4.1 Consider the model (1 – B)Z, = (1 - 3B – .5B2)aş. (a) Is the model for Ze stationary? Why? (b) Let W, = (1 - B)?Zz....
4.1 Consider the model (1 B)PZ, (1 .3B - 5B2)a (a) Is the model for Z, stationary? Why? (b) Let W (1 - BYZ. Is the model for W, stationary? Why? (c) Find the ACF for the second-order differences W'.
Consider the MA(1) model x5 wt 0.6W-1 with the w assumed to be jid N0,02). A. Give a numerical value for the first lag autocorrelation. B. Give a numerical value for the second lag autocorrelation. C. Describe the appearance of the ACF for this model. D. Use R to sketch the ACF for this model. The commands are: acfprob3-ARMAacfíma-c(.6), lag max-10) plot seal0,10), acfprob3, xlm-c(1.10), lab-"lags", type-"h") (In the plot command, the type-"h" causes projections from the value to the...
QUESTION 3 (a) Consider the ARMA (1, 1) process -Bat-1-where o and θ are model parame- are independent and identically distributed random variables with mean 0 z, oz,-1 ters, and a1, a2, and variance σ (i) Show that the variance of the process is γ,- (ii) Using (i) or otherwise, show that the autocorrelation function (ACF) of the process is: ifk=0. (b) Let Y be an AR(2) process of the special form Y-2Y-2e (i) Find the range of values of...
4. Consider the ARMA (2, 3) process, I( 0.1%-1 +0.12%-2 + Ze + 0.3Zn-1-0.045-2-0.012Zt-3, where fZ) is a white noise process with unit variance. It is known that the above process is overestimated [4 marks (b) Hence, determine the stationarity and invertibility of the process. [4 marks (c) Find the first three lags of the autocorrelation function (ACF) for the process. [12 marks) (5 marks] (a) Suggest a parsimonious model for the above process. (d) Find the first three lags...
1. Consider the vector field z, y, z) = 〈re,zz,H) and the surface s in the figure below oriented outward. Unit circle Use Stokes' Theorem in two different ways to find/curl F dS, by: (a) [7 pts.] evaluating ф F-dr where C in the positively oriented unit circle in the figure (which is the boundary of S), (b) [7 pts.] evaluating curl F dS, where Si is the upward oriented unit disc bounded by C
1. Consider the vector field...
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...
a) Let z,w ∈ C, prove or disprove: Ln(z/w) = Lnz − Lnw b) Find all values in C and the principal value of j^j and ln(-3) c) Find all z ∈ C such that i. tanh z = 2 ii. e^z = 0 iii. Ln(Ln(z)) = −jπ
4. Let (Yi] be a stationary process with mean zero and let a, b and c be constants. Let st be a seasonal with period 4, that is, st-st+4, t-1, 2, . . . , and Xt = a + bt + ct2 + st + Y. (i) Let (ho, do )-min( (k, d)such that k > 0, d 0, and the proces s W t ▽k▽dX,-(1 B)a Find ko and do. For W, (with k = ko and d...
6.Let W={(a +b-c,2a +3b, -a +3c,-b-2c): a,b, CER) a) For what value of n is W isomorphic to R"?, clear answer the question and justify your answer b) Find an isomorphism T:R" W for the value of n you found in part (a).Please make it clear from your work that your function T is really an isomorphism. 7.Let T:P2(R) – P2(R) be a linear transformation such that T(x2)=x2 a) Prove that if there exist distinct linearly independent polynomials 2,9 €...
2. Let [et be a zero mean white noise process with variance 0.25. Suppose that the observed process is k = et + 0.5e-2. a. Explain why {Yt) is stationary. b. Compute yo-V(Y.) c. Compute the autocorrelation pkY, kl-0,1,2,... for Y) d. Let Wt = 3 + 4t + h. i. Find the mean of {W) ii. Is W3 stationary? Why or why not? iii. Let Z Vw, W,- W,_1. Is {Z.1 stationary? Why or why not?