The following solutions address various statistical and regression-related concepts. These explanations and calculations aim to clarify key ideas and principles in statistics, econometrics, and data analysis.
Solution 1: Uncorrelated but Dependent Random Variables
Answer: Yes, the random variables X and Y are uncorrelated and dependent because the covariance of X and Y can be expressed as:
Cov(X,Y) = E(X - E(X))(Y - E(Y)) = E(XY) - E(X)E(Y)
By substituting the mean values, Cov(X,Y) = E(XY).
If E(XY) equals E(X)E(Y), the random variables are independent or dependent. In this case, E(XY) = E(X)E(Y) = 0, demonstrating that the variables are uncorrelated (Cov(XY) = 0).
Solution 2: Bias in OLS Regression
Answer: The direction of the bias in OLS regression depends on the estimators and the covariance between the regressors and omitted variables. When the omitted variable ρ_Xu has a positive covariance with both a regressor and the dependent variable, the OLS estimate of the included regressor's coefficient will exceed the true value of that coefficient. Thus, the direction of bias in the wage equation's coefficient β_1 is greater than the true value of the coefficient of EDU.
Solution 3: R-squared and Cross Validation
Answer: The statement is false. R-squared is a statistic used to measure the amount of variation in the dependent variable explained by the independent variables. It does not indicate whether relevant variables are omitted from the regression equation. To address the omission of relevant variables, cross-validation can be used to identify and evaluate their impact.
Solution 4: Percentage Increase in Earnings
Answer: Given the equation Ln(Earnings) = 3 + 0.01*Experience, we can calculate Earnings as follows:
Ln(Earnings) = 3 + 0.01*3 Ln(Earnings) = 3.03 Earnings = Exp(3.03) Earnings ≈ 20.70
The percentage increase in earnings for 3 additional years of experience is approximately 21%.
Solution 5: Regression Analysis of Price and Quantity
Answer: In the regression of P and Q, since the mean and covariance of the error term u and v are both zero, it can be deduced that, on average, for every unit increase in Quantity, the Price will increase by β_1 units.
Solution 6: Population of Interest vs. Population Studied
Answer: A population of interest is the group from which a researcher aims to draw conclusions. On the other hand, the population studied is a subset of the general population, consisting of individuals with a specific trait, such as age, gender, or health condition. In the context provided, the population studied comprises graduates from Columbia, while the population of interest focuses on understanding the differences in learning across various Ivy League schools.
Solution 7: Convergence of Regression Coefficients
Answer: The estimate of β_1 converges in probability to α_1 when the limit of the difference between the two coefficients approaches zero as the sample size (n) increases.
Solution 8: Convergence of Regression Coefficients
Answer: The true value of α_1 and β_1 is 1.5 because they both converge in probability to the same value.
Solution 9: Convergence of OLS Estimator
Answer: It is reasonable to conclude that the OLS estimator δ1 converges in probability to α1. As the sample size (n) increases, the probability of the random variable approaching zero suggests that the estimator becomes more precise.