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Chapter I begins by describing the salient parts of a formal science of economics that the author has put together in this book and in earlier works (B.P. Stigum 1990 and 2003). The parts comprise a unitary methodological basis for a science, an explication of the meaning of facts and fiction in econometrics, and a confrontation of the methods of present-day applied econometrics with the methods that a formal science advocates. Next the chapter discusses interesting ways in which the axiomatic method and the model-theoretic method of developing theories are used in mathematical economics and statistics, and describes the underlying ideas of a formal theory-data confrontation. Thereafter follows a discussion of three controversial aspects of formal econometrics: (1) the essence of an economic theory; (2) the double role of theory in applied econometrics and the need for bridge principles; and (3) why theory is required both for the design of an empirical analysis and for the interpretation of its results. The chapter ends with a brief description of the contents of the remaining nine chapters.

Chapter VI begins with a discussion of the axioms of a formal theory-data confrontation in which the data appear as vector-valued sequences of observations of a vector-valued random process. Then it describes important characteristics of I(1) ARIMA processes, one of which is their tendency to display long positive and negative sojourns. This property of the process can be used to carry out meaningful empirical analyses of positively valued time series; e.g., spot and forward exchange rate: Let the observations of the exchange rates be observations of an auxiliary I(1) ARIMA process, analyse them with currently available software programs, and check if long sequences of the exchange rates have the characteristics of an I(1) ARIMA process. A second characteristic of an I(1) ARIMA process is that any multivariate version of such a process can be written as an error correction model. In present-day econometrics, statistical analyses of error correction models of vector-valued time series are used to determine the degree of cointegration of the time series. The import of such an analysis hinges on the theoretical meaningfulness of the pertinent error correction model. The chapter demonstrates that an empirically relevant error correction model need not be theoretically meaningful.

The chapter presents Ragnar Frisch and Trygve Haavelmo’s vision of a science of economics, explicates in simple terms the meaning of a formal theory-data confrontation, and describes and resolves an interesting riddle in applied econometrics. Frisch and Haavelmo’s idea of a unified theoretical-quantitative and empirical-quantitative approach to economic problems can be realized in two ways. In one the researcher identifies his theory variables with Haavelmo’s true variables and carries out his empirical analysis the way present-day econometricians do. In the other the researcher identifies his theory variables with variables in Frisch’s model world and carries out his empirical analysis the way formal econometrics prescribes. As exemplified in a formal data confrontation of a simultaneous-equations model of a perfectly competitive commodity market, the empirical analyses in the two scenarios may end in different descriptions of the dynamics of data variables and yield different statistical inferences about social reality. The chapter concludes with a discussion of the mentioned riddle in applied econometrics. The riddle elicits a clear view of the kind of knowledge about social reality that an applied econometrician can obtain.

Chapter V begins with a discussion of formal theory-data confrontations in which the sample population plays a significant role. The formalism differs from the theory-data confrontations in Chapters III and IV, but the fundamental ideas of the empirical analysis are the same. The chapter presents an example in which the formal theory-data confrontation prescribes a factor-analytic test of Milton Friedman’s Permanent Income Hypothesis. This test is then contrasted with a factor-analytic test of Friedman’s hypothesis based on ideas that Ragnar Frisch developed in his 1934 treatise on confluence analysis. In Frisch’s confluence analysis Friedman’s permanent components of income and consumption become so-called systematic variates, and Friedman’s transitory com-ponents become accidental variates. Both the systematic and the accidental variates are unobservables that live and function in the real world - here the data universe. The two tests provide an extraordinary example of how different the present-day-econometrics treatment of errors in variables and errors in equations is from the formal-econometrics treatment of inaccurate observations of variables in Frisch’s model world.

Chapter VII has two purposes. One is to study the methodological problems that arise in analysing positively valued time series in foreign exchange. The other is to contrast the analysis of time series in formal econometrics with the analysis of such data in present-day econometrics. The chapter presents an axiomatic data confrontation of a theory of spot and forward exchange in foreign currency markets. In the formulation of the axioms, actual and auxiliary theory and data variables interact in such a way that the problem that usually arise in the analysis of positively valued time series disappears. The data for the empirical analysis comprise observations on spot and forward exchange rates in the market for Swiss Francs and US Dollars. In the empirical analysis, the given data are analysed, first, with the prescriptions of formal econometrics and, then, with the prescriptions on which present-day econometric time-series analysis insist. The statistical results yield different descriptions of the dynamics of foreign exchange and different inferences about the economics of social reality. In doing that the two contrasting empirical analyses provide interesting ingredients for the discussion of how best to incorporate economic theory in empirical analyses.

Chapter III explains and illustrates what it means for an economic theory to be empirically relevant. A theory’s empirical relevance depends on the empirical context in which it is tested and on the explication of empirical relevance. In present-day econometrics the empirical context of a test is determined by the characteristics of the data generating process. Also, the theory is empirically relevant if and only if the test gives the researcher no reason to reject the theory. In formal econometrics the empirical context of a test is determined by a probability distribution of the data - the MPD - that is induced by the probability distribution of the theory variables and the bridge principles. Also, the theory is empirically relevant if and only if there is a model of its axioms and a model of the bridge principles that determine a model of the MPD that belongs to a 95% confidence band around a meaningful statistical estimate of the MPD. Examples from experimental economics, consumer choice, simultaneous equations, and choice among safe and risky assets illustrate the contrasting characteristics of the notion of empirical relevance in present-day and formal econometrics.

Chapter IV discusseseconometric analyses of qualitative response models. A qualitative response model is an econometric modelin which the dependent variable is either discrete or half continuous and half discrete. Each proto-type of such models appears in two forms – a standard model and a latent-variable model. Both forms suffer from a serious defect: they fail to provide 1a researcher with the means he needs to ascertain the empirical relevance of his parameter estimates. That is a serious defect since the relations between dependent and explanatory variables that an empirically irrelevant estimate of a qualitative response model describes are meaningless. In a case study of 1980 US females’ participation in the labor force the chapter uses a Probit version of the two models to contrast present-day econometric analysis of the standard model with a formal econometric analysis of the latent-variable model. The empirical analysis reveals that the Probit versions of the two qualitative response models are not empirically relevant in the given empirical context. The empirical analysis also reveals that a present-day analysis of the standard model may deem it empirically relevant while a formal econometric analysis of the latent-variable model deems the latter empirically irrelevant and vice versa.

Chapter VIII presents a novel mathematical theory of non-linear cointegration among second-order random processes. It begins by explaining why the accepted characterization of integrated second-order processes is inadequate for the analysis of non-linearly cointegrated economic systems, proposes alternative characterization of integrated processes, and develops novel ideas of non-linearly cointegrated second-order random processes. Thereafter, it presents a data confrontation of an economic theory about the dynamics of spot rates in foreign exchange whose variables share the behavior characteristics of the random processes in the mathematical theory. The data consist of weekly observations of a triple of exchange rates, and the empirical analysis is carried out in two ways – one by the methods of formal econometrics and another by the methods of present-day econometrics. The results of the two empirical analyses differ in interesting ways. Both agree that the behavior of the three exchange rates has the characteristics on which the mathematical theory insists, but their description of the dynamics of foreign exchange differ. Also, the present-day econometrics analysis rejects the empirical relevance of the given economic theory while the formal econometrics analysis accepts it. The acceptance of the theory carries interesting information about the dynamics of foreign exchange in social reality.