hwk4soln(1) - STSCI 4550 ILRST 4550 ORIE 5550 Applied Time...

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STSCI 4550 / ILRST 4550 / ORIE 5550 Applied Time Series Analysis, Spring 2012 Professor David S. Matteson Assignment #4 Suggested Solutions Out of 24 possible points Note: BJR refers to class text: Time Series Analysis: Forecasting and Control, 4th Edition by Box, Jenkins, Reinsel (2008). 1.a (2 pts) Based on the code, the two specific models are: (1 - 0 . 95 B + 0 . 35 B 2 ) x t = a t , a t iid N (0 , 1) y t = 2 cos(0 . 5 t + 0 . 6 π ) + b t , b t iid N (0 , 1) 1.b (2 pts) 1
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Figure 1: Time series plots and ACF plots of x t and y t . The process x t is a simulated AR(2) model. There is some some oscillating pattern in the time series plot, but it isn’t exact. When we examine the ACF, we also see an oscillating pattern, but it dampens at the higher lags. The “pseudoperiodic behavior” arises for an AR model with complex roots in the respective characteristic equation (see part 1.c below). The time series plot for y t exhibit exact cyclic patterns, which can be verified by the ACF plot. Note the pattern persists for all lags. This is because the process y t is constructed as a sum of a deterministic process with a period of 2 π/ 0 . 5 = 4 π and some additive Gaussian white noise. 1.c (2 pts) The function polyroot() takes in a vector of polynomial coefficients in increasing order and nu- merically approximates the complex roots. For example, suppose suppose y t = n i =0 a i x i , if we were to give polyroot() the vector of coefficients [ a 0 , ...a n ], then the output will be numerical approximations to n zeroes of y t . 1.d (2 pts) x t will be stationary if the roots to the equation 1 - 0 . 95 B + 0 . 35 B 2 = 0 lie outside the unit circle. Using polyroot() , the roots can be found to be 1 . 36 + 1 . 01 i and 1 . 36 - 1 . 01 i , both of which lie outside the unit circle as p (1 . 36 2 + 1 . 01 2 ) > 1. To check if a complex number lies outside the unit circle, suppose we have z = x + yi , then we need to compare the magnitude of z , p ( x 2 + y 2 ), to 1.
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