Herman Wold had a long and productive career, spanning six decades.
Studying with Harald Cramér In 1927 Wold enrolled at the
Stockholm University to study
mathematics. It was an opportune time, for
Harald Cramér had been appointed Professor of Actuarial Mathematics and Mathematical Statistics. Wold would later write, "To belong to Cramér's first group of students was good luck, an advantage that simply cannot be exaggerated." After graduating in 1930 Wold worked for an insurance company; he also did work on mortality data with Cramér and later designed a tariff for the insurance companies. He started work on a PhD on stochastic processes with Cramér as supervisor. Away from the thesis Wold and Cramér did some joint work, their best known result being the
Cramér–Wold theorem (1936).
Time series and the Wold decomposition Wold's thesis,
A Study in the analysis of stationary time series, was an important contribution. The main result was the "
Wold decomposition" by which a
stationary series is expressed as a sum of a deterministic component and a stochastic component which can itself be expressed as an infinite
moving average. Beyond this, the work brought together for the first time the work on individual processes by English statisticians, principally
Udny Yule, and the theory of stationary stochastic processes created by Russian mathematicians, principally
A. Ya. Khinchin. Wold's results on univariate time series were generalized to multivariate time series by his student
Peter Whittle. The
Wold decomposition and the related
Wold's theorem inspired
Beurling's
factorization theorem in
harmonic analysis and related work on
invariant subspaces of
linear operators.
Theory of consumer demand In 1938 a government committee appointed Wold to do an econometric study of consumer demand in Sweden. The results were published in 1940. In parallel, he worked on the
theory of demand. His book
Demand Analysis (1952) brought together his work on the theory of demand, the theory of stochastic processes, the
theory of regression and his work on Swedish data.
Systems of simultaneous equations and causal inference In 1943 and 1944,
Trygve Haavelmo put forward his ideas on the
simultaneous equations model, arguing that
systems of simultaneous equations should be central in econometric research. Wold noted some limitation of the maximum-likelihood approach favoured by Haavelmo and the
Cowles Commission; Wold cautioned that the literature contained some exaggerated claims for the superiority of maximum-likelihood estimation. In 1945 to 1965, Wold proposed and elaborated on his "recursive causal chain" model, which was more appropriate for many applications, according to Wold: For such "recursive causal chain" models, the
least squares method was computationally efficient and enjoyed superior
theoretical properties, which it lacked for general time-series models. Wold's writings on causality and recursive-chain models have been recognized as scientific inventions by recent work on
causality and
graphical models in statistics, especially by
Judea Pearl and Nanny Wermuth.
Multivariate analysis and partial least squares At the end of his career, Wold turned away from econometric modelling and developed multivariate techniques for what he called "soft" modelling. Some of these methods were developed through interactions with his student
K. G. Jöreskog, although the latter's focus was primarily on maximum likelihood methods. His son
Svante Wold applied these techniques in chemistry and developed the field of chemometrics. ==Appointments==