Answer :
### Question:
Given the regression model based on a random sample of 200 persons:
[tex]\[ \begin{array}{llll} \text{LnINC} = & 0.12 \text{LnFin} + 0.05 \text{Educ} - 0.15 \text{Female} \\ \hline (t\text{-ratios}) & (3.28) & (2.50) & (3.05) \\ (p\text{ values }) & (0.001) & (0.006) & (0.001) \\ R^2=0.82 & \end{array} \][/tex]
Where:
- [tex]\(\text{LnINC}\)[/tex] = natural log of the person's income
- [tex]\(\text{LnFin}\)[/tex] = natural log of the father's income
- \text{Educ} = number of years in education
- [tex]\(\text{Female} = 1\)[/tex] if female; 0 if male.
Interpret the following:
i. The regression coefficients.
ii. The [tex]\(p\)[/tex]-values for the [tex]\(t\)[/tex]-statistics.
iii. The [tex]\(R^2\)[/tex].
iv. Mention three determinants of supply.
### Answer:
#### i. Interpretation of the Regression Coefficients:
(3 Marks)
- Coefficient of [tex]\(\text{LnFin} = 0.12\)[/tex]:
- This coefficient indicates that for a 1% increase in the father's income, the person's income is expected to increase by 0.12%, holding other factors constant. The interpretation here uses the logarithmic transformation properties, where changes in independent variables translate to percentage changes in the dependent variable.
- Coefficient of \text{Educ} = 0.05:
- This coefficient means that each additional year of education results in a 5% increase in the person's income, holding other variables constant. This suggests an increase in education can positively affect a person's potential earnings.
- Coefficient of [tex]\(\text{Female} = -0.15\)[/tex]:
- This negative coefficient implies that being female is associated with a decrease in the natural log of income by 0.15 units compared to being male, holding other factors constant. This suggests a gender disparity where, on average, females earn less than males for the same levels of parental income and education.
#### ii. The [tex]\(p\)[/tex]-values for the [tex]\(t\)[/tex]-statistics:
(1 Mark)
- [tex]\(p\)[/tex]-value of [tex]\(\text{LnFin} = 0.001\)[/tex]:
- This low [tex]\(p\)[/tex]-value suggests there is strong evidence against the null hypothesis that the coefficient is zero; thus, we conclude that the father's income significantly affects the person's income.
- [tex]\(p\)[/tex]-value of \text{Educ} = 0.006:
- This [tex]\(p\)[/tex]-value also suggests sufficient evidence to reject the null hypothesis. Even though it's slightly higher than that of [tex]\(\text{LnFin}\)[/tex], it still indicates a significant positive effect of education on the person's income.
- [tex]\(p\)[/tex]-value of [tex]\(\text{Female} = 0.001\)[/tex]:
- Like [tex]\(\text{LnFin}\)[/tex], this [tex]\(p\)[/tex]-value is very low, indicating strong evidence against the null hypothesis. We conclude that gender significantly influences the natural log of the person's income.
#### iii. The [tex]\(R^2\)[/tex]:
(1 Mark)
- [tex]\(R^2 = 0.82\)[/tex]:
- This [tex]\(R^2\)[/tex] value of 0.82 indicates that 82% of the variability in the dependent variable ([tex]\(\text{LnINC}\)[/tex]) is explained by the regression model. This is a high value, suggesting that the model fits the data well and the predictors ([tex]\(\text{LnFin}\)[/tex], \text{Educ}, and [tex]\(\text{Female}\)[/tex]) together explain a large proportion of the variation in the person's income.
#### iv. Three Determinants of Supply:
(3 Marks)
- Price of Inputs:
- The cost of resources used in the production process can affect the supply of goods. Higher input prices can decrease the supply as production becomes more expensive.
- Number of Firms:
- The number of producers in the market impacts the total supply. More firms typically mean a higher overall supply of the goods or services they produce.
- Price of Substitute Products:
- If the price of substitutes changes, it can affect the supply of a good. For instance, if the price of a substitute rises, producers might switch to produce more of the higher-priced goods, thereby affecting supply dynamics.
By interpreting the regression coefficients, analyzing the [tex]\(p\)[/tex]-values, understanding the [tex]\(R^2\)[/tex] value, and discussing key supply determinants, this comprehensive analysis provides insights into the statistical model and economic principles at play.
Given the regression model based on a random sample of 200 persons:
[tex]\[ \begin{array}{llll} \text{LnINC} = & 0.12 \text{LnFin} + 0.05 \text{Educ} - 0.15 \text{Female} \\ \hline (t\text{-ratios}) & (3.28) & (2.50) & (3.05) \\ (p\text{ values }) & (0.001) & (0.006) & (0.001) \\ R^2=0.82 & \end{array} \][/tex]
Where:
- [tex]\(\text{LnINC}\)[/tex] = natural log of the person's income
- [tex]\(\text{LnFin}\)[/tex] = natural log of the father's income
- \text{Educ} = number of years in education
- [tex]\(\text{Female} = 1\)[/tex] if female; 0 if male.
Interpret the following:
i. The regression coefficients.
ii. The [tex]\(p\)[/tex]-values for the [tex]\(t\)[/tex]-statistics.
iii. The [tex]\(R^2\)[/tex].
iv. Mention three determinants of supply.
### Answer:
#### i. Interpretation of the Regression Coefficients:
(3 Marks)
- Coefficient of [tex]\(\text{LnFin} = 0.12\)[/tex]:
- This coefficient indicates that for a 1% increase in the father's income, the person's income is expected to increase by 0.12%, holding other factors constant. The interpretation here uses the logarithmic transformation properties, where changes in independent variables translate to percentage changes in the dependent variable.
- Coefficient of \text{Educ} = 0.05:
- This coefficient means that each additional year of education results in a 5% increase in the person's income, holding other variables constant. This suggests an increase in education can positively affect a person's potential earnings.
- Coefficient of [tex]\(\text{Female} = -0.15\)[/tex]:
- This negative coefficient implies that being female is associated with a decrease in the natural log of income by 0.15 units compared to being male, holding other factors constant. This suggests a gender disparity where, on average, females earn less than males for the same levels of parental income and education.
#### ii. The [tex]\(p\)[/tex]-values for the [tex]\(t\)[/tex]-statistics:
(1 Mark)
- [tex]\(p\)[/tex]-value of [tex]\(\text{LnFin} = 0.001\)[/tex]:
- This low [tex]\(p\)[/tex]-value suggests there is strong evidence against the null hypothesis that the coefficient is zero; thus, we conclude that the father's income significantly affects the person's income.
- [tex]\(p\)[/tex]-value of \text{Educ} = 0.006:
- This [tex]\(p\)[/tex]-value also suggests sufficient evidence to reject the null hypothesis. Even though it's slightly higher than that of [tex]\(\text{LnFin}\)[/tex], it still indicates a significant positive effect of education on the person's income.
- [tex]\(p\)[/tex]-value of [tex]\(\text{Female} = 0.001\)[/tex]:
- Like [tex]\(\text{LnFin}\)[/tex], this [tex]\(p\)[/tex]-value is very low, indicating strong evidence against the null hypothesis. We conclude that gender significantly influences the natural log of the person's income.
#### iii. The [tex]\(R^2\)[/tex]:
(1 Mark)
- [tex]\(R^2 = 0.82\)[/tex]:
- This [tex]\(R^2\)[/tex] value of 0.82 indicates that 82% of the variability in the dependent variable ([tex]\(\text{LnINC}\)[/tex]) is explained by the regression model. This is a high value, suggesting that the model fits the data well and the predictors ([tex]\(\text{LnFin}\)[/tex], \text{Educ}, and [tex]\(\text{Female}\)[/tex]) together explain a large proportion of the variation in the person's income.
#### iv. Three Determinants of Supply:
(3 Marks)
- Price of Inputs:
- The cost of resources used in the production process can affect the supply of goods. Higher input prices can decrease the supply as production becomes more expensive.
- Number of Firms:
- The number of producers in the market impacts the total supply. More firms typically mean a higher overall supply of the goods or services they produce.
- Price of Substitute Products:
- If the price of substitutes changes, it can affect the supply of a good. For instance, if the price of a substitute rises, producers might switch to produce more of the higher-priced goods, thereby affecting supply dynamics.
By interpreting the regression coefficients, analyzing the [tex]\(p\)[/tex]-values, understanding the [tex]\(R^2\)[/tex] value, and discussing key supply determinants, this comprehensive analysis provides insights into the statistical model and economic principles at play.
