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...
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...
Suppose that we believe a weakly stationary return sequence r following the model, where at ls the 1.1.d. noise sequence with mean 0 and variance σ. and at s independent of rt-1,Tt-2. (a) Express the mean μ of the return sequence rt using φο, φι, φ2 and σ (lag-0 autocovariance) of r) (d) Express the lag-1 autocorrelation ρι using φο, φι, φ2 and σ
Exercise 4: Heavy-tails (a) Consider the time series {Y determined by the equation 1t-1 where ao >0 and a20. Give a necessary condition for Y in this case find its mean and autocovariance function to be stationary, and (3 marks)
Exercise 4: Heavy-tails (a) Consider the time series {Y determined by the equation 1t-1 where ao >0 and a20. Give a necessary condition for Y in this case find its mean and autocovariance function to be stationary, and (3 marks)
Let wt for t = . . .,-2,-1, 0, 1, 2, . . . be an independent and identically distributed process with wt ~ M0, σ2). and consider the time series Determine the mean and the autocovariance function of xt and state whether it is stationary
2.6 Consider a process consisting of a linear trend with an additive noise term consisting of independent random variables wt with zero means and variances o that is where Bo, B1 are fixed constants (a) Prove t is nonstationary. (b) Prove that the first difference series Vxt finding its mean and autocovariance function Xt t-s stationary by
Problem 1: Consider the model Y = BO + Bi X+e, where e is a N(0,02) random variable independent of X. Let also Y = Bo + B1X. Show that E[(Y - EY)^3 = E[(Ỹ – EY)^3 + E[(Y – Y)1.
ne 10. 2019 4. A random process Z(t) is given by, Z(t) = Kt, where K is a random variable The probability dessity function for K is given below. Use this information to answer the questions below (20 points k <-1 0 fK(k)=-k-1sks k> 1 0 (a) Find the mean function for Z(t). (b) Find the autocovariance function for Z(e). (c) Is this process wide sense stationary (WSS)? Explain your answer in 2-3 sentences.
ne 10. 2019 4. A random...
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...
1. Consider the simple linear regression model: Ү, — Во + B а; + Ei, where 1, . . , En are i.i.d. N(0,02), for i1,2,... ,n. Let b1 = s^y/8r and bo = Y - b1 t be the least squared estimators of B1 and Bo, respectively. We showed in class, that N(B; 02/) Y~N(BoB1 T;o2/n) and bi ~ are uncorrelated, i.e. o{Y;b} We also showed in class that bi and Y 0. = (a) Show that bo is...
Consider a Diamond model, where we set the productivity factor At to unity (1) in all periods. The working population. Lt, grows at rate n, i.e., Lt+1-(1 + n) Lt. Lower-case letters denote per-worker terms, e.g earned (from labor) in the first period of life (wi) is spent on saving (St) and first-period consumption (C1t). The first-period budget constraint can thus be written Ki/Lt. Agents live for two perioo In retirement, the same agent consumes C2t+1, consisting of savings from...