Quantile regression with nonadditive fixed effects

I use capital letters to designate random variables and lower case letters to denote potential values that those random variables may take.

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.

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.

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

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

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.

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

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.

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.

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