ECO82001: Economics & Quantitative Analysis

In your role as an economic analyst you have been asked the following question:

• How much does education affect wage rates?

The Excel data file (wage) contains 100 observations for each of the following variables:

Instructions

Conduct a simple linear regression analysis to examine the relationship between ‘education’ (theindependent variable) and ‘wage’ (the dependent variable). Using the Excel data file, prepare a 2000 word report using the following structure:

• Purpose (2 marks)
• Background (2 marks)
• Method (4 marks)
• Results (4 marks)
• Discussion (5 marks)
• Recommendations (3 marks)

In preparing your report you must address the following questions:

1. Obtain summary statistics and histograms for the variables WAGE and EDUC. Discuss the data characteristics (2 marks).
2. Estimate the linear regression WAGE = ?1 + ?2EDUC + ? and interpret the slope (3 marks).
3. Calculate the residuals and plot them against EDUC. Are any patterns evident and, if so, what does this mean? (3 marks)
4. Estimate the quadratic regression WAGE = ?1 + ?2EDUC2 + ? and interpret the results. Estimate the marginal effect of another year of education on wage for a person with 12 years of education,and for a person with 14 years of education. Compare these values to the estimated marginaleffect of education from the linear regression in part (b). (4 marks).
5. Construct a histogram of ln(WAGE). Compare the shape of this histogram to that for WAGE frompart (a). Which appears to be more symmetric and bell-shaped? (2 marks).
6. Estimate the log-linear regression ln(WAGE) = ?1 + ?2EDUC + ? and interpret the slope. Estimatethe marginal effect of another year of education on wage for a person with 12 years of education,and for a person with 14 years of education. Compare these values to the estimated marginaleffects of education from the linear regression in part (b) and quadratic in part (d) (6 marks).

Your report must be submitted as a single file (maximum of 2000 words) with an appendix that contains the details of your statistical analyses.

Table of Contents:

Introduction:

What this report talks about is the relationship between wage level of employees as affected bytheir education levels as the data that has been provided has two basic variables of wage levelsand education of employees in years. We use statistical tools to establish the relationshipbetween education level and wage level and to find out the degree of relationship (cause and

effect) that exists between education level and wage level. The report has various sections, thebackdrop of the study which describes the purpose of the report, the method of statistical analysisused in this report, the discussion of various results of descriptive analysis and regressionanalysis and also the recommendations that are drawn out of this analysis.

Purpose of the report:

There has been many earlier studies that has focused on the relationship between the education of employees and their wage levels. (Connolly & Gottschalk, 2006) Though there is a positive

relationship between education level and wage level of workers, and that this education has ancausal relationship with the wages of data, there has also been other factors that affect the wageas an dependent variables and two such variables are the gender (as wages tend to lower forfemale sections) and also class to which these employees belong and most importantly theexperience of workers. Keeping the gender and class as constant, the main purpose of this reportis to establish the relationship between education and wage level of employees and to seewhether as the education level increases in years, wages increase or not and to what extent do

they increase.

Background of the report:

Most of the economists in today’s world tend to establish cause and effect relationship betweentwo variables that would enable us to predict the dependent variable based on the independentvariable. In our analysis the dependent variable is the education and the independent variable isthe wage level of employees. The variables that are provided to us as data is the Wage –that indicates the earnings per hour for the employees, the ln-wage- that is the log of wage earningsper hour, educ-which provides the eudation in years of the employees and the educ2 whichindicates the years of education squared to establish a quadratic regression. This report is basedon the analysis that has been conducted with the help of these four variables.

Methods used in the report:

The statistical tools used in this report are

1. Summary statistics of the wage and education levels which describe the measures of central tendency for the two variables and the histograms of educ and wage levels whichwould indicate the distribution of the data values of the two variables (whether they areskewed or peaked)
2. Linear regression establishes relationship between wage and education levels and theequation would give the intercept and the slope of the regression line. The slopecoefficient is the degree of relationship between the education levels and how it affectsthe wage level. And the intercept is the constant component of the equation which lies inthe y-axis.
3. A residual plot of regression equation establishes the differences between predictedvalues of the dependent variable and how far it varies from the actual variables.
4. Quadratic regression between wage level and educ squared is a form of linear regression between the wage level and the education level squared. The reasoning behind quadraticregression is the most common way to fit curves to the data using linear regression is toinclude polynomial terms, such as squared predictors.
5. Histogram of ln(wage) and comparison of the same with histogram of wage is to providea graphical representation of the log of wage variable which will depict the normaldistribution plot of the variable.
6. Log linear regression between ln(wage) and education level which establishes the causalrelationship between log of wage level and the education level.

The various results of the above analysis and their discussion along with is reported in the nextsection.

Results and Discussion:

Descriptive statistics and Histograms:

The summary statistics of the wage and education variables are given in the table below:

In the wage variable, skewness is higher than zero and positive which is equal to 1.5 which is greater than zero, and hence it is a asymmetric distribution with deviations from a normal distribution. And since it is positive and greater than zero, the values are mostly concentrated to the left of the mean of the distribution at 22.3, while the extreme values are placed to the right of the mean at 22.3.The maximum wage rate per hour is at 76.4, while the minimum wage is as low as 4.3. The median wage rate is 19.4 and the mode wage of the employees at 38.5 is higher than the mean 22.3. Kurtosis is an indicator of peakedness of a distribution or flattening of a distribution. As Kurtosis is calculated as 2.61 which is lesser than 3., indicates that the distribution is not a peaked distribution.

The descriptive statistics of education level in years of education is having a mean of 13.8, median of 13 and a mode of 12 years of education, which shows that there is certain level normality among the measures of central tendency(Barrow, 2006). This is also shown by the skewness value of 0.4 which is almost near to zero and indicates a almost symmetrical distribution and normal distribution. This is shown in the histogram that follows.

Linear regression between wage level and education level:

The linear regression with wage as the dependent variable and education level as independent variable is conducted and we have the following regression output.

WAGE = ?1 + ?2EDUC + ?

WAGE = -6.9 + 2.1 EDUC + ?

The slope of the equation is 2.1 which imply that one unit increase in education increases the wage by 2.1 units. As the sign of the slope is positive it indicates that as years of education increases, the wage rate also increases.

Predictions of the wage with 12 and 14 years of education:

When a person has 12 years of education, the predicted wage rate according to this model is given as

WAGE = -6.9 + 2.1 (12) = 18.3

Similarly, when a person has 14 years of education, the predicted wage rate according to this model is given as

WAGE = -6.9 + 2. (14) = 22.5

But the R² value is 0.2 which means that only 20% of the changes in the dependent variable (wage) is explained by the changes in the independent variable education.

Residual plot of the linear regression:

Aresidual plotis a graph that shows theresidualson the vertical axis and the independent variable on the horizontal axis. If the points in a residual plot are randomly dispersed around the horizontal axis, a linear regression model is appropriate for the data; otherwise, a non-linear model is more appropriate. Since the residual plot exhibits a random pattern, the linear fit of the model is appropriate(Cooper & Schindler, 2012).

Quadratic Regression:

When we conduct a quadratic regression with wage as dependent variable and education squared as independent variable the regression equation is of the form,

WAGE = ?1 + ?2EDUC2 + ?

We have the quadratic regression equation as

WAGE = 8 + 0.1EDUC2 + ?

The slope of the equation is 0.1 which implies that one unit increase in education squared increases the wage by 0.1units. As the sign of the slope is positive it indicates that as years of education squared increases, so does the wage rate. But the R² value again is 0.2 which means that only 20% of the changes in the dependent variable (wage) is explained by the changes in the independent variable education squared.

A person with 12 years of education would have 144 as the value of education squared and his predicted wage according to model is given as

WAGE = 8 + 0.1(144) = 22.4

A person with 14 years of education would have 196 as the value of education squared and his predicted wage according to model is given as

WAGE =8 + 0.1(196) = 27.6

The marginal effect of increase in two years of additional education on wage is given as 27.6-22.4= 5.2. the margianl effect of increase in one year of additional education on wage is $2.6.

Histogram of ln_wage:

The histogram of ln_wage is given as above and the skewness value of the ln_wage variable is given as 0.04 in the descriptive statistics. This shows that ln_wage is more symmetrical than the wage as an variable as the skewness of wage is given as 1.5 which shows asymmetry. The measures of median and mode are 3.0 and 3.6 which is higher than the mean of ln_wage at 2.9 which means that the measures of central tendency are more or less equal(Freund, Mohr, & Wilson, 2010).

Log-linear regression:

Log-linear regression of ln(wage) as the dependent variable and education as the indepdent variable would give us a regression equation as follows

ln(WAGE) = ?1 + ?2EDUC + ?

From the log-linear regression output we have

ln(WAGE) =1.6+ 0.1 EDUC + ?

A person with 12 years of education, the predicted ln_wage according to model is given as

ln(WAGE) =1.6 + 0.1 (12) = 2.8

A person with 14 years of education, the predicted ln_wage according to model is given as

n(WAGE) =1.6 + 0.1 (14) + ? =3

The marginal effect of increase in two years of additional education on ln (wage) is given as 3-2.8 = 0.2. the margianl effect of increase in one year of additional education on ln(wage) is 0.1.

The three models compared:

The marginal effect of education on the wage level is higher in the quadratic regression model than in the linear and log-linear model as can be seen from the above table.

Recommendations:

We can see from the above examination that training positively affects wages of representatives, Higher the years of instruction, higher is the pay rate.

The circumstances and end results relationship between training level and wage rate demonstrates that higher the instruction, higher will be the pay rate of representatives. What's more, the relationship table above likewise demonstrates that the connection in the midst of pay and instruction is sure at 0.4 which speaks to a moderate level of connection between's the two variables. So when we give more years of education to representatives or laborers, we can enhance their pay rate and consequently their way of life over the long haul. However all the three models have a low coefficient of determination suggesting that there are different variables influencing the pay rate separated from training and these models have excluded them in their examination.