Econometrics

Econometrics is a rapidly developing branch of economics which, broadly speaking, aims to give empirical content to economic relations. The term ‘econometrics’ appears to have been first used by Pawel Ciompa as early as 1910; although it is Ragnar Frisch, one of the founders of the Econometric Society, who should be given the credit for coining the term, and for establishing it as a subject in the sense in which it is known today (see Frisch, 1936, p. 95). Econometrics can be defined generally as ‘the application of mathematics and statistical methods to the analysis of economic data’, or more precisely in the words of Samuelson, Koopmans and Stone (1954),

… as the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference (p. 142).

A simplification or aggregation problem is faced by a research worker whenever he finds that his data are too numerous or in too much detail to be manageable and he feels the need to reduce or combine the detailed data in some manner. He will want to use aggregative measures for his new groups. But what will be the effect of this procedure on his results? How can he choose among alternative procedures? In grouping his data and/or relations he must also decide how many groups to use; a smaller number is more manageable but will cause more of the original information to be lost. The research worker seeks a solution to this problem that will best serve his objectives, or those of some decision-maker who will use his results.

The Almon distributed lag, due to Shirley Almon (1965), is a technique for estimating the weights of a distributed lag by means of a polynomial specification.

Bunch maps were developed by Ragnar Frisch (1934) to deal with the problems of confluence analysis. By ‘confluence analysis’ he meant the study of several variables in some sets of which a regression equation might have a meaning, while in others it might not because of the existence of more than one relation between the variables. Frisch’s exposition of bunch maps was based on a situation where each variable in a set could be split into two components: one, the systematic component, was connected with the other variables; the other, the disturbance, was not so connected. The method was used to try to determine sets of variables in which one, and only one, exact linear relation held between the systematic components of the variables. Examples of the use of the method were given for constructed data where exact relations did exist. It is less clear whether they were assumed to exist in examples of applications to actual economic data. The other major applications of bunch maps were in Richard Stone’s work on consumers’ expenditure (Stone, 1945, 1954), but he did not consider an assumption of exact linear relations between systematic components as satisfactory.

When a particular event is observed, such as an economic variable taking a value in some region of the set of all possible values, it is natural to ask why that event occurred rather than some other. If, just earlier, some other event was observed to occur, it is also natural to ask if the joint observation of the two events indicates a relationship and possibly one that could be called an influence of one event by another, or even a causation. For a unique, or very rare event, such as the start of a world war, it will be very difficult to present more than sensible and suggestive statistical evidence about causation. However, in economics, values for many variables are observed with great regularity, such as daily stock market prices or monthly production figures and so a generating mechanism can be postulated that produces these values and the investigation and understanding of this mechanism is obviously one of the main tasks for the economist. In such studies, ideas such as theories, laws and causation arise very naturally, and economists in their workings use such words very frequently. It is unfortunately true that not all writers give the same meanings to these words. The understanding of causality is not the same for all economists, but this is hardly surprising as statisticians and philosophers are also not in agreement among themselves.

Causal notions arise when we week to understand the workings of a complex system by analysing it into component subsystems and mechanisms. Thus, if we wish to understand the quantities of strawberries that are produced and consumed and the prices at which they are exchanged, we may consider a number of mechanisms that affect quantity and price. What mechanisms we will include depends on how widely we draw the boundaries of the system to be examined.

The censored normal regression model considered by Tobin (1958), also commonly known as the ‘tobit’ model, is the following: <math display='block'> <mrow> <msubsup> <mi>y</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>=</mo><mi>&#x03B2;</mi><msub> <mi>x</mi> <mi>i</mi> </msub> <mo>+</mo><msub> <mi>u</mi> <mi>i</mi> </msub> <mo>.</mo><mtext>&#x2003;</mtext><msub> <mi>u</mi> <mi>i</mi> </msub> <mo>&#x223C;</mo><mi>I</mi><mi>N</mi><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><msup> <mi>&#x03C3;</mi> <mn>2</mn> </msup> <mo stretchy='false'>)</mo> </mrow> </math>$$y_i^* = \beta {x_i} + {u_i}.\quad {u_i} \sim IN(0,{\sigma ^2})$$ The observed y _i are related to y _i * according to the relationship (1)<math display='block'> <mtable columnalign='left'> <mtr> <mtd> <msub> <mi>y</mi> <mi>i</mi> </msub> <mo>=</mo><msubsup> <mi>y</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mtext>&#x2003;</mtext><mi>i</mi><mi>f</mi><mtext>&#x2009;</mtext><msubsup> <mi>y</mi> <mi>i</mi> <mo>*</mo> </msubsup> <mo>&#x003E;</mo><msub> <mi>y</mi> <mn>0</mn> </msub> </mtd> </mtr> <mtr> <mtd> <mo>=</mo><msub> <mi>y</mi> <mn>0</mn> </msub> <mtext>&#x2003;</mtext><mtext>&#x2009;</mtext><mtext>&#x2003;</mtext><mi>o</mi><mi>t</mi><mi>h</mi><mi>e</mi><mi>r</mi><mi>w</mi><mi>i</mi><mi>s</mi><mi>e</mi> </mtd> </mtr> </mtable> </math>$$\eqalign{ & {y_i} = y_i^*\quad if\,y_i^* > {y_0} \cr & = {y_0}\quad \;\quad otherwise \cr} $$ where y₀ is a prespecified constant (usually zero). The y _i * could take values <y₀. The only thing is that they are not observed. Thus, y _i is set equal to y₀ because of non-observability. The values x _i are observed for all the observations. If both y _i and x _i are unobserved for y _i ⩽ y₀ then we have what is known as a truncated regression model.

These are those statistical models which specify the probability distribution of discrete dependent variables as a function of independent variables and unknown parameters. They are sometimes called qualitative response models, and are relevant in economics because the decision of an economic unit frequently involves discrete choice: for example, the decision regarding whether a person joins the labour force or not, the decision as to the number of cars to own, the choice of occupation, the choice of the mode of transportation, etc.

In economics, as well as in other disciplines, qualitative factors often play an important role. For instance, the achievement of a student in school may be determined, among other factors, by his father’s profession, which is a qualitative variable having as many attributes (characteristics) as there are professions. In medicine, to take another example, the response of a patient to a drug may be influenced by the patient’s sex and the patient’s smoking habits, which may be represented by two qualitative variables, each one having two attributes. The dummy-variable method is a simple and useful device for introducing, into a regression analysis, information contained in qualitative or categorical variables; that is, in variables that are not conventionally measured on a numerical scale. Such qualitative variables may include race, sex, marital status, occupation, level of education, region, seasonal effects and so on. In some applications, the dummy-variable procedure may also be fruitfully applied to a quantitative variable such as age, the influence of which is frequently U-shaped. A system of dummy variables defined by age classes conforms to any curvature and consequently may lead to more significant results.

Endogeneity and exogeneity are properties of variables in economic or econometric models. The specification of these properties for respective variables is an essential component of the entire process of model specification. The words have an ambiguous meaning, for they have been applied in closely related but conceptually distinct ways, particularly in the specification of stochastic models. We consider in turn the case of deterministic and stochastic models, concentrating mainly on the latter.

This essay surveys the history and recent developments on economic models with errors in variables. These errors may arise from the use of substantive unobservables, such as permanent income, or from ordinary measurement problems in data collection and processing. The point of departure is the classical regression equation with random errors in variables: <math display='block'> <mrow> <mi>y</mi><mo>=</mo><msup> <mi>X</mi> <mo>*</mo> </msup> <mi>&#x03B2;</mi><mo>+</mo><mi>u</mi><mo>,</mo> </mrow> </math>$$y = {X^*}\beta + u,$$ where y is a n × 1 vector of observations on the dependent variable, X* is a n × k matrix of unobserved (latent) values on the k independent variables, ß is a k × 1 vector of unknown coefficients, and u is a n × 1 vector of random disturbances. The matrix of observed values on X* is <math display='block'> <mrow> <mi>X</mi><mo>=</mo><msup> <mi>X</mi> <mo>*</mo> </msup> <mo>+</mo><mi>V</mi><mo>,</mo> </mrow> </math>$$X = {X^*} + V,$$ where V is the n × k matrix of measurement errors. If some variables are measured without error, the appropriate columns of V are zero vectors. In the conventional case the errors are uncorrelated in the limit with the latent values X* and the disturbances u; and the errors have zero means, constant variances, and zero autocorrelation. In observed variables the model becomes <math display='block'> <mrow> <mi>y</mi><mo>=</mo><mi>X</mi><mi>&#x03B2;</mi><mo>+</mo><mo stretchy='false'>(</mo><mi>u</mi><mo>&#x2212;</mo><mi>V</mi><mi>&#x03B2;</mi><mo stretchy='false'>)</mo><mo>.</mo> </mrow> </math>$$y = X\beta + (u - V\beta ).$$

Econometricians have developed a number of alternative methods for estimating parameters and testing hypotheses in simultaneous equations models. Some of these are limited information methods that can be applied one equation at a time and require only minimal specification of the other equations in the system. In contrast, the full information methods treat the system as a whole and require a complete specification of all the equations.

In economic analysis we often assume that there exists an underlying structure which generated the observations of real-world data. However, statistical inference can relate only to characteristics of the distribution of the observed variables. A meaningful statistical interpretation of the real world through this structure can be achieved only if there is no other structure which is also capable of generating the observed data.

Information theory is a branch of mathematical statistics and probability theory. Thus, it can and has been applied in many fields, including economics, that rely on statistical analysis. As we are concerned with it, the technical concept of ‘information’ must be distinguished from the semantic concept in common parlance. The simplest and still the most widely used technical definitions of information were first introduced (independently) by Shannon and Wiener in 1948 in connection with communication theory. Though decisively and directly related, these definitions must also be distinguished from the definition of ‘information’ introduced by R.A. Fisher in 1925 for estimation theory.

In one of its simplest formulations the problem of estimating the parameters of a system of simultaneous equations with unknown random errors reduces to finding a way of estimating the parameters of a single linear equation of the form Y = Xβ₀+ ε where β_o is unknown, Y and X are vectors of data on relevant economic variables and ε is the vector of unknown random errors. The most common method of estimating β₀ is the method of least squares: <math display='block'> <mrow> <msub> <mover accent='true'> <mi>&#x03B2;</mi> <mo>&#x005E;</mo> </mover> <mrow> <mi>O</mi><mi>L</mi><mi>S</mi> </mrow> </msub> <mo>&#x2261;</mo> </mrow> </math>$${\hat \beta _{OLS}} \equiv $$ argmin ε(β)′ε(β), where ε(β) ≡ Y - Xβ. Under fairly general assumptions <math display='block'> <mrow> <msub> <mover accent='true'> <mi>&#x03B2;</mi> <mo>&#x005E;</mo> </mover> <mrow> <mi>O</mi><mi>L</mi><mi>S</mi> </mrow> </msub> </mrow> </math>$${\hat \beta _{OLS}}$$ is an unbiased estimator of β₀ provided E(ε_t|X) = 0 for all t, where ε_t is the tth-coordinate of ε.

A cursory reading of recent textbooks on econometrics shows that historically the emphasis in our discipline has been placed on models that are without measurement error in the variables but instead have stochastic ‘shocks’ in the equations. To the extent that the topic of errors of measurement in variables (or latent variables) is treated, one will usually find that for a classical single-equation regression model, measurement error in the dependent variable, y, causes no particular problem because it can be subsumed within the equation’s disturbance term. But when it comes to the matter of measurement errors in the independent variables, the argument will usually be made that consistent parameter estimation is unobtainable unless repeated observations on y are available at each data point, or strong a priori information can be employed. The presentation usually ends there, leaving us with the impression that the errors-in-variables ‘problem’ is bad enough in the classical regression model and surely must be worse in more complicated models.

The notion that an economic variable leads or lags another variable is an intuitive and simple notion. Nevertheless, it has proven difficult to go from this intuitive notion to a precise, empirically testable, definition.

The term limited dependent variable was first used by Tobin (1958) to denote the dependent variable in a regression equation that is constrained to be non-negative. In Tobin’s study the dependent variable is the household’s monetary expenditure on a durable good, which of course must be non-negative. Many other economic variables are non-negative. However, non-negativity alone does not invalidate standard linear regression analysis. It is the presence of many observations at zero which causes bias to the least squares estimator and requires special analysis. For example, Tobin’s data contain many households for which the expenditure on a durable good in a given year is zero.

Two convenient classifications for variables which are not amenable to treatment by the principal tool of econometrics, regression analysis, are quantal responses and limited responses. In the quantal response (all or nothing) category are dichotomous, qualitative and categorical outcomes, and the methods of analysis identified as probit and logit are appropriate for these variables. Illustrative applications include decisions to own or rent, choice of travel mode, and choice of professions. The limited response category covers variables which take on mixtures of discrete and continuous outcomes, and the prototypical model and analysis technique is identified as tobit. Examples are samples with both zero and positive expenditures on durable goods, and models of markets with price ceilings including data with both limit and non-limit prices. While the tobit model evolved out of the probit model and the limited and quantal response methods share many properties and characteristics, they are sufficiently different to make separate treatment more convenient.

If there is a number, θ, such that Y = log_e(X - θ) is normally distributed, the distribution of X is lognormal. The important special case of θ = 0 gives the two-parameter lognormal distribution, X ~ Λ(μ,σ2) with Y ~ N(μ,σ2) where μ and σ2 denote the mean and variance of log_e X. The classic work on the subject is by Aitchison and Brown (1957). A useful survey is provided by Johnson and Kotz (1970). They also summarize the history of this distribution: the pioneer contributions by Galton (1879) on its genesis, and by McAlister (1879) on its measures of location and dispersion, were followed by Kapteyn (1903), who studied its genesis in more detail and also devised an analogue machine to generate it. Gibrat’s (1931) study of economic size distributions was a most important development and his law of proportionate effect is given in equation (1) below.

The topic ‘macroeconometric models’ is very broad. Conceivably it could include any study in which at least one equation was estimated using macroeconomic data. I will limit my discussion to structural models that try to explain the overall economy, although I will also have a few things to say about vector autoregressive models.

Exact multicollinearity means that there is at least one exact linear relation between the column vectors of the n × k data matrix of n observations on k variables. More commonly, multicollinearity means that the variables are so intercorrelated in the data that the relations are ‘almost exact’. The term was used by Frisch (1934) mainly in the context of attempts to estimate an exact relation between the systematic components of variables whose observed values contained disturbances or errors of measurement but where there might also be other relations between the systematic components which made estimates dangerous or even meaningless. In more recent work the data matrix has usually been the matrix X of regressor values in the linear regression model Y = X ß + ε with no measurement errors. Confusion between the two cases led at one time to some misunderstanding in the literature. Other terms used for the same phenomenon are collinearity and ill-conditioned data — although the latter may contain aspects of the scaling of variables which are irrelevant to multicollinearity.

Economic theory guides empirical research primarily by suggesting which variables ought to enter a relationship. But as to the functional form that this relationship ought to take, it only gives general information such as stating that certain first and second partial derivatives of a relationship must be positive or such as ruling out certain functional forms. In some applications, notably consumer demand systems, the theory rules out models that are linear in the parameters such as <m:math display='block'> <m:mrow> <m:mi>y</m:mi><m:mo>=</m:mo><m:mstyle displaystyle='true'> <m:mo>&#x2211;</m:mo> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mstyle><m:msub> <m:mi>&#x03B2;</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>+</m:mo><m:mi>e</m:mi></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$y = \sum {{x_i}} {\beta _i} + e$$ and thus provides a natural impetus to the development of statistical methods for models that are non-linear in the parameters such as <m:math display='block'> <m:mrow> <m:mi>y</m:mi><m:mo>=</m:mo><m:mo stretchy='false'>(</m:mo><m:mstyle displaystyle='true'> <m:mo>&#x2211;</m:mo> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:mstyle><m:msub> <m:mi>&#x03B2;</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy='false'>)</m:mo><m:mo>/</m:mo><m:mo stretchy='false'>(</m:mo><m:mstyle displaystyle='true'> <m:mo>&#x2211;</m:mo> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mi>i</m:mi> </m:msub> <m:msub> <m:mi>y</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>&#x2212;</m:mo><m:mn>1</m:mn></m:mrow> </m:mstyle><m:mo stretchy='false'>)</m:mo><m:mo>+</m:mo><m:mi>e</m:mi></m:mrow> </m:math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$y = (\sum {{x_i}} {\beta _i})/(\sum {{x_i}{y_i} - 1} ) + e$$

In recent years considerable attention has been paid to the role of hypothesis testing in econometrics and its links with the problem of model selection in econometrics. One important topic considered in this recent work is the testing of ‘non-nested’ or separate models. Broadly speaking, two models (or hypotheses) are said to be ‘non-nested’ if neither can be obtained from the other by the imposition of appropriate parametric restrictions or as a limit of a suitable approximation; otherwise they are said to be ‘nested’. (A more formal definition can be found in Pesaran, 1987.) Non-nested models can arise from differences in the underlying theoretical paradigms and/or from differences in the way a particular relationship suggested by economic theory is modelled. Examples of non-nested econometric models abound in the literature: demand systems (Deaton, 1978; Murray, 1984), Keynesian and new classical models of unemployment (Pesaran, 1982a; Dadkhah and Valbuena, 1985), effects of dividend taxes on corporate investment decisions (Poterba and Summers, 1983), money demand functions (McAleer, Fisher and Volker, 1982) and empirical models of exchange rate determination (Backus, 1984), to mention just a few. Other examples of non-nested hypotheses arise when the probability distributions under consideration belong to separate parametric families, such as log-normal versus exponential, or Poisson versus geometric distributions.

Path analysis is a method for estimating and testing the internal consistency of models with a postulated causal structure. The postulated structure is displayed in the form of path diagrams, where one-way arrows link causal variables to their outcomes, and curved two-headed arrows connect related variables whose causal links are not under study. Estimation proceeds along the lines of method of moments and instrumental variables theory: the causal ordering of variables along distinct paths are exploited to express the unknown structural parameters in terms of the population moments of the observed and the unobserved variables. Estimating equations are obtained by replacing the population moments of the observed variables by their sample counterparts, which are then solved for the unknown parameters and the estimates of the moments of the unobservables (which themselves can be thought of as structural parameters).

Random coefficients models generalize conventional fixed coefficients models to avoid inconsistent and inaccurate assessments of relationships among variables.

It has long been recognized that forecasts affect outcomes. Similarly, outcomes affect expectations. Thus, there is a mapping from expectations to outcomes and back to expectations and so from expectations to expectations. A rational expectations equilibrium is a fixed point of this mapping in which expectations generate outcomes which confirm the original expectations. A rational expectations equilibrium is a natural solution concept in a model with expectations. The heuristic reasoning is that outside rational expectations equilibria agents make systematic mistakes; expectations are not confirmed by outcomes in that the expectations are not correct on the average. Consequently, it is very plausible that outside rational expectations equilibria, agents will eventually notice that they are making systematic mistakes and attempt to revise the way they forecast in order to eliminate the sources of the systematic errors. This suggests that agents are not in equilibrium until they have learned to form rational expectations.

When observations are taken at regular intervals within a year (by month or by quarter), most economic time series are likely to exhibit some degree of seasonal variation. An obvious example, known to everyone, is the existence of a ‘high’ and ‘low’ season for air transportation and other recreational activities. Perhaps less obvious, but equally important, is the presence of a seasonal pattern in most economic aggregates such as the index of production, price indices, the unemployment rate and so on.

The problem of selection bias in economic and social statistics arises when a rule other than simple random sampling is used to sample the underlying population that is the object of interest. The distorted representation of a true population as a consequence of a sampling rule is the essence of the selection problem. Distorting selection rules may be the outcome of decisions of sample survey statisticians, self-selection decisions by the agents being studied or both.

To clarify the concept of simulation requires placing simulation models in the context of available modelling approaches. One can distinguish models by their structure, size, complexity, purpose, solution technique and probabilistic specification. Generally, the purpose for which a model is specified, the state of knowledge in the area, and the relative importance of indirect effects should guide model specification (Robinson, 1987).

Models that attempt to explain the workings of the economy typically are written as interdependent systems of equations describing some hypothesized technological and behavioural relationships among economic variables. Supply and demand models, Walrasian general equilibrium models, and Keynesian macromodels are common examples. A large part of econometrics is concerned with specifying, testing, and estimating the parameters of such systems. Despite their common use, simultaneous equations models still generate controversy. In practice there is often considerable disagreement over their proper use and interpretation.

A lengthy list of implicit and explicit assumptions is required to draw inferences from a data set. Substantial doubt about these assumptions is a characteristic of the analysis of non-experimental data and much experimental data as well. If this doubt is left unattended, it can cause serious doubt about the corresponding inferences.

If a theory suggests that there is a linear relationship between a pair of random variables X and Y, then an obvious way to test the theory is to estimate a regression equation of form <mrow> <mi>Y</mi><mo>=</mo><mi>&#x03B1;</mi><mo>+</mo><mi>&#x03B2;</mi><mi>X</mi><mo>+</mo><mi>e</mi><mo>.</mo></mrow> </math>]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$Y = \alpha + \beta X + e.$$ Estimation could be by least-squares and the standard diagnostic statistics would be a t-statistic on β, the R2 value and possibly the Durbin-Watson statistic d. With such a procedure there is always the possibility of a type ii error, that is accepting the relationship as significant when, in fact, X and Y are uncorrelated. This possibility increases if the error term e is autocorrelated, as first pointed out by Yule (1926). As the autocorrelation structure of e is the same as that for Y, when the true β = 0, this problem of ‘nonsense correlations’ or ‘spurious regressions’ is most likely to occur when testing relationships between highly autocorrelated series.

Getting facts, expectations, reasons or attitudes by interviewing people has a long history, but scientific survey research required three innovations which only came in the 20th century — scientific probability sampling, controlled question stimuli and answer categorization, and multivariate analysis of the rich resulting data sets. Textbooks abound (Moser and Kalton, 1971; Lansing and Morgan, 1971; Sonquist and Dunkelberg, 1977; Rossi et al., 1983).

Transformations of many kinds are used in statistical method and theory including simple changes of unit of measurement to facilitate computation or understanding, and the linear transformations underlying the application and theory of multiple regression and the techniques of classical multivariate analysis. Nevertheless the word transformation in a statistical context normally brings to mind a non-linear transformation (to logs, square roots, etc.) of basic observations done with the objective of simplifying analysis and interpretation. This essay focuses on that aspect.

Economic theory usually fails to describe the functional relationship between variables (the CES production function being an exception). In econometrics, implications of simplistic choice of functional form include the danger of misspecification and its attendant biases in assessing magnitudes of effects and statistical significance of results, It is safe to say that when functional form is specified in a restrictive manner a priori before estimation, most empirical results that have been debated in the professional literature would have had a modified, even opposite, conclusion if the functional relationship had not been restrictive (see Zarembka, 1968, p. 509, for an illustration; also, Spitzer, 1976).

Two-stage least squares (TSLS) is a method of estimating the parameters of a single structural equation in a system of linear simultaneous equations. The TSLS estimator was proposed by Theil (1953a, 1961) and independently by Basmann (1957). The early work on simultaneous equation estimation was carried out by a group of econometricians at the Cowles Commission. This work was based on the method of maximum likelihood. Anderson and Rubin (1949) proposed the limited information maximum likelihood (LIML) estimator for the parameters of a single structural equation.

Brownian motion is the most renowned, and historically the first stochastic process that was thoroughly investigated. It is named after the English botanist, Robert Brown who in 1827 observed that small particles immersed in a liquid exhibited ceaseless irregular motion. Brown himself mentions several precursors starting at the beginning with Leeuwenhoek (1632–1723). In 1905 Einstein, unaware of the existence of earlier investigations about Brownian motion, obtained a mathematical derivation of this process from the laws of physics. The theory of Brownian motion was further developed by several distinguished mathematical physicists until Norbert Wiener gave it a rigorous mathematical formulation in his 1918 dissertation and in later papers. This is why the Brownian motion is also called the Wiener process. For a brief history of the scientific developments of the process see Nelson (1967).