Exercise 4: Heavy-tails (a) Consider the time series {Y determined by the equation 1t-1 where ao ...
Consider the following model on a return series rt=t+ at +0.25at-1, where at riid N(0,02), t = 1, ... ,T. (a) What are the mean function and autocovariance function for this return series? Is this return series {rt} weakly stationary? Justify your answer. (b) Consider first differences of the return series above, that is, consider wt = Vrt=rt – Pt-1. What are the mean function and autocovariance function for this time series? Is this time series {wt} weakly stationary? Justify...
2. (a) Consider the following process: where {Z) is a white noise process with unit variance. [1 mark] ii. Find the infinite moving average representation of X,i.e., find the scquence [6 marks] i. Explain why the process is stationary. (6) such that Xt = Σ b,2-j. iii. Calculate the mean and the autocovariance "Yo, γι and 72 of the process. 7 marks iv. Given 40 = 0.1 and Xo = 1.8, find the 2-step ahead forecast of the time series...
can anyone answere these with workings please i i Consider an equation of the form y = f(x) with sets defined as follows: A contains all maxima in the function B contains all points of inflection in the function C contains all stationary points in the function D contains all points where the second derivative of the function equals zero. Which of the following statements are true, and which are false: Ais a necessary condition for C ii is a...
10. (4 pts) In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (x2 +1)y" – 6y = 0 subject to the initial condition y(0) = 3, y'(0) = 3. Since the equation has an ordinary pts at x = 0 and it has a power series solution in the form y = {cnt" no (1) Insert the formal power series into the differential equation and derive the...
In this exercise we consider finding the first five coefficients in the series solution of the first order linear initial value problem (+3)y' 2y 0 subject to the initial condition y(0) 1. Since the equation has an ordinary point at z 0 it has a power series solution in the form We learned how to easily solve problems like this separation of variables but here we want to consider the power series method (1) Insert the formal power series into...
2. Consider the time series X, = 2 + 0.5t +0.8X1-1 + W, where W N(0.1). (a) (8 points) Calculate E(X2) Is this process weakly stationary? Give reasons for your answer. Hint: Find the mean function of {X) and then substitute t = 20. (b) (3 points) Calculate Var(X20) Question 2 continues on the next page... Page 4 of 12 c)(4 points) Consider the first differences of the time series above, that is Is {%) a weakly stationary process. Prove...
(1 point) In this exercise we consider the second order linear equation y" + series solution in the form y = 0. This equation has an ordinary point at x = 0 and therefore has a power y = cmx". n=0 We learned how to easily solve problems like this in several different ways but here we want to consider the power series method. (1) Insert the formal power series into the differential equation and derive the recurrence relation Cn...
Exercise 1: The Taylor series for In(y) about y = 1 is (4) In(y) = 9 (-1)"+(v - 1) n=1 for y-1€ (-1,1] (that is, y E (0,2]). What polynomials do we get if we truncate this series at n = 1? n = 2? n = 0 (hint: the n = Oth approximation is defined!)? Compare the value of each of these with that of In(y) at y = 1.1 and y = 1.75. Note how the error differs...
Name: 3) Bessel's Functions. Consider the differential equation y xy+y- power series solution of y +xy+y- Section: 003 402 404 406 a) Use the method of Frobenius (which we learned in 7.3) to find a recurrence relation for the b) Find a general form of the answer, using only factorials (not the Gamma function), c) Determine the radius of convergence of your power series answer d) This is called a Bessel function of order zero. What is the differential equation...
Exercise 4: (5 points) consider the following differential equation 3y - y Let = f(ty) be the right-hand side of the above equation. a. Compute a/ay. b. Determine and sketch the region in the ty-plane where functions. and array are both continuous C. For the initial condition y(0) = 1 (i.e.to = 0, y = 1), would a unique solution of the equation exist? Explain.