ECON 103, Winter 2025
Assignment 2

Question 1

(a) For the regression, the 95% confidence interval for the intercept is [72.5,78.1]. Calculate the standard error of the estimated intercept.

(b) From the regression output, the standard error for the slope coefficient is 0.32. Test the null hypothesis that the true slope, β2 is 1.2 (or less) against the alternative that the true slope is greater than 1.2 at the 5% level of significance. Show all steps of this hypothesis test, including the null and alternative hypotheses, and state your conclusion.

(c) On the regression output, the automatically provided p-value for the estimated slope is 0.003. What is the meaning of this value? Use a sketch to illustrate your answer.

(d) A CEO claims that increasing investments in DEI sustainability programs has no impact on retention rates. Using the estimated equation and the information in parts (a)–(c), test the null hypothesis that the slope parameter, β2 is zero or less, against the alternative hypothesis that it is positive. Use the 10% level of significance. Show all steps of this hypothesis test, including the null and alternative hypotheses, and state your conclusion. Is there any statistical support for the CEO’s claim?

(e) In 2022, a University’s Econ department had an average monthly investment of $15,000. Based on the estimated equation, the least squares estimate E(RET|SUST INC) = 15 is 97.05%, with a standard error of 1.2. The actual retention rate for the department that year was 95%. Would you say this value is surprising or not surprising? Explain.

Question 2

(a) If we estimate the regression W AGE = β1 + β2EDUC + e for individuals living in a metropolitan area, where MET RO = 1, is there a statistically significant positive relationship between expected wages and education at the 1% level for these individ uals? Clearly state the test statistic used, the rejection region, and the test p-value.

What do you conclude? How much of an effect is there and what does it mean?

(b) Estimate the elasticity of expected WAGE with respect to EDUC, evaluated at the sample means. Construct a 95% interval estimate for the elasticity, treating the sample means as if they are given (not random) numbers. What is the interpretation of the interval?

(c) Test the null hypothesis that the elasticity, calculated in part (b), is one against the alternative that the elasticity is not one. Use the 1% level of significance. Clearly state the test statistic used, the rejection region, and the test p-value. What do you conclude?

(d) Estimate the quadratic regression W AGE = α1 + α2EDUC2 + e and discuss the results. Estimate the marginal effect of another year of education on wage for a person with 12 years of education and for a person with 16 years of education. Compare these values to the estimated marginal effect of education from the linear model.

(e) Plot the fitted linear model from (a) and the fitted values from the quadratic model from part (e) in the same graph with the scatterplot of WAGE and EDUC. Which model appears to fit the data better? Does your R2 align with this deduction?

(f) Estimate the expected wage, E(W AGE|EDUC) = b1 + b2EDUC, for an individual with 16 years of education. Construct a 95% interval estimate of the expected wage.

(h) Construct a residual histogram of the regression in (a) and carry out the Jarque–Bera test for normality. Is it more important for the variables EDUC and WAGE to be normally distributed, or that the random error e be normally distributed? Explain your reasoning.

Question 3