Problem 1 An analyst believes that an increase in the price of Tim Horton’s medium coffee by 2$ decreases the demand for Tim Horton’s medium coffee by 15%. At the same time, the analyst believes that an increase in the price of Tim Horton’s large coffee by 3$ increases the demand for Tim Horton’s medium coffee by 20%. i. Under the analyst’s hypothesis, are the medium and large coffee net substitutes or net complements? ii. Write down a regression model that allows you to investigate the two claims of the analyst. iii. Formalize a joint hypothesis test that allows you to test both hypothesis together. How would you conduct such a test? Be precise. I want to know what test statistic you would use and how you would use this test statistic to influence your decision. iv. What assumptions are required for you to conduct this hypothesis test? Problem 2 In a study relating college grade point average to time spent in various activities, you distribute a survey to several students. The students are asked how many hours they spend each week in four activities: studying, sleeping, working and leisure. Any activity is put into one of the four categories, so that for each student, the sum of hours in the four activities must be 24 ∗ 7 = 168 i. In the model GP A = β0 + β1study + β2sleep + β3work + β4leisure + u does it make sense to hold sleep, work and leisure fixed, while changing study? ii. Explain why this model violates Assumptions MLR.3. iii. How could you reformulate the model so that its parameters have a useful interpretation and it satisfies Assumption MLR.3? Why does this work? 1 Problem 3 Consider the following baseline regression, where wages are in $ per hour and education is in years: wagei = β0 + β1educi + ui Using the OLS estimator for this simple linear regression, a researcher finds: βˆ 0 = 2.12 Use this model as a reference point to answer the following questions: i. Using the same dataset, a researcher adds experience to the wage regression wagei = β0 + β1educi + β2expi + ϵi and estimates βˆ 1 = 1.98 a. if Cov(educ \i , expi) > 0, what is the sign of βˆ 2? Show your work (Is it positive or negative?) b. if Cov(educ \i , expi) < 0,="" what="" is="" the="" sign="" of="" βˆ="" 2?="" show="" your="" work="" ii.="" using="" the="" same="" dataset,="" a="" researcher="" instead="" adds="" age="" to="" the="" wage="" regression="" wagei="β0" +="" β1educi="" +="" β3age="" +="" νi="" and="" estimates="" βˆ="" 1="2.4," βˆ="" 3="−0.4" what="" is="" the="" sign="" of="" cov(educ="" \i="" ,="" agei)?="" explain="" intuitively="" the="" mechanism="" through="" which="" age="" affects="" the="" estimated="" relatonship="" between="" education="" and="" wages.="" problem="" 4="" a="" researcher="" wants="" to="" investigate="" how="" the="" relationship="" between="" education="" and="" wages="" differs="" between="" men="" and="" women.="" he="" records="" the="" variable="" genderi="0" if="" the="" individual="" is="" a="" man="" and="" genderi="1" if="" the="" individual="" is="" a="" woman.="" he="" estimates="" the="" following="" regression="" model:="" wagei="β0" +="" β1genderi="" +="" β2educi="" +="" β3educi="" ∗="" genderi="" +="" ui="" where="" e[ui="" |genderi="" ,="" educi="" ]="0" 2="" i.="" provide="" an="" expression="" for="" e[wagesi="" |genderi="man," educi="" ]="" and="" e[wagesi="" |genderi="woman," educi="" ].="" ii.="" what="" parameter="" represents="" a="" change="" in="" the="" intercept="" of="" wages="" between="" men="" and="" women?="" what="" parameter="" represents="" a="" change="" in="" the="" slope="" of="" education="" on="" wages="" between="" men="" and="" women?="" iii.="" if="" βˆ="" 1="">< 0="" and="" βˆ="" 3=""> 0, graphically show the relationship between education (x-axis) and wages (y-axis) for men and women separately. iv. Intuitively, how to men and women differ? Problem 5 A researcher is not convinced that increasing years of education always causes an increase in wages. He thinks getting a two-years professional college degree is better than a four-years university degree. In a dataset, he records individuals with 12 years of education as high school graduates, 14 years of education as professional degrees holders and 16 years of education as university-degree holders. These are the only possible categories: educi = 12, if Highschool 14, if P rofessional 16, if University To separately investigate the effect of different education levels on wages, a researcher proposes the following regression: wagei = β0 + β1HSi + β2P rofi + β3Unii + ui (1) Where HSi = ( 1, if educi = 12 0 otherwise Where P rofi and Unii are defined similarly. i. Show that the regression in (1) has too many parameters (MLR.3 is not satisfied). hint: you can rely on E[wagei | educi ] ii. Propose an alternative regression model that captures all the information in (1) while satisfiying MLR.3. hint: there is more than one correct answer. iii. Propose a null hypothesis and a test-statistics to investigate whether increasing education by one year always (weakly) increases wages. iv. To compute the test-statistic you provided in (iii), what regression would you specify to impose the null hypothesis? 3 (Bonus) Problem 6 An analyst believes the quantity demanded for Starbucks coffee (Qd) is a linear function of its price (P): Qd = β0 + β1P + u (2) At the same time, another clever analyst tells you this regression won’t allow you to recover the causal effect of prices on quantity demanded (β1) because prices are also a function of quantity demanded. The analyst also tells you that Starbucks marginal costs of producing coffee (MC) affects the price it charges independently of quantity demanded: P = α0 + α1Qd + α2MC + ϵ (3) i. How can you use information from the equation (1) and (2) to identify the causal effect of prices on quantity demanded (β1)? Be precise. ii. Given your answer in (i), you are able to identify β1 given data on quantity demanded Qd, prices P and Starbucks’ marginal costs MC. Can you use quantity purchased as a proxy for quantity demanded? Why or why not?