Narrow Results

In this chapter, we will show that the asymptotic theory for linear regression models with IID observations carries over to ergodic stationary linear time series regression models with Martingale Difference Sequence (MDS) disturbances. Some basic concepts in time series analysis are introduced, and some tests for serial correlation are described.

This chapter aims to summarize the theories, models, methods and tools of modern econometrics which we have covered in the previous chapters. We first review the classical assumptions of the linear regression model and discuss the historical development of modern econometrics by various relaxations of the classical assumptions. We also discuss the challenges and opportunities for econometrics in the Big data era and point out some important directions for the future development of econometrics.

Electrochemical corrosion processes can be investigated by observation of charge flows between the electrolyte and the corroding metal. Usually, the charge flows are observed as spontaneous current and voltage fluctuations (electrochemical noise) in a three electrode setup. Different types of corrosion processes can be recognized by electrochemical noise analysis. Uniform corrosion rate can be evaluated by estimation of polarization resistance between the metal and electrolyte. Local corrosion events (breakdowns of the passive layer) that produce characteristic transients observed in noise can be detected as well. Different methods of electrochemical noise analysis are presented in a brief review. The limitations and advantages of the proposed methods for corrosion monitoring and research are underlined. The experimental results are also discussed.

We have shown that the least squares estimator for a non-ergodic, first order, self-exciting, threshold autoregressive model is strongly consistent under quite general conditions.

In this chapter, we shall build on the fundamental notions of probability distribution and statistics in the last chapter, and extend consideration to a sequence of random variables. In financial application, it is mostly the case that the sequence is indexed by time, hence a stochastic process. Interesting statistical laws or mathematical theories result when we look at the relationships within a stochastic process. We introduce an application of the Central Limit Theorem to the study of stock return distributions.

Our goal in this paper is to look at stochastic financial models and especially on those properties of the involved distributions which are expressed in terms of the moments. The questions discussed and the results presented reveal the role which the moments play in the analysis of distributions. Interesting conclusions can be derived in both cases when available are finitely many moments and when we know all moments.

Among the results included in the paper are sharp lower and/or upper bounds for option prices in terms of a finite number of moments. However the main attention is paid to the determinacy of distributions by their moments. While any light tailed distribution is uniquely determined by its moments, the uniqueness may fail for heavy tailed distributions. And, here is the point. Heavy tailed distributions are essentially involved in stock market modelling, and most of them are non-unique in terms of the moments. This is why the phenomenon non-uniqueness of distributions deserves a special attention.

We treat distributions on the positive half-line used to describe, e.g., stock prices and option prices, and distributions on the whole real line describing log-returns. For reader's convenience we give a brief and unified general picture of existing results about uniqueness and non-uniqueness of distribution in terms of their moments. Some of the results are classical and well-known. In several cases we provide here new arguments along with presenting new recent results on the moment determinacy of random variables and their non-linear transformations and also of stochastic processes which are solutions to SDEs.