4. [30 Points] A random variable y is related to (dependent on) three other random variables 2, z and w. Data on y, t, z and w is collected for a random sample of n individuals (i = 1, ...., n). Assume that n > 30. The relationship of y with t, z and w from the data of these n individuals is shown in the three graphs below: Z w (a) (10 Points) Specify an appropriate OLS regression model (write the regression equa- tion) to quantify the dependence of y on x, z and w. Include all appropriate regressors on the right hand side of the equation and carefully include the subscripts and associ- ated coefficients. (b) (10 Points] What are the expected signs of ALL coefficients on the right hand side of the regression equation? (NOTE: Include all coefficients that show the relationship of y with r, z and w, except the constant.] (c) (5 Points) Suppose that the relationship of y with 1, 2 and w is not the one as demonstrated by the graphs, but is now expressed as follows: y = 80+8sz:+822+3+ Specify an appropriate OLS regression model (write the regression equation) which estimates the value of the parameters B1, B. & B; and is linear in parameters. (HINT: If a = e, then In(a) = b.] 7 (d) [5 Points] Suppose that the relationship of y with I, z and w is not the one as demonstrated by the graphs, but is now expressed as follows: y; = e(Buz: 292 293 ) Is it possible to estimate the value of the parameters B1, B,& B, using an OLS regression model which is linear in parameters? If yes, specify the model (write the equation). If not, provide an argument for your answer. 4. [30 Points] A random variable y is related to (dependent on) three other random variables 2, z and w. Data on y, t, z and w is collected for a random sample of n individuals (i = 1, ...., n). Assume that n > 30. The relationship of y with t, z and w from the data of these n individuals is shown in the three graphs below: Z w (a) (10 Points) Specify an appropriate OLS regression model (write the regression equa- tion) to quantify the dependence of y on x, z and w. Include all appropriate regressors on the right hand side of the equation and carefully include the subscripts and associ- ated coefficients. (b) (10 Points] What are the expected signs of ALL coefficients on the right hand side of the regression equation? (NOTE: Include all coefficients that show the relationship of y with r, z and w, except the constant.] (c) (5 Points) Suppose that the relationship of y with 1, 2 and w is not the one as demonstrated by the graphs, but is now expressed as follows: y = 80+8sz:+822+3+ Specify an appropriate OLS regression model (write the regression equation) which estimates the value of the parameters B1, B. & B; and is linear in parameters. (HINT: If a = e, then In(a) = b.] 7 (d) [5 Points] Suppose that the relationship of y with I, z and w is not the one as demonstrated by the graphs, but is now expressed as follows: y; = e(Buz: 292 293 ) Is it possible to estimate the value of the parameters B1, B,& B, using an OLS regression model which is linear in parameters? If yes, specify the model (write the equation). If not, provide an argument for your