Econometrics
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 β 1 {\displaystyle \beta _{1}} measures the increase in the natural log of the wage attributable to one more year of education. The term ε {\displaystyle \varepsilon } is a random variable representing all other factors that may have direct influence on wage. The econometric goal is to estimate the parameters, β 0 and β 1 {\displaystyle \beta _{0}{\mbox{ and }}\beta _{1}} under specific assumptions about the random variable ε {\displaystyle \varepsilon }. For example, if ε {\displaystyle \varepsilon } 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 ϵ {\displaystyle \epsilon } 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 β 1 {\displaystyle \beta _{1}} identifiable. An overview of econometric methods used to study this problem were provided by Card (1999).
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