Herman Wold | |
---|---|

Born | |

Died | 16 February 1992 83) | (aged

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 Econometrics |

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.

- Early life
- Scientific achievements
- Studying with Harald Cramér
- Time series and the Wold decomposition
- Theory of consumer demand
- Systems of simultaneous equations and causal inference
- Multivariate analysis and partial least squares
- Appointments
- Selected writings by H. O. A. Wold
- See also
- Notes
- References
- External links

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] }

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.

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

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).

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.

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] }

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.

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.

- 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).

- Causality
- Cramér, Harald
- Convex preferences
- Cramér–Wold theorem
- Demand (economics)
- Directed acyclic graph
- Econometrics
- Graphical model
- Jöreskog, Karl Gustav
- Latent variable
- Least squares
- Moving average
- Multivariate analysis
- Multivariate statistics
- Observational study
- Partial least squares regression
- Stationary process
- Structural equation model
- Time series
- Uppsala University
- Utility theory
- Whittle, Peter
- Wold decomposition
- Wold's theorem

- ↑
*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.

- "A synthesis of pure demand analysis

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**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 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.

**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.

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 *X**causes* Y, the Granger causality tests whether X *forecasts**Y.*

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**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.

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- 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.*

- ET Interviews: Professor H. O. A. Wold on the Econometric Theory page.
- New School: Herman Wold, 1908-1992
- Herman Wold at the Mathematics Genealogy Project
- Newspaper clippings about Herman Wold in the 20th Century Press Archives of the ZBW

- There is a photograph at

- H. O. A. Wold on the Portraits of Statisticians page.

- See also (external links, again)

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