Err. z P> z [95% Conf. Interval] mdterm gsp regdead WNCentral South Border _cons sgma_u sgma_e rho (fracton of varance due to u_). estmates store re Note that ths dffers from the fxed effects estmates for several reasons. Frst, we get estmates for the tme-nvarant varables. Second, these results use the varaton between states as well as the varaton wthn states across tme, although the effect of usng ths varaton s pretty small on both the magntude of the coeffcents and ther standard errors.. xttest0 Breusch and Pagan Lagrangan multpler test for random effects vaprate[stcode,t] = Xb + u[stcode] + e[stcode,t] Estmated results: Var sd = sqrt(var) vaprate e u Test: Var(u) = 0 ch2(1) = Prob > ch2 = Ths ndcates that we can reject the null that var(u) = 0 wth a hgh degree of confdence (p value =.000). Ths mples that there s seral correlaton n our errors, and that the random effects model s more approprate than an OLS model. (v) Whch model s more approprate, FE or RE? What s your underlyng testng hypothess? What mplcaton does the null hypothess have?

7 . hausman fe re Dscuss the tradeoffs between usng pooled OLS, fxed-effects, and random-effects for ths model Coeffcents ---- (b) (B) (b-b) sqrt(dag(v_b-v_b)) fe re Dfference S.E. mdterm gsp b = consstent under Ho and Ha; obtaned from xtreg B = nconsstent under Ha, effcent under Ho; obtaned from xtreg Test: Ho: dfference n coeffcents not systematc ch2(2) = (b-b)'[(v_b-v_b)^(-1)](b-b) = 0.18 Prob>ch2 = Ths test shows us whether the coeffcents under the random effects model dffer statstcally from the fxed effects model. The null hypothess s that the coeffcents are not dfferent. In ths case, they are not the p-value assocated wth a test that they are the same s.9158, so we cannot reject the null hypothess. The mplcaton s that whle dfferent states may have systematcally dfferent errors, the errors are not correlated wth the X varables. As a result, we would prefer the somewhat more effcent estmates from the random effects model. (Note, however, that there s lttle dfference between the standard errors of the fxed and random effects models.) Some useful STATA commands for ths problem: Declare the data to be panel data before usng the xtreg commands: s stcode declares the cross sectonal unts are ndcated by stcode ts year declares tme perods are ndcated by year. One way to run a fxed-effects model : xtreg vaprate mdterm gsp regdead WNCentral South Border, fe Store the estmaton results wth estmates store fe (You can replace fe n ths last command wth whatever name you want to use) The bottom lne of the estmaton results gve us an F test for poolng. Another way to run a fxed effects model: x: reg vaprate mdterm gsp regdead WNCentral South Border.stcode Yet another way to run a fxed effects model: areg vaprate mdterm gsp regdead WNCentral South Border, absorb(stcode)

8 Random effects model: xtreg vaprate mdterm gsp regdead WNCentral South Border, re Store the estmaton results wth estmates store re (You can replace re n ths last command wth whatever name you want to use) Testng commands xttest0 xthausman Replcaton Exercse One mportant queston n publc fnance s whether the structure of the welfare system affects marrage and famly formaton. Ths assgnment examnes the mpact of welfare wavers (IMPDUM) on the propensty to be a never marred woman (NEVERMAR). Pror to the 1996 federal Welfare Reform Act, states could apply for wavers to experment wth alternatve systems here related to marrage. For more nformaton, see Maranne Btler, Jonah Gelbach, Hlary Hoynes, and Madelne Zovodny (2004) The Impact of Welfare Reform on Marrage and Dvorce. Ths assgnment s focused on dentfyng and mplementng a model usng varaton across states and over tme n a partcular polcy. Download the dataset wr-nevermar.dta. The dataset conssts of a sample of black women ages wth a hgh school educaton or less from The data come from the March CPS. 1. Summarze and descrbe the data. Graph the mean of NEVERMAR by year. Descrbe the trend n ths varable. (use the egen command to create the mean of ths varable by year). sum Varable Obs Mean Std. Dev. Mn Max state year rmaxpay urate empgrat mpdum everwav black msadum age age age age nevermar des

9 0 Percent states wth waver state byte %9.0g State d number year byte %8.0g Survey year rmaxpay float %9.0g Max bens for fam of 3, n 1000s of 1997$ urate float %9.0g State unemployment rate empgrat float %9.0g State empl growth rate mpdum byte %9.0g =1 f waver mplemented before march of ths year, & tanf not yet everwav byte %9.0g state ever had a major waver black byte %8.0g non-hspanc black msadum byte %9.0g =1 f n msa age1625 byte %8.0g age between 16 and 25 age2634 byte %8.0g age between 26 and 34 age3544 byte %8.0g age between 35 and 44 age4554 byte %8.0g age between 45 and 54 nevermar float %9.0g =1 f never marred egen meannevermar = mean(nevermar), by(year) twoway (lne meannevermar year, sort), yttle(percent women never marred--mean across states) Survey year 2. Now graph the mean of the polcy varable IMPDEM over tme. How s that varable trendng over tme? Dscuss the trends n (1) and (2) together. egen meanmpdum = mean(mpdum), by(year) twoway (lne meanmpdum year, sort), yttle(percent states wth waver) Survey year

10 Both varables are trendng upward. The percent never marred ncreased, and then dropped. After 1993, t pcked up agan. State wavers began n 1993 as well and expanded rapdly. 3. Collapse the data to the year level and estmate the model: NEVERMAR t = α +γimpdum t + e t Where s the dentfcaton comng from n ths tme seres model? Why mght ths be a based estmate of γ? If t s based, what do you thnk the key omtted varable s? Why don t you control for t? Does ths emprcal dentfcaton strategy make sense? Don t forget to preserve the data before you do ths and restore after. Read up on the collapse command you want to collapse by year.. preserve. collapse nevermar mpdum, by(year). reg nevermar mpdum Source SS df MS Number of obs = F( 1, 6) = Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = nevermar Coef. Std. Err. t P> t [95% Conf. Interval] mpdum _cons restore The dentfcaton here s comng from the tme seres varaton essentally, ths s a regresson correlatng nformaton n the two graphs we just produced. Ths ndcates what we saw--that wavers and percent never marred both ncreased over ths tme perod. However, there are many reasons why ths s lkely to be based. A major concern s that states may have chosen to use wavers because they were experencng the rse n percent unmarred. The omtted varable would be anythng that s leadng to hgher rates of never marred women. Ths s hard to control for because there are many reasons why ths mght be on the rse, ncludng some cultural explanatons that are hard to quantfy. Ths emprcal dentfcaton strategy would NOT make sense f the trends n never marred are themselves causng the adopton of wavers. Note that ths observaton of concurrent tme trends strongly suggests that these wll need to be controlled for n the analyss. 4. Now estmate a cross sectonal model for 1995: NEVERMAR,s,95 = α +γimpdum s,95 + ε,s,95

11 Where s the dentfcaton comng from n ths cross sectonal model? Why mght ths be a based estmate of γ? If t s based, what do you thnk the key omtted varable s? Why don t you control for t? Does ths emprcal dentfcaton strategy make sense? What f you had estmated the model usng only data from 1991?. reg nevermar mpdum f year==95 Source SS df MS Number of obs = F( 1, 1317) = 0.22 Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = nevermar Coef. Std. Err. t P> t [95% Conf. Interval] mpdum _cons Here the dentfcaton s comng from cross state dfferences comparng states n 1995 that have and have not adopted wavers. Agan, ths may be based f states that adopted wavers are systematcally dfferent from states that dd not. See above for the same sorts of ssues about omtted varables. If we had estmated ths n 1991, there would have been no estmate on IMPDUM because no states had wavers n that year there was no cross sectonal varaton. 5. Now return to the full model. Estmate the smplest model NEVERMAR,s,t = α +γimpdum s,t + ε,s,t Interpret the magntude of the coeffcent on IMPDUM. (Remember when you do that, NEVERMAR s 0/1). Is ths what you expected to fnd? Relate your answer to (3).. reg nevermar mpdum Source SS df MS Number of obs = F( 1, 11365) = 3.71 Model Prob > F = Resdual R-squared = Adj R-squared = Total Root MSE = nevermar Coef. Std. Err. t P> t [95% Conf. Interval] mpdum _cons These results that adoptng a waver leads to a 2.7 percentage pont reducton n the fracton of women who have never marred. No ths s not what we expected to fnd. Note that ths result s between the tme seres and cross sectonal results the tme seres varaton appears to be partly responsble for these results.

12 6. Reestmate the model n (5) accountng for the fact that IMPDUM vares only at the state level, whle the data s at the ndvdual level. In STATA, use reg yvar xvars, cluster(state) You should cluster your standard errors for the rest of ths problem.. reg nevermar mpdum, cluster(state) Lnear regresson Number of obs = F( 1, 50) = 1.02 Prob > F = R-squared = Root MSE = (Std. Err. adjusted for 51 clusters n state) Robust nevermar Coef. Std. Err. t P> t [95% Conf. Interval] mpdum _cons Note that only the standard errors change, and that they are larger as we mght expect. 7. Add demographc varables and economc varables to the regresson (age varables, MSADUM, URATE, EMPGRAT, RMAXPAY). Interpret the coeffcent sand relate them to what you mght have predcted based on economc theory. Be careful wth unts when you nterpret the varables.. reg nevermar mpdum age2634 age3544 age4554 msadum urate empgrat rmaxpay, cluster(state) Lnear regresson Number of obs = F( 8, 50) = Prob > F = R-squared = Root MSE = (Std. Err. adjusted for 51 clusters n state) Robust nevermar Coef. Std. Err. t P> t [95% Conf. Interval] mpdum age age age msadum urate empgrat rmaxpay _cons Ths ndcates that women are 37 percentage ponts less lkely to have never marred that women (the omtted category). Lvng n a MSA s assocated wth a 4 percent hgher probablty of beng never marred. A one percentage pont ncrease n the unemployment rate s assocaton wth.3 percentage pont ncrease n the probablty of beng never marred, although ths s not statstcally sgnfcant. A

13 one thousand dollar ncrease n the state maxmum benefts for a famly of three s assocated wth a.08 percentage pont ncrease n the probablty of beng never marred; agan ths s not sgnfcant. 8. Add year fxed effects to the model (keep n the demographc and economc varables). That s, estmate NEVERMAR,s,t = α +γimpdum s,t + βx,s,t + η t + ε,s,t Use the x: reg command to do ths so you can see the coeffcents on the fxed effects. How does the dentfcaton n THIS model dffer from that n (3)? What happens to the coeffcent on IMPDUM? Explan why the coeffcent changed as t dd usng reasonng related to omtted varable bas.. x: reg nevermar mpdum age2634 age3544 age4554 msadum urate empgrat rmaxpay.year, cluster(state).year _Iyear_89-96 (naturally coded; _Iyear_89 omtted) Lnear regresson Number of obs = F( 15, 50) = Prob > F = R-squared = Root MSE = (Std. Err. adjusted for 51 clusters n state) Robust nevermar Coef. Std. Err. t P> t [95% Conf. Interval] mpdum age age age msadum urate empgrat rmaxpay _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _cons Now the dentfcaton s comng from average cross-state dfferences n the facton never marred-- ths s comparng the average dfference between states wth and wthout wavers n each year and averagng that dfference for all years. Note that the coeffcent s much less postve the sgn has swtched. Ths s because of the overall effects of tme the percent never marred was rsng n ALL states over tme, just as the number of wavers was rsng. Tme (or tme proxyng for tme-varyng factors that rased the percent never marred) s the omtted varable that s now controlled for. 9. Add state fxed effects to the model along wth the year fxed effects.

14 NEVERMAR,s,t = α +γimpdum s,t + βx,s,t + ξ s + η t + ε,s,t Descrbe how the dentfcaton n THIS model dffers from that n (3). What happens to the coeffcent on IMPDUM? Explan why the coeffcent changed as t dd usng reasonng related to omtted varable bas.. x: reg nevermar mpdum age2634 age3544 age4554 msadum urate empgrat rmaxpay.year.state, cluster(state).year _Iyear_89-96 (naturally coded; _Iyear_89 omtted).state _Istate_11-95 (naturally coded; _Istate_11 omtted) Lnear regresson Number of obs = F( 14, 50) =. Prob > F =. R-squared = Root MSE =.4036 (Std. Err. adjusted for 51 clusters n state) Robust nevermar Coef. Std. Err. t P> t [95% Conf. Interval] mpdum age age age msadum urate empgrat rmaxpay _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Istate_ _Istate_ _Istate_ _Istate_ Etc. By addng state fxed effects, ths model s now essentally a dfference n dfference model. The dentfcaton s comng from the change n the percent never marred n states that passed wavers compared to the change n states that dd not. Ths accounts for the fact that states that passed wavers may always have had percent never marred that dffered from states that dd not pass wavers. It also accounts for the fact that the percent never marred was rsng over tme n all states. The omtted varables here may be anythng that both caused hgh percent never marred and a hgh probablty of passng a waver that would have led the coeffcents to be postvely based. Indeed, we see here that accountng for ths leads to much more negatve coeffcents. 10. Add state specfc tme trends to the model: NEVERMAR,s,t = α +γimpdum,s,t + βx,s,t + Θ s Tme + ξ s + η t + ε,s,t

15 How do the results for IMPDUM change wth the tme trends? Explan why the results changed. To do ths, use ths command: x: reg yvar xvars.year.state*year, cluster(state). x: reg nevermar mpdum age2634 age3544 age4554 msadum urate empgrat rmaxpay.year.state*year, cluster(state).year _Iyear_89-96 (naturally coded; _Iyear_89 omtted).state _Istate_11-95 (naturally coded; _Istate_11 omtted).state*year _IstaXyear_# (coded as above) Lnear regresson Number of obs = F( 13, 50) =. Prob > F =. R-squared = Root MSE = (Std. Err. adjusted for 51 clusters n state) Robust nevermar Coef. Std. Err. t P> t [95% Conf. Interval] mpdum age age age msadum urate empgrat rmaxpay _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_ _Iyear_96 (dropped) _Istate_ _Istate_ _Istate_ _Istate_ _Istate_ _Istate_ etc _Istate_ _Istate_ year _IstaXyea~ _IstaXyea~ Etc After addng tme trends, the results are even larger a waver s assocated wth a 6.6 percentage pont reducton n the percent never marred. Ths specfcaton controls for the fact that states may have had dfferent trends n the rates of never marred. Lke the last specfcaton, ths s essentally a dfference n dfference specfcaton. Instead of essentally just comparng the average dfference n never

16 marred before and after the polcy waver, ths specfcaton compares the dfference relatve to the average trend for that state.