The Time Series and Cross-Section Asymptotics of Empirical Likelihood Estimator in Dynamic Panel Data Models

Transcription

1 The Time Series and Cross-Section Asymptotics of Empirical Likelihood Estimator in Dynamic Panel Data Models Günce Eryürük August 200 Centro de Investigación Económica, Instituto Tecnológico Autónomo de México, 0700 México DF, México.

2 Abstract Dynamic panel data models have attracted lots of attention because of their great flexibility and naturally these models provide a large number of moment conditions. Typically these models are estimated by generalized method of moments GMM estimators. Empirical likelihood EL is shown to have various advantages over other methods like GMM. One of its advantages arises in the many moments settings. However, little is known, yet, on the relative merits of the EL estimator in dynamic panel data models. In this article, we try to fill that gap by establishing the asymptotic properties of the EL estimator for a dynamic panel data model with individual effects when both N and T tend to infinity. We give the relative rates of N and T for which this estimator is consistent and asymptotically normal. Monte Carlo experiments are conducted to compare the performance of EL estimator with that of commonly used GMM and LIML estimators. The results from the Monte Carlo study show that as the instruments increase in number and get weaker EL outperforms the other two estimators in terms of median bias. JEL Classification: C3, C23. Keywords: Dynamic panel data model, empirical likelihood, fixed effects, limited information maximum likelihood, generalized method of moments.

3 Introduction Dynamic panel data models offer great flexibility to empirical researchers. Many economic phenomena are dynamic in nature. Examples include household consumption, firms factor demands, and countries economic growth. Dynamic panel data models allow researchers to control for unobserved heterogeneity in adjustment dynamics between different individual units and thereby provide improved insights in such models. Generalized method of moments GMM estimators are widely used in estimation of these models see, among others, Anderson & Hsiao 982, Holtz-Eakin, Newey, & Rosen 988, and Arellano & Bond 99. However, the standard GMM estimator obtained after first differencing has been found to suffer from substantial finite sample bias, especially when the instruments are weak and the number of moments is large relative to the cross section sample size see Alonso-Borrego & Arellano 999. This low precision of GMM is also evident in more general contexts. To improve the small sample properties of GMM estimators, a number of alternative estimators have been suggested, including, among others, EL, continuous updating CU, and exponential tilting ET estimators. Newey & Smith 2004 show that these estimators are members of a class of generalized empirical likelihood GEL estimators. They use this structure to compare their higher order asymptotic properties with those of GMM. They find that EL has two theoretical advantages. First, its asymptotic bias does not grow with the number of moment restrictions, while the bias of GMM often does. This, as a result, suggests that in estimation of models with many moment conditions, the bias of EL will be less than the bias of GMM. Consequently, EL can be an important alternative to GMM in such applications. The second theoretical advantage of EL estimator is that after it is bias-corrected using probabilities obtained from EL, it is higher order efficient relative to other bias-corrected This occurs when the series are highly autoregressive, i.e., the autoregressive parameter is close to one, and the relative variance of fixed effects to the variance of idiosyncratic shocks is large see Blundell & Bond 998.

4 estimators. It is natural to expect that the GEL estimators possess similar advantages in dynamic panel data models, as well. Unfortunately, little is known on the relative merits of these estimators in these models. The purpose of this article is to provide further insight into the asymptotic properties of the EL estimator in dynamic panel data framework under the double asymptotics, i.e., asymptotics taken as both T and N going to infinity and to study its behavior for alternative relative rates of increase for N and T. This asymptotics is motivated by the increased availability of panel data sets covering different individuals, regions, and countries over a relatively long time period. Among the important examples of these data sets are the PSID household panel in the US, Penn World table and the balance sheet-based company panels. For panels in which T is not negligible relative to N, the analysis of the asymptotic behavior of the estimators as both T and N tend to infinity may provide better approximations to the finite sample behavior of the estimators and hence may be useful in assessing alternative methods. Previously, this type of asymptotics is used by Alvarez & Arellano They derive the asymptotic properties of commonly used panel data estimators among them includes within groups WG, GMM, and limited information maximum likelihood LIML estimators. It is also the case that for dynamic panel data models, the number of available lags which can be used as instruments for the equations in the first differences is of order OT 2, hence when T is large the number of moment conditions is also large. This corresponds to what is known in the literature as the many moment conditions situation. It is well known that in the linear instrumental variable regression models, using many moments causes the usual Gaussian asymptotic approximation to be poor. To address this problem alternative asymptotics, i.e., number of moment conditions increasing as well as the sample size, is suggested Bekker 994. Hence in dynamic panel data models letting T grow corresponds to this many moments asymptotics. In a more general context Newey & Windmeijer 200 consider this asymptotics for GEL and GMM type estimations. However, in dynamic panel 2

5 data models the conditions, especially the relative rates of T and N for which EL estimator is consistent and asymptotically normal can not be driven straightforwardly from their results. To drive those conditions under these settings is the purpose of this article. Examples concerning EL estimator in dynamic panel data models are few. In a Monte Carlo study, Oĝuzoĝlu 2006 compares performance of a number of estimators including GMM, EL, transformed ML, minimum distance and bias corrected LSDV estimators in an autoregressive panel model for various parameter combinations. The results show that the biases of all estimators considered tend to increase as the autoregressive parameter gets larger. The increase in bias is the highest for LSDV, whereas EL is the least sensitive to changes in this parameter. Moreover, the bias of GMM does not decrease much as T increases. When the overall performances are concerned, i.e., in terms of comparisons based on biases, standard deviations, and root mean square errors, EL performs the best. In the same framework, Gonzalez 2007 considers the finite-sample size properties of the overidentification tests for a hybrid of EL and bootstrap estimators. Previously, a similar study was carried out by Brown & Newey 200 and Bowsher 2002 for the GMM estimator. Gonzalez 2007 investigates whether the limitations encountered within GMM estimation are extended to EL-bootstrap estimator. Her results show that EL-bootstrap may be a good alternative to GMM estimator within this setting. She also applies this estimator using the cash-flow series data for 74 US firms. Although a few studies considered the finite sample performance of the EL estimator in dynamic panel data models, none of them, to our knowledge, analyze its asymptotic performance explicitly under the aforementioned settings. This article tries to fill this gap. Specifically, we establish the asymptotic properties of the EL estimator for a firstorder autoregressive model with individual effects when both N and T tend to infinity. We show that this estimator is consistent and asymptotically normal. We also compare the asymptotic properties of EL estimator with those of the GMM and LIML estimators, which are popular in empirical research. 3

6 The paper is organized as follows. Section 2 presents the model and the estimators. In section 3 we establish the asymptotic properties of the EL estimator. For comparison purposes we give those for the LIML and GMM estimators, as well. A comparison of these estimators in finite samples using Monte Carlo simulations is given in section 4. Section 5 concludes and states plans for future work. Proofs are relegated to the Appendix. 2 The Model and The Estimators 2. The Model We consider a first order univariate autoregressive panel data model given by y it = α 0 y i,t + η i + v it, for t =,..., T ; i =,..., N where y it is the observable variable whose dynamics are of interest; for example, local government expenditure variable, α 0 <, η i is the fixed effect representing the unobserved heterogeneity among individuals, and v it is the idiosyncratic variable with zero mean and variance σ 2 given η i, y i0,..., y i,t and has no autocorrelation. We assume that y i0 is observed. Define x it y i,t. The parameter of interest is α 0. Our goal is to analyze the asymptotic properties of EL estimator of this parameter. For comparison purposes we are going to consider that of GMM and LIML estimators. Next we shall define these estimators. 2.2 The Estimators The GMM Estimator. The GMM estimator considered here is a version developed by Arellano & Bover 995, which simplifies characterization of the weight matrix in GMM estimation. Arellano & Bover 995 eliminate the fixed effect η i in by applying Helmert s transformation. For example, the t-th element of transformed v it can be written 4

7 as: v it = c t v it T t v i,t v it ] t =,..., T where c 2 t = T /T t +. That is, to each of the first T observations the mean of the remaining future observations available in the sample is subtracted. The weighting c t is introduced to equalize the variances. This transformation can be applied by using the forward orthogonal deviations operator, A, where T T... T T T 0 A = diag T T,..., T 2... T 2 T 2 T 2. 2 ]/ Equation with the variables stacked over t can be written as y i = α 0 x i + η i ı T + v i where y i = y i,..., y it ], x i = x i,..., x it ], v i = v i,..., v it ], and ı dimension T vector of ones. Operating A on this equation produces the transformed model: 2 y i = α 0 x i + v i where yi = Ay i, x i = Ax i, vi = Av i. Note that the fixed effect are eliminated because Aı = 0. Also, A A = Q T I T ı T ı T /T Q T is known as WG operator and AA = I T. Thus, if V arv i = σ 2 I T, the vector of errors in orthogonal deviations also has V arvi = σ 2 I T. Let z it = x i,..., x it ]. The model 2 and the stated conditions imply the following moment conditions 3 Ez it v it] = 0 t =,..., T. 5

8 There are m T T /2 orthogonality conditions. These moment conditions can be written, more compactly, as EZ iv i ] = 0, where Z i = T m z i z i z it T 2 y i y = i0 y i y i0... y it 2 The constant variance of v it given η i, y i0,..., y i,t implies that. 4 EZ iv i v i Z i = σ 2 EZ iz i. Therefore, letting x = x,... x N and y = y,... y N, an asymptotically efficient GMM estimator of α 0 based on the moment conditions in 3 is given by where Z = Z,..., Z N. α GMM = x ZZ Z Zy x ZZ Z Zx The LIML Estimator. A non-robust analog of the LIML estimator of the simultaneous equations literature solves the following problem: y αx ZZ Z Z y αx α LIML = arg min α y αx y αx. The robust LIML analog, or continuously updated GMM estimator in the terminology of Hansen, Heaton & Yaron 996, can be written as N α CU = arg miny αx Z Z α iyi αx i yi αx i Z i Z y αx. In the non-robust version, instead of keeping σ 2 fixed in the weighting matrix of the GMM criterion, it is continuously updated by making it a function of the arguments in the estimating criterion. 6

9 The EL Estimator. Empirical Likelihood estimation Qin & Lawless 994 and Imbens 997 is a one-step method that achieves the same first-order asymptotic efficiency as robust GMM. The empirical likelihood estimator maximizes a multinomial pseudo likelihood or empirical likelihood function subject to the orthogonality conditions. Letting p i be the probability of observation i, the multinomial log likelihood of the data is given by the empirical likelihood estimator: L = ln p i. The EL estimator maximizes this function subject to the restrictions p i 0, The Lagrangian is given by L = p i = and ln p i + φ p i Z iy i αx i = 0. p i Nλ N p i Z iy i αx i, where λ and φ are Lagrange multipliers. Taking the derivative of L with respect to p i we obtain the following first-order conditions p i φ Nλ Z iy i αx i = 0. Multiplying by p i and adding equations we get φ = N. Hence, p i = N + λ Z i y i αx i. The multipliers of the moment conditions can be determined as implicit functions λα solving for a given value of α: N such that + λ Z i y i αx i /N. + λ Z i y i αx i The concentrated likelihood function for α: N L c α = N Z iy i αx i = 0 + λ Z i y i αx i 7.

10 Therefore, the EL estimator is given by α EL = arg min α ln + λα Z iy i αx i ]. A computationally useful alternative expression for α EL is α EL = arg min α Qα, where Qα = max λ ln + λ Z iy i αx i ]. 3 The Asymptotic Properties Of The Estimators In this section we derive the asymptotic properties of the previous estimators when both N and T tend to infinity. Following Alvarez & Arellano 2003, we make the following assumptions: Assumption. {v it } t =,... T ; i =,... N are i.i.d across time and individuals and independent of η i and y i0 with Ev it ] = 0, V arv it ] = σ 2, and finite moments up to fourth order. Assumption 2. The initial observations satisfy y i0 = η i α 0 + ω i0 where ω i0 is independent of η i and i.i.d. with the steady state distribution of the homogenous process, so that ω i0 = j=0 αj 0 v i, j. Assumption 3. η i are i.i.d. across individuals with Eη i ] = 0, V arη i ] = σ 2 η and finite fourth order moment. Note that under these assumptions, the moment conditions given in 3 do not represent all the available moment conditions available. Ahn & Schmidt 995 present additional moment conditions and argue that they are important in improving the GMM estimation in highly persistent samples. However, we focus only on the moment conditions in 3 as they remain valid under much weaker assumptions. 8

11 3. The GMM and the LIML Estimators Alvarez & Arellano 2003 show that under the stated assumptions as both N and T tend to infinity, provided log T /N 0, α GMM is consistent for α 0 : α GMM p α0 Moreover, provided T/N c, 0 c α GMM α 0 N + α 0 ] d N 0, α 2 0. Although, number of moment conditions, m tend to infinity at the rate T 2, their result show that α GMM remains consistent. However, in the structural equation setting, when both the number of instruments and the sample size tend to infinity, while their ratio tends to a positive constant, the two-stage least squares estimator is shown to be inconsistent Kunitomo 980, Morimune 983, and Bekker 994. The intuition for this consistency of α GMM is defined by Alvarez & Arellano 2003 as in structural equation setting too many instruments produces over fitting and undesirable closeness to the OLS coefficients. Here a large number of instruments is associated with larger values of T and in such a case closeness to OLS, which is the WG estimator, becomes increasingly desirable because endogeneity bias tends to zero as T. For, LIML estimator, they show that, under the stated assumptions, as both N and T tend to infinity, provided T/N c, 0 c 2, α LIML is consistent for α 0 : α LIML p α0 Moreover, α LIML α 0 ] 2N T + α 0 d N 0, α 2 0. Note that GMM and LIML estimators are both asymptotically normal with the same asymptotic variance, however, unless T/N 0, they exhibit a bias term in their asymptotic distributions differing in its order of magnitude: +α/n for GMM and +α/2n T for LIML. Provided T < N, the LIML has a smaller asymptotic bias. 9

12 3.2 The EL Estimator For consistency and asymptotic normality of EL estimator some additional assumptions are needed. Let λ min S and λ max S denote the smallest and the largest eigenvalues of a symmetric matrix S, respectively. Assumption 4. i There is C > 0 such that /C λ min EZ i y i αx i y i αx i Z i ], λ max EZ i y i αx i y i αx i Z i ] C, and λ max EZ i x i x i Z i] C; ii sup N N Z i y i αx i y i αx i Z i EZ i y i αx i y i αx i Z i ] p 0. α Assumption 5. For a constant C and γ > 2, E v it 2γ < C and E η i 2γ < C. Assumption 4 puts restrictions on the rate at which T and hence the number of moment conditions, m, can grow relative to N. Assumption 5 puts further moment restriction on the error and the fixed effect terms. Note that Assumption 5 along with Assumption 2 imply that E y i0 2γ < C. The following is a consistency result for the EL estimator. Theorem. Let Assumptions -5 hold. Then as both N and T tend to infinity, provided T 4 2 γ /N 2 γ 0, γ > 2, α EL is consistent for α 0 : α EL p α0. For the consistency of the EL estimator a further restriction is need on the relative rates at which T and N can grow. This is also the case in Newey & Windmeijer 200 for a general cross sectional model. Next we give the asymptotic normality result. This result is parallel to Theorem 3 of Newey & Windmeijer 200. Theorem 2. Let Assumptions -5 hold. Then as both N and T tend to infinity, provided T /N 0, αel α 0 d N0, α

13 In their article Newey & Windmeijer 200 give an asymptotic variance of GEL estimator as a summation of two terms. The first term corresponds to the conventional asymptotic variance term of GMM. The additional term can be considered as a higher order variance term in asymptotic theory with fixed number of moment conditions. They note that this term can be important even when the sample size is large under certain conditions, which includes weak moments. Under the restrictions on the relative rates on N and T given by Theorem 2, the additional terms, in our case, tend to zero 2. Hence, the gradient of the EL objective function at the true parameter, α 0, scaled by takes the following form: Qα 0 α = x Z Z v v Z Z v + o p. The asymptotic variance of Qα 0 α converges in probability to x Z Z v v Z Z x p α0 2. Then, as shown in Appendix, theorem 2 follows from theorem 3 of Newey & Windmeijer 200. Theorem 2 requires that for asymptotic normality T has to grow much slower than N does. More specifically, it is required that limt/n c = 0. This condition is much more strict than 0 c 2 requirement which is needed for asymptotic normality of LIML. Moreover, for LIML, when c = 0, the asymptotic bias disappears and the two estimators become asymptotically equal. To compare the relative performances of these estimators in the finite samples we conduct a Monte Carlo study in the next section. 4 Monte Carlo Study In this section we report some Monte Carlo simulations of EL, GMM, and LIML estimators for various combinations of T and N values. Our focus is specifically on large T and 2 The proof is available upon request from the author.

14 moderated N values. As T gets large the number of orthogonality conditions also gets large it grows at the rate of T T /2. Hence, the purpose of these experiments is to compare the biases of these estimators for different values of T and N when many moment conditions exist. As an extension we consider the case where the moment conditions are also weak besides being many. Weak moment conditions case occurs when the lagged levels of the series are only weakly correlated with subsequent orthogonal deviations, i.e., the series y i0,..., y i,t ] is weakly correlated with y i,t for t =,..., T. In our model, instruments available for the transformed equations become weak either as the autoregressive parameter α approaches unity or as the variance of the individual effects η i increases relative to the variance of the transient shocks v it Bond, Hoeffler & Temple, 200. For all cases we conducted 000 replications from the model specified in section 2 under normality, i.e. each sample consists of N independent observation of y i0, y i,..., y it generated from the process y i0 = α η i + α 2 /2 v i0, y it = αy i,t + η i + v it for t =,..., T with v i = v i0, v i,..., v it N0, σ 2 I and η i N0, ση 2 independent of v i. For the first set of results the many strong instruments case, σ 2 and ση 2 are and 0, respectively. We let α to take the values of 0.2, 0.5 and 0.8. This design follows Alvarez & Arellano 2003 closely. For the second set of results, we increase ση 2 from 0 to and 4 while holding σ 2 at and let α to take low 0.2 and high 0.9 values. The first set of results are summarized in tables and 2. In Table we report median, interquartile range iqr, and median absolute error mae of the EL, GMM, and LIML estimators for N = 00 with T 0 = 0, 25, and 50, where T 0 = T + the actual number of time series observations in the data. Table 2 reports the similar results for N = 50. Table and 2 reveal that in all cases the median bias of GMM estimator is always larger than the median biases of the EL and LIML estimators. The EL and LIML biases are both very small, however, the ranking between the two is not obvious. When T is small relative 2

15 to N and α is small EL bias is smaller than the LIML bias. The difference between them gets smaller for a square panel with T 0 = 50 and N = 50. When it comes to dispersion, GMM always has a smaller interquartile range than the other two estimators. Again, the ranking between EL and LIML is not clear. When T 0 = 25 LIML interquartile range is always smaller, however, for the other cases there is not an obvious order. Finally, for median absolute errors, Table and 2 show that except one case the case when T 0 = 50, N = 00, and α = 0.2 the GMM median absolute error is always the smallest. The ranking is less obvious between EL and LIML for this comparison criterion as well. In Table for T 0 = 0 and 50 and α = 0.2 and 0.8 EL median absolute error is smaller than LIML. In Table 2 it is smaller for T 0 = 0 and α = 0.2, however as T gets larger and closer to N, LIML median absolute error becomes smaller, although the difference between them is very small especially when α is large. For the weak instruments setup, we first let the autoregressive parameter α take a value close to unity while keeping the relative variance of individual effects to the variance of idiosyncratic errors low as in the previous cases, i.e., σ 2 = and ση 2 = 0. The results for this case are reported in Table 3. To see the effect of increasing α on the performances of the estimators more conveniently we reproduce part of Table here as well. Second, we let the relative variance of individual effects to increase, specifically σ 2 = ση 2 =, and α take low 0.2 and high 0.9 values. The results for this case are reported in Table 4. Third, we let the variance of individual effects increase to 4 while holding the variance of idiosyncratic errors still at and α take low and high values. Table 5 reports the results for this case. For all these cases we let N = 00 and T = 0, 25, and Relative performances of EL, GMM and LIML estimators do not change when α is close to unity either. GMM is outperformed by EL and LIML estimators in terms of median, but 3 We run the same simulations for N = 50 as well, however, those results are not included here since the main conclusions did not change. 3

16 outperforms in terms of interquartile range and median absolute error measures. Although, LIML performs slightly better than EL, the difference in terms of all three measures between them is very small. Table 4 and 5 reveal interesting changes in the relative performances of the estimators. In Table 4, we give the results for the case when the variance of individual effects takes a positive value while the relative variance is not high. In terms of median, for α = 0.2, LIML still performs better than the other two estimators. However, when the persistence in the series increases and the number of instruments gets large EL performs better than the rest. EL performs better than LIML also in terms of interquartile range and median absolute error when α approaches to unity or the number of instruments gets larger no matter if α is small or large. The deterioration in the performance of LIML estimator due to high persistency becomes more obvious when the relative variance of individual effects gets larger Table 5. When α = 0.9 for all values of T EL dominates GMM and LIML estimators in terms of median. It is interesting to note that in this setting, when the number of instruments along with persistency increase the deterioration in the performance of LIML estimator in terms of all three measures becomes severe. On the other hand, the performance of EL estimator stays robust to the existence of weak instruments even in the extreme case. Although EL and LIML both perform well in settings where instruments are strong, LIML performs slightly better than EL. However, when the over all performances are concerned in both strong and weak instrument cases, we conclude that EL is more reliable. Especially, when the instruments are many and weak LIML performs a lot worse as compared to EL and GMM. Panel data sets with these features are common see Blundell & Bond 998 for an example. Hence, these results suggest that when LIML is used in estimation of panel data models one should be cautious. If the data is long and persistent it would be better to use EL instead of LIML. 4

17 5 Conclusion In this paper we show that in autoregressive panel data models, the EL estimator that uses all the moment conditions based on all the available lags at each period are consistent and asymptotically normal when both N and T tend to infinity. When showing normality, we applied Newey & Windmeijer 200 method that they use for a general cross sectional model. For the EL estimator, for normality, the required condition on the relative rates at which N and T can grow turns out to be much more strict than that of the LIML and GMM estimators. Under this restriction, the LIML and GMM asymptotic biases disappear. Therefore, all three estimators that we consider have the same asymptotic distribution. To be able to distinguish the finite sample performances of these estimators we consider a Monte Carlo study. This study reveals that GMM always has the largest median bias, although it has the smallest dispersion in all the scenarios we consider. The ranking between EL and LIML is not obvious when the instruments are not weak. However, when the instruments get weaker and larger in number, EL estimator clearly dominates LIML estimator. In future work, we plan to extend the current results by looking at the second order asymptotics of the EL estimator. This helps us to understand the nature of the asymptotic bias of EL and hence enables us to compare the three estimators analytically. As a second extension, we plan to relax the assumptions on the initial conditions and the homoscedasticity and study the properties of robust LIML estimator. 5

18 Appendix Through out the Appendix, we let C to denote a generic positive constant that may be different in different uses. Also, let w.p.a. stand for with probability approaching one. To show the order of each term, we frequently employ results in Alvarez & Arellano AA refers to the paper of Alvarez & Arellano 2003; also AA. reads the formula. in Alvarez & Arellano Lemma. N N Z i v i = O p T N. Proof. By independence across individuals we have E Z N ivi 2 = E N 2 vi Z i Z iv i A- = N 2 = σ2 N tr { E Z iz i } = σ2 N T t E vi Z i Z iv i 2 + N 2 E yis 2. t= s=0 E vi Z i E Z j vj j>i Recall z i z i E Z iz 0 z i = E i z i z i,t z i,t q T T q z i z i z = E i2 z i , z i,t z i,t 6

19 where z it z it = yi y i0 y i,t y i y i0... y i y i,t..... y i,t y i0... yi,t 2. Let ω it = y it η i α 0. Then we have t yis 2 = s=0 where ω i = t t s= ω i,s. t ωi,s 2 η i 2t ω i α 0 + t ηi 2 α 0 2, s= Under Assumptions 3, we have E ωi,s 2 σ 2 = α0 2, η i E = 0, E ω i η 2 i α 0 2 α 0 = σ 2 η α 0 2. Therefore, t E yis 2 σ 2 = t α0 2 s=0 + σ 2 η α 0 2, and for C = σ 2 σ 2 + σ2 η α 2 α 0 0, A- becomes 2 T T T Ct = C N 2N. t= Hence the conclusion follows by Markov s Inequality. Lemma 2. E sup Z iy i αx i γ] = OT 3γ 2. α 7

20 Proof. Consider Z iy i αx i γ = T t yit αx it 2 t= s= y 2 i,s ] γ 2 T T γ 2 y it αx γ t it t= s= γ yi,s 2 2 T = T γ 2 α0 αx it vit γ t t= s= y 2 i,s The inequality in the second line is obtained using Loève s c r inequality. From AA A43, we can write γ 2. A-2 A-3 A-4 A-5 x it = ψ t y i,t η i c t ṽ itt, where α T t c t = T t +, ψ t = c t αφ T t T t, φ j = αj α, and ṽ itt = T t φ T tv it φ v i,t. Using this expression for x it, we have, for β = α α 0, T Z iy i αx i γ = T γ 2 v it + βc t ṽ itt βψ t yi,t η i t γ α t= T CT γ 2 vit γ + βc t ṽ itt γ + βψ t y i,t γ t= + η i βψ t γ] t α where we used the fact that m A-6 a i s= γ yi,s 2 2, γ c m γ a i γ with c γ being m γ. Taking the expectation of sup Z i y i αx i γ we get α E sup Z iy i αx i γ] T { CT γ 2 E v γ t it α t= + sup βc t γ t E ṽ itt γ α 8 s= s= γ ] yi,s 2 2 s= γ ] yi,s 2 2 γ yi,s 2 2

21 + sup βψ t γ t E y i,t γ α s= + sup βψt γ η i E γ t α α s= γ ] yi,s 2 2 y 2 i,s γ 2 ]}. Now, we consider the first, second, and fourth elements on the right hand side. Note that v it and ṽ itt includes the present and future legs of the error terms, i.e., v it, v i,t+,..., v it, hence by Assumption we have v t E it γ s= t E ṽitt γ s= y 2 i,s y 2 i,s γ γ ] 2 = E t v it γ E s= 2 ] = E ṽ itt γ E t s= γ yi,s 2 2 y 2 i,s η i E γ t γ ] y 2 2 i,s = E η i t γ γ E y 2 2 i,s. α α s= s= Under Assumption 5 E v it γ, E ṽ itt γ, and E η i γ are constant. Therefore, the order ] of magnitude of E sup Z i y i αx i γ is determined by the third term in A-6. α We have A-7 t E y it γ s= y 2 i,s γ where we have used Loève s c r inequality. A-8 Solving recursively we obtain 2 ] E y i,t γ t γ 2 t s=, γ 2, y ] i,s γ t = t γ 2 E ] y i,t y i,s γ, s= y i,t = α t y i0 + αt α η t 2 i + α k v i,t k and using A-8 along with Loève s c r inequality in A-7, and under Assumption 2 and Assumption 5 we get k=0 A-9 t γ 2 t E α s+t 2 yi0 2 + αt t 2 α αs η i y i0 + α k α s v i,t k y i0 + αs α αt y i0 η i s= 9 k=0

22 + αs α t t 2 α α η2 i + α k αs α v s 2 i,t kη i + α k α t y i0 v i,s k k=0 k=0 j=0 k=0 k=0 s 2 + α k αt α v t 2 s 2 i,s kη i + α k+j v i,t j η i v i,s k γ Ct γ 2 t α γs+t 2 E y 2 i0 γ + α t α αs γ E ηi γ E yi0 γ s= + α s γ E y i0 γ t 2 E α k α s v i,t k γ + α s α αt γ E y i0 γ E η i γ + αs α k=0 α t α γ E η i 2 γ + αs α γ E η i γ t 2 E α k v i,t k γ + α t γ E y i0 γ s 2 E α k v i,s k γ + α t γ E η i γ s 2 E α k v i,s k γ α k=0 + E t 2 s 2 α k+j v it j η i v i,s k γ j=0 k=0 Ct γ { α γ t 2 α γ + α t α γ t α γ t α γ t α α γ + t γ α γ 2 + α γ t t α γ + αγ t α γ + α t γ α α γt + αγ t α γ + t α γ + αγ t α γ t α γ α γ t γ + α t t α γ s γ s= + α t t α α γ s γ + t γ t α γ 2 s γ. s= ] Hence, to determine the order of magnitude of E sup Z i y i αx i γ we consider α T CT γ 2 sup βψ t γ t E ω i,t γ t= T CT γ 2 t E ω i,t γ α t= s= s= y 2 i,s k=0 γ ] yi,s 2 2 γ 2 ] T CT γ 2 Ct γ { 2 α γ + α α γ + t γ α t= γ 2 + α γ t + α γ + α 2γ t + α γ k=0 s= 20

23 + + α γ t + α α γ = OT 3γ 2, α γ t s= α γ t γ + α γ s γ + t γ α γ 2 t s γ s= t s γ } where the first inequality uses the fact that βψ t γ is bounded in t and α and the second inequality uses the fact that α γ t, α t and α t are bounded in t. For the last result we used the fact that T v+ T t= tv /v + cf. Hamilton 994, Proposition 7.4 h. s= Proof of Theorem. Combining the result in Lemma 2 with the first result in Appendix of Guggenberger & Smith 2005, we have A-0 max sup i N α Z iy i αx i = O p N /γ Esup Z iy i αx i γ ] /γ = O p N /γ T 3 2/γ. By the hypothesis of the theorem, namely N /γ T 3 2/γ T 2 /N 0 for γ > 2, there exists τ N such that T N = oτ N and τ N = on /γ T 3+2/γ. Let L N = {λ : λ τ N }. Note that sup λ L N, α <,i N λ Z iy i αx i τ N max i N sup α < α Z iy i αx i = O p τ N N /γ T 3 2/γ 0. Note that the multipliers of the moment conditions, λ, have to satisfy the condition λ Z i y i αx i >, for all i =,..., N. Let Lα be the set of λs that satisfies this condition, i.e. Lα = {λ : λ Z i y i αx i >, i =,..., N}. Therefore, there exists a C such that w.p.a., for all α, λ L N, i N A- L N Lα, C + λ Z i y i αx i ]2 C, + λ Z i y i αx i ]3 C. Let P α, λ = N ln + λ Z i y i αx i ]. By a Taylor expansion around λ = 0 with Lagrange remainder for all λ L N P α, λ = λ Z iyi αx i λ T N Z i y i αx i y i αx i Z i + λ Z i y i αx i ] 2 λ 2

24 where λ lies between λ and 0. By Assumption 4 i, LemmaA0 of Newey & Windmeijer 200, we have, w.p.a. λ min N N Z i y i αx i y i αx i Z i C and λ max N N Z i y i αx i y i αx i Z i C. Then, w.p.a. for all α < and λ LN, A-2 λ Z iy i αx i C T λ 2 P α, λ λ λ Z iy i αx i C T λ 2 Z iy i αx i C T λ 2 Let λ = arg max λ L N P α0, λ. By the right hand inequality in eq. A-2, 0 = P α 0, 0 P α 0, λ λ Z iv i C T λ 2. Subtracting C T λ 2 from both sides and dividing through by C T λ and using the result of Lemma gives λ C N Z iv i = O p T N. Now, following the same arguments in the proof of Lemma A3 of Newey & Windmeijer 200, it can be shown that λ = O p T N. Now, expanding around λ = 0 note that we let α α EL for simplification to obtain Q α = P α, λ = ln + yi αx i Z i + λ Z i y i αx i ] λ=0 λ N ] 2 λ Z i y i αx i y i αx i Z i + λ Z i y i λ=0 λ αx i ]2 + 3 = + ˆr + λ Z i y i αx i ] 3 ] 3 yi αx i Z iˆλ yi αx i Z i λ 2 λ Z iy i αx i yi αx i Z i λ where ˆr = 3 ] 3 + λ Z i y i αx i ] 3 yi αx i Z iˆλ 22

25 and λ λ. We have, w.p.a. r 3 λ max sup i N α C λ max sup i N α Z iy i αx i T λ N Z iy i αx i T λ N Z i y i αx i y i αx i Z i + λ Z i y i αx i ] 3 ] Z iy i αx i yi αx i Z i λ ] λ O p T 2 /NN /γ T 3 2/γ C T λ 2 = o p T N. Also, λ solves the equations: Expanding around λ = 0: where 0 = = R = 2 + λ Z i y i αx i ]Z iy i αx i = 0. + λ Z i y i αx i ]Z iy i αx i λ=0 + λ Z i y i αx i ]2 Z iy i αx i y i αx i Z i λ=0 λ 2 + λ Z i y i αx i ] 3 yi αx i Z iˆλ] Z i yi αx i Z iy i αx i Z iy i αx i yi αx i Z i λ + R 2 + λ Z i y i αx i ] 3 yi αx i Z iˆλ] Z i yi αx i and λ lies in between 0 and λ. Note that max λ Z i i N y i αx i max λ Z i i N y i αx i λ max sup i N α Then we have Z i y i αx i 0. R C max sup Z iy i αx i i N α + λ Z i y i αx i 3 T λ N C max Z iy i αx i yi αx i Z i λ sup i N α Z iy i αx i T λ 2 = O p N /γ T 3 2/γ T/N = o p T/N. 23

26 Solving for λ: λ = ] Z iy i αx i yi αx i Z i ] Z iy i αx i + R. Plugging into Q α: A-3 ] ] Q α = yi αx i Z i Z iyi αx i yi αx i Z i ] ] Z iyi αx i + R Z 2 iyi αx i + R ] ] Z iyi αx i yi αx i Z i Z iy i αx i + R + o p T N ] ] = Z 2 iyi αx i Z iyi αx i yi αx i Z i ] ] Z iyi αx i 2 R Z iyi αx i yi αx i Z i R + o p T N ] ] Z iyi αx i Z iyi αx i yi αx i Z i ] Z iyi αx i R 2 C + o p T N ] ] = Z iyi αx i Z iyi αx i yi αx i Z i ] Z iyi αx i + o p T N. The first term on the right side of A-3 is the objective function of continuously updating GMM estimator CUE. Hence the last conclusion shows that the difference of the CUE and EL objective functions converges uniformly to zero in α. The remainder of the proof 24

27 then follows from the proof for of Theorem 3 the consistency of the LIML estimator of AA upon noting that in a linear model under homoskedasticity CUE is the LIML estimator as mentioned in section 2.2. Lemma 3. If Assumption 4 i holds, Assumption 8 ii of Newey & Windmeijer 200 also holds. Proof. In Newey & Windmeijer 200 notation a Ω k α Ω k α ] b = a EZ iy i αx i x i Z i ] EZ iy i αx i x i Z i ] ] b = a EZ iw i α = a 0 EZ iw i 0 α α Wi Z i ] EZ iw i α W i Z i ] ] b 0 W i Z i ] ] b { 0 0 = tr 0 α α E Wi Z i ba Z iw i ] } = α α Ex i Z i ba Z ix i ] = α α Ea Z ix i b Z ix i ] α α Ea Z ix i 2 ] /2 Eb Z ix i 2 ] /2 C α α a b, where for the second to last inequality, we used Cauchy-Schwartz inequality and the last inequality is obtained by noting that Ea Z ix i 2 ] = a EZ ix i x i Z i ]a a 2 λ max {EZ ix i x i Z i ]} C a 2, by Rayleigh quotient and Assumption 4. The other parts of 8 ii follow by Ω k,l α and Ω kl α not depending on α. 25

28 Proof of Theorem 2. We proceed by verifying all the hypotheses of Theorem 3 of Newey & Windmeijer 200. Note that Z i y i αx i is twice continuously differentiable and that its first derivative does not depend on α, so Assumption 7 is satisfied. Also, for our case, Assumption 9 i of Newey & Windmeijer 200 holds from the hypothesis of the theorem, namely N /γ T 3 2/γ T 2 /N 0 for γ > 2. Now, we show that their Assumption 6 and Assumption 9 ii also hold. In Newey & Windmeijer 200 notation, we have m T E g i 4 ] + E G i 4 ] n = E Z iv i 4 ] + E Z ix i 4 ] N T CE Z i 4 ] + E x i 4 Z i 4 ] N. The order of magnitude of the second term in the summation dominates that of the first term. Therefore, it is sufficient to show that E Z i x i 4 T ] N 0. Following the similar steps as in Lemma 2, we note that E Z i x i γ ] = E Z i y i αx i γ ] = OT 3γ 2. Hence, for γ = 4, we have E Z ix i 4 ] = OT 0 Hence, the first part of Assumption 6 and Assumption 9 ii of Newey & Windmeijer 200 hold since T /N 0. The second part of their Assumption 6 follows from Assumption 4 i, and the rest holds by the model being linear in α. The parts of Assumption 8 i of Newey & Windmeijer 200 follow similarly upon noting that, for W i sup α N = yi : x i by Assumption 4 i we have Z iy i αx i yi αx i Z i EZ iy i αx i yi αx i Z i ] = sup α α N W i Z i Z iw i EW i Z i Z iw i ] α 0. 26

29 Hence, using Newey & Windmeijer 200 notation, we have Ω k α Ω k α = N = α N Z iy i αx i x i Z i EZ iy i αx i x i Z i ] Ω k,l α Ω k,l α = N = 0 N Ω kl α = Ω kl α = 0. W i Z i Z iw i EW i Z i Z iw i ] 0 Z ix i x i Z i EZ ix i x i Z i ] W i Z i Z iw i EW i Z i Z iw i ] 0,, and Finally, assumption 8 ii is satisfied by Lemma 3. References Ahn, S. C. & Schmidt, P Efficient estimation of models for dynamic panel data. Journal Econometrics, 68, Alonso-Borrego, C. & Arellano, M Systematically normalized instrumental-variable estimation using panel data. Journal of Business & Econometric Statistics, 7, Alvarez, J. & Arellano, M The time series and cross-section asymptotics of dynamic panel data estimators. Econometrica, 7, Anderson, T. W. & Hsiao, C Formulation and estimation of dynamic models using panel data. Journal of Econometrics, 8, Arellano, M Panel Data Econometrics. Oxford University Press, New York, NY, U.S.A. 27

30 Arellano, M., & Bond, S. R., 99. Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Review of Economic Studies, 58, Arellano, M., & Bover, O., 995. Another look at the instrumental variable estimation of error-components models. Journal of Econometrics, 68, Bekker, P. A Alternative approximations to the distributions of instrumental variable estimators. Econometrica, 62, Blundell, R. & Bond, S Initial conditions and moment restrictions in dynamic panel data models. Journal od Econometrics, 87, Bond, S. R Dynamic panel data models: a guide to micro data method and practice. Portuguese Economic Journal,, Bowsher, C. G On testing overidentifying restrictions in dynamic panel data models. Economics Letters, 77, Brown, B. & Newey, W GMM, efficient bootstrapping, and improved inference. Journal of Business & Econometric Statistics, 20, Gonzalez, A Empirical likelihood: improved inference within dynamic panel data models. ESE Discussion Papers 54, Edinburgh School of Economics, University of Edinburgh. Guggenberger, P. & Smith, R. J Generalized empirical likelihood estimators and test under partial, weak, and strong identification. Econometric Theory, 2, Hamilton, J. D Time Series Analysis. Princeton University Press, Princeton, NJ, U.S.A. Han, C. & Phillips, P C. B GMM with many moment conditions. Econometrica, 74,

31 Hansen, L. P., Heaton, J., & Yaron, A Finite sample properties of some alternative GMM estimators obtained from financial market data. Journal of Business & Econometric Statistics, 4, Holtz-Eakin, D., Newey, W., & Rosen, H Estimating vector autoregressions with panel data. Econometrica, 56, Imbens, G. W One step estimators for over-identified generalized method of moments models. Review of Economics Studies, 64, Kiviet, J. F On bias, inconsistency, and efficiency of various estimators in dynamic panel data models. Journal of Econometrics, 68, Kunitomo, N Asymptotic expansions of the distributions of estimators in a linear functional relationship and simultaneous equations. Journal of the American Statistical Association, 75, Morimune, K Approximate distribution of k-class estimators when the degree of overidentifiability is large compared with the sample size. Econometrica, 5, Newey, W. & Smith, R. J Higher-order properties of GMM and generalized empirical likelihood estiamtors. Econometrica, 72, Newey, W. & Windmeijer, F GMM with many weak moment conditions. Econometrica, 77, Nickell, S. 98. Biases in dynamic models with fixed effects. Econometrica, 49, Oĝuzoĝlu, U Empirical likelihood estimation of dynamic panel data models with fixed effects. Unpublished manuscript. Qin, J. & Lawless, J Empirical likelihood and generalized estimating equations. Annals of Statistics, 22,

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33 Table : Properties of EL, GMM & LIML Estimators: Many Instruments Case N = 00 α = 0.2 α = 0.5 α = 0.8 EL GMM LIML EL GMM LIML EL GMM LIML T 0 = 0 median iqr mae T 0 = 25 median iqr mae T 0 = 50 median iqr mae iqr is the 75th-25th interquartile range; mae denotes the median absolute error; σ 2 = and σ 2 η = 0. 3

34 Table 2: Properties of EL, GMM & LIML Estimators: Many Instruments Case N = 50 α = 0.2 α = 0.5 α = 0.8 EL GMM LIML EL GMM LIML EL GMM LIML T 0 = 0 median iqr mae T 0 = 25 median iqr mae T 0 = 50 Median iqr mae iqr is the 75th-25th interquartile range; mae denotes the median absolute error; σ 2 = and σ 2 η = 0. 32

35 Table 3: Properties of EL, GMM & LIML Estimators: Many Weak Instruments Case N = 00; σ 2 = ; σ 2 η = 0 α = 0.2 α = 0.9 EL GMM LIML EL GMM LIML T 0 = 0 median iqr mae T 0 = 25 median iqr mae T 0 = 50 median iqr mae iqr is the 75th-25th interquartile range; mae denotes the median absolute error. 33

36 Table 4: Properties of EL, GMM & LIML Estimators: Many Weak Instruments Case N = 00; σ 2 = ; σ 2 η = α = 0.2 α = 0.9 EL GMM LIML EL GMM LIML T 0 = 0 median iqr mae T 0 = 25 median iqr mae T 0 = 50 median iqr mae iqr is the 75th-25th interquartile range; mae denotes the median absolute error. 34

37 Table 5: Properties of EL, GMM & LIML Estimators: Many Weak Instruments Case N = 00; σ 2 = ; σ 2 η = 4 α = 0.2 α = 0.9 EL GMM LIML EL GMM LIML T 0 = 0 median iqr mae T 0 = 25 median iqr mae T 0 = 50 median iqr mae iqr is the 75th-25th interquartile range; mae denotes the median absolute error. 35