Intercept (C(1)): The coefficient is 9.227823, and its p-value is 0.0000. The intercept represents the expected value of lnearnings when all other independent variables are zero. The low p-value suggests that this constant term is statistically significant.
Education (C(2)): The coefficient is 0.097694, and its p-value is 0.0000. This indicates that for each additional year of education, lnearnings are expected to increase by approximately 0.0977. Education is statistically significant in predicting lnearnings.
Experience (C(3)): The coefficient is 0.008724, and its p-value is 0.0080. This suggests that for each additional year of experience, lnearnings are expected to increase by approximately 0.0087. Experience is also statistically significant in predicting lnearnings.
Self-Employment (C(4)): The coefficient is -0.043752, and its p-value is 0.6844. Being self-employed (dself=1) is associated with a decrease in lnearnings by approximately 0.0438 compared to being a salaried worker (dself=0). However, the high p-value indicates that self-employment status is not statistically significant in predicting lnearnings in this model.
d)
The regression equation is: lnearnings = 9.227823 + 0.097694 * education + 0.008724 * exper - 0.043752 * dself
Here are the predictions for the specified scenarios:
For a worker with 15 years of education, 10 years of experience, and a salaried worker (dself=0):
The coefficient associated with "Log of Hours (C(5))" in the regression model is 1.116486, and its p-value is 0.0000.
Interpretation:
The coefficient of 1.116486 for "Log of Hours" suggests that for each additional unit increase in the natural logarithm of hours worked, lnearnings is expected to increase by approximately 1.1165 units, holding all other variables constant.
P-value:
The p-value of 0.0000 indicates that the coefficient for "Log of Hours" is statistically significant. In other words, there is strong evidence to suggest that the number of hours worked, as captured by its natural logarithm, is a significant predictor of lnearnings in the model.
g)
Null and Alternative Hypotheses:
Null Hypothesis (H0):
The sector of employment (dself) and the natural logarithm of hours worked (LNHOURS) do not jointly affect lnearnings (LNEARNINGS). In other words, the coefficients of dself and LNHOURS are jointly equal to zero.
Alternative Hypothesis (HA):
The sector of employment (dself) and the natural logarithm of hours worked (LNHOURS) jointly affect lnearnings (LNEARNINGS). At least one of the coefficients of dself and LNHOURS is not equal to zero.
h)
Joint Test:
To test whether the sector of employment and the natural logarithm of hours worked jointly matter for lnearnings, you need to conduct a joint test using the F-statistic. This test evaluates the null hypothesis that all the coefficients associated with dself and LNHOURS are equal to zero (i.e., they have no joint effect).
F-statistic: 12.61965
Prob(F-statistic): 0.000000 (very low, indicating the overall joint model is significant)
Conclusion:
Based on the joint test results:
The F-statistic is highly significant (Prob(F-statistic) = 0.000000), indicating that at least one of the coefficients for dself and LNHOURS is not equal to zero, meaning they have a joint effect on lnearnings.
Therefore, you should reject the null hypothesis that the sector of employment and the natural logarithm of hours worked have no joint effect on lnearnings.
The joint test supports the researcher's claim that these two variables jointly matter for lnearnings.
In summary, the sector of employment (self-employed or not) and the natural logarithm of hours worked do jointly matter for lnearnings, as evidenced by the significant F-statistic. This suggests that both of these factors combined have a statistically significant effect on lnearnings.
Bonus question:
Regression (b) - Initial Model:
In (b), you performed a regression of lnearnings on education, experience, and self-employment status (dself).
The coefficients for education and experience were found to be statistically significant, while the coefficient for self-employment was not significant (high p-value).
The model had an R-squared of 0.179259, indicating that about 17.9% of the variation in lnearnings was explained by the model.
Regression (e) - Expanded Model:
In (e), you extended the model by including the natural logarithm of hours worked (lnhours) as an additional predictor.
This model included education, experience, self-employment status, and lnhours as independent variables.
The coefficients for all variables were estimated, and the coefficients for education, experience, and lnhours were found to be statistically significant (low p-values).
The R-squared increased to 0.179259, indicating that the new model explains a similar amount of variation in lnearnings.
Discussion:
Similarities:
Both regressions estimate the relationship between lnearnings and various predictors, and they share the same R-squared value, indicating a similar goodness of fit.
Differences:
In regression (e), the inclusion of lnhours as a predictor resulted in significant coefficients for education, experience, and lnhours, whereas the self-employment status (dself) remained statistically insignificant.
Potential Issue:
One potential issue that could explain the difference in the significance of dself between the two regressions is multicollinearity. Multicollinearity occurs when two or more independent variables are highly correlated, which can make it difficult for the model to distinguish their individual effects. In regression (e), the inclusion of lnhours may have reduced multicollinearity, allowing the model to better capture the effect of dself.
Preference:
Given that regression (e) provides a more comprehensive model with additional significant predictors, it would generally be preferred over regression (b). The inclusion of lnhours allows you to capture another dimension of the relationship between hours worked and lnearnings, which is valuable in understanding the factors affecting earnings.
In summary, the expanded regression (e) is preferred due to its more comprehensive set of significant predictors and the potential improvement in addressing multicollinearity. This model provides a more complete understanding of the factors influencing lnearnings.
2.
a)
infrate(Sample average)
ph(Sample average)
(Nr of observations)
Cities without lead pipes
(if lead=0)
0.381
7.153
55
Cities with lead pipes (if lead=1)
0.403
7.402
117
b)
lead=0
Lead=1
c)
It appears that cities with and without lead pipelines differ in terms of infant mortality (infrate) and the pH-index of water (ph). Here's a description of these differences and their potential implications:
Infant Mortality (infrate): Cities with lead pipelines seem to have higher infant mortality rates (infrate) compared to cities without lead pipelines. This suggests that the presence of lead water pipes may be associated with an increase in infant mortality.
Water pH-Index (ph): Cities without lead pipelines tend to have higher pH levels in their water, indicating a more alkaline environment. Higher pH levels may be an indicator of better water quality. Conversely, cities with lead pipelines may have lower pH levels, potentially indicating water quality issues associated with lead pipes.
d)
e)
Intercept (C(1)):
The estimated intercept, C(1), is 0.701.
This represents the expected infant mortality rate (INFRATE) when all other predictor variables are set to zero.
The coefficient is statistically significant at the 1% significance level (p-value = 0.0001).
Interpretation: When all other factors are held constant, the estimated intercept represents the baseline infant mortality rate.
Temperature (C(5)):
The coefficient associated with temperature is not explicitly specified in the output.
We cannot determine its interpretation or statistical significance without the coefficient value and p-value. Please check the output for these details.
Typhoid Rate (C(6)):
The estimated coefficient for the typhoid rate, C(6), is 0.006.
The coefficient is statistically significant at the 1% significance level (p-value = 0.004).
Interpretation: For a one-unit increase in the typhoid death rate (TYPHOID_RATE), the infant mortality rate (INFRATE) is expected to increase by approximately 0.006 deaths per 100 in population.
Statistical Significance at the 10% Level:
The intercept and the coefficient for the typhoid rate are statistically significant at the 1% significance level (p-value < 0.01), indicating strong evidence that they have an impact on infant mortality.
The statistical significance at the 10% level is not directly provided in the output for the temperature coefficient. To determine its significance, you need the p-value associated with the temperature coefficient.
f)
The marginal effect of lead on infant mortality in the population model is simply the coefficient C(2).
The marginal effect of lead on infant mortality is approximately 0.373 which represents the change in infant mortality rate (in deaths per 100 in population) associated with a one-unit change in the lead variable, all other factors being constant.
g)
To estimate the marginal effect of lead on infant mortality for different pH levels using the regression results, we can use the coefficient associated with "LEAD" (C(2)) and the pH values of interest. The marginal effect is simply the coefficient for "LEAD," which represents the change in infant mortality rate (INFRATE) associated with a one-unit change in the lead variable.
Estimated marginal effect of lead
ph=6
0.373
ph=6.5
0.373
ph=7
0.373
ph=7.5
0.373
h)
The research claim suggests that the marginal effect of lead on infant mortality may be mitigated by the pH-index of the water. In other words, the effect of lead on infant mortality might vary depending on the pH level of the water. To test this claim, we can set up a hypothesis test. Here are the null and alternative hypotheses for a two-sided test:
Null Hypothesis (H0):
The marginal effect of lead on infant mortality is constant across different pH levels. In other words, there is no interaction between lead and pH in affecting infant mortality.
Alternative Hypothesis (HA):
The marginal effect of lead on infant mortality varies with changes in the pH level of the water. In other words, there is an interaction effect between lead and pH in affecting infant mortality.
i)
Here's a summary of the relevant statistics:
The coefficient C(2) for LEAD is 0.461798, indicating the effect of lead on infant mortality.
The coefficient C(3) for PH is -0.075179, indicating the effect of pH on infant mortality.
The coefficient C(4) for the interaction between PH and LEAD is -0.056862.
To test the interaction effect, we need to examine the p-value associated with the coefficient for the interaction term (C(4)). Here's the relevant information:
C(4) (Interaction Term) p-value: 0.0631
Given the p-value of 0.0631 for the interaction term, it is not statistically significant at the conventional 5% significance level. Therefore, you fail to reject the null hypothesis (H0) that the effect of lead on infant mortality does not depend on the pH level of the water.
In conclusion:
Based on these results, it does not appear that the effect of lead on infant mortality is significantly influenced by the acidity level of the water (pH).
The p-value for the interaction term (C(4)) is greater than 0.05, suggesting that the interaction is not statistically significant.
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