! SOLVE THIS PLEASE
\fEXERCISE 4.36. In Example 4.20, use Eq. 4.16 to verify that the fake spheres with a = 2 and a = , have no umbilical points. (4.16) T. EXAMPLE 4.20 (Fake Spheres). We wish to construct surfaces of revolution with constant Gaussian curvature equal to 1. If K = 1, then Eq. 4.15 says that x" = -x. The following is a solution for every a > 0: (4.17) x(t) = a cos(t). The hypothesis that y is parametrizationgth, so (x')2 + (2')2 = 1, means that z is determined by x as follows: (4.18) = (t) = VI - x'(s)2 ds = 1 - a2 sin? (s) ds. (4.15) KIt is postulated that a reasonable demand-supply model for the wine industry in Australia, under market clearing assumption, would be given by Qt = ontop" +azpi tash toA tu demand Qt = Po+ BIP"" + ByS + the supply where Q: = real per capita consumption of wine, Po = price of wine relative to CPI, Ph = price of beer relative to CPI, Y, = real per capita disposable income, A, = real per capital advertising expenditure, and S, = storage cost. CPI is the Consumer Price Index. The endogenous variables in this model are @ and P", and the exogenous variables are p, Y, A and S. The variance of wt and t are, respectively on, and of, and Cov(us, ") = Our # 0. The errors do not exhibit any correlation over time. (a) (5 marks) Provide the reduced form for Pu. (b) (5 marks) The OLS estimation of the demand function, based on annual data from 1955-1975 (7 = 20), gave the following results (all variables are in logs and figures in parentheses are t-ratios). Q+ = -23.651 + 1.158PM -0.275 P +3.212 Y, -0.603 At (-6.04) (4.0) (-0.45) (4.5) (-1.3) All the coefficients except that of Y have the wrong signs. The coefficient of P" (price elasticity of demand, o) not only has the wrong sign but also appears significant. Explain why the OLS parameter estimator may give rise to these counter-intuitive results. You are expected to use your results in answer (a) to support your answer.end Run the above script yourself in MATLAB, and report what you observe. Finally, explain why one among the two numerical least squares solutions xhat_QR and xhat_normal, is more accurate (with respect to the known analytical solution Canalytical from part (a)). Copy/paste code: close all; clear; clc; for k=5:8 A = [1 1; 10^(-k) 0; 0 101-k)]; b = [-10^(-k); 1 + 10^(-k); 1 - 10^(-k)]; xhat_QR = A\\b xhat_normal = (A*A)MA'*b) endLeast Square - I need help with only part (b) . I got x_analytical = [1 -1 ] . Thank you! -10-k Let k be any positive integer, and A = 10-k 0 , b= 1+ 10-4 0 10-k - 10-k (a) By hand, analytically compute the least squares solution Zanalytical for the above A, b from the normal equation. You may use the formula for the inverse of a 2 x 2 matrix: a ad - bc Show all the steps in your hand calculations. (b) Consider the following MATLAB script that numerically computes the least squares solution for k = 5,6, 7, 8, in two different ways: close all; clear; clc; format long for k=5:8 A = [1 1; 10*(-k) 0; 0 10*(-k)]; b = [-10*(-k); 1 + 10*(-k); 1 - 10"(-k)]; xhat_QR = A\\b xhat_normal - (A' *A)\\(A'*b)
