Data on 1500 purchases of canned lite tuna are in the data file tunafish. There are four brands of tuna (Starkist - water, Starkist - oil, Chicken of the Sea - water, Chicken of the Sea - oil). The A.C. Nielsen data were made available through the University of Chicago's Graduate School of Business. The data file tunafish_small is a smaller dataset with 250 purchases. The data are in "stacked" format with four data lines per purchase, one for each tuna brand. The consumer choice is indicated by the indicator variable CHOICE. Relevant variables are NETPRICE = price minus coupon value, if used; DISPLAY \(=1\) if product is on display, FEATURE \(=1\) if item is featured, and INCOME \(=\) household income.

a. What is the primary variable-type distinction between NETPRICE and INCOME?

b. What is the sample percentage of purchases for each brand? What do you observe about consumer preferences for these product choices?

c. Using the conditional logit model, write the probability of choosing each brand using as explanatory variables NETPRICE, DISPLAY, and FEATURE, plus an alternative specific constant using Starkist packed in water as the base category.

d. Estimate the model specified in part (c).

e. For the model in (d) find the marginal effect of NETPRICE on the probability of choice of each brand, using for all brands DISPLAY \(=\) FEATURE \(=1\). Do these marginal effects have the signs you anticipate? Are the marginal effects statistically significant?

f. Add the variable INCOME to the model specified in (c). Perform a likelihood ratio test of its significance.

g. For the model in (f) find the marginal effect of NETPRICE on the probability of choice of each brand, using for all brands DISPLAY \(=\) FEATURE \(=1\) and \(I N C O M E=30\).