We characterize the asymptotic behavior of conventional variance estimators in linear regression with high-dimensional fixed effects under a drift in which both the proportional fixed-effect dimension $\rho_n = d_{K_n}/n \to \rho \in [0,1)$ and the residual treatment variance $\tau_n^2 = nQ_{K_n} \to \tau^2 \in (0, \infty]$ are non-degenerate. Three findings emerge. First, under strict exogeneity and conditional homoskedasticity, the Cattaneo--Jansson--Newey-corrected $t$-statistic is asymptotically exact for any $\tau^2>0$: there is no Stock--Yogo-style threshold in $\tau^2$. Second, the Eicker--White HC0 estimator is biased downward by a fixed factor $(1-\rho)$, producing over-rejection that grows with saturation. Third, HC3 over-corrects in the opposite direction by a factor $1/(1-\rho)$. The leave-one-out estimator (HC2) removes the first-order leverage distortion and is asymptotically exact under homoskedasticity or design-balanced heteroskedasticity; under general heteroskedasticity with non-uniform leverage, HC2 retains an additional bias of order $\rho|\mu - \omega^2|$ that we characterize. An empirical application to Piotroski F-Score returns in CEE markets illustrates the predicted variance hierarchy in real data.
Fixed-effect saturation does not itself distort conventional inference, but classical measurement error does. Under a local noise drift $\sigma_\nu^2=c^2/n$, the FE-OLS $t$-statistic converges to a non-central normal; saturation contributes a common $\sqrt{1-\rho}$ scaling rather than preferentially destroying signal or noise. Inverting the size distortion gives a Stock--Yogo-style critical value. Self-consistency of the within-reliability-corrected pilot yields a breakdown reliability $\lambda^{\dagger}=|t|/(|t|+\eta^{\dagger})$ --- the minimum within reliability at which conventional inference retains nominal size within the chosen tolerance --- computable from the reported $t$-statistic alone and algebraically $\rho$-free conditional on it; $\eta^{\dagger}\approx0.65$ at $5\%$ size and a 5-point tolerance. Replacing $|t|$ by $|t|+z_{1-\gamma_\beta}$ gives a certified breakdown reliability; with a lower-reliability bound whose coverage error is $\gamma_\lambda$, false certification is at most $\gamma_\beta+\gamma_\lambda$. Under a checkable projection-compatibility condition, a cluster-level score CLT and consistency of the Arellano variance estimator in the many-fixed-effect regime justify applying the same map to the reported cluster-robust $t$-statistic; clustering can reverse a verdict. In a saturated democracy--growth panel, aggregate V-Dem polyarchy is certified at $\gamma_\beta=0.05$, conditional on the supplied measurement model, while its judicial-constraints sub-index is flagged under i.i.d.\ and clustered standard errors. In a twin-pair wage design, the specification is flagged under both independent and correlated reporting-error models, although implied coverage of the nominal-$95\%$ interval ranges from $8\%$ to $68\%$. The diagnostic covers classical error in a continuous regressor, not binary-treatment misclassification.
A (computationally inefficient) adaptive estimator that, so long as $p$ is a mixture of $k$ symmetric log-concave densities, achieves error comparable with the optimal estimator that knows $p$ and has $\tilde\Theta(n/k)$ samples.
We study estimation of a constant conditional variance $\sigma^2$ in nonparametric regression with a $d$-dimensional random design. This is an important problem, and similar questions arise in causal inference. The regression function is $\beta_b$-H\"older smooth, the design density is $\beta_g$-H\"older smooth and bounded above and away from zero, and we consider the nonparametric regime $\beta_b>1$ and $d>4\beta_b$. Set $\beta_g^\star=\beta_b(1-4\beta_b/d)/\{1+2\beta_b/d+8(\beta_b/d)^2\}$. We give an estimator whose mean squared error is upper bounded by $Cn^{-4(\beta_b+1)/(d+4)}$ in the low-regularity regime when $0<\beta_g\leq\beta_g^\star$. The low-regularity branch is based on a new two-scale construction: the covariate space is partitioned into cells, the local polynomial trend is projected out within each suitable cell, and the squared normalized contrast from one eligible close pair per cell is averaged across cells. In the high regularity regime when $\beta_g>\beta_g^\star$, a higher-order influence function estimator of Robins, Li, Tchetgen Tchetgen, and van der Vaart (2008) provides the rate $Cn^{-8\beta_b/(d+4\beta_b)}$. We also give an all-pairs ridge extension, which achieves the same two-scale rate, and evaluate the methods alongside a range of existing estimators in simulations.
Edgar Dobriban, Rajarshi Mukherjee, James M. Robins et al.· 0 citations
We study finite-sample valid tests of the one-sided mean hypothesis $H_0:\mu\leq 1$ against $H_1:\mu>1$ for nonnegative random variables. To do so, we develop a leave-one-out dual certificate framework, where certain pointwise inequalities imply p-value validity under the conditional mean null $\mathbb{E}[X_i\mid\mathbf{X}_{-i}]\leq 1$, and which also gives conditions that allow combining dual certificates for p-values to show that their pointwise minimum is also a valid p-value. The framework proves finite-sample validity of Wang and Zhao's nonparametric likelihood-ratio statistic $T_{\mathrm{nplr}}$, yields a new p-value $T_{\mathrm{bin}+}$ extending the Clopper--Pearson binomial test to general nonnegative random variables, and shows that the pointwise minimum $\min\{T_{\mathrm{nplr}},T_{\mathrm{bin}+}\}$ is itself a valid and more powerful p-value. We establish sharp optimality results for such testing problems in two regimes: both $T_{\mathrm{nplr}}$ and $T_{\mathrm{bin}+}$ attain a universal detectability boundary for the null $H_0$ without moment or tail assumptions, and $T_{\mathrm{bin}+}$ attains a nonparametric power lower bound under $n^{-1/2}$-local alternatives to $H_0$. Efficient algorithms and numerical experiments demonstrate substantial finite-sample power gains over existing valid methods.
Classical likelihood-ratio tests and $\Delta$AIC exacerbate the statistical significance crisis by scaling with sample size, often flagging negligible improvements as highly significant. While causal estimands like the average treatment effect (ATE) quantify practical magnitude, their reliance on the expectation operator ties them to the data's original coordinate scale. Furthermore, existing pseudo-$R^2$ metrics are inadequate: variance-based measures ignore higher-order distributional changes, and current formulations lack invariance to monotone transformations. We resolve these limitations by introducing Entropic Variance (EV) as a rigorous, scale-independent generalization of error variance in ordinary least squares. We define the population EV-based parameter, $\rho^2_V$, which projects unbounded cross-entropy onto a standardized $[0,1]$ scale, and establish that the EV-based $F_\text{V}$ statistic asymptotically follows an $F$-distribution. Building on these distributional properties, we propose two estimators: the empirical population $R^2_{\text{SV}}$ and the out-of-sample predictive $R^2_{\text{SVP}}$. Both are derived by exponentiating per-observation cross-entropy and incorporate a degrees-of-freedom correction for training optimism. Leveraging the $F_\text{V}$-distribution, we derive refined $p$-values and confidence intervals for $\rho^2_V$ without requiring intractable Fisher information matrices. Simulation studies and a Parkinson's disease microbiome application demonstrate the superiority of variable selection via these EV-$R^2$ metrics. Notably, evaluating the $R^2_{\text{SVP}}$ of a LASSO path via data-splitting reduced false discovery rates from 80% to 6% in simulations while fully preserving signal recall.
We study identification of two-way unobserved heterogeneity in the nonparametric panel regression $G_{it}=g(\alpha_i,\gamma_t)+\varepsilon_{it}$, where identification of the latent types reduces to constructing identified, \emph{injective} proxies for them. To this end we consider the singular value decomposition (SVD) of the bivariate regression function $g(\alpha,\gamma)$ on a product domain $\Omega_\alpha\times\Omega_\gamma$, whose left singular functions $\{u_r\}$ serve as proxies for the unobserved heterogeneity parameter $\alpha$. The arguments are symmetric for $\{v_r\}$ vis-\`a-vis $\gamma$. We work under an \emph{observational-equivalence simplification}: two values of $\alpha$ that induce the same conditional response $g(\alpha,\cdot)$ are identified, so that the response map $\alpha\mapsto g(\alpha,\cdot)$ is injective by construction. We show two things. First, this reduction is \emph{equivalent} to injectivity of the full collection of left singular eigenfunctions, so no further condition is needed over the infinite collection $\{u_r\}_{r\ge1}$. Second, under a single additional \emph{local injectivity} condition, a finite collection of leading eigenfunctions $U_R=(u_1^{\top},\dots,u_R^{\top})^{\top}$ is injective for all sufficiently large $R$. The proof reduces a global univalence question to a local first-order condition plus a topological compactness argument, bypassing the global Jacobian conditions usually required.
Hugo Freeman, Dennis Kristensen· 0 citations
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