Introduction

Simulation methods have in common the use of two random samples: the first is constituted of true observations of the studied phenomenon whereas the second is artificially generated by a computer. The distribution of the first sample is unknown and its size is determined by the available databases. The second sample obeys the will of the statistician, who chooses the generating process and its size depends on the available computing power and time.

There are many different reasons for using simulated data in statistical inference. We restrict our presentation to three uses of simulated data. The first one is motivated by numerical problems in an estimation procedure which we illustrate by providing a method for computing integrals. The other two methods pertain to the analysis of the small sample properties of estimators. On one hand, we may look for the small sample properties of an estimator or a test for a given value of the parameters. On the other hand, we can put forward another distribution of the estimators by the method called bootstrap which we will try to justify by various arguments. The three sections about the uses of simulations are preceded by a brief introduction to techniques for generating random scalars and vectors.

Random Number Generators

Consider a probability distribution on ℝp characterized by its distribution function F. We propose to construct algorithms that allow us to generate sequences of independent vectors with distribution F.

Introduction

In contrast to other fields, the economic theory rarely specifies functional forms but usually specifies only a list of relevant variables to explain a phenomenon. The specification of the form of the relationship largely results from the empirical study which yields a “good” model which “works well.” A first level of the analysis consists in writing a (linear, loglinear, nonlinear, …) model and in estimating it without taking the approximating nature of the model into account. This approach was explained in Chapters 7, 8, and 9. A second approach consists in specifying a parametric model whose incorrect specification is explicit. This leads for instance to correct the expression for the variances or to choose robust methods to the misspecification. This methodology was the topic of Chapter 9.

Finally, it is possible to free oneself of all specification constraints by adopting a nonparametric approach to the estimation of the regression, an approach in which the data itself chooses the form of the function of interest.

Various estimation methods of the nonparametric regression have been developed and are now commonly employed. We will present here the kernel method (corresponding to what we presented in Chapter 5). This method is simple but is dominated in some cases by other approaches (local polynomial, Fourier series expansion, wavelets …).

Introduction

This chapter is dedicated to the study of linear dynamic models, more precisely to the study of the temporal evolution of one or several variables. It is organized in the following way. The first part provides the definitions of stochastic processes in discrete time. Next, we study models that have a particular representation, namely the univariate ARMA(p, q) models. In the last part we extend these models to the multivariate setting.

Generally speaking, a stochastic process is a family of random variables on a common probability space indexed by the elements of an ordered set T which is the time index set. The random variable indexed by an element i ∈ T describes the state of the process at time i. The stochastic processes considered here are defined in the following way.

Definition 12.1A stochastic process is a family of random variables {xi, i ∈ T} where the time index set T is a subset of the real line ℝ.

We could denote by T the set of all parameters, but to avoid the confusion with the parameters in a statistical sense, we call it the time index set. It is often called the domain of the definition of the stochastic process {xi, i ∈ T}. If T is an interval of the real line, then the process is said to be a continuous-time process.

Introduction

The past 15 years have witnessed important developments in the modelling of nonlinear time series. This modelling opens the path to a variety of different models. It poses multiple difficulties, both from a theoretical point of view relative to the basic properties of the models and the estimators, and from an empirical point of view, for instance in the choice of the specification of nonlinearity.

The theoretical basis that is necessary to prove consistency and asymptotic normality of the moment estimators has been treated in Chapter 2, including for the non i.i.d. case. Indeed, Chapter 2 introduced tools that allow us to apply the law of large numbers and the central limit theorem to dynamic models. In particular, we have seen the concept of statistical models that are uniformly mixing stationary and the concept of near epoch dependence (NED). This enabled us to state Theorem 2.6 in Chapter 2, which is nothing but a central limit theorem for stationary mixing processes. We will return to some of these concepts again in this chapter.

Here, we are not attempting to provide a general and comprehensive presentation of nonlinear dynamic models because each model has characteristics and properties that result in complexity and difficulties that are specific to those. Moreover, the study of these models is expanding fast. Therefore, we will settle for the presentation of some examples only.

Introduction

In the previous chapters, we presented a set of statistical methods that were initially general and then specific to particular classes of statistical models. These methods and models have been chosen because they were suited to treat economic data but they belonged essentially to mathematical statistics. It is sufficient to look at some textbooks for the application of statistics in scientific areas other than economics (physics, biology, sociology, psychology and so on) to find an important part of the tools presented here, sometimes with a different relative importance or a different terminology, but fundamentally similar regarding the modes of thought and the results. The recent development in econometrics attests of the increasing diversity of the methods it uses. Formerly centered around a small number of models, econometrics has seized more and more statistical techniques; there are few statistical methods that have not found a use in the treatment of economic data.

Does the econometric method coincide with statistics? Most econometricians would answer no to this question. Most likely, the econometricians would consider that the autonomy of their field results from the approach of structural modeling. In broad outline, we mean by structural modeling a specification of the statistical model starting from theoretical relations that describe the phenomenon under study. These relations result from the “micro” or “macro” economic analysis and are essentially the mathematical expression of behavioral rules of agents or groups of agents involved in the examined phenomenon.