计量经济分析 = Econometric analysis

Meyn & Tweedie is back! The bible on Markov chains in general state spaces has been brought up to date to reflect developments in the field since 1996 - many of them sparked by publication of the first edition. The pursuit of more efficient simulation algorithms for complex Markovian models, or algorithms for computation of optimal policies for controlled Markov models, has opened new directions for research on Markov chains. As a result, new applications have emerged across a wide range of topics including optimisation, statistics, and economics. New commentary and an epilogue by Sean Meyn summarise recent developments and references have been fully updated. This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.

Markov Chains and Stochastic Stability

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

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

This purpose of this introductory paper is threefold. First, it introduces the Monte Carlo method with emphasis on probabilistic machine learning. Second, it reviews the main building blocks of modern Markov chain Monte Carlo simulation, thereby providing and introduction to the remaining papers of this special issue. Lastly, it discusses new interesting research horizons.

/pdf/an-introduction-to-mcmc-for-machine-learning-22slt5b6x5.pdf

An introduction to MCMC for machine learning

Empirical and dual likelihood are two of a growing array of artificial or approximate likelihoods currently in use in statistics. A question of major interest focuses on the performance of these new constructs relative to ordinary parametric likelihoods. We consider here two criteria, local power and conditional properties. Looking at tests of a scalar parameter, we show that there is no loss of efficiency in using a dual or empirical likelihood model, to second order. To third order, either the artificial likelihood or the true likelihood test could be more efficient; this is determined by the Fisher information, the distance between the null and the alternative hypotheses, and the statistical curvature of the models. Conditionality properties are assessed by comparing empirical likelihood with quasilikelihood. If there is overdispersion present, or more generally an unknown amount of dispersion, then there is little difference in the ancillaries for conditional inference based on empirical likelihood or on other possible likelihoods, such as a quasilikelihood. If the amount of dispersion is known, however, it is correct to base inference on the quasilikelihood.

https://www.jstor.org/stable/pdfplus/2337383.pdf

An evaluation of the power and conditionality properties of empirical likelihood

SUMMARY Empirical likelihood was introduced as a nonparametric analogue of ordinary parametric likelihood. It is well known that the empirical likelihood ratio statistic inherits a number of properties of the parametric likelihood ratio statistic, such as the asymptotic chi-squared distribution and Bartlett correctability. This raises the question of whether or not the same is true in the presence of nuisance parameters. Recent work by Qin & Lawless (1994) indicates that the chi-squared distribution is still valid to first order. We show that, when nuisance parameters are present, as introduced via a system of estimating equations, the asymptotic expansion for the signed square root of the empirical likelihood ratio statistic has a nonstandard form. This implies that the empirical likelihood ratio statistic itself does not permit a Bartlett correction.

Empirical likelihood in the presence of nuisance parameters

Financial prices are often discretized—with smallest tick size of one cent, for example. Thus prices involve rounding errors. Rounding errors affect the estimation of volatility, and understanding them is critical, particularly when using high frequency data. We study the asymptotic behavior of realized volatility (RV), which is commonly used as an estimator of integrated volatility. We prove the convergence of the RV and scaled RV under varous conditions on the rounding level and the number of observations. A bias-corrected volatility estimator is proposed and an associated central limit theorem is shown. The simulation and empirical results demonstrate that the proposed method can yield substantial statistical improvement. ( JEL: C02, C13,C14)

/pdf/rounding-errors-and-volatility-estimation-5fv4fwtag3.pdf

Rounding Errors and Volatility Estimation

This chapter reviews our recent work on disentangling high frequency volatility estimators from market microstructure noise, based on maximum-likelihood in the parametric case and two (or more) scales realized volatility (TSRV) in the nonparametric case. We discuss the basic theory, its extensions and the practical implementation of the estimators.

Estimating Volatility in the Presence of Market Microstructure Noise: A Review of the Theory and Practical Considerations

http://galton.uchicago.edu/~mykland/paperlinks/Jacod-Mykland2015-1-s2.0-S0304414915000411-main.pdf

Per A. Mykland

Papers

An evaluation of the power and conditionality properties of empirical likelihood

Empirical likelihood in the presence of nuisance parameters

Rounding Errors and Volatility Estimation

Estimating Volatility in the Presence of Market Microstructure Noise: A Review of the Theory and Practical Considerations

Microstructure noise in the continuous case: Approximate efficiency of the adaptive pre-averaging method