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

This paper presents a generalized pre-averaging approach for estimating the integrated volatility. This approach also provides consistent estimators of other powers of volatility in particular, it gives feasible ways to consistently estimate the asymptotic variance of the estimator of the integrated volatility. We show that our approach, which possess an intuitive transparency, can generate rate optimal estimators (with convergence rate n-1/4).

Microstructure noise in the continuous case: the pre-averaging approach

: The signed square root statistic R typically has cumulants on the form cum(p)(R) = theta(2,p) + n(-p/2)Kp + O(n(-(p+2)/2)). This paper shows how to compute Kp without invoking the Bartlett identities. As an application, we show how the family of alternatives influences the coverage accuracy of R, and in particular that a bad choice of family can lead to arbitrary undercoverage for confidence intervals based on R.

/pdf/likelihood-computations-without-bartlett-identities-1l5c22srht.pdf

Likelihood Computations without Bartlett Identities.

We would like to congratulate Jianqing Fan with an excellent and well written survey of some of the literature in this area. We will here focus on some of the issues which are at the reserach frontiers in financial econometrics but are not covered in the survey. Most importantly, we consider the estimation of actual volatility. Related to this is the realization that financial data is actually observed with error (typically called market microstructure), and that one needs to consider a hidden semimartingale model. This has implications for the Markov models discussed above.

Discussion of paper "A selective overview of nonparametric methods in financial econometrics" by Jianqing Fan

In this paper, we investigate the implication of non-stationary market microstructure noise to integrated volatility estimation, provide statistical tools to test stationarity and non-stationarity in market microstructure noise, and discuss how to measure liquidity risk using high frequency financial data. In particular, we discuss the impact of non-stationary microstructure noise on TSRV (Two-Scale Realized Variance) estimator, and design three test statistics by exploiting the edge effects and asymptotic approximation. The asymptotic distributions of these test statistics are provided under both stationary and non-stationary noise assumptions respectively, and we empirically measure aggregate liquidity risks by these test statistics from 2006 to 2013. As byproducts, functional dependence and endogenous market microstructure noise are briefly discussed. Our empirical study indicates the prevalence of non-stationary market microstructure noise in the New York Stock Exchange.

Discerning Non-Stationary Market Microstructure Noise and Time-Varying Liquidity in High Frequency Data

In this paper we introduce a general method for estimating the quadratic covariation of one or more spot parameters processes associated with continuous time semimartingales. This estimator is applicable to a wide range of spot parameter processes, and may also be used to estimate the leverage effect of stochastic volatility models. The estimator we introduce is based on sums of squared increments of second differences of the observed process, and the intervals over which the differences are computed are rolling and overlapping. This latter feature lets us take full advantage of the data, and, by sufficiency considerations, ought to outperform estimators that are only based on one partition of the observational window. The main result of the paper is a central limit theorem for such triangular array rolling quadratic variations. We highlight the wide applicability of this theorem by showcasing how it might be applied to a novel leverage effect estimator. The principal motivation for the present study, however, is that the discrete times at which a continuous time semimartingale is observed might depend on features of the observable process other than its level, such as its (non-observable) spot-volatility process. As the main application of our estimator, we therefore show how it may be used to estimate the quadratic covariation between the spot-volatility process and the intensity process of the observation times, when both of these are taken to be semimartingales. The finite sample properties of this estimator are studied by way of a simulation experiment, and we also apply this estimator in an empirical analysis of the Apple stock. Our analysis of the Apple stock indicates a rather strong correlation between the spot volatility process of the log-prices process and the times at which this stock is traded (hence observed).

Per A. Mykland

Papers

Microstructure noise in the continuous case: the pre-averaging approach

Likelihood Computations without Bartlett Identities.

Discussion of paper "A selective overview of nonparametric methods in financial econometrics" by Jianqing Fan

Discerning Non-Stationary Market Microstructure Noise and Time-Varying Liquidity in High Frequency Data

A CLT for second difference estimators with an application to volatility and intensity