1. [42 points] Suppose that you have an iid random sample {Xi, Y} of three data points (X1, Yl) = (0, 0) ; (X2, Y2) = (2,3); (X3, Y3) = (4,3). (a) (2 point) Calculate the sample mean of X; and Yi. (b) (6 points) You run a simple linear regression of Y, on Xi. Calculate the OLS estimates of the slope and intercept. (c) (3 points) Calculate the predicted value and residuals for each of three data points. (d) (6 points) Calculate the TSS, ESS and SSR. (e) (4 points) Check whether TSS=ESS+SSR and interpret the equation. (f) (3 points) Calculate the R for the regression. (g) (3 points) Interpret R2 in words. (h) (4 points) Calculate the sample covariance between the residuals and Xi, which is defined in general as 1 Et-, (Xi - X)(ui - u), where n is the sample size and u; is the residual. (i) (4 points) Calculate the sample covariance between the residuals and predicted value, which is defined in general as I EL, (Yi - Y)(ui - u), where n is the sample size and Y; is the predicted value. (j) (4 points) Draw a graph to show (1) the three data points (2) the estimated regression line and (3) their residuals. (k) (3 points) Draw the point (X, Y) in the graph above and explain why the point (X, Y) is always on the regression line.Problem 4 [32 points]: The table below shows the response variable y to be explained by explanatory variables x and u: Observation i 1 2 3 4 5 6 7 8 9 10 Response Variable y 22 44 18 22 3 35 34 20 8 5 Explanatory Variable x 3 9 5 3 7 8 4 Explanatory Variable u 2 5 3 8 4 -2 5 5 (a) [6 points] Suppose a simple linear regression is performed based on EV x only. That is, Vi = a, tax, te,, i=1,2,..., 10 Using the "Im" function in R, obtain the OLS estimates a,, a, and R2. (b) [6 points] Suppose a simple linear regression is performed based on EV u only. That is, Vi = Yo + yu; +e,, i=1,2,...,10 Using the "Im" function in R, obtain the OLS estimates yo, 7, and R2. (c) [13 points] Suppose a multiple linear regression is used to explain y by both x and u. That is, Vi = Bot Bix; + Bu, te,, i=1,2, ...,10 Based on matrix operations in R (i.e. A%*%B, t(A), solve(A) on Ch3 page 32), (i) show that OLS estimates B =(Bo, B,, B2)' = (4.225, 3.96875, 0.33333)', (ii) compute the values of SYY, RSS, o' and Var(B). (iii) compute the value of R', and show that it's equal to the sum of R2 from part (a) and (b). [Note: The relationship is true only if vectors x and u are orthogonal. That is, I'm = Corr(x,u) = _ _(x, -x)(u, -1) =0] (d) [7 points] Consider a new data point with (x, u) = (1, 1). What is the best point estimator for the response, and a 95% prediction interval for the response?One particular company over which an investment bank writes European call options has experienced a severe fall in its share price. However, analysts have not revised their expectation that the share price will grow in f4 in six months. The table below shows the share price together with the price of the options. Date Share price Option price I November E3.00 10.90 2 November (2.00 $0.60 You may assume that the basic Black-Scholes framework is used to price the options. (i) Explain why the option price has fallen even though the expected return has increased according to the analysts. [3] (Hi) State any requirements for the option price to have fallen to its level on ? November. [3] [Total 6] (i) Outline the approach adopted by Shiller to test for "excessive volatility" and state the criticisms of his work. [7] State one difficulty of testing the strong form of the efficient market hypothesis and state the general conclusion of studies carried out on it. [2] [Total 91 (1) State the martingale representation theorem, including conditions for its application, defining all terms used. [3] Let S, denote the price of an underlying security at time f r denotes the risk free rate of return expressed in continuously compounded form, B, represents an accumulated "bank account" at time f that earns the risk free rate of return Let Y he any derivative payment contingent on Fy payable at some fixed future time T. where For is the sigma algebra generated by S, for O S & S T. You may assume that, under the equivalent measure O, the process D=eh S, is a martingale and that as, = B(rD dt + dDJ (Hi) Show that the value of this derivative at time /