Exogeneity. Preliminary and Incomplete

Transcription

1 Estimation of Fixed Effects Models of Panel Data wh Sequential Exogeney Preliminary and Incomplete Valentin Verdier Michigan State Universy April 19, 2013 Abstract This paper considers the Fixed Effects estimation of non-linear models of panel data wh multiplicative unobserved effects and where instrumental variables are predetermined as opposed to strictly exogenous. Existing estimators for these models suffer from a weak instrumental variable problem which can cause them to be too inaccurate to be reliable. In this paper we present addional sets of restrictions which can be used for more precise estimation. Monte Carlo simulations show that using these addional moment condions improves the precision of the estimators significantly and hence should facilate the use of these models. 1

2 1 Introduction The models we consider are such that for each observation i of a random sample of large size n and each time period t of a fixed number of time periods T we can specify: E(y x,u,z )=h 0 (x,β 0 )+h 1 (x,β 0 )u (1.1) E(u z )=E(u +1 z ) t T 1 (1.2) In this model x are observed covariates and u captures the effect of unobserved covariates. z are observed instruments that do not belong to the mean equation for y once we condion on the observed and unobserved covariates x, u.weconsidercaseswherez i1 z i2... z it so that we have sequential exogeney, also called predetermined instruments. (1.2) restricts the effects of unobserved covariates to have a Fixed Effects (FE) structure in their condional mean. Indeed requires that at each time period the effects of unobserved covariates have the same mean condional on the instruments as the effects of unobserved covariates for future time periods. Dynamic linear models wh addive heterogeney are a special case of the group of models described by (1.1) and (1.2) wh x = y 1, z =[y 0,...,y 1 ], h 0 (x, β) =βx, h 1 (.,.) =1: y = β 0 y 1 + c i + ν (1.3) E(c i + ν y 1,...,y 0 )=E(c i y 1,...,y 0 ) (1.4) Here we wrote u = c i +ν. Tradionally unobserved effects wh a Fixed Effects structure have been explicely decomposed between a time constant part, sometimes called unobserved heterogeney, and a transory part. In this paper we keep a more general notation asin(1.1) and(1.2) formoreflexibily. Other special cases of the models we consider have been used to modelcountdependentvariables, such as the linear feedback model presented in Blundell, Griffh and Windmeijer (2002): 2

3 E(y y 1,...,y i0,x,...,x i1,c i )=γ 0 y 1 + exp(x θ 0 )c i t =1,...,T (1.5) In this case u = y γ 0 y 1 exp(x θ 0 ) so that y = h 0 (x,β 0 )+h 1 (x,β 0 )u which of course implies E(y x,u,z )= h 0 (x,β 0 )+h 1 (x,β 0 )u where z = {y 1,...,y i0,x,...,x i1 }. In addion E(u z ) = E(c i z ) = E(u +1 z ). Our specification also includes models for count dependent variables where covariates cannot be used as instruments, i.e. x / z,butwhereenoughinstrumentsareavailabletoidentifytheparameters of the model. An example where instruments available are lagged covariates is presented in Windmeijer (2000) 1 : y = exp(x β 0 )c i ν (1.6) E(c i ν x 1,...,x i0 )=E(c i x 1,...,x i0 ) (1.7) Multiplicative FE models have been used first to analyze count dataandadescriptionofthestate of the lerature on dynamic models of count data wh unobserved heterogeney is given in Windmeijer (2008). However the class of models we consider in this paper is very appropriate for the analysis of any data that requires the specification of a non-linear response function like binary, fractional, ordered, non-negative, corner solution response data and so on. For a binary dependent variable for instance, a dynamic prob model wh sequential exogeney in the explanatory variables could be specified: E(y y 1,...,y i0,c i,x,...,x i1 )=c i Φ(γ 0 y 1 + x θ 0 ) (1.8) Here c i /2isthecondionalprobabilyofbeinginstatey =1wheny 1 =0,x =0andalso captures a time constant unobserved propensy to be in state y =1. It is important to note as well that the generaly of our chosen specification also allows us to use 1 We present slightly different assumptions here than in Windmeijer (2000) since Windmeijer (2000) goes to great lengths to avoid making assumptions on condonal means and only consider assumptions of uncorrelation but does so at the expense of making two mistakes. One of which being that assuming that x 1 is uncorrelated wh ν does not imply E(x 1c i(ν +1 ν )) = 0 as is claimed in Windmeijer (2000). The other one will be mentioned in Section

4 models where some of the explanatory variables are endogenous but where instruments are available: E(y y 1,x,u,z )=Φ(γ 0 y 1 + x θ 0 )u (1.9) E(u z )=E(u +1 z ) (1.10) Where u is a random variable between zero and one and captures the effect onthemeanofy of unobserved explanatory variables which are not independent fromx but have the same mean condional on instruments as effects on the mean in future time periods. In comparison, non-linear models wh endogenous covariates and predetermined instruments but where unobserved effects are not multiplicative have to impose very strong parametric assumptions notonlyonthedistributionofobservedand unobserved effects condional on the instruments but also on the stochastic process that the instruments themselves follow. An example of such an approach can be foundingilesandmurtazashvili(2013). In the simpler case where one considers a dynamic model wh strictly exogenous covariates, an other group of models that allow for non-lineary and unobserved heterogeney is the group of dynamic correlated random effect (CRE) models, as discussed in Wooldridge (2005) and used in Browning, Ejraes and Alvarez (2010) for instance. The main advantage of these models is that they allow for the effects of explanatory variables to depend on unobserved heterogeney in very flexible ways. The advantage of the multiplicative unobserved effects models we present in this paper compared to dynamic CRE models is that they are not restricted to dynamic models and do not make strong distributional assumptions. In this paper we consider Fixed Effects estimation of all models thatcanbedescribedby(1.1)and (1.2). Here by FE estimation we understand any estimation method that does not make use of any other restriction on the relationship between instruments and unobserved effects than the Fixed Effects structure, i.e. E(u z ) = E(u +1 z ). FE estimators for multiplicative heterogeney models wh sequential exogeney have already been proposed based on condional moment condions derived in Chamberlain (1992) and Wooldridge (1997). However these estimators are well known for suffering from aweakinstrumentalvariableproblemwhichmakesthemtooinaccurate to be widely used in practice. In this paper we examine one solution to this problem which is to impose addional assumptions in order to obtain useful addional moment condions. The addional moment condions that we consider in 4

5 this paper are generalized versions of the addional moment condionsforthelineardynamicmodel wh addive unobserved effects presented in Arellano and Bover (1995), Ahn and Schmidt (1995) and Blundell and Bond (1998). Windmeijer (2000) considered some oftheaddionalmomentcondionswe present here, namely uncorrelation of the transory shocks, for a special case of our model. However in Windmeijer (2000) these addional moment condions are attributed to a set of assumptions which is not strong enough to support them. Hence seems useful to present them in this paper again as part of aunifyingframework. In Section 2 we will present multiplicative Fixed Effects models in more detail. In Section 3 we will discuss the estimator that is currently used and discuss addional sets of restrictions that can be used for estimation when instruments are stationary or when transory shocks are serially uncorrelated. In Section 4wewillshowusingMonteCarlosimulationsthattheproposions to address the weak instrumental variable problem of these models result in significant improvements in accuracy and hence effectively migate the weak instrumental variable problem. In Section 5wewillshowhowtoestimateandperform inference on measures of interest of the effect of covariates on themeanofy. 2 The Model We assume that we have a random sample of panel data, and that wecanwrethefollowingmodelfor arandomdrawi at a given time period t between 1 and T : E(y x,u,z )=h 0 (x,β 0 )+h 1 (x,β 0 )u (2.1) E(u z )=E(u +1 z ) (2.2) Where z is a set of instruments that increases wh time, i.e. z i1 z i2... z it so that we have sequential exogeney. For simplicy we will consider the case where y, h 0 (.,.), h 1 (.,.) andc i are scalars but all the results can be generalized to systems of equations if needed. Following the argument made in Chamberlain (1992), the modeldescribedby(2.1),(2.2) isstatistically 5

6 indistinguishable from: E( y it h 0 (x it,β 0 ) z it )=E(u it z it ) h 1 (x it,β 0 ) (2.3) E( y h 0 (x,β 0 ) z 1 )=0 t =2,...,T h 1 (x,β 0 ) (2.4) Where denotes the difference operator. Since E(u it z it )isunknownandunrestricted,(2.3)does not participate in estimating β 0.Thereforewecanrestrictourattentiontoestimatingβ 0 from (2.4). For notation we will wre: ρ t (w i,β) y h 0 (x,β 0 ) h 1 (x,β) t =2,...,T (2.5) Where w i {y,x } t=1,...,t.sothecondionalmomentrestrictionsavailableforestimation are: E(ρ t (w i,β 0 ) z 1 )=0 t =2,...,T (2.6) Chamberlain (1992) has shown that an optimal estimator would be ˆβ opt that solves: n D 1 Σ ρ t (w i,z i, ˆβ opt )=0 (2.7) Such an estimator would achieve the asymptotic information bound for estimating β 0 from these condional moment restrictions which is J = E( T D 1 t=2 Σ D ) where D Σ V ar( ρ t (w i,z i,β 0 ) z 1 ), ρ t () is defined by: E( ρt β (w i,z i,β 0 ) z 1 ), ρ T (w i,z i,β)=ρ T (w i,β) ρ t (w i,z i,β)=ρ t (w i,β) Γ,t+1 ρ t+1 (w i,z i,β)... Γ,T ρ T (w i,z i,β) t =2,...,T 1 where z i = {z 1 } t=2,...,t,γ,s Cov(ρ, ρ is z is 1 )V ar( ρ is z is 1 ) 1 s>twhere ρ = ρ (β 0 ), ρ (β) = ρ t (w i,β)and ρ = ρ (β 0 ), ρ (β) = ρ t (w i,z i,β). The intuion behind this result is that the asymptotic 6

7 information bound from all the sequential condional moment restrictions is the sum of the information bounds for each condional moment restriction once these restrictions have been orthogonalized. However such an estimator is usually not feasible whout addional assumptions since D, Σ and ρ are not observed, i.e. they are not known functions of the data andofβ 0. One could think about approximating such moment condions arbrarily well as suggested in Chamberlain (1992) or partially studied in Hahn (1997) but this introduces several new problems and therefore is left for future research. In the next section we present the estimator that is currently usedandwhataddionalassumptionscan be made to obtain useful addional moment condions. 3 Estimation of the Parameters Under the condional moment restrictions given in (2.6), any function of z 1 can be used as instruments for ρ (β) toestimateβ 0.Windmeijer(2008)forinstancerecommendstheuseofallavailable lags of the instruments in levels, in our notation this is just z t 1.Sotheestimatorthatiscommonlyusedtoestimate β 0 from the model given in (2.1) and (2.2) is: ˆβ = argmin β (Z i ρ i (β)) (Z i ρ i ( β)ρ i ( β) Z i) 1 i i Z i ρ i (β) (3.1) Where ρ i (β) =[ρ (β)] t=2,...,t, Z i =[Z i2,...,z it ], Z =[0, 0,...,z 1, 0,...,0] where the number of zeros preceding z 1 is equal to the added dimensions of z i1,...,z 2 and β is a preliminary n-consistent estimator of β 0.Theasymptoticvarianceofof n( ˆβ β 0 )is: Avar =(E(Z i ρ i (β 0 ) β It is shown in Appendix A, that this asymptotic variance is equal to: ) E(Z i ρ i (β 0 )ρ i (β 0 ) Z ρ i (β 0 ) i ) 1 E(Z i )) 1 (3.2) β Avar =(E( T t=2 D 1 Σ D ) E( T e e )) 1 (3.3) t=2 1/2 Where e t is the error term from the linear projection of D Σ on z 1 where z 1 = {z i1 Γ 1,t Σ1/2,z i2 Γ i2,t Σ1/2,...,z Σ1/2 1 }, sothatˆβ can be seen as the estimator resulting from 7

8 alinearapproximationoftheoptimalmomentcondions. This estimator is often found to be too imprecise to be reliable. In this section we show two sets of assumptions that when true can be very useful to improve the accuracy of the estimator of the parameters. 3.1 Estimation wh Stationary Instruments For the models described in (2.1), is possible in some applications that part of the instruments, denote z stat, has a time constant covariance wh the unobserved effects and atimeconstantmeansothatin addion to (2.2) we can assume: E(z stat )=µ z stat (3.4) Cov(z stat,u )=γ (3.5) (3.4) and (3.5) imply that E(z stat u )istimeconstantaswellsincee(z stat u )=γ + µ z state(u )and E(u )istimeconstantbythelawoferatedexpectationsande(u i1 z i1 )=... = E(u it z i1 )whichis implied (2.2). This in turn implies that E(z stat u )=E(z 1 stat u ) sincee(z 1 stat u 1) =E(z 1 stat u ) from (2.2). Let K stat be the dimension of z stat. We can use for estimation the (T 1) K stat addional moment condions 2 : E((z stat z stat 1) y h 1 )=0 t =2,...,T (3.6) (x,β 0 ) Example of the Linear Feedback Model An example of a model where such addional moment condions canbeusedbuthavenotbeenused in previous studies is the linear feedback model (LFM) presented in Blundell, Griffh and Windmeijer (2002). For γ 0 < 1: E(y y 1,...,y i0,x,...,x i1,c i )=γ 0 y 1 + c i µ(x,θ 0 ) (3.7) 2 Note that the moment condions E((z s stat z s 1)u stat ) =0fors 1donotconstuteusefuladdionalmoment condions since they are implied by the moment condions E((z s stat z s 1)u stat s) =0andE(z s(u stat τ u τ 1)) = 0 τ =0,...,s 1andE(z s 1(u stat τ u τ 1)) = 0 τ =0,...,s 1. 8

9 This model implies that for estimation we can use the sequence ofcondionalmomentcondionscorresponding to the condional moment condions (2.4) considered in the previous section: E( y γ 0 y 1 µ(x,θ 0 ) y 1 γ 0 y 2 y 2,...,y i0,x 1,...,x i1 )=0 t =2,...,T (3.8) µ(x 1,θ 0 ) So for this specific model we have u = y γ 0 y 1 µ(x,θ 0 ). Blundell, Griffh and Windmeijer (2002) also assumes that x is strictly stationary condional on c i. 3 This implies that E(µ(x,θ 0 ) c i )=g 1 (c i )forsomefunctiong 1 (c) thatisconstantacrosstime. Consider the difference equation given by: y = γ 0 y 1 + µ(x,θ 0 )c i ɛ. (3.9) Where ɛ = y γ 0 y 1 c i µ(x,θ 0 ).Theassociatedstationaryprocessisdefinedbys = s=1 γs 0 c iµ(x s,θ 0 )ɛ s. Then E(s c i ) = c ig 1 (c i ) 1 γ 0 since E(ɛ c i,x,x 1,...) = E( y γ 0 y 1 c i µ(x,θ 0 ) ) = 1. So if we simply assume that the deviation of y i0 from s i0 has mean zero condional on c i,wehavee(y i0 c i )= c ig 1 (c i ) 1 γ 0 E(y c i )= c ig 1 (c i ) 1 γ 0 so that t =1,...,T.Thisassumptionisthegeneralizationoftherestrictiononinialcondions made in Blundell and Bond (1998) for dynamic linear models wh addive unobserved effects. It results in the addional over-identifying moment condions: E((y 1 y 2 ) y γ 0 y 1 )=E((y 1 y 2 )c i ) t =2,...,T µ(x,θ 0 ) =0 t =2,...,T Since for this specific model these condions would not be plausible whout the stationary of x, we can also add the moment condions: 3 They do so in a different attempt to migate the weak IV problem offeestimatorsforthelfm.blundell,griffhand Windmeijer (2002) proposes a so called pre-sample mean estimator which attempts to control for unobserved heterogeney by using the average of observations on the dependent variable for many periods before the rest of the sample started as a proxy for time constant unobserved heterogeney. However this estimator suffers from two severe drawbacks which make unusable in practice: supposes one has many observations on the dependent variable before the start of the rest of the sample but most importantly the assumptions under which the pre-sample average is a good proxy for unobserved heterogeney are highly unrealistic, in particular supposes that the covariates x have a mean that is proportional to c i and restricts µ() to be the linear index exponential function. 9

10 3.1.2 Time Demeaned Instruments E((x x 1 ) y γ 0 y 1 )=0t =2,...,T 1 (3.10) µ(x,θ 0 ) In some applications might not be plausible to assume that part of the instruments is mean stationary. However similar addional moment condions as (3.6) can be obtained after time demeaning of the instruments if Cov(z stat,u )istimeconstant. Inthissectionweconsiderthecondionsnecessaryfor this to be true when z stat wre: From (2.1) and (2.2), Cov(z stat is not self mean stationary.,u )=Cov(z stat,u it ). This quanty will be time constant if we can z stat Cov(e,u it )=0 = d i + e Where d i is an unobserved time constant random vector. Indeed in this case Cov(z stat,u ) = Cov(z stat,u it )=Cov(d i,u it )whichistimeconstant. To provide more intuion regarding what assumptions could imply such a result, we can use the unobserved heterogeney decomposion of unobserved effects to wre u = c i ν wh c i and ν such that E(u z )=E(c i z )=E(u +1 z ). Therefore Cov(z stat,u )=Cov(z stat,c i ). Cov(z stat,c i ) Cov(z stat is,c i )=E(z stat = E((z stat c i ) E(z stat )E(c i ) (E(zis stat c i ) E(zis stat )E(c i )) zis stat )c i ) E(z stat zis stat )E(c i ) Hence Cov(z stat,c i )willbetimeconstantifthechangeinz stat over time is uncorrelated wh the time constant part of the unobserved effects. If this is true we can use for estimation the following addional moment condions: E(( z stat z stat 1) y h 1 )=0 t =2,...,T (3.11) (x,β 0 ) 10

11 Where z stat = z stat E(z stat ). Asimpleinformaltestofwhetherthechangeinz stat part of the unobserved effects could be to regress the change in z stat over time is uncorrelated wh the time constant over time on time period dummies and as many time constant explanatory variables as available and test the joint significance of the time constant covariates in the regression. 3.2 Serially Uncorrelated Transory Shocks In some applications might be unlikely that instruments or functionsoftheinstrumentshaveatimeconstant covariance wh unobserved effects. For instance consider the case of the linear feedback model where only time period dummy variables are used as covariates so that x = D t.thene(y y 1,...,y i0,c i )= γ 0 y 1 + µ t c i where µ t is a deterministic constant that depends on t. Even if we assume that y i0 does not deviate from the stationary process s i0 = s=1 γs 0 c iµ s ɛ i s, E(y i1 y i0 c i )= s=1 γs 0 c i(µ s+1 µ s ) so that in general y y 1 will be correlated wh c i and therefore y 1 y 2 can not be used as an instrument for the equation in level even if is time demeaned. However in such cases other addional restrictions might beavailable that would come fromrestrictions on the variance covariance matrix of u i =[u i1,...,u it ]. It is sometimes plausible to assume that the only source of serial correlation in the unobserved effects is time constant unobserved effects so that Cov(u,u is ) = Cov(u iq,u ir ) s < t,q < r 4. In general such restrictions imply T (T 1)/2 1 addional overidentifying moment restrictions which can be wrten as: y y is E( h 1 (x,β 0 ) h 1 (x is,β 0 ) )=τ 0 t, s =1,...,T, s<t (3.12) Where τ 0 is an addional parameter added to β 0 defined by τ 0 = Cov(u,u is )+E(u )E(u is ) t s which doesn t depend on t or s since E(u )isconstantby(3.5). Thisishowevernottrueinthecase of dynamic models since then some of these moment condions are already implied by (3.4) and (3.5). For dynamic models, u = y /h 1 (y 1,x )andy 1,x z so u z. Hence (3.4) and (3.5) imply E(u u is )=E(u u ir ) s, r < t so that Cov(u u is )=Cov(u u ir ) s, r < t. Therefore assuming 4 One could also consider addional restrictions of the type E(u u s) =τ(s) sothatserialcorrelationintheunobserved effects only depends on the number of lags s separating these unobserved effects and not on the chosen time periodt. We do not consider this possibily in this paper for simplicy. 11

12 Cov(u,u is )=Cov(u iq,u ir ) t<s,q<rin the case of dynamic models will imply the addional T 2 over-identifying restrictions: y it y it s E( h 1 (x it,β 0 ) h 1 (x it s,β 0 ) )=τ 0 s =1,...,T 1 (3.13) These moment restrictions are the generalization of the addional moment condions derived for linear dynamic models in Ahn and Schmidt (1995). In the case of dynamicmodelsanassumptionsuchas Cov(u,u is )=Cov(u iq,u ir ) t s, q r can be very plausible since is possible to argue that modeling dynamics will also account for all serial correlation in unobserved effects other than serial correlation due to the time constant part of unobserved effects. The linear feedback model presented in (3.7) for instance implies such addional moment restrictions eventhough they have not been used for estimation in previous studies.indeed: E( y it γ 0 y it 1 µ(x it,θ 0 ) y it s γ 0 y it s 1 )=E(c 2 i ) s =1,...,T 1 (3.14) µ(x it s,θ 0 ) Windmeijer (2000) has derived similar moment condions for the model presented in(1.6) and(1.7) but under a set of assumptions that was too weak. Windmeijer (2000) only assumes that c 2 i is uncorrelated wh ɛ and that ɛ is uncorrelated wh ɛ is for t s which doesn t imply E(c 2 i ɛ ɛ is )=E(c 2 i )E(ɛ )E(ɛ is )= E(c 2 i )henceseemsthataspecificationofmodelsintermsofcondional expectations and unobserved effects as in (2.1) and (2.2) is more straightforward than the specification of the model in terms of uncorrelation found in Windmeijer (2000). 4 Monte Carlo Evidence To study the small sample performance of the estimators we present in this paper, we consider estimating the Linear Feedback model presented in Blundell, Griffh and Windmeijer (2002). 12

13 y P oisson(γy 1 + exp(βx + η i )) t =1,...,T x = ρx 1 + τη i + ɛ x i0 = τ 1 ρ η i + ξ i y i0 P oisson( exp(βx i0 + η i ) ) 1 γ η i N(0,σ 2 η) ɛ N(0,σ 2 ɛ ) ξ i N(0, σ 2 ɛ 1 ρ 2 ) The only difference wh the data generating process of Blundell, Griffh and Windmeijer (2002) is that we do not obtain y i0 as the last draw from fifty draws starting at y i 49 P oisson(exp(βx i0 + η i )) but instead impose E(y i0 c i )=c i E(exp(βx ))/(1 γ). Since we will restrict our attention to γ<1, both data generating processes will be very similar even though not exactly equivalent. Wh this model, we will consider using for estimation the sequence of moment condions: E(z ( y γ 0 y 1 h 1 (x,β 0 ) y 1 γ 0 y 2 )) = 0 t =2,...,T (4.1) h 1 (x 1,β 0 ) Where z =(y 2,...,y i0,x 1,...,x i1 )orz =(y 2,x 1 ). The addional condions that arise from serial uncorrelation of the transory shocks are: E( y it γ 0 y it 1 h 1 (x it,β 0 ) (y γ 0 y 1 h 1 (x,β 0 ) y 1 γ 0 y 2 )) = 0 t =2,...,T 1 (4.2) h 1 (x 1,β 0 ) And the addional condions that arise from the restriction imposed on the inial condions are: E((y 1 y 2 ) y γ 0 y 1 )=0t =2,...,T h 1 (x,β 0 ) E((x x 1 ) y γ 0 y 1 )=0t =2,...,T h 1 (x,β 0 ) 13

14 We will consider four groups of estimators: using no addional moment condions, using the addional moment condions from serially uncorrelated transory shocks, using the addional moment condions from the restrictions on the inial condions and using both sets of addional moment condions. Whin each group we will consider the GMM estimator that uses all available lags of the instruments for the condional moment condions and the GMM estimator that uses only one lag of the instruments for the condional moment condions. For each estimator we will also consider the two-step GMM estimator wh the identy matrix as inial weighting matrix and the erated GMM estimator which is a multiple step GMM estimator that takes as many steps as are needed for the estimates to converge 5. Therefore we will be considering a total of sixteen estimators. Table 1 and Table 2 report the bias and root Mean Squared Error (MSE) of the estimators of γ 6.Table 3 and Table 4 report the ratio of the mean of standard errors of the estimators of γ over the standard deviations of these estimators. Therefore these tables capture the bias of estimators of the variance of the estimators of γ. Table 5 and Table 6 report the coverage rate of the 95%confidence intervals created from the estimators of γ and their associated standard errors. All results are from 1,000 replications. The first conclusion from Table 1 and Table 2 is that using the addional moment condions presented in the previous section results in large efficiency gains wh very sizeable decreases in both bias and standard deviations. This gain is especially noticeable when eher set of addional moment condions is used compared to not using any set of addional moment condions. Using an addional set of addional moment condions usually represents a more modest marginal gaininefficiencyorsometimesnogainat all. Bias is almost always smaller when using only one lag of the instruments instead of all available lags. When all available lags of the instruments are used, erated GMM seems to perform better than two step GMM. When erated GMM is used, using all lags of the instruments is usually slightly preferable in terms of root MSE to using only one lag but not in terms of bias and when two step GMM is used using only one lag of the instruments can sometimes achieve both a lower bias and root MSE. Table 3 and Table 4 show a severe downward bias in standard errors for small n and large T and when 5 We do not present the results of erated GMM estimation for n =100becauseforthissmallsamplesizetheerated GMM algorhm failed to converge in less than 400 erations in 25% of the bootstrap draws when T =4and50%ofthe bootstrap draws when T =8. CondionalonhavingtheeratedGMMalgorhmconverging for n =100,eratedGMM seemed to provide some efficiency gain and significantly better inference when many moment condions are used compared to two step GMM in a similar way as for larger sample sizes. 6 We only show results for the estimation of γ here but results for estimation of β exhib similar patterns. 14

15 all available lags of the instruments are used. This problem is alleviated by using erated GMM especially when T is large. However even doing so standard deviations can be significantly under-estimated. This bias in standard errors is due to the use of many over-identifying moment condions. The same problem of downward biased standard errors has been studied for the special case of linear models in Windmeijer (2005) and for models of count data in Windmeijer (2008). However these two papers concentrate on the bias originated from using a preliminary estimator to compute the optimal weighting matrix, whereas we see that using erated GMM instead of two step GMM helps but does not solve completely the problem of downward biased standard errors. Asymptotic analysis under manymomentcondionsseemstoindicate that most of the bias comes from the correlation between the gradient of the moment functions and the moment functions themselves. This result has been presented inamoregeneralsettinginneweyand Windmeijer (2009) and applying a higher order asymptotic analysis wh many moment condions to the specific models we consider in this paper is the subject of ongoing research. Bootstrapped standard errors might also be a solution. Table 5 and Table 6 show the effect of both downard biased standard errors and bias in the estimator of γ on inference. For small n or large T the coverage of confidence intervals is significantly lower than the confidence level of 95%, particularly when all available lags of the instruments are used. This problem is alleviated by using erated GMM but not completely solved. Corrected standard errors should participate in constructing better confidence intervals as could bias correction, particularly in the case where no addional moment condion is available. Similarly as for correction of the standard errors, bias correction could be based on higher order asymptotic analysis. The first conclusion of this section is that using addional restrictions on the stationary of the instruments or serial uncorrelation of transory shocks can make a big difference in terms of the precision of the point estimates. It does not solve however the problem of inference which was already present wh previous estimators and is due to the poor properties of GMM standard errors in cases where many over-identifying condions are used. Using erated GMM can improve the qualy of inference compared to two step GMM especially when T is relatively large whout solving the problem completely. Theresultspresentedinthissectionalso suggest that using only one lag of the instruments can result in much better inference especially when T 15

16 is relatively large. Previous studies of FE estimation of models similar to the ones we consider in this paper, such as Arellano and Bond (1991) or Windmeijer (2008), recommendedtheuseofalllagsofthe instruments in (4.1). However the Monte Carlo evidence we presented indicates that using only one lag of the instruments causes only a modest loss in accuracy, especially when addional moment condions are available, but results in significantly lower bias and significantly better inference compared to using all available lags of the instruments. 5 Average Partial Effects Wh multiplicative heterogeney models, Average Partial Effects (APE) are very simple to compute. Average Partial Effects are defined by: AP E fw = E fw ( y x ) (5.1) Where f w is some distribution over the domain of w and y x denotes the change in y caused by a small change in x. 7 So AP E fw = E fw ( h 1(x, β 0 ) cu) x = E fw ( h 1(x, β 0 ) y x h 1 (x, β 0 ) ) Many applications are interested in the average effect across an observed subset of the population, denote A. Thiscorrespondstousingf w = f(w A) sothatap E A = E( y x A) =E( h 1(x,β 0 ) y x h 1 (x,β 0 ) A). For instance we could be interested in the average effect of x on y across the entire population in some given time period tape t = E( h 1(x,β 0 ) x y h 1 (x,β 0 ) ). Or in the case of a binary explanatory variable x1, wh x =(x 1,x 1 ), we could be interested in Average Treatment Effect on the Treated at a given time period AT ET t = E(y(1,x 1) y(0,x 1) x1 =1)=E((h1 ((1,x 1),β 0) h 1 ((1,x 1),β 0)) h 1 (x,β 0 ) x1 =1). Estimation and inference are straightforward in this case once a consistent estimator ˆβ for β 0 is defined. 7 Here we use the notation for partial derivatives but in case of discretechangesinx of xwe could use the counterfactuals notation and use y(x) =y (x + x) y x interchangeably. y 16

17 Since E(1(i A)( h 1(x,β 0 ) y x h 1 (x,β 0 ) AP E At)) = 0 where 1(.) istheindicativefunction,wecanjustadd this moment condion to the moment condions used to estimate β 0 and obtain an addional estimator of AP E At as well an estimator for the asymptotic variance of and ˆβ where ˆβ denotes the estimator of β 0 we will be using. ˆ AP E At and covariance between ˆ AP E At Since we are adding one moment condion for one new parameter, the estimator ˆβ will not be affected by estimation of Average Partial Effects. In addion the GMM estimator of APE will be given by: Where n A = n 1(i A). APˆ E At = 1 n A n 1(i A) h 1(x, ˆβ) x y h 1 (x, ˆβ) If we think that APE should be equal across time periods, we can imposethisrestrictioninthegmm estimation by adding the moment restrictions {E(1 A ( h 1(x t,β 0 ) x y t (5.2) h 1 (x t,β 0 ) AP E A)) = 0} t=1,...,t which might affect estimation of β 0 or we can estimate average partial effects for each time period and combine them using Minimum Distance Estimation which will not affect estimation of β 0. In other suations, if f w can be consistently estimated by f w (ˆη) whereˆη is a vector of estimators of nuisance parameters η 0,then APˆ E fw = E fw(ˆη) ( h 1(x, ˆβ) y ) (5.3) x h 1 (x, ˆβ) is consistent for AP E fw. If (ˆβ, ˆη) arejointlyasymptoticallynormalandaconsistentestimator of their asymptotic variance-covariance matrix is available then inference can be performed using the deltamethod. 6 Conclusion These results hopefully constute a first step towards a more widespreaduseofnon-linearmodelsofpanel data wh unobserved effects in applications where only sequential exogeney is available. The problem of weak instrumental variables seems to be migated significantly by the use of addional moment condions originating from addional restrictions of stationary of the instruments or serial uncorrelation of the transory shocks. Monte Carlo evidence also seems to suggest that is preferrable to use only one or a 17

18 few lags of the instruments compared to all available lags since this results in much better inference at the expense of only small losses in efficiency. Two directions are available in order to obtain estimators wh better inference. One consists in studying the higher order properties of the GMM estimator wh many over-identifying restrictions, the other consists in finding good exactly identifying moment condions. Both of these approaches are left for future research. 18

19 A GMM Estimation and Efficiency Bound Define Σ i =[Cov(ρ,ρ is z imax(t,s) )] s=1,...,t t=1,...,t.definea and Σ 1 to be the terms of the LDL decomposion of Σ 1 : Σ 1 = A Σ 1 A where Σ isdiagonalanda is upper-triangular wh only ones on the diagonal. We can show that A =[1(s t)( 1) 1(s t) Γ t,s ] s=2,...,t t=2,...,t where 1(.) istheindicativefunctionand Σ = diag({v ar( ρ t z t 1 )} t=2,...,t )sothat J = E( T t=2 D 1 t Σ t D t ) = E(Ë(A ρ i β z i ) Σ 1 Ë(A ρ i β z i )) Where Ë([y t ] t=2,...,t [x 1,...,x T 1 ]) is a matrix operator that returns E(y t x t 1 )ass(t 1) th element, t =2,...,T,wherey t are row vectors. Using standard results of GMM estimation we can wre: n( ˆβLin β 0 )=W 1 ( 1 n ρ i Z i ) ( 1 n Z n β i ρ i ρ n iz i) 1 1 n Z i ρ i + o p (1) n W =( 1 n ρ i Z i ) ( 1 n Z n β i ρ i ρ n iz i) 1 1 n ρ i Z i n β Where ρ i = ρ i (β 0 )and ρ i β = ρ i(β 0 ) β. Applying the WLLN, 1 n n Z i ρ i β = O p (1). Also 1 n n Z i(ρ i ρ i Σ i)z i = o p(1) since E(Z i (ρ i ρ i Σ i )Z i )=0fromhowZ i and Σ i where defined. Using the CLT, 1 n n Z iρ i = O p (1). Using Slutsky s theorem, assuming plim 1 n n Z iρ i ρ i Z i is p.d., we have ( 1 n n Z iρ i ρ i Z i ) 1 ( 1 n n Z iσ i Z i ) 1 = o p (1). So W = V + o p (1) where V =( 1 n n Z i ρ i β ) ( 1 n n Z iσ i Z i ) 1 1 n n Z i ρ i β and using Slutsky s theorem again, assuming plimw is fine and p.d., W 1 = V 1 + o p (1). Therefore we can rewre: 19

20 In addion: n( ˆβLin β 0 )=V 1 ( 1 n ρ i Z i ) ( 1 n Z n β i Σ i Z 1 n i n ) 1 Z i ρ i + o p (1) n V =( 1 n ρ i Z i ) ( 1 n Z n β i Σ i Z 1 n ρ i i n ) 1 Z i n β n( ˆβ Lin β 0 )( ˆβ Lin β 0 ) = V 1 + o p (1) (A.1) Since: 1 n n Z i ρ i ( 1 n n Z i ρ i ) = 1 n = 1 n = 1 n j n Z i ρ i ρ j Z j j=1 j Z i ρ i ρ i Z i + o p(1) j Z i Σ i Z i + o p (1) Where the second equaly follows from random sampling and the WLLN. We can rewre V as: V = ρ Z ( Z Z ) 1 Z ρ β 0 β 0 (A.2) Where ρ =[ ρ β 1 β,..., ρ n β ], ρ i β Consider the matrix linear projection of ρ i β deterministic matrix defined by the moment condions: = Σ 1/2 A ρ i β, Z =[ Z 1,..., Z n ], Z i = Z i A 1 Σ1/2. on Z i, LP ( ρ i β Z i )= Z i C,whereC is a dim(z i) dim(β) E( Z i ( ρ i β Z ic)) = 0 (A.3) It is a standard result that as long as E( Z i Z i )isfineandp.d. projection is consistently estimated by: ande( Z i ρ i β )exists,thislinear 20

21 LP ˆ ( ρ β Z i )= Z i ( Z 1 Z) Z ρ β 0 = LP ( ρ i β Z i )+o p (1) Define the stacked estimated linear projections by Z( Z Z) 1 Z. SinceP Z is idempotent, we have: ˆ LP ( ρ β Z) = Z( Z Z) 1 Z ρ β 0, denote P Z = V = ρ ρ P β Z /n 0 β 0 = ρ ρ P β ZP Z /n 0 β 0 = LP ˆ ( ρ β Z) ˆ ρ LP ( β Z)/n = 1 n LP ˆ ( ρ n β Z i ) LP ˆ ρ ( β Z i ) = E(LP ( ρ β Z i ) LP ( ρ β Z i )) + o p (1) Where the last equaly follows from Newey and McFadden (1994) for instance. In addion, the matrix linear projection of Ë( ρ β Z i )on Z i defined by: ρ β on Z i is the same as the matrix linear projection of E( Z i (Ë( ρ i β Z i ) Z ic)) = 0 (A.4) Since the t th vector of Z i, Z,isafunctionofZ since z contains z i1,...,z 1.Therefore: V = E(LP (Ë( ρ i β Z i ) Z i ) LP (Ë( ρ i β Z i ) Z i )) + o p (1) (A.5) So using the standard results on linear projection: 21

22 V = E(Ë( ρ i Z β i ) Ë( ρ i Z β i )) E(e i e i)+o p (1) T = E( D Σ T t t Dt ) E( e e )+o p (1) t=2 t=2 Where e i = E( ρ i β Z i ) LP (Ë( ρ i β Z i ) Z i )ande = E( ρ β Z ) LP (E( ρ β Z ) Z ). 22

23 References Ahn, Seung C. and Peter Schmidt Efficient estimation of models for dynamic panel data. Journal of Econometrics 68(1):5 27. URL: Arellano, Manuel and Olympia Bover Another look at the instrumental variable estimation of error-components models. Journal of Econometrics 68(1): URL: Arellano, Manuel and Stephen Bond Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations. The Review of Economic Studies 58(2): ArticleType: research-article / Full publication date: Apr., 1991 / Copyright  c 1991 The Review of Economic Studies, Ltd. URL: Blundell, Richard, Rachel Griffh and Frank Windmeijer Individual effects and dynamics in count data models. Journal of Econometrics 108(1): URL: Blundell, Richard and Stephen Bond Inial condions and moment restrictions in dynamic panel data models. Journal of Econometrics 87(1): URL: Browning, Martin, Mette Ejraes and Javier Alvarez Modelling Income Processes wh Lots of Heterogeney. The Review of Economic Studies 77(4): URL: Chamberlain, Gary Comment: Sequential Moment Restrictions in Panel Data. Journal of Business & Economic Statistics 10(1): URL: Giles, John and Irina Murtazashvili A Control Function Approach to Estimating Dynamic Prob 23

24 Models wh Endogenous Regressors. Journal of Econometric Methods 0(0):1 19. URL: Hahn, Jinyong Efficient estimation of panel data models wh sequential moment restrictions. Journal of Econometrics 79(1):1 21. URL: Newey, Whney K. and Daniel McFadden Chapter 36 Large sampleestimationandhypothesis testing. In Handbook of Econometrics, ed.robertf.engleanddaniell.mcfadden. Vol.Volume4 Elsevier pp URL: Newey, Whney K. and Frank Windmeijer Generalized Method of Moments Wh Many Weak Moment Condions. Econometrica 77(3): URL: Windmeijer, Frank Moment condions for fixed effects count data models wh endogenous regressors. Economics Letters 68(1): URL: Windmeijer, Frank A fine sample correction for the variance of linear efficient two-step GMM estimators. Journal of Econometrics 126(1): URL: Windmeijer, Frank GMM for Panel Data Count Models. In The Econometrics of Panel Data, ed. Laszlo Matyas and Patrick Sevestre. Number 46 in Advanced Studies in Theoretical and Applied Econometrics Springer Berlin Heidelberg pp URL: Wooldridge, Jeffrey M Multiplicative Panel Data Models whout the Strict Exogeney Assumption. Econometric Theory 13(5): URL: 24

25 Wooldridge, Jeffrey M Simple solutions to the inial condions problem in dynamic, nonlinear panel data models wh unobserved heterogeney. Journal of Applied Econometrics 20(1). URL: 25

26 Table 1: T=4 γ =0.5; β =0.5; ρ =0.5; τ =0.1; ση 2 =0.5; σ2 ɛ =0.5 N=100 N=500 N=1000 N=2000 gmm method Condions Used Instruments parameter bias rmse bias rmse bias rmse bias rmse twostep no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ igmm no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ

27 Table 2: T=8 γ =0.5; β =0.5; ρ =0.5; τ =0.1; ση 2 =0.5; σ2 ɛ =0.5 N=100 N=500 N=1000 N=2000 gmm method Condions Used Instruments parameter bias rmse bias rmse bias rmse bias rmse twostep no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ igmm no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ

28 Table 3: T=4 γ =0.5; β =0.5; ρ =0.5; τ =0.1; ση 2 =0.5; σ2 ɛ =0.5 N=100 N=500 N=1000 N=2000 gmm method Condions Used Instruments parameter ratio E(se)/sd ratio E(se)/sd ratio E(se)/sd ratio E(se)/sd twostep no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ igmm no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ

29 Table 4: T=8 γ =0.5; β =0.5; ρ =0.5; τ =0.1; ση 2 =0.5; σ2 ɛ =0.5 N=100 N=500 N=1000 N=2000 gmm method Condions Used Instruments parameter ratio E(se)/sd ratio E(se)/sd ratio E(se)/sd ratio E(se)/sd twostep no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ igmm no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ

30 Table 5: T=4 γ =0.5; β =0.5; ρ =0.5; τ =0.1; ση 2 =0.5; σ2 ɛ =0.5 N=100 N=500 N=1000 N=2000 gmm method Condions Used Instruments parameter coverage coverage coverage coverage twostep no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ igmm no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ

31 Table 6: T=8 γ =0.5; β =0.5; ρ =0.5; τ =0.1; ση 2 =0.5; σ2 ɛ =0.5 N=100 N=500 N=1000 N=2000 gmm method Condions Used Instruments parameter coverage coverage coverage coverage twostep no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ igmm no addional condions All Lags γ One Lag γ inial condions All Lags γ One Lag γ serial uncorrelation All Lags γ One Lag γ both sets of condions All Lags γ One Lag γ