Mathematics for Econometrics

Frontmatter

Chapter 1. Vectors and Vector Spaces

Abstract

In nearly all of the discussion in this volume, we deal with the set of real numbers. Occasionally, however, we deal with complex numbers as well.

Phoebus J. Dhrymes

Chapter 2. Matrix Algebra

Abstract

Definition 2.1. Let \(a_{ij} \in \mathcal{F},\) i = 1,2,…,m, j = 1,2,…,n, where \(\mathcal{F}\) is a suitable space, such as the one-dimensional Euclidean or complex space.

Phoebus J. Dhrymes

Chapter 3. Systems of Linear Equations

Abstract

Consider the system of linear equations

$$\displaystyle{ Ax = b, }$$

(3.1)

where A is m × n and b is an m-element vector.

Phoebus J. Dhrymes

Chapter 4. Matrix Vectorization

Abstract

It is frequently more convenient to write a matrix in vector form. For lack of a suitable term, we have coined for this operation the phrase “vectorization of a matrix”.

Phoebus J. Dhrymes

Chapter 5. Vector and Matrix Differentiation

Abstract

Frequently, we need to differentiate quantities like tr(AX) with respect to the elements of X, or quantities like Ax, z′Ax with respect to the elements of (the vectors) x and/or z.

Phoebus J. Dhrymes

Chapter 6. DE Lag Operators GLSEM and Time Series

Abstract

In this chapter we deal with econometric applications of (vector) difference equations with constant coefficients, as well as with aspects of the statistical theory of time series and their application in econometrics.

Phoebus J. Dhrymes

Chapter 7. Mathematical Underpinnings of Probability Theory

Abstract

The purpose of this chapter is to provide a background on the results from probability and inference theory required for the study of several of the topics of contemporary econometrics.

An attempt will be made to give proofs for as many propositions as is consistent with the objectives of this chapter which are to provide the tools deemed necessary for the exposition of several topics in econometric theory; it is clearly not our objective to provide a substitute to a mathematical textbook of modern probability theory.

Phoebus J. Dhrymes

Chapter 8. Foundations of Probability

Abstract

Consider the problem of constructing a model of the process (experiment) of throwing a die and observing the outcome; in doing so, we need to impose on the experiment a certain probabilistic framework since the same die thrown under ostensibly identical circumstances, generally, yields different outcomes. The framework represents, primarily, the investigator’s view of the nature of the process, but it must also conform to certain logical rules.

Phoebus J. Dhrymes

Chapter 9. LLN, CLT and Ergodicity

Abstract

We recall from Chap. 8 that discussion of random variables (r.v.) takes place in a probability space (Ω, \(\mathcal{A}\), \(\mathcal{P}\)), where Ω is the sample space, \(\mathcal{A}\) is the σ-algebra and \(\mathcal{P}\) is the probability measure.

Phoebus J. Dhrymes

Chapter 10. The General Linear Model

Abstract

In this chapter, we examine the General Linear Model (GLM), an important topic for econometrics and statistics, as well as other disciplines. The term general refers to the fact that there are no restrictions in the number of explanatory variables we may consider, the term linear refers to the manner in which the parameters enter the model. It does not refer to the form of the variables. This is often termed in the literature the regression model, and analysis of empirical results obtained from such models as regression analysis.

Phoebus J. Dhrymes

Chapter 11. Panel Data Models

Abstract

The study of empirical models based on cross-section times series data dates back well into the early 1950s. However, the first modern attempt to consistently model the behavior of agents in such contexts can be traced to Balestra and Nerlove (1966), hereafter referred to as BN, who studied the demand for natural gas by state in the US, over the period 1957–1962.

Phoebus J. Dhrymes

Chapter 12. GLSEM and TS Models

Abstract

In this chapter, we take up two important applications, involving simultaneous equation, AR, and ARMA models, which are very important in several disciplines, such as economics, other social sciences, engineering and statistics. We partially discussed such topics in Chap. 4, in the context of difference equations, since simultaneous equation and AR models involve the use of difference equations and it is important to establish the nature and the properties of their solutions.

Phoebus J. Dhrymes

Chapter 13. Asymptotic Expansions

Abstract

This chapter deals with situations in which we wish to approximate the limiting distribution of an estimator. As such it is different from other chapters in that it does not discuss topics in core econometrics and the ancillary mathematics needed to develop and fully understand them. Moreover, its purpose is different from that of the earlier (theoretical) chapters. Its aim is not only to introduce certain (additional) mathematical concepts but also to derive certain results that may prove useful for econometric applications involving hypothesis testing.

Phoebus J. Dhrymes

Backmatter