The mathematicization of economics
Over the course of the 20th century, articles in "core journals" in economics have been almost exclusively written by economists in Academia. As a result, much of the material transmitted in those journals relates to economic theory, and "economic theory itself has been continuously more abstract and mathematical." A subjective assessment of mathematical techniques employed in these core journals showed a decrease in articles that use neither geometric representations nor mathematical notation from 95% in 1892 to 5.3% in 1990A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 both lacked statistical analysis of data and lacked displayed mathematical expressions that were indexed with numbers at the margin of the page
There are complications even in this simple example, and it is often easy to mistake statistical significance with economic significance. Statistical significance is neither necessary nor sufficient for economic significance. In order to estimate the theoretical demand relationship, the observations in the data set must be price and quantity pairs that are collected along a demand schedule that is stable. If those assumptions are not satisfied, a more sophisticated model or econometric method may be necessary to derive reliable estimates and tests.
Data sets to which econometric analyses are applied can be classified as time-series data, cross-sectional data, panel data, and multidimensional panel data. Time-series data sets contain observations over time; for example, inflation over the course of several years. Cross-sectional data sets contain observations at a single point in time; for example, many individuals' incomes in a given year. Panel data sets contain both time-series and cross-sectional observations. Multi-dimensional panel data sets contain observations across time, cross-sectionally, and across some third dimension. For example, the Survey of Professional Forecasters contains forecasts for many forecasters (cross-sectional observations), at many points in time (time series observations), and at multiple forecast horizons (a third dimension).
Econometric analysis may also be classified on the basis of the number of relationships modeled. Single-equation methods model a single variable (the dependent variable) as a function of one or more explanatory (or independent) variables. In many econometric contexts, the commonly-used ordinary least squares method may not recover the theoretical relation desired or may produce estimates with poor statistical properties, because the assumptions for valid use of the method are violated. One widely-used remedy is the method of instrumental variables (IV). For an economic model described by more than one equation, simultaneous-equation methods may be used to remedy similar problems, including two IV variants, Two-Stage Least Squares (2SLS), and Three-Stage Least Squares (3SLS).
Other important unifying or distinguishing methods include the Method of Moments, Generalized Method of Moments (GMM), time series analysis, and Bayesian methods
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 ε is uncorrelated with education produces a misspecified model. A second technique for dealing with omitted variables is instrumental variables estimation. Still a third technique is to include in the equation additional set of measured covariates which are not instrumental variables, yet render β1 identifiable. An overview of econometric methods used to study this problem can be found in Card (1999)....
Over the course of the 20th century, articles in "core journals" in economics have been almost exclusively written by economists in Academia. As a result, much of the material transmitted in those journals relates to economic theory, and "economic theory itself has been continuously more abstract and mathematical." A subjective assessment of mathematical techniques employed in these core journals showed a decrease in articles that use neither geometric representations nor mathematical notation from 95% in 1892 to 5.3% in 1990A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 both lacked statistical analysis of data and lacked displayed mathematical expressions that were indexed with numbers at the margin of the page
Purpose
Two main purposes of econometrics are to give empirical content to economic theory by formulating economic models in testable form and to estimate those models and test them as to acceptance or rejection.
For example, consider one of the basic relationships in economics: the relationship between the price of a commodity and the quantities of that commodity that people wish to purchase at each price (the demand relationship). According to economic theory, an increase in the price would lead to a decrease in the quantity demanded, holding other relevant variables constant so as to isolate the relationship of interest. A mathematical equation can be written that describes the relationship between quantity, price, other demand variables like income, and a random term ε to reflect simplification and imprecision of the theoretical model:
There are complications even in this simple example, and it is often easy to mistake statistical significance with economic significance. Statistical significance is neither necessary nor sufficient for economic significance. In order to estimate the theoretical demand relationship, the observations in the data set must be price and quantity pairs that are collected along a demand schedule that is stable. If those assumptions are not satisfied, a more sophisticated model or econometric method may be necessary to derive reliable estimates and tests.
Methods
One of the fundamental statistical methods used by econometricians is regression analysis. For an overview of a linear implementation of this framework, see linear regression. 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.Data sets to which econometric analyses are applied can be classified as time-series data, cross-sectional data, panel data, and multidimensional panel data. Time-series data sets contain observations over time; for example, inflation over the course of several years. Cross-sectional data sets contain observations at a single point in time; for example, many individuals' incomes in a given year. Panel data sets contain both time-series and cross-sectional observations. Multi-dimensional panel data sets contain observations across time, cross-sectionally, and across some third dimension. For example, the Survey of Professional Forecasters contains forecasts for many forecasters (cross-sectional observations), at many points in time (time series observations), and at multiple forecast horizons (a third dimension).
Econometric analysis may also be classified on the basis of the number of relationships modeled. Single-equation methods model a single variable (the dependent variable) as a function of one or more explanatory (or independent) variables. In many econometric contexts, the commonly-used ordinary least squares method may not recover the theoretical relation desired or may produce estimates with poor statistical properties, because the assumptions for valid use of the method are violated. One widely-used remedy is the method of instrumental variables (IV). For an economic model described by more than one equation, simultaneous-equation methods may be used to remedy similar problems, including two IV variants, Two-Stage Least Squares (2SLS), and Three-Stage Least Squares (3SLS).
Other important unifying or distinguishing methods include the Method of Moments, Generalized Method of Moments (GMM), time series analysis, and Bayesian methods
Example
A simple example of a relationship in econometrics from the field of labor economics is:
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 ε is uncorrelated with education produces a misspecified model. A second technique for dealing with omitted variables is instrumental variables estimation. Still a third technique is to include in the equation additional set of measured covariates which are not instrumental variables, yet render β1 identifiable. An overview of econometric methods used to study this problem can be found in Card (1999)....
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