- HW 1 (Due 1/22/2004) From Chapter 1: 1. Problem 1 page 192. Problem 4 page 203. Discuss equations 1.15 and 1.16 commenting on the statement on page 11 “conditional distributions are more relevant than marginal distributions in studying asset returns.”4. Verify the statements about the return and log return made on page 11 and continued on page 12 of the text under the assumption that the log returns of assets are independent and identically distributed as a normal distribution (section titled “Lognormal Distribution” beginning on page 11.5. Verify equation 1.18 on page 14.- HW 2 (Due 2/3/2004) From Chapter 1 and 2:1. Describe and plot simulated realizations from both a covariance stationary process and a nonstationary process (your choice of nonstationarities). Examine the autocorrelation function of each and comment.2. Derive the autocorrelation function and the spectral density function for a MA(1) model. Comment on the behavior of the partial autocorrelation function. How does the behavior differ from that of an AR(1) model? Explain.3. Simulate a realization from an AR(5) and an MA(5) model. For each series plot the sample ACF, PACF and spectral density. What do these plots tell you about the process?4. Consider the U.S. monthly interest rates from April 1953 to Janurary 2001, of the 1-year Treasury constant maturity rate and the 3-year rate. Examine the marginal histogram or nonparametric density estimate, the lagged scatterplots (up to lag 15), the sample ACF, PACF and spectral density of each series. Alsoexamine lagged cross-scatterplot and the corresponding cross-correlations between the series. Comment on what you see. Conjecture a relationship between the two series based on what you see.5. Problem 5 page 76.- HW3 (Due 2/24/2004) From Chapters 1, 2 and 8 plus notes.1. Return to problems 3 and 4 from HW2 and investigate the sample spectral densities as requested earlier.2. Problem 6 page 773. Problem 7 page 774. Problem 9 page 775. Problem 3 page 3556. Show that the first autocorrelation coefficient of a fractional ARMA model is d/(1-d).7. For each of the following models AR(2), MA(2), ARMA(2,2), ARFIMA(2,1/3,0) plot a realization of length 200 and the respective sample spectral density of this realization (you choose the coefficients). Comment on the behavior of the sample spectral density and how the information containedin this function assists you with model formulation or understanding of thestochastic process. Also, what would you expect to see in the spectral density for a multiplicative seasonal ARMA model with seasonal AR order of