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[8 points] If E(ut) = 0 but V(ut) =  EMBED Equation.3  is replaced by  EMBED Equation.3  = EMBED Equation.3 , t =1, , n, what are the problems with the OLS estimator of  EMBED Equation.3 ? How would you estimate  EMBED Equation.3  in this model taking the above heteroskedasticity problem into account? Show all the necessary steps. [8 points] Consider the model in (1). If E(ut) = 0 but the assumption that  EMBED Equation.3 s are independent over observation is replaced by  EMBED Equation.3 <1 and  EMBED Equation.3 ~i.i.d. EMBED Equation.3 Is the OLS estimator of  EMBED Equation.3  unbiased? How would you estimate the parameters in (1) correcting for autocorrelation if (i)  EMBED Equation.3  is known, and (ii)  EMBED Equation.3  is unknown? Show all the necessary steps in each case. [5 points] If X is measured with error, is there any problem with the OLS estimators? Explain. 2. A researcher has data on years of schooling, S, weekly earnings in dollars, W, hours worked per week, H, and hourly earnings, E (computed as W/H) for a sample of 1755 white males in the United States in the year 2000. She calculates LW, LE, and LH as the natural logarithms of W, E, and H, respectively, and fits the following regressions. The results from these regressions shown in the table below (standard errors in parentheses; RSS = residual sum of squares): Column 1: a regression of LE on S. Column 2: a regression of LW on S and LH, and Column 3: a regression of LE on S and LH. The correlation between S and LH was 0.06. (1)(2)(3)(4)(5)Respondents All All All FT PT Dependent variable LE LW LE LW LW S 0.099 (0.006) 0.098 (0.006) 0.098 (0.006) 0.101 (0.006) 0.030 (0.049) LH  1.190 (0.065) 0.190 (0.065) 0.980 (0.088) 0.885 (0.325) constant 6.111 (0.082) 5.403 (0.254) 5.403 (0.254) 6.177 (0.345) 7.002 (1.093) Residual SS 741.5 737.9 737.9 626.1 100.1 observations 1755 1755 1755 1669 86  [6 points] Explain why specification (1) is a restricted version of specification (2), stating and interpreting the restriction. [4 points] Supposing the restriction to be valid, explain whether you expect the coefficient of S and its standard error to differ, or be similar, in specifications (1) and (2). [4 points] Supposing the restriction to be invalid, how would you expect the coefficient of S and its standard error to differ, or be similar, in specifications (1) and (2)? [7 points] Perform an F test of the restriction. [4 points] Perform a t test of the restriction. [4 points] Explain whether the F test and the t test could lead to different conclusions. [5 points] At a seminar, a commentator says that part-time workers tend to be paid worse than full-time workers and that their earnings functions are different. Defining full-time workers as those working at least 35 hours per week, the researcher divides the sample and fits the earnings functions for full-time workers (column 4) and part-time workers (column 5). Test whether the commentators assertion is correct. [3 points] What are the implications of the commentators assertion for the test of the restriction? 3. Consider the following system of demand and supply equations:  EMBED Equation.DSMT4  (1)  EMBED Equation.DSMT4  (2)  EMBED Equation.DSMT4 where Q and P are random endogenous variables. 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