Question No 1. Give the definition of stationarity in strong and weak sense. Are the following processes stationary? Why? White noise (give the definition), Random walk (give the definition), MA(n) process (give the definition).
Question No. 2 How to test stationarity of an empirically given time series? (you can use textbooks and lectures with references).
Question No. 3 Give the definition of the autocorrelation function of a stationary time series.
Question No. 4 Derive a formula for the autocorrelation function of a MA(n) process (you can use textbooks and lectures with references).
Question No. 5 Derive autocorrelation function of the following MA(2) process:
X(t) = Z(t) + 0.5Z(t-1) + 0.25Z(t-2)?
Question No 6 Under which condition a general AR(n) process is stationary?
Question No 7 For which a the following processes are stationary:
X(t) = aX(t-1) + 2aX(t-2) + Z(t); X(t) = 2aX(t-1) + aX(t- 2) + Z(t)?
Question No 8 Under which condition a general MA(n) process is invertible?
Question No 9 For which a the following processes are invertible:
X(t) = Z(t) + aZ(t-1)+2aZ(t-2); X(t)=Z(t) + 2aZ(t- 1) + aZ(t-2)?
Question No. 10 Under which conditions a general ARMA(m,n) process is stationary and invertible? For which a the following process is stationary and invertible
X(t) = aX(t-1) + 2aX(t-2) + Z(t) + aZ(t-1) + 2aZ(t-2)?