Problem 3.1: Error autotocorrelation and heteroskedasticity Standard variance components model:

Transcription

1 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

2 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 Sde av 0

3 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 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 ρ 1 ρ 1 ρ Sde 3 av 0

4 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 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

5 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

6 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 = Group varable (): dcode Number of groups = 4134 R-sq: whn = Obs per group: mn = 1 between = avg = 4.6 overall = max = 1 F(6,14867) = corr(u_, Xb) = Prob > F = ln_wage Coef. Std. Err. t P> t [95% Conf. Interval] age age tenure not_smsa uno south _cons sgma_u sgma_e rho (fracton of varance due to u_) F test that all u_=0: F(4133, 14867) = 8.31 Prob > F = (We note that only 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

7 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 and u between and unon s 0,079. The correlatons between the endogenous varable and the nstruments are and So they are not very good nstruments.. corr tenure south unon (obs=19007) Sde 7 av 0

8 tenure south unon tenure south uno corr res3 not_smsa unon (obs=19007) res3 not_smsa unon res not_smsa uno IV-modell 3: xtvreg ln_wage age* not_s (tenure = south unon), fe Fxed-effects (whn) IV regresson Number of obs = Group varable: dcode Number of groups = 4134 R-sq: whn =. Obs per group: mn = 1 between = avg = 4.6 overall = max = 1 F(4138,14869) = corr(u_, Xb) = Prob > F = ln_wage Coef. Std. Err. t P> t [95% Conf. Interval] tenure age age not_smsa _cons sgma_u sgma_e rho (fracton of varance due to u_) F test that all u_=0: F(4133,14869) = 1.44 Prob > F = Instrumented: tenure Instruments: age age not_smsa south unon. correlate, _coef tenure age age not_smsa _cons tenure age age not_smsa _cons Note: corr(u_, Xb) = 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

9 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, 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 = Group varable: dcode Number of groups = 4134 R-sq: whn = Obs per group: mn = 1 between =. avg = 4.6 overall = max = 1 ch(4) = sd(u_ + avg(e_.))= Prob > ch = ln_wage Coef. Std. Err. z P> z [95% Conf. Interval] tenure age age not_smsa _cons Instrumented: tenure Instruments: age age not_smsa unon south Sde 9 av 0

10 . correlate, _coef tenure age age not_smsa _cons tenure age age not_smsa _cons 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 = Group varable: dcode Number of groups = 4134 R-sq: whn = Obs per group: mn = 1 between = avg = 4.6 overall = max = 1 Wald ch(4) = corr(u_, X) = 0 (assumed) Prob > ch = ln_wage Coef. Std. Err. z P> z [95% Conf. Interval] Sde 10 av 0

11 tenure age age not_smsa _cons sgma_u sgma_e rho (fracton of varance due to u_) Instrumented: tenure Instruments: age age not_smsa south unon. correlate, _coef tenure age age not_smsa _cons tenure age age not_smsa _cons Densy 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) Model 3..B xtabond n w k ys yr1980-yr1984, lags(1) robust Model 3..-B3 xtabond n w k ys yr1980-yr1984, lags(1) twostep Model 3..-B4 xtabond n w k ys yr1980-yr1984, lags() Model 3..-B5 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) Model 3..-B6 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) robust Model 3..-B7 xtabond n l(0/1).w l(0/1).k l(0/1).ys yr1980-yr1984, lags(1) twostep Model 3..-B8 xtabond n l(0/).w l(0/).k l(0/).ys yr1980-yr1984, lags() Model 3..-B9 xtabond n l(0/).w l(0/).k l(0/).ys yr1980-yr1984, lags() robust Model 3..-B10 xtabond n l(0/).w l(0/).k l(0/).ys yr1980-yr1984, lags() twostep 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

12 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

13 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) = Tme varable (t): year Obs per group: mn = 5 avg = max = 7 One-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD D D D D D yr198 D D Sde 13 av 0

14 D _cons Sargan test of over-dentfyng restrctons: ch(7) = Prob > ch = Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -.79 Pr > z = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.7 Pr > z = 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) = Tme varable (t): year Obs per group: mn = 5 avg = max = 7 One-step results Robust D. Coef. Std. Err. z P> z [95% Conf. Interval] LD D D D D D yr198 D D D _cons Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -. Pr > z = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = Pr > z = 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) = Tme varable (t): year Obs per group: mn = 5 avg = max = 7 Two-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD D D Sde 14 av 0

15 D D D yr198 D D D _cons 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 = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.3 Pr > z = 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) = Tme varable (t): year Obs per group: mn = 4 avg = max = 6 One-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD LD D D D D D yr198 D D D _cons Sargan test of over-dentfyng restrctons: ch(5) = Prob > ch = Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = Pr > z = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = Pr > z = 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

16 Wald ch(1) = Tme varable (t): year Obs per group: mn = 5 avg = max = 7 One-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD D LD D LD D LD D D yr198 D D D _cons Sargan test of over-dentfyng restrctons: ch(7) = Prob > ch = Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = Pr > z = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -1.3 Pr > z = 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) = Tme varable (t): year Obs per group: mn = 5 avg = max = 7 One-step results Robust D. Coef. Std. Err. z P> z [95% Conf. Interval] LD D LD D LD D LD D D yr198 Sde 16 av 0

17 D D D _cons Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -3.3 Pr > z = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -1.5 Pr > z = 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) = Tme varable (t): year Obs per group: mn = 5 avg = max = 7 Two-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD D LD D LD D LD D D yr198 D D D _cons Warnng: Arellano and Bond recommend usng one-step results for nference on coeffcents Sargan test of over-dentfyng restrctons: ch(7) = Prob > ch = Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -.55 Pr > z = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -1.0 Pr > z = 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) = Tme varable (t): year Obs per group: mn = 4 avg = max = 6 One-step results Sde 17 av 0

18 D. Coef. Std. Err. z P> z [95% Conf. Interval] LD LD D LD LD D LD LD D LD LD D D yr198 D D D _cons Sargan test of over-dentfyng restrctons: ch(5) = 59.5 Prob > ch = Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = -4.6 Pr > z = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = Pr > z = 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) = Tme varable (t): year Obs per group: mn = 4 avg = max = 6 One-step results Robust D. Coef. Std. Err. z P> z [95% Conf. Interval] LD LD D LD LD D LD LD D LD LD D D yr198 D Sde 18 av 0

19 D D _cons Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = Pr > z = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = Pr > z = 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) = Tme varable (t): year Obs per group: mn = 4 avg = max = 6 Two-step results D. Coef. Std. Err. z P> z [95% Conf. Interval] LD LD D LD LD D LD LD D LD LD D D yr198 D D D _cons Warnng: Arellano and Bond recommend usng one-step results for nference on coeffcents Sargan test of over-dentfyng restrctons: ch(5) = Prob > ch = Arellano-Bond test that average autocovarance n resduals of order 1 s 0: H0: no autocorrelaton z = Pr > z = Arellano-Bond test that average autocovarance n resduals of order s 0: H0: no autocorrelaton z = -0.5 Pr > z = 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

20 Densy res year res Medan splne Lnear predcton res Medan splne. correlate, _coef LD. LD. D. LD. LD. D. LD. LD. n n w w w k k k LD LD D LD LD D LD LD D LD LD D D yr198 D D D _cons D. LD. LD. D. D. D. D. D. ys ys ys yr1980 yr1981 yr198 yr1983 yr D LD LD D D yr198 D D D _cons _cons _cons Sde 0 av 0