Econometrics

Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships.[1] More precisely, it is "the quantitative analysis of actual economic phenomena based on the concurrent development of theory and observation, related by appropriate methods of inference".[2] An introductory economics textbook describes econometrics as allowing economists "to sift through mountains of data to extract simple relationships".[3] The first known use of the term "econometrics" (in cognate form) was by Polish economist Paweł Ciompa in 1910.[4] Jan Tinbergen is considered by many to be one of the founding fathers of econometrics.[5][6][7] Ragnar Frisch is credited with coining the term in the sense in which it is used today.[8]

A basic tool for econometrics is the multiple linear regression model.[9] Econometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods.[10][11] Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. Applied econometrics uses theoretical econometrics and real-world data for assessing economic theories, developing econometric models, analysing economic history, and forecasting.

Basic models: linear regression

A basic tool for econometrics is the multiple linear regression model.[9] In modern econometrics, other statistical tools are frequently used, but linear regression is still the most frequently used starting point for an analysis.[9] Estimating a linear regression on two variables can be visualised as fitting a line through data points representing paired values of the independent and dependent variables.

Okuns law differences 1948 to mid 2011
Okun's law representing the relationship between GDP growth and the unemployment rate. The fitted line is found using regression analysis.

For example, consider Okun's law, which relates GDP growth to the unemployment rate. This relationship is represented in a linear regression where the change in unemployment rate () is a function of an intercept (), a given value of GDP growth multiplied by a slope coefficient and an error term, :

The unknown parameters and can be estimated. Here is estimated to be −1.77 and is estimated to be 0.83. This means that if GDP growth increased by one percentage point, the unemployment rate would be predicted to drop by 1.77 points. The model could then be tested for statistical significance as to whether an increase in growth is associated with a decrease in the unemployment, as hypothesized. If the estimate of were not significantly different from 0, the test would fail to find evidence that changes in the growth rate and unemployment rate were related. The variance in a prediction of the dependent variable (unemployment) as a function of the independent variable (GDP growth) is given in polynomial least squares.

Theory

Econometric theory uses statistical theory and mathematical statistics to evaluate and develop econometric methods.[10][11] Econometricians try to find estimators that have desirable statistical properties including unbiasedness, efficiency, and consistency. An estimator is unbiased if its expected value is the true value of the parameter; it is consistent if it converges to the true value as the sample size gets larger, and it is efficient if the estimator has lower standard error than other unbiased estimators for a given sample size. Ordinary least squares (OLS) is often used for estimation since it provides the BLUE or "best linear unbiased estimator" (where "best" means most efficient, unbiased estimator) given the Gauss-Markov assumptions. When these assumptions are violated or other statistical properties are desired, other estimation techniques such as maximum likelihood estimation, generalized method of moments, or generalized least squares are used. Estimators that incorporate prior beliefs are advocated by those who favour Bayesian statistics over traditional, classical or "frequentist" approaches.

Methods

Applied econometrics uses theoretical econometrics and real-world data for assessing economic theories, developing econometric models, analysing economic history, and forecasting.[12]

Econometrics may use standard statistical models to study economic questions, but most often they are with observational data, rather than in controlled experiments.[13] In this, the design of observational studies in econometrics is similar to the design of studies in other observational disciplines, such as astronomy, epidemiology, sociology and political science. Analysis of data from an observational study is guided by the study protocol, although exploratory data analysis may be useful for generating new hypotheses.[14] Economics often analyses systems of equations and inequalities, such as supply and demand hypothesized to be in equilibrium. Consequently, the field of econometrics has developed methods for identification and estimation of simultaneous-equation models. These methods are analogous to methods used in other areas of science, such as the field of system identification in systems analysis and control theory. Such methods may allow researchers to estimate models and investigate their empirical consequences, without directly manipulating the system.

One of the fundamental statistical methods used by econometricians is regression analysis.[15] Regression methods are important in econometrics because economists typically cannot use controlled experiments. Econometricians often seek illuminating natural experiments in the absence of evidence from controlled experiments. Observational data may be subject to omitted-variable bias and a list of other problems that must be addressed using causal analysis of simultaneous-equation models.[16]

In addition to natural experiments, quasi-experimental methods have been used increasingly commonly by econometricians since the 1980s, in order to credibly identify causal effects.[17]

Example

A simple example of a relationship in econometrics from the field of labour economics is:

This example assumes that the natural logarithm of a person's wage is a linear function of the number of years of education that person has acquired. The parameter measures the increase in the natural log of the wage attributable to one more year of education. The term is a random variable representing all other factors that may have direct influence on wage. The econometric goal is to estimate the parameters, under specific assumptions about the random variable . For example, if is uncorrelated with years of education, then the equation can be estimated with ordinary least squares.

If the researcher could randomly assign people to different levels of education, the data set thus generated would allow estimation of the effect of changes in years of education on wages. In reality, those experiments cannot be conducted. Instead, the econometrician observes the years of education of and the wages paid to people who differ along many dimensions. Given this kind of data, the estimated coefficient on Years of Education in the equation above reflects both the effect of education on wages and the effect of other variables on wages, if those other variables were correlated with education. For example, people born in certain places may have higher wages and higher levels of education. Unless the econometrician controls for place of birth in the above equation, the effect of birthplace on wages may be falsely attributed to the effect of education on wages.

The most obvious way to control for birthplace is to include a measure of the effect of birthplace in the equation above. Exclusion of birthplace, together with the assumption that is uncorrelated with education produces a misspecified model. Another technique is to include in the equation additional set of measured covariates which are not instrumental variables, yet render identifiable.[18] An overview of econometric methods used to study this problem were provided by Card (1999).[19]

Journals

The main journals that publish work in econometrics are Econometrica, the Journal of Econometrics, the Review of Economics and Statistics, Econometric Theory, the Journal of Applied Econometrics, Econometric Reviews, the Econometrics Journal,[20] Applied Econometrics and International Development, and the Journal of Business & Economic Statistics.

Limitations and criticisms

Like other forms of statistical analysis, badly specified econometric models may show a spurious relationship where two variables are correlated but causally unrelated. In a study of the use of econometrics in major economics journals, McCloskey concluded that some economists report p-values (following the Fisherian tradition of tests of significance of point null-hypotheses) and neglect concerns of type II errors; some economists fail to report estimates of the size of effects (apart from statistical significance) and to discuss their economic importance. She also argues that some economists also fail to use economic reasoning for model selection, especially for deciding which variables to include in a regression.[21][22]

In some cases, economic variables cannot be experimentally manipulated as treatments randomly assigned to subjects.[23] In such cases, economists rely on observational studies, often using data sets with many strongly associated covariates, resulting in enormous numbers of models with similar explanatory ability but different covariates and regression estimates. Regarding the plurality of models compatible with observational data-sets, Edward Leamer urged that "professionals ... properly withhold belief until an inference can be shown to be adequately insensitive to the choice of assumptions".[23]

See also

Notes

  1. ^ M. Hashem Pesaran (1987). "Econometrics," The New Palgrave: A Dictionary of Economics, v. 2, p. 8 [pp. 8–22]. Reprinted in J. Eatwell et al., eds. (1990). Econometrics: The New Palgrave, p. 1 [pp. 1–34]. Abstract Archived 18 May 2012 at the Wayback Machine (2008 revision by J. Geweke, J. Horowitz, and H. P. Pesaran).
  2. ^ P. A. Samuelson, T. C. Koopmans, and J. R. N. Stone (1954). "Report of the Evaluative Committee for Econometrica," Econometrica 22(2), p. 142. [p p. 141-146], as described and cited in Pesaran (1987) above.
  3. ^ Paul A. Samuelson and William D. Nordhaus, 2004. Economics. 18th ed., McGraw-Hill, p. 5.
  4. ^ "Archived copy". Archived from the original on 2 May 2014. Retrieved 1 May 2014.CS1 maint: Archived copy as title (link)
  5. ^ "1969 - Jan Tinbergen: Nobelprijs economie - Elsevierweekblad.nl". elsevierweekblad.nl. 12 October 2015. Archived from the original on 1 May 2018. Retrieved 1 May 2018.
  6. ^ Magnus, Jan & Mary S. Morgan (1987) The ET Interview: Professor J. Tinbergen in: 'Econometric Theory 3, 1987, 117–142.
  7. ^ Willlekens, Frans (2008) International Migration in Europe: Data, Models and Estimates. New Jersey. John Wiley & Sons: 117.
  8. ^ • H. P. Pesaran (1990), "Econometrics," Econometrics: The New Palgrave, p. 2, citing Ragnar Frisch (1936), "A Note on the Term 'Econometrics'," Econometrica, 4(1), p. 95.
       • Aris Spanos (2008), "statistics and economics," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. Archived 18 May 2012 at the Wayback Machine
  9. ^ a b c Greene, William (2012). "Chapter 1: Econometrics". Econometric Analysis (7th ed.). Pearson Education. pp. 47–48. ISBN 9780273753568. Ultimately, all of these will require a common set of tools, including, for example, the multiple regression model, the use of moment conditions for estimation, instrumental variables (IV) and maximum likelihood estimation. With that in mind, the organization of this book is as follows: The first half of the text develops fundamental results that are common to all the applications. The concept of multiple regression and the linear regression model in particular constitutes the underlying platform of most modeling, even if the linear model itself is not ultimately used as the empirical specification.
  10. ^ a b Greene, William (2012). Econometric Analysis (7th ed.). Pearson Education. pp. 34, 41–42. ISBN 9780273753568.
  11. ^ a b Wooldridge, Jeffrey (2012). "Chapter 1: The Nature of Econometrics and Economic Data". Introductory Econometrics: A Modern Approach (5th ed.). South-Western Cengage Learning. p. 2. ISBN 9781111531041.
  12. ^ Clive Granger (2008). "forecasting," The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. Archived 18 May 2012 at the Wayback Machine
  13. ^ Wooldridge, Jeffrey (2013). Introductory Econometrics, A modern approach. South-Western, Cengage learning. ISBN 978-1-111-53104-1.
  14. ^ Herman O. Wold (1969). "Econometrics as Pioneering in Nonexperimental Model Building," Econometrica, 37(3), pp. 369-381.
  15. ^ For an overview of a linear implementation of this framework, see linear regression.
  16. ^ Edward E. Leamer (2008). "specification problems in econometrics," The New Palgrave Dictionary of Economics. Abstract. Archived 23 September 2015 at the Wayback Machine
  17. ^ Angrist, Joshua D; Pischke, Jörn-Steffen (May 2010). "The Credibility Revolution in Empirical Economics: How Better Research Design is Taking the Con out of Econometrics". Journal of Economic Perspectives. 24 (2): 3–30. doi:10.1257/jep.24.2.3. ISSN 0895-3309.
  18. ^ Pearl, Judea (2000). Causality: Model, Reasoning, and Inference. Cambridge University Press. ISBN 978-0521773621.
  19. ^ Card, David (1999). "The Causal Effect of Education on Earning". In Ashenfelter, O.; Card, D. (eds.). Handbook of Labor Economics. Amsterdam: Elsevier. pp. 1801–1863. ISBN 978-0444822895.
  20. ^ "The Econometrics Journal – Wiley Online Library". Wiley.com. Retrieved 8 October 2013.
  21. ^ McCloskey (May 1985). "The Loss Function has been mislaid: the Rhetoric of Significance Tests". American Economic Review. 75 (2).
  22. ^ Stephen T. Ziliak and Deirdre N. McCloskey (2004). "Size Matters: The Standard Error of Regressions in the American Economic Review," Journal of Socio-economics, 33(5), pp. 527-46 Archived 25 June 2010 at the Wayback Machine (press +).
  23. ^ a b Leamer, Edward (March 1983). "Let's Take the Con out of Econometrics". American Economic Review. 73 (1): 31–43. JSTOR 1803924.

Further reading

External links

Applied economics

Applied economics is the application of economic theory and econometrics in specific settings. As one of the two sets of fields of economics (the other set being the core), it is typically characterized by the application of the core, i.e. economic theory and econometrics, to address practical issues in a range of fields including demographic economics, labour economics, business economics, industrial organization, agricultural economics, development economics, education economics, engineering economics, financial economics, health economics, monetary economics, public economics, and economic history. The process often involves a reduction in the level of abstraction of this core theory. There are a variety of approaches including not only empirical estimation using econometrics, input-output analysis or simulations but also case studies, historical analogy and so-called common sense or the "vernacular". This range of approaches is indicative of what Roger Backhouse and Jeff Biddle argue is the ambiguous nature of the concept of applied economics. It is a concept with multiple meanings. Among broad methodological distinctions, one source places it in neither positive nor normative economics but the art of economics, glossed as "what most economists do".

Computational economics

Computational economics is a research discipline at the interface of computer science, economics, and management science. This subject encompasses computational modeling of economic systems, whether agent-based, general-equilibrium, macroeconomic, or rational-expectations, computational econometrics and statistics, computational finance, computational tools for the design

of automated internet markets, programming tool specifically designed for computational economics and the teaching of computational economics. Some of these areas are unique, while others extend traditional areas of economics by solving problems that are tedious to study without computers and associated numerical methods.Computational economics uses computer-based economic modelling for the solution of analytically and statistically- formulated economic problems. A research program, to that end, is agent-based computational economics (ACE), the computational study of economic processes, including whole economies, as dynamic systems of interacting agents. As such, it is an economic adaptation of the complex adaptive systems paradigm. Here the "agent" refers to "computational objects modeled as interacting according to rules," not real people. Agents can represent social, biological, and/or physical entities. The theoretical assumption of mathematical optimisation by agents in equilibrium is replaced by the less restrictive postulate of agents with bounded rationality adapting to market forces, including game-theoretical contexts. Starting from initial conditions determined by the modeler, an ACE model develops forward through time driven solely by agent interactions. The scientific objective of the method is "to ... test theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each research building on the work before."Computational solution tools include for example software for carrying out various matrix operations (e.g. matrix inversion) and for solving systems of linear and non-linear equations. For a repository of public-domain computational solutions, visit here.

The following journals specialise in computational economics: ACM Transactions on Economics and Computation, Computational Economics, Journal of Applied Econometrics, Journal of Economic Dynamics and Control and the Journal of Economic Interaction and Coordination.

David Forbes Hendry

Sir David Forbes Hendry, FBA CStat (born 6 March 1944) is a British econometrician, currently a professor of economics and from 2001–2007 was head of the Economics Department at the University of Oxford. He is also a professorial fellow at Nuffield College, Oxford.He obtained an M.A. in economics with first class honours from the University of Aberdeen in 1966. He then went to the London School of Economics and completed an MSc (with distinction) in Econometrics and Mathematical Economics in 1967.

He received his PhD from the London School of Economics under the supervision of John Denis Sargan in 1970, and until joining the University of Oxford as professor of economics in 1982, was a lecturer, then reader and finally professor of economics at the LSE. Hendry also served as a research professor at Duke University from 1987 until 1991.

His work is predominantly on time series econometrics and the econometrics of the demand for money. In recent years he has worked on the theory of forecasting and also on automated model building. He also studies the econometrics of climate change as co-director of the Climate Econometrics research centre at Nuffield College, Oxford.He was elected a fellow of the British Academy, a fellow of the Econometric Society, honorary member of the American Economic Association and foreign honorary member of the American Academy of Arts and Sciences.

In 2001 he received an honorary doctorate (dr. philos. h.c.) from The Norwegian University of Science and Technology (NTNU).He was knighted in the 2009 Birthday Honours.His most recent book is Hendry, D.F. and B. Nielsen (2007), Econometric Modeling: A Likelihood Approach (Princeton University Press).

"The Methodology and Practice of Econometrics: A Festschrift in Honour of David F. Hendry", edited by Jennifer L Castle and Neil Shephard, was published by Oxford University Press in 2009.

EViews

EViews (Econometric Views) is a statistical package for Windows, used mainly for time-series oriented econometric analysis. It is developed by Quantitative Micro Software (QMS), now a part of IHS. Version 1.0 was released in March 1994, and replaced MicroTSP. The TSP software and programming language had been originally developed by Robert Hall in 1965. The current version of EViews is 10, released in June 2017.

Econometric Society

The Econometric Society is an international society of academic economists interested in applying statistical tools to their field. It is an independent organization with no connections to societies of professional mathematicians or statisticians. It was founded on December 29, 1930, at the Stalton Hotel in Cleveland, Ohio. As of 2014, there are about 700 Elected Fellows of the Econometric Society, making it one of the most prevalent research affiliations.The sixteen founding members were Ragnar Frisch, Charles F. Roos, Joseph A. Schumpeter, Harold Hotelling, Henry Schultz, Karl Menger, Edwin B. Wilson, Frederick C. Mills, William F. Ogburn, J. Harvey Rogers, Malcolm C. Rorty, Carl Snyder, Walter A. Shewhart, Øystein Ore, Ingvar Wedervang and Norbert Wiener. The first president was Irving Fisher.The Econometric Society sponsors the Economics academic journal Econometrica and publishes the journals Theoretical Economics and Quantitative Economics.

Econometric Theory

Econometric Theory is an economics journal specialising in econometrics, published by Cambridge Journals. Its current editor is Peter Phillips.

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 Joel Sobel.

Econometrica was established in 1933. Its first editor was Ragnar Frisch, recipient of the first Nobel Memorial Prize in Economic Sciences in 1969, who served as an editor from 1933 to 1954. Although Econometrica is currently published entirely in English, the first few issues also contained scientific articles written in French.

The Econometric Society aims to attract high-quality applied work in economics for publication in Econometrica through the Frisch Medal. This prize is awarded every two years for an empirical or theoretical applied article published in Econometrica during the past five years.

Economic methodology

Economic methodology is the study of methods, especially the scientific method, in relation to economics, including principles underlying economic reasoning. In contemporary English, 'methodology' may reference theoretical or systematic aspects of a method (or several methods). Philosophy and economics also takes up methodology at the intersection of the two subjects.

Endogeneity (econometrics)

In econometrics, endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term. The distinction between endogenous and exogenous variables originated in simultaneous equations models, where one separates variables whose values are determined by the model from variables which are predetermined; ignoring simultaneity in the estimation leads to biased estimates as it violates the exogeneity assumption of the Gauss–Markov theorem. The problem of endogeneity is unfortunately, oftentimes ignored by researchers conducting non-experimental research and doing so precludes making policy recommendations. Instrumental variable techniques are commonly used to address this problem.

Besides simultaneity, correlation between explanatory variables and the error term can arise when an unobserved or omitted variable is confounding both independent and dependent variables, or when independent variables are measured with error.

Gretl

gretl is an open-source statistical package, mainly for econometrics. The name is an acronym for Gnu Regression, Econometrics and Time-series Library.

It has both a graphical user interface (GUI) and a command-line interface. It is written in C, uses GTK+ as widget toolkit for creating its GUI, and calls gnuplot for generating graphs. The native scripting language of gretl is known as hansl (see below); it can also be used together with TRAMO/SEATS, R, Stata, Python, Octave, Ox and Julia.

gretl can output models as LaTeX files.

Besides English, gretl is also available in Albanian, Basque, Bulgarian, Catalan, Chinese, Czech, French, Galician, German, Greek, Italian, Polish, Portuguese (both varieties), Romanian, Russian, Spanish, Turkish and Ukrainian.

Gretl has been reviewed several times in the Journal of Applied Econometrics and, more recently, in the Australian Economic Review.A review also appeared in the Journal of Statistical Software in 2008. Since then, the journal has featured several articles in which gretl is used to implement various statistical techniques.

Journal of Applied Econometrics

The Journal of Applied Econometrics is a peer-reviewed academic journal covering econometrics, published by John Wiley & Sons. It focuses on applications rather than theoretical issues. It was established in 1986 and is published seven times per year. Its editor-in-chief is Barbara Rossi. Since 1994 it has required its authors to deposit a complete set of data (provided they are non-confidential) into the journal's Data Archive, in order to enable the replication of empirical results published in the journal.

Journal of Econometrics

The Journal of Econometrics is a scholarly journal in econometrics. It was first published in 1973. Its current editors are A. Ronald Gallant, John Geweke, Cheng Hsiao, and Peter M. Robinson.

The journal publishes work dealing with estimation and other methodological aspects of the application of statistical inference to economic data, as well as papers dealing with the application of econometric techniques to economics.

The journal also publishes a supplement to the Journal of Econometrics which is called "Annals of Econometrics". Each issue of the Annals includes a collection of papers on a single topic selected by the editor of the issue.

List of statistical packages

Statistical software are specialized computer programs for analysis in statistics and econometrics.

RATS (software)

RATS, an abbreviation of Regression Analysis of Time Series, is a statistical package for time series analysis and econometrics. RATS is developed and sold by Estima, Inc., located in Evanston, IL.

SHAZAM (software)

SHAZAM is a comprehensive econometrics and statistics package for estimating, testing, simulating and forecasting many types of econometrics and statistical models. SHAZAM was originally created in 1977 by Kenneth White.

Simultaneous equations model

Simultaneous equation models are a type of statistical model in the form of a set of linear simultaneous equations. They are often used in econometrics. One can estimate these models equation by equation; however, estimation methods that exploit the system of equations, such as generalized method of moments (GMM) and instrumental variables estimation (IV) tend to be more efficient.

TSP (econometrics software)

TSP is a programming language for the estimation and simulation of econometric models. TSP stands for "Time Series Processor", although it is also commonly used with cross section and panel data. The program was initially developed by Robert Hall during his graduate studies at Massachusetts Institute of Technology in the 1960s. The company behind the program is TSP International which was founded in 1978 by Bronwyn H. Hall, Robert Hall's wife. After their divorce in April 1983, the asset of TSP was split into two versions, and subsequently the two versions have diverged in terms of interface and types of subroutines included. One version is TSP, still developed by TSP International. The other version, initially named MicroTSP, is now named EViews, developed by Quantitative Micro Software.

The Review of Economics and Statistics

The Review of Economics and Statistics is a peer-reviewed academic journal covering applied quantitative economics. It was established in 1919 as The Review of Economic Statistics and obtained its current name in 1948.The inaugural issue stated the purpose of the journal to be to:

promote the collection, criticism, and interpretation of economic statistics ... by investigation of the sources and probable accuracy of existing statistics ... and by developing the application to economic statistics of modern methods of statistical analysis which have hitherto been utilized more extensively in other sciences than in economics.The journal is edited at Harvard University's Kennedy School of Government and published by MIT Press. Its current editors-in-chief are Amitabh Chandra, Yuriy Gorodnichenko, Bryan S. Graham, Amit K. Khandelwal, Asim Ijaz Khwaja, and Brigitte C. Madrian.

Time series

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. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.

Time series are very frequently plotted via line charts. Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.

Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test theories that the current values of one or more independent time series affect the current value of another time series, this type of analysis of time series is not called "time series analysis", which focuses on comparing values of a single time series or multiple dependent time series at different points in time. Interrupted time series analysis is the analysis of interventions on a single time series.

Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as deriving in some way from past values, rather than from future values (see time reversibility.)

Time series analysis can be applied to real-valued, continuous data, discrete numeric data, or discrete symbolic data (i.e. sequences of characters, such as letters and words in the English language).

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