Mid Term Exam MSDA3055 Linear Regression and Time Series Oct 25, 2021 Question 1 Consider the regression model Y, = 1 + B2X; + U; Suppose you know that B1 = 4. Derive a formula for the least squares estimator of B2. Answer: Question 2 A researcher has data for 50 countries on N (the average number of newspapers purchased per adult in one year) and G (GDP per capita) measured in US $, and fits the following regression: N = 25 + 0.02 G Where r2 = 0.06 and RSS = 4000. The researcher realizes that GDP has been underestimated by $100 in every country and that N should have been regressed on G*, where G* = G + 100 Explain, with mathematical proofs, how the following components of the output would have differed: a. The coefficient of GDP b. The intercept Answer: Question 3 Suppose that log Y = B2 log X + U where Y is the number of crimes committed in town and X is the number of police officers working in town. You, the researcher, wish to interpret this regression as a causal model of the effect of X on Y. a. How would you interpret B2? (Hint: You may need to use the derivative chain rule.) b. How would your answer to part (a) change if the model had been defined with X instead of log X? Answer: Question 4 Consider the California Test Score Data Set (given in excel file in Question 4 sheet paper.) a. How many observations do you have in the data set? b. Consider the variable avginc (the average district income) in column O, measured in 1000s of dollars: i. Define a new variable, income, which is the variable avginc multiplied by 1000. ii. What is the mean and standard deviation of avginc? iii. What is the mean and standard deviation of income? c. What is the correlation between avginc and math score in column R? What is correlation between income and math score in column R? Are the two correlations the same or different? Explain. d. Run a simple OLS linear regression with income as dependent variable and math score as independent variable and interpret its coefficient estimates. e. Interpret goodness-of- fit for OLS linear regression. Answer: Question 5 Consider the regression model Y, = 2.5 + 31.5 X; + U; Where the total number of observations is 25, standard deviation of the int nd th deviation of the slope is 3.2. Test the claim that the slope must be less than 30 in a significant level of 1%. Answer:Question 6 Which of the following statements is TRUE concerning 0L5 estimation? a. 0L5 minimizes the sum of the vertical distances from the points to the line b. OLS minimizes the sum of the square of the vertical distances from the points to the line. c. 0L5 minimizes the sum of the horizontal distances from the points to the line (1. OLS minimizes the sum of the square of the horizontal distances from the points to the line. Answer: Question 7 The residual from a standard regression model is dened as The difference between the actual value, Y, and the mean, 17. The difference between the tted value, I7, and the mean, 17. The difference between the actual value, Y, and the tted value, 17'. The square ofthe difference between the tted value, 17', and the mean, I7. Flange Answer: Question 8 Which of the following statements best describes the algebraic representation of the tted regression line? a. =ii+ ag-+17, b. =+xi c. zd+3xi+Ui d. t1:&+,t?x,+, Question') In a multiple regression model, the adjusted R2 Cannot be negative. Will never be greater than the regression R2. Equals the square of correlation coefcient r (:1. Cannot decrease when an additional explanatory variable is added. 319'?\" Answer: Question 10 If we multiply both Y and X by 1000 and re-estimate the regression, the intercept coefcient and its standard error will a. Increase by 1000 times b. Decrease by 1000 times c. Remain same (1. Increase by [101?] times Answer: Question 11 In a simple linear regression model, the elasticity of Y with respect to X is given by: C. 32X 01- .32 [1/Y] Answer: Question 12 Given I",- = 131 + 13'sz + ,8ng; + 11,-, state which of the following statements is true: a. 32 measures the change in Y per unit change in X1, holding the value of X3 constant b. 33 measures the change in Y per unit change in X3, and not of any effect that X 2 may have on Y. c. Both [a] and [b] are true. d. Neither {a} nor [b] is true. Answer: Question 13 When comparing r2 of two regression models, the models should have the same X variables. Y variables. Error term. Beta coefcients. 9-!" 9"!\" Question 14- If in Y;- = 51 + [3in + Up both Y and X are standardized variables, the intercept term will be Positive Negative Between -1 and +1 Equal to zero P-F'F'?' Answer: Question 15 Which one of the following is NOT an assumption of the classical linear regression model? a. The explanatory variables are uncorrelated with the error terms. b. The disturbance terms have zero mean. c. The dependent variable is not correlated with the disturbance terms. d. The disturbance terms are independent of one another