4. Suppose the following environments and solve the given questions. (28 pts) (X, Y is a set of two random variables such that X and Y stand for student's gender and height (in cm), respectively. Here, we let 1, clo X = if male student; if female student. Suppose that the empirical researcher is interested in finding out the relationship between X and Y by the ordinary least squares estimator. We assume the following conditions: There are n observations in total. The following model is estimated by the ordinary least squares estimator: Y = a, +3, + U. Here, the subscript") stands for the index for the i-th student. We suppose that the above model does not satisfy the classical linear model condition, although it is known that the error satisfies the conditional homoskedasticity. 1 If we let up and um be the population means of Y for female and male students, respectively, rewrite a. and B. as functions of up and ym. (14 pts) 2 The researcher wishes to test a hypothesis given as Ho: a. = 150 and . = 0 using Wald-test statistic. Provide the formula of the Wald-test statistic and its asymptotic behaviors under the null and alternative hypotheses. (14 pts) 4. Suppose the following environments and solve the given questions. (28 pts) (X, Y is a set of two random variables such that X and Y stand for student's gender and height (in cm), respectively. Here, we let 1, clo X = if male student; if female student. Suppose that the empirical researcher is interested in finding out the relationship between X and Y by the ordinary least squares estimator. We assume the following conditions: There are n observations in total. The following model is estimated by the ordinary least squares estimator: Y = a, +3, + U. Here, the subscript") stands for the index for the i-th student. We suppose that the above model does not satisfy the classical linear model condition, although it is known that the error satisfies the conditional homoskedasticity. 1 If we let up and um be the population means of Y for female and male students, respectively, rewrite a. and B. as functions of up and ym. (14 pts) 2 The researcher wishes to test a hypothesis given as Ho: a. = 150 and . = 0 using Wald-test statistic. Provide the formula of the Wald-test statistic and its asymptotic behaviors under the null and alternative hypotheses. (14 pts)