# Herman Wold

Herman Wold
Born December 25, 1908
Skien, Norway
Died February 16, 1992 (aged 83)
Nationality Swedish
Fields Statistics
Econometrics
Institutions Uppsala University
Alma mater University of Stockholm
Doctoral students Sten Malmquist
Peter Whittle
Karl Jöreskog
Known for Wold's theorem
MA($\infty$) representation of stationary stochastic processes
Wold decomposition (Operator theory)
Cramér–Wold theorem
Time series analysis
Utility theory
Consumer demand
Latent variable
Structural equation models
Consistent least squares estimation of recursive triangular systems
Causal inference in observational studies
Partial least squares regression

Herman Ole Andreas Wold (December 25, 1908 – February 16, 1992) was a Norwegian-born econometrician and statistician who had a long career in Sweden. Wold was known for his work in mathematical economics, in time series analysis, and in econometric statistics.

In mathematical statistics, Wold contributed the Cramér–Wold theorem characterizing the normal distribution and developed the Wold decomposition in time series analysis. In microeconomics, Wold advanced utility theory and the theory of consumer demand. In multivariate statistics, Wold contributed the methods of partial least squares (PLS) and graphical models. Wold's work on causal inference from observational studies was decades ahead of its time, according to Judea Pearl.[1]

## Scientific Achievements

### Studying with Harald Cramér

In 1927 Wold enrolled at the University of Stockholm 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. Y. 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.[2]

### Systems of simultaneous equations and causal inference

In 1943-4, 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 the years 1945-65, 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.

## Appointments

The story of Wold's academic appointments is briefly told. In 1942 he became professor of statistics at Uppsala University and he stayed there until 1970. He then moved to Gothenburg, retiring from there in 1975.

In 1960 Wold became a member of the Royal Swedish Academy of Sciences. He was a member of the Prize Committee for the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel from 1968 to 1980.

## Early life

Herman Wold was born in Skien, Southern Norway the youngest in a family of six brothers and sisters. In 1912 the family moved to Sweden and became Swedish citizens. Herman's father had a small fur and hide business.

## Selected writings by H. O. A. Wold

• 1938. A Study in the Analysis of Stationary Time Series, Almqvist & Wiksell
• 1949. Statistical Estimation of Economic Relationships, in Econometrica,
• 1952. Demand Analysis: A Study in Econometrics, with Lars Juréen.
• 1954. "Causality and Econometrics," Econometrica, 22(2), p p. 162-177.
• 1964.Econometric model building : essays on the causal chain approach (edited by Herman O.A. Wold).
• 1969. "Econometrics as Pioneering in Nonexperimental Model Building," Econometrica, 37(3), p p. 369-381.
• 1980. The Fix-Point Approach to Interdependent Systems (edited by Herman Wold), North-Holland.
• 1982. Systems under Indirect Observation: Causality, Structure, Prediction (edited by K. G. Jöreskog and H. Wold), North-Holland.

There is an extensive bibliography published with the ET interview (below).

## Notes

1. ^ Causality, 2nd ed.
2. ^
• "A synthesis of pure demand analysis I (Dedicated to Professor Harald Cramér on the occasion of his 50th birthday September 25th 1943)". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal] 26. 1943a. pp. 85–118. MR 9296.
• "A synthesis of pure demand analysis II". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal] 26. 1943b. pp. 220–263. MR 11939.
• "A synthesis of pure demand analysis III". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal] 27. 1944. pp. 69–120. MR 11940.)
• Wold, Herman; Juréen, Lars (in association with Wold) (1953). Demand analysis: A study in econometrics. Wiley publications in statistics. New York: John Wiley and Sons, Inc. Stockholm: Almqvist and Wiksell. pp. xvi+358. MR 64385.

## References

Autobiography
• J. Gani, ed. (1982). The Making of Statisticians, New York: Springer-Verlag. This has a chapter in which Wold tells the story of his life.
• See the ET interview under External Links.
Discussion
• Svante Wold "Wold, Herman Ole Andreas";, pp. 213–4 in Leading Personalities in Statistical Sciences from the Seventeenth Century to the Present, (ed. N. L. Johnson and S. Kotz) 1997. New York: Wiley. Originally published in Encyclopedia of Statistical Science.