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Quantile regression with nonadditive fixed effects

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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 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 produces consistent estimates for small T. I estimate the effect of the 2008 tax rebates on the short-term household consumption distribution.

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Notes

  1. I 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 “individuals" in this paper as the observed unit, though the estimator is also useful for state fixed effects, fixed effects based on demographics, and so on.

  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. It is also sometimes assumed that \(\alpha _i(U_{it})= \alpha _i\) such that the fixed effects do not vary based on \(U_{it}\).

  5. This framework intentionally follows Chernozhukov and Hansen (2005) closely.

  6. For GQR, any non-additive fixed effects would need to be included in the estimation of the conditional probability that the outcome is smaller than the quantile function. With small T, this approach will not generate consistent predictions. This limitation partially motivates the introduction of QRPD.

  7. With no time effects and only a constant, then \({\mathcal {B}} \equiv \left\{ {\mathbf {b}} \quad | \quad \tau -\frac{1}{NT} < \frac{1}{NT} \sum \nolimits _{i=1} \sum \nolimits _{t=1} {\mathbf {1}}\!\left( Y_{it} \le {\mathbf {D}}_{it}'{\mathbf {b}}\right) \le \tau \right\} \). Note that \({\mathcal {B}}\) is guaranteed to be non-empty.

  8. With exact identification, the moment conditions hold with equality (or as close to equality as possible) regardless so there is no loss in efficiency.

  9. 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.

  10. 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) control for quarter fixed effects, number of adults, and number of dependents. The interaction terms correspond to the “time fixed effects” discussed in Sect. 3.2. They are important for the estimation of QTEs. If, for example, I did not allow the consumption distribution to shift based on number of adults and children, then the “high quantiles" would primarily refer to large households. We are likely interested in how the top of the household consumption—given its size—responds to transitory income.

  11. Estimates greater than one are consistent with households using transitory income shocks to supplement savings to make larger purchases.

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  1. RAND, Arlington, USA

    David Powell

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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, Lane Burgette, 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 Stata Conference. Funding from the CDC (R01CE02999) is gratefully acknowledged.

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Powell, D. Quantile regression with nonadditive fixed effects. Empir Econ 63, 2675–2691 (2022). https://doi.org/10.1007/s00181-022-02216-6

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