Herman Wold

Last updated
Herman Wold
Born(1908-12-25)25 December 1908
Died16 February 1992(1992-02-16) (aged 83)
Nationality Swedish
Alma mater University of Stockholm
Known for Wold's theorem
MA() 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
Scientific career
Fields Statistics
Institutions Uppsala University
Doctoral advisor Harald Cramér
Doctoral students Sten Malmquist
Peter Whittle
Karl Jöreskog

Herman Ole Andreas Wold (25 December 1908 – 16 February 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]

Early life

Herman Wold was born in Skien, Southern Norway. He was 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.

Scientific achievements

Herman Wold had a long and productive career, spanning six decades.

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. 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. [2]

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[ citation needed ] 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.


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.

Selected writings by H. O. A. Wold

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

See also


  1. Causality, 2nd ed.
    • "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   0009296.
    • "A synthesis of pure demand analysis II". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]. 26. 1943b. pp. 220–263. MR   0011939.
    • "A synthesis of pure demand analysis III". Skandinavisk Aktuarietidskrift [Scandinavian Actuarial Journal]. 27. 1944. pp. 69–120. MR   0011940.)
    • 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   0064385.

Related Research Articles

Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. More precisely, it is "the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference". An introductory economics textbook describes econometrics as allowing economists "to sift through mountains of data to extract simple relationships". The first known use of the term "econometrics" was by Polish economist Paweł Ciompa in 1910. Jan Tinbergen is one of the two founding fathers of econometrics. The other, Ragnar Frisch, also coined the term in the sense in which it is used today.

Harald Cramér

Harald Cramér was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of statistical theory".

Trygve Haavelmo Norwegian economist and econometrician

Trygve Magnus Haavelmo, born in Skedsmo, Norway, was an economist whose research interests centered on econometrics. He received the Nobel Memorial Prize in Economic Sciences in 1989.

C. R. Rao Indian-American mathematician

Calyampudi Radhakrishna Rao, FRS known as C R Rao is an Indian-American mathematician and statistician. He is currently professor emeritus at Pennsylvania State University and Research Professor at the University at Buffalo. Rao has been honoured by numerous colloquia, honorary degrees, and festschrifts and was awarded the US National Medal of Science in 2002. The American Statistical Association has described him as "a living legend whose work has influenced not just statistics, but has had far reaching implications for fields as varied as economics, genetics, anthropology, geology, national planning, demography, biometry, and medicine." The Times of India listed Rao as one of the top 10 Indian scientists of all time. Rao is also a Senior Policy and Statistics advisor for the Indian Heart Association non-profit focused on raising South Asian cardiovascular disease awareness.

Granger causality Statistical hypothesis test for forecasting

The Granger causality test is a statistical hypothesis test for determining whether one time series is useful in forecasting another, first proposed in 1969. Ordinarily, regressions reflect "mere" correlations, but Clive Granger argued that causality in economics could be tested for by measuring the ability to predict the future values of a time series using prior values of another time series. Since the question of "true causality" is deeply philosophical, and because of the post hoc ergo propter hoc fallacy of assuming that one thing preceding another can be used as a proof of causation, econometricians assert that the Granger test finds only "predictive causality". Using the term "causality" alone is a misnomer, as Granger-causality is better described as "precedence", or, as Granger himself later claimed in 1977, "temporally related". Rather than testing whether Xcauses Y, the Granger causality tests whether X forecastsY.

Structural equation modeling Form of causal modeling that fit networks of constructs to data

Structural equation modeling (SEM) is a label for a diverse set of methods used by scientists in both experimental and observational research across the sciences, business, and other fields. It is used most in the social and behavioral sciences. A definition of SEM is difficult without reference to highly technical language, but a good starting place is the name itself.

Peter Whittle was a mathematician and statistician from New Zealand, working in the fields of stochastic nets, optimal control, time series analysis, stochastic optimisation and stochastic dynamics. From 1967 to 1994, he was the Churchill Professor of Mathematics for Operational Research at the University of Cambridge.

In statistics, Wold's decomposition or the Wold representation theorem, named after Herman Wold, says that every covariance-stationary time series can be written as the sum of two time series, one deterministic and one stochastic.

Sir David Forbes Hendry, FBA CStat is a British econometrician, currently a professor of economics and from 2001 to 2007 was head of the Economics Department at the University of Oxford. He is also a professorial fellow at Nuffield College, Oxford.

Karl Gustav Jöreskog is a Swedish statistician. Jöreskog is a Professor Emeritus at Uppsala University, and a co-author of the LISREL statistical program. He is also a member of the Royal Swedish Academy of Sciences. Jöreskog received his bachelor's, master's, and doctoral degrees in Uppsala University. He is also a former student of Herman Wold. He was a statistician at Educational Testing Service (ETS) and a visiting professor at Princeton University.

Ulf Grenander Swedish American mathematician

Ulf Grenander was a Swedish statistician and professor of applied mathematics at Brown University.

Paris Institute of Statistics

Institut de Statistiques de l'Université de Paris is a graduate school of statistics based in Paris, in the fifth arrondissement. It offers specializations in actuarial sciences, Biostatistics as well as industry and services. The ISUP is considered as one of the most prestigious centers of learning of statistics in France, reflected in the number of job offers received regularly, the strength of its alumni network, and wages offered to its students out of school, which place it in the "top 15-ranked" French Grandes Ecoles Founded in 1922 by the mathematician Émile Borel, is the oldest and one of the most prestigious schools for statistics in France. The institute is currently affiliated to Sorbonne University and located on the campus of Jussieu.

The methodology of econometrics is the study of the range of differing approaches to undertaking econometric analysis.

Georges Darmois was a French mathematician and statistician. He pioneered in the theory of sufficiency, in stellar statistics, and in factor analysis. He was also one of the first French mathematicians to teach British mathematical statistics.

In economics, non-convexity refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences and convex budget sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood. When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failures, where supply and demand differ or where market equilibria can be inefficient. Non-convex economies are studied with nonsmooth analysis, which is a generalization of convex analysis.

Gustav Elfving Finnish mathematician and statistician

Erik Gustav Elfving was a Finnish mathematician and statistician. In statistics, he wrote pioneering papers about the optimal design of experiments. He made other notable contributions to the mathematical sciences and to Finnish universities.

Teunis (Teun) Kloek is a Dutch economist and Emeritus Professor of Econometrics at the Erasmus Universiteit Rotterdam. His research interests centered on econometric methods and their applications, especially nonparametric and robust methods in econometrics.

Siddhartha Chib is a statistician and econometrician and Professor of Econometrics and Statistics at Washington University in St. Louis. His work is primarily in Bayesian statistics, econometrics, and Markov chain Monte Carlo methods.


There is a photograph at
See also (external links, again)