The starting point for the analysis of series of observations indexed by time {yt, t ≥ 0} is to examine their graphical representations. Oftentimes it happens that the series exhibit an explosive pattern, that is, they give the impression of tending toward infinity with t. In such a case, the attention is focused on the dominant components of the series which are smoother than the original ones, but asymptotically equivalent to them. In this chapter we will mainly stress the importance of such components, which we will assume are diverging in a polynomial fashion. It is possible to obtain such a behavior through a nonlinear transformation of the original series most of the time. We can ask a number of questions about this trend component, according to whether we examine the series separately or jointly.

What is the rate of divergence of the various series? What are the differences among them?

What happens to the series once the dominant component is eliminated; are there still some diverging components and how important are they?

The joint plot of two series sometimes shows fairly strong links among the trend components of the series. Is it possible to make these links explicit, to study the cases where they are particularly strong, and to compare the strength of these links with those of other components of the series?

Most of the models and results described in the previous chapters are based on the notion of an optimal forecast of a variable given its past. For example, this notion has been useful to define the autoregressive form of a process, its innovation, and so on, but also to analyze in greater details the links between processes (partial correlation, causality measurement, etc.) or to develop estimation methods (maximum likelihood in conditional form).

In this chapter we are interested in a more practical aspect of this optimal forecast.

Economic agents, e.g., firms, financial operators, consumers, must in general decide their current behavior taking into consideration the ideas they have about the future. Since this future is partially unknown, they have to forecast it in a proper way. This leads us to study two questions which are linked to each other:

1. (i) How do agents foresee their future?

2. (ii) How do their expectations influence their current behavior?

In the first section we will start by recalling a number of results on forecasts which have been introduced in the previous chapters. We will treat the updating problem in great detail and specify the economic terminology which is slightly different from the probabilistic terminology used up to now.

The other sections are all related to the analysis of explanatory models containing expectations among the explanatory variables. More precisely, we will look at the properties of these models in the case of optimal expectations (or rational in the economic terminology).

In preceding chapters, especially in Chapters 6, 15, and 16, we studied the optimal properties of general estimation and testing procedures in finite samples. On the other hand, in large samples the optimality problem was discussed in some special cases only. The goal of this chapter is to study such a problem more systematically.

The theory of asymptotic efficiency is technically difficult. It is also based on tools that are relatively different from those used up to now. For these two reasons we shall present the main results of this theory on an intuitive basis. In the same spirit of simplification the theoretical aspects will be discussed within a sampling framework. Then we shall consider successively the asymptotic optimality of estimators and the asymptotic optimality of tests. For each of these problems there exist different optimality concepts depending on the order of approximation retained in the asymptotic expansions. These concepts correspond to the properties of consistency, first-order asymptotic efficiency and secondorder asymptotic efficiency.

Asymptotic Comparison of Estimators

Hereafter we consider a random sample (Y1,…, Yn) drawn from a common probability distribution that belongs to a family of probability distributions parameterized by a parameter vector θ ∈ θ. We assume that the statistical model is homogenous and we let f(y;θ) denote the marginal probability density for one observation. The object of interest is the parameter vector θ.

It is natural to restrict a priori the study to estimators Tn(Y) that are consistent for θ. There exist, however, an infinite number of consistent estimators. To choose among those estimators, it is therefore necessary to invoke additional criteria based on the concept of asymptotic precision.