Quantile Regression with Nonadditive Fixed Effects

RAND Corporation From the SelectedWorks of David Powell

2015

Quantile Regression with Nonadditive Fixed Effects David Powell, Rand Corporation

Available at: http://works.bepress.com/david_powell/1/

Quantile Regression with Nonadditive Fixed Effects∗ David Powell RAND † March 2015 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. The estimator can be used to evaluate the impact of exogenous or endogenous treatment variables on the outcome distribution, while conditioning on fixed effects for identification purposes. Many quantile panel data estimators include an additive fixed effect which separates the disturbance term and assumes the parameters do not vary based on the fixed effect. QRPD allows the parameters to vary based on a nonadditive disturbance term. The estimator is consistent for small T and straightforward to implement. Keywords: fixed effects, instrumental variables, panel data, quantile regression, nonseparable disturbance

∗

I am very grateful to Jerry Hausman and Whitney Newey for their guidance. I also want to thank Abby Alpert, David Autor, Marianne Bitler, Yingying Dong, Tal Gross, Jon Gruber, Amanda Pallais, Christopher Palmer, Jim Poterba, Nirupama Rao, Jo˜ao M.C. Santos Silva, and Hui Shan 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. † [email protected]

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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, Gelbach and Hoynes (2006)). Quantile estimation, such as quantile regression (QR) introduced by Koenker and Bassett (1978), allows for the impact of the covariates to vary with a nonseparable disturbance term. In many empirical applications, researchers use panel data to difference their variables or include fixed effects to account for unobserved heterogeneity and aid identification. 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 existing quantile panel data techniques include an additive fixed effect term. This term alters the interpretation of the parameters of interest (relative to cross-sectional QR) by separating the disturbance term into different components and assuming that the parameters do not vary based on the fixed effect. In this paper, I introduce an estimator which uses within-group 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 . The estimation technique is straightforward to implement with standard statistical software. Quantile estimation techniques are typically used to estimate the distribution of

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the outcome, represented by Yit , for given values of the treatment variables, represented by Dit .1 In mean regression, panel data allow for the inclusion of fixed effects to identify off of within-group variation. Many quantile panel data estimators use an analogous method and include additive fixed effects. This paper uses nonadditive fixed effects, maintaining the nonseparable disturbance term commonly associated with quantile estimation. I model outcomes as Yit = D′it β(Uit∗ ),

Uit∗ ∼ U (0, 1),

(1)

where D′it β(τ ) is strictly increasing in τ . I use a linear-in-parameters framework due to its popularity in applied work and relative ease in implementing. Uit∗ represents ability or proneness for the outcome (Doksum (1974)) and may be a function of several disturbance terms. It is a rank variable and the assumption of a uniform distribution is a normalization. For comparisons with other quantile estimators, let Uit∗ = f (αi , Uit ) where Uit ∼ U (0, 1). In words, proneness for the outcome is an unknown function of both an individual fixed effect2 and an observation-specific disturbance term. I will place no structure on the function f (·). The QTEs represent the causal effect of a change of the treatment variables from d1 to d2 on Yit , holding τ fixed: d′2 β(τ ) − d′1 β(τ ).

(2)

I introduce a quantile estimator for panel data (QRPD) which estimates QTEs for the outcome variables Yit . To adopt similar terminology as Chernozhukov and Hansen (2008), 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 refer to the fixed effects as “individual fixed effects,” though the estimator is also applicable in other contexts, such as repeated cross-sections where fixed effects are based on cells.

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the structural quantile function (SQF) of interest for equation (1) is SY (τ |d) = d′ β(τ ),

τ ∈ (0, 1).

(3)

The SQF defines the quantile of the latent outcome variable Yd = d′ β(U ∗ ) for a fixed d and a randomly-selected U ∗ ∼ U (0, 1). Estimation of the SQF is possible using QR when U ∗ and D are independent. QRPD will relax this independence assumption. QRPD is useful when U ∗ and D are not independent, but we are still interested in the outcome distribution if d did not provide information about U ∗ (i.e., if policy imposed d). To highlight, QR and QRPD are both useful for estimating equation (3), but QRPD relaxes the assumptions of QR to estimate the SQF by conditioning on individual fixed effects. Additive fixed effect models do not estimate the SQF represented in equation (3). I develop QRPD in an instrumental variables framework and discuss it relative to instrumental variable quantile regression (IVQR, Chernozhukov and Hansen (2006), Chernozhukov and Hansen (2008)), which relies on the conditional restriction P (Yit ≤ D′it β(τ )|Zit ) = τ.

(4)

This condition states that the probability the outcome variable is smaller than the quantile function is the same for all Zit and equal to τ . QRPD lets this probability vary by individual and even within-individual. 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 . Some people may be very “prone” to large values of the outcome variables while others are less prone, and this information is useful to relax the restriction in equation (4). In the next section, I provide more background about quantile models with fixed effects. In Section 3, I introduce the model and estimation technique. Section 4 discusses

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identification, consistency, asymptotic normality, and inference. Section 5 provides simulations and uses the QRPD estimator on an application. Section 6 concludes.

2

Background

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), and Rosen (2012). The literature on quantile estimation with fixed effects is primarily concerned with the difficulties in estimating 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 (αi ), let us assume that αi is instead known. Quantile estimators with additive fixed effects provide estimates of the distribution of (Yit −αi )|Dit instead of estimating the distribution of Yit |Dit . In many empirical applications, this may be undesirable. Observations at the top of the (Yit −αi ) distribution may be at the bottom of the Yit distribution. Consequently, additive fixed effect models cannot provide information about the effects of the policy variables on the outcome distribution. The motivation for QRPD is that there are many cases when researchers are interested in estimating QTEs for the outcome variable Yit , 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 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 heterogenous effects (i.e., the effect is constant throughout the distribution). This argument likely does not extend generally to the case of heterogenous effects. Ponomareva (2011) introduces an additive effects estimator that is consistent for small T .

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changes the interpretation of the QTEs. In an additive fixed effect models, the model is Yit = αi + D′it β(Uit ) where the parameters vary based only on Uit , not Uit∗ .4 In many applications, the motivation for using quantile regression is to allow the parameters of interest to vary based on the nonseparable disturbance term Uit∗ . Separating αi in these cases partially undermines this original motivation and there is frequently5 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 over time. The corresponding SQF for additive fixed effect quantile models is ˜ τ ), SY (τ |d, αi ) = αi + d′ β(˜ ˜ τ ) is used to highlight that these parameters are different than those in equation where β(˜ (3). Even when the conditions for QR are met, the estimates resulting from QR and additive fixed effect quantile models are not comparable. Table 1 lists the differences between the three types of available quantile estimators with panel data. 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 SQF (and, consequently, the QTEs) and the conditional outcome distribution that is being studied. QRPD relaxes the assumptions of IVQR. Instead of assuming that Uit∗ |Zit ∼ U (0, 1), Uit∗ |Zit is allowed to have an unspecified distribution. Related to the motivation for QRPD, Chernozhukov et al. (2013) discusses identifi4

Some additive fixed effect models also allow the fixed effect to vary with Uit (i.e., αi (Uit )). It is possible that one might be interested in the distribution of the outcome variable given a fixed αi 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. 5

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cation of bounds on quantile effects in nonseparable panel models with exogenous variables. They show that these bounds tighten as T increases. The interpretation of QRPD parallels the interpretation of the bounds using the Chernozhukov et al. (2013) framework. 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 estimator to provide consistent estimates for small T and one of the few instrumental variables quantile panel data estimators. 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, as any set of exogenous fixed effects which saturate the model (e.g., time fixed effects) can be solved for given estimates for the other parameters.

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2.1

Simulated Data Example

I will briefly motivate QRPD through a simple example. In Section 5.1, I perform Monte Carlo simulations with similarly-generated data:

t ∈ {0, 1} Disturbance: Uit∗ = f (αi , Uit ) ∼ U (0, 1) ψit ∼ U (0, 1) Policy Variable: Dit = αi + ψit Outcome: Yit = Uit∗ (1 + Dit )

For a given Dit , the SQF is (1 + Dit )τ . Notice that Dit is a function of αi such that Uit∗ |Dit ̸∼ U (0, 1), implying that QR produces inconsistent estimates since the level of the policy variable is correlated with the disturbance term. The generated data are simple but should illustrate 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 because within-individual changes in the policy variables are independent of within-individual changes in ability. However, an additive fixed effect would be inappropriate with these data. Given that the effect of Dit varies based on a function of both αi and Uit , the appropriate estimation technique must allow the parameters of interest to vary based on Uit∗ . Despite the simplicity of this data generating process, existing quantile estimation methods cannot estimate the relevant SQF or QTEs.

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2.2

Motivating Example: Tax Rebates and the Distribution of Consumption

Johnson, Parker and Souleles (2006) studies whether fiscal policy can increase consumer spending by examining the relationship between tax rebate receipt and household consumption of non-durable goods. Using the randomized timing of the 2001 tax rebates, this paper implements an instrumental variable strategy to estimate the mean increase in non-durable good consumption. They find evidence that fiscal policy can increase short-term spending. Tax rebates can be and frequently are targeted, implying that understanding the distributional impacts are important to policy. For example, low consumption households may be more responsive to transitory income than high consumption households, but mean estimates mask this possible heterogeneity. Furthermore, consumption is heavily-skewed and “censored” at 0, implying that mean techniques may be inappropriate for such analysis while QTEs will provide useful evidence. While estimation of QTEs in this context is policy-relevant, the empirical strategy requires conditioning on household fixed effects since households receiving rebates in any specific month are different than those not receiving rebates. Additive fixed effect quantile models are also inappropriate given that we are interested in the distribution of Cit |Rit , where C represents household consumption and R represents the total rebate amount. Additive fixed effect models would provide information about the distribution of (Cit − αi )|Rit . However, households at the bottom of the (Cit − αi )|Rit distribution may be near the top of the Cit |Rit distribution. Consequently, estimates from an additive fixed effect estimator provide little information about the responsiveness of the bottom of the consumption distribution to tax rebates. Similarly, Misra and Surico (2011) studies the effects of the 2001 tax rebates on consumption using a quantile framework, but they difference their consumption measure first and then perform IVQR. These estimates must be interpreted as the impact 9

of rebates on the distribution of changes in household consumption. In Section 5.2, I use QRPD to estimate the impact of rebates on the consumption distribution. The estimator conditions on household fixed effects but still provides QTEs referring to consumption, not consumption relative to an additive fixed effect.

3

Model

I develop the estimator in an instrumental variables context given instruments Zi = (Zi1 , · · · , ZiT ). If the treatment variables are exogenous then Zi = Di and many of the identification conditions (discussed later) are met trivially. All conditions in this paper are assumed to hold jointly with probability one. A1 Potential Outcomes and Monotonicity: Yit = D′it β(Uit∗ ) where D′it β(Uit∗ ) is strictly increasing in Uit∗ ∼ U (0, 1). A1 is a standard monotonicity condition for quantile estimators (e.g., Chernozhukov and Hansen (2006)). Uit∗ may be a function of several unobserved disturbance terms, summarizing these terms into one rank variable. Alternatively, one can imagine using an unrestricted disturbance term ϵ∗ in the equation of interest Yit = D′it β(ϵ∗it ). There exists a mapping of ϵ∗ to U ∗ . [ A2 Conditional Independence: E

1(Uit∗

≤ τ) −

1(Uis∗

]

≤ τ )|Zi = 0 for all s, t.

A2 is a conditional independence assumption. A2 can be replaced by a stronger assumption of stationarity such that Uit∗ |Zi ∼ Uis∗ |Zi . Instead, the distribution of Uit∗ |Zi can change over time as long as Zi does not predict this change. To aid intuition, let us return to using ϵ∗ . Assumption A2 puts no structure on the overall mean or variance of ϵ∗it for any t. The assumption allows ϵ∗it = at + ct × ϵ∗i1 (Chernozhukov et al. (2013) includes a 10

similar assumption) for some time-varying constants at , ct .6 A2 simply requires Zi to not be systematically related to changes in the distribution of ϵ∗it , using the panel nature of the data to relax the corresponding assumption for IVQR. Note that no restrictions have been placed on the relationship between Uit∗ and αi (i.e., Uit∗ = f (αi , Uit )). Furthermore, there are no assumptions on the relationship between αi and Zi , paralleling fixed effect mean regression models. These assumptions lead to two separate moment conditions. Both conditions will be important for identification. Theorem 3.1 (Moment Conditions). Suppose A1 and A2 hold. Then for each τ ∈ (0, 1), {

] [ 1 ∑∑ ′ ′ β(τ )) ≤ D E (Z − Z ) 1(Y ≤ D β(τ )) − 1(Y is it is it is it 2T 2 t s

} = 0,

(5)

E[1(Yit ≤ D′it β(τ )) − τ ] = 0.

(6)

Proofs are included in Appendix Section A.1. Equation (5) 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-individuals). When discussing identification and other properties, it is easiest to use the following equivalent formulation7

where Zi =

} ] )[ 1 ∑( Zit − Zi 1(Yit ≤ D′it β(τ )) = 0, E T t {

1 T

∑T t=1

(7)

Zit . Equation (7) illustrates that though the quantile function itself is

6

More relaxed error structures than this one are also allowed by A2, but this example is a useful illustration. 7 This condition is equivalent to equation (5) through a straightforward rearrangement of terms.

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the same as one that would be estimated by IVQR (if the IVQR assumptions were met), identification is originating from within-person variation in the instruments. Equation (6) holds with traditional quantile estimators such as QR and IVQR. With IVQR, one assumes both P (Yit ≤ D′it β(τ )) = τ and P (Yit ≤ D′it β(τ )|Zit ) = τ . The QRPD estimator replaces the latter assumption with a weaker one. This is the gain from employing panel data.

3.1

Estimation

Estimation uses Generalized Method of Moments (GMM). Sample moments are defined by

N 1 ∑ 1 gˆ(b) = gi (b) with gi (b) = N i=1 T

{ T ∑(

Zit − Zi

)[

} ] 1(Yit ≤ D′it b) .

(8)

t=1

For identification, it is also necessary to use a sample equivalent of equation (6). To simplify estimation, I constrain the sample version of this equation to hold with equality. When time fixed effects (or any set of dummy variables, indexed by t, which saturate the space) are included, equations (5) and (6) imply P (Yit ≤ D′it β(τ )) = τ

for all t.

(9)

Consequently, I can define the parameter set such that { B≡

N 1 1 ∑ < 1(Yit ≤ D′it b) ≤ τ b | τ− N N i=1

12

} for all t .

(10)

Constraining the parameters to B is a simple way to force Yit ≤ D′it b to hold for 100τ % of the observations in each time period.8 Then, [) = arg min gˆ(b)′ Aˆ ˆg (b) β(τ

(11)

b∈B

ˆ Aˆ can simply be the identity matrix and two-step GMM for some weighting matrix A. estimation can be used. Alternatively, one can minimize the quadratic form using sample moments corresponding to equations (5) and (6) simultaneously. The approach discussed in this paper is only a suggestion but has significant computational advantages. In the application in Section 5.2, this insight reduces the number of parameters that need to be independently estimated from 162 to 1. There is some potential loss in efficiency by implementing the estimator in this manner, but the computational gains are substantial. This simplification follows directly from the moment conditions and makes the estimator straightforward to implement using standard statistical software. 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 (9), 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. Without equation (9), P (Yit ≤ D′it β(τ )) ̸= τ for some t and the estimates cannot be interpreted in the same manner as cross-sectional QR estimates. In the application of this paper, I account for time fixed effects. If consumption patterns changed dramatically over time, these fixed effects are necessary for the 95th quantile { With no time effects and only a constant, then B ≡ b Note that B is guaranteed to be non-empty. 8

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| τ−

1 NT


0. ) ] [ ∑ ( T ∂γt (τ,ϕ) 1 ′ ′ fY (Dit β(τ )|Zi ) exists such that G′ AG nonA8 G ≡ E T t=1 (Zit − Zi ) Xit + ∂ϕ′ singular. 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) ∂ϕ′

term can be excluded if this procedure

is not used. The other regularity conditions are standard. p p d) −→ Theorem 4.2 (Consistency). If A1 - A7 hold and Aˆ −→ A positive definite, then β(τ

β(τ ). A discussion is included in Appendix Section A.2.

4.3

Asymptotic Normality

Stochastic equicontinuity is an important condition for asymptotic normality of GMM es{ } timators and follows here from the fact that the functional class 1(Yit ≤ D′it b), b ∈ Rk is Donsker and the Donsker property is preserved when the class is multiplied by a bounded random variable. Thus, {

T ] )[ 1 ∑( ′ Zit − Zi 1(Yit ≤ Dit b) , b ∈ Rk T t=1

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}

is Donsker with square integrable envelope 2 max(i,t) |Zit − Zi |. Stochastic equicontinuity follows from Theorem 1 in Andrews (1994). Define Σ ≡ E[gi (β(τ ))gi (β(τ ))′ ]. p Theorem 4.3 (Asymptotic Normality). If A1 - A8 hold and Aˆ −→ A positive definite, √ d d) − β(τ )) −→ N [0, (G′ AG)−1 G′ AΣAG(G′ AG)−1 ]. then N (β(τ

A discussion is included in Appendix Section A.2.

4.4

Inference

I adopt an approach similar to the histogram estimation technique suggested in Powell (1986) to obtain consistent estimates of G: [ ( ) ] N T ( ) \ ∑ ∑ 1 1 ∂γ (τ, ϕ) t [) ≤ h , ˆ= G (Zit − Zi ) X′it + 1 Yit − D′it β(τ 2N h i=1 T t=1 ∂ϕ′ b= for small h. For a consistent estimate of Σ, I use Σ

18

1 N

∑ i

[))gi (β(τ [))′ . gi (β(τ

5 5.1

Applications Simulations

To illustrate the usefulness of the QRPD estimator, I generate the following data. F (·) represents the CDF of αi + Uit : t ∈ {0, 1} Fixed Effect: αi ∼ U (0, 1) Uit ∼ U (0, 1) Total Disturbance: Uit∗ ≡ F (αi + Uit ) ⇒ Uit∗ ∼ U (0, 1) Year Effect: δ0 = 1, δ1 = 2 ψit ∼ U (0, 1) Instrument: Zit = αi + ψit Policy Variable: Dit = Zit + Uit Outcome: Yit = Uit∗ (δt + Dit )

Note that Dit is a function of Uit so IV is necessary. Zit is exogenous conditional on αi so IVQR estimates would be inconsistent and conditioning on fixed effects is necessary. The impact of Dit is a function of Uit∗ and varies by observation. The coefficient on Dit in the SQF is equal to τ . Additive fixed effect quantile models assume that the coefficient only depends on Uit and should also produce inconsistent estimates. Year fixed effects are crucial as the distribution changes (differentially) across years. I generate these data for N = 500, T = 2. Grid-searching is used to minimize the GMM objective function. 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., 19

Uit∗ |Zit ∼ U (0, 1)) required to use cross-sectional quantile estimation techniques. It should not be surprising that IVQR performs poorly. 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. Note that IVQRFE fares well near the median because, for this data generating process, the median of Uit and the median of Uit∗ are equal. 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 uniform. Yet, existing quantile estimators do not allow one to estimate quantile treatment effects 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 Non-Durable Consumption

In this section, I use the QRPD estimator to estimate the effect of tax rebates on short-term non-durable consumption using data from Johnson, Parker and Souleles (2006). Tax rebates in 2001 were provided to a majority of the population, and the timing of the rebates was based on the second-to-last digit of the tax filer’s Social Security number. Johnson, Parker and Souleles (2006) study how tax rebates affect mean changes in non-durable consumption using the Consumer Expenditure Survey, which provides information on rebates and consumption for the same household for multiple quarters.

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I estimate a similar specification:13

Cit = αi + γht + βRit + ϵit

(13)

where Cit represents non-durable consumption of household i in quarter t. R represents the size of the rebate received (0 if no rebate). I include interactions based on household size and month.14 Johnson, Parker and Souleles (2006) note that households receiving rebates in any specific month are different than those not receiving rebates and, consequently, their empirical strategy requires accounting for fixed household differences. Furthermore, they argue that the actual amount of the rebate is a function of income and other factors, which may be correlated with other determinants of consumption, but the timing of a rebate is plausibly exogenous. They use 1(Rit > 0) as an instrument for Rit so that identification only originates from changes in rebate receipt and not the rebate amount. Because timing is random given receipt of a rebate, this instrument is plausibly exogenous conditional on household fixed effects. Mean-IV regression of equation (13) estimates β = 0.24 with a standard error of 0.10. This result implies that for every $1 received, the household spends an additional 24 cents on non-durable goods. Because we are also interested in whether the lower part of the consumption distribution is more or less responsive to tax rebates, QTEs provide useful information about the effects of this type of fiscal policy. Consumption is heavily-skewed such that it is possible that the mean estimate provides little information about the impact 13

There are some differences between this specification and the one used in Johnson, Parker and Souleles (2006) to facilitate the comparison between the mean and quantile estimates. First, I do not use differences because we are not interested in the distribution of the change in consumption. Instead, I include household fixed effects. Second, I include household size × time interactions because these will be important for proper interpretation of the QTEs. The mean-IV results using this specification are similar to those found in Johnson, Parker and Souleles (2006). 14 Instead of just month fixed effects, it is also important to account for household size. Without accounting for household size, the quantile estimates at the top of the distribution would primarily refer to large households since they tend to consume more. Instead, we are likely interested in how tax rebates affect households at the top of the distribution given their size.

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at any part of the distribution. Traditional quantile estimation methods are incapable of providing these estimates in this context. Location-shift quantile models would provide us with the impact of tax rebates on the τ th quantile of Cit − αi . I use the QRPD of this paper to estimate QTEs. The results are presented in Table 3. I find significant effects for the bottom of the consumption distribution. The effect appears to decrease to 0 in the upper part of the distribution until the very top. I cannot reject a null effect above quantile 55. However, it is important to highlight that at no point in the distribution can I reject the mean estimate. The estimates in Table 3 show the distributional impact of tax rebates on non-durable consumption. These estimates are not possible given existing quantile panel data methods.

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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 fixed effects. This estimator should be useful in contexts where identification requires differences and it is believed that the effects of the variables are heterogenous 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. I also apply the estimator to the analysis of Johnson, Parker and Souleles (2006). 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 requires estimation of the distribution of (Yit − αi )|Zit , even when the distribution of interest 22

is Yit |Zit . Additive fixed effect quantile models do not allow the parameters of interest to vary based on αi . The QRPD estimator introduced in this paper allow the parameters to vary based on an unknown function of the fixed effect and an observation-specific disturbance term while relaxing the identification assumptions required for QR and IVQR.

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A

Appendix

A.1

Moment Conditions and Identification

Theorem 3.1 (Moment Conditions). Suppose A1 and A2 hold. Then for each τ ∈ (0, 1), {

1 ∑∑ E (Zit − Zis ) [1(Yit ≤ D′it β(τ )) − 1(Yis ≤ D′is β(τ ))] 2 2T t s

Proof of (5):

{ E

} for all s, t,

(5)

E[1(Yit ≤ D′it β(τ )) − τ ] = 0.

(6)

[

=0

(Zit − Zis ) 1(Yit ≤ D′it β(τ )) − 1(Yis ≤ D′is β(τ ))

} ]

}] [ { = E E (Zit − Zis ) [1(Yit ≤ D′it β(τ )) − 1(Yis ≤ D′is β(τ ))] Zi }] [ { = E (Zit − Zis )E 1(D′it β(Uit∗ ) ≤ D′it β(τ )) − 1(D′is β(Uis∗ ) ≤ D′is β(τ )) Zi [ ] } { = E (Zit − Zis )E 1(Uit∗ ≤ τ ) − 1(Uis∗ ≤ τ ) Zi =0

by A2

Proof of (6): E[1(Yit ≤ D′it β(τ ))] = E[1(D′it β(Uit∗ ) ≤ D′it β(τ ))] = P [Uit∗ ≤ τ ] = τ

by A1

24

by A1

by A1

by A1 by A1

Theorem 4.1 (Identification). If (i) A1 - A4 hold; { ∑ ( ]} [ ] )[ ˘ ˘ = τ , then β˘ = (ii) E T1 Tt=1 Zit − Zi 1(Yit ≤ D′it β) = 0; (iii) E 1(Yit ≤ D′it β) β(τ ). First, some notation: 

(1)′

 P (Yit ≤ d β|Zi )  .. Γ(Zi , β) ≡  .   ′ P (Yit ≤ d(M ) β|Zi )

   .  

Initially, I assume that the policy variables are discrete such that Ψ includes all possible values. The extension is straightforward and included after the proof. ˘ = Proof. Starting with (ii) and the Law of Iterated Expectations, E[(Zit −Zi )Π(Zi )Γ(Zi , β)] 0. ′ ˘ i ) = P (Yit ≤ d(1)′ β(˜ Without loss of generality, assume that P (Yit ≤ d(1) β|Z τ )|Zi )

for some τ˜ ∈ (0, 1). ′ ˘ i ) = P (Yit ≤ d(m)′ β(˜ By A3, we know that P (Yit ≤ d(m) β|Z τ )|Zi ) for all m. ′ ′ A4 implies that d(m) β˘ = d(m) β(˜ τ ) for all m. By the full rank assumption in A3 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 Ψ 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

25

is that there exists a subset of values that have positive probability.15 Here, I simply add a term to Π(Zi ) for the probability that Dit does not equal one of the values in Ψ and a corresponding term to Γ(Zi , β): 



P (Di1 = ̸ d , · · · , Di1 ̸= d )|Zi )   .. ΠF (Zi ) ≡  .  Π(Zi )  P (DiT = ̸ d(1) , · · · , DiT ̸= d(M ) )|Zi ) (1)

(M )

    

  ΓF (Zi , β) ≡ 

 Γ(Zi , β) P (Yit ≤ D′it β|Zi , Dit ̸= d(1) , · · · , Dit ̸= d(M ) )

 .

˘ = Note that the above proof does not change when we analyze E[(Zit − Zi )ΠF (Zi )ΓF (Zi , β)] 0.

A.2

QRPD Properties

These properties are discussed for small T as N → ∞. This discussion will assume that B is defined by equation (10). Let ||·|| be the Euclidean norm and fY (·) represent the pdf of Yit conditional on Zi .

A.2.1

Consistency

p p d) −→ Theorem 4.2 (Consistency). If A1 - A7 hold and Aˆ −→ A positive definite, then β(τ

β(τ ). Note that the following conditions hold: 15

Importantly, this assumption holds in the application of this paper since many households received rebates that were multiples of $300.

26

1. Identification holds by Theorem 4.1. 2. Compactness of B holds by assumption A6. 3. gi (b) is continuous at each b with probability one under A4. 4. E ||gi (b)|| ≤ supt E Zit − Zi < ∞ by A7. Under these conditions, consistency follows immediately from Theorem 2.6 of Newey and McFadden (1994).

A.2.2

Asymptotic Normality

p Theorem 4.3 (Asymptotic Normality). If A1 - A8 hold and Aˆ −→ A positive definite, √ d d) − β(τ )) −→ then N (β(τ N [0, (G′ AG)−1 G′ AΣAG(G′ AG)−1 ].

[). Define β0 ≡ β(τ ), βˆ ≡ β(τ [ ∑ ( ) ] (τ,ϕ) Also, G(β) ≡ E T1 Tt=1 (Zit − Zi ) D′it + ∂γt∂ϕ fY (D′it β|Zi ) . ′ [ ∑ ( ]] )[ T 1 ′ g(β) ≡ E T t=1 Zit − Zi 1(Yit ≤ Dit β) . Proof: g(β)′ Ag(β) is minimized at β0 implying that G(β0 )′ Ag(β0 ) = 0.

Expanding each element of g(β0 ) around βˆ and multiplying by √ √ √ ˆ − G(β) ˜ N (βˆ − β0 ) N g(β0 ) = N g(β)

√ N gives

(14)

ˆ ˜ ˜ where β meets the condition β − β0 ≤ β − β0 and takes on different values for each 27

˜ column of G(β). p

˜ −→ G(β0 ). By assumption of i.i.d and continuity, G(β) Focus on the

√ ˆ term: N g(β)

[√ ] √ √ √ ˆ = ˆ − N g(β) ˆ − N gN (β) ˆ − N g(β) N gN (β) ] √ √ [ √ ˆ ˆ ˆ . = N gN (β) − g(β) − gN (β0 ) + N gN (β0 ) − N gN (β) | {z } | {z } | {z } (2)

(1)

(1): Define empirical process vN (β) =

√1 N

∑N i=1

(3)

[gN (β) − g(β)].

{ } The functional class 1(Yit ≤ D′it b), b ∈ Rk is Donsker and the Donsker property is preserved when the class is multiplied by a bounded random variable (see Theorem 2.10.6 in van der Vaart and Wellner (1996)16 ). Thus, {

T ] )[ 1 ∑( Zit − Zi 1(Yit ≤ D′it b) , b ∈ Rk T t=1

}

is Donsker with envelope 2 max(i,t) |Zit − Zi |. Stochastic equicontinuity of vN (·) follows from A7 and Theorem 1 in Andrews (1994). Stochastic equicontinuity and consistency of βˆ implies that part (1) is op (1). (2): By the Central Limit Theorem, ˆ (3): By consistency of β,

√ d N gN (β0 ) −→ N (0, Σ) where Σ = E[g(β0 )g(β0 )′ ].

√ ˆ = op (1). N gN (β)

Plugging into (14) and using the assumption that G′ AG nonsingular √ [ ] d N (βˆ − β(τ )) −→ N 0, (G′ AG)−1 G′ AΣAG(G′ AG)−1 16

See Example 2.10.10 as well.

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A.3

Nonseparable Time Fixed Effects

The primary motivation of this paper is to introduce a quantile estimator which conditions on fixed effects – relaxing the assumptions required for QR and IVQR – but can still be interpreted in the same manner as cross-sectional quantile estimates. Additive time effects are important for this interpretation and act as a simple way of enforcing equation (9). Equation (9) holds with cross-sectional QR (performed one year at a time). Without these time fixed effects, the estimates referring to the top of the distribution may correspond primarily to observations in specific time periods. For example, if one is studying the wage distribution during a period of high inflation, then the top quantile estimates will primarily refer to later years in the sample. If one does not want to include additive time fixed effects, the framework of this paper does not require them. Alternatively, one may want to condition on nonseparable time ∑ ∑N ∑T 1 fixed effects such that Uit∗ = f (αi , Uit , γt ). Let Zt = T1 N i=1 Zit and Z = N T i=1 t=1 Zit . We can replace assumption A2 with [ E

1(Uit∗

≤ τ) −

1(Uis∗

]

≤ τ )|Zi , Zt , Z = 0.

Under this assumption, we get the following condition: { E

)[ ] 1 ∑( Zit − Zi − Zt + Z 1 (Yit ≤ D′it β(τ )) |Zi , Zt , Z T t

Assuming a constant, estimation can proceed as before.

29

} = 0.

References Andrews, Donald W.K., “Empirical process methods in econometrics,” Handbook of Econometrics, 03 1994, 4, 2247–2294. 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), 988–1012. Canay, Ivan A., “A Note on Quantile Regression for Panel Data Models,” The Econometrics Journal, 2011, 14, 368–386. Chernozhukov, Victor and Christian Hansen, “Instrumental quantile regression inference for structural and treatment effect models,” Journal of Econometrics, 2006, 132 (2), 491–525. and , “Instrumental variable quantile regression: A robust inference approach,” Journal of Econometrics, January 2008, 142 (1), 379–398. and Han Hong, “An MCMC approach to classical estimation,” Journal of Econometrics, August 2003, 115 (2), 293–346. , Iv´ an Fern´ andez-Val, Jinyong Hahn, and Whitney Newey, “Average and Quantile Effects in Nonseparable Panel Models,” Econometrica, 2013, 81 (2), 535–580. Doksum, Kjell, “Empirical Probability Plots and Statistical Inference for Nonlinear Models in the Two-Sample Case,” Ann. Statist., 1974, 2 (2), 267–277. Galvao Jr., Antonio F., “Quantile regression for dynamic panel data with fixed effects,” Journal of Econometrics, September 2011, 164 (1), 142–157. 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), 181–182. Harding, Matthew and Carlos Lamarche, “A quantile regression approach for estimating panel data models using instrumental variables,” Economics Letters, 2009, 104 (3), 133 – 135. 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), 1589–1610. Koenker, Roger, “Quantile regression for longitudinal data,” Journal of Multivariate Analysis, October 2004, 91 (1), 74–89. Koenker, Roger W and Gilbert Bassett, “Regression Quantiles,” Econometrica, January 1978, 46 (1), 33–50.

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Lamarche, Carlos, “Robust penalized quantile regression estimation for panel data,” Journal of Econometrics, August 2010, 157 (2), 396–408. Misra, Kanishka and Paolo Surico, Heterogeneous responses and aggregate impact of the 2001 income tax rebates, Citeseer, 2011. Newey, Whitney K. and Daniel McFadden, “Large sample estimation and hypothesis testing,” Handbook of Econometrics, 1994, 4, 2111–2245. Ponomareva, Maria, “Quantile Regression for Panel Data Models with Fixed Effects and Small T: Identification and Estimation,” Working Paper, University of Western Ontario May 2011. Powell, James L., “Censored regression quantiles,” Journal of Econometrics, June 1986, 32 (1), 143–155. Rosen, Adam M., “Set identification via quantile restrictions in short panels,” Journal of Econometrics, 2012, 166 (1), 127 – 137. van der Vaart, A.W. and Jon A. Wellner, Weak Convergence and Empirical Processes, Springer, 1996.

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Tables Table 1: Comparison of Estimators Assumption Underlying Model Outcome Distribution Structural Quantile Function Interpretation for τ th quantile

Pooled IVQR

Additive Fixed Effects

QRPD

Uit∗ |Zit ∼ U (0, 1) Yit = D′it β(Uit∗ ) Yit d′ β(τ ) τ th quantile of U ∗

Uit |Zit , αi ∼ U (0, 1) Yit = αi + D′it β(Uit ) Yit − αi ˜ τ) αi + d′ β(˜ τ th quantile of U

∗ Uit∗ |Zi ∼ Uis |Zi ′ Yit = Dit β(Uit∗ ) Yit d′ β(τ ) 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 ˜ τ ) notation for the additive fixed effects model is used since τ˜ is mentioned in the text. The β(˜ ∗ different from τ . Uit = f (αi , Uit ). Differences in the Structural Quantile Function imply differences in the quantile treatment effects.

Table 2: IVQRPD Simulation (N=500, T=2) IVQR Quantile 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

Mean Bias 0.56057 0.70229 0.80304 0.87783 0.93577 0.98169 1.01647 1.04178 1.06114 1.06906 1.06489 1.04540 1.00899 0.96410 0.91812 0.86625 0.79638 0.70683 0.58787

MAD 0.55 0.70 0.80 0.88 0.93 0.98 1.02 1.04 1.06 1.07 1.07 1.05 1.01 0.96 0.92 0.87 0.80 0.71 0.59

IVQRFE RMSE 0.56753 0.70723 0.80664 0.88058 0.93802 0.98365 1.01806 1.04303 1.06216 1.06987 1.06563 1.04602 1.00952 0.96461 0.91867 0.86687 0.79722 0.70813 0.59085

Mean Bias 0.39750 0.34740 0.29736 0.24750 0.19762 0.14765 0.09748 0.04731 -0.00259 -0.05266 -0.10259 -0.15269 -0.20252 -0.25235 -0.30238 -0.35251 -0.40264 -0.45260 -0.50250

MAD 0.41 0.36 0.31 0.26 0.21 0.16 0.13 0.10 0.09 0.10 0.11 0.15 0.19 0.24 0.29 0.34 0.39 0.44 0.49

IVQRPD RMSE 0.42170 0.37478 0.32898 0.28468 0.24270 0.20403 0.17123 0.14851 0.14093 0.15030 0.17430 0.20768 0.24663 0.28898 0.33360 0.37954 0.42653 0.47395 0.52185

Mean Bias -0.00544 -0.01025 -0.00941 -0.01046 0.00099 0.00181 0.00337 0.00291 0.00773 0.00852 0.00442 0.00167 -0.00151 -0.00279 -0.00361 -0.00390 -0.00539 -0.00672 -0.01127

MAD 0.05 0.06 0.08 0.09 0.11 0.11 0.12 0.12 0.13 0.13 0.13 0.13 0.12 0.12 0.12 0.12 0.12 0.10 0.09

RMSE 0.07027 0.09861 0.11788 0.13316 0.14822 0.16042 0.16867 0.17832 0.18106 0.18329 0.18429 0.18474 0.18685 0.18217 0.18069 0.17601 0.16687 0.15145 0.12454

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).

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Table 3: IVQRPD Estimates of the Impact of Tax Rebates on Consumption Quantile IVQRPD Quantile IVQRPD 5 10 15 20 25 30 35 40 45 50

0.22* (0.12) 0.29** (0.13) 0.28*** (0.08) 0.25** (0.10) 0.29*** (0.10) 0.27*** (0.08) 0.28*** (0.09) 0.26*** (0.10) 0.22* (0.12) 0.30* (0.18)

55 60 65 70 75 80 85 90 95

N

0.29** (0.14) 0.25 (0.16) 0.21 (0.14) 0.21 (0.15) 0.08 (0.23) -0.04 (0.27) -0.14 (0.31) 0.23 (0.35) 0.32 (0.62) 20,975

Significance levels: *10%, **5%, ***1%

33