Please answer the following questions.,,,
1. Consider a system of the demand and supply equations: Demand: Q = do + BP + uj (1) Supply: Q = 2P + vZ + U2 ( 2 ) where Q and P are endogenous variables; Z is an exogenous variable such that Elu, [Z] = 0 and E[uz |Z] = 0; and uj and uz are structural errors. (a) Show that the reduced-form equations can be expressed in the following form: Q = [10 + 1 12+ V1 P = M20 + #212+ V2 where f10, 11, 120 and #21 are the reduced-form parameters, and v, and vz are the reduced-form errors. Express Win, Hin, #20 and My in terms of the parameters of the structural eqs. (1) and (2).(b) Use the estimates of 1710, 120, w11, and 121 in (3) and (4) to solve for the parameters of the structural equations in (1) and (2). (c)What is the problem with the estimation of (1) or (2) directly using OLS? (d) Is the demand equation (1) identified, je can eq. (1) be estimated consistently? What about the supply equation (2)?4. Consider the following system of demand and supply equations: where Q and P are random endogenous 1variables. {a} Show that the reduced form equations can be expressed as Q=W10+W11Z+1 (3} P = 7720475le +'U2 (4) Express 1r\"), 1:20, 11, and 21 in terms of the parameters of the structural equations in (1) and [2). (h) Use the estimates of 1710., 2\Consider the single index model of investment returns in which for any security i : where E(2.) =0, E(e,8;) =0 for i# j, E(R,, 2,) =0 and R., is the return on the market. (i) Assuming that this model applies, derive expressions for the mean investment return on security i, and the mean investment return on a portfolio P, containing a securities, with a proportion x, invested in security i. [3] (ii) Show that Cip = Ex, Cy, where Cip and Cy are the covariance of investment returns between security / and portfolio P and securities i and j respectively. [2] (iii) State a general expression for the variance of of portfolio P in terms of the covariances Ci . [1] (iv) Use your results from (ii) and (iii) to show that: dop 1 Bip = dx, OF where Pip =- iP and comment briefly on this result. [7] [Total 13]Question One Let X be a random variable that is uniformly distributed on [0, 1] (i.e. f(x) = 1 on that interval and zero elsewhere). In Problem Set #4, you use the "2-step"/CDF technique and the transformation method to determine the PDF of each of the following transformations, Y = g(X). Now that you have the PDFs, compute (a) Elg(X)], (b) g(E[X]), (c) Var(s(X)) and (d) g(Var(X)) for each of the following transformations: 1. Y = Xi, fr(y) = 4y' on [0, 1] and zero otherwise. 2. Y = e- *, fr(y) = 4 on [!, 1] and zero otherwise. 3. Y = 1-e-*, fv(y) = by on [0, 1 - !] and zero otherwise. 4. How does (a) Elg(X)] compare to (b) g(E[X]) and (c) Var(g(X)) to (d) g(Var(X)) for each of the above transformations? Are there any generalities that can be noted? Explain. Question Two Compute the expectation and the variance for each of the following PDF's. 1. fx(x) = ar"-1, 0
0. 2. fx(x) = 1, x = 1,2, ..., n, where n is an integer. 3. fx(1) = (2 -1)', 0
