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Variance Estimation with Dependence and Heterogeneous Means

Luther Yap

Abstract

This paper considers the problem of estimating the variance of a sum of a triangular array of random vectors with heterogeneous means. When random vectors exhibit two-way cluster dependence or weak dependence, standard variance estimators designed under homogeneous means can underestimate the true variance, which results in subsequent tests being oversized. To restore validity, this paper proposes a simple conservative variance estimator robust to heterogeneous means and shows its asymptotic validity.

Variance Estimation with Dependence and Heterogeneous Means

Abstract

This paper considers the problem of estimating the variance of a sum of a triangular array of random vectors with heterogeneous means. When random vectors exhibit two-way cluster dependence or weak dependence, standard variance estimators designed under homogeneous means can underestimate the true variance, which results in subsequent tests being oversized. To restore validity, this paper proposes a simple conservative variance estimator robust to heterogeneous means and shows its asymptotic validity.
Paper Structure (9 sections, 6 theorems, 46 equations, 3 tables)

This paper contains 9 sections, 6 theorems, 46 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Setting
  3. Main Theoretical Results
  4. Numerical Illustrations
  5. Simulation
  6. Empirical Application
  7. Proofs
  8. Proofs of Theorems and Propositions
  9. Auxiliary Lemmas and Proofs

Key Result

Theorem 1

Suppose Assumptions asmp:psi_dependence and asmp:clt hold a.s. Let $\nu$ be a nonstochastic vector with $\dim\left(\nu\right)=\dim\left(Y_{n,i}\right)$ and $\left\Vert \nu\right\Vert =1$. Then, for $S_{n}:=\sum_{i\in N_{n}}\nu^{\prime}\left(Y_{n,i}-E\left[Y_{n,i}\right]\right)$ and $\sigma_{n}^{2}:= where $\Phi$ denotes the distribution function of $N(0,1)$.

Theorems & Definitions (17)

  • Example 1
  • Definition 1
  • Example 2
  • Theorem 1
  • Example 3
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Example 4
  • proof : Proof of \ref{['thm:clt']}
  • ...and 7 more