Method of Moments parameter estimation. Consider a probability model characterizing the return on the market. The model corresponded to a version of the Poisson mixture of normals we have been studying. Denote the return in time period t as rt. The arrival of an extraordinary event or "disaster" is governed by a Poisson random variable, j = {0, 1, 2, 3, . . .}, with an intensity parameter . When j = 0 the disaster is avoided, and we interpret larger values of j as the arrival of disasters of increasing magnitude, i.e., j = 2 is a bigger disaster than j = 1. Assume that at each time period, rt follows a normal distribution with mean j, and variance 2 (i.e., the mean shifts with the value of j, but the variance stays the same), for constant parameters > 0, , and > 0. (a) What are the first four cumulants of rt as a function of the parameters > 0, , , and ? (b) We will now combine these theoretical values with the corresponding sample values you calculated in Questions 1(b) to calibrate or estimate the unknown parameters of the model.

(c) Repeat this exercise but assume that extraordinary events are more frequent and can happen once per year (i.e., once every 12 months) on average. How do the parameter estimates change? How does the implied theoretical value for excess kurtosis compare to the one you found in part (b)? How does it compare to the sample estimate? Interpret your results. (d) Which assumption about makes the model seem closer to the evidence in the sample? (e) Optional bonus question. Use the Poisson mixture of normals model from Assignment #2. Recall that model had the feature that conditional on the disaster event, the value of both the mean and the variance of the normal distribution changed to j and 2 + j2, respectively. Given a value of = 1/120, can you find values for the 4 parameters , > 0, > 0, and > 0 such that the four theoretical cumulants are exactly equal to the four values of those cumulants in the sample? Repeat this exercise assuming = 1/12. What changes? Interpret your results.