A researcher is studying the effect of parents’ level of education on educational achievement of individuals. Using data on completed years of education (educ), mother’s level of education (motheduc), father’s level of education (fatheduc), a measure of cognitive ability (abil), and logarithm of family income (l.income), the researcher estimates 4 models. Table (1) shows the OLS estimates for each model, with standard errors in parentheses underneath each coefficient. The dependent variable in all the models is educ. Model 1 Model 2 Model 3 Model 4 Independent variables l.income – – – 5 (0.2) motheduc 0.27 (0.02) 0.18 (0.03) 0.17 (0.03) 0.15 (0.03) fatheduc – 0.11 (0.02) 0.11 (0.02) 0.10 (0.02) abil 0.53 (0.03) 0.52 (0.03) 0.39 (0.03) 0.35 (0.03) abil_squared – – 0.05 (0.01) 0.09 (0.01) Intercept 6.94 (0.32) 8.4 (0.12) 8.74 (0.31) 8.74 (0.31) N 1230 1230 1230 1210 SSR 3999.24 3899.97 3785.24 3541.2 R-squared 0.35 0.38 0.45 0.49 Table (1) Interpret the R-squared of the regression in model 1. [5 marks] Can you reject the claim that the returns to ability are linear? Be explicit about the hypothesis and any assumptions you make in order to answer the question. [6 marks] Carefully interpret the effect of increasing of l.income by 10. [6 marks] How the degrees of freedom in model (3) compare to the degrees of freedom in model (4)? Explain. [5 marks] Why did the coefficient of motheduc decrease after adding variable fatheduc to the regression? Discuss and if necessary, incorporate the Gauss-Markov assumptions in your answer. [8 marks] Use the Stata outputs presented in handout (1) to answer the following questions. Write the results from specification (1) that are presented in figure (2) in the format of a fitted line (round the decimals places up to 2). [6 marks] After adding variable abil to the model (Figure 3), what happened to the standard error of the coefficient of educ? Discuss. [10 marks] Are the coefficients of fatheduc and motheduc individually significant at 5% level? [4 marks] Will you keep the two variables that control for parental education in the model? Explain your answer. [10 marks] A researcher is collecting individual data by asking visitors in a train station in a medium sized town to fill in a questionnaire. It takes nearly 30 minutes to complete the survey questions. To encourage higher participation rate, the researcher provides a small financial incentive that is enough to buy a sandwich. The researcher wants to use this sample to make predictions about effect of using social media on health among the residents of the town. Explain the random sampling assumption, that is one of the Gauss-Markov assumptions for linear regressions. [5 marks] Do you think this assumption is satisfied in the sample collected by the researcher? Explain your answer in no more than 85 words. [5 marks] What are the possible consequences of violation of random sampling assumption for predictions based on this sample? Explain your answer in no more than 85 words [5 marks] Section B Attempt one question from this section for maximum of 25 marks The model below explains CEO salary (salary) expressed in thousands of pounds as a function of a firm’s sales in million pounds (sales), the return on a firm’s equity (roe), and the industry in which the firm is operating. There are two industries in the sample, services and consumer goods. These industries are captured respectively by two dummy variables, service and goods. The regression based on a sample of 209 firms is as follows: (salary) ̂ ＝ 4.5 + 0.27sales + 0.015roe – 0.08service (0.24) (0.03) (0.05) (0. 05) R-squared ＝ 0.35 SSR(Residual Sum of Squares) ＝ 42.9 Interpret the coefficient of service. [3 marks] In the regression model, the consumer goods as an industry is omitted from the model. Explain the implications of adding the binary variable goods to the regression model. [5 marks] The model is re-estimated with an interaction between the dummy variable service and the two continuous variables of the model. (salary) ̂ ＝ 4.9 + 0.22sales + 0.015roe – 5. 8service + 0.86 sales×service (0.24) (0.03) (0.003) (0. 5) (0.08) – 0.009 roe×service (0.008) R-squared ＝ 0.51 SSR＝ 31 Interpret the effect of roe on CEO’s salary and explain whether this differs between service and consumer goods industries. [7 marks] Implement an F-test for whether the determinants of CEO salaries differ significantly between service and consumer goods industries. What do you conclude? [10 marks] Suppose we are interested in examining the effect of pollution on housing prices. We estimate a model with the log of housing price as the dependent variable defined as log(price). Our main independent variable is nox, which is a measure of pollutants in the atmosphere. The variable ranges from 3 to 8, and higher values indicate a higher concentration of pollutants in the air. We also control for the number of bedrooms in a house (rooms) and the property’s proximity to public transport (dist). Our regression model yields the following estimates with standard errors in parentheses: (log(price)) ̂ ＝ 12.2 – 0.14nox – 0.01dist – 0.731rooms + 0.08rooms2 (0.57) (0.017) (0.01) (0.17) (0.01) Sample size: 506 R-squared ＝ 0.54 What is the purpose the quadratic term in rooms here (rooms2)? [3 marks] Interpret the coefficient on noxs. [4 marks] Write down an expression that shows the effect of a change in rooms on log(price), and use this to show how the effect of an additional room on house prices is different between a house with 2 bedrooms and a house with 3 bedrooms. [6 marks] Sketch a graph illustrating the relationship between log(price) and rooms. Make sure to label your graph and find the value of a possible turning point [6 marks] Test the overall significance of the regression in this question at the 5% level of significance. What do you conclude? [6 marks]
