Economics Homework: Regression Analysis of Wage Determinants

This homework assignment delves into regression analysis, exploring the relationship between wages and various factors such as beauty, IQ, height, and experience. The solution presents a detailed analysis of multiple regression models, interpreting coefficients, and assessing statistical significance through hypothesis testing. It covers topics like the impact of predictor variables on earnings, the use of R-squared, and the interpretation of p-values. The assignment also examines the effect of interaction terms and the use of quadratic terms in the regression model. The document provides a comprehensive understanding of how different variables influence earnings and the statistical methods used to analyze these relationships, including t-tests and the assessment of statistical significance at different levels.

Answers
Question 1
Our model in column 2 is;
Ln wi = β0 +β1 (Beauty) + β2 (IQ) + β3 (Height) +μi
E [Ln wi | x] = β0 +β1 (Beauty) + β2 (IQ) + β3 (Height)
E [Ln wi | x] = β0 +0.029 (Beauty) + 0.166 (IQ) + 0.047(Height)
β1 =0.029. It shows that log wage increases by 0.029 when beauty increases by one unit.
Β2 = 0.166. The estimate indicates that the log wage increases by 0.166 when the IQ increases
by a unit.
Β3 =0.047. The estimate shows that the log wage increases by 0.047 when the height increases
by a unit.
Question 2
Ho: β1 ≥ 0 vs.
H1: β1 < 0
β1 is the regression estimate of the independent variable beauty. Therefore, the null hypothesis
states that there is a positive linear relationship between beauty and the log of annual earnings.
The alternative hypothesis states that there is a negative linear relationship between beauty and
the log of annual earnings.
Question 3
The estimates β1 and β2 of in column 3 are;
β1= 0.028
β2 =0.070

It is true to claim that IQ has more than twice the effect on earnings as beauty. The estimate of
IQ is greater than the estimate of beauty. IQ has more than twice the effect on earnings as
beauty. In the model, we determine how IQ influences earnings holding the variable beauty
constant and also the influence of beauty on earnings holding the variable IQ constant.
Question 4
Conducting a t-test
Ho: β2 ≥ 0.17 vs.
H1: β2 < 0.17
The p-value is 0.065. At a 1% level of significance, we fail to reject the null hypothesis since
0.065 > 0.01. Therefore, there is no enough evidence to conclude that the estimate of IQ is less
than 0.17.
At a 10% level of significance, we shall reject the null hypothesis and accept the alternative
hypothesis. Therefore, there is enough evidence to conclude that the regression estimate of IQ is
less than 0.17. It is because the p-value is less than 0.1 (0.065 <0.1).
Question 5
Usually, R-squared increases when we increase another predictor variable in the regression
model. In the first column, we had two predictor variables (beauty and IQ). In the second
column, we have three predictor variables (beauty, IQ and height). The R-squared of column
three is thus greater than the R-squared of column 2 (0.113 < 0.121). However, R-squared
increases only If the added variable has some statistically significant effect on the model.
Question 6
Ho: β1 = 0 vs.
H1: β1≠ 0
P-value is 0.006, which is less than 0.05.

At a 5% level of significance, we shall reject the null hypothesis and accept the alternative
hypothesis. Therefore, there is enough evidence to conclude that there is a linear relationship
between beauty and the log of the earnings.
Question 7
When we use another scale, the estimates will decrease while the standard errors will increase. In
a regression equation, the coefficients of the predictor variables are estimated jointly. When one
variable changes, other variables in the model will change. The R-squared will increase due to
the increase of the score of beauty by 9 using a 20-point scale.
Question 8
β1 is the coefficient estimate that captures the effect of beauty on wages.
It is useful to add the interaction term because it tells us how beauty and the height work together
to influence the response variable log of earnings. It helps to comprehend more about the
relationship between earnings and the interaction between beauty and height.
Question 9
The type of test to use is the test of significance. We determine if the variable is statistically
significant at a certain level of significance to decide if it should belong to the model. If the test
is less than the level of significance, we include the variable in the model because it is
statistically significant.
Question 10
The regression model is;
Ln wi = β0 +β1 (Beauty) + β2 (IQ) + β3 (Height) + β4 (exp) + β5 (exp2) + μi
E [Ln wi | x] == β0 +0.028 (Beauty) + 0.070(IQ) + 0.011 (Height) -0.016 (exp) + 0.001 (exp2)
The estimate of the quadratic term is 0.001
Ho: β5 = 0 vs.
H1: β5≠ 0
t calculated / t statistic = 0.001-0/ s.e = 0.001-0/0.000 =0.000

n = 1978, n-1 =1977
t critical at 5% level of significance from the t-table = 1.962
t calculated is less than t critical (0.000 < 1.962). We shall fail to reject the null hypothesis and
accept the alternative hypothesis. Therefore, there is no enough evidence to conclude that there
exists a linear relationship between the quadratic term (exp2) and the log of earnings. The
quadratic term is thus not statistically significant at 5% level of significance.

References
Frost, J. (2017, October 31). Understanding the Interaction Effects in Statistics - Statistics By
Jim. Statistics By Jim. https://statisticsbyjim.com/regression/interaction-effects/
Mckee, D. (2015). How to interpret regression tables [YouTube Video]. In YouTube.
https://www.youtube.com/watch?v=o86xvmUYo-Q
Minitab Blog Editor. (2015). Understanding Hypothesis Tests: Significance Levels (Alpha) and
P values in Statistics. Minitab.Com. https://blog.minitab.com/blog/adventures-in-
statistics-2/understanding-hypothesis-tests-significance-levels-alpha-and-p-values-in-
statistics
Multiple Regression Coefficient output. (2020, April). Multiple Regression Coefficient output.
Cross Validated. https://stats.stackexchange.com/questions/457814/multiple-regression-
coefficient-output