ECON 510: Panel data econometrcs Semnar 3: October., 007 Problem 3.1: Error autotocorrelaton and heteroskedastcy Standard varance components model: (0.1) y = k+ x β + + u, ε = + u, IID(0, ), u Rewrng the model n matrx form gves: y = etk + X + ε IID(0, ), x u, = 1,..., N; t = 1,..., T. β, ε = et + u, = 1,..., N ε,..., 1 εn are uncorrelated and have zero expectaton, wh covarance matrx [ ] V ( ) E( ) E ( )( ) ( ε = εε = e + u e + u = e e ) + I. Model (0.1) can thus be T T T T wrten compactly as: (0.) y = etk + Xβ + ε, ε = et + u IID( 0T,1, Ω T), = 1,..., N, T where (0.3) Ω + L (, ) ( ) + L = Ω = ee + I = M M O M L + T T T T T Models wh autocorrelaton: Model A1: u ε = + u, IID(0, ), = ρu + η, ρ < 1, η IID(0, η x,., t 1 ), η Assume that the AR(1) process started nfnetly long back n tme. Then We derve the varance of ε : u t = ρ η. r= 0 r t r Sde 1 av 0
r ( r t r) r ( r= 0 ) var( ε ) = var( + u ) = var + ρ η t r 1 = + ρη = + ρ η = + 1 ρ r var ( ) var( ) = 0 r= 0 t r η From Lllard and Wlls (1978) p.989 1, we know that η + u = + =, ε = j t = s (1 ρ ) τ τ η τ E( εε js ) = + ρu = + ρ = 1 j, t s 0 ε τ (1 ρ ) + = = > ε ε 0 j The frst lne s the varance, the second the covarance between nventons n dfferent perods, for the same ndvdual. Thus, the covarance of ε s: τ η cov( ε, ε js ) = + ρ, = j, t s = τ > 0 (1 ρ ) We can wre the resdual covarance matrx of the genune dsturbance, u = ( u,..., u ) : 1 T Ω T 1 1 ρ ρ K ρ ρ 1 ρ. L = ρ 1 L. 1 ρ M M M O M T 1 ρ.. 1 L * η ρ and the covarance matrx of ε = ( ε 1,..., εt ) = (,..., ) + ( u 1,..., ut ) as: η η η T 1 ρ ρ ρ η + + + K + 1 ρ 1 ρ 1 ρ 1 ρ η η η + ρ. + + ρ L 1 ρ 1 ρ 1 ρ * ΩT = Ω + ee T T = η η η + ρ. + ρ + L 1 ρ 1 ρ 1 ρ M M M O M T 1 η η + ρ.. L + 1 ρ 1 ρ 1 Lllard, Lee A. and Robert J. Wlls (1978). Dynamc Aspects of Earnng Mobly. Econometrca, Vol. 46, No. 5, pp. 985-101. Sde av 0
In model A1, there s homoschedascy and autocorrelaton, but not equcorrelaton unless ρ = 0. Ths can be seen from the fact that : t s η + ρ 1 ρ corr( ε, εs ) = η + 1 ρ The correlaton s ndependent of but vares wh t s for ρ 0. Model A: u ε = + u IID(0, ), = ρu + η, ρ < 1, η (0, ), η, t 1 η x,., We go through the same steps as for model A1, the only dfference beng that η s now ndvdual specfc (has subscrpt ). The aggregate covarance matrx s thus: Ω * T T T η η η ρ ρ η η η η T 1 + + ρ + ρ K + ρ 1 ρ 1 ρ 1 ρ 1 ρ η η η + ρ. + + ρ L 1 ρ 1 ρ 1 ρ = Ω + ee = + + + L. 1 ρ 1 ρ 1 ρ M M M O M T 1 η η + ρ.. L + 1 ρ 1 ρ In ths model, ε s both heteroskedastc and autocorrelated. We now have: whch vares wh both and t t s η + ρ 1 ρ corr( ε, εs ) = η + 1 ρ s for ρ 0. Note: t s η t s η + ρ + ρ cov( ε, εs ) 1 ρ 1 ρ corr( ε, εs ) = = = st. dev( ε )* st. dev( εs ) η η η + + + 1 ρ 1 ρ 1 ρ Sde 3 av 0
Model A3: u ε = + u IID(0, ), = ρu + η, ρ < 1, η (0, ), η, t 1 η x,. We go through the same steps as for models A1 and A, the only dfference beng that now both η and ρ are ndvdual specfc (have subscrpt ). The aggregate covarance matrx s thus:, Ω η η η T 1 + + ρ + ρ K + ρ 1 ρ 1 ρ 1 ρ 1 L. L. M M M O M T 1 η η + ρ.. L + 1 ρ 1 ρ η η η + ρ + + ρ 1 ρ 1 ρ 1 ρ * T = Ω + ee T T = η η η + ρ + ρ + 1 ρ 1 ρ 1 ρ η ρ In ths model, as n A, ε s both heteroskedastc and autocorrelated. We now have: whch vares wh both and t t s + ρ 1 corr( ε, εs ) = + 1 s for ρ 0. η ρ η ρ To sum up, the models A1-A3 all have autocorrelaton, and A-A3 also have heteroskedastcy. However, none of them have equcorrelaton, as the correlaton of ε vares wh t s for ρ 0. Models wh error heteroskedastcy: Model H1: ε = + u (0, ), = 1,..., u, IID(0, ) η x,. N Sde 4 av 0
The covarance matrx for ths model s: V( ε) = ( ete T ) + IT just as (0.3), only wh ndvdual. We thus have: corr( ε, εs ) = + whch can be wrten out ε n ths model s heteroskedastc, and s equcorrelated for ndvdual. Model H: u ε = + u IID(0, ), (0, ) = 1,..., η x,., N The covarance matrx for ths model s: V( ε ) = ( e e ) + I T T just as (0.3), only wh ndvdual. We thus have: corr( ε, εs ) = + T whch can be wrten out ε n ths model s heteroskedastc, and s equcorrelated for ndvdual. Model H3: u ε = + u (0, ), = 1,..., (0, ), = 1,..., η x,., N N The covarance matrx for ths model s: V( ε) = ( ete T ) + IT just as (0.3), only wh ndvdual and. We thus have: corr( ε, εs ) = + whch can be wrten out ε n ths model s heteroskedastc, and s equcorrelated for ndvdual. All of the models H1-H3 have heteroskedastc dsturbances, ε s, whch are equcorrelated for ndvdual. Would you consder any of these extensons as mprovements of the model? That depends on the real structure of the data. For nstance, there s no pont n usng an autocorrelaton covarance matrx for estmaton unless there actually s autocovarance n the dsturbances. We lose more perods when we have to account for autocorrelaton. Sde 5 av 0
Problem 3. A: Instrument varables If we treat all varables as exogenous, we can use the one-stage whn estmator. Xtreg We assume that the model can be wrten y = Y γ + X β + μ + ν = Z + u ν μ + = u u ~IID( 0, ) δ. xtreg ln_wage age* ten not_s un so, fe (dcode) Fxed-effects (whn) regresson Number of obs = 19007 Group varable (): dcode Number of groups = 4134 R-sq: whn = 0.1333 Obs per group: mn = 1 between = 0.375 avg = 4.6 overall = 0.031 max = 1 F(6,14867) = 381.19 corr(u_, Xb) = 0.074 Prob > F = 0.0000 ln_wage Coef. Std. Err. t P> t [95% Conf. Interval] age.0311984.003390 9.0 0.000.045533.0378436 age -.0003457.0000543-6.37 0.000 -.00045 -.000393 tenure.017605.0008099 1.76 0.000.0160331.019079 not_smsa -.097535.015377-7.76 0.000 -.11889 -.07678 uno.097567.0069844 13.97 0.000.0838769.111576 south -.06093.01337-4.66 0.000 -.088158 -.0359706 _cons 1.09161.05316 0.87 0.000.989079 1.194151 sgma_u.3910683 sgma_e.5545969 rho.70091004 (fracton of varance due to u_) F test that all u_=0: F(4133, 14867) = 8.31 Prob > F = 0.0000 (We note that only 19007 obs are used n the regresson, due to mssng varables n UNION) The F test s a test for absence of fxed effects. We can assume fxed effects. We can make a plot to look at the resduals. Frst we predct Xb, We have named yhatt We then compute the resduals: y- yhatt: We have named res, or we can type: predct <varname>, ue Eher way we obtan ) μ + v ), the combned resdual.. predct yhatt. gen res= ln_wage- yhatt (or predct res_,ue). hst res. hst res, normal. twoway (scatter res yhatt) (msplne res yhatt). twoway (scatter res year) (msplne res year) Ths gves the plots below. By lookng at the plot seems that our model seems to f assumptons on ~IID( 0, ) u Sde 6 av 0
We can also predct the frst dfferenced overall component ε = u u 1, by typng: predct res,e We then obtan these plots. IV-modell : xtvreg ln_wage age* not_s (tenure = south unon), fe If we beleve that tenure s an endogenous varable, we can try to handle ths wh nstruments. It s suggested that we use unon and south as nstruments for tenure. We then need another specfcaton of the model. y = Y γ + X β + μ + ν = Z + μ + ν u = 1) δ μ + ν ~IID( 0, ) u X s stll a vector of exogenous varables; Y s a vector of observatons of endogenous varables, that are allowed to correlate whν. N s the number of observatons, and n s the number of grls.(groups) We then use xtvreg whch s a twostage estmator. Frst we estmate It s mportant to construct nstruments that are strongly correlated wh the endogenous varable, but not u. We fnd that the correlaton between u and not_smsa s -0.1451 and u between and unon s 0,079. The correlatons between the endogenous varable and the nstruments are -0.066 and -0.131. So they are not very good nstruments.. corr tenure south unon (obs=19007) Sde 7 av 0
tenure south unon -------------+--------------------------- tenure 1.0000 south -0.066 1.0000 uno 0.1600-0.131 1.0000. corr res3 not_smsa unon (obs=19007) res3 not_smsa unon -------------+--------------------------- res3 1.0000 not_smsa -0.1451 1.0000 uno 0.079-0.0693 1.0000 IV-modell 3: xtvreg ln_wage age* not_s (tenure = south unon), fe Fxed-effects (whn) IV regresson Number of obs = 19007 Group varable: dcode Number of groups = 4134 R-sq: whn =. Obs per group: mn = 1 between = 0.1304 avg = 4.6 overall = 0.0897 max = 1 F(4138,14869) = 74.14 corr(u_, Xb) = -0.6843 Prob > F = 0.0000 ln_wage Coef. Std. Err. t P> t [95% Conf. Interval] tenure.403531.0373419 6.44 0.000.1671583.3135478 age.0118437.009003 1.3 0.188 -.0058037.09491 age -.001145.0001968-6.17 0.000 -.0016003 -.000886 not_smsa -.0167178.033936-0.49 0.6 -.08313.0497767 _cons 1.67887.166657 10.3 0.000 1.35944 1.99713 sgma_u.70661941 sgma_e.6309359 rho.55690561 (fracton of varance due to u_) --------------------------------------------------------------- --------------- F test that all u_=0: F(4133,14869) = 1.44 Prob > F = 0.0000 Instrumented: tenure Instruments: age age not_smsa south unon. correlate, _coef tenure age age not_smsa _cons -------------+--------------------------------------------- tenure 1.0000 age -0.3709 1.0000 age -0.7337-0.3543 1.0000 not_smsa 0.4131-0.156-0.3054 1.0000 _cons 0.617-0.9515 0.0637 0.079 1.0000 Note: corr(u_, Xb) = -0.6843 s hgh, the model seems to be even worse than before (note: ths does not happen f we use another nstrument than unon, for nstance hours) Lookng at the resduals y y = Z ) δ y we see that they do not seem to f assumptons on IID (0, ). Our model specfcaton wh nstrument varables does not mprove our estmaton. Sde 8 av 0
By usng tenure as a endogenous varable, usng south and unon as nstruments, we fnd that age and not_smsa are no longer sgnfcant. If we beleve for nstance from other studes that these should be sgnfcant, we should use a dfferent model specfcaton. IV-modell 4: We are asked to use a between estmaton. After passng 1) trough the between estmator we are left wh y = + δ + μ + ν Z Where w 1 = T T t= 1 w for w { y, Z, v} We smlarly defne X as the matrx of nstruments X after they have passed trough the between transformaton. These nstruments are used to correct the bases on the coeffcents. We do not succeed, we fnd that sd(u_ + avg(e_.))= 0,4445007. The resdual plots also show that the coeffcents are not constant for dfferent values of X β. xtvreg ln_wage age* not_smsa (tenure= unon south), be (dcode) Between-effects IV regresson: Number of obs = 19007 Group varable: dcode Number of groups = 4134 R-sq: whn = 0.0881 Obs per group: mn = 1 between =. avg = 4.6 overall = 0.1483 max = 1 ch(4) = 51.38 sd(u_ + avg(e_.))=.4445007 Prob > ch = 0.0000 ln_wage Coef. Std. Err. z P> z [95% Conf. Interval] tenure.1349486.010693 13.14 0.000.11481.155076 age.044055.0135488 3.13 0.00.0158503.0689607 age -.0008035.000143-3.75 0.000 -.00135 -.0003835 not_smsa -.619405.016784-16.09 0.000 -.938457 -.300354 _cons.8430686.0967 4.15 0.000.4453395 1.40798 Instrumented: tenure Instruments: age age not_smsa unon south Sde 9 av 0
. correlate, _coef tenure age age not_smsa _cons -------------+--------------------------------------------- tenure 1.0000 age -0.1843 1.0000 age 0.083-0.9898 1.0000 not_smsa 0.0096-0.00 0.0156 1.0000 _cons 0.1310-0.991 0.9767 0.0016 1.0000 IV-modell 5: If we beleve, or are wllng to assume, that all μ s are uncorrelated wh the other covarates, we can f the random-effects model. There are two varance components to estmate, the varance of μ and ν. Snce the varance components are unknown, the consstent estmates are requred to mplement feasble GLS. ˆ 1 gls 1 1 A consstent estmator s obtaned by β = ( X ' Σ X ) ( X ' Σ y) ) ) ) u ' u j The resduals n estmatng Σ, j = are frst obtaned form OLS regresson. T 1 The estmates and ther standard errors are calculated usng Σ ). (note: We are not que sure about ths, and hope that ths can be commented on at the semnar.) xtvreg ln_wage age* not_s (tenure = south unon), re (dcode). xtvreg ln_wage age* not_s (tenure = south unon), re (dcode) GSLS random-effects IV regresson Number of obs = 19007 Group varable: dcode Number of groups = 4134 R-sq: whn = 0.060 Obs per group: mn = 1 between = 0.1745 avg = 4.6 overall = 0.106 max = 1 Wald ch(4) = 941.5 corr(u_, X) = 0 (assumed) Prob > ch = 0.0000 ln_wage Coef. Std. Err. z P> z [95% Conf. Interval] Sde 10 av 0
tenure.177948.011174 15.87 0.000.155397.199194 age.0191674.0066388.89 0.004.0061555.03179 age -.0008496.0001057-8.04 0.000 -.0010567 -.000645 not_smsa -.11993.0130456-16.5 0.000 -.3756 -.186443 _cons 1.4761.1037797 13.76 0.000 1.405 1.631014 sgma_u.33156584 sgma_e.6309359 rho.1674808 (fracton of varance due to u_) --------------------------------------------------------------- --------------- Instrumented: tenure Instruments: age age not_smsa south unon. correlate, _coef tenure age age not_smsa _cons -------------+--------------------------------------------- tenure 1.0000 age -0.370 1.0000 age -0.199-0.8895 1.0000 not_smsa 0.0847-0.047-0.0171 1.0000 _cons 0.3147-0.9874 0.887-0.0007 1.0000 Densy 0.5 1 1.5-0 4 res 3..B The STATA output from runnng the regressons can be found on the pages below. We have gven the models numbers, and comment on all the models frst. Model 3..B1. xtabond n w k ys yr1980-yr1984, lags(1)... 13 Model 3..B xtabond n w k ys yr1980-yr1984, lags(1) robust... 14 Model 3..-B3 xtabond n w k ys yr1980-yr1984, lags(1) twostep... 14 Model 3..-B4 xtabond n w k ys yr1980-yr1984, lags()... 15 Model 3..-B5 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1)... 15 Model 3..-B6 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) robust... 16 Model 3..-B7 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) twostep... 17 Model 3..-B8 xtabond n l(0/).w l(0/).k l(0/).ys yr1980-yr1984, lags()... 17 Model 3..-B9 xtabond n l(0/).w l(0/).k l(0/).ys yr1980-yr1984, lags() robust... 18 Model 3..-B10 xtabond n l(0/).w l(0/).k l(0/).ys yr1980-yr1984, lags() twostep... 19 Dynamc panel data models allow past realsatons of the dependent varable to affect s current level. 1) y = y 1 + X β + + μ μ ρμ, t 1 + η, η ~ (0, η ~IID(o, ) = ) and μ are assumed to be ndependent for each over all t. β s a vector of parameters to be estmated Sde 11 av 0
Var( ( μ ) = 1 ρ A rellano and Bond derve a generalzed method-of-moments estmator for, β usng lagged levels of the dependent varable as nstruments. Ths method assumes that there s no second-order autocorrelaton n the μ. xtabond ncludes the test for autocorrelaton and the Sargan test of over-dentfyng restrctons for ths model. W e do not know the AR structure but can be dfferent for each ndvdual ( η ~ (0, ) ) and the varances ( IID ) ) may be dfferent for dfferent ndvdual A A3) ~ (0, s. (se Model 1- Frst dfferencng of the equaton removes the and produces an equaton that can be estmated usng nstrumental varables. In all the models we use the lagged dependent varable as an nstrument varable. We have then lost the three frst observatons, to lags and dfferencng. Snce x contans only strctly exogenous covarates, Δ x wll serve as s own nstrument. The nstrument matrx has one row for each tme perod we are nstrumentng. η Z y 0 =. 0 0K y 3 K. K 0K 0 0. O y Δx Δx. Δx 4 5, T T The dffcult part s to defne and mplement ths knd of nstrument matrx for each. We have tred dfferent methods of ths n model B1-B10, whout much success. It mght be that we have omted varables, and that our attempts wll be no use wh ths model. We have an unbalanced panel. Ths makes the algebra more dffcult as we can not use kroneker products. But stata handles ths. Mssng observatons are handled by droppng the rows for whch there are no data, and fllng nn zeroes n columns where mssng data would be requred. I t =Index set of ndvduals whch are observed n perod t. t =1,.T P=Index set of perods where ndvdual s observed =1, N T the number of perods when at least one ndvdual s observed. N s the number of ndvduals whch are observed at least one perod. V defne D as a (NxT) matrx whose element (,t) s Sde 1 av 0
D = 1f 0 f ndvdual s observed n perod t ndvdual s not observed n perod t t = 1,..N = 1,...T In model Model 3..B1 the genune dsturbance follows an AR(1) prosess. Sargan test of over-dentfyng restrctons s rejected. Possbly due to heteroskedastcy. The presence of second order autocorrelaton would mply that the estmates are nconsstent. Model 3..B s smlar to B1 but we now have computed robust standard errors, taken nto account that we suspect heteroskedastcy.we see that the coeffcents are the same, as they should be, and the (robust) standard errors are larger. But we stll suspect that the estmates are nconsstent, because of the presence of second order autocorr. Model 3..B3. Areallo Bond recommends one step, but we see that Sagran test s not rejected and the autocorrelaton test says there s no frst order autocorrelaton. But the estmates may stll be nconsstent, because of the presence of second order autocorrelaton. We also note that several of the coeff. have changed, one has even swched sgn. Model 3..-B4 We use two lags of the dependent varable, but s not sgnfcant for lag. The other results do not dffer much. Model 3..-B5 and B6 Here we use both the 1 dfference and the lagged varable. The results do not dffer much. Model 3..-B7-10 We now use two lags of the dependent varable. But the estmates may stll be nconsstent, because of the presence of second order autocorrelaton. xtabond n w k ys yr1980-yr1984, lags(1) Arellano-Bond dynamc panel-data estmaton Number of obs = 751 Group varable (): d Number of groups = 140 Wald ch(9) = 645.91 Tme varable (t): year Obs per group: mn = 5 avg = 5.36486 max = 7 One-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..356604.0761519 4.68 0.000.07349.505859 D1. -.511453.056485-9.71 0.000 -.6146145 -.408361 D1..3086461.08417 10.93 0.000.53934.3639988 D1..503803.0958316 5.5 0.000.3154537.6911069 D1..019560.0143097 1.37 0.17 -.0084863.0476067 D1..005486.06508 0.91 0.364 -.038461.064943 yr198 D1..043438.096175 1.46 0.144 -.0148054.10199 D1..074359.0370875.00 0.045.0015457.146961 Sde 13 av 0
D1..0918581.0444807.07 0.039.0046775.1790387 _cons -.014838.0056797 -.61 0.009 -.05970 -.003706 Sargan test of over-dentfyng restrctons: ch(7) = 83.97 Prob > ch = 0.0000 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -.79 Pr > z = 0.005 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.7 Pr > z = 0.4745 Model 3..B xtabond n w k ys yr1980-yr1984, lags(1) robust Arellano-Bond dynamc panel-data estmaton Number of obs = 751 Group varable (): d Number of groups = 140 Wald ch(9) = 433.33 Tme varable (t): year Obs per group: mn = 5 avg = 5.36486 max = 7 One-step results Robust D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..356604.1371188.60 0.009.087856.6535 D1. -.511453.1701517-3.01 0.003 -.8449164 -.177934 D1..3086461.05345 5.77 0.000.038817.4134105 D1..503803.1513647 3.3 0.001.066109.7999496 D1..019560.013986 1.40 0.16 -.0078518.04697 D1..005486.0303305 0.68 0.498 -.038898.079995 yr198 D1..043438.039568 1.09 0.74 -.03473.107148 D1..074359.0459919 1.61 0.107 -.0159065.1643784 D1..0918581.0573505 1.60 0.109 -.005468.0463 _cons -.014838.0061046 -.43 0.015 -.068031 -.008734 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -. Pr > z = 0.063 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.61 Pr > z = 0.5443 Model 3..-B3 xtabond n w k ys yr1980-yr1984, lags(1) twostep Arellano-Bond dynamc panel-data estmaton Number of obs = 751 Group varable (): d Number of groups = 140 Wald ch(9) = 618.30 Tme varable (t): year Obs per group: mn = 5 avg = 5.36486 max = 7 Two-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..65143.0559895 4.74 0.000.1554058.3748807 D1. -.410314.0430384-9.53 0.000 -.494668 -.359604 D1..563969.0351334 7.30 0.000.1875368.35571 Sde 14 av 0
D1..543633.0916815 5.93 0.000.3639307.733158 D1..003073.0099064.05 0.040.0008911.039736 D1..003441.019103 0.18 0.858 -.034105.041095 yr198 D1..0051906.06486 0.0 0.844 -.0466085.0569898 D1..005109.031803 0.64 0.519 -.0418.08844 D1..0190986.036079 0.53 0.596 -.051609.0898001 _cons -.0119679.00448 -.67 0.008 -.00753 -.0031834 Warnng: Arellano and Bond recommend usng one-step results for nference on coeffcents Sargan test of over-dentfyng restrctons: ch(7) = 3. Prob > ch = 0.4 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -1.4 Pr > z = 0.165 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.3 Pr > z = 0.7473 Model 3..-B4 xtabond n w k ys yr1980-yr1984, lags() Arellano-Bond dynamc panel-data estmaton Number of obs = 611 Group varable (): d Number of groups = 140 Wald ch(10) = 49.41 Tme varable (t): year Obs per group: mn = 4 avg = 4.36486 max = 6 One-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..3809966.0913604 4.17 0.000.019335.5600597 LD. -.0314535.037183-0.85 0.398 -.1044.041493 D1. -.558806.0595507-9.37 0.000 -.674998 -.4415633 D1..3604439.033473 10.77 0.000.948394.460483 D1..506865.110165 4.60 0.000.909451.77848 D1..0058845.0194738 0.30 0.763 -.03833.044054 D1. -.001017.03771-0.03 0.975 -.06547.06317 yr198 D1..0158584.045833 0.35 0.76 -.078953.104611 D1..0370505.0581743 0.64 0.54 -.0769689.15107 D1..047605.071393 0.60 0.549 -.097167.186881 _cons.0009947.014716 0.08 0.936 -.034491.054385 Sargan test of over-dentfyng restrctons: ch(5) = 74.97 Prob > ch = 0.0000 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -3.13 Pr > z = 0.0017 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.39 Pr > z = 0.6973 Model 3..-B5 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) Arellano-Bond dynamc panel-data estmaton Number of obs = 751 Group varable (): d Number of groups = 140 Sde 15 av 0
Wald ch(1) = 813.95 Tme varable (t): year Obs per group: mn = 5 avg = 5.36486 max = 7 One-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..5630709.109438 5.15 0.000.348604.7775376 D1. -.5534161.0561889-9.85 0.000 -.663544 -.443879 LD..309860.0751487 4.1 0.000.165714.4571489 D1..3063797.097639 10.9 0.000.480435.3647159 LD. -.05857.0503968-1.04 0.300 -.1510616.0464901 D1..68309.1195031 5.1 0.000.388609.857056 LD. -.597117.1434-4.17 0.000 -.8778661 -.3163679 D1..00449.015664 0.8 0.777 -.067.0351304 D1. -.037774.04408-1.55 0.1 -.085601.0100561 yr198 D1. -.0710787.03883 -.16 0.031 -.13558 -.00669 D1. -.081401.045751-1.91 0.056 -.1646857.00055 D1. -.080054.0513176-1.56 0.119 -.1806347.00567 _cons.0050601.0078047 0.65 0.517 -.010369.00357 Sargan test of over-dentfyng restrctons: ch(7) = 77.00 Prob > ch = 0.0000 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -3.39 Pr > z = 0.0007 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -1.3 Pr > z = 0.03. Model 3..-B6 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) robust Arellano-Bond dynamc panel-data estmaton Number of obs = 751 Group varable (): d Number of groups = 140 Wald ch(1) = 64.34 Tme varable (t): year Obs per group: mn = 5 avg = 5.36486 max = 7 One-step results Robust D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..5630709.119788 4.70 0.000.383009.7978409 D1. -.5534161.177358-3.1 0.00 -.9007918 -.060403 LD..309860.10698.58 0.010.0741357.5455846 D1..3063797.0547069 5.60 0.000.199156.413603 LD. -.05857.067917-0.77 0.441 -.1854099.0808384 D1..68309.1694083 3.68 0.000.907968.954865 LD. -.597117.187489-3.19 0.001 -.964118 -.301159 D1..00449.0144535 0.31 0.759 -.03899.037575 D1. -.037774.060604-1.45 0.147 -.0888499.013305 yr198 Sde 16 av 0
D1. -.0710787.0357855-1.99 0.047 -.14117 -.0009405 D1. -.081401.0470945-1.73 0.085 -.1735437.0110635 D1. -.080054.0564116-1.4 0.156 -.1906187.0305108 _cons.0050601.00944 0.54 0.591 -.0134075.03576 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -3.3 Pr > z = 0.001 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -1.5 Pr > z = 0.099 Model 3..-B7 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) twostep Arellano-Bond dynamc panel-data estmaton Number of obs = 751 Group varable (): d Number of groups = 140 Wald ch(1) = 1060.48 Tme varable (t): year Obs per group: mn = 5 avg = 5.36486 max = 7 Two-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..467055.0745046 6.7 0.000.30999.6130518 D1. -.4870518.0470883-10.34 0.000 -.579343 -.3947605 LD..39611.0611649 3.9 0.000.1197401.359501 D1..9986.040350 5.53 0.000.1439136.300836 LD..05494.0500748 1.05 0.94 -.0456507.1506391 D1..600489.1031834 5.8 0.000.39853.80747 LD. -.43655.118918-3.55 0.000 -.6554304 -.1893007 D1..0081.0107633 0.6 0.794 -.018834.039079 D1. -.043003.001913 -.13 0.033 -.085945 -.003446 yr198 D1. -.065143.07014 -.41 0.016 -.1181041 -.01183 D1. -.067189.0310614 -.16 0.031 -.180081 -.006497 D1. -.0738373.0354618 -.08 0.037 -.1433411 -.0043335 _cons.0007818.0053805 0.15 0.884 -.0097637.011374 Warnng: Arellano and Bond recommend usng one-step results for nference on coeffcents Sargan test of over-dentfyng restrctons: ch(7) = 37.14 Prob > ch = 0.095 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -.55 Pr > z = 0.0108 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -1.0 Pr > z = 0.3076 Model 3..-B8 xtabond n l(0/).w l(0/).k l(0/).ys yr1980-yr1984, lags() Arellano-Bond dynamc panel-data estmaton Number of obs = 611 Group varable (): d Number of groups = 140 Wald ch(16) = 549.88 Tme varable (t): year Obs per group: mn = 4 avg = 4.36486 max = 6 One-step results Sde 17 av 0
D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..7590458.1534595 4.95 0.000.458706 1.05981 LD. -.118499.0491858 -.40 0.016 -.14653 -.018474 D1. -.664705.0683354-9.17 0.000 -.7604053 -.495357 LD..4450418.1093473 4.07 0.000.30751.6593584 LD. -.1459958.0759505-1.9 0.055 -.94856.008644 D1..355865.0379609 9.36 0.000.808846.496884 LD. -.0810551.0601376-1.35 0.178 -.19896.036814 LD. -.0184798.04759-0.44 0.66 -.101339.0643794 D1..6353047.1386783 4.58 0.000.3635001.907109 LD. -.8009587.1938173-4.13 0.000-1.180834 -.410837 LD..040576.1563103 1.31 0.19 -.10305.51040 D1..0108957.0159 0.49 0.63 -.03531.0543146 D1. -.07497.0370657-0.61 0.539 -.095397.0498978 yr198 D1. -.0338001.050975-0.66 0.507 -.1337044.0661041 D1. -.0194175.0673381-0.9 0.773 -.1513978.11568 D1. -.0011615.084187-0.01 0.989 -.166165.1638419 _cons -.0004955.0150878-0.03 0.974 -.0300669.09076 Sargan test of over-dentfyng restrctons: ch(5) = 59.5 Prob > ch = 0.0001 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -4.6 Pr > z = 0.0000 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.11 Pr > z = 0.9096 Model 3..-B9 xtabond n l(0/).w l(0/).k l(0/).ys yr1980-yr1984, lags() robust Arellano-Bond dynamc panel-data estmaton Number of obs = 611 Group varable (): d Number of groups = 140 Wald ch(16) = 647.69 Tme varable (t): year Obs per group: mn = 4 avg = 4.36486 max = 6 One-step results Robust D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..7590458.134198 5.66 0.000.4961561 1.01935 LD. -.118499.0457147 -.59 0.010 -.078491 -.086506 D1. -.664705.190668-3.9 0.001-1.000173 -.57678 LD..4450418.1795079.48 0.013.09318.7968707 LD. -.1459958.0873153-1.67 0.095 -.3171306.051389 D1..355865.0601116 5.91 0.000.374698.4731031 LD. -.0810551.074481-1.09 0.76 -.70374.06497 LD. -.0184798.03538-0.57 0.570 -.08531.045934 D1..6353047.177370 3.58 0.000.876654.989439 LD. -.8009587.6686-3.05 0.00-1.315814 -.861035 LD..040576.16445 1.4 0.14 -.117857.5597 D1..0108957.0175574 0.6 0.535 -.035161.0453075 D1. -.07497.031617-0.73 0.467 -.084016.0385 yr198 D1. -.0338001.041608-0.81 0.417 -.1153503.0477501 Sde 18 av 0
D1. -.0194175.0558735-0.35 0.78 -.18974.090095 D1. -.0011615.073711-0.0 0.987 -.145635.1433095 _cons -.0004955.016406-0.04 0.969 -.05707.04797 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -3.95 Pr > z = 0.0001 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.10 Pr > z = 0.906 Model 3..-B10 xtabond n l(0/).w l(0/).k l(0/).ys yr1980-yr1984, lags() twostep Arellano-Bond dynamc panel-data estmaton Number of obs = 611 Group varable (): d Number of groups = 140 Wald ch(16) = 1059.4 Tme varable (t): year Obs per group: mn = 4 avg = 4.36486 max = 6 Two-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD..719585.08744 8.8 0.000.550963.89954 LD. -.0968684.077448-3.49 0.000 -.15147 -.044896 D1. -.554483.0568186-9.75 0.000 -.6656107 -.448859 LD..408884.0935179 4.31 0.000.195968.5861801 LD. -.133653.053101 -.51 0.01 -.373413 -.09189 D1..791604.045515 6.13 0.000.18995.3683685 LD. -.0196619.055974-0.36 0.7 -.18047.0887189 LD. -.04709.063051-1.79 0.073 -.098649.0044649 D1..586981.1170406 4.98 0.000.353308.810935 LD. -.6633483.143779-4.61 0.000 -.94515 -.3815466 LD..19541.1191 1.79 0.074 -.007148.44669 D1..004616.013031 0.35 0.77 -.01616.0304936 D1. -.04347.047054-1.76 0.079 -.091849.0049945 yr198 D1. -.054147.033349-1.6 0.105 -.1157899.0109604 D1. -.030466.0419099-0.76 0.444 -.1141886.0500954 D1. -.034731.05315-0.65 0.514 -.1388994.069453 _cons.00866.00907 0.5 0.801 -.0154946.000678 Warnng: Arellano and Bond recommend usng one-step results for nference on coeffcents Sargan test of over-dentfyng restrctons: ch(5) = 31.68 Prob > ch = 0.1673 Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -3.48 Pr > z = 0.0005 Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.5 Pr > z = 0.8048 Lookng at the resduals=predcted_y - y, does not seem lke we can assume standard assumptons of normaly and constant varance. Sde 19 av 0
Densy 0.1..3.4.5-4 - 0 res -4-0 1976 1978 1980 198 1984 year res Medan splne -4-0 -1 -.5 0.5 Lnear predcton res Medan splne. correlate, _coef LD. LD. D. LD. LD. D. LD. LD. n n w w w k k k -------- LD. 1.0000 LD. -0.4966 1.0000 D1. 0.0185-0.676 1.0000 LD. 0.488-0.088-0.769 1.0000 LD. -0.1588-0.0786 0.6073-0.544 1.0000 D1. -0.0416-0.0503-0.0785 0.100 0.0040 1.0000 LD. -0.5603 0.779 0.03-0.3463 0.0884-0.639 1.0000 LD. -0.603-0.597 0.0791-0.135-0.036 0.655-0.1719 1.0000 D1. 0.0966-0.0180-0.538 0.503-0.1663 0.0837-0.1507-0.0715 LD. -0.3557-0.006 0.7719-0.8535 0.5145-0.0846 0.683 0.198 LD. 0.1330 0.034-0.5510 0.537-0.540 0.005-0.1195-0.073 D1. -0.0800 0.176-0.3169 0.1438-0.3610 0.0955-0.037 0.096 D1. -0.0505 0.1136-0.3418 0.11-0.4099 0.153-0.075 0.1953 yr198 D1. -0.0930 0.1934-0.635 0.0739-0.443 0.0434 0.086 0.0797 D1. -0.0177 0.1196-0.1840-0.0008-0.4918-0.0791 0.0910 0.081 D1. -0.073 0.188-0.3305 0.093-0.5857-0.0790 0.14 0.0951 _cons 0.1179-0.78 0.353-0.0558 0.5535 0.314-0.538-0.045 D. LD. LD. D. D. D. D. D. ys ys ys yr1980 yr1981 yr198 yr1983 yr1984 -------- D1. 1.0000 LD. -0.6909 1.0000 LD. 0.148-0.6316 1.0000 D1. 0.3594-0.63 0.160 1.0000 D1. 0.368-0.1718 0.0343 0.8633 1.0000 yr198 D1. 0.0774-0.019 0.1796 0.8038 0.8670 1.0000 D1. -0.1406 0.0445 0.605 0.693 0.73 0.919 1.0000 D1. -0.135-0.0656 0.3313 0.6807 0.6959 0.8886 0.9577 1.0000 _cons 0.14 0.0330-0.90-0.6376-0.64-0.8055-0.8860-0.900 _cons -------------+--------- _cons 1.0000. Sde 0 av 0