arXiv subjects

Jonathan B. Hill

Publications and source records attributed to Jonathan B. Hill.

13 recordsLinked to original sources

arXiv2026

A High Dimensional Wild Bootstrap Max-Test for Detecting the Presence of Significant Predictors

We construct a block bootstrap max-test for detecting the presence of significant predictors in a high dimensional setting, allowing for weakly dependent and heterogeneous (possibly non-stationary) data. The number of covariates to be screened may be large $p$ $>>$ $n$, and growing at an exponential rate, provided $\ln (p)$ $=$ $o(n^{a})$ for some $a$ $>$ $0$ that depends on memory decay and the growth of higher moments. We study the problem of correlation screening in a high dimensional marginal regression setting, assuming so-called \textit{physical dependence} in a time series setting. We entirely sidestep covariance matrix estimation and adaptive re-sampling by working with a max-statistic over the many computed parameters. Thus we do not need endogenous selection of the most relevant predictor index yielding non-uniform asymptotics, nor do we need a post-estimation Bonferroni correction. The non-standard limit distribution arising from the maximum of an increasing number of estimators is easily approximated by a multiplier (wild) block bootstrap. The max-test controls for size well, performs well against various deviations from the null, including very slight deviations with a weak or sparse signal. A numerical experiment is performed and an empirical example with the VIX volatility index is provided.

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arXiv2026

Max-laws of large numbers for weakly dependent high dimensional arrays with applications

We derive so-called weak and strong \textit{max-laws of large numbers} for $% \max_{1\leq i\leq k_{n}}|1/n\sum_{t=1}^{n}x_{i,n,t}|$ for zero mean stochastic triangular arrays $\{x_{i,n,t}$ $:$ $1$ $\leq $ $t$ $\leq n\}_{n\geq 1}$, with dimension counter $i$ $=$ $1,...,k_{n}$ and dimension $% k_{n}$ $\rightarrow $ $\infty $. Rates of convergence are also analyzed based on feasible sequences $\{k_{n}\}$. We work in three dependence settings: independence, Dedecker and Prieur's (2004) $τ$-mixing and Wu's (2005) physical dependence. We initially ignore cross-coordinate $i$ dependence as a benchmark. We then work with martingale, nearly martingale, and mixing coordinates to deliver improved bounds on $k_{n}$. Finally, we use the results in three applications, each representing a key novelty: we ($i$) bound $k_{n}$\ for a max-correlation statistic for regression residuals under $α$-mixing or physical dependence; ($ii$) extend correlation screening, or marginal regressions, to physical dependent data with diverging dimension $k_{n}$ $\rightarrow $ $\infty $; and ($iii$) test a high dimensional parameter after partialling out a fixed dimensional nuisance parameter in a linear time series regression model under $τ$% -mixing.

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arXiv2025

Symmetrization for high dimensional dependent random variables

We establish a generic symmetrization property for dependent random variables $\{x_{t}\}_{t=1}^{n}$ on $\mathbb{R}^{p}$, where $p$ $>>$ $n$ is allowed. We link $\mathbb{E}ψ(\max_{1\leq i\leq p}|1/n\sum_{t=1}^{n}(x_{i,t}$ $-$ $\mathbb{E}x_{i,t})|)$ to $\mathbb{E}ψ(\max_{1\leq i\leq p}|1/n$ $\sum_{t=1}^{n}η_{t}(x_{i,t}$ $-$ $\mathbb{E}% x_{i,t})|)$ for non-decreasing convex $ψ$ $:$ $[0,\infty )$ $\rightarrow $ $\mathbb{R}$, where $\{η_{t}\}_{t=1}^{n}$ are block-wise independent random variables, with a remainder term based on high dimensional Gaussian approximations that need not hold at a high level. Conventional usage of $% η_{t}(x_{i,t}$ $-$ $\tilde{x}_{i,t})$ with $\{\tilde{x}% _{i,t}\}_{t=1}^{n} $ an independent copy of $\{x_{i,t}\}_{t=1}^{n}$, and Rademacher $η_{t}$, is not required in a generic environment, although we may trivially replace $\mathbb{E}x_{i,t}$ with $\tilde{x}_{i,t}$. In the latter case with Rademacher $η_{t}$ our result reduces to classic symmetrization under independence. We bound and therefore verify the Gaussian approximations in mixing and physical dependence settings, thus bounding $\mathbb{E}ψ(\max_{1\leq i\leq p}|1/n\sum_{t=1}^{n}(x_{i,t}$ $-$ $\mathbb{E}x_{i,t})|)$; and apply the main result to a generic % Nemirovski (2000)-like $\mathcal{L}_{q}$-maximal moment bound for $\mathbb{E}\max_{1\leq i\leq p}|1/n\sum_{t=1}^{n}(x_{i,t}$ $-$ $\mathbb{E}x_{i,t})|^{q}$, $q$ $\geq $ $1$.

math.PR↗

arXiv2025

$\mathcal{L}_{q}$-maximal inequality for high dimensional means under dependence

We derive an $\mathcal{L}_{q}$-maximal inequality for zero mean dependent random variables $\{x_{t}\}_{t=1}^{n}$ on $\mathbb{R}^{p}$, where $p$ $>>$ $% n $ is allowed. The upper bound is a familiar multiple of $\ln (p)$ and an $% l_{\infty }$ moment, as well as Kolmogorov distances based on Gaussian approximations $(ρ_{n},\tildeρ_{n})$, derived with and without negligible truncation and sub-sample blocking. The latter arise due to a departure from independence and therefore a departure from standard symmetrization arguments. Examples are provided demonstrating $(ρ_{n},% \tildeρ_{n})$ $\rightarrow $ $0$ under heterogeneous mixing and physical dependence conditions, where $(ρ_{n},\tildeρ_{n})$ are multiples of $\ln (p)/n^{b}$ for some $b$ $>$ $0$ that depends on memory, tail decay, the truncation level and block size.

math.PR↗

arXiv2024

Heavy Tail Robust Estimation and Inference for Average Treatment Effects

We study the probability tail properties of Inverse Probability Weighting (IPW) estimators of the Average Treatment Effect (ATE) when there is limited overlap between the covariate distributions of the treatment and control groups. Under unconfoundedness of treatment assignment conditional on covariates, such limited overlap is manifested in the propensity score for certain units being very close (but not equal) to 0 or 1. This renders IPW estimators possibly heavy tailed, and with a slower than sqrt(n) rate of convergence. Trimming or truncation is ultimately based on the covariates, ignoring important information about the inverse probability weighted random variable Z that identifies ATE by E[Z]= ATE. We propose a tail-trimmed IPW estimator whose performance is robust to limited overlap. In terms of the propensity score, which is generally unknown, we plug-in its parametric estimator in the infeasible Z, and then negligibly trim the resulting feasible Z adaptively by its large values. Trimming leads to bias if Z has an asymmetric distribution and an infinite variance, hence we estimate and remove the bias using important improvements on existing theory and methods. Our estimator sidesteps dimensionality, bias and poor correspondence properties associated with trimming by the covariates or propensity score. Monte Carlo experiments demonstrate that trimming by the covariates or the propensity score requires the removal of a substantial portion of the sample to render a low bias and close to normal estimator, while our estimator has low bias and mean-squared error, and is close to normal, based on the removal of very few sample extremes.

stat.ME↗

arXiv2024

A Global Wavelet Based Bootstrapped Test of Covariance Stationarity

We propose a covariance stationarity test for an otherwise dependent and possibly globally non-stationary time series. We work in a generalized version of the new setting in Jin, Wang and Wang (2015), who exploit Walsh (1923) functions in order to compare sub-sample covariances with the full sample counterpart. They impose strict stationarity under the null, only consider linear processes under either hypothesis in order to achieve a parametric estimator for an inverted high dimensional asymptotic covariance matrix, and do not consider any other orthonormal basis. Conversely, we work with a general orthonormal basis under mild conditions that include Haar wavelet and Walsh functions; and we allow for linear or nonlinear processes with possibly non-iid innovations. This is important in macroeconomics and finance where nonlinear feedback and random volatility occur in many settings. We completely sidestep asymptotic covariance matrix estimation and inversion by bootstrapping a max-correlation difference statistic, where the maximum is taken over the correlation lag $h$ and basis generated sub-sample counter $k$ (the number of systematic samples). We achieve a higher feasible rate of increase for the maximum lag and counter $\mathcal{H}_{T}$ and $\mathcal{K}_{T}$. Of particular note, our test is capable of detecting breaks in variance, and distant, or very mild, deviations from stationarity.

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arXiv2023

Testing Many Zero Restrictions in a High Dimensional Linear Regression Setting

We propose a test of many zero parameter restrictions in a high dimensional linear iid regression model with $k$ $>>$ $n$ regressors. The test statistic is formed by estimating key parameters one at a time based on many low dimension regression models with nuisance terms. The parsimoniously parametrized models identify whether the original parameter of interest is or is not zero. Estimating fixed low dimension sub-parameters ensures greater estimator accuracy, it does not require a sparsity assumption nor therefore a regularized estimator, it is computationally fast compared to, e.g., de-biased Lasso, and using only the largest in a sequence of weighted estimators reduces test statistic complexity and therefore estimation error. We provide a parametric wild bootstrap for p-value computation, and prove the test is consistent and has non-trivial $\sqrt{n/\{\ln (n)\mathcal{M}% _{n}\}}$-local-to-null power where $\mathcal{M}_{n}$ is the $l_{\infty }$ covariate fourth moment.

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arXiv2022

A Smoothed P-Value Test When There is a Nuisance Parameter under the Alternative

We present a new test when there is a nuisance parameter under the alternative hypothesis. The test exploits the p-value occupation time [PVOT], the measure of the nuisance parameter subset on which a p-value test based on a a test statistic rejects the null hypothesis. Key contributions are: (i) An asymptotic critical value upper bound for our test is the significance level α, making inference easy. (ii) We only require the test statistic to have a known or bootstrappable limit distribution, hence we do not require root(n)-Gaussian asymptotics, allowing for weak or non-identification, boundary values, heavy tails, infill asymptotics, and so on. (iii) A test based on the sup-p-value may be conservative and in some cases have trivial power, while the PVOT naturally controls for this by smoothing over the nuisance parameter space. Finally, (iv) the PVOT uniquely allows for bootstrap inference in the presence of nuisance parameters when some estimated parameters may not be identified.

stat.ME↗

arXiv2022

Testing (Infinitely) Many Zero Restrictions

This paper proposes a max-test for testing (possibly infinitely) many zero parameter restrictions in an extremum estimation framework. The test statistic is formed by estimating key parameters one at a time based on many empirical loss functions that map from a low dimension parameter space, and choosing the largest in absolute value from these individually estimated parameters. The parsimoniously parametrized loss identify whether the original parameter of interest is or is not zero. Estimating fixed low dimension sub-parameters ensures greater estimator accuracy, does not require a sparsity assumption, and using only the largest in a sequence of weighted estimators reduces test statistic complexity and therefore estimation error, ensuring sharper size and greater power in practice. Weights allow for standardization in order to control for estimator dispersion. In a nonlinear parametric regression framework we provide a parametric wild bootstrap for p-value computation without directly requiring the max-statistic's limit distribution. A simulation experiment shows the max-test dominates a conventional bootstrapped test.

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arXiv2020

Weak-Identification Robust Wild Bootstrap applied to a Consistent Model Specification Test

We present a new robust bootstrap method for a test when there is a nuisance parameter under the alternative, and some parameters are possibly weakly or non-identified. We focus on a Bierens (1990)-type conditional moment test of omitted nonlinearity for convenience, and because of difficulties that have been ignored to date. Existing methods include the supremum p-value which promotes a conservative test that is generally not consistent, and test statistic transforms like the supremum and average for which bootstrap methods are not valid under weak identification. We propose a new wild bootstrap method for p-value computation by targeting specific identification cases. We then combine bootstrapped p-values across polar identification cases to form an asymptotically valid p-value approximation that is robust to any identification case. The wild bootstrap does not require knowledge of the covariance structure of the bootstrapped processes, whereas Andrews and Cheng's (2012, 2013, 2014) simulation approach generally does. Our method allows for robust bootstrap critical value computation as well. Our bootstrap method (like conventional ones) does not lead to a consistent p-value approximation for test statistic functions like the supremum and average. We therefore smooth over the robust bootstrapped p-value as the basis for several tests which achieve the correct asymptotic level, and are consistent, for any degree of identification. They also achieve uniform size control. A simulation study reveals possibly large empirical size distortions in non-robust tests when weak or non-identification arises. One of our smoothed p-value tests, however, dominates all other tests by delivering accurate empirical size and comparatively high power.

math.ST↗

arXiv2019

A Max-Correlation White Noise Test for Weakly Dependent Time Series

This paper presents a bootstrapped p-value white noise test based on the maximum correlation, for a time series that may be weakly dependent under the null hypothesis. The time series may be prefiltered residuals. The test statistic is a normalized weighted maximum sample correlation, where the maximum lag increases at a rate slower than the sample size. We only require uncorrelatedness under the null hypothesis, along with a moment contraction dependence property that includes mixing and non-mixing sequences. We show Shao's (2011) dependent wild bootstrap is valid for a much larger class of processes than originally considered. It is also valid for residuals from a general class of parametric models as long as the bootstrap is applied to a first order expansion of the sample correlation. We prove the bootstrap validity without exploiting extreme value theory (standard in the literature) or recent Gaussian approximation theory. Finally, we extend Escanciano and Lobato's (2009) automatic maximum lag selection to our setting with an unbounded choice set, and find it works strikingly well in controlled experiments. Our proposed test achieves accurate size under various white noise null hypotheses and high power under various alternative hypotheses including distant serial dependence.

stat.ME↗

arXiv2018

Asymptotic Theory for the Maximum of an Increasing Sequence of Parametric Functions

\cite{HillMotegi2017} present a new general asymptotic theory for the maximum of a random array $\{\mathcal{X}_{n}(i)$ $:$ $1$ $\leq $ $i$ $\leq $ $\mathcal{L}\}_{n\geq 1}$, where each $\mathcal{X}_{n}(i)$ is assumed to converge in probability as $n$ $\rightarrow $ $\infty $. The array dimension $\mathcal{L}$ is allowed to increase with the sample size $n$. Existing extreme value theory arguments focus on observed data $\mathcal{X}_{n}(i)$, and require a well defined limit law for $\max_{1\leq i\leq \mathcal{L}}|\mathcal{X}_{n}(i)|$ by restricting dependence across $i$. The high dimensional central limit theory literature presumes approximability by a Gaussian law, and also restricts attention to observed data. \cite{HillMotegi2017} do not require $\max_{1\leq i\leq \mathcal{L}_{n}}|\mathcal{X}_{n}(i)|$ to have a well defined limit nor be approximable by a Gaussian random variable, and we do not make any assumptions about dependence across $i$. We apply the theory to filtered data when the variable of interest $\mathcal{X}_{n}(i,θ_{0})$ is not observed, but its sample counterpart $\mathcal{X}_{n}(i,\hatθ_{n})$ is observed where $\hatθ_{n}$ estimates $θ_{0}$. The main results are illustrated by looking at unit root tests for a high dimensional random variable, and a residuals white noise test.

math.ST↗

arXiv2015

Robust estimation and inference for heavy tailed GARCH

We develop two new estimators for a general class of stationary GARCH models with possibly heavy tailed asymmetrically distributed errors, covering processes with symmetric and asymmetric feedback like GARCH, Asymmetric GARCH, VGARCH and Quadratic GARCH. The first estimator arises from negligibly trimming QML criterion equations according to error extremes. The second imbeds negligibly transformed errors into QML score equations for a Method of Moments estimator. In this case, we exploit a sub-class of redescending transforms that includes tail-trimming and functions popular in the robust estimation literature, and we re-center the transformed errors to minimize small sample bias. The negligible transforms allow both identification of the true parameter and asymptotic normality. We present a consistent estimator of the covariance matrix that permits classic inference without knowledge of the rate of convergence. A simulation study shows both of our estimators trump existing ones for sharpness and approximate normality including QML, Log-LAD, and two types of non-Gaussian QML (Laplace and Power-Law). Finally, we apply the tail-trimmed QML estimator to financial data.

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