1 - Statistical Models

Summary

Introduction

In this chapter, we review the usual statistical terminology that introduces the fundamental notions of sample, parameters, statistical model, and likelihood function. Our presentation avoids all technical developments of probability theory, which are not strictly necessary in this book. For example, σ-fields (or σ-algebras) are not introduced, nor are measurability conditions. The mathematical rigor of the exposition is necessarily weakened by this choice, but our aim is to focus the interest of the reader on purely statistical concepts.

It is expected that the reader knows the usual concepts of probability as well as the most common probability distributions and we refer to various reference books on this theme in the bibliographic notes.

At the end of this chapter, we emphasize conditional models, whose importance is fundamental in econometrics, and we introduce important concepts such as identification and exogeneity.

Sample, Parameters, and Sampling Probability Distributions

A statistical model is usually defined as a triplet consisting of a sample space, a parametric space and a family of sampling probability distributions.

We denote by x the realization of a sample. It is always assumed that x is equal to a finite sequence (x_i)i=1,…,n where n is the sample size and x_i is the ith observation. We limit ourselves to the case where x_i is a vector of m real numbers (possibly integers) belonging to a subset X of ℝ^m. Hence, the sample space is Xⁿ ⊂ ℝ^mn.