A dataset contains information on 4000 full-time workers from the 1998 Current Population Survey in the United States.
The relevant variables are as follows:
β’ AHE stands for average hourly earnings (in 1998 dollars)
β’ College=1 if has a college education, 0 if just has a high school education
β’ Female = 1 if female, 0 if male (in years)
The following is the regression model estimated using this data:
(1) π΄π»πΈπ = π½0 + π½1πΆπππππππ + π½2πΉππππππ + π½3π΄πππ + ππ
The corresponding calculated coefficients are shown in Table 1, with standard errors in parentheses.
Table 1: Regression Findings for Average Hourly Earnings Determinants
Average hourly earnings (AHE) (1)
College 5.48 (0.21)
Female -2.62 (0.20)
Age 0.29 (0.04)
Northeast/ Midwest/ South Constant 4.40 (1.05)
R2 0.190
N 4000
a) Identify the dependent and independent variables in the regression model (1). Is AHE a continuous or discrete variable? Explain your answer.
The dependent variable is the one which outputβs variation is studied in the model. Its values depend on the independent variables, and the parameters of such dependence is a matter for research. While independent variable is an input that is a potential cause for dependent variable values variation. The dependent variable in the regression model (1) is average hourly earnings (AHE). Independent variables include age, college, and female (last two are dummies).
AHE is a continuous variable that makes possible to apply to it more mathematical operations. Variables counted in dollars or other currency may be considered discrete as money physically and by law has its particular finite scale, for example, for dollar, the minimal portion is a 0.01. However, here applied average values that in statistic can be counted with any required precision and to count the variable as continuous is more convenient for mathematical purposes.
b) What is a dummy variable? Of the list of relevant variables given above, identify the dummy variables.
The dummy variable or binary variable is a variable that contains values 1 and 0 and reflects an effect of some category that can be only present or absent (therefore it is binary). Application of dummy variables allows the researcher to sort data dividing it to opposite categories (2 per variable), at the same time, the number of dummies in one model is not limited and therefore the number of subcategories. They play a role of proxy allowing to code qualitative information into the quantitative form.
The dummy variables in the model are college and female. Both have only two values, 0 and 1. For college 1 means presence of college education, while 0 its absence (just a high school education). For female one means a person is female, while 0 means he is not (therefore male.). In both cases, it is impossible to add more values signing master degree, or PhD, or neutral gender as soon as it conflicts with the nature and purpose of dummies.
c) In Table 1 column (1), interpret the coefficient on College. Is it statistically significant at the 5% level? Explain your answer.
The coefficient on college is 5.48 that means people are having a college education get in average 5.48 dollars per hour more than people finished only high school (by point estimation) or the wage gap by education is 5.48 dollars.
The estimation is statistically significant at the 5% level. To test the significance of the coefficient estimation and therefore wage gap significance we will use t-test. H0 is the education wage gap is zero; H1 is the education wage gap is statistically significantly different from zero. The t-statistic for the education wage gap is 5.48/0.21 (ratio of the point estimate and the standard error of the college variable, under the null of no effect) = 26.095. The 5% critical value for the two-tailed t-test (with 4000-4=3996 degrees of freedom or about β) is 1.96. The counted for education wage gap t-statistic is much higher than the critical value. Hence, we should reject the H0 and accept H1. The education wage gap is statistically significantly different from zero at the 5% level.
d) In Table 1 column (1), interpret the coefficient on Female. Is it statistically significant at the 5% level? Explain your answer.
The coefficient on a female is -2.62 that mean female worker get in average 2.62 dollars per hour less than the male (by point estimation) or the wage gap by gender is 2.62 dollars.
The estimation is statistically significant at the 5% level. To test the significance of the coefficient estimation and therefore wage gap significance we will use t-test. H0 is the gender wage gap is zero; H1 is the gender wage gap is statistically significantly different from zero. The t-statistic for the gender wage gap is -2.62/0.2 (ratio of the point estimate and the standard error of the female variable, under the null of no effect) = -13.1. The 5% critical value for the two-tailed t-test (with 4000-4=3996 degrees of freedom or about β) is 1.96. The counted for gender wage gap t-statistic in the value of modulus is much higher than the critical value. Hence, we should reject the H0 and accept H1. The gender wage gap is statistically significantly different from zero at the 5% level.
e) Does your answer in part (d) above suggest a causal relationship between Female and AHE? Explain your answer.
The answer in part (d) suggest a probably existing causal relationship between variables Female and AHE as soon as the coefficient estimation is statistically significant at the 5% level (and 99% level too, 13.1 is higher than critical value 2.58) and the gender wage gap is different from zero. Therefore, an expected wage for workers of a different gender will have in average the gap (2.62 dollars per hour) and this gap will be attributable to this particular input change. Within the model, the Female is an independent variable, and AHE is a dependent one, that state a direction of causation while coefficient itself only reflect an existing relation between variables variance. On the other hand, the real reason for such gap existence may lay outside of this particular formal model and causation should be questioned, never taken just from the formal equation. For example, female workers often chose jobs with the psychological profile different from male, more socially oriented; they are rarely presented on the top-management positions. Therefore, the reason for lower wages may be not a gender itself but the structure of jobs and wages in the economy, asymmetry coming from social traditions and so on.
