what problems are created when we assume the errors are homoskedastic when in fact they are heteroskedastic

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1) (5 points) Read “EC295_assign2_DataDesc.pdf”. Produce a table of summary statistics (mean, standard deviation, min, max) for all the variables in the dataset. For each variable, describe what type of variable it is, and interpret its mean.

2) Estimating the relationship between course evaluations and beauty

a. (5 points) Create a scatterplot between course evaluations and beauty, with course evaluations on the vertical axis. Comment on the direction of the relationship between the two.

b. (5 points) Suppose you are interested in knowing whether beauty increases course evaluations. You propose the following linear regression model 𝑐𝑜𝑢𝑟𝑠𝑒_𝑒𝑣𝑎𝑙 = 𝛽0 + 𝛽1𝑏𝑒𝑎𝑢𝑡𝑦 + 𝑢 Estimate and interpret the two parameters of the model using the robust option (i.e., reg y x, robust). Does the intercept have a useful interpretation in this context?

c. (5 points) What is the predicted course evaluation for someone of average beauty? What is the difference in course evaluations between the most and least beautiful people in the sample? Show your work.

d. (5 points) Construct a 90% confidence interval for the effect a 2 unit increase in beauty on course evaluations

e. (5 points) State the first three assumptions of OLS. For each, state whether you think it is true or false and explain why.

3) (20 points) Testing hypotheses about the relationship between course evaluations and beauty a. (5 points) Based on the regression results from question 2

b, does beauty have a statistically significant effect on course evaluations? Explain.

4 b. (5 points) Compute the p-value for the null hypothesis that the slope equals 0.20 versus the alternative that it does not equal 0.20. Do you reject the null hypothesis at the 5% level? At the 1% level?

c. (5 points) Compute the residual for each person in the sample using the regression results from 2b. Then, plot these residuals against beauty in a scatterplot. Using this scatterplot, comment on whether you think the errors are homoskedastic or heteroskedastic. [Note: While the residuals do not definitively answer this question, we can look at the spread of the residuals at each level of classize for clues.]

d. (5 points) What problems are created when we assume the errors are homoskedastic when in fact they are heteroskedastic?

4) Dummy variables and multiple regression

a. (5 points) Suppose you think there is a gender gap in course evaluations, and model that relationship as 𝑐𝑜𝑢𝑟𝑠𝑒_𝑒𝑣𝑎𝑙 = 𝛽0 + 𝛽1𝑓𝑒𝑚𝑎𝑙𝑒 + 𝑢 Estimate and interpret the slope and intercept in this model using the robust option.

b. (5 points) Test the null hypothesis that women have lower course evaluations than men, versus the alternative that women score higher, at the 5% significance level.

c. (5 points) Construct and interpret a 95% confidence interval for the gender gap in course evaluations.

d. (5 points) Suppose you instead estimated the following model: 𝑐𝑜𝑢𝑟𝑠𝑒_𝑒𝑣𝑎𝑙 = 𝛾0 + 𝛾1𝑚𝑎𝑙𝑒 + 𝑢 where male is a dummy variable that takes the value 1 for male and 0 otherwise (i.e., male=1-female). Derive by hand the slope and intercept in this model.

e. Suppose you now add beauty as an additional independent variable in the regression model 𝑐𝑜𝑢𝑟𝑠𝑒_𝑒𝑣𝑎𝑙 = 𝛽0 + 𝛽1𝑓𝑒𝑚𝑎𝑙𝑒 + 𝛽2𝑏𝑒𝑎𝑢𝑡𝑦 + 𝑢 i) (5 points) Estimate the parameters of this new model using the robust option, and interpret them precisely. ii) (5 points) Using the formula for omitted variables bias, explain why the estimate of 𝛽1in this regression is smaller than the estimate from 4a. 5

f. (5 points) Using measures of fit we have discussed in class, compare the fit of the model in 4d to that of 4a.

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