Asymptotic distributions of the quadratic GMM estimator in linear dynamic panel data models

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1 Asymptotic distributions of the quadratic GMM estimator in linear dynamic panel data models By Tue Gørgens Chirok Han Sen Xue ANU Working Papers in Economics and Econometrics # 635 May 2016 JEL: C230 ISBN:

2 Asymptotic distributions of the quadratic GMM estimator in linear dynamic panel data models Tue Gørgens Chirok Han Sen Xue Abstract: This paper establishes asymptotic distributions of the quadratic GMM estimator of the autoregressive parameter in simple linear dynamic panel data models with fixed effects under standard minimal assumptions. The number of time periods is assumed to be small. Focusing on settings where autoregressive parameter is uniquely identified, nonstandard convergence rates and limiting distributions arise in the well-known random walk case, as well as in other previously unrecognized cases. The paper finds that the convergence rates are slow in the nonstandard cases, and the limiting distributions are a mixture of two nonnormal distributions. The findings are illustrated using Monte Carlo simulations. Keywords: Dynamic panel data models, fixed effects, generalized method of moments, nonstandard limiting distributions. JEL classification codes: C230. Acknowledgment: This research was supported in part by a grant from the Korean Government (NRF-2014S1A2A ). The Australian National University. tue.gorgens@anu.edu.au. Korea University. chirokhan@korea.ac.kr. The Australian National University. sen.xue@anu.edu.au.

3 1 1 Introduction In this paper we establish the asymptotic distributions of the GMM estimator of the autoregressive parameter in simple linear AR(1) dynamic panel data models with fixed effects. We consider the case where the number of individuals is large and the number of time periods is small. We impose only the standard minimal assumptions that the idiosyncratic errors are serially uncorrelated and uncorrelated with the initial data values and with the fixed effects. The assumptions imply a number of linear and quadratic moment restrictions on the first and second moments of the data, which can be used to identify and estimate the autoregressive parameter. We focus on the GMM estimator that is based on all moment restrictions available under the standard minimal assumptions. We refer to this as the quadratic GMM estimator. We establish the rates of convergence and the asymptotic distributions of the quadratic GMM estimator under general conditions. In many cases, standard regularity conditions are satisfied and standard root-n convergence and asymptotic normality prevail. However, we show that nonstandard asymptotic results arise when the linear moment restrictions alone do not identify the autoregressive parameter and the quadratic moment restrictions have a double root. Most notably, this can happen in the well-known random walk case where the autoregressive parameter is 1 and there are no fixed effects (i.e. the time series for each individual is a random walk), provided the variances of the idiosyncratic errors are constant over time. We find that the asymptotic distribution is very unusual in the nonstandard cases, in that it consists of two parts depending on the sign of a particular statistic. If the statistic is negative, the rate of convergence is n 1/2 and the limiting distribution is highly skewed. If the statistic is positive, the rate of convergence is n 1/4, which is very slow, and simulation results indicate that the limiting distribution is bimodal. The quadratic GMM estimator is important because it is efficient under the standard minimal assumptions, provided an optimal weight matrix is used. The quadratic GMM estimator can be contrasted with the inefficient difference GMM estimator, which is based on linear moment restrictions only (e.g. Holtz-Eakin, Newey, and Rosen, 1988; Arellano and Bond, 1991). The system GMM estimator is generally not valid under the standard

4 2 minimal assumptions, but requires an additional stationarity assumption (e.g. Arellano and Bover, 1995; Ahn and Schmidt, 1995; Blundell and Bond, 1998). The stationarity assumption may not be appropriate in all empirical applications (e.g. Hayakawa, 2009; Calzolari and Magazzini, 2013). The random walk case has attracted considerable interest in the literature, because data often exhibit substantial persistence and asymptotic distributions typically provide poor approximations to the finite-sample distributions in these situations (e.g. Blundell and Bond, 1998). For example, Bond and Windmeijer (2005) studied the Wald test when the datagenerating process is near the random walk. Kruiniger (2009) and Bun and Kleibergen (2013) derived local asymptotic approximations to the finite-sample distributions of certain well-known linear GMM estimators when the data are close to the random walk case. These studies maintain the stationarity assumption. The present paper contributes to this literature by establishing the asymptotic distribution of the (nonlinear) quadratic GMM estimator, by showing that there are more nonstandard cases than the random walk, and by avoiding the stationarity assumption. Our results also show that standard asymptotic results are usually applicable in the random walk case if the idiosyncratic errors are heteroskedastic. Hence, the quadratic GMM estimator can be a solution to the identification and weak IV problems associated with other estimators when the autoregressive parameter is 1 or close to 1. Throughout this paper, we focus on cases where the autoregressive parameter is uniquely identified. In other recent work, we showed that the identification of the autoregressive parameter cannot be taken for granted, even when it is assumed to be less than 1 (Gørgens, Han, and Xue, 2016). In particular, we showed that there are three possibilities when the linear moment restrictions do not provide identification. Either the quadratic moment restrictions uniquely identify the autoregressive parameter, or they partially identify it, or it is fully unidentified. The latter occurs if and only if there is no individual time variation in the data for t = 0,..., T 2, and is not of much practical interest. Partial identification comes in the form of precisely two candidate values, one of which is the true value. This paper focuses on the case where the autoregressive parameter is uniquely identified, and we defer

5 3 distributional issues related to partial identification to future research. The paper is organized as follows. Section 2 presents the model and defines the quadratic GMM estimator. Section 3 establishes consistency and limiting distributions. Section 4 presents simulations illustrating the nonstandard results, and Section 5 concludes. Appendix A contains proofs of theorems. 2 The model and the quadratic GMM estimator Suppose we have two-dimensional panel data. Following common practice, we shall refer to the first dimension as individuals and the second as time periods. For i = 1,..., n individuals and t = 0,..., T times, let y it be scalar random variables which are observed and available for analysis. 1 Note that y i0 is observed. Assumption 1 The random variables {(y i0,..., y it ) : i = 1,..., n} are independent across individuals, have finite means, and satisfy y it = y it 1 α 0 + c i + v it, t = 1,..., T, (1) with E(v it ) = 0 for t = 1,..., T. In Assumption 1, α 0 is an unknown (true) parameter to be estimated, and the sum c i +v it is an unobserved term decomposed into an individual-specific random variable, c i, which is constant over time, and an individual- and time-specific term, v it. The model does not have an explicit constant term; this is subsumed into c i. The parameter of interest is α 0. Assumption 1 postulates that the joint distribution of the observed variables derives from the joint distribution of y i0, c i, and v i1,..., v it. Restrictions on the latter are needed to identify α 0, and the standard set are stated as Assumption 2 (Ahn and Schmidt, 1995). 1 For simplicity, the expression i = 1,..., n is suppressed in the remainder of this paper.

6 4 Assumption 2 The random variables in Assumption 1 have finite variances and satisfy E(v it y i0 ) = 0, t = 1,..., T; (2) E(v it c i ) = 0, t = 1,..., T; (3) E(v is v it ) = 0, s = 1,..., t 1, t = 2,..., T. (4) The restrictions imposed by Assumptions 1 and 2 can be utilized to estimate α 0. The first moment restrictions (Han and Kim, 2014; Gørgens, Han, and Xue, 2016) are E( y it y it 1 α) = 0, t = 2,..., T. (5) The linear AB moment restrictions (Holtz-Eakin, Newey, and Rosen, 1988; Arellano and Bond, 1991) are E(y is ( y it y it 1 α)) = 0, s = 0,..., t 2, t = 2,..., T. (6) The quadratic AS moment restrictions (Ahn and Schmidt, 1995) are E((y it y it 1 α)( y it y it 1 α)) = 0, t = 2,..., T 1. (7) There are many other valid moment restrictions, but Ahn and Schmidt (1995) showed that under Assumption 2 they cannot contribute to identification or efficiency over the set given above. We refer to the GMM estimator based on the complete set of moment restrictions as the quadratic GMM estimator. The empirical moment restrictions are defined by replacing expectations in (5), (6), and (7) with sample averages. To simplify the discussion, it is convenient to express the empirical moment restrictions in matrix form. Define z it =

7 5 (1, y i0,..., y it ) and the vectors z i0 y i2. z it 2 y it h 0i =, h y it y 1i = i2. z i0 y i1. z it 2 y it 1 y it y i1 + y it 1 y i2., h 2i = 0. 0 y it 1 y i1.. (8) y it y it 1 y it y it 2 + y it 1 y it 1 y it 1 y it 2 Define the moment function, m, by m(z it, α) = h 2i α 2 h 1i α + h 0i. Define g n by n g n (α) = n 1 m(z it, α). (9) i=1 Then the empirical moment restrictions can be written as g n (α) = 0. For a given weight matrix, W n, the GMM loss function, Q n, is defined by Q n (α) = g n (α) W n g n (α). (10) This is a quartic polynomial in α, which can have at most three distinct local extrema; namely, two local minima and one local maximum. The first-order condition for a minimum is DQ n (α) = 0, where 1 2 DQ n(α) = D α g n (α) W n g n (α). (11) There are either 1 or 3 real-valued solutions to the first-order condition. If there are 3 distinct real roots, one of them corresponds to a local maximum while the others correspond to local minima. In this case, the GMM loss function can be evaluated at the roots to identify the global minimum. We define the quadratic GMM estimator, α n, as the global minimizer of Q n. The quadratic GMM estimator is not unique since it depends on the weight matrix.

8 6 3 Asymptotic theory In this section, we focus on cases where α 0 is uniquely identified, and consider the asymptotic properties of the quadratic GMM estimator. When standard regularity conditions are satisfied, the global minimizer of Q n, i.e. α n, converges to the true value, α 0, and its limiting distribution is normal. However, the standard regularity conditions are not satisfied in all cases. In the following, references to results discussed by Newey and McFadden (1994) are prefixed by NM. Define the nonstochastic function g by g(α) = E(m(z it, α)) = E(h 2i )α 2 E(h 1i )α + E(h 0i ). (12) Then α 0 is uniquely identified by the moment restrictions if and only if α 0 is the unique solution to g(α 0 ) = 0. While all elements of g may contribute to identification, there are cases where the elements corresponding to the linear moment restrictions are constant functions and identification stems entirely from the quadratic moment restrictions (Gørgens, Han, and Xue, 2016). In choosing a weight matrix it is therefore essential to ensure identification is preserved. Accordingly, let W be a nonstochastic positive semidefinite matrix such that W g(α) = 0 holds if and only if g(α) = 0. When α 0 is uniquely identified, we have the usual consistency result (NM-Theorem 2.6), stated here as Theorem 1 for convenience. Theorem 1 Suppose Assumptions 1 and 2 hold, W n p W, W is positive semidefinite and satisfies W g(α) = 0 if and only if g(α) = 0, and α 0 is a unique solution to g(α 0 ) = 0. If α 0 Θ where Θ is known and compact, then α n p α 0. Given consistency, the next question concerns the rate of convergence and the limiting distribution of the estimator, α n. If key regularity conditions are satisfied, then asymptotic normality and consistency of the estimator of the asymptotic variance may follow from standard theorems. However, it turns out that there are cases where standard asymptotics break down. In the remainder of this section, we treat first the standard, then the nonstandard

9 7 cases. If the right-hand sides are well defined, define the following nonstochastic matrices G = E(D α m(z it, α 0 )), (13) Ω = E(m(z it, α 0 )m(z it, α 0 ) ), (14) and V = (G W G) 1 G W ΩW G(G W G) 1. (15) Define also the sample analogues G n = n 1 Ω n = n 1 and n D α m(z it, α n ), (16) i=1 n m(z it, α n )m(z it, α n ), (17) i=1 V n = (G n W ng n ) 1 G n W nω n W n G n (G n W ng n ) 1. (18) Theorem 2 provides asymptotic normality and a consistent estimator of the asymptotic variance under standard regularity conditions. Theorem 2 Suppose the assumptions of Theorem 1 are satisfied. If α 0 is in the interior of Θ, and if G W G is nonsingular, then n 1/2 (α n α 0 ) N(0, V ). Suppose further that for a neighborhood of α 0 we have E(sup α m(z it, α) 2 ) <. Then V n p V. The proof of Theorem 2 is omitted, since the conclusions follow almost immediately from NM-Theorem 3.4 and NM-Theorem 4.5. The last part of Theorem 2 implies that the asymptotic variance of α n can be estimated by Avar(α n ) = n 1 V n. The complicated regularity condition essentially assumes finite and bounded fourth moments; this is used to ensure

10 8 consistency of Ω n. The critical regularity condition for ensuring standard asymptotic results is the nonsingularity of G W G. Since W is a positive semidefinite matrix which preserves identification, the singularity of G W G depends on whether or not G = 0. Recall that g(α) = E(h 2i )α 2 E(h 1i )α + E(h 0i ) and that G = 2E(h 2i )α 0 E(h 1i ). Each element of g(α) is a polynomial in α of degree at most 2, say p 2 α 2 + p 1 α + p 0. The corresponding element of G is 2p 2 α 0 + p 1. Since identification of α 0 means that g(α) = 0 if and only if α = α 0, the degree of the polynomial and the value of G is closely related to the identification. If the degree of a particular element of g is 0 (i.e. p 2 = 0, p 1 = 0, p 0 = 0), the corresponding element of G is 0. In this case, the corresponding moment restriction does not identify α 0. If the degree is 1 (i.e. p 2 = 0, p 1 0), the corresponding element of G is not 0. In this case, the corresponding (linear) moment restriction uniquely identifies α 0. If the degree is 2 (i.e. p 2 0), the corresponding element of G is 2p 2 α 0 + p 1. In this case, the corresponding (quadratic) moment restrictions uniquely or partially identifies α 0. In particular, it is not difficult to show that 2p 2 α 0 + p 1 = 0 if and only if the element of g(α) uniquely identifies α 0 and has a double root at α 0. To sum up, G = 0 if either α 0 is fully unidentified, or if the linear (first or AB) moment restrictions fail to identify α 0 but the quadratic AS moment restrictions provide unique identification in the form of a common double root. Two question arise. First, is the case where α 0 is uniquely identified but G = 0 compatible with data-generating processes given by Assumptions 1 and 2? That is, is it possible to observe such data? Second, what is the rate of convergence and the limiting distribution of the quadratic GMM estimator in such cases? In Theorem 3, we give an affirmative answer to the first question. Theorem 3 Suppose Assumptions 1 and 2 hold, and α 0 is uniquely identified. Then G 0 except the following situations: (i) T 3, α 0 = 1, E(c 2) = 0, and i E(v2 ) = it 1 E(v2 ) for all t = 2,..., T 1. it (ii) T = 3, ( 5 1)/2 α 0 < 1, (1 α 0 )E(y i0 ) = E(c i ), (1 α 0 )E(y 2 i0 ) = E(y i0c i ), E(v 2 i1 ) = α 0E(c i y i0 ) + α 0 (1 α 0 ) 1 E(c 2 i ), and E(v2 i2 ) = (α2 0 + α 0 1)α 1 0 E(v2 i1 ).

11 9 A proof is provided in Appendix A.1. Interestingly, Theorem 3 shows that heteroskedasticity is important for obtaining standard root-n asymptotic results in the well-known random walk case where α 0 = 1 and E(c 2 i ) = 0. The answer to the second question is provided in Theorem 4. Briefly, when G = 0 the GMM estimator converges at a slow O p (n 1/4 ) rate, and the asymptotic distribution is highly nonstandard. For convenience, define δ = α α 0 and δ n = α n α 0, and consider the reparameterized loss function g n (δ) W n g n (δ), where g n is defined by g n (δ) = g n (δ + α 0 ). Then g n (δ) W n g n (δ) is globally minimized at δ = δ n. To state the results some additional notation is needed. First, define h 0i = h 2i α 2 0 h 1iα 0 + h 0i, h 1i = 2h 2i α 0 + h 1i, and h 2i = h 2i. Then we have g n (δ) = n 1 n i=1 ( h 2i δ 2 h 1i δ+ h 0i ). Second, define H jn = n 1 n i=1 h ji for j = 0, 1, 2. Third, define A 0n = H 0n W nh 0n, A 1n = 2H 0n W nh 1n, A 2n = H 1n W nh 1n + 2H 0n W nh 2n, A 3n = 2H 1n W nh 2n, and A 4n = H 2n W nh 2n. Then we have g n (δ) W n g n (δ) = A 4n δ 4 + A 3n δ 3 + A 2n δ 2 + A 1n δ + A 0n. (19) Finally, define L n = A 3n 4A 4n, (20) and B 1n = 4A 4nL 3 n + 3A 3nL 2 n + 2A 2nL n + A 1n 4A 4n, (21) B 2n = 6A 4nL 2 n + 3A 3nL n + A 2n 2A 4n. (22) Let { } denote the indicator function. Then we have the following result. Theorem 4 Suppose the assumptions of Theorem 1 are satisfied. If α 0 is in the interior of Θ and if G = 0, then n 1/2 (α n α 0 ){B 2n 0} (w L w 1 /w 2 ){w 2 0} and n 1/4 (α n α 0 ){B 2n > 0} sgn(w 1 )w 1/2 2 {w 2 > 0}, where (w L, w 1, w 2 ) is the weak limit of (n 1/2 L n, nb 1n, n 1/2 B 2n ). Also P(B 2n 0) 1/2.

12 10 Theorem 4 covers both exceptions given in Theorem 3; in particular, it provides the asymptotic distribution for the quadratic GMM estimator in the random walk case where α 0 = 1, c i = 0 for all i, and the variances of the idiosyncratic errors are constant over time. A proof of Theorem 4 can be found in Appendix A.2. The proof does not use the explicit formula for α n. Instead, it is based on first establishing the rate of convergence, and then establishing that a normalized version of the loss function weakly converges locally uniformly; the limiting distribution of the estimator is then the same as the limiting distribution of the (unique) minimizer of the normalized loss function. Theorem 4 shows that the asymptotic distribution in the nonstandard cases is a mixture of two nonnormal distributions. Different scale normalizations are needed depending on the sign of B 2n, and the overall rate of convergence is very slow. Moreover, working directly with the explicit formula for α n, it can be shown that the rate of convergence of the normalized estimator n 1/2 (α n α 0 ){B 2n 0} to (w L w 1 /w 2 ){w 2 0} is also slow. That is, both the first-order and the higher-order approximation errors vanish only slowly as the sample size increases. Division by the mean-zero normal random variable w 2 in (w L w 1 /w 2 ){w 2 0} results in a limiting distribution with a very long and thick left tail. Since n 1/2 B 2n = O p (1), we may normalize by nb 2n instead of n 1/2. The asymptotic result is then nb 2n (α n α 0 ){B 2n 0} (w L w 2 w 1 ){w 2 0}. This is likely to provide a better approximation, especially in the left tail of the distribution. In the next section we verify the theoretical results in a small simulation study. The slow rate of convergence in nonstandard cases means that only when enormous sample sizes are employed is there good agreement between the asymptotic and the finite-sample distributions. (This is also likely to be true for data-generating processes which are near a nonstandard case.) In sample sizes typically encountered in practice, the approximation appears to be poor.

13 11 4 Simulation evidence In this short section we present simulation evidence to confirm the theoretical results in three cases. 2 First we confirm the asymptotic distribution for the nonstandard random walk case. Specifically, for DGP1 we set T = 4, α 0 = 1, E(y 2 ) = 1, i0 E(c2) = 0, E(y i i0c i ) = 0, and E(v 2 ) = it 1 for t = 1,..., T. This design satisfies part (i) of Theorem 3. Figure 1 shows kernel density estimates for nb 2n (α n α 0 ){B 2n 0}. (The mass point at 0 is omitted.) Asymptotically, the density is unimodal and somewhat skewed to the right. Figure 2 shows the difference between the finite-sample and the limiting CDF, and confirms convergence in distribution. Figures 3 and 4 provide similar graphs for n 1/4 (α n α 0 ){B 2n > 0}. Asymptotically, the density is bimodal, with a large mode between 1 and 0 and a smaller model between 0 and 1. The difference between the finite-sample and the limiting CDFs obviously vanishes as the sample size increases, confirming convergence in distribution. Figures 2 and 4 show that the first-order asymptotic distributions provide poor approximations to the finite-sample distributions for DGP1. Indeed, enormous sample sizes are required for the finite-sample distributions to resemble the limiting distributions. Interestingly, the magnitude of the differences is similar, despite very different rates of convergence. This is further evidence that the order of the approximation errors are relatively higher than usual. DGP2 is chosen to illustrate that the nonstandard limiting theory applies in cases other than the random walk. Here we set T = 3, α 0 = 0.8, E(y 2 i0 ) = 1, E(c2 i ) = 1, E(y i0c i ) = 0.2, E(v 2 ) = 1.96, i1 E(v2 ) = 1.45, i2 E(v2 ) = 1, and i2 E(v2 ) = 1. This design satisfies part (ii) of i2 Theorem 3. Figures 5 and 6 show results for nb 2n (α n α 0 ){B 2n 0}, and Figures 7 and 8 show results for n 1/4 (α n α 0 ){B 2n > 0}. The patterns are similar to those for DGP1. The last experiment illustrates that standard asymptotics apply in the random walk case when the idiosyncratic errors are heteroskedastic. In DGP3, we set T = 4, α 0 = 1, E(y 2 i0 ) = 1, E(c 2) = 0, E(y i i0c i ) = 0, E(v 2 ) = 1, i1 E(v2 ) = 2, i2 E(v2 ) = 0.5, and i2 E(v2 ) = 1. Figure 9 i2 shows kernel density estimates of the finite-sample distributions of n 1/2 V 1/2 (α n α 0 ) as 2 The basic variables y i0, c i, and v i1,..., v it have mean 0 and are normally distributed in all experiments. All simulations have Monte Carlo replications and use the optimal weight matrix.

14 12 well as the theoretical N(0, 1) limiting distribution. The agreement is obvious, even for moderate sample sizes. Figure 10 shows the difference between the finite-sample CDF and the standard normal CDF, and the convergence is clear. 5 Concluding remarks The literature concerned with the linear AR(1) dynamic panel data model with fixed effects is large. Authors have considered different assumptions and different estimators. In this paper, we focused on the quadratic GMM estimator, which is based on all moment restrictions available under the standard minimal assumptions. If optimal weight matrices are used, the quadratic GMM estimator is more efficient than the popular difference GMM estimator, which is based on the linear moment restrictions only. We investigated the asymptotic properties of the quadratic GMM estimator, under the maintained assumption that the autoregressive parameter is uniquely identified. We characterized the data-generating processes for which standard asymptotic results are and are not applicable. The nonstandard cases include the well-known random walk case when the variances of the idiosyncratic errors are constant over time, as well as other cases that have not previously been discussed in the literature. We derived the limiting distribution for the nonstandard cases, and found that the rate of convergence is very slow and the limiting distribution is not normal. Finally, we verified the theoretical limiting distributions is a small set of Monte Carlo experiments. In other recent work, we investigated identification of the autoregressive parameter, both in terms of observed data and in terms of the underlying data-generating process (Gørgens, Han, and Xue, 2016). The autoregressive parameter is uniquely identified in most cases. It is fully unidentified only when there is no time variation in the data, which is not particularly interesting case from a practical point of view. However, there are also cases where the autoregressive parameter is partially identified. When it is partially identified, there are precisely two solutions to the moment restrictions. In the present paper, we have assumed the autoregressive parameter is known to be uniquely identified. If it is not possible to rule out partial identification a priori, then the analysis is more complicated. Intuitively,

15 13 in case of partial identification, both empirical roots converge to the respective population roots. However, it does not follow that the global minimizer converges to the true parameter value. Moreover, the finding of two roots in a particular empirical study does not necessarily indicate partial identification. We defer the question of inference in the presence of partial identification to future research. A Proofs A.1 Proof of Theorem 3 To prove Theorem 3, we need to describe data-generating processes for which the linear (first moment or AB) moment restrictions fail to identify α 0 but the quadratic AS moment restrictions provide unique identification in the form of a common double root. Identification of α 0 under Assumptions 1 and 2 was characterized by Gørgens, Han, and Xue (2016), and the proof here borrows heavily from the arguments in our earlier work. For convenience, define Ψ = (α 0 1)E(c i y i0 ) + E(c 2). i Case Ψ = 0 and E(v 2 ) = 0 for all t = 1,..., T 2 it In their Appendix A.3, Gørgens, Han, and Xue (2016) showed that when the linear AB moment restrictions do not identify α 0, then the quadratic AS moment restrictions for t = 1,..., T 2 are satisfied for any value of α, while the restriction for t = T 1 becomes a linear equation, α 0 E(v 2 ) it 1 E(v2 )α = 0. (23) it 1 It follows that the quadratic AS moment restrictions uniquely identify α 0 if and only if E(v 2 it 1 ) 0. Therefore, if α 0 is uniquely identified, then G is a vector of 0s except for the last element which is nonzero, so G W G is nonsingular.

16 14 Case Ψ = 0 and E(v 2 ) 0 for some t = 1,..., T 2 it In their Appendix A.3, Gørgens, Han, and Xue (2016) showed that the linear AB moment restrictions do not identify α 0 if and only if α 0 = 1 and E(c 2 ) = 0. Moreover, the quadratic i AS moment restrictions become E(v 2 it 1 )α2 + [E(v 2 ) + it 1 E(v2 )]α it E(v2 ) = 0, t = 2,..., T 1. (24) it At least one of these equation is of degree 2, because E(v 2 ) 0 for some t = 1,..., T 2. it If T = 3, then there are roots at α = 1 and α = E(v 2 i1 )/E(v2 ). Hence, there is a double i1 root (and G W G is singular) if and only if E(v 2 i2 ) = E(v2 i1 ). If T 4, then the equations have two common roots at α = 1 and α = λ if and only if E(v 2 ) = it λe(v2 ) for all t = 2,..., T 1. Hence, there is a double root if and only if it 1 E(v 2 it ) = E(v2 it 1 ) for all t = 2,..., T 1. That is, if α 0 is uniquely identified, then G W G is singular if and only if E(v 2 ) = it E(v2 ) for all t = 2,..., T 1. it 1 Case Ψ 0 In their Theorem 3 and Appendix A.3, Gørgens, Han, and Xue (2016) showed that the linear AB moment restrictions do not identify α 0 if and only if 0 α 0 < 1, (α 0 1)E(y 2 ) + i0 E(y i0 c i ) = 0, and E(v 2 ) = it αt (1 α 0 0) 1 Ψ for t = 1,..., T 2. Moreover, the quadratic AS moment restrictions can be written α t 1 0 (α t α t 2 0 )α + α t 2 0 α 2 = 0, t = 2,..., T 2, (25a) (α T α 0 Σ) (α T α T Σ)α + α T 3 0 α 2 = 0, (25b) where Σ = (1 α 0 )Ψ 1 E(v 2 it 1 ) αt 1 0 and where we use the convention 0 0 = 1. If T = 3, then (25a) is empty and (25b) has roots α = α 0 and α = (1 α 2 0 ) + E(v2 i2 )(1 α 0 )Ψ 1. Suppose 0 < α 0 < 1. Then E(v 2 ) 0. By definition of Ψ, (25b) has roots at i1 α = α 0 and α = (1 α 2) + α 0 0E(v 2 i2 )/E(v2 ). Hence there is a double root if and only if i1 E(v 2 i2 ) = (α2 0 + α 0 1)α 1 0 E(v2 i1 ). This equation has a solution for α 0 in [0, 1) when ( 5

17 15 1)/2 α 0 < 1. Suppose α 0 = 0. Then E(v 2 ) = 0. By definition of Ψ, (25b) has roots i1 at α = 0 and α = 1 + E(v 2 i2 )/(E(c2 i ) E(y i0c i )). Hence there is a double root if and only if E(v 2 i2 ) = (E(c2 i ) E(y i0c i )). The requirements Ψ 0 and E(v 2 i2 ) 0 imply E(y i0c i ) > E(c 2 i ). However, the first equation in the text above (25) implies E(y i0 c i ) = E(y 2 ), and then the i0 Cauchy-Schwarz inequality E(y i0 c i ) 2 E(y 2 i0 )E(c2) simplifies to E(y i i0c i ) E(c 2 ). Since the i two inequalities are incompatible, a double root is not possible, so G W G is nonsingular. If T 4, then (25a) and (25b) have two common roots if and only if Σ = 0. The roots are α = α 0 and α = 1. Since α 0 1, a double root is not possible, so G W G is nonsingular. A.2 Proof of Theorem 4 As mentioned, we work with reparameterized GMM loss function g n (δ) W n g n (δ), which is minimized at δ n = α n α 0 as noted before. Depress the cubic term by the further reparameterization from δ to γ where γ = δ L n, where L n = A 3n /(4A 4n ) is given in (20). Then g n (δ) W n g n (δ) = 4 A jn δ j = A 4n γ 4 + K 2n γ 2 + K 1n γ + K 0n, γ = δ L n, (26) j=0 where K 0n = A 4n L 4 n + A 3nL 3 n + A 2nL 2 n + A 1nL n + A 0n, (27) K 1n = 4A 4n L 3 n + 3A 3nL 2 n + 2A 2nL n + A 1n, (28) K 2n = 6A 4n L 2 n + 3A 3nL n + A 2n. (29) Dividing by 4A 4n and ignoring the constant term (which does not affect the minimization problem), we can write the objective function as Q n (γ) = 1 4 γ4 1 2 B 2nγ 2 B 1n γ, (30) where B 1n = K 1n /(4A 4n ) and B 2n = K 2n /(2A 4n ) are given in (21) and (22). This Q n is globally minimized at γ n where γ n = δ n L n = α n α 0 L n.

18 16 Before embarking on the main proof, we need some lemmas. The first two are used to sign and bound the global minimizer γ n of the depressed quartic polynomial Q n for given n. Lemma 1 characterizes the sign of γ n. Lemma 1 Consider equation (30). If B 1n = 0 and B 2n 0, then γ n = 0. If B 1n = 0 and B 2n > 0, then γ n = ±B 1/2 2n. If B 1n 0, then sgn(γ n ) = sgn(b 1n ). PROOF Suppose that B 1n = 0. Then Q n (γ) = 1 4 γ4 1 2 B 2nγ 2. If B 2n 0, this is globally minimized at zero; otherwise, the global minimum is attained when γ 2 = B 2n. Next, note that Q n (γ) Q n ( γ) = 2B 1n γ. Suppose that B 1n > 0. Then for any γ < 0, we have Q n (γ) > Q n ( γ). It follows that the minimum of Q n occurs on [0, ); that is, γ n 0. Also γ n 0 because D Q n (0) = B 1n 0. Thus, γ n > 0 if B 1n > 0. We can similarly show that γ n < 0 if B 1n < 0. Lemma 2 provides bounds for γ n. Lemma 2 Consider equation (30). If B 2n 0, then B 2n γ n B 1n. If B 2n > 0, then B 1/2 2n γ n B 1/2 2n + B 1n 1/3. PROOF Consider the case B 2n 0. Clearly there is a unique solution to the first-order condition, D Q n (γ) = γ 3 B 2n γ B 1n = 0, because γ 3 is increasing and B 2n γ+b 1n is decreasing. Obviously that unique solution must be the global minimizer, γ n. Suppose B 1n > 0. By Lemma 1, we have γ n > 0 and then, since B 2n γ n + B 1n = γ 3 n 0, we have B 2nγ n B 1n. Since B 2n 0, we have B 1n B 2n γ n 0, and so B 2n γ n B 1n. The case B 1n < 0 is similar. The case B 1n = 0 is trivial. Consider the case B 2n > 0. First, suppose B 1n = 0. By Lemma 1, there are two global minimizers, B 1/2 2n and B1/2 2n, and obviously B1/2 2n γ n B 1/2 2n. Now suppose B 1n > 0. The first-order condition D Q n (γ) = γ 3 B 2n γ B 1n = 0 may have 1 or 3 real solutions. By Lemma 1, the global minimizer satisfies γ n > 0. Fortunately, since γ 3 is increasing at an increasing rate and is 0 at γ = 0, while B 2n γ + B 1n is increasing at a constant rate and is strictly positive at γ = 0, there is a unique positive solution to the first-order condition. Furthermore, since D Q n (B 1/2 2n ) = B 1n < 0 and D Q n (B 1/2 2n +B1/3 1n ) = 2B 2nB 1/3 1n +3B1/2 2n B2/3 1n > 0, we have B 1/2 2n γ n B 1/2 2n + B1/3 1n. The case B 1n < 0 is similar.

19 17 Define w Ln = n 1/2 L n, w 1n = nb 1n, and w 2n = n 1/2 B 2n. Lemma 3 Under the assumptions of Theorem 4, there is a random variable (w L, w 1, w 2 ) such that (w Ln, w 1n, w 2n ) (w L, w 1, w 2 ). The lemma implies that w Ln = O p (1), w 1n = O p (1), and w 2n = O p (1) or, equivalently, that L n = O p (n 1/2 ), B 1n = O p (n 1 ), and B 2n = O p (n 1/2 ). Moreover, these rates are sharp in the sense that generally Var(w L ) > 0, Var(w 1 ) > 0, and Var(w 2 ) > 0. PROOF Recall from Section 3 that g(δ) = g(δ +α 0 ) and g(δ) = E( h 2i )δ 2 E( h 1i )δ +E( h 0i ). As argued in Section 3, when α 0 is uniquely identified and G = 0, then the nonconstant elements of g are all second-degree polynomials whose derivative is 0 at 0. Since also g(0) = 0 by the assumption of unique identification, and G = 2E(h 2i )α 0 E(h 1i ) = E( h 1i ) by definition, it follows that E( h 0i ) = 0, E( h 1i ) = 0 and E( h 2i ) 0. By Assumptions 1 and 2, the data are iid across individuals and all variances exist. Hence, it follows by the Lindeberg-Lévy central limit theorem and the Cramér-Wold device that n 1/2 H 0n e 0 and n 1/2 H 1n e 1, where e 0 and e 1 are normal random variables. Furthermore, it follows by Khinchin s law of large numbers that H 2n p E( h 2i ). Therefore, A 1n = 2H W 0n nh 1n = O p (n 1 ), A 2n = H W 1n nh 1n + 2H W 0n nh 2n = O p (n 1/2 ), A 3n = 2H W 1n nh 2n = O p (n 1/2 ), and plim A 4n = plim H W 2n nh 2n p E( h 2i ) WE( h 2i ). Since W is positive semidefinite and preserves identification, we have E( h 2i ) WE( h 2i ) > 0. Finally, L n = O p (n 1/2 ), K 1n = O p (n 1 ), K 2n = O p (n 1/2 ), B 1n = O p (n 1 ), and B 2n = O p (n 1/2 ). For simplicity, define A 4 = E( h 2i ) WE( h 2i ). It then follows that and L n = A 3n 4A 4 + o p (n 1/2 ) = H 1n WE( h 2i) 2A 4 + o p (n 1/2 ), B 1n = 2A 2nL n + A 1n 4A 4 W + o p (n 1 ) = H WE( h 2i )E( h 2i ) W 0n 2A 4 2A 2 4 H 1n + o p (n 1 ), B 2n = A 2n 2A 4 + o p (n 1/2 ) = H 0n WE( h 2i) A 4 + o p (n 1/2 ).

20 18 Define w L, w 1, and w 2 as the leading terms on the right-hand sides in the above equations with e 0 and e 1 replacing H 0n and H 1n. Then (w Ln, w 1n, w 2n ) p (w L, w 1, w 2 ), which in turn implies (w Ln, w 1n, w 2n ) (w L, w 1, w 2 ). Note that the distribution of (w L, w 1, w 2 ) follows from the underlying normality of (e 0, e 1 ) and the continuous mapping theorem. The asymptotic behavior of α n is very different when B 2n 0 and when B 2n > 0. Lemma 4 provides rates of convergence. Lemma 4 Under the assumptions of Theorem 4, we have nb 2n (α n α 0 L n ){B 2n 0} = O p (1), n 1/4 (α n α 0 L n ){B 2n > 0} = O p (1), nb 2n (α n α 0 ){B 2n 0} = O p (1) and n 1/4 (α n α 0 ){B 2n > 0} = O p (1). PROOF Write γ n = γ n {B 2n 0} + γ n {B 2n > 0}. By Lemmas 2 and 3, B 2n γ n {B 2n 0} B 1n {B 2n 0} = O p (n 1 ) and γ n {B 2n > 0} (B 1/2 2n + B 1n 1/3 ){B 2n > 0} = O p (n 1/4 ). The conclusion now follows since α n α 0 = γ n + L n, and L n = O p (n 1/2 ) by Lemma 3. As mentioned in Section 3, nb 2n can be replaced with n 1/2 when B 2n 0. However, normalizing by nb 2n is likely to yield a more useful asymptotic distribution. Also, notice that it is not possible to improve on the slow convergence rate when B 2n > 0, because B 1/2 2n {B 2n > 0} γ n {B 2n > 0} and B 1/2 2n = O p(n 1/4 ) by Lemmas 2 and 3. That is, the rate n 1/4 is sharp. For the proof of Theorem 4, we consider the cases B 2n 0 and B 2n > 0 separately. First consider the case B 2n 0. We reparameterize from γ to θ where θ = nb 2n γ. Define N n by N n (θ) = n 1/2 (nb 2n ) 4 Q n (n 1 θ/b 2n ) = 1 4 n 1/2 θ w3 θ 2 w 2n 1n w 3 θ. (31) 2n Let θ N n denote the minimizer of N n, and note that θ N n = nb 2n γ n = nb 2n (α n α 0 L n ). Define N n and N by N n (θ) = N n(θ){b 2n 0} + θ 2 {B 2n > 0} (32)

21 19 and N (θ) = ( 1 2 w3 2 θ 2 w 1 w 3 2 θ){w 2 0} + θ 2 {w 2 > 0}. (33) Since N n (0) = 0, we have min θ N n (θ) 0, and therefore the minimizer of N n equals θ N n when B 2n 0 and 0 when B 2n > 0. Hence, the minimizer θ N n of N n can be written θ N n = θ N n{b 2n 0} = nb 2n (α n α 0 L n ){B 2n 0}. (34) Similarly the minimizer of N (θ) is unique and can be written θ N = w 1{w 2 0}. Now, since θ N n = O p(1) by Lemma 4, and since N n (θ) N (θ) locally uniformly, we have that θ N n θ N. That is, nb 2n(α n α 0 L n ){B 2n 0} w 1 {w 2 0}. Since nl n B 2n {B 2n 0} w L w 2 {w 2 0} by Lemma 3, we also have nb 2n (α n α 0 ){B 2n 0} (w L w 2 w 1 ){w 2 0}. Finally, since n 1/2 B 2n has a nondegenerate limiting distribution, we have n 1/2 (α n α 0 ){B 2n 0} (w L w 1 /w 2 ){w 2 0}. It is notable that fast (root-n) pointwise convergence of the quartic polynomial N n (θ) to N (θ) does not necessarily imply fast convergence of the minimizer of N (θ) to the min- n imizer of N (θ). Informal direct evaluation of the cubic roots shows that the convergence of the normalized estimator n 1/2 (α n α 0 ){B 2n 0} to its weak limit is at a slow n 1/4 rate. Now consider B 2n > 0. We reparameterize from γ to θ where θ = n 1/4 sgn(b 1n )γ. The term sgn(b 1n ) will be used to ensure that the limit function has a unique minimum. Define P n by P n (θ) = n Q n (n 1/4 sgn(b 1n )θ) = 1 4 θ w 2nθ 2 n 1/4 w 1n θ. (35) Let θ Pn denote the minimizer of P n, then θ Pn = n 1/4 sgn(b 1n )γ n = n 1/4 sgn(b 1n )(α n α 0 L n ). Importantly, θ Pn 0 since sgn(b 1n ) = sgn(γ n ) by Lemma 1. Define P n and P by P n (θ) = P n(θ){b 2n > 0, θ 0} + θ 2 (1 {B 2n > 0, θ 0}) (36)

22 20 and P (θ) = ( 1 4 θ w 2θ 2 ){w 2 > 0, θ 0} + θ 2 (1 {w 2 > 0, θ 0}). (37) Since P n (0) = 0, we have min θ P n (θ) 0, and therefore the minimizer of P n equals θ Pn when B 2n > 0 (the minimizer of P n is positive, so the factor {θ 0} does not matter) and equals 0 when B 2n 0. Hence, the minimizer θ Pn of P n can be written θ Pn = θ Pn{B 2n > 0} = n 1/4 sgn(b 1n )(α n α 0 L n ){B 2n > 0}. (38) The expression ( 1 4 θ w 2θ 2 ) in (37) has three roots; namely, w 1/2 2, 0, and w1/2 2. The root at w 1/2 2 is cancelled in P (θ) by the factor {θ 0}, and the root at 0 corresponds to a local maximum. Therefore the global minimizer of P (θ) is unique and can be written θ P = w1/2 2 {w 2 > 0}. Now, since θ Pn = O p(1) by Lemma 4, and since P n (θ) P (θ) locally uniformly, we have that θ Pn θ P. That is, n1/4 sgn(b 1n )(α n α 0 L n ){B 2n > 0} w 1/2 2 {w 2 > 0}. Finally, because L n = O p (n 1/2 ) and sgn(b 1n ) sgn(w 1 ), we have n 1/4 (α n α 0 ){B 2n > 0} sgn(w 1 )w 1/2 2 {w 2 > 0} as claimed. References Ahn, S. C. and P. Schmidt (1995). Efficient estimation of models for dynamic panel data. Journal of Econometrics 68, Arellano, M. and S. Bond (1991). Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations. Review of Economic Studies 58(2), Arellano, M. and O. Bover (1995). Another look at the instrumental variable estiamtion of error-components models. Journal of Econometrics 68, Blundell, R. and S. Bond (1998). Initial conditions and moment restrictions in dynamic panel data models. Journal of Econometrics 87, Bond, S. and F. Windmeijer (2005). Reliable inference for GMM estimators? Finite sam-

23 21 ple properties of alternative test procedures in linear panel data models. Econometric Reviews 24, Bun, M. and F. Kleibergen (2013). Identification and inference in moments based analysis of linear dynamic panel data models. Discussion paper 2013/07, Amsterdam School of Economics. Calzolari, G. and L. Magazzini (2013). A powerful test of mean stationarity in dynamic models for panel data: Monte Carlo evidence. Working paper 14, Department of Economics, University of Verona. Gørgens, T., C. Han, and S. Xue (2016). Moment restrictions and identification in linear dynamic panel data models. ANU Working Papers in Economics and Econometrics #633, Australian National University. Han, C. and H. Kim (2014). The role of constant instruments in dynamic panel estimation. Economics Letters 124, Hayakawa, K. (2009). On the effect of nonstationary initial conditions in dynamic panel data models. Journal of Econometrics 153, Holtz-Eakin, D., W. Newey, and H. S. Rosen (1988). Estimating vector autoregressions with panel data. Econometrica 56(6), Kruiniger, H. (2009). GMM estimation and inference in dynamic panel data models with persistent data. Econometric Theory 25, Newey, W. K. and D. McFadden (1994). Large sample estimation and hypothesis testing. In R. F. Engle and D. L. McFadden (Eds.), Handbook of econometrics. Volume 4. Amsterdam; London and New York: Elsevier, North-Holland.

24 22 PDF n=10 4 n=10 5 n=10 6 n=10 7 Limit θ Figure 1: DGP1: Density of scaled estimator when B 2n 0 CDFn CDF n=10 4 n=10 5 n=10 6 n= θ Figure 2: DGP1: CDF convergence for scaled estimator when B 2n 0

25 23 PDF n=10 4 n=10 5 n=10 6 n=10 7 Limit θ Figure 3: DGP1: Density of scaled estimator when B 2n > 0 CDFn CDF n=10 4 n=10 5 n=10 6 n= θ Figure 4: DGP1: CDF convergence for scaled estimator when B 2n > 0

26 24 PDF n=10 4 n=10 5 n=10 6 n=10 7 Limit θ Figure 5: DGP2: Density of scaled estimator when B 2n 0 CDFn CDF n=10 4 n=10 5 n=10 6 n= θ Figure 6: DGP2: CDF convergence for scaled estimator when B 2n 0

27 25 PDF n=10 4 n=10 5 n=10 6 n=10 7 Limit θ Figure 7: DGP2: Density of scaled estimator when B 2n > 0 CDFn CDF n=10 4 n=10 5 n=10 6 n= θ Figure 8: DGP2: CDF convergence for scaled estimator when B 2n > 0

28 26 PDF n=10 2 n=10 3 n=10 4 n=10 5 N(0,1) θ Figure 9: DGP3: Density of scaled estimator CDFn CDF n=10 2 n=10 3 n=10 4 n= θ Figure 10: DGP3: CDF convergence for scaled estimator