Question 3 (a) Write down an MA(5) model. (b) Derive the unconditional mean and variance for the MA(5) model. (You need to specify the conditions used and show all the workings) (c) The dataset sp500.csv contains the weekly price of the Standard & Poors 500 (S&P 500) index. Read the data into R and obtain the continuously compounded returns (in % by 100). Construct one-line plot (make sure the x-axis is in actual date NOT time index) and one histogram of the index (make sure you include appropriate labels and title). In addition, make some comments on two graphs. (d) Briefly discuss how a random walk model can be used to describe/examine the efficient market hypothesis (EMH). (e) Do you think the Standard & Poors 500 index follows the random walk model? Explain. (NO computation, estimation and hypothesis required in this question). (f) Use no more than two sentences to explain why the variance ratio can be used to examine the random walk hypothesis. (g) Obtain the continuously compounded return of the Standard & Poors 500 index. Is the return consistent with the EMH? Why? (h) Estimate the MA(5) model and write down the estimated model, is it a reasonable fit? Explain why or why not. (i) Compute the unconditional mean and variance based on the estimated results in (g). Discuss the results. (j) If we wish to improve the model, provide three possible models and explain the reason why you think these three models are reasonable. (k) Estimate the models you suggest in (h) and write down the estimated models. (l) Compare all the estimated models and discuss the best model to the data.