. Help me to answer the following attachments.
A researcher is interested in estimating the relation between the total house rent a family pays (Y, measured in dollars) as a function of the size of the house (X, , measured in square feet) and the number of rooms in the house (X2 ) by surveying 350 prospective locations. Both X, and X2 are standard normal variables. The estimated regression function is as follows: Y= 10.85 + 1.25X, + 1.98X2. The correlation between X, and X, was calculated to be 0.64. Suppose the size of the house is 3.25 and the number of rooms in the house is 5. Let PC, and PC denote the first and the second principal components of the variables X, and X2, respectively. The variance of the first principal component, var (PC, ) , will be The variance of the second principal component, var (PC2 ) , will be (Round your answers to two decimal places.) The first principal component explains | % of the variance of X, and X2- The second principal component explains |% of the variance of X, and X2. (Round your answers to whole numbers.)The government of a country suffering from hyperinflation has sponsored an economist to monitor the price of a "basket" of items in the population's staple diet over a one-year period. As part of his study, the economist selected six days during the year and on each of these days visited a single nightclub, where he recorded the price of a pint of lager. His report showed the following prices: Day (i ) 8 29 57 92 141 148 Price ( P;) 15 17 22 51 88 95 In P 2.7081 2.8332 3.0910 3.9318 4.4773 4.5539 [i= 475 [i =54,403 [InP, = 21.5953 _(InP,)2 = 81.1584 Liln P; = 1,947.020 The economist believes that the price of a pint of lager in a given bar on day i can be modelled by: In P, = a + bite; where a and b are constants and the e; 's are uncorrelated M(0,o') random variables. (i) Estimate a , b and oz. [5] (ii) Calculate the linear correlation coefficient r. [1] (iii) Obtain a 99% confidence interval for b . [2] (iv) Determine a 95% confidence interval for the average price of a pint of lager on day 365: (a) in the country as a whole (b) in a randomly selected bar. [7] [Total 151Q.2) (i) What are the different types of loans? Describe in brief. (3) (ii) A loan is being repaid with 25 annual payments of Rs.300/ each. With the 10" payment, the borrower pays an extra Rs. 1000/-, and then repays the balance over 10 years with a revised annual payment. The effective rate of interest is 8%. Calculate the amount of the revised annual payment. (3) (iii) An investor borrows an amount at an annual effective interest rate of 5% and will repay all interest and principal in a lump sum at the end of 10 years. She uses the amount borrowed to purchase a Rs.1000/- par value 10-year bond with 8% semiannual coupons bought to yield 6% convertible semiannually. All coupon payments are reinvested at a nominal rate of 4% convertible semiannually Calculate the net gain to the investor at the end of 10 years after the loan is repaid. (4) (iv) A loan is repaid with level annual payments based on an annual effective interest rate of 7%. The 8th payment consists of Rs.789/- of interest and Rs.211/- of principal. Calculate the amount of interest paid in the 18" payment. (5) (v) Define the characteristics of government index linked bonds. Explain in practice why most index linked securities carry some inflation risk in practice. (3) [18]3) This problem is a game theory version of the same ideas at work in 2e. Suppose that two firms sell identical output. Each firm has two possible actions: it can produce a low level of output, or a high level of output. If both firms produce a low level of output, a high price results, and each firm gets $5 million in profits. If both firms produce a high level of output, a lower price results, and each firm only earns $3 million. If one firm produces low output but the other firm produces high output, the firm producing more output will get a large benefit from the other firm restricting quantity and driving up the price: the high-quantity firm will earn $6 million, while the low-quantity firm will earn only $2 million. These payoffs are a simplified version of the Cournot story in question 2. a) Use this information to construct a payoff matrix for this game. Use the following format. I've filled in one of the hard boxes to get you started: (JA, (B) B: High B: Low A: High ??? ??? A: Low ($2m, $6m) ??? b) What is firm A's best response if firm B produces High? What is A's best response if firm B produces Low? c) Use Nash equilibrium to explain why both firms end up producing the High level of output, even though they can earn higher profits if both produce low. d) Does either firm have a dominant strategy? Explain
