Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This paper is concerned with the estimation of covariance matrices in the presence of heteroskedasticity and autocorrelation of unknown forms. Currently available estimators that are designed for this context depend upon the choice of a lag truncation parameter and a weighting scheme. Results in the literature provide a condition on the growth rate of the lag truncation parameter as T \rightarrow \infty that is sufficient for consistency. No results are available, however, regarding the choice of lag truncation parameter for a fixed sample size, regarding data-dependent automatic lag truncation parameters, or regarding the choice of weighting scheme. In consequence, available estimators are not entirely operational and the relative merits of the estimators are unknown. This paper addresses these problems. The asymptotic truncated mean squared errors of estimators in a given class are determined and compared. Asymptotically optimal kernel/weighting scheme and bandwidth/lag truncation parameters are obtained using an asymptotic truncated mean squared error criterion. Using these results, data-dependent automatic bandwidth/lag truncation parameters are introduced. The finite sample properties of the estimators are analyzed via Monte Carlo simulation.

/pdf/heteroskedasticity-and-autocorrelation-consistent-covariance-3tz1r06ty1.pdf

Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation

We propose a nonparametric method for automatically selecting the number of autocovariances to use in computing a heteroskedasticity and autocorrelation consistent covariance matrix. For a given kernel for weighting the autocovariances, we prove that our procedure is asymptotically equivalent to one that is optimal under a mean squared error loss function. Monte Carlo simulations suggest that our procedure performs tolerably well, although it does result in size distortions.

https://papers.ssrn.com/sol3/Delivery.cfm/nber_t0144.pdf?abstractid=225129&mirid=1

Automatic Lag Selection in Covariance Matrix Estimation

Authors proposes a nonparametric method for automatically selecting the number of autocovariances to use in computing a heteroskedasticity and autocorrelation consistent covariance matrix. For a given kernel for weighting the autocovariances, we prove that our procedure is asymptotically equivalent to one that is optimal under a mean-squared error loss function. Monte Carlo simulations suggest that our procedure performs tolerably well, although it does result in size distortions.

/pdf/automatic-lag-selection-in-covariance-matrix-estimation-27kis435ax.pdf

This paper considers a new class of heteroskedasticity and autocorrelation consistent (HAC) covariance matrix estimators. The estimators considered are prewhitened kernel estimators with vector autoregressions employed in the prewhitening stage. The paper establishes consistency, rate of convergence, and asymptotic truncated mean squared error (MSE) results for the estimators when a fixed or automatic bandwidth procedure is employed. Conditions are obtained under which prewhitening improves asymptotic truncated MSE. Monte Carlo results show that prewhitening is very effective in reducing bias, improving confidence interval coverage probabilities, and rescuing over-rejection of t-statistics constructed using kernel-HAC estimators. On the other hand, prewhitening is found to inflate variance and MSE of the kernel estimators. Since confidence interval coverage probabilities and over-rejection of t-statistics are usually of primary concern, prewhitened kernel estimators provide a significant improvement over the standard non-prewhitened kernel estimators.

/pdf/an-improved-heteroskedasticity-and-autocorrelation-4vqnzatu8s.pdf

An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator

A two parameter family of distributions on partitions is obtained by mixing a multinomial with a symmetric Dirichlet distribution. Estimates for the parameters are proposed and studied in various asymptotic limits.

/pdf/distributions-on-partitions-48x16r1ig1.pdf

Distributions on Partitions

Soit S n , n≥1, une marche aleatoire et t=t x =inf {n≥1:nf(f n /n)>α. On obtient des developpements asymptotiques, a deux termes lorsque α→α pour les distributions marginales de tα et une somme partielle normalisee

/pdf/asymptotic-expansions-in-boundary-crossing-problems-5889qiq7bx.pdf

Asymptotic expansions in boundary crossing problems

This paper deals with the problem of estimating the covariance matrix of the least-squares regression coefficients under heteroskedasticity and/or autocorrelation of unknown form. We consider an estimator proposed by White [17] and give a relatively simple proof of its consistency. Our proof is based on more easily verifiable conditions than those of White. An alternative estimator with improved small sample properties is also presented.

Estimation of the Covariance Matrix of the Least-Squares Regression Coefficients When the Disturbance Covariance Matrix Is of Unknown Form

In the sequential design of experiments an experimenter performs experiments sequentially to make an eventual inference about the true state of nature. A Bayesian formulation of this problem is considered. The parameter space is assumed finite and there are a finite number of repeatable experiments. Sufficient conditions are given for a procedure to be second order efficient as the sampling costs approach zero. The asymptotic analysis of a related Markov control problem is also presented.

/pdf/second-order-efficiency-in-the-sequential-design-of-36b4pkh9nc.pdf

Second Order Efficiency in the Sequential Design of Experiments

Let $\{S_n\}$ be a random walk on the integers with negative drift, and let $A_n = \{S_k \geq 0, 1 \leq k \leq n\}$ and $A = A_\infty$. Conditioning on $A$ is troublesome because $P(A) = 0$ and there is no natural sigma-field of events "like" $A. A$ natural definition of $P(B\mid A)$ is $\lim_{n\rightarrow\infty}P(B\mid A_n)$. The main result here shows that this definition makes sense, at least for a large class of events $B$: The finite-dimensional conditional distributions for the process $\{S_k\}_{k\geq 0}$ given $A_n$ converge strongly to the finite-dimensional distributions for a measure $\mathbf{Q}$. This distribution $\mathbf{Q}$ is identified as the distribution for a stationary Markov chain on $\{0,1,\ldots\}$.

/pdf/limit-theorems-for-random-walks-conditioned-to-stay-positive-csme7f1x9n.pdf

Robert W. Keener

Papers

Distributions on Partitions

Asymptotic expansions in boundary crossing problems

Estimation of the Covariance Matrix of the Least-Squares Regression Coefficients When the Disturbance Covariance Matrix Is of Unknown Form

Second Order Efficiency in the Sequential Design of Experiments

Limit Theorems for Random Walks Conditioned to Stay Positive