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

The paper considers Edgeworth expansions for estimators of volatility. Unlike the usual exapsions, we have found that in order to obtain meaningful terms, one needs to let the size of the noise to go zero asymptotically. This is reflected in our expansions. The results have application to Cornish-Fisher inversion and bootstrapping.

http://www.stat.cmu.edu/tr/tr821/tr821.pdf

Edgeworth expansions in small noise asymptotics

In the setting of high-frequency data, where a continuous time process is only observed at discrete times, simplifying assumptions are often imposed on the relation between the observation times and the process being monitored. As is well known, these assumptions often do not hold in practice, and many authors have studied models for, and the consequences of, the observation times being endogenous in the sense that they depend on the observable process itself. The impetus for the current paper is that the observation times might depend on other features of the observable process than its level, such as its (non-observable) spot-volatility process. In this paper we introduce an estimator of the quadratic covariation between the spot-volatility process and the intensity process of the observation times, both taken to be semimartingales. The estimator we introduce is not limited to estimating the covariation of the spot-volatility and observation time intensity, but applies under mild regularity conditions to the quadratic covariation between any two spot-process semimartingales. We derive a convergence rate for our estimator, and prove a central limit theorem. Both these results apply quite generally to estimators of quadratic (co)variations that are based on rolling and overlapping sums of squared semimartingale increments. We illustrate the use of the estimator by a simulation study, and conduct an empirical analysis of the Apple stock over $21$ trading days. This analysis indicates a rather strong correlation between the spot volatility and the observation times.

Volatility and intensity

: It is explored how incomplete markets can be studied with the help of asymptotics. A compound Poisson model for the stock price is assumed and an expansion for the price of a European option is obtained as the stock price process converges to a geometric Brownian motion. This formulation also permits one to confront statistical uncertainty in the volatility of the stock price, and we show how this uncertainty impacts on the value of the option.

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA316737&Location=U2&doc=GetTRDoc.pdf

Options Pricing in Incomplete Markets: An Asymptotic Approach.

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).

/pdf/microstructure-noise-in-the-continuous-case-the-pre-2pbv5yk65a.pdf

Microstructure Noise in the Continuous Case: The Pre-Averaging Approach - JLMPV-9

We develop a principal component analysis (PCA) for high frequency data. As in Northern fairly tales, there are trolls waiting for the explorer. The first three trolls are market microstructure noise, asynchronous sampling times, and edge effects in estimators. To get around these, a robust estimator of spot covariance matrix is developed based on the Smoothed TSRV (Mykland et al. (2017)). The fourth troll is how to pass from estimated time-varying covariance matrix to PCA. Under finite dimensionality, we develop this methodology through the estimation of realized spectral functions. Rates of convergence and central limit theory, as well as an estimator of standard error, are established. The fifth troll is high dimension on top of high frequency, where we also develop PCA. With the help of a new identity concerning the spot principal orthogonal complement, the high-dimensional rates of convergence have been studied by freeing several strong assumptions in classical PCA. As an application, we show that our first principal component (PC) potentially outperforms the S&P 100 market index, while three of the next four PCs are cointegrated with two of the Fama-French non-market factors.

Per A. Mykland

Papers

Edgeworth expansions in small noise asymptotics

Volatility and intensity

Options Pricing in Incomplete Markets: An Asymptotic Approach.

Microstructure Noise in the Continuous Case: The Pre-Averaging Approach - JLMPV-9

The Five Trolls Under the Bridge: Principal Component Analysis with Asynchronous and Noisy High Frequency Data