# (Solved) Problem 1. Policymakers are interested in how the number of crimes in a large city depends on the unemployment rate and the number of TV advertisements

Problem 1. Policymakers are interested in how the number of crimes in a large city depends on the unemployment rate and the number of TV advertisements which discourage criminal acts. Using the data for the period from January 2011 to July 2019, the following OLS estimates are obtained (standard errors are in parentheses):

Population regression function (PRF): Yt = β0 + β1Xt + β2At + ut

Sample regression function (SRF): Yt = b0 + b1Xt + b2At + et

OLS SRF estimates: Ŷt = 151.0 + 8.899 Xt – 20.01 At

(40.51) (4.589) (8.031)

R^2 = 0.845, S = 0.472, n = 103

where Yt = the number of crimes in month t

Xt = the unemployment rate (in percentage points) in month t

At = the number of TV advertisements which discouraged criminal acts in month t

(A) What are the population parameters in this example? What are the point estimators in this example? Clearly explain the difference between a population parameter and a point estimator.

(B) Referring back to part (a), what are the two components of Yt in the population regression function, and what are the two components of Yt in the sample regression function? Explain clearly how each component in the PRF relates to each component in the SRF.

(C) Clearly interpret the estimates 151.0, 8.899, and 20.01.

(D) Clearly interpret the coefficient of determination (R^2).

(E) Test whether the unemployment rate explains the behavior of the number of crimes. In answering, write the null and alternative hypotheses and the decision rule. Show your calculations. State your conclusion.

(F) Test whether the unemployment rate and the number of TV advertisements jointly explain the behavior of the number of crimes. In answering, write the null and alternative hypotheses and the decision rule. Show your calculations. State your conclusion. Use Excel (as explained in Unit 6) to find the critical values.

(G) State the assumption(s) that are needed for your answer to parts (e and f) to be valid.

(H) Calculate the 95% confidence interval estimate of the population parameter of the number of TV advertisements. Show your calculations. Interpret your results.

(I) It is claimed that the number of TV advertisements (which discourage criminal acts) does not explain the behavior of the number of crimes. Use your results in part (h) to determine whether you reject this claim. Clearly explain.

(J) Calculate the 90% confidence interval estimate of error variance (σ^2). Show your calculations. Interpret your results. Use Excel (as explained in Unit 6) to find the chi-square values.

(K) Use your result in part (j) to explain whether you reject the claim that the true standard error of regression is 0.45 against the alternative that it is bigger than 0.45. In answering this question, write the null and alternative hypotheses.

1) List the steps in regression analysis. Have we taken all these steps in this example? Explain clearly.