Exercises

Part I (Probability) Q1. Random Variables a. Gender: random, no underlying covariate/biased introduced; b. No. crash: random, depending on unpredictable factors: hardware failures, software malfunction, etc.; c. Commute time: random, influenced by unpredictable variables: transport delays, traffic jams, accidents; d. Computer assignment: random, no obvious assignment process known; e. Rainfall: random due to randomness of atmostpheric system. Temperature, wind speed, humidity, and other factors may vary randomly. Q4. Sample Mean There is a thin chance that the sample average weight of four students is exactly equal to the class (population) mean. However, the sample mean is generally close to but not equal to the population average due to sample error. ...

Logit Model Exercises Question 1 a. Describe ROC Curve ROC curves usually look like a bow reaching out to the top left corner of the chart. The more the curve bow out from the diagonal, the better the credit scoring model. b. Estimate of the Probability of External Financing 1 2 3 mdl2_zi <- \(def, ass, iag, p_fin){-0.72 + 0.02*def + 0.0003*ass - 0.002*iag + 0.79*p_fin} mdl2_prob <- \(z){exp(z)/(1+exp(z))} prob_firm1 <- mdl2_prob(mdl2_zi(1.10, 1.00, 0.00, 0.00)) The probability of external financing of Firm 1 using Model 2 is 33.23. c. Marginal Effect Explanation 1 2 3 mdl2_me_pf <- \(z){(exp(z)/(1+exp(z))^2)*0.79} me_f1 <- mdl2_me_pf(mdl2_zi(1.10, 1.00, 0.00, 0.00)) me_f2 <- mdl2_me_pf(mdl2_zi(0.13, 1.00, 0.00, 0.50)) The marginal effect generally varies across observations because it depends not only on the estimated coefficients but also the evaluation point through \(z_i\), i.e. \(f(z_i)\). For instance, the marginal effects shrink at extreme probabilities as \(f(z_i)\) becomes smaller. ...

Panel Data, Endogeneity, and Diff-in-Diff Exercises Question 4 a. Endogeneity Issue The regression model omits a dummy variable of Industry (1=Creative, 0=Non-creative), i.e. \(\epsilon_{it} = c D_i^{CRE} + u_{it}\), which may create endogeneity problem if \(Cov(F_{it}, D_i^{CRE}) \neq 0\). b. Sign of Biased Coefficient Assume that \(F_{it}<0.5\) on average, then \(Cov(F_{it}, D_i^{CRE}) > 0\) and \(Cov(R_{it}, D_i^{CRE}) > 0\), leading to a positively biased estimate of \(b\), i.e. \(\hat b > b\). ...