Consider ARMA(1,1) modelxt = 0.5xt1 + wt + 0.4wt1.Assume that the noise wt has variance 2 = 1.(a) derive a causal representation for xtin terms of wt, wt1, . . . . That is,obtain the coefficients j in the expression xt =j=0 jwtj .(b) Use the representation in part (a) to derive the autocovariance function(h) of xt. (Note that (h) = (h). Thus, you only need to considerh 0.)(c) Use part (b) to derive the spectral density. (Hint: when computethe spectral density, you need to use the identity h=0 h1/(1 e2i) for ||

2. Consider AR.MA(1,1) model xt = -0.5xt-l + wt + 0.4wt-1. Assume that the noise wt has variance 02 = 1. (a) derive a causal representation for st, in terms of wt,wt-1, . . . That is, . - .- . , - - oo , . obtain the coefficients sz in the expression xt - j=0 z/Jjwt-3 (b) Use the representation in part (a) to derive the autocovariance function 7(h) of an. (Note that 'y(h) = "y(-h). Thus, you only need to consider h 2 0) (c) Use part (b) to derive the spectral density. (Hint: when compute the spectral density, you need to use the identity Xiao phc=','27""ih = 1/(1 - pte-hm) for IpI