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1 EBS Business School Prof. Dr. Mehdi Hosseinkouchack Fall 2023 BSc Econometrics 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) states that μ_female = μ_male, indicating that there is no gender difference in average hourly earnings. Hypothesis 1 (alternative): μ_female ≠ μ_male (The average hourly wage differs across genders in a statistically meaningful way.) 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]

2 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

4 independent variables, Part E postulated a linear connection. On the other hand, the natural logarithms of age (ln_age) and earnings (ln_ahe) were shown in Parts F and G, respectively. R- squared, a measure of the models' goodness of fit, went up somewhat from 0.180 in Part E to 0.197 in Part G. Depending on the underlying nature of the relationship, one can choose between logarithmic and linear transformations. Coefficients in Part E are easily interpreted, but non- linear patterns may be better captured in Parts F and G, which include logarithmic adjustments. Part G's inclusion of the natural logarithm of age implies that potential non-linearities or diminishing returns with age may be taken into account. In general, the choice of model is determined by the particular properties of the data and the intended level of interpretability of the outcomes. To validate model assumptions, more diagnostics and residual checks should be carried out. the preference between linear and logarithmic specifications depends on the underlying relationship in your data and the assumptions you have about how earnings change with age. If there is evidence of non-linearities, logarithmic transformations might better capture the pattern. The R-squared values can also be considered to assess the overall goodness of fit. It's recommended to assess the residuals and conduct additional diagnostic tests to validate model assumptions. 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]

5 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 I1 Report the regression results and provide interpretations for each estimated coefficient. Answer: The association between many predictor factors and the natural logarithm of average hourly earnings (ln_ahe) is revealed by the regression analysis. The intercept, at 1.9061, represents the baseline value and represents the expected ln_ahe when all other predictors are zero. The female variable's positive coefficient (0.0528) indicates that, on average, being a woman is linked to a little rise in ln_ahe. The positive coefficient of 0.3071 for the bachelor variable indicates that those with a bachelor's degree have a greater ln_ahe than people without one. According to the age coefficient (0.0267), ln_ahe often rises with age. The interaction terms provide more detailed information. For example, the negative coefficient for female_age (-0.0083) indicates that the increase in ln_ahe with age for females is less marked than. I2 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 5% type I error. Answer: Null Hypothesis (H0): Gender and bachelor status have no bearing on the effect of age on wages (both interaction variables are jointly equal to zero).