Instrumental Variable Estimation of Panel Data Models with Weakly Exogenous Variables

Transcription

1 Instrumental Variable Estimation of Panel Data Models with Weakly Exogenous Variables Jörg Breitung Center of Econometrics and Statistics University of Cologne Kazuhiko Hayakawa Department of Economics, Hiroshima University Meng Qi Department of Economics, Hiroshima University March 20, 205 Preliminary & Incomplete Abstract In this paper, we propose an instrumental variables estimator for panel data models with weakly exogenous variables he model is allowed to include heterogeneous time trends besides the standard fixed effects he proposed instrumental variable estimator is constructed by removing the fixed effects (and time trends from both the model and instruments by a variant of GLS transformation We show that the proposed estimator has the same asymptotic distribution as the bias corrected fixed effects estimator when both N and, the dimensions of cross section and time series, are large Monte Carlo simulation results reveal that the proposed estimator performs well in finite samples and outperforms the conventional IV/GMM estimators using instruments in levels in many cases his research is supported by KAKENHI(

2 Introduction Using panel data in empirical studies has become much more popular than before since many panel data sets are available in these days Accordingly, many types of panel data models and estimation procedures have been proposed Among them, most basic approach is the fixed effects (FE regression model where unobserved individual specific effects are allowed to be correlated with regressors However, consistency of fixed effects estimator relies on the strict exogeneity assumption, ie, the regressors and idiosyncratic errors are uncorrelated for all periods when the length of panel data, denoted as is small Unfortunately, there are many cases in which the strict exogeneity assumption is violated A leading example is a dynamic panel model Regardless of whether the regressors besides the lagged dependent variables are strictly or weakly exogenous, or endogenous, the lagged dependent variable is correlated with the idiosyncratic errors by construction, and hence the fixed effect estimator is inconsistent when is small (cf Nickell, 98 o address this problem, estimation procedures using instrumental variables (IV have been extensively considered since the work of Anderson and Hsiao (98 hese include, among others, Holtz-Eakin, Newey and Rosen (988, Arellano and Bond (99, Arellano and Bover (995, Ahn and Schmidt (995 and Blundell and Bond (998 etc While most of these studies focus on short panels, there are cases where long panel data are available, typically in macro panels Inspired by the availability of long panel data, several papers study large N and large asymptotic properties of aforementioned estimators where N is the number of cross-sectional units Earlier papers that considered large N and large dynamic panels are Hahn and Kuersteiner (2002 and Alvarez and Arellano (2003 Hahn and Kuersteiner (2002 and Alvarez and Arellano (2003 demonstrate that, when and N are large, the fixed effect estimator is consistent but its asymptotic distribution is not centered around the true value in the context of (vector autoregressive models o correct for the bias, Hahn and Kuersteiner (2002 also proposes a bias-corrected fixed effects estimator More recently, an alternative instrumental variables estimator has been proposed by Hayakawa (2009 he novel feature of his instrumental variables estimator is that it has the same asymptotic distribution as the bias-corrected FE estimator when and N are large Since that IV estimator simply uses variables deviated from past means as instruments, as opposed to the commonly used level variables, it is quite easy to use in practice 2 From the theoretical point of view, that IV estimator has addressed the trade-off problem of using many instruments Although many instruments are required to improve efficiency, the IV estimator becomes efficient despite the same number of instruments as the parameters is used Hence, the IV estimator becomes efficient with the minimal number of instruments his property has an advantage that it does not cause a large finite sample bias induced by using many instruments hereby, the trade-off problem between the bias and efficiency of the generalized method of moments (GMM estimator is addressed: both the bias and variance of the IV estimator become small simultaneously However, while there are several nice features as above, unfortunately, the asymptotic equivalence between the IV and bias-corrected FE estimators are only proved in the context of (VAR models, which are somewhat restrictive in practice One of the purposes of this paper is to demonstrate that this equivalence result holds for more general case with additional regressors Specifically, we demonstrate that the asymptotic distributions of the IV and bias-corrected fixed See also Hayakawa (205b and Lee, Okui and Shintani (204 for its extension 2 In the Stata command xtabond2 by David Roodman, which is routinely used in empirical studies, we can estimate dynamic panel data models by that instruments 2

3 effect estimator with large N and are identical for linear panel data models including dynamic models as well as static panel data models with weakly exogenous regressors Moreover, we demonstrate that this equivalence result holds even when the errors are heteroskedastic and heterogeneous time trends are included in the model, which are not allowed in Hayakawa (2009, 205b We conduct Monte Carlo simulation to investigate the finite sample behavior of estimators Concequently, we find that the IV/GMM estimators using new instruments tend to outperform the fixed effects estimator and IV/GMM estimators using instruments in levels he rest of this paper is organized as follows In Section 2, we introduce the models and estimators In Section 3, the large N and asymptotic properties of estimators introduced in Section 2 are derived In Section 4, we carry out Monte Carlo simulation to investigate the finite sample behavior of estimators, and in Section 5, we conclude With regard to the notation, we define j j For a matrix A {a ij }, a ij denotes the (i, j element of A A 2 tr (A A ij a2 ij denotes the Euclidean norm of a matrix A 2 Model and estimators In this section, we introduce models and estimators We first consider a model with fixed effects and then consider a model with heterogeneous time trends 2 Fixed effects model Consider a panel data model with fixed effects, given by y it w itδ + η i + v it (i,, N; t,, ( where δ and w it are k vectors Errors v it are serially and cross-sectionally uncorrelated Fixed effects η i can be correlated with the regressor w it his model includes several models as special cases Static model: y it x itβ + η i + v it where δ β and w it x it and β and x it are r vectors with k r AR(p model: y it α y i,t + + α p y i,t p + η i + v it, where δ (α,, α p, w it (y i,t,, y i,t p, and δ and w it are p vectors with k p ARX(p model: y it α y i,t + + α p y i,t p + x itβ + η i + v it where δ ( α,, α p, β, wit (y i,t,, y i,t p, x it, and δ and w it are (p + r vectors with k p + r In a matrix form, the model ( can be written as y i W i δ + η i ι + v i, (i,, N (2 where y i (y i,, y i, W i (w i,, w i, ι (,, and v i (v i,, v i Define the following matrix that can be used to remove fixed effects: 0 F ι diag ( c ι, c ι 2,, c ι [ ] F ι F ι 2 F3 ι ( 0 2 F ι 22 F ι

4 0 ( if s > t {fst} ι c ι t + O t if s t ( c ι s s O s if s < t (3 where c ι2 t ( t/( t +, F ι and F 3 ι are scalars, Fι 2 is 2, F ι 22 is 2 2, and F ι 23 is 2 Multiplying F ι to (2, the model to be estimated becomes y ι i W ι iδ + v ι i, (i,, N (4 where y ι i Fι y i ( yi ι,,, yι i W ι i F ι W i ( w ι i,, wι i and v ι i F ι v i ( vi ι,, vι i with yit ι c ι t [y it (y i,t+ + + y i /( t], w ι it c ι t [w it (w i,t+ + + w i /( t] and vit ι cι t [v it (v i,t+ + + v i /( t] for t,, Note that the fixed effects η i is removed by taking a deviation from future means Next, we introduce an instrumental variable In empirical studies, (a subset of lagged level variables w i,, w it are commonly used as instruments Instead of using variables in levels, Hayakawa (2009, 205b suggest to use variables deviated from past means o introduce variables deviated from past means, let us define B ι diag ( c ι,, c ι 2, c ι ( {b ι st} (5 B ι is obtained by rotating Fι he mathematical relationship between Fι and Bι is given in (37 in the appendix Using this, we define an instrumental variable Z ι i B ι W i (z ι i2,, zι i where 3 [ z ι it c ι t+ w it w ] i,t + + w i, (i,, N; t 2,, (6 t Note that the first period is lost due to the difference property of the transformation matrix (5 Since E(z ι is vι it 0 for 2 s t holds, we can ( construct moment conditions from this Specifically, we consider the moment conditions E t2 zι it vι it 0 he corresponding instrumental variable estimator is given by ( N δ ι ( N IV (7 t2 z ι itw ι it t2 How the moment conditions derived? z ι ity ι it In Hayakawa (2009, it is shown in the context of AR(p models that z ι it has the same structure as the infeasible optimal instruments which leads to efficient estimation Here, we provide an alternative explanation how the moment conditions E (z ι it vι it 0 are derived For this, let us 3 Compared with the form given in Hayakawa (2009, the coefficient c t+ is slightly different However, this is inconsequential and does not affect the main result of this paper 4

5 define two variables ṙ it and r it for some r it such that ṙ it r it (r i,t + + r i /(t and r it r it (r i,t+ + + r i /( t Note that ṙ it is a variable deviated from past means while r it is a variable deviated from future means Hence, when r it v it, r it and vit ι are related such that r it vit ι /cι t We demonstrate that the moment conditions E (z ι it vι it 0 can be obtained from the fixed effects model: (y it ȳ i (w it w i δ + (v it v i, (i,, N; t,, (8 where ȳ i t y it, w i t w it and v i t v it Note that, after some algebra, (w it w i and (v it v i can be written as w it w i t v it v i t ẇ it + t ẅ it, v it + t v it Hence, the covariance between the regressors and error term in (8 becomes E [(w it w i (v it v i ] (t 2 ( t(t 2 E(ẇ it v it + 2 E(ẅ it v it ( t(t ( t2 + 2 E(ẇ it v it + 2 E(ẅ it v it 0 his non-zero correlation is the reason why the fixed effect estimator is inconsistent when is small However, among the four terms, the third term has zero mean E(ẇ it v it 0, which can be used to consistently estimate δ even when is small Multiplying c ι t+ cι t to this moment condition in order to account for time series heteroskedasticity, we obtain c ι t+ cι te(ẇ it v it E (z ι it vι it 0 his indicates that the proposed moment conditions are derived from the valid part (ie, no correlation of the moment conditions of the fixed effects estimator 22 rend model Next, we consider a panel data model with usual fixed effects and heterogeneous time trends, given by y it w itδ + η i + λ i t + v it, (i,, N; t,, In this model, both η i and λ i can be correlated with w it Panel data models with heterogeneous time trends are studied by, say, Wansbeek and Knaap (999, and Phillips and Sul (2007 etc In a matrix form, this model can be written as y i W i δ + η i ι + λ i τ + v i, (i,, N (9 where τ (, 2,, o remove both η i and λ i, we need to multiply a matrix that is orthogonal to both ι and τ While there are several matrices that achieves this (eg the second differences, we consider the following matrix: 2( 2 2 2( ( ( ( ( ( ( ( ( 2 F τ F τ ( ( ( 2 0 2( 4 2( 4+3 2( ( 2 2 2( 2+3 2

6 [ ] F τ F τ 2 F τ F τ 22 F τ 23 {fst} τ 2c τ s [ 2( s +3(t s ] ( s( s O 0 ( if s > t c τ t + O t if s t ( ( s + O t s if s < t ( s 2 (0 ( where F τ diag ( c τ, cτ 2,, cτ 2, c τ t ( t ( t / ( t + ( t + 2, F τ and F τ 3 are 2 2, F τ 2 is 2 4, F τ 22 is 4 4, and F τ 23 is 4 2 he matrix F τ is obtained as a GLS transformation of second differences he formal derivation of F τ is provided in appendix Multiplying (0 to (9, we have the following transformed model y τ i W τ i δ + v τ i, (i,, N (2 where y τ i F τ y i (y τ i,, yτ i 2, W τ i F τ W i (w τ i,, wτ i 2, and v τ i F τ v i (v τ i,, vτ i 2 he tth row of y τ i, Wτ i and v τ i are given by y τ it f τ tty it + f τ t,t+y i,t+ + + f τ t y i, w τ it f τ ttw it + f τ t,t+w i,t+ + + f τ t w i, v τ it f τ ttv it + f τ t,t+v i,t+ + + f τ t v i, where fst τ is defined in ( Next, to introduce an instrumental variables, we define 2( 2+3 2( ( 4+6 2( 4+3 2( B τ B τ ( 2 2( ( ( ( ( ( ( {b τ st} 2( ( ( ( ( where B τ diag ( c τ 2, c τ 2, cτ Note that B τ can be obtained by rotating F τ (see (37 in the appendix Using this, we define an instrumental variable Z τ i B τ W i (z τ i3,, zτ i where its tth row is given by z τ it b τ t 2,tw it + b τ t 2,t w i,t + + b τ t 2,w i, (i,, N; t 3,, (4 with b τ st being defined in (3 Note that the first two periods are lost due to the difference property of the transformation matrix B τ Since E (z τ is vτ it 0, (3 s t 2 holds, we can ( construct moment conditions from 2 them Specifically, we consider the moemnt conditions E t3 zτ it vτ it 0 he corresponding instrumental variable estimator is given by ( N ( 2 N 2 δ τ IV t3 z τ itw τ it t3 z τ ity τ it (3 (5 6

7 23 Unified model o derive the asymptotic properties of the proposed IV estimators (7 and (5 in the next section, we formulate the above two models ( and (9 in a unified framewrok For this, let us define a variable d such that d corresponds to the FE model while d 2 corresponds to the trend model Also, let us define C and F such that C ι and F F ι for the FE model, and C (ι, τ and F F τ for the trend model hereby, the case (d, d, F, C (,, F ι, ι corresponds to the FE model while (d, d, F, C (2, 2, F τ, (ι, τ corresponds to the trend model Note that F has the properties such that F C 0, F F I d and F ( F Q I C C C C I R, (6 { R ι ι FE model 2(2 + ( ι ι + 2 ( ( + τ τ 6 ( (ι τ + τ ι trend model Using these, the models ( and (9 can be written as y i W i δ + C η i + v i, (i,, N (8 Multiplying F to (8, we have the following transformed model (7 y i W i δ + v i (9 where y i F y i (y i,, y id W i F W i (w i,, w id and v i F v i (v i,, v id he tth row of (9 can be written as y it w itδ + v it, (i,, N; t,, d (20 Note that the models (4 and (2 are the special cases of (20 Similarly, let z it, (i,, N; t d +,, denote zι it ( for FE model given by (6, and z τ it for the trend model given by (4, respectively Since E d td+ z it v it 0, we have the following instrumental variable estimator δ IV ( N d td+ z itw it ( N d td+ z ity it (2 Note that the previous two estimators (7 and (5 are the special case of (2 In order to compare the IV estimator with the FE estimator in the next section, we further reformulate the above model and estimator Since the first and last d periods are lost due to difference properties of F and B, the middle 2d periods are used in estimation Hence, the model in a matrix form becomes ẏi Ẇ i δ + v i, (i,, N (22 ( ( where ẏi yi,d+,, y i d, Ẇ i (w i,d+,, w i d and v i vi,d+,, v i d We further reformulate (22 in terms of y i, W i and v i For this, let us define L (0 d d, I d hen, by noting that L y i (y i,d+,, y i and L W i (w i,d+,, w i, the model (22 can be written as F d L y i F d L W i δ + F d L v i, (i,, N (23 7

8 Similarly, by using K (I d, 0 d d and K W i (w i,, w id, we have Ż i (z i,d+,, z i d B d K W i where B ( denotes B ι for the FE model and Bτ for trend model Using these, the d moment conditions E td+ z it v it 0 can be written as E(W i K B d F d L v i 0, and the IV estimator (2 can be written as ( N ( N [ N ] δ IV Z i W i Z i y i W ik B d F d L W i N W ik B d F d L y i (24 In the next section, we compare the asymptotic properties of this IV estimator with that of the FE estimator given by [ N ] N δ F E W iq W i W iq y i (25 where Q is defined in (6 3 Asymptotic properties In this section, we derive the asymptotic properties of the IV estimator (24 and FE estimator (25 when N and are large We first make the following assumptions Assumption he error term v it are serially and cross-sectionally uncorrelated and satisfy E(v it w it,, w i, η i 0 (26 Assumption 2 he regressor w it follows the process: { µ w it i + ξ it FE model µ i + κ i t + ξ it trend model (27 where E (ξ it 0, E ( ξ it ξ i,t+s Γi,s and l Γ i,l < for all i Also, for all i, assme that E(ξ it v is 0 for t s and E(ξ it v is ϕ i,t s 0 for t > s where l ϕ i,l < µ i and κ i are uncorrelated with v it for all i and t, but can be correlated with η i and λ i in an unrestrected manner Assumption 3 As N,, N N N d td+ N d td+ ξ it ξ it p Γ 0, (28 ξ it v it d N (0, Ω (29 where Γ 0 lim N N N Γ i0 and Ω lim N N N E(v2 it ξ itξ it Assumption indicates that the regressor w it is weakly exogenous he correlation structure between regressors and errors are specified in Assumption 2 Assumption 3 is a high-level assumption that can be used to derive the large N and asymptotic properties More primitive assumptions can be found in Phillips and Moon (999 he following Lemma is useful to understand the relationship between δ F E and δ IV 8

9 Lemma Let Assumptions and 2 hold hen, N (a W N ik B d F d L W i N ( log W N iq W i + O p N (b W iq v i N d ( N ξ it v it O p N N (c N N td+ W ik B d F d L v i N N d td+ (30 (3 ξ it v it + o p ( (32 Lemma (a indicates that the denominators of δ F E and δ IV are asymptotically equivalent when is large Also, comparing (b and (c, we find that the second term of the right-hand side makes a significant difference in IV and FE estimators When N/ converges to a non-zero constant, the second term becomes O p (, and because of this, the asymptotic distribution of δ F E is not centered around the true value as shown in heorem below his bias is due the incidental parameter problem Contrary to the FE estimator, the second term of (c vanishes asymptotically Hence, as shown in heorem 3 below, the asymptotic distribution of δ IV is centered around the true value Specifically, the asymptotic distributions of δ F E and δ IV are given in the following theorems heorem Let Assumptions,2 and 3 hold Also assume that N/ κ, (0 < κ < hen the asymptotic distribution of δ F E as N, is given by N ( δf E δ d N (b, Γ 0 ΩΓ 0 where b κγ 0 h and h plim N, N N E(W iq v i his result implies that the asymptotic distribution of the FE estimator is not centered around the true value due to the bias caused by the incidental parameter problem o correct for this bias, we consider a bias-corrected FE estimator: δ BCF E δ F E Γ 0 h (33 where Γ 0 N N W i Q W i and h is a consistent estimtor of h his bias correction is not always possible in practice and feasibility depends on the model specification For instance, if the model is assumed to be AR(, then, it is possible to correct the bias as proposed in Hahn and Kuersteiner (2002 However, for other cases, say, for a model with weakly exogenous regressors, bias-correction is infeasible unless a specific form is assumed for the regressors, which is undesirable in practice, since the form of bias depends on the correlation structure between the regressors and errors Apart from the feasibility, the asymptotic distribution of bias-corrected FE estimator is given in the following theorem heorem 2 Let Assumptions,2 and 3 hold hen the asymptotic distribution of δ BCF E as N, is given by N ( δbcf E δ d N (0, Γ 0 ΩΓ 0 Finally, the asymptotic distribution of the IV estimator is given in the following theorem heorem 3 Let Assumptions,2 and 3 hold hen, the asymptotic distribution of δ IV as N, is given by N ( δiv δ d N (0, Γ 0 ΩΓ 0 9

10 his theorem implies that δ IV has the same asymptotic distribution δ BCF E when both N and are large Moreover, although we consider large N and large panels, δ IV is still consistent (though not efficient for large N and small panels whereas δ F E is inconsistent in such cases 4 Monte Carlo simulation In this section, we investigate the finite sample properties of the proposed estimators in the context of dynamic panel data models with/without time trends 4 Design he data are generated as y it αy i,t + βx it + η i + φλ i t + v it, x it ρx i,t + τ η η i + φτ λ λ i t + θv i,t + e it Note that the case with φ 0 corresponds to the FE model while that with φ corresponds to the trend model In a matrix form, this can be written as ( ( ( ( ( ( y it α βρ y i,t ( + βτ η ( + βτ + η i +φ λ v it + βθv i,t + βe it λ i t+ x it 0 ρ x i,t τ η τ λ θv i,t + e it or p it Φp i,t + c η η i + φc λ λ i t + ε it (34 where p it (y it, x it, c η (+βτ η, τ η, c λ ( +βτ λ, τ λ, ε it (v it +βθv i,t +βe it, θv i,t + e it and ( α βρ Φ (35 0 ρ Alternatively, p it can be written as in a component form: where p it a i + φb i t + ζ it, (36 ζ it Φζ i,t + ε it a i (I Φ c η η i (I Φ Φ (I Φ c λ λ i, b i (I Φ c λ λ i, [ ( + β 2 θ 2 ] σv 2 + β 2 σe 2 βθ 2 σv 2 + βσe 2 V ar (ε it βθ 2 σv 2 + βσe 2 θ 2 σv 2 + σe 2 Data for y it and x it are generated from (36 For the sample size, we consider 0, 25, 50, 00 and N 00, 250 For parameter values, we consider α 04, 08, β, ρ 05, θ 02, τ η 05, τ λ 05 v it, e it, η i and λ i are independently generated as v it N (0, σv, 2 e it N (0, σe, 2 η i N (0, ση 2 and λ i N (0, σλ 2 with σ2 v σe 2 06, ση 2, 5 and σλ 2 We report the median bias, interquartile range (IQR and median absolute error (MAE based on 2,000 replications 0

11 42 Estimators to be compared For the estimators, in addition to the FE and IV estimators given in (24 and (25, respectively, we also consider another estimators including IV estimator using instruments in levels and the GMM estimators using instruments in levels and new instruments since we can expecet efficiency gain when is not so large 4 he IV estimator using instruments in levels can be written as ( N ( d N d δ IV (L w it w it w it yit t t With regard to the GMM estimator, we consider the moment conditions given by E(Z iv i 0 where Z i diag(z i,d 0 +,, z i d and v i (v i,d 0 +,, v i d he corresponding one-step GMM estimator is given by ( N ( N ( N δ GMM W i Z i Z iz i Z iw i ( N d td 0 + W W ( N ( N i Z i Z iz i Z iy i ( t Z t Z t Z t Z t W t d td 0 + W ( t Z t Z t Z t Z t y t where W t (w t,, w Nt, Z t (z t,, z Nt and y t (yt,, y Nt For the choice of z it, we consider two types he first is to use variables in levels such that z it (w i,t l,, w it with d 0 0 he second is to use the new instruments transformed by a matrix B ι or Bτ such that z it (w i,t l,, w it with d 0 d For the choice of the lag of instruments, we consider l, 3 In the tables, instruments in levels with l, 3 are denoted as LEV and LEV3, respectively, while the new instruments with l, 3 are denoted as BOD and BOD3, respectively For the computation of standard errors, we use those obtained under large N and fixed since they are more accurate than those obtained under large N and large (see Hayakawa, 205a 43 Results Simulation results are provided in ables -4 We first consider the model with fixed effects only From ables and 2, we find that the FE estimator for α is severly biased when 0 However, as gets larger, the bias becomes small as expected since the FE estimator is consistent when is large However, in terms of accuracy of inference, the sizes are severly distorted even when is large, say, 00 his is because the asymptotic distribution of the FE estimator is not centered around the true value due to the incidental parameter problem Also, note that increase in N does not reduce the bias since the bias of FE estimator does not depend on N With regard to the FE estimator of β, the performance is better than those of α However, it still shows some bias and size distortions his result implies that the FE estimator does not work even when is large Also, note that a widely acceptable bias-correction method is not 4 A bias corrected FE estimator is not compared since it is not available in the current case where the regressors is weakly exogenous

12 available since the regressor is weakly exogenous 5 With regard to the IV and GMM estimators, in terms of MAE, the IV estimators using instruments in levels perform worst among the four estimators mainly due to the large dispersions With regard to the remaining three estimators, they perform very similarly in terms of MAE when 0 However, as gets larger, IV and GMM estimators using new instruments outperform the GMM estimator using instruments in levels With regard to the choice fo IV or GMM estimators using new instruments, it is observed that GMM estimator tends to slightly smaller MAEs than IV estimator In terms of accuracy of inference, IV and GMM estimators using new instruments have almost correct empirical sizes in all cases while the GMM estimator using instruments in levels have large size distortions especially when 0 and α 08 With regard to the effects of lags of instruments l, we find that the efficiency of GMM estimator using instruments in levels substantially depends on l Comparing the IQRs with l and 3, the reduction of dispersion with l 3 is substantial though it induces many instruments Contrary to IV/GMM with instruments in levels, the effcts of l in GMM with new instruments are minor and the IQRs are relatively smaller than those of GMM with instruments in levels his result is consistent with the theoretical implication that using new instruments leads to efficient estimation Considering overall performance, we may conclude that the IV estimator or GMM estimator using new instruments with l tend to perform best in many cases Next, we consider the models with both fixed effects and heterogeneous time trends he results are provided in ables 3 and 4 Compared with the models with fixed effects only, the FE estimator is severly biased when is small in this model too, and the magnitude of bias is larger his also can be seen in the substantial size distortions even for a large 00 his implies that the FE estimator deteriorates further if time trends are included in the model With regard to the IV and GMM estimators, IV estimator using instruments in levels perform poorly compared with other estimators However, contrary to the previous model, other three IV and GMM estimators perform poorly when 0 Compared with the previous model with fixed effects, the dispersion is much larger when 0 However, the performances of these estimators improve as gets larger When 25 or larger, three estimators perform reasonably well when α 04 while more than 50 is required when α 08 For the relative performance among the three estimators, we find that the GMM estimator using instruments in levels perform best when 0 However, for all other cases, the GMM estimator using new instruments perform best 5 Conclusion In this paper, we have proposed a new instrumental variable estimator for panel data models including static and dynamic models with weakly exogenous variables and with fixed effects and/or heterogeneous time trends We showed that the new IV estimator is consistent regardless of the size of as long as N is large Furthermore, we showed that the new IV estimator has the same asymptotic distribution as the bias-corrected fixed effects estimator, which is sometimes infeasible, when both N and are large his implies that the new IV estimator is as efficient as the fixed effects estimator Monte Carlo simulation results revealed that the GMM estimator with new instruments tends to perform best in almost all cases for dynamic panel data models with fixed effects only When the model contains heterogeneous time trends, all the estimators 5 If the regressors are strictly exogenous, bias-corrected FE estimators such as Bun and Carree (2005 or Breitung and Hayakawa (205 can be used 2

13 do not perform well when is small, and a large is requried for reliable performance Appendix Proof of Lemma First, we decompose d matrix F as [ ] F F 2 F 3 F {f st }, (s,, d ; t,, 0 2d d F 22 F 23 where F is d d, F 2 is d 2d, F 3 is d d, F 22 is 2d 2d, and F 23 is 2d d Note that B and F have the following relationship B I d F I (37 where I 0 0 and I 2 I I I Furthermore, using (27, W i can be written as W i ι µ i + τ κ i + Ξ i C Ψ i + Ξ i where Ξ i (ξ i,, ξ i and Ψ i (µ i, κ i (a: Note the following decomposition: N N W ik B d F d L W i N N W iq W i + N N W i ( K B d F d L Q Wi (38 Using F d L C B d K C Q C 0 and (6, the second term of (38 can be further decomposed as N N W i ( K B d F d L Q Wi N N Ξ ( i K B d F d L I Ξi + N N Ξ ir Ξ i o consider the first term of right-hand side of (39, we derive the explicit form of A K B d F d L I Using (37, F d L d F L and [ ] K I d L, L d I 2d L d, L L 0 d d 0 d 2d I d 0 2d d I 2d 0 2d d 0 d d 0 d 2d 0 d d 0 d d 0 d 2d 0 d d 0 2d d I 2d 0 2d d 0 d d 0 d 2d I d, 0 d d 0 d 2d 0 2d d I 2d (39 3

14 we have A K B d L d F L L I K I d F d I 2d L d F L L I ( K ( ( I d L F L d I 2d L d F L L I I d I d F 23 I 2d F 22 I d F 23 I 2d F d d I 2d F 22 I 2d F 22 I 2d I 2d F 22 I 2d F 23 (40 0 d d 0 d 2d I d {A ij }, (i, j, 2, 3 Next, we derive the form of each A ij Using I 2d F 22 we have I 2d F f d d 0 f d+2,d+2 f d+2,d f d+,d+ f d+,d+2 f d+,d f d, d + f d, f d+2,d + f d+,d + f d+2, f d+, [, I df 23, I 2d F 22 f d+,d f d+2,d f d d 0 f d+,d+2 f d+2,d+2 f d+,d+ 0 0 f d+, f d+2, f d, f d+,d + f d+2,d + f d, d + ] A 2 (d 2d I d F 23I 2d F 22 [ { f d, f d+,d+ f d, f d+2,d+2 + f d, f d+,d+2 f d, d +f d+,d+ f d, d +f d+2,d+2 + f d, d +f d+,d+2 a jk 2 } k l k l 2d l f d+l, f d l+, d 2d l f d+l, d +f d l+, d k k f d l+, j+f d+l,d+k f d l+, j+f d+l,d+k + f d k+, j+f d+k,d+k l ( log O, c ι l cι l+ l( l cι k cι k+ k < 3 l l c ι l cι l+ l( l (j ; k,, 2d 4c τ l+2 cτ l (3j l 3(2 3k+l 3 l(l+( l 2( l 3 2cτ k+2 cτ k (3j k 3 k(k+ (j, 2; k,, 2d for FE model for trend model ] A 3 (d d a jk 3 [ 2d I d F 23I 2d F 23 l f d+l, f d l+, d + 2d l f d+l, d +f d l+, d + { } 2d f d+l, j+ f d l+, d +k ( log O l 2 c ι l+ cι l l ( l l 2d 4c τ l+2 cτ l (3k+l 6( 3j l l, 2d l f d+l, f d l+, 2d l f d+l, d +f d l+,, (j, k for FE model l(l+( l 2( l 3, (j, 2; k, 2 for trend model ] 4

15 A 22 ( 2d 2d I 2d F 22I 2d F 22 I 2d { a jk 22 } a 22 a 2 22 a, 2d 22 0 a a 2, 2d a 2d, 2d 22 0 if j > k f d j+, d j+f d+j,d+j if j k k l f d l+, d j+f d+l,d+k if j < k ( 0 ( ( if j > k c ι j cι j+ O j + O ( j + O + if j k j 2 ( j ( 2 k j+ c ι l+ cι l l ( l l O log if j < k (j, k,, 2d 0 ( ( if j > k c τ j cτ j+2 O j + O if j k ( j k 4c τ l+2 cτ l (3j l+3(2 3k+l 3 l,l>j,l<k l(l+( l 2( l 3 2cτ j+2 cτ j (2 +j 3k 3 ( j 2( j 3 2cτ k+2 cτ k (3j k+3 k(k+ O ( log (j, k,, 2d if j < k for FE model for trend model, A 23 ( 2d d I 2d F 22I 2d F 23 2d l f d+l, d f d l+, d + 2 l f d+l,d+2f d l+, d + l f d+l,d+f d l+, d + 2d l f d+l, d f d l+, 2 l f d+l,d+2f d l+, l f d+l,d+f d l+, { a jk 23 2d j } 2d j+ l f d+l,d j+f d l+, d +k f d+l,d j+f d l+, d +k + f d j+, d j+f d+j,d +k l 2d j c ι l+ cι l l ( l l cι d j+ cι d+j 2d j 4c τ l+2 cτ l (3k+l 6( 3j l 6 l l(l+( l 2( l 3 + 2cτ j+2 cτ j ( j+3k 9 ( log O ( d jl, (j,, 2d ; k for FE model ( j 2( j 3 for trend model (j,, 2d ; k, 2 We now assess the first term of (39 Using Ξ i (Ξ i, Ξ 2i, Ξ 3i where Ξ i is d k, Ξ 2i is 2d k, and Ξ 3i is d k, we have where S i Ξ ia Ξ i S i + S 2i + S 3i + S 4i + S 5i + S 6i S i Ξ iξ i, S 2i Ξ ia 2 Ξ 2i, S 3i Ξ ia 3 Ξ 3i, S 4i Ξ 2iA 22 Ξ 2i, S 5i Ξ 2iA 23 Ξ 3i, S 6i Ξ 3iΞ 3i We now evaluate each term We consider the FE model and trend model separately below 5

16 FE model From the definition of Ξ i and Assumption 2, we have E(S i E ( ξ i ξ i Γi0 O ( ( ( Using a,t 2 O log for all t, a 3 O log and Assumption 2, we have E(S 2i d a,t 2 E ( ( ξ i ξ log ( log it O Γ i,t O, t2 E(S 3i a, 3 E ( ( ( ξ i ξ log log i O Γ i, O Similarly, using a t,t 22 O(/(t + + O(/( t and a s,t 22 O(log / for all s t, we have E(S 4i d d a s,t 22 E ( ξ is ξ d it s2 t2 d t2 [ O O (log ( ( + O t + t t2 t2 a t,t 22 E ( ξ it ξ d it + ] E ( ( ξ it ξ log it + O s2 d ts+ d s2 ts+ Finally, using a t, 23 O(log / for all t, and the definition of Ξ 3i, we have E(S 5i d a t, 23 E ( ξ it ξ d ( ( log log i O Γ i, t O, t2 t2 E(S 6i E ( ξ i ξ i Γi0 O ( hus, for the FE model, we have S i 6 l S li O(log for all i and obtain N N Ξ ia Ξ i O p ( log a s,t 22 E ( ξ is ξ it d Γ t s,i (4 rend model From the definition of Ξ i, we have E(S i E ( ξ i ξ ( i E ξi2 ξ i2 2Γi0 O ( ( Since a,t 2, a 2,t 2, a k 3 and a2k 3 are O log for all t and k, using Assumption 2, we have E(S 2i E(S 3i d t2 ( log O t ( log O [ a,t 2 E ( ξ i ξ it + a 2,t 2 E ( ξ i2 ξ ] it ( log ( log Γ i,t + O Γ i,t 2 O, t2 t2 [( a, 3 + a,2 3 E ( ξ i ξ it + (a 2, 3 + a2,2 3 E ( ξ i2 ξ ] it ( log (Γ i, + 2Γ i, 2 + Γ i, 3 O 6

17 Similarly, using a t,t 22 O(/(t + + O(/( t and a s,t 22 O(log / for all s t, we have E(S 4i d d a s,t 22 E ( ξ is ξ d it s3 t3 d t3 [ O O (log ( ( + O t + t t3 a t,t 22 E ( ξ it ξ d it + ] E ( ( ξ it ξ log it + O s3 d ts+ d s3 ts+ a s,t 22 E ( ξ is ξ it d Γ t s,i Finally, using a t, 23 O(log / and a t,2 23 O(log / for all t, and the definition of Ξ 3i, we have E(S 5i d a t, 23 E ( ξ it ξ d i + a t,2 23 E ( ξ it ξ i t2 d t2 t2 ( log O (Γ i, t + Γ i, t O ( log E(S 6i E ( ξ i ξ i E ( ξi ξ i 2Γi0 O ( hus, for the trend model, we have S i 6 l S li O(log for all i and obtain N (42 N Ξ ia Ξ i O p ( log, Next, we consider the second term of (39 Let us define H i Ξ ir Ξ i hen, for the FE model, using (7, we have E(H i E(Ξ iι ι Ξ i Γ i0 + s ts+ t s E ( ξ it ξ is (Γ i,t s + Γ i,t s O ( (43 where we used ts+ Γ i,t s s l Γ i,l s l Γ i,l < l Γ i,l < For the trend model, using (7, we have H i 2 (2 + Ξ iι ι Ξ i ( H i + H 2i + H 3i + 2Ξ iτ τ Ξ i ( ( + 6 (Ξ iι τ Ξ i + Ξ iτ ι Ξ i ( Using Assumption 2 and (43, we have [( ( 2 (2 + E(H i ( E ξ it E(H 2i t 2 ( ( + E 2 ( ( + s ξ is ] (( ( tξ it s t t ( t ( s 7 s O (, sξ is E(ξ it ξ is O (,

18 E(H 3i ( 6 se(ξ ( it ξ is + te(ξ it ξ is t s t s ( 6 ( s ( t E(ξ ( it ξ is + E(ξ it ξ is O ( s t s s where we used 0 < t/ and 0 < s/ for all s and t E(H i O( for both FE and trend models, and we obtain N N ( Ξ ir Ξ i O p By combining (4, (42, and (44, we obtain N N W ik B d F d L W i Hence, for each i, we have (44 N N W iq W i + O p ( log (b: Using Q (ι, τ 0, we have N N W iq v i N N Ξ iq v i N N Ξ iv i N N Ξ ir v i he first term converges in distribution to N (0, Ω by Assumption 3 o assess the second term, let us define h i Ξ i R v i hen, for the case of FE model, using Assumption 2, we have [( E(h i ( E ] ξ it v is E (ξ it v is + E (ξ it v is t s ts+ s s ts+ t st ϕ i,t s O ( (45 where we used ts+ ϕ i,t s s l ϕ i,l s l ϕ i,l < l ϕ i,l < For trend model, we have h i 2 (2 + Ξ iι ι v i ( h i h 2i + h 3i 2Ξ iτ τ v i ( ( (Ξ iι τ v i + Ξ iτ ι v i ( Using Assumption 2 and (45, we have [( ( 2 (2 + E(h i ( E ] 2 (2 + ξ it v is ( t s (( ( 2 E(h 2i ( ( + E tξ it sv is t s ( 2 ( ( t s ϕ ( + i,t s O (, E(h 3i 6 ( s ts+ ( t s se(ξ it v is + t s te(ξ it v is s ts+ ϕ i,t s O (, 8

19 6 s ts+ ( s ϕ i,t s + 6 ( s st+ ( t ϕ i,t s O ( where we used 0 < t/ and 0 < s/( for all s and t hus, both for FE and trend models, E(h i O( and obtain N ( N Ξ ir v i N h N N O p where h N N N h i (c Noting B d K C 0, we have the following decomposition: N N W ik B d F d L v i N N N N Ξ ik B d F d L v i Ξ iv i + N N Ξ ia v i he first term converges in distribution to N (0, Ω by Assumption 3 o derive the order of the second term, let us define s i Ξ ia v i hen, using (40, s i can be decomposed as where s i s i + s 2i + s 3i + s 4i + s 5i + s 6i s i Ξ iv i, s 2i Ξ ia 2 v 2i, s 3i Ξ ia 3 v 3i, s 4i Ξ 2i A 22 v 2i, s 5i Ξ 2iA 23 v 3i, s 6i Ξ 3iv 3i o derive the variance of s i, we need to calculate V ar(s ki and Cov(s ki, s li, (k l for k, l,, 6 We consider the FE and trend models separately FE model We have V ar(s i V ar(ξ i v i O( Using a,t 2 O ( log V ar(s 2i V ar for all t, a 3 O ( log ( d t2 a,t 2 ξ i v it, we have d ( t2 a,t 2 V ar(s 3i V ar ( a 3ξ i v i (a 3 2 V ar (ξ i v i O 2 V ar (ξi v it O ( (log 2 ( (log 2 Similarly, using a t,t 22 O(/(t + + O(/( t and a s,t 22 O(log / for all s t, we have V ar(s 4i V ar [ d s2 d ts d (a t,t t2 a s,t 22 ξ is v it ] d 22 2 V ar (ξ it v it + 9 d s2 d 2 d (a s,t ts s2 ts V ar (ξ is v it (a s,t 22 2 V ar (ξ is v it,

20 O( + O ( (log 2 Finally, using a t, 23 O(log / for all t, and the definition of Ξ 3i, we have ( d d ( V ar(s 5i V ar a t, 23 ξ it v i (a t, (log V ar (ξ it v i O, t2 V ar(s 6i V ar (ξ i v i O( For the covariances, we have t2 Cov(s i, s 2i Cov(s i, s 3i Cov(s i, s 4i Cov(s i, s 5i Cov(s i, s 6i Cov(s 2i, s 3i Cov(s 2i, s 5i Cov(s 2i, s 6i Cov(s 3i, s 4i Cov(s 4i, s 5i Cov(s 4i, s 6i 0, Cov(s 2i, s 4i E d ( d t2 d t 2 t 2 2 d t2 ( d a,t 2 ξ i v it t2 d a t,t 22 ξ it v it + a,t 2 a t 2,t 2 22 E(ξ i ξ it 2 v it v it2 + d a,t 2 a t,t 22 E(ξ i ξ itvit 2 + ( (log 2 O + O ( (log 2, d Cov(s 3i, s 5i a 3a t, 23 E(ξ i ξ itvi 2 O t2 ( log Cov(s 3i, s 6i a 3E(ξ i ξ i vi 2 O, d t2 d s2 ts+ ( (log 2 Cov(s 5i, s 6i a t, 23 E(ξ it ξ i vi 2 O(log d s2 ts+ d t 2 herefore, for FE model, we have V ar(s i O ( (log 2, and ( N V ar Ξ ia v i N ( (log 2 V ar(s i O N N Hence, it follows that N N Ξ ia v i O p (log / o p ( rend model V ar(s i V ar( Using a s,t 2 O 2 s t d s2 22 ξ is v it a s,t d t 2 s+ a,t 2 a s,t 22 E(ξ i ξ isv 2 it, 2 ξ it v is V ar(ξ i v i + V ar(ξ i2 v i2 O( ( ( log, s, 2 for all t and a jk 3 O log ( 2 2d ( V ar(s 2i V ar a j,t (log 2 2 ξ i v it O s t3 20, we have, a,t 2 a s,t 2 22 E(ξ i ξ isv it v it2

21 V ar(s 3i V ar ( 2 2 s t a st 3ξ it v is s t (a st 3 2 V ar (ξ it v is + O ( (log 2 Similarly, using a t,t 22 O(/(t + + O(/( t and a s,t 22 O(log / for all s t, we have V ar(s 4i V ar [ d s3 d ts d (a t,t t3 a s,t 22 ξ is v it ] d 22 2 V ar (ξ it v it + O( + O ( (log 2 d s3 d d (a s,t ts s3 ts V ar (ξ is v it (a s,t 22 2 V ar (ξ is v it Finally, using a t,s 23 O(log /, s, 2 for all t, and the definition of Ξ 3i, we have ( 2 d V ar(s 5i V ar a t,s 23 ξ it v is + 2 d s t2 V ar(s 6i V ar s t3 ( (log V ar (ξ it v is + O (a t,s ( 2 s t For the covariances of trend model, 2 ξ it v is O( Cov(s i, s 2i Cov(s i, s 3i Cov(s i, s 4i Cov(s i, s 5i Cov(s i, s 6i Cov(s 2i, s 3i Cov(s 2i, s 5i Cov(s 2i, s 6i Cov(s 3i, s 4i Cov(s 4i, s 5i Cov(s 4i, s 6i 0,, 2 ( 2 ( d d Cov(s 2i, s 4i E a s,t 2 ξ is v it + 2 s t2 d d s t 3 t d d s t 3 s 2 3 t 2 s d s t3 ( (log 2 O t3 d a t,t 22 ξ it v it + a,t 2 a t 2,t 2 22 E(ξ is ξ it 2 v it v it2 d d s3 ts+ a,t 2 a s 2,t 2 22 E(ξ is ξ is 2 v it v it2 a s,t 2 a t,t 22 E(ξ is ξ itv 2 it + + O ( (log 2, 2 d d s s 2 2 ts ξ is v it a s,t a s,t 2 a s 2,t 22 E(ξ is ξ is 2 v 2 it Cov(s 3i, s 5i 2 2 d a s s 2 3 a t,s 2 23 E(ξ is ξ itvi 2 s + O s s 2 t3 ( (log 2, 2

22 Cov(s 3i, s 6i Cov(s 5i, s 6i 2 2 s s 2 s t3 ( a s s 2 log 3 E(ξ is ξ i s2 +vi 2 O s, + 2 d a t,s 23 E(ξ it ξ i s +vi 2 s + O(log herefore, for trend model, we have var(s i O ( (log 2, and ( N var Ξ ia v i N ( (log 2 var(s i O N N Hence, it follows that N N Ξ ia v i O p (log / o p ( Proof of heorems and 2 We first provide a proof of heorem Using (44 and Assumption 3, we have N N W iq W i N N d td+ Next, we have the following decomposition N N W iq v i N N d td+ ( ξ it ξ it + O p p Γ 0 (46 ξ it v it + N h N where h N N N h i Hence, using Assumption 3, as N, with N/ κ, (0 < κ <, we obtain N ( δf E δ d ( N ( N N W iq W i N N W iq W i N N ( κγ 0 h, Γ 0 ΩΓ 0 N N W iq v i d td+ ( ξ it v it + N N N W iq W i h N where h plim N, hn heorem 2 can be proved by noting that Γ 0 and h are consistent estimators of Γ 0 and h with large N and Proof of heorem 3 Using Lemma (a, (46 and Assumption 3, we have N N W ik B d F d L W i N N Also, using Lemma (c and Assumption 3, we have N N W ik B d F d L v i N Combining (47 and (48, we obtain the result N W iq W i + O p ( log d td+ 22 p Γ 0 (47 ξ it v it + o p ( d N (0, Ω (48

23 Derivation of F We derive the form of F ι and Fτ Although a brief derivation of Fι is given in Arellano (2003, a complete derivation is not provided Hence, we fill that gap Let us define the following matrix that takes the first difference: D 0 Multiplying D by ( and noting that D ι 0, we have D y i D W i + D v i where it is simply assumed that V ar(v i σvi 2 Since V ar(d v i σvd 2 D, the transformed error is serially correlated o correct for the serial correlation, we use the following transformation matrix, which is a GLS transformation: F ι ( D D /2 D, where (D D /2 is the upper triangular Cholesky factorization of (D D with D D o compute (D D /2, we need to derive the inverse matrix Φ ι (D D {ϕ ι st} Using the results by El-Mikkawy and Karawia (2006, we have n n+ if s t or s t n s(n s+ ϕ ι n+ if s t < n st s(n t+ n+ if s < t t(n s+ n+ if s > t where n Next, we need to compute the Cholesky factorization to Φ ι For a K K matrix A {a ij }, its Cholesky factorization is given by A LL where L (l ij is the lower triangular matrix hen using l ij, we can write the elements of A as follows: a l 2, a 2 l 2 l, a 22 l l 2 22, a 3 l 3 l, a 32 l 3 l 2 + l 32 l 22, l 33 l l l 2 33, 6 A matrix with this structure is called tridiagonal matrix 7 Arellano (2003 does not provide the details how the upper triangular Cholesky factorization can be computed 23

24 a K l K l, a K2 l K l 2 + l K2 l 22,, a KK l 2 K + + l 2 KK l ij can be solved sequentially as follows: l a, l 2 a 2 /l, l 22 a 22 l 2 2, l 3 a 3 /l, l 32 (a 32 l 3 l 2 /l 22, l 33 a 33 l 2 3 l2 32, l K a K /l, l K2 (a K2 l K l 2 /a K2, l KK a KK l 2 K l2 K,K he explicit form of F ι is obtained by letting A Φι Next, we consider a model with individual effects and heterogeneous time trends given by (9 o remove both η i and λ i from the model, we need to take second differences In terms of a model in matrix, this corresponds to multiplying by D D, we have D D y i D D W i δ + D D v i Since the transformed error is serially correlated, we consider the following GLS-type transformation matrix: F τ ( D D D D /2 D D, where ( D D D D /2 is the upper triangular Cholesky factorization of ( D D D D o compute F τ, we need to derive the inverse matrix Φτ ( D D D D {ϕ τ st } with D D D D Using the results by Chen (203, after a lengthy calculation, we have t(n t+(n t+2 (n+2(n+3 if s, t,, n 2 6(n (n+2(n+3 if s, t n 2n (n+2(n+3 if s, t n (n t+(n t+2(n t+3 3nt (n+(n+2(n+3 if s 2, t,, n 2 6( 3n+3n 2 4 ϕ τ (n+(n+2(n+3 if s 2, t n st 6(n if s 2, t n (n+2(n+3 t(n s+(n s+2(t+(n+5s 3t+3ns nt 2st+3 6(n+3(n+2(n+ if s 3,, n, t,, s 2 s(s+(n t+(n t+2( n+3s 5t+ns 3nt+2st 3 6(n+(n+2(n+3 if s 3,, n, t s,, n 2 s(s+(3n s 3ns+3n 2 2 (n+(n+2(n+3 if s 3,, n, t n s(s+(n s+ (n+2(n+3 if s 3,, n, t n where n 2 Using these and applying the algorithm of Cholesky factorization introduced above where A Φ τ, after a lengthy calculation, we obtain the explicit expression of F τ as in (0 8 A matrix with this structure is called pentadiagonal matrix 24

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27 able : Fixed effects model: α 04, β 0 N 00, ση 2 α β FE IV GMM GMM IV GMM GMM FE IV GMM GMM IV GMM GMM LEV LEV LEV3 BOD BOD BOD3 LEV LEV LEV3 BOD BOD BOD3 0 Bias IQR MAE Size Bias IQR MAE Size Bias IQR MAE Size Bias IQR MAE Size N 00, ση 2 5 α β FE IV GMM GMM IV GMM GMM FE IV GMM GMM IV GMM GMM LEV LEV LEV3 BOD BOD BOD3 LEV LEV LEV3 BOD BOD BOD3 0 Bias IQR MAE Size Bias IQR MAE Size Bias IQR MAE Size Bias IQR MAE Size

28 able (cont: Fixed effects model: α 04, β 0 N 250, ση 2 α β FE IV GMM GMM IV GMM GMM FE IV GMM GMM IV GMM GMM LEV LEV LEV3 BOD BOD BOD3 LEV LEV LEV3 BOD BOD BOD3 0 Bias IQR MAE Size Bias IQR MAE Size Bias IQR MAE Size Bias IQR MAE Size N 250, ση 2 5 α β FE IV GMM GMM IV GMM GMM FE IV GMM GMM IV GMM GMM LEV LEV LEV3 BOD BOD BOD3 LEV LEV LEV3 BOD BOD BOD3 0 Bias IQR MAE Size Bias IQR MAE Size Bias IQR MAE Size Bias IQR MAE Size