4. [30 Points] A random variable y is related to (dependent on) three other random variables x, z and w. Data on y, x, z and w is collected for a random sample of n individuals (i = 1, ...., n). Assume that n > 30. The relationship of y with x, z and w from the data of these n individuals is shown in the three graphs below: X Y Z (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 x, z and w, except the constant.] W (c) [5 Points] Suppose that the relationship of y with z, z and w is not the one as demonstrated by the graphs, but is now expressed as follows: Y = 0+++Bsw+ui Specify an appropriate OLS regression model (write the regression equation) which estimates the value of the parameters 3, 3 & 33 and is linear in parameters. [HINT: If a e, then In(a) = b.] 7 (d) [5 Points] Suppose that the relationship of y with z, z and w is not the one as demonstrated by the graphs, but is now expressed as follows: = e(Buz122) Vi = Is it possible to estimate the value of the parameters 3, 3 & 3, 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.