Herman Wold
Encyclopedia
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 model
s. Wold's work on causal
inference
from observational studies was decades ahead of its time, according to Judea Pearl
.
. 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).
" 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 subspace
s of linear operators.
. 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.
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.
.
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.
, 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.
There is an extensive bibliography published with the ET interview (below).
There is a photograph at
See also (external links, again)
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...
and statistician
Statistician
A statistician is someone who works with theoretical or applied statistics. The profession exists in both the private and public sectors. The core of that work is to measure, interpret, and describe the world and human activity patterns within it...
who had a long career in Sweden
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
. Wold was known for his work in mathematical economics
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...
, in time series
Time series
In statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
analysis, and in econometric statistics.
In mathematical statistics
Mathematical statistics
Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis...
, Wold contributed the Cramér–Wold theorem characterizing the normal distribution and developed the Wold decomposition
Wold decomposition
In operator theory, the Wold decomposition, named after Herman Wold, or Wold-von Neumann decomposition, after Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space...
in time series analysis. In microeconomics
Microeconomics
Microeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...
, Wold advanced utility theory and the theory of consumer demand. In multivariate statistics
Multivariate statistics
Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis...
, Wold contributed the methods of partial least squares (PLS)
Partial least squares regression
Partial least squares regression is a statistical method that bears some relation to principal components regression; instead of finding hyperplanes of maximum variance between the response and independent variables, it finds a linear regression model by projecting the predicted variables and the...
and graphical model
Graphical model
A graphical model is a probabilistic model for which a graph denotes the conditional independence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning....
s. Wold's work on causal
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
inference
Statistical inference
In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...
from observational studies was decades ahead of its time, according to Judea Pearl
Judea Pearl
Judea Pearl is a computer scientist and philosopher, best known for developing the probabilistic approach to artificial intelligence and the development of Bayesian networks ....
.
Studying with Harald Cramér
In 1927 Wold enrolled at the University of Stockholm to study mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. It was an opportune time, for Harald Cramér
Harald Cramér
Harald Cramér was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. He was once described by John Kingman as "one of the giants of statistical theory".-Early life:Harald Cramér was born in Stockholm, Sweden on September...
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 decompositionWold decomposition
In operator theory, the Wold decomposition, named after Herman Wold, or Wold-von Neumann decomposition, after Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space...
" by which a stationary series
Stationary process
In the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
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
Udny Yule
George Udny Yule FRS , usually known as Udny Yule, was a British statistician, born at Beech Hill, a house in Morham near Haddington, Scotland and died in Cambridge, England. His father, also George Udny Yule, and a nephew, were knighted. His uncle was the noted orientalist Sir Henry Yule...
, and the theory of stationary stochastic processes created by Russian mathematicians, principally A. Y. Khinchin
Aleksandr Yakovlevich Khinchin
Aleksandr Yakovlevich Khinchin was a Soviet mathematician and one of the most significant people in the Soviet school of probability theory. He was born in the village of Kondrovo, Kaluga Governorate, Russian Empire. While studying at Moscow State University, he became one of the first followers...
. Wold's results on univariate time series were generalized to multivariate time series by his student Peter Whittle
Peter Whittle
Peter Whittle is a mathematician and statistician, working in the fields of stochastic nets, optimal control, time series analysis, stochastic optimization and stochastic dynamics...
.
The Wold decomposition
Wold decomposition
In operator theory, the Wold decomposition, named after Herman Wold, or Wold-von Neumann decomposition, after Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space...
and the related Wold's theorem inspired Beurling
Arne Beurling
Arne Carl-August Beurling was a Swedish mathematician and professor of mathematics at Uppsala University and later at the Institute for Advanced Study in Princeton, New Jersey....
's factorization theorem in harmonic analysis and related work on invariant subspace
Invariant subspace
In mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...
s 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 demandSupply and demand
Supply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...
. His book Demand Analysis (1952) brought together his work on the theory of demand, the theory of stochastic processes, the theory of regression
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
and his work on Swedish data.
Systems of simultaneous equations and causal inference
In 1943-4, Trygve HaavelmoTrygve Haavelmo
Trygve Magnus Haavelmo , born in Skedsmo, Norway, was an influential economist with main research interests centered on the fields of econometrics and economics theory. During World War II he worked with Nortraship in the Statistical Department in New York City. He received his Ph.D...
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
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
method was computationally efficient and enjoyed superior theoretical
Estimation theory
Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the...
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
Causality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
and graphical models in statistics, especially by Judea Pearl
Judea Pearl
Judea Pearl is a computer scientist and philosopher, best known for developing the probabilistic approach to artificial intelligence and the development of Bayesian networks ....
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 by his student K. G. JöreskogKarl Jöreskog
*Jöreskog, K. G., & Sörbom, D. . Advances in factor analysis and structural equation models. New York: University Press of America.*Jöreskog, K. G., & Moustaki, I. . Factor analysis of ordinal variables: A comparison of three approaches. Multivariate Behavioral Research, 36, 347–387.- Festschrift...
.
Appointments
The story of Wold's academic appointments is briefly told. In 1942 he became professor of statistics at Uppsala UniversityUppsala University
Uppsala University is a research university in Uppsala, Sweden, and is the oldest university in Scandinavia, founded in 1477. It consistently ranks among the best universities in Northern Europe in international rankings and is generally considered one of the most prestigious institutions of...
and he stayed there until 1970. He then moved to Gothenburg
Gothenburg
Gothenburg is the second-largest city in Sweden and the fifth-largest in the Nordic countries. Situated on the west coast of Sweden, the city proper has a population of 519,399, with 549,839 in the urban area and total of 937,015 inhabitants in the metropolitan area...
, retiring from there in 1975.
In 1960 Wold became a member of the Royal Swedish Academy of Sciences
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the Royal Academies of Sweden. The Academy is an independent, non-governmental scientific organization which acts to promote the sciences, primarily the natural sciences and mathematics.The Academy was founded on 2...
. 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 SkienSkien
' is a city and municipality in Telemark county, Norway. It is part of the traditional region of Grenland. The administrative centre of the municipality is the city of Skien. Skien is also the administrative centre of Telemark county....
, Southern Norway
Norway
Norway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...
the youngest in a family of six brothers and sisters. In 1912 the family moved to Sweden
Sweden
Sweden , officially the Kingdom of Sweden , is a Nordic country on the Scandinavian Peninsula in Northern Europe. Sweden borders with Norway and Finland and is connected to Denmark by a bridge-tunnel across the Öresund....
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 EconometricaEconometricaEconometrica is a peer-reviewed academic journal of economics, publishing articles not only in econometrics but in many areas of economics. It is published by the Econometric Society and distributed by Wiley-Blackwell. Econometrica is one of the most highly ranked economics journals in the world...
, - 1952. Demand Analysis: A Study in Econometrics, with Lars Juréen.
- 1954. "Causality and Econometrics," EconometricaEconometricaEconometrica is a peer-reviewed academic journal of economics, publishing articles not only in econometrics but in many areas of economics. It is published by the Econometric Society and distributed by Wiley-Blackwell. Econometrica is one of the most highly ranked economics journals in the world...
, 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. JöreskogKarl Jöreskog*Jöreskog, K. G., & Sörbom, D. . Advances in factor analysis and structural equation models. New York: University Press of America.*Jöreskog, K. G., & Moustaki, I. . Factor analysis of ordinal variables: A comparison of three approaches. Multivariate Behavioral Research, 36, 347–387.- Festschrift...
and H. Wold), North-Holland.
There is an extensive bibliography published with the ET interview (below).
See also
- CausalityCausalityCausality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
- Cramér, HaraldHarald CramérHarald Cramér was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. He was once described by John Kingman as "one of the giants of statistical theory".-Early life:Harald Cramér was born in Stockholm, Sweden on September...
- Convex preferencesConvex preferencesIn economics, convex preferences refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes"...
- Cramér–Wold theorem
- Demand (economics)Supply and demandSupply and demand is an economic model of price determination in a market. It concludes that in a competitive market, the unit price for a particular good will vary until it settles at a point where the quantity demanded by consumers will equal the quantity supplied by producers , resulting in an...
- Directed acyclic graphDirected acyclic graphIn mathematics and computer science, a directed acyclic graph , is a directed graph with no directed cycles. That is, it is formed by a collection of vertices and directed edges, each edge connecting one vertex to another, such that there is no way to start at some vertex v and follow a sequence of...
- EconometricsEconometricsEconometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...
- Graphical modelGraphical modelA graphical model is a probabilistic model for which a graph denotes the conditional independence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning....
- Jöreskog, Karl Gustav
- Latent variableLatent variableIn statistics, latent variables , are variables that are not directly observed but are rather inferred from other variables that are observed . Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models...
- Least squaresLeast squaresThe method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
- Moving average
- Multivariate analysisMultivariate analysisMultivariate analysis is based on the statistical principle of multivariate statistics, which involves observation and analysis of more than one statistical variable at a time...
- Multivariate statisticsMultivariate statisticsMultivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis...
- Observational studyObservational studyIn epidemiology and statistics, an observational study draws inferences about the possible effect of a treatment on subjects, where the assignment of subjects into a treated group versus a control group is outside the control of the investigator...
- Partial least squares regressionPartial least squares regressionPartial least squares regression is a statistical method that bears some relation to principal components regression; instead of finding hyperplanes of maximum variance between the response and independent variables, it finds a linear regression model by projecting the predicted variables and the...
- Stationary processStationary processIn the mathematical sciences, a stationary process is a stochastic process whose joint probability distribution does not change when shifted in time or space...
- Structural equation model
- Time seriesTime seriesIn statistics, signal processing, econometrics and mathematical finance, a time series is a sequence of data points, measured typically at successive times spaced at uniform time intervals. Examples of time series are the daily closing value of the Dow Jones index or the annual flow volume of the...
- Uppsala UniversityUppsala UniversityUppsala University is a research university in Uppsala, Sweden, and is the oldest university in Scandinavia, founded in 1477. It consistently ranks among the best universities in Northern Europe in international rankings and is generally considered one of the most prestigious institutions of...
- Utility theory
- Whittle, PeterPeter WhittlePeter Whittle is a mathematician and statistician, working in the fields of stochastic nets, optimal control, time series analysis, stochastic optimization and stochastic dynamics...
- Wold decompositionWold decompositionIn operator theory, the Wold decomposition, named after Herman Wold, or Wold-von Neumann decomposition, after Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space...
- Wold's theorem
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.
External links
- ET Interviews: Professor H. O. A. Wold on the Econometric Theory page.
- New School: Herman Wold, 1908-1992
- Mathematics Genealogy Project: Herman Wold
There is a photograph at
- H. O. A. Wold on the Portraits of Statisticians page.
See also (external links, again)