1. The minimum (legal) drinking age in the United States is age 21 before age 21, individuals are not permitted to purchase or publicly consume alcohol. Underage drinking still occurs through either illegal purchases or consumption of alcohol purchase by older individuals. Suppose that you are interested in determining the effect of the minimum drinking age law upon alcohol consumption. The variables of interest are alcohol (the average number of alcoholic drinks that an individual drinks in a week; continuous variable) and age (an individual's age in years; discrete variable). The general form of the conditional expectation is E'(alcahal|age) = m(age), and you are particularly interested in the difference between what happens at age 21 and what happens at age 20, Le. m(21) m(20), but at the same time trying to control for a possible relationship between alcohol and age (a relationship that would exist even without a drinking age law). Assume that we focus on the population of ages between 16 and 25 (age E {16,17,...,25}). (a) Explain why the linear model E(alcahal|age) : {31 + gage is too simplistic here. (b) Dene the variable average : 1(age 2 21) (equal to one if age 2 21, equal to zero otherwise). i. Utilizing the average variable, propose a linear model for E (alcohol (age) that is a step function with one value for underage individuals and one value for average individuals (with a jump in the conditional expectation at age 21). ii. Utilizing the average variable, propose a linear model for E(alcoh,allage) that has a common slope for both underage and average individuals but that allows for a jump in the conditional expectation at age 21. iii. Show that your model in (i) is nested within (or a special case of) your model in (ii) by stating which parameter restriction(s) of the model in (ii) yields the model in (i). iv. In terms of the parameters from your model in (ii), what is the effect of the drinking age law? What value(s) of the parameter(s) would correspond to a situation in which the drinking law has no effect? v. In terms of the parameters from your model in (ii), what is the expected difference in average alcohol consumption between a 25 year old and a 22 year old? How about between a 22 year old and a 19 year old? (c) Propose a new linear model that now allows for a different slope for underage and average individuals (but still allows the jump at age 21). How do your answers to parts iv. and v. from the previous question change