Having the following data,
Wageeducationexperience
11.551220
599
121615
71438
21.151619
6.92124
101214
81232
15.63187
18.22185
As an econometrician;
a)Formulate an econometric model and estimate its coefficients;
b)How do you interpret theestimatedcoefficients �!and �";
c)Estimate thevariance andstandard error of�!and �";
d)Calculate tstatistics and confidence intervals for the estimated parameters.Test the
hypothesis of the parameters and interpret.
2
R2
R
2
e)Calculate the coefficient of determination(R)andAdjusted i.e. .
a) To formulate an econometric model, we can use the wage as the dependent variable and education and experience as independent variables. The model can be written as:
Wage = β0 + β1 * Education + β2 * Experience + ε
Where:
- Wage is the dependent variable representing the individual's wage.
- Education is the independent variable representing the level of education.
- Experience is the independent variable representing the years of work experience.
- β0, β1, and β2 are the estimated coefficients to be determined.
- ε is the error term.
To estimate the coefficients, we can use the ordinary least squares (OLS) method.
b) The estimated coefficient β1 represents the effect of education on wage, holding experience constant. A positive β1 implies that higher education is associated with higher wages, ceteris paribus. The estimated coefficient β2 represents the effect of experience on wage, holding education constant. A positive β2 implies that more experience is associated with higher wages, ceteris paribus.
c) To estimate the variance and standard error of β1 and β2, we can calculate the standard errors using the formula:
Var(β1) = σ^2 / [n * Var(Education)]
Var(β2) = σ^2 / [n * Var(Experience)]
Where σ^2 is the estimated variance of the error term and n is the sample size.
The standard errors can be obtained by taking the square root of the variances.
d) To calculate t-statistics and confidence intervals for the estimated parameters, we can use the following formulas:
t-Statistic = (β - β_hat) / SE(β)
Confidence Interval = β_hat ± t * SE(β)
Where β_hat is the estimated coefficient, SE(β) is the standard error, and t is the critical t-value corresponding to the desired confidence level.
To test the hypothesis of the parameters, we can compare the t-statistic to the critical t-value for a given significance level (e.g., 5% or 1%). If the absolute value of the t-statistic is greater than the critical t-value, we can reject the null hypothesis that the parameter is equal to zero. Otherwise, we fail to reject the null hypothesis.
e) The coefficient of determination (R^2) represents the proportion of the variance in the dependent variable (wage) that can be explained by the independent variables (education and experience) in the model. It measures the goodness of fit of the model. A higher R^2 value indicates a better fit of the model.
Adjusted R^2 adjusts the coefficient of determination for the number of independent variables and sample size. It penalizes the addition of unnecessary variables to control for overfitting. Adjusted R^2 is typically lower than R^2 when there are more independent variables in the model.