Determine which of the following ARMA processes are causal and which are invertible. In each case,w_tdenotes white noise with mean 0 and variance 1.

(a)x_t=0.2x_t1+ 0.48x_t2+w_t

(b)x_t=1.9x_t10.88x_t2+w_t+ 0.2w_t1+ 0.7w_t2

(c)x_t=0.6x_t1+w_t+ 1.2w_t1

(d)x_t=1.8x_t10.81x_t2+w_t

(e)xt=1.6x_t1+w_t0.4w_t1+ 0.04w_t2

2. There should be three causal processes in problem 1. For each causal process from problem 1, compute the ACF

3. Consider the MA(2) processx_t=w_t+w_t2, wherew_tis white noise with mean 0 and variance

a. Compute the PACF values₁₁and₂₂