Herman Ole Andreas Wold (25 December 1908 – 16 February 1992) was a
Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
*Norwegians, both a nation and an ethnic group native to Norway
*Demographics of Norway
*The Norwegian language, including the ...
-born
econometrician
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8†...
and
statistician
A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors.
It is common to combine statistical knowledge with expertise in other subjects, and statisticians may wor ...
who had a long career in
Sweden
Sweden, formally the Kingdom of Sweden,The United Nations Group of Experts on Geographical Names states that the country's formal name is the Kingdom of SwedenUNGEGN World Geographical Names, Sweden./ref> is a Nordic country located on ...
. Wold was known for his work in
mathematical economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference an ...
, in
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
analysis, and in
econometric
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
statistics.
In
mathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical an ...
, Wold contributed the
Cramér–Wold theorem In mathematics, the Cramér–Wold theorem in measure theory states that a Borel measure, Borel probability measure on \mathbb^k is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint conv ...
characterizing the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
and developed the
Wold decomposition
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states ...
in
time series analysis
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
. In
microeconomics
Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...
, Wold advanced
utility theory
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
and the theory of
consumer demand
In economics, demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given time. The relationship between price and quantity demand is also called the demand curve. Demand for a specific item ...
. In
multivariate statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the dif ...
, Wold contributed the methods of
partial least squares (PLS) and
graphical model
A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a Graph (discrete mathematics), graph expresses the conditional dependence structure between random variables. They are ...
s. Wold's work on
causal
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
from
observational studies
In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws inferences from a sample (statistics), sample to a statistical population, population where the dependent and independent variables, independ ...
was decades ahead of its time, according to
Judea Pearl
Judea Pearl (born September 4, 1936) is an Israeli-American computer scientist and philosopher, best known for championing the probabilistic approach to artificial intelligence and the development of Bayesian networks (see the article on belief ...
.
Early life
Herman Wold was born in
Skien Skien () is a city and municipality in Vestfold og Telemark county in Norway. In modern times it is regarded as part of the traditional region of Grenland, although historically it belonged to Grenmar/Skiensfjorden, while Grenland referred the Norsj ...
, Southern
Norway
Norway, officially the Kingdom of Norway, is a Nordic country in Northern Europe, the mainland territory of which comprises the western and northernmost portion of the Scandinavian Peninsula. The remote Arctic island of Jan Mayen and t ...
. He was the youngest in a family of six brothers and sisters. In 1912 the family moved to
Sweden
Sweden, formally the Kingdom of Sweden,The United Nations Group of Experts on Geographical Names states that the country's formal name is the Kingdom of SwedenUNGEGN World Geographical Names, Sweden./ref> is a Nordic country located on ...
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
Stockholm University ( sv, Stockholms universitet) is a public research university in Stockholm, Sweden, founded as a college in 1878, with university status since 1960. With over 33,000 students at four different faculties: law, humanities, soci ...
to study
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. It was an opportune time, for
Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) 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 statist ...
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 In mathematics, the Cramér–Wold theorem in measure theory states that a Borel measure, Borel probability measure on \mathbb^k is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint conv ...
(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
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states ...
" 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
In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
. Beyond this, the work brought together for the first time the work on individual processes by English statisticians, principally
Udny Yule
George Udny Yule FRS (18 February 1871 – 26 June 1951), usually known as Udny Yule, was a British statistician, particularly known for the Yule distribution.
Personal life
Yule was born at Beech Hill, a house in Morham near Haddington, ...
, 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
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states ...
and the related
Wold's theorem
In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wold, says that every covariance-stationary ...
inspired
Beurling's
factorization theorem in
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
and related work on
invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''.
General descrip ...
s of
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s.
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
Trygve Magnus Haavelmo (13 December 1911 – 28 July 1999), born in Skedsmo, Norway, was an economist whose research interests centered on econometrics. He received the Nobel Memorial Prize in Economic Sciences in 1989.
Biography
After attendi ...
put forward his ideas on the
simultaneous equations
In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought. An equation system is usually classified in the same manner as single e ...
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
The Cowles Foundation for Research in Economics is an economic research institute at Yale University. It was created as the Cowles Commission for Research in Economics at Colorado Springs in 1932 by businessman and economist Alfred Cowles. In 193 ...
; 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
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
method was computationally efficient and enjoyed superior
theoretical
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
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 (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
and
graphical models
''Graphical Models'' is an academic journal in computer graphics and geometry processing publisher by Elsevier. , its editor-in-chief is Jorg Peters of the University of Florida.
History
This journal has gone through multiple names. Founded in 1 ...
in statistics, especially by
Judea Pearl
Judea Pearl (born September 4, 1936) is an Israeli-American computer scientist and philosopher, best known for championing the probabilistic approach to artificial intelligence and the development of Bayesian networks (see the article on belief ...
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
Uppsala University ( sv, Uppsala universitet) is a public university, public research university in Uppsala, Sweden. Founded in 1477, it is the List of universities in Sweden, oldest university in Sweden and the Nordic countries still in opera ...
and he stayed there until 1970. He then moved to
Gothenburg
Gothenburg (; abbreviated Gbg; sv, Göteborg ) is the second-largest city in Sweden, fifth-largest in the Nordic countries, and capital of the Västra Götaland County. It is situated by the Kattegat, on the west coast of Sweden, and has ...
, retiring from there in 1975.
In 1960 Wold became a member of the
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special ...
. He was a member of the Prize Committee for the
from 1968 to 1980.
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
''Econometrica'' is a peer-reviewed academic journal of economics, publishing articles in many areas of economics, especially econometrics. It is published by Wiley-Blackwell on behalf of the Econometric Society. The current editor-in-chief is Gui ...
,
* 1952. ''Demand Analysis: A Study in Econometrics,'' with Lars Juréen.
* 1954. "Causality and Econometrics," ''
Econometrica
''Econometrica'' is a peer-reviewed academic journal of economics, publishing articles in many areas of economics, especially econometrics. It is published by Wiley-Blackwell on behalf of the Econometric Society. The current editor-in-chief is Gui ...
'', 22(2),
p. 162177.
* 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. 369381.
* 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).
See also
*
Causality
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
*
Cramér, Harald
*
Convex preferences In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly ...
*
Cramér–Wold theorem In mathematics, the Cramér–Wold theorem in measure theory states that a Borel measure, Borel probability measure on \mathbb^k is uniquely determined by the totality of its one-dimensional projections. It is used as a method for proving joint conv ...
*
Demand (economics)
In economics, demand is the quantity of a good that consumers are willing and able to purchase at various prices during a given time. The relationship between price and quantity demand is also called the demand curve. Demand for a specific item ...
*
Directed acyclic graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one ve ...
*
Econometrics
Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem Pesaran (1987). "Econometrics," ''The New Palgrave: A Dictionary of ...
*
Graphical model
A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a Graph (discrete mathematics), graph expresses the conditional dependence structure between random variables. They are ...
*
Jöreskog, Karl Gustav
*
Latent variable
In statistics, latent variables (from Latin: present participle of ''lateo'', “lie hidden”) are variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or me ...
*
Least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
*
Moving average
In statistics, a moving average (rolling average or running average) is a calculation to analyze data points by creating a series of averages of different subsets of the full data set. It is also called a moving mean (MM) or rolling mean and is ...
*
Multivariate analysis
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the dif ...
*
Multivariate statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the dif ...
*
Observational study
In fields such as epidemiology, social sciences, psychology and statistics, an observational study draws inferences from a sample (statistics), sample to a statistical population, population where the dependent and independent variables, independ ...
*
Partial least squares regression
Partial least squares regression (PLS 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 li ...
*
Stationary process
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
*
Structural equation model
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 scienc ...
*
Time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
*
Uppsala University
Uppsala University ( sv, Uppsala universitet) is a public university, public research university in Uppsala, Sweden. Founded in 1477, it is the List of universities in Sweden, oldest university in Sweden and the Nordic countries still in opera ...
*
Utility theory
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
*
Whittle, Peter
*
Wold decomposition
In mathematics, particularly in operator theory, Wold decomposition or Wold–von Neumann decomposition, named after Herman Wold and John von Neumann, is a classification theorem for isometric linear operators on a given Hilbert space. It states ...
*
Wold's theorem
In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wold, says that every covariance-stationary ...
Notes
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. ''
External links
ET Interviews: Professor H. O. A. Woldon th
Econometric Theorypage.
*
*
;There is a photograph at
H. O. A. Woldon th
page.
;See also (external links, again)
{{DEFAULTSORT:Wold, Herman
Members of the Royal Swedish Academy of Sciences
Swedish statisticians
Time series econometricians
Mathematical statisticians
20th-century Swedish mathematicians
Uppsala University faculty
University of Gothenburg faculty
1908 births
1992 deaths
Fellows of the Econometric Society
Presidents of the Econometric Society
20th-century Swedish economists
People from Skien
Norwegian emigrants to Sweden