answer all the questions please

d. (7 marks) Demonstrate that the two stage least squares (2SLS) estimate of ag is identical to the IV estimate in c. Recall that the 2SLS estimate is the OLS estimate of ag in model (8) when Y is replaced by its predicted (fitted) values by OLS in the reduced form equation Y = VI+72+W. where W is an error term (the Greek letters continue to be parameters). e. (6 marks) Suppose that on = 0. Determine whether X/ Y is a consistent estimate of 02. Justify your answer.c. (5 marks) Explain with detailed arguments whether there is any valid instrument to estimate on. If your answer is positive, demonstrate that the corresponding instru- mental variable (IV) estimate is consistent. oz is identified because the exogenous variable Z is omitted in equation (8), so it can be used as instrument as it is independent of V. Since it is present in equation (7) it is correlated with Y (relevant) if Bs / 0. E (Z, - 2) ( X, - X) E(Zi - 2) Xi E ( Zi - 2 ) ( Y - Y) E ( Z: - Z) ( Y - Y) and substituting (8) we find thatQUESTION 3 (34 marks): Consider an econometric model where two variables Y and X are jointly determined by the following equations Y = BitB,X+BZ+U (7) X = antaY +V. (8) The Greek letters denote unknown parameters, U and V are model errors, mutually un- correlated, with zero mean, and Z is an exogenous variable, independent of the errors. a. (4 marks) Explain with detailed arguments, whether B, are o, identified under the information provided in the problem presentation. b. (12 marks) Demonstrate that the OLS estimate of on in the equation (8) is asymp- totically biased in general. Which would be your conclusion if By = 0? Which would be your conclusion if 02By = 1? We have that AOLS E ( Y - P ) ( X, - X) E ( Y - P) X. E ( Y - P) ( Y - P) and substituting (8) we find thatd. (9 marks) Write the restriction over the parameters in equation (5) that the regres- sion equations are identical for male and female employees, that is, the mean log-wage of female employees with any given values of S, A and T equals the mean log-wage of male employees with the same values of S, A and T. Provide an expression for the null hypothesis (Ho) of the mentioned restriction and for the alternative (H,). The re- stricted OLS estimation provides RSS = 93.1805. Use this information, together with the information provided in the problem formulation, to calculate the test statistic, in- dicating which is its approximate distribution under the null hypothesis. Establish a decision rule, using a significance level of 5%, and the corresponding conclusion on the inference provided by the test. The hypotheses arec. (9 marks) write the restriction over the parameters in equation (5) that the marginal effects of S over In W and of T over In W are equal for male and female employees. Provide an expression of equation (5) imposing these two restrictions. The restricted OLS estimation provides RSS = 80.8747. Use this information, together with the information provided in the problem formulation, to calculate the test statistic, in- dicating which is its approximate distribution under the null hypothesis. Establish a decision rule, using a significance level of 5%, and the corresponding conclusion on the inference provided by the test. The marginal effects of S areb. (9 marks) Write the expression (or formula) for the marginal effect of A over In W for male employees in terms of the unknown parameters, implied by regression equa- tion (5). Repeat the exercise for female employees. Provide the null hypothesis that the marginal effect of A on In W for male employees is equal to the marginal effect of A on In W for female employees. Provide also the expression for the alternative hy- pothesis. Give the equation for the restricted regression implied by Ho. The restricted OLS estimation provides RSS = 81.7242. Use this information, together with the information provided in the problem formulation, to calculate the test statistic, in- dicating which is its approximate distribution under the null hypothesis. Establish a decision rule, using a significance level of 5%, and the corresponding conclusion on the inference provided by the test.h. (3 marks) Explain with detailed arguments whether R' would be the same in the two regressions. QUESTION 2 (33 marks): You are conducting an econometric investigation into the hourly wage rates of male and female employees. Your particular interest is in comparing the determinants of wage rates for female and male workers. The sample data consist of a random sample of observations on 526 paid employees, 252 of whom are females and 274 of whom are males. The sample data provide observations on the following variables: W hourly wage rate, measured in dollars per hour S = number of years of formal education completed, in years A = age, in years; T = firm tenure of employee, in years;. F = 1 if employee is female and =0 if employee is male a. (6 marks) Use the estimation results to compute the OLS estimates of the slope coef- ficients of the regressors S, A and T when only the 274 observations corresponding to male employees are used. Repeat the exercise when only the 252 observations cor- responding to female employees are used.c. (4 marks) Write S, in terms of an. d. (5 marks) Demonstrate that This is y - r = 2, where e. (3 marks) Demonstrate that the residuals of (3) are identical to those of (4). f. (3 marks) Demonstrate that the standard errors of By and ag are identical. g. (4 marks) Determine the relationship between the t statistic using By and the t sta- tistic using a2. The t statistic for Ho : 8, = 0 isQUESTION 1 (33 marks): A researcher is considering two regression specifications to esti- mate the relationship between a variable X and a variable Y, a. (4 marks) Determine whether (2) can be expressed as a restricted version of (1). (2) can be rewritten as b. (7 marks) Using the same n observations of the variables Y and X, the researcher fits the two specifications using ordinary least squares (OLS). The fits are c. (4 marks) Write B, in terms of a. d. (5 marks) Demonstrate that