Economics 326 — Introduction to Econometrics II
An interval variable is one where the difference between two values is meaningful. Example: “Education” when measured in years. The difference between 12 and 10 years of education is meaningful.
In some data sets, education is reported as an ordinal variable: only the order of its values matters, but the difference between values has no meaning. The following two variables are equivalent:
\text{Education}_{i}=\left\{ \begin{array}{ll} 1 & \text{if high-school graduate,} \\ 2 & \text{if college graduate,} \\ 3 & \text{if advanced degree.} \end{array} \right.
\text{Education}_{i}=\left\{ \begin{array}{ll} 1 & \text{if high-school graduate,} \\ 10 & \text{if college graduate,} \\ 234 & \text{if advanced degree.} \end{array} \right.
A categorical variable has one or more categories, but there is no natural ordering to the categories. Examples: gender, race, marital status, geographic location.
The following two variables are equivalent:
\text{Gender}_{i}=\left\{ \begin{array}{ll} 1 & \text{if observation } i \text{ corresponds to a woman,} \\ 2 & \text{if observation } i \text{ corresponds to a man.} \end{array} \right.
\text{Gender}_{i}=\left\{ \begin{array}{ll} 1 & \text{if observation } i \text{ corresponds to a man,} \\ 2 & \text{if observation } i \text{ corresponds to a woman.} \end{array} \right.
Categorical and ordinal variables are also called qualitative.
Qualitative variables cannot simply be included in a regression because the regression technique assumes that all variables are interval.
A dummy variable is a binary zero-one variable that takes on the value one if some condition is satisfied and zero if that condition fails:
\text{Married}_{i}=\left\{ \begin{array}{ll} 1 & \text{if individual } i \text{ is married,} \\ 0 & \text{if individual } i \text{ is not married.} \end{array} \right.
\text{Unmarried}_{i}=\left\{ \begin{array}{ll} 1 & \text{if individual } i \text{ is not married,} \\ 0 & \text{if individual } i \text{ is married.} \end{array} \right.
Note that \text{Married}_{i}+\text{Unmarried}_{i}=1 for all observations i.
Preview of the wage1 data from the wooldridge package:
library(wooldridge)
data(wage1)
head(wage1[, c("wage", "female", "educ", "exper", "tenure")], n = 10) wage female educ exper tenure
1 3.10 1 11 2 0
2 3.24 1 12 22 2
3 3.00 0 11 2 0
4 6.00 0 8 44 28
5 5.30 0 12 7 2
6 8.75 0 16 9 8
7 11.25 0 18 15 7
8 5.00 1 12 5 3
9 3.60 1 12 26 4
10 18.18 0 17 22 21Consider the following regression:
\text{Wage}_{i}=\beta_{0}+\delta_{0}\text{Female}_{i}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i},
and assume that, conditional on all independent variables, \mathrm{E}\left[U_{i} \mid \mathbf{X}\right]=0.
Here, tenure refers to the number of years the worker has been employed at their current firm.
If observation i corresponds to a woman, \text{Female}_{i}=1, and
\begin{aligned} &\mathrm{E}\left[\text{Wage}_{i} \mid \text{Female}_{i}=1, \text{Educ}_{i}, \text{Exper}_{i}, \text{Tenure}_{i}\right] \\ &= \beta_{0}+\delta_{0}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}. \end{aligned}
If observation i corresponds to a man, \text{Female}_{i}=0, and
\begin{aligned} &\mathrm{E}\left[\text{Wage}_{i} \mid \text{Female}_{i}=0, \text{Educ}_{i}, \text{Exper}_{i}, \text{Tenure}_{i}\right] \\ &= \beta_{0}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}. \end{aligned}
Thus,
\begin{aligned} \delta_{0} &= \mathrm{E}\left[\text{Wage}_{i} \mid \text{Female}_{i}=1, \text{Educ}_{i}, \text{Exper}_{i}, \text{Tenure}_{i}\right] \\ &\quad - \mathrm{E}\left[\text{Wage}_{i} \mid \text{Female}_{i}=0, \text{Educ}_{i}, \text{Exper}_{i}, \text{Tenure}_{i}\right]. \end{aligned}
The model:
\text{Wage}_{i}=\beta_{0}+\delta_{0}\text{Female}_{i}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}.
For men (\text{Female}_{i}=0):
\text{Wage}_{i}^M=\beta_{0}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}.
For women (\text{Female}_{i}=1):
\text{Wage}_{i}^W=\left(\beta_{0}+\delta_{0}\right)+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}.
In this case, men play the role of the base group.
\delta_{0} measures the wage difference relative to the base group.
Estimated equation:
\begin{aligned} \widehat{\text{Wage}}_{i} &= \underset{(0.72)}{-1.57} - \underset{(0.26)}{1.81}\, \text{Female}_{i} + \underset{(0.049)}{0.572}\, \text{Educ}_{i} \\ &\quad + \underset{(0.012)}{0.025}\, \text{Exper}_{i} + \underset{(0.021)}{0.141}\, \text{Tenure}_{i}. \end{aligned}
The dependent variable is the wage per hour.
\hat{\delta}_{0}=-1.81 implies that a woman earns $1.81 less per hour than a man with the same level of education, experience, and tenure. (These are 1976 wages.)
The difference is also statistically significant.
The model:
\ln\left(\text{Wage}_{i}\right)=\beta_{0}+\delta_{0}\text{Female}_{i}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}.
In this case,
\begin{aligned} \delta_{0} &= \ln\left(\text{Wage}^{F}\right)-\ln\left(\text{Wage}^{M}\right) \\ &= \ln\left(\frac{\text{Wage}^{F}}{\text{Wage}^{M}}\right) \\ &= \ln\left(\frac{\text{Wage}^{M}+\left(\text{Wage}^{F}-\text{Wage}^{M}\right)}{\text{Wage}^{M}}\right) \\ &= \ln\left(1+\frac{\text{Wage}^{F}-\text{Wage}^{M}}{\text{Wage}^{M}}\right) \\ &\approx \frac{\text{Wage}^{F}-\text{Wage}^{M}}{\text{Wage}^{M}}. \end{aligned}
Estimated equation:
\begin{aligned} \widehat{\ln\left(\text{Wage}_{i}\right)} &= \underset{(0.099)}{0.417} - \underset{(0.036)}{0.297}\, \text{Female}_{i} + \underset{(0.007)}{0.080}\, \text{Educ}_{i} \\ &\quad + \underset{(0.005)}{0.029}\, \text{Exper}_{i} - \underset{(0.00010)}{0.00058}\, \text{Exper}_{i}^{2} \\ &\quad + \underset{(0.007)}{0.032}\, \text{Tenure}_{i} - \underset{(0.00023)}{0.00059}\, \text{Tenure}_{i}^{2}. \end{aligned}
\hat{\delta}_{0}=-0.297 implies that a woman earns 29.7% less than a man with the same level of education, experience, and tenure.
Instead of
\ln\left(\text{Wage}_{i}\right)=\beta_{0}+\delta_{0}\text{Female}_{i}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i},
consider:
\ln\left(\text{Wage}_{i}\right)=\theta_{0}+\gamma_{0}\text{Male}_{i}+\theta_{1}\text{Educ}_{i}+\theta_{3}\text{Exper}_{i}+\theta_{4}\text{Tenure}_{i}+U_{i}.
Since \text{Male}_{i}=1-\text{Female}_{i},
\begin{aligned} \ln\left(\text{Wage}_{i}\right) &= \theta_{0}+\gamma_{0}\text{Male}_{i}+\theta_{1}\text{Educ}_{i}+\theta_{3}\text{Exper}_{i}+\theta_{4}\text{Tenure}_{i}+U_{i} \\ &= \theta_{0}+\gamma_{0}\left(1-\text{Female}_{i}\right)+\theta_{1}\text{Educ}_{i}+\theta_{3}\text{Exper}_{i}+\theta_{4}\text{Tenure}_{i}+U_{i} \\ &= \left(\theta_{0}+\gamma_{0}\right)-\gamma_{0}\text{Female}_{i}+\theta_{1}\text{Educ}_{i}+\theta_{3}\text{Exper}_{i}+\theta_{4}\text{Tenure}_{i}+U_{i}. \end{aligned}
We conclude that \delta_{0}=-\gamma_{0}, \beta_{0}=\theta_{0}-\delta_{0}, \beta_{1}=\theta_{1}, etc.:
\ln\left(\text{Wage}_{i}\right)=\left(\beta_{0}+\delta_{0}\right)-\delta_{0}\text{Male}_{i}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}.
Thus, changing the base group has no effect on the conclusions.
In this dataset, gender is recorded as a binary variable (female/male). The dummy variable approach shown here applies to any binary grouping.
Consider the equation:
\begin{aligned} \ln\left(\text{Wage}_{i}\right) &= \beta_{0}+\delta_{0}\text{Female}_{i}+\gamma_{0}\text{Male}_{i} \\ &\quad +\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}. \end{aligned}
Recall that the intercept is a regressor that takes the value one for all observations.
In this dataset, \text{Female}_{i}+\text{Male}_{i}=1 for all observations i, so we have perfect multicollinearity. Such an equation cannot be estimated.
One cannot include an intercept and dummies for all the groups!
One of the dummies has to be omitted and the corresponding group becomes the base group:
Men are the base group: \ln\left(\text{Wage}_{i}\right)=\beta_{0}+\delta_{0}\text{Female}_{i}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}.
Women are the base group: \ln\left(\text{Wage}_{i}\right)=\theta_{0}+\gamma_{0}\text{Male}_{i}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}.
Alternatively, one can include both dummies without the intercept:
\ln\left(\text{Wage}_{i}\right)=\pi_{0}\text{Female}_{i}+\pi_{1}\text{Male}_{i}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}.
In R, a regression without an intercept can be estimated by adding + 0 or - 1 to the formula:
or equivalently:
The coefficients on the dummy variables lose the difference interpretation.
We can also allow the returns to education to be different for men and women:
\begin{aligned} \ln\left(\text{Wage}_{i}\right) &= \beta_{0}+\delta_{0}\text{Female}_{i}+\beta_{1}\text{Educ}_{i}+\delta_{1}\left(\text{Female}_{i}\cdot \text{Educ}_{i}\right) \\ &\quad +\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}. \end{aligned}
The variable \left(\text{Female}_{i}\cdot \text{Educ}_{i}\right) is called an interaction.
The equation for men (\text{Female}_{i}=0):
\ln\left(\text{Wage}_{i}^{M}\right)=\beta_{0}+\beta_{1}\text{Educ}_{i}+\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}.
The equation for women (\text{Female}_{i}=1):
\begin{aligned} \ln\left(\text{Wage}_{i}^{F}\right) &= \left(\beta_{0}+\delta_{0}\right)+\left(\beta_{1}+\delta_{1}\right)\text{Educ}_{i} \\ &\quad +\beta_{3}\text{Exper}_{i}+\beta_{4}\text{Tenure}_{i}+U_{i}. \end{aligned}
\delta_{1} can be interpreted as the difference in the return to education between women and men (the base group) after controlling for experience and tenure.
Estimated equation:
\begin{aligned} \widehat{\ln\left(\text{Wage}_{i}\right)} &= \underset{(0.119)}{0.389} - \underset{(0.168)}{0.227}\, \text{Female}_{i} + \underset{(0.008)}{0.082}\, \text{Educ}_{i} - \underset{(0.0131)}{0.0056}\, \text{Female}_{i}\cdot \text{Educ}_{i} \\ &\quad + \underset{(0.005)}{0.029}\, \text{Exper}_{i} - \underset{(0.00011)}{0.00058}\, \text{Exper}_{i}^{2} \\ &\quad + \underset{(0.007)}{0.032}\, \text{Tenure}_{i} - \underset{(0.00024)}{0.00059}\, \text{Tenure}_{i}^{2}. \end{aligned}
\hat{\delta}_{1}=-0.0056, suggesting that the return to education for women is 0.56 percentage points less than for men; however, this difference is not statistically significant. We cannot reject the hypothesis that the return to education is the same for men and women.
In the previous examples, \text{Educ} was a quantitative variable: years of education.
Suppose now that instead the education variable is ordinal:
\text{Education}_{i} = \left\{ \begin{array}{ll} 1 & \text{if high-school dropout,} \\ 2 & \text{if high-school graduate,} \\ 3 & \text{if some college,} \\ 4 & \text{if college graduate,} \\ 5 & \text{if advanced degree.} \end{array} \right.
Only the order is important, and there is no meaning to the distance between the values.
Adding such a variable to the regression will give a meaningless result.
\text{Education}_{i} = \left\{ \begin{array}{ll} 1 & \text{if high-school dropout,} \\ 2 & \text{if high-school graduate,} \\ 3 & \text{if some college,} \\ 4 & \text{if college graduate,} \\ 5 & \text{if advanced degree.} \end{array} \right.
Define 5 new dummy variables:
\begin{aligned} E_{1,i} &= \left\{ \begin{array}{cc} 1 & \text{if high-school dropout,} \\ 0 & \text{otherwise.} \end{array} \right. \quad & E_{2,i} &= \left\{ \begin{array}{cc} 1 & \text{if high-school graduate,} \\ 0 & \text{otherwise.} \end{array} \right. \\ E_{3,i} &= \left\{ \begin{array}{cc} 1 & \text{if some college,} \\ 0 & \text{otherwise.} \end{array} \right. \quad & E_{4,i} &= \left\{ \begin{array}{cc} 1 & \text{if college graduate,} \\ 0 & \text{otherwise.} \end{array} \right. \\ E_{5,i} &= \left\{ \begin{array}{cc} 1 & \text{if advanced degree,} \\ 0 & \text{otherwise.} \end{array} \right. \end{aligned}
To avoid the dummy variable trap, one of the dummies has to be omitted:
\text{Wage}_{i}=\beta_{0}+\delta_{0}\text{Female}_{i}+\delta_{2}E_{2,i}+\delta_{3}E_{3,i}+\delta_{4}E_{4,i}+\delta_{5}E_{5,i}+\text{Other Factors}
Group 1 (high-school dropout) becomes the base group.
\delta_{2} measures the wage difference between high-school graduates and high-school dropouts.
\delta_{3} measures the wage difference between individuals with some college education and high-school dropouts.
The previous definitions compare each group to the base group (high-school dropouts). Alternatively, we can define dummies that compare each group to the previous one:
\begin{aligned} D_{2,i} &= \left\{ \begin{array}{cc} 1 & \text{if high-school graduate or higher,} \\ 0 & \text{otherwise.} \end{array} \right. \\ D_{3,i} &= \left\{ \begin{array}{cc} 1 & \text{if some college or higher,} \\ 0 & \text{otherwise.} \end{array} \right. \\ D_{4,i} &= \left\{ \begin{array}{cc} 1 & \text{if college graduate or higher,} \\ 0 & \text{otherwise.} \end{array} \right. \\ D_{5,i} &= \left\{ \begin{array}{cc} 1 & \text{if advanced degree,} \\ 0 & \text{otherwise.} \end{array} \right. \end{aligned}
The model:
\text{Wage}_{i}=\beta_{0}+\delta_{0}\text{Female}_{i}+\gamma_{2}D_{2,i}+\gamma_{3}D_{3,i}+\gamma_{4}D_{4,i}+\gamma_{5}D_{5,i}+\text{Other Factors}
\gamma_{2} measures the wage difference between high-school graduates and high-school dropouts.
\gamma_{3} measures the wage difference between individuals with some college and high-school graduates.
\gamma_{4} measures the wage difference between college graduates and individuals with some college.
\gamma_{5} measures the wage difference between individuals with advanced degrees and college graduates.
