5. Note: The individual parts of this problem are independent and have nothing to do with each other Hint: For each of the proofs, start with a model with starting values for the variables, then consider the value of the left-hand-side when X changes by one percent (linear-log and log-log model) or one unit (log-linear model). Finally, take the difference between those two equations and use the fact that differences in logs can be converted into percentage changes to prove the interpretation of the coefficient. (a) A student wants to study the impact of the number of kilometers (K) run by a car on its resale price (P, measured in dollars). For the study, she selects a random sample of 125 second hand car sellers from her city and estimates the following regression equation: P=57.85-250.25ln(K) What is the interpretation of the slope coefficient? Prove mathematically why the coefficient has the interpretation you provided (b) A researcher is interested in finding out the relationship between the price of a house (H, mea- sured in hundred dollars) and its distance from the highway (D, measured in kilometers) passing through her district. She estimates the following regression using observations on prices of houses and their corresponding distance from the highway: In(H) 68.56.07D What is the interpretation of the slope coefficient? Prove mathematically why the coefficient has the interpretation you provided (c) A survey was conducted to study the impact of per capita GNP (X, measured in thousand dol- lars) on life expectancy at birth (Y). Data across 100 countries was collected and the following regression was estimated: In(Y) 70.52+1.19ln(X) What is the interpretation of the slope coefficient? Prove mathematically why the coefficient has the interpretation you provided