Quantile Regression with Nonadditive Fixed Effects

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1 RAND Corporation From the SelectedWorks of David Powell 2016 Quantile Regression with Nonadditive Fixed Effects David Powell, Rand Corporation Available at:

2 Quantile Regression with Nonadditive Fixed Effects David Powell RAND April 2016 Abstract This paper introduces a quantile regression estimator for panel data (QRPD) with nonadditive fixed effects, maintaining the nonseparable disturbance term commonly associated with quantile estimation. QRPD estimates the impact of exogenous or endogenous treatment variables on the outcome distribution using within variation in the treatment variables or instruments for identification purposes. Most quantile panel data estimators include additive fixed effects which separates the disturbance term and assumes the parameters vary based only on the time-varying components of the disturbance term. QRPD is consistent for small T and straightforward to implement. The nonadditive fixed effects are never estimated or even specified. Simulation results show that QRPD works well even when instrumental variables quantile regression and additive fixed effect quantile estimators are biased. I estimate the effect of the 2008 tax rebates on the distribution of consumption, finding evidence of substantial heterogeneity in the household spending response to transitory income. Keywords: nonadditive fixed effects, instrumental variables, panel data, quantile treatment effects, nonseparable disturbance JEL classification: C13, C31, C33, C51 I am very grateful to Jerry Hausman and Whitney Newey for their guidance. I also want to thank Abby Alpert, David Autor, Matt Baker, Marianne Bitler, Yingying Dong, David Drukker, Tal Gross, Jon Gruber, Amanda Pallais, Christopher Palmer, Jim Poterba, Nirupama Rao, João M.C. Santos Silva, Hui Shan, and Travis Smith for helpful discussions. I am grateful for helpful comments received from seminar participants at the North American Summer Meeting of the Econometric Society, RAND, UCI, and the 2014 Stata Conference. dpowell@rand.org 1

3 1 Introduction Many empirical applications use quantile regression analysis when the variables of interest potentially have varying effects at different points in the conditional distribution of the outcome variable. These heterogenous effects have proven to provide useful information missed by mean regression techniques (Bitler et al. (2006)). Additionally, it is common in applied work to use panel data to account for unobserved heterogeneity and aid identification through the inclusion of fixed effects. With the popularity of both fixed effect and quantile regression models, there has been a growing literature at the intersection of these two methods. Most of the literature extending quantile methods to panel data include additive fixed effects in the quantile function. This paper introduces a method which permits nonadditive fixed effects, maintaining the nonseparable disturbance term commonly associated with quantile estimation. The model and estimator are developed in an instrumental variables framework and the fixed effects are allowed to have an arbitrary relationship with the instruments. The estimates are consistent for T 2 and the estimator is straightforward to implement. For outcome Y it (i indexes individual and t indexes time) and treatment variables D it, 1 cross-sectional quantile regression assumes that the treatment variables or instruments (Z it ) do not predict the probability that the outcome is smaller than the quantile function, represented by q(d it, τ). For instrumental variables quantile regression (IVQR), the assumption is P (Y it q(d it, τ) Z it ) = τ for some τ (0, 1). In this paper, I relax this assumption in the presence of panel data to allow for individual-level 2 heterogeneity such that E [1(Y it q(d it, τ)) 1(Y is q(d it, τ)) Z i ] = 0, (1) where Z i = (Z i1,..., Z it ). The probability that the outcome is less than the quantile function is not specified and varies. Since we observe the same person multiple times in panel data, we can use this information to learn that the probability that a person has a low value of the outcome variable given their treatment variables may not be τ. This is the gain from employing panel data. In mean regression, panel data allow for the inclusion of fixed effects to identify off 1 I will use capital letters to designate random variables and lower case letters to denote potential values that those random variables may take. 2 I will refer to individuals in this paper as the observed unit, though the estimator is also useful for repeated cross-sections at the state-level and other similar data. 2

4 of within-group variation. Many quantile panel data estimators use an analogous method and include additive fixed effects. Additive fixed effects alter the interpretation of the parameters of interest (relative to cross-sectional quantile regression (QR)) by separating the disturbance term into different components and assuming that the parameters vary based solely on the time-varying components of the disturbance term. In this paper, I introduce an estimator which uses within-individual variation for identification purposes, but maintains the nonseparable disturbance property which typically motivates the use of quantile estimators. The resulting estimates can be interpreted in the same manner as cross-sectional quantile estimates (i.e., the impact of the explanatory variables on the τ th quantile of the outcome distribution) while using the panel nature of the data to relax the typical assumptions required to estimate quantile treatment effects. The fixed effects are never estimated and the coefficient estimates are consistent for small T. This paper models outcomes similar to Chernozhukov and Hansen (2005) using a potential outcomes framework: Y d it U d it = q(d, Uit d ), where q(d, τ) is strictly increasing in τ and represents ability or proneness for the outcome (Doksum (1974)) and U(0, 1). Uit d may be a function of several disturbance terms, some fixed and some time-varying. The relationships between Uit d and the fixed effects are never specified and can be arbitrary. Furthermore, the instruments and fixed effects can also be arbitrarily correlated which distinguishes the corresponding estimator from quantile estimators with correlated random effects. The quantile treatment effects (QTEs) represent the causal effect of a change of the treatment variables from d 1 to d 2 on Y it, holding τ fixed: q(d 2, τ) q(d 1, τ). I introduce a quantile regression for panel data (QRPD) which estimates QTEs even when the instruments are correlated with Uit d of interest is such that U d it Z it U(0, 1). The structural quantile function (SQF) S Y (τ d) = q(d, τ), τ (0, 1). (2) Additive fixed effect models do not estimate the SQF represented in equation (2) while IVQR requires more restrictive assumptions to estimate equation (2). Recent work has developed inference procedures for quantile regression methods for clustered data (see Hagemann (2016) for a discussion). This paper uses a simple clusterrobust inference method paralleling the estimator introduced in Parente and Santos Silva (2016). QRPD produces consistent and asymptotically normal estimates under standard regularity conditions. I also present simulations which show that QRPD performs well even 3

5 in cases in which additive fixed effect models and IVQR perform poorly. Finally, I apply QRPD to estimate the distributional effects of the 2008 economic stimulus payments on short-term household consumption. In the next section, I provide more motivation for conditioning on nonadditive fixed effects and additional background about quantile models with fixed effects. In Section 3, I introduce the model and estimation technique, while also discussing identification. Section 4 considers consistency, asymptotic normality, and inference. Section 5 provides simulation results and applies the QRPD estimator to the empirical application. Section 6 concludes. 2 Background In this section, I further motivate the usefulness of QRPD while also discussing existing quantile panel data methods. I start by introducing the empirical application of this paper before presenting a simple data-generating process that I will later extend for my simulations. 2.1 The Effects of Tax Rebates on the Consumption Distribution Parker et al. (2013) studies whether fiscal policy can increase consuming spending by examining the relationship between tax rebates and short-term changes in household consumption. Using the randomizing timing of the 2008 economic stimulus payments, they implement an instrumental variable strategy to estimate the mean increase in quarterly household spending for each rebate dollar received. Distributional effects are also important for policy as low-consuming households may be especially responsive to transitory income relative to high-consuming households. Mean regression masks this heterogeneity. Additive fixed effect quantile models also will not provide evidence about this heterogeneity. In Section 5.2, I apply QRPD to study the effects of tax rebates on the distribution of short-term consumer spending. Following Johnson, Parker and Souleles (2006) and Parker et al. (2013), it is necessary to condition on household fixed effects to estimate causal parameters. However, we are likely interested in the distribution of C it R it, where C represents household consumption and R represents the total rebate amount. Additive fixed effect 4

6 models would provide information about the distribution of (C it α i ) R it. Households at the bottom of the (C it α i ) R it distribution may be near the top of the C it R it distribution. Thus, existing quantile methods will not estimate the distribution of C it R it, but QRPD is designed to estimate this heterogeneity. The results in Section 5.2 suggest that there are large differences in responsiveness to rebates across low- and high-consuming households. 2.2 Simulated Data Example In Section 5.1, I perform Monte Carlo simulations with similarly-generated data as the following example. Consider the model: Y it = U it(1 + D it ), D it = α i + ψ it, where Uit = f(α i, U it ) for some unknown function f( ) such that Uit U(0, 1). For a given d it, the structural quantile function is (1 + d it )τ. Since D it is a function of α i, then Uit D it U(0, 1). Consequently, QR produces inconsistent estimates given that the level of the policy variable is correlated with the disturbance term. This data generating process is simple but illustrates common themes in applied work. The variable of interest provides information about the individual s ability so identification cannot originate from cross-sectional variation. Individual fixed effects can be used to aid identification. However, additive fixed effects would be inappropriate with these data. Given that the effect of D it varies based on a function of both α i and U it, the appropriate estimation technique must allow the treatment effects to vary based on Uit. Despite the simplicity of this data generating process, existing quantile estimation methods cannot estimate the corresponding QTEs. 2.3 Quantile Panel Data Methods A growing literature has developed quantile panel data estimators with additive fixed effects, including Koenker (2004), Harding and Lamarche (2009), Lamarche (2010), Canay (2011), Galvao Jr. (2011), Ponomareva (2011), Kato et al. (2012), and Rosen (2012). The literature on quantile estimation with fixed effects is primarily concerned with the difficulties in esti- 5

7 mating a large number of fixed effects in a quantile framework and considering incidental parameters problems when T is small. 3 The primary motivation for QRPD is conceptual so I discuss the existing quantile panel data estimators in this spirit. While existing quantile panel data methods focus on estimation of the fixed effects (α i ), let us assume that α i is instead known. Quantile estimators with additive fixed effects provide estimates of the distribution of (Y it α i ) D it instead of estimating the distribution of Y it D it. In many empirical applications, this is undesirable. Observations at the top of the (Y it α i ) distribution may be at the bottom of the Y it distribution. Consequently, additive fixed effect models cannot provide information about the effects of the policy variables on the outcome distribution (only the outcome-relative-to-fixed-effect distribution). A related literature develops quantile panel data methods with correlated random effects (Geraci and Bottai (2007); Abrevaya and Dahl (2008); Graham et al. (2015); Arellano and Bonhomme (2016)). These estimators do not permit an arbitrary relationship between the treatment variables and the individual effects. The motivation for QRPD is that there are many cases when researchers are interested in estimating QTEs for the outcome variable Y it, but they do not believe that they are identified cross-sectionally. Conditioning on individual fixed effects should be helpful in relaxing the identification assumptions but using an additive fixed effect quantile model changes the interpretation of the QTEs. For the following discussion, assume that α i and U it do not depend on d for an individual (i.e., U d it = U it for all d). With additive fixed effects, the model is Y it = α i + D itβ(u it ), where the parameters vary based only on U it, not U it. 4 The motivation for using quantile regression is often to allow the parameters of interest to vary based on the nonseparable disturbance term U it. Separating α i in these cases partially undermines this original motivation and there is frequently 5 little economic justification to allow the parameters to vary based only on part of the disturbance term and exclude the other part simply because it is fixed 3 Graham et al. (2009) shows that there is no incidental parameters problem in a quantile model with additive fixed effects when there are no heterogeneous effects (i.e., the effect is constant throughout the distribution). This argument likely does not extend generally to the case of heterogeneous effects. Ponomareva (2011) introduces an additive effects estimator that is consistent for small T. 4 Some additive fixed effect models also allow the fixed effect to vary with U it (i.e., α i (U it )). 5 It is possible that one might be interested in the distribution of the outcome variable given a fixed α i and this may support using an additive fixed effect model. The framework used in this paper is not intended to nest additive fixed effect quantile frameworks. 6

8 over time. The corresponding SQF for additive fixed effect quantile models is S Y (τ d, α i ) = α i + d β( τ), where β( τ) is used to highlight that these parameters are different than those in equation (2). Even when the conditions for QR are met, the estimates resulting from QR and additive fixed effect quantile models are not comparable. Table 1: Comparison of Estimators Pooled IVQR Additive Fixed Effects QRPD Assumption Underlying Model Uit Z it U(0, 1) Y it = D it β(u it ) U it Z it, α i U(0, 1) it = α i + D it β(u it) Uit Z i Uis Z i Y it = D it β(u it ) Outcome Distribution Y it Y it α i Y it Structural Quantile Function d β(τ) α i + d β( τ) d β(τ) Interpretation for τ th quantile τ th quantile of U τ th quantile of U τ th quantile of U Notes: Pooled IVQR refers to IVQR without conditioning on fixed effects. Additive fixed effects includes the fixed effects additively in the spirit of Harding and Lamarche (2009) and other papers mentioned in the text. The β( τ) notation for the additive fixed effects model is used since τ is different from τ. U it = f(α i, U it ). Differences in the Structural Quantile Function imply differences in the quantile treatment effects. Table 1 lists the differences between the three types of available quantile estimators with panel data using a linear-in-parameters specification in each case and assuming that the disturbance terms do not vary within-individual based on the policy variables. First, I list pooled IVQR, which does not condition on individual fixed effects. Second, the table includes a quantile panel data estimator with additive fixed effects. Third, I include the QRPD estimator. Note that the additive fixed effects change the structural quantile function (and, consequently, the QTEs) and the conditional outcome distribution that is being studied. QRPD relaxes the assumptions of IVQR. Instead of assuming that Uit Z it U(0, 1), Uit Z i is allowed to have an unspecified distribution. The QRPD estimator is, to my knowledge, the first quantile panel data estimator to provide point estimates which can be interpreted in the same manner as cross-sectional regression results while allowing an arbitrary correlation between the fixed effects and the instruments. It is also one of the few quantile (additive or nonadditive) fixed effects estimators to provide consistent estimates for small T and one of the few instrumental variables quantile panel data estimators. Similar to the motivation for QRPD, Chernozhukov et al. (2013) discusses identification of bounds on quantile effects in nonseparable panel models with exogenous variables. They show that these bounds tighten as T increases. The inter- 7

9 pretation of QRPD parallels the interpretation of the bounds using the Chernozhukov et al. (2013) framework. A further advantage of QRPD is that the moment conditions are simple to interpret and implement. Because the individual fixed effects are never estimated or even specified, the number of parameters that need to be estimated is small relative to most quantile panel data estimators and implementation of this estimator is simple compared to those found in the literature. I also use the properties of the moment conditions to reduce the number of parameters that need to be independently estimated even further. 3 Model The model is developed in a potential (latent) outcome framework and intentionally follows Chernozhukov and Hansen (2005) closely. With panel data, the assumptions necessary for IVQR can be relaxed with significant gains. 3.1 Main Conditions of Model All conditions in this paper are assumed to hold jointly with probability one. A1 Potential Outcomes and Monotonicity: Yit d = q(d, Uit d ), Uit d U(0, 1), where q(d, τ) is strictly increasing in τ. A1 is a standard monotonicity condition for quantile estimators (e.g., Chernozhukov and Hansen (2006)). Uit d may be a function of several unobserved disturbance terms, summarizing these terms into one rank variable. Alternatively, one can imagine using a non-normalized disturbance term ϵ d in the equation of interest Yit d ϵ d to U d. = d β(ϵ d it ). There exists a mapping of [ ] A2 Independence: E 1(U it d τ) 1(U is d τ) Z i = 0 for all s, t and for each d. [ A2 is the primary ] departure from Chernozhukov and Hansen (2005, 2006), replacing E 1(U it d τ) Z it = τ. A2 simply requires Z i to not be systematically related to changes in the distribution of Uit d over time, using the panel nature of the data to relax the corre- 8

10 sponding assumption for IVQR. 6 A2 allows for individual-specific probabilities that U it is less than or equal to τ. A3 Selection: D it = δ t (Z i, V i ) for some unknown function δ t ( ) and random vector V i. A3 places little structure on the relationship between the instruments and treatment variables. The function varies by time period which generates within-individual variation. Correlations between V i and Uit D framework potentially necessary. A4 Rank Similarity: U d it Z i, V i U d it Z i, V i for each d, d. are the driving force that make an instrumental variables A4 allows the ranks of individuals within the outcome distribution to change based on the treatment variables, though these changes cannot be systematic. This assumption relaxes rank invariance. A5 Observables: The observed random vector consists of Y it := Y D it, D it, Z it. No restrictions in the above conditions have been placed on the relationship between U d it and α d i. Furthermore, there are no assumptions on the relationship between α d i and Z i, paralleling fixed effect mean regression models. These conditions lead to the main results of this paper. Theorem 3.1 (Main Results). Suppose A1-A5 hold. Then, the following three conditions hold with probability 1: 1. For Uit := Uit D, Y it = q(d it, Uit), Uit U(0, 1). ] 2. For each τ (0, 1), E [1(Y it q(d it, τ)) 1(Y is q(d is, τ)) Z i = 0 for all s, t. [ ] 3. For each τ (0, 1), P Y it q(d it, τ) = τ. Proofs are included in Appendix Section A.1. Condition 1 states that A1-A5 generate a 6 A2 can be replaced by a stronger assumption of stationarity such that Uit d Z i Uis d Z i. Instead, the distribution of Uit d Z i can change over time as long as Z i does not predict this change. To aid intuition, consider q(d, ϵ d it ) where a map exists between ϵd it and normalized Uit d. Assumption A2 puts no structure on the overall mean or variance of ϵ d it for any t. The assumption allows ϵ d it = a d t +c d t ϵ d i1 (Chernozhukov et al. (2013) includes a similar assumption) for some time-varying constants a d t, c d t. More relaxed error structures than this one are also allowed by A2, but this example is a useful illustration. 9

11 model with a nonadditive disturbance term. Conditions 2 and 3 imply two moment conditions which will be useful for estimation. The moment conditions are stated below: Corollary 3.2 (Moment Conditions). Suppose A1-A5 hold. Then for each τ (0, 1), { 1 [ ] } E (Z 2T 2 it Z is ) 1(Y it q(d it, τ)) 1(Y is q(d is, τ)) = 0, (3) t s E[1(Y it q(d it, τ) τ] = 0. (4) These conditions follow immediately from Theorem 3.1. Equation (3) is a useful formulation since it shows that the estimator is simply a series of within-individual comparisons. The estimator uses the panel nature of the data to allow the probability that Y is smaller than the quantile function to vary across individuals (and even within-individual). When discussing identification and other properties, it is easiest to use the following equivalent formulation 7 of equation (3): { 1 E T t ( ) [ ] } Zit Z i 1(Y it q(d it, τ)) = 0, (5) where Z i = 1 T T t=1 Z it. This formulation provides similar intuition. Identification originates solely from within-individual variation in the instruments (Z it Z i ) and there are no assumptions on the individual-specific probabilities that the outcome is smaller than the quantile function. This probability varies across individuals and even within-individuals. On average, it is equal to τ due to equation (4), but the conditions above permit individual-specific heterogeneity. With IVQR, one assumes both P (Y it q(d it, τ)) = τ and P (Y it q(d it, τ) Z it ) = τ. The QRPD estimator replaces the latter restriction with a weaker one. 7 This condition is equivalent to equation (3) through a straightforward rearrangement of terms. 10

12 3.2 Identification This section discusses the uniqueness of q(d it, τ) and then alternative assumptions for the linear case where q(d it, τ) = D itβ(τ). Let D it take on M possible values with positive probability. I define a T M matrix of the relationship between Z i (L T ) and (d (1),, d (M) ): Π(Z i ) P (D i1 = d (1) Z i ) P (D i1 = d (M) Z i ) P (D it = d (1) Z i ) P (D it = d (M) Z i ) Identification requires two additional assumptions: A6 First Stage: E[Z i Π(Z i )] is rank M. A7 Continuity: Y it continuously distributed conditional on Z i. E[Z i Π(Z i )] is an L M matrix, and the requirement in A6 assumes that the instruments have a rich relationship with the policy variables. A7 is a typical assumption for quantile models. Given conditions A1-A7, uniqueness of q(d it, τ) follows. To illustrate, I consider an alternative function denoted q(d it, τ). { 1 Theorem 3.3 (Identification). If (i) A1 - A7 hold; (ii) E T ( ) [ ]} T t=1 Zit Z i 1(Y it q(d it, τ) = 0; (iii) E [1(Y it q(d it, τ)] = τ, then q(d it, τ) = q(d it, τ). A discussion is included in Appendix Section A.1. Theorem 3.3 provides conditions under which non-parametric identification holds. Because implementation will consider the linearin-parameters case, it is useful to discuss identification when q(d it, τ) = D itβ(τ). For the linear case, I designate the number of policy variables by k and define Π(Z i ) (same as above) as a matrix of the relationship between the instruments and (d (1),, d (M) ), which may include only a subset of possible values of D it. Assumption A6 is replaced by A6 Full Rank and First Stage: There exists (d (1),, d (M) ) such that E[Z i Π(Z i )] is rank M and (d (1),, d (M) ) is rank k. Identification in the linear-in-parameters case is formalized in Appendix Section A.1. The benefits of A6 relative to A6 is that only a subset of values of the treatment variable 11

13 need to meet the conditions in order for identification to hold Estimation Estimation uses Generalized Method of Moments (GMM). Sample moments are defined by ĥ(b) = 1 N 1 N T ( ) [ ] T t=1 Zit Z i 1(Y it D h i (b) with h i (b) [ ] itb) 1(Y it D itb) τ i=1 1 T T t=1 The estimated parameters are β(τ) = arg min b ĥ(b) Ŵ ĥ(b) for some weighting matrix Ŵ. Ŵ can simply be the identity matrix and two-step GMM estimation can be used in the overidentified case. The minimization of ĥ(b) Ŵ ĥ(b) may be computationally difficult. I propose a simplification to aid estimation. Define sample moments using ĝ(b) = 1 N N g i (b) with g i (b) = 1 T i=1 { T t=1 ( ) [ ] } Zit Z i 1(Y it D itb). (6) To simplify estimation, I constrain the sample version of equation (4) to hold. When time fixed effects (or any set of dummy variables, indexed by t, which saturate the space) are included, equations (3) and (4) imply P (Y it D itβ(τ)) = τ for all t. Consequently, I can define the parameter set such that B { b τ 1 N < 1 N } N 1(Y it D itb) τ for all t. (7) i=1 Constraining the parameters to B is a simple way to force Y it D itb to hold for (approximately) 100τ% of the observations in each time period. 9 Then, β(τ) = arg min b B ĝ(b) Âĝ(b) (8) for some weighting matrix Â. Two-step GMM estimation can be used in the overidentified 8 In the Section 5.2 application, many households received rebates that were multiples of $600 given the rebate formula so there are discrete values with positive probabilities and condition A6 holds. 9 With no time effects and only a constant, then B { b τ 1 i=1 t=1 1(Y it D it b) τ}. Note that B is guaranteed to be non-empty. NT < 1 NT 12

14 case. In the application in Section 5.2, this simplification reduces the number of parameters that need to be independently estimated from 923 to 1. There is some potential loss in efficiency by implementing the estimator in this manner in the overidentified case, but the computational gains are substantial. I emphasize the role of time fixed effects given that they are routinely included in applied work with panel data and because they provide a simple way to enforce equation (4), a condition which holds for single year cross-sectional QR estimation. Cross-sectional QR (with one year of data) provides estimates referring to the τ th quantile of the distribution within that year. Including time fixed effects with panel data preserves this interpretation. 10 It is straightforward to confine all possible b to the set B. Let D (X, 1(t = 1),..., 1(t = T )) where X are the policy variables of interest and 1(t = s) is an indicator variable equal to 1 for time period s. Given parameters associated with X, it is simple to find the year fixed effects which constrain estimates to B. Let b represent coefficients on X such that D itb = γ t + X it b. Define γ t (τ, b) as the τ th quantile of the distribution of Y it X it b at time t: Set γ t (τ, b) such that τ 1 N < 1 N N 1(Y it X it b γ t (τ, b)) τ. (9) i In finite samples, ˆγ t (τ, b) is not necessarily unique. The proposed estimation steps are the following. Define a grid of values for the parameters associated with the variables in X. For each value: 1. Calculate the year fixed effects to constrain the parameter set to B. 2. Evaluate ĝ(b) Âĝ(b) where g i (b) is defined in equation (6). The b that minimizes this condition is β(τ). In many economic applications, it is typical to have only one or two treatment variables (not counting the time fixed effects). In these cases, grid-searching is appropriate. With more treatment variables, other optimization methods are necessary, such as Markov Chain Monte Carlo (MCMC). In simulations with several 10 As an example, consider studying policy effects on the distribution of consumption over a long time period with high inflation. Without year fixed effects (which shift the entire distribution of consumption), the high quantiles would primarily refer to later time periods. Including time fixed effects allows for the estimates to be interpreted as the effects of the policies on the outcome distribution within a year, equivalent to the interpretation of cross-sectional quantile regression estimates. 13

15 treatment variables, MCMC worked well in implementing QRPD. 4 Properties This section briefly discusses consistency, asymptotic normality, and inference. These properties are discussed for fixed T as N. This discussion will assume that B is defined by equation (7). I account for the proposed estimation strategy which considers the year fixed effect estimates as functions of the coefficient estimates on X. Let be the Euclidean norm and f Y ( ) represent the conditional pdf of Y it. 4.1 Consistency The following assumptions are necessary to discuss consistency and asymptotic normality. As before, D (X, 1(t = 1),..., 1(t = T )) where X are the policy variables of interest and 1(t = s) is an indicator variable equal to 1 for time period s (i.e., a time fixed effect). Let ϕ represent the true coefficients on X such that D itβ(τ) = γ t + X itϕ. The assumptions are: A8 (Y i, D i, Z i ) i.i.d. A9 B is compact. A10 sup t E Zit Z i <, supt E Zit Z i 2+δ < for some δ > 0. [ ( ) ] 1 A11 G E T T t=1 (Z it Z i ) X it + γ t(τ,ϕ) f ϕ Y (D itβ(τ) Z i ) exists such that G AG nonsingular. The formula for G accounts for the recommended estimation procedure which links the coefficients on X to the time fixed effects. The γt(τ,ϕ) term can be excluded if this procedure ϕ is not used, but the gradient must then involve the derivative of equation (4). The other regularity conditions are standard. Consistency follows immediately from these conditions and Theorem 2.6 in Newey and McFadden (1994). Theorem 4.1 (Consistency). If A1 - A10 hold and  β(τ). A discussion is included in Appendix Section A p A positive definite, then β(τ) p

16 4.2 Asymptotic Normality Stochastic equicontinuity is an important condition for asymptotic normality of GMM estimators and follows here from the fact that the functional class { 1(Y it D itb), b R k} is Donsker and the Donsker property is preserved when the class is multiplied by a bounded random variable. Thus, { 1 T T t=1 ( Zit Z i ) [ 1(Y it D itb)], b R k } is Donsker with square integrable envelope 2 max (i,t) Z it Z i. Stochastic equicontinuity follows from Theorem 1 in Andrews (1994). Define Σ E[g i (β(τ))g i (β(τ)) ]. Theorem 4.2 (Asymptotic Normality). If A1 - A11 hold and  then N( β(τ) d β(τ)) N [0, (G AG) 1 G AΣAG(G AG) 1 ]. A discussion is included in Appendix Section A.2. p A positive definite, 4.3 Inference I adopt an approach similar to the histogram estimation technique suggested in Powell (1986) to obtain consistent estimates of G. The variance estimates adjust for within-individual clustering by the definition of Σ. Recent work has introduced cluster-robust inference procedures (Hagemann (2016); Parente and Santos Silva (2016)) for quantile regression. It is important in this context to adjust for clustering at the individual-level. To estimate G, I use Ĝ = 1 2Nh [ N 1 T i=1 ( T (Z it Z i ) X it + t=1 γ t (τ, ϕ) ϕ ) ( Yit 1 D it β(τ) h) ], for small h. For a consistent estimate of Σ, I use Σ = 1 N i g i( β(τ))g i ( β(τ)) γ. t (τ,ϕ) represents an estimate of the relationship between small differences in the estimated ϕ parameters on the policy variables and the estimated year fixed effects. 15

17 5 Applications 5.1 Simulations To illustrate the usefulness of the QRPD estimator, I generate the following data, similar to the example in Section 2.2: Y it = U it(δ t + D it ), Z it = α i + ψ it, D it = Z it + U it, with α i, U it, ψ it U(0, 1). U it is the CDF of α i + U it such that it is distributed U(0, 1). D it is a function of U it so instrumental variables are necessary. Z it is exogenous conditional on α i. The impact of D it is a function of U it and varies by observation. The coefficient on D it in the SQF is equal to τ. I generate these data for N = 500, T = 2 with δ 0 = 1, δ 1 = 2. Table 2 presents the results of the simulation for the coefficient of interest. I first show results using IVQR (Chernozhukov and Hansen (2006)) to illustrate that the generated data do not meet the assumptions (e.g., U it Z it U(0, 1)) required to use cross-sectional quantile estimation techniques. I also show results using IVQRFE (Harding and Lamarche (2009)), which assumes an additive fixed effect. Again, the assumptions for this estimator are not met and the estimator, as expected, performs poorly on the generated data. The data generating process above is simple and includes properties that are likely common in applied work. First, it is important to account for individual-level heterogeneity. Second, the impact of the policy variable is not constant. Yet, existing quantile estimators do not allow one to estimate QTEs for this data generating process. The final set of results in Table 2 uses QRPD. The QRPD estimator performs well throughout the distribution. 5.2 QTEs of Tax Rebates on Short-Term Household Spending I use the QRPD estimator to estimate the effect of tax rebates on quarterly consumption using data from Parker et al. (2013). Tax rebates in 2008 were provided to a majority of the population and were distributed over the span of several weeks. Parker et al. (2013) uses the differential timing of rebate receipt to study how tax rebates affect mean changes 16

18 Table 2: IVQRPD Simulation (N=500, T=2) IVQR IVQRFE IVQRPD Quantile Mean Bias MAD RMSE Mean Bias MAD RMSE Mean Bias MAD RMSE MAD=Median Absolute Deviation, RMSE=Root Mean Squared Error. IVQR refers to the estimator introduced in Chernozhukov and Hansen (2006). IVQRFE uses Harding and Lamarche (2009). in quarterly consumption using household longitudinal data in the Consumer Expenditure Survey. It is important to account for household fixed effects since rebate receipt was not random. Instead, it was a function of household income and the number of dependents in the previous year. To identify solely off of timing, Parker et al. (2013) uses 1(R it > 0) as an instrument for R it, conditioning on household fixed effects. They estimate the relationship between consumption and rebates using first-differences to eliminate the additive fixed effect. parallel the use of first-differences, I treat each (i, t 1) and (i, t) as a household pair and condition on fixed effects for each pair. For inference, I modify estimation of Σ to account for households having multiple household pairs. The estimate generated from mean-iv regression is presented in Table 3, Column (1). This estimate implies that each rebate dollar increased household expenditures in the quarter of rebate receipt by 51 cents. 11 While distributional estimates are interesting, additive fixed effect models would provide 11 This estimate is slightly different from the results presented in Parker et al. (2013) because I include interactions based on quarter, number of adults, and number of dependents. Number of adults and number of dependents are held constant within a household pair, using the values in the initial quarter of the pair. Parker et al. (2013) controls for quarter fixed effects, number of adults, and number of dependents. To 17

19 Figure 1: QTE Estimates Quantile Treatment Effect Quantile Coefficient Estimate 90% Confidence Interval Notes: Confidence intervals are truncated at -4 and 1. The structural quantile function also includes interactions based on quarter, number of adults, and number of dependents. estimates of the distribution of (C it α i ) R it where C it represents consumption of household i in quarter t. However, we are likely interested in the distribution of C it R it. It should be noted that Misra and Surico (2014) uses quantile regression to estimate heterogeneous consumption responses to tax rebates. However, they first-difference their data and then perform quantile regression, estimating the distribution of C it C i,t 1 R it, which does not provide information about whether high-consuming or low-consuming households are responsive to rebates. Furthermore, while differencing is valid in mean regression, it does not necessarily account for fixed effects in a quantile framework even if those fixed effects are additive. 12 The estimates displayed in Figure 1 show the distributional impact of tax rebates on consumption using QRPD. The QTE estimates are centered around zero until close to the median of the consumption distribution, and then steadily increase. I estimate especially large effects at the top of the distribution. These heterogeneous responses are important since the estimates imply that the lower half of the consumption distribution is generally 12 If the treatment effects are homogenous, then this approach is valid. Other restrictions may also permit differencing. 18

20 Table 3: Average Treatment Effect Estimates (1) (2) Rebate ($) 0.507** 0.455*** (0.226) (0.103) N Number of Households Estimator Mean-IV QRPD (quantiles 1-99) Notes: ***Significance 1%, ** Significance 5%, * Significance 10%. Standard errors in parentheses adjusted for clustering at household level. Subsampling used to generate standard errors for Column (2). Interactions based on quarter, number of children, and number of adults included. Column (2) is the estimate from integrating over the QTE estimates for quantiles unresponsive to transitory income while the upper part of the distribution is especially responsive. Uncovering this heterogeneity is not possible with existing quantile panel data methods. Another advantage of estimating QTEs is that it is straightforward to estimate average treatment effects using 1 β(τ)dτ. Integrating over the QTE estimates, I present average 0 treatment effects in the presence of nonadditive fixed effects in Table 3, Column (2). I estimate a smaller effect than the mean-iv estimate. The estimate implies that each rebate dollar increases mean household spending by 46 cents. 6 Conclusion In this paper, I introduce a quantile estimator for panel data which maintains the nonseparable disturbance term traditionally associated with quantile estimation. The instruments can be arbitrarily correlated with the nonadditive fixed effects. This estimator should be useful in contexts where identification requires differences and it is believed that the effects of the variables are heterogeneous throughout the outcome distribution. The resulting estimates can be interpreted in the same manner as traditional cross-sectional quantile estimates. This estimator performs well in simulations. The estimator is consistent for small T and straightforward to use with standard statistical software. The estimator in this paper contrasts with existing quantile panel data estimators which typically include a separate additive term for the fixed effect. This additive fixed effect 19

21 requires estimation of the distribution of (Y it α i ) D it, not Y it D it. Additive fixed effect quantile models assume that the parameters of interest vary based solely on the time-varying components of the disturbance term. The QRPD estimator introduced in this paper allow the parameters to vary based on an unknown function of the fixed effect and an observationspecific disturbance term while relaxing the identification assumptions required for QR and IVQR. I apply the estimator to study the heterogeneous effect of tax rebates on quarterly household spending. I find that high-consuming households are especially responsive to transitory income while I cannot statistically reject that low-consuming households do not respond to transitory income. Estimating the relationship between rebates and the distribution of consumption is not possible using additive fixed effect models, but QRPD provides a simple method to estimate the quantile treatment effects. 20

22 References Abrevaya, Jason and Christian M Dahl, The Effects of Birth Inputs on Birthweight, Journal of Business & Economic Statistics, 2008, 26, Andrews, Donald W.K., Empirical process methods in econometrics, Handbook of Econometrics, , 4, Arellano, Manuel and Stephane Bonhomme, Nonlinear Panel Data Estimation via Quantile Regressions, The Econometrics Journal, Bitler, Marianne P, Jonah B Gelbach, and Hilary W Hoynes, What Mean Impacts Miss: Distributional Effects of Welfare Reform Experiments, The American Economic Review, 2006, 96 (4), Canay, Ivan A., A Note on Quantile Regression for Panel Data Models, The Econometrics Journal, 2011, 14, Chernozhukov, Victor and Christian Hansen, An IV Model of Quantile Treatment Effects, Econometrica, 2005, 73 (1), and, Instrumental quantile regression inference for structural and treatment effect models, Journal of Econometrics, 2006, 132 (2), , Iván Fernández-Val, Jinyong Hahn, and Whitney Newey, Average and Quantile Effects in Nonseparable Panel Models, Econometrica, 2013, 81 (2), Doksum, Kjell, Empirical Probability Plots and Statistical Inference for Nonlinear Models in the Two-Sample Case, Ann. Statist., 1974, 2 (2), Galvao Jr., Antonio F., Quantile regression for dynamic panel data with fixed effects, Journal of Econometrics, September 2011, 164 (1), Geraci, Marco and Matteo Bottai, Quantile regression for longitudinal data using the asymmetric Laplace distribution, Biostatistics, 2007, 8 (1), Graham, Bryan S, Jinyong Hahn, Alexandre Poirier, and James L Powell, A Quantile Correlated Random Coefficients Panel Data Model,

23 Graham, Bryan S., Jinyong Hahn, and James L. Powell, The incidental parameter problem in a non-differentiable panel data model, Economics Letters, November 2009, 105 (2), Hagemann, Andreas, Cluster-robust bootstrap inference in quantile regression models, Journal of the American Statistical Association, Harding, Matthew and Carlos Lamarche, A quantile regression approach for estimating panel data models using instrumental variables, Economics Letters, 2009, 104 (3), Johnson, David S., Jonathan A. Parker, and Nicholas S. Souleles, Household Expenditure and the Income Tax Rebates of 2001, American Economic Review, December 2006, 96 (5), Kato, Kengo, Antonio F Galvao, and Gabriel V Montes-Rojas, Asymptotics for panel quantile regression models with individual effects, Journal of Econometrics, 2012, 170 (1), Koenker, Roger, Quantile regression for longitudinal data, Journal of Multivariate Analysis, October 2004, 91 (1), Lamarche, Carlos, Robust penalized quantile regression estimation for panel data, Journal of Econometrics, August 2010, 157 (2), Misra, Kanishka and Paolo Surico, Consumption, income changes, and heterogeneity: Evidence from two fiscal stimulus programs, American Economic Journal: Macroeconomics, 2014, 6 (4), Newey, Whitney K. and Daniel McFadden, Large sample estimation and hypothesis testing, Handbook of Econometrics, 1994, 4, Parente, Paulo MDC and João M.C. Santos Silva, Quantile regression with clustered data, Journal of Econometric Methods, 2016, 5 (1), Parker, Jonathan A, Nicholas S Souleles, David S Johnson, and Robert McClelland, Consumer Spending and the Economic Stimulus Payments of 2008, The American Economic Review, 2013, 103 (6),

24 Ponomareva, Maria, Quantile Regression for Panel Data Models with Fixed Effects and Small T: Identification and Estimation, Working Paper, University of Western Ontario May Powell, James L., Censored regression quantiles, Journal of Econometrics, June 1986, 32 (1), Rosen, Adam M., Set identification via quantile restrictions in short panels, Journal of Econometrics, 2012, 166 (1), van der Vaart, A.W. and Jon A. Wellner, Weak Convergence and Empirical Processes, Springer,

25 A Appendix A.1 Moment Conditions and Identification Theorem 3.1 (Main Results). Suppose A1-A5 hold. Then, the following three conditions hold with probability 1: 1. For Uit := Uit D, Y it = q(d it, Uit), Uit U(0, 1). ] 2. For each τ (0, 1), E [1(Y it q(d it, τ)) 1(Y is q(d is, τ)) Z i = 0 for all s, t. [ ] 3. For each τ (0, 1), P Y it q(d it, τ) = τ. Proof of Condition 2: ] E [1(Y it q(d it, τ)) Z i = z i [ = E ( 1 ( q(d it, Uit D ) q(d it, τ) ) ] Z i = z i by A1, A5 ) = P Uit D τ Z i = z i by A1 ( = P Uit D τ Z i = z i, V i = v i )dp [V i = v i Z i = z i ] by definition ( = P U δt(z i,v i ) it τ Z i = z i, V i = v i )dp [V i = v i Z i = z i ] by A3 ( = P Uit 0 τ Z i = z i, V i = v i )dp [V i = v i Z i = z i ] by A4 ( ) = P Uit 0 τ Z i = z i by definition, represents the distribution for any fixed value of D (implied by A4). Equiva- where Uit 0 lently, By A2, ] ( ) E [1(Y is q(d is, τ)) Z i = z i = P Uis 0 τ Z i = z i ( ) ( P Uit 0 τ Z i = z i = P Uis 0 τ Z i = z i ), completing the proof. 24

26 Proof of Condition 3: [ ] P Y it q(d it, τ) [ ] = P q(d it, Uit D ) q(d it, τ) = P [ U D it τ ] by A1 = τ by A1 by A1, A5 Together, the proofs of Conditions 2 and 3 prove Condition 1 in Theorem 3.1. { 1 Theorem 3.3 (Identification). If (i) A1 - A7 hold; (ii) E T ( ) [ ]} T t=1 Zit Z i 1(Y it q(d it, τ) = 0; (iii) E [1(Y it q(d it, τ)] = τ, then q(d it, τ) = q(d it, τ). First, some notation: Γ(Z i, q) P (Y d(1) it q(d (1), τ) Z i ). P (Y d(m) it q(d (M), τ) Z i ) Proof. Starting with (ii) and the Law of Iterated Expectations, E[(Z it Z i )Π(Z i )Γ(Z i, q)] = 0.. Without loss of generality, assume that P (Yit d(1) for some τ (0, 1). q(d (1), τ) Z i ) = P (Y d(1) it q(d (1), τ) Z i ) By A6, we know that P (Y d(m) it q(d (m), τ) Z i ) = P (Y d(m) it q(d (m), τ) Z i ) for all m. A7 implies that q(d (m), τ) = q(d (m), τ) for all m. Because of (iii), we know that τ = τ, implying that q(d (m), τ) = q(d (m), τ). Theorem A.1 { (Identification: Linear Case). If (i) A1, A2, A3, A4, A5, A6, A7 1 hold; (ii) E T ( ) [ T t=1 Zit Z i 1(Y it D β) ]} [ it = 0; (iii) E 1(Y it D β) ] it = τ, then β = β(τ). Define Γ(Z i, β) P (Y d(1) it d (1) β Z i ). P (Y d(m) it d (M) β Z i ). 25

27 Initially, I assume that the policy variables are discrete such that (d (1),, d (M) ) includes all possible values. The extension to the case where this matrix only includes a subset of possible values is straightforward and included after the proof. Proof. Starting with (ii) and the Law of Iterated Expectations, E[(Z it Z i )Π(Z i )Γ(Z i, β)] = 0. Without loss of generality, assume that P (Y it d (1) β Zi ) = P (Y it d (1) β( τ) Z i ) for some τ (0, 1). By A6, we know that P (Y it d (m) β Zi ) = P (Y it d (m) β( τ) Z i ) for all m. A7 implies that d (m) β = d (m) β( τ) for all m. By the full rank assumption in A6 then, β = β( τ). Because of (iii), we know that τ = τ, implying that β = β(τ). Extension: The proof is straightforward to extend when (d (1),, d (M) ) only includes a subset of possible values of the policy variables. This is useful for cases where one or more variables can take on numerous values and, potentially, are continuous at some points. The necessary assumption is that there exists a subset of values that have positive probability. Here, I simply add a term to Π(Z i ) for the probability that D it does not equal one of the values in (d (1),, d (M) ) and a corresponding term to Γ(Z i, β):, Π F (Z i ) Π(Z i ) P (D i1 d (1),, D i1 d (M) ) Z i ). P (D it d (1),, D it d (M) ) Z i ) Γ F (Z i, β) [ Γ(Z i, β) P (Y d(o) it D itβ Z i, D it d (1),, D it d (M) ) ]. The above proof does not change when we analyze E[(Z it Z i )Π F (Z i )Γ F (Z i, β)] = 0. 26

28 A.2 QRPD Properties These properties are discussed for small T as N. This discussion will assume that B is defined by equation (7). Let be the Euclidean norm and f Y ( ) represent the pdf of Y it conditional on Z i. A.2.1 Consistency Theorem 4.1 (Consistency). If A1 - A10 hold and  β(τ). p A positive definite, then β(τ) p Note that the following conditions hold: 1. Identification holds by Theorem A Compactness of B holds by A9. 3. g i (b) is continuous at each b with probability one under A7. 4. E g i (b) sup t E Zit Z i < by A10. Under these conditions, consistency follows immediately from Theorem 2.6 of Newey and McFadden (1994). A.2.2 Asymptotic Normality Theorem 4.2 (Asymptotic Normality). If A1 - A11 hold and  then N( β(τ) d β(τ)) N [0, (G AG) 1 G AΣAG(G AG) 1 ]. Define g(β) E [ 1 T ( ) [ ]] T t=1 Zit Z i 1(Y it D itβ). p A positive definite, Proof: The result follows from Theorem 7.2 in Newey and McFadden (1994). The following conditions necessary for Theorem 7.2 hold immediately from the assumptions: 1. Consistency of the estimates is established in Theorem 4.1 and identification is shown in Theorem A.1. 27