We study inference for linear quantile regression with two-way clustered data. Using a separately exchangeable array framework and a projection decomposition of the quantile score, we characterize regime-dependent convergence rates and establish a self-normalized Gaussian approximation. We propose a two-way cluster-robust sandwich variance estimator with a kernel-based density ``bread'' and a projection-matched ``meat'', and prove consistency and validity of inference in Gaussian regimes. We also show an impossibility result for uniform inference in a non-Gaussian interaction regime.
Quantile regression (QR), introduced by Koenker and Bassett Jr (1978), is a widely used tool for studying heterogeneous effects and tail risks in economics and finance. In many empirical environments, however, observations are indexed by multiple clustering dimensions and exhibit dependence along each of them. A canonical example is a two-way array {(y gh , X gh ) : g = 1, . . . , G, h = 1, . . . , H} in which observations can be correlated within the g-dimension and within the h-dimension because of latent shocks shared by units in the same row or column (e.g., worker × firm, exporter × destination).
This paper develops a unified large-sample theory and feasible inference procedures for linear QR under two-way clustering. We study the conditional quantile regression model and allow for rich two-way dependence using an Aldous-Hoover-Kallenberg (AHK) representation for separately exchangeable arrays (Aldous, 1981;Hoover, 1979;Kallenberg, 1989). This framework has become a standard device for modeling multi-way clustered dependence and for deriving projection-based asymptotics for array data (e.g., Davezies et al., 2021;Menzel, 2021;Chiang et al., 2024;Graham, 2024). Building on this structure, we establish a selfnormalized central limit theorem that accommodates regime-dependent rates and delivers asymptotic normality.
We then propose a feasible two-way cluster-robust variance estimator (CRVE) for QR of the familiar sandwich form Σ(τ ) = D(τ ) -1 Ω(τ ) D(τ ) -1 .
The “bread” D(τ ) is a kernel-based estimator of the conditional density at the target quantile, adapted here to two-way clustering, while the “meat” Ω(τ ) aggregates row-and columncluster covariance contributions along with a residual component in a manner that mirrors the underlying projection decomposition.
Four features fundamentally complicate establishing the consistency of Σ(τ ) relative to standard two-way clustered mean regression (e.g., Cameron et al. 2011;MacKinnon et al. 2021) and to one-way clustered quantile regression (e.g., Parente and Santos Silva 2016;Hagemann 2017). First, unlike mean regression, the quantile score is non-smooth, which makes uniform control of score fluctuations in neighborhoods of β 0 (τ ) more delicate. Second, the Jacobian depends on the conditional density at zero and is estimated nonparametrically, so the proof must control the bias and stochastic error of a kernel-based “bread” under clustering. Third, unlike one-way clustered quantile regression, the effective convergence rate of β(τ ), denoted r GH , can vary across dependence regimes: depending on the relative magnitudes of the row, column, and interaction components of the score, different projection terms may dominate the leading stochastic fluctuation. Fourth, two-way dependence precludes re-CRVE and pigeonhole bootstrap results for GMM (e.g. quantile IV) under multiway clustering (Davezies et al., 2018), and recent advances in weak-dependence-robust covariance estimation for quantile regression (Galvao and Yoon, 2024).
Relative to these papers, our contributions are threefold. First, we establish asymptotic normality of Powell’s kernel estimator under two-way clustering, and derive a feasible optimal bandwidth rule. Second, to the best of our knowledge, we provide the first feasible two-way cluster-robust variance estimator for quantile regression at a fixed τ ∈ (0, 1), and prove its uniform validity whenever the Gaussian limit arises. Although Davezies et al. (2018) propose multiway variance estimation for GMM, their approach is not directly applicable here because it relies on a plug-in Jacobian that requires knowledge of the true conditional density, which is typically unavailable in practice. Moreover, unlike the setting emphasized in Davezies et al. (2018), we do not impose nondegeneracy of the asymptotic variance: the rate of convergence is allowed to vary with the strength of clustering, and our variance estimator is designed to adapt across these regimes. Third, when the limiting distribution in two-way clustered quantile regression is non-Gaussian, we show that uniform consistency of inference is impossible.
The remainder of the paper is organized as follows. Section 2 introduces the two-way QR framework and the projection decomposition and develops the regime-adaptive limit theory for β(τ ). Section 3 proposes D(τ ) and Ω(τ ) and establishes the consistency of Σ(τ ) and the validity of inference based on the t-statistic. Section 4 presents Monte Carlo evidence. Section 5 studies an application to teacher licensing. Section 6 concludes. Technical proofs and additional results are deferred to the appendix.
2 Two-Way Clustering in Quantile Regression
Let {(y ghi , X ⊤ ghi ) : g = 1, . . . , G, h = 1, . . . , H, i = 1, . . . , N gh } be an array of observations, where y ghi ∈ R is the scalar response and X ghi ∈ R d is a vector of regressors. The first index g identifies the cluster in the first dimension (the g-cluster), and the second
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