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D Compute and report the average hourly earnings (ahe) across genders. Is there a statistically significant difference in the averages? In your solution you should write out the null and alternative hypotheses and explain the testing procedure before conducting the corresponding testing procedure; in doing so, follow the discussions from section 3.4 of SW (the reference book of the course). Use a type I error of 5%. Answer: The null hypothesis (H0) posits that the mean hourly earnings for females (μ_female) are equal to the mean hourly earnings for males (μ_male), suggesting the absence of any statistically significant gender-based difference. In contrast, Hypothesis 1 (alternative) proposes that μ_female is not equal to μ_male, implying a statistically meaningful disparity in average hourly wages between genders. E Run a regression of ahe on a constant, age, female and bachelor and provide the regression Answer: OLS Regression Results Dep. Variable: ahe R-squared: 0.18 Model: OLS Adj. R- squared: 0.18 Method: Least Squares F-statistic: 544.5 Date: Sun 12 Nov 2023 Prob (F- statistic): 6.51e -320 Time: 2:33:35 Log- Likelihood: -27443 No. Observations: 7440 AIC: 5.49E+04 Df Residuals: 7436 BIC: 5.49E+04 Df Model: 3 Covariance Type: nonrobus t coef std err t P>|t| [0.025 0.975] intercept 1.8662 1.188 1.571 0.116 -0.462 4.194 age 0.5103 0.04 12.912 0 0.433 0.588 female -3.8103 0.23 -16.596 0 -4.26 -3.36 bachelor 8.3186 0.227 36.584 0 7.873 8.764 Omnibus: 1975.582 Durbin- Watson: 1.935

Prob(Omnibus) : 0 Jarque- Bera (JB): 6089.399 Skew: 1.36 Prob(JB): 0 Kurtosis: 6.499 Cond. No. 316 Notes: [1] Standard Errors assume that the covariance matrix of the errors is correctly specified. F Run a regression of ln(ahe) on a constant, age, female and bachelor. Answer: OLS Regression Results Dep. Variable: ln_ahe R-squared: 0.196 Model: OLS Adj. R- squared: 0.196 Method: Least Squares F-statistic: 605.7 Date: Sun 12 Nov 2023 Prob (F- statistic): 0 Time: 2:34:23 Log- Likelihood: -5066.6 No. Observations: 7440 AIC: 1.01E+04 Df Residuals: 7436 BIC: 1.02E+04 Df Model: 3 Covariance Type: nonrobust coef std err t P>| t| [0.025 0.975] intercept 1.9414 0.059 33.083 0 1.826 2.056 age 0.0255 0.002 13.067 0 0.022 0.029 female -0.1923 0.011 -16.953 0 -0.215 -0.17 bachelor 0.4378 0.011 38.964 0 0.416 0.46 Omnibus: 316.825 Durbin- Watson: 1.936 Prob(Omnibus) : 0 Jarque-Bera (JB): 508.141 Skew: -0.375 Prob(JB): 4.56E-111 Kurtosis: 4.037 Cond. No. 316

employed natural logarithms for age (ln_age) and earnings (ln_ahe), respectively. The R-squared, a metric gauging model fit, exhibited a modest increase from 0.180 in Part E to 0.197 in Part G. The choice between linear and logarithmic transformations hinges on the inherent nature of the relationship being investigated. While Part E's coefficients offer straightforward interpretations, the inclusion of logarithmic adjustments in Parts F and G is designed to potentially capture non-linear patterns. Notably, Part G's introduction of the natural logarithm of age suggests an acknowledgment of potential non-linearities or diminishing returns with age. The selection of the most suitable model depends on the unique characteristics of the dataset and the desired interpretability of the results. To validate model assumptions, it is advisable to conduct thorough diagnostics and checks on residuals. Assessing the residuals and performing additional diagnostic tests becomes crucial, especially if there are indications of non-linearities. The consideration of R-squared values aids in evaluating the overall goodness of fit, but a comprehensive assessment involves scrutinizing residuals and conducting diagnostic tests to validate the underlying assumptions of the chosen model. I Run a regression of ln(ahe) on female, bachelor, and the interaction terms female × age and bachelor × age. OLS Regression Results Dep. Variable: ln_ahe R-squared: 0.197 Model: OLS Adj. R- squared: 0.196 Method: Least Squares F-statistic: 364.6 Date: Sun 12 Nov 2023 Prob (F- statistic): 0 Time: 2:35:51 Log- Likelihood: -5064.2 No. Observations: 7440 AIC: 1.01E+04 Df Residuals: 7434 BIC: 1.02E+04 Df Model: 5 Covariance Type: nonrobust coef std err t P>| t| [0.025 0.975]

intercept 1.9061 0.094 20.343 0 1.722 2.09 female 0.0528 0.119 0.444 0.65 7 -0.181 0.286 bachelor 0.3071 0.118 2.607 0.00 9 0.076 0.538 age 0.0267 0.003 8.495 0 0.021 0.033 female_age -0.0083 0.004 -2.067 0.03 9 -0.016 0 bachelor_age 0.0044 0.004 1.112 0.26 6 -0.003 0.012 Omnibus: 314.174 Durbin- Watson: 1.937 Prob(Omnibus) : 0 Jarque-Bera (JB): 503.182 Skew: -0.373 Prob(JB): 5.44E-110 Kurtosis: 4.032 Cond. No. 916 11 Report the regression results and provide interpretations for each estimated coefficient. Answer: The regression analysis sheds light on the relationship between various predictor factors and the natural logarithm of average hourly earnings (ln_ahe). The intercept, set at 1.9061, signifies the baseline value, representing the expected ln_ahe when all other predictors are zero. The positive coefficient of 0.0528 for the female variable suggests that, on average, being a woman is associated with a slight increase in ln_ahe. Similarly, the positive coefficient of 0.3071 for the bachelor variable indicates that individuals with a bachelor's degree tend to have a higher ln_ahe compared to those without one. The age coefficient (0.0267) implies that ln_ahe generally increases with age. Further insights are provided by the interaction terms. For instance, the negative coefficient for female_age (-0.0083) suggests that the rise in ln_ahe with age is less pronounced for females compared to other groups. 12 Test if the effect of age on earnings depends on other factors or not. Write out the null and alternative hypotheses and conduct a relevant statistical test; use a 5%