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This chapter introduces extended asymptotic theories on the unit root developed in Fuller (1976), Dickey and Fuller (1979), Phillips (1987), and Phillips and Perron (1988) among others. The theories are explained in two steps. The first deals with the elementary but fundamental case where Δx_t is i.i.d with zero mean. The second step is given in Chapter 6. It explains more advanced aspects including the case where Δx_t is an ARMA.

This chapter describes the augmented Dickey-Fuller method for the case where Δx_t is i.i.d. Difference stationarity is tested as the null hypothesis against trend stationarity, assuming that {x_t} may possibly contain a linear deterministic trend. It also presents a method that does not work.

Addresses the issue of testing, and reveals some intrinsic problems pertaining to hypothesis testing beneath the achievements of formalizing econometrics. Theory verification through applied studies forms one of the main motives for formalizing methods of model estimation and identification, and the statistical theory of hypothesis testing was accepted without much dispute quite early as the technical vehicle to fulfil this desire. However, during the adoption of the theory into econometrics in the 1940s and 1950s, the achievable domain of verification turned out to be considerably reduced, as testing in econometrics proper gradually dwindled into part of the modelling procedure and pertained to model evaluation using statistical testing tools; in the applied field, empirical modellers took on the task of discriminating between and verifying economic theories against the model results, and carried this out in an ad hoc and often non‐sequitur manner. Describes how the desire to test diverged into model evaluation in econometric theory on the one hand, and economic theory verification in practice on the other, as econometric testing theory took shape. The story begins with the early period prior to the formative movement in the first section of the chapter; the following section looks at the period in which the theme of hypothesis testing was introduced, and the first test emerged in econometrics; the last two sections report, respectively, on how model testing in applied econometrics and test design in theoretical econometrics developed and moved apart.

Narrates the process of how identification was formalized in econometrics. The issue of identification stemmed from the quest to know the attainability of economically meaningful relationships from statistical analysis of economic data in early estimation attempts. It arose out of the ‘correspondence’ problems ‘between economic activity, the data that activity generates, the theoretical economic model and the estimated relationship’ (Morgan, 1990). When identification theory was eventually formalized, its purpose became focused on the conditions under which a certain set of values of structural parameters could be uniquely determined from the data among all the permissible sets embodied in a mathematically complete theoretical model, usually composed of a simultaneous‐equations system. These conditions are most commonly known as the order and rank conditions in the context of linear, simultaneous‐equations models in present‐day econometrics textbooks. The emergence of identification theory played a key role in the formal establishment of the structural approach of orthodox econometrics through its links to model testing and model specification. Traces the formalization of identification theory around two interwoven themes: how the identification problem was perceived and described in connection with the other issues in econometric modelling; and how the problem was formalized and tackled with mathematical and statistical means. The first section outlines the early appearance of the identification problem and some ad hoc solutions for particular cases and model forms before the mid‐1930s; the second centres upon the initial systematic work on the issue around 1940; the third is devoted to the contribution of the Cowles group; and the completion of the theoretical framework and its overlaps with other modelling issues form the subject of the last section.

This chapter recounts the evolution of econometric models up to the 1940s, discussing the common criteria and principles used for model choice, and the generalization of model construction as econometrics focused on the structural modelling procedure. The first section reviews the pre‐model period, and the second looks at the emergence of models and the structural method of model construction. The initial generalization (formalization) efforts of the model‐building strategy and criteria are dealt with in the third section. Concludes with the establishment of the structural modelling procedure (the maturity of simultaneous‐equations model formulation).

Narrates the process of how estimation was formalized. Estimation can be seen as the genesis of econometrics, since finding estimates for coefficients of economically meaningful relationships has always been the central motive and fulfilment of applied modelling activities. The process therefore became separated out as one of the basic steps along with model construction, identification, and testing. Subsequent research activities in estimation were confined to technical development of new optimal estimators for increasingly complicated model forms. The first section of the chapter describes the early developments in estimation methods centring around the least squares (LS) principle; how this led to the maximum‐likelihood (ML) method in a simultaneous‐equations system is the content of the second section; the third section turns to look at special problems in the context of time‐series analysis; other developments concerning errors‐in‐variables models are summed up in the fourth section; and the final completion of basic estimation theory in orthodox econometrics takes up the final section.