2. Given a random sample, {(yi,xi),i=1,,n} from a bivariate random variable (Y,X): {(y1,x1),,(y,x)} where Y is continuous random variable and X is a Bernoulli random variable with 0 and 1 realizations, the location regression model is considered as yi=0+0xi+i,i=1,,n and ^n and ^n are the OLS estimators for 0 and 0. Please answer the following questions given all classical assumptions are hold. (a) Show the null H0:0=0 is equivalent to H0:E(YX=0)=E(YX=1). (b) What is ^n ? 1 (c) Show ^n is equal to the difference between yn1 and yn2, where yn1=i=1nI(xi=1)i=1nyiI(xi=1)=n1i=1nyiI(xi=1),yn2=i=1nI(xi=0)i=1nyiI(xi=0)=n2i=1nyiI(xi=0) and I() denotes an indicatot function which equals 1 if statement in the parentheses is true and 0 if is false. (d) How do you test H0:E(YX=0)=E(YX=1) using yn1 and yn2 ? (e) How do you test H0:0=0 using ^n ? (f) Are the test statistics for H0:E(YX=0)=E(YX=1) using yn1 and yn2 and for H0:0=0 using ^n equivalent? 2. Given a random sample, {(yi,xi),i=1,,n} from a bivariate random variable (Y,X): {(y1,x1),,(y,x)} where Y is continuous random variable and X is a Bernoulli random variable with 0 and 1 realizations, the location regression model is considered as yi=0+0xi+i,i=1,,n and ^n and ^n are the OLS estimators for 0 and 0. Please answer the following questions given all classical assumptions are hold. (a) Show the null H0:0=0 is equivalent to H0:E(YX=0)=E(YX=1). (b) What is ^n ? 1 (c) Show ^n is equal to the difference between yn1 and yn2, where yn1=i=1nI(xi=1)i=1nyiI(xi=1)=n1i=1nyiI(xi=1),yn2=i=1nI(xi=0)i=1nyiI(xi=0)=n2i=1nyiI(xi=0) and I() denotes an indicatot function which equals 1 if statement in the parentheses is true and 0 if is false. (d) How do you test H0:E(YX=0)=E(YX=1) using yn1 and yn2 ? (e) How do you test H0:0=0 using ^n ? (f) Are the test statistics for H0:E(YX=0)=E(YX=1) using yn1 and yn2 and for H0:0=0 using ^n equivalent