1. Derive the autocovariance functions for AR(2), MA(2) and AMRA(2,2)
processes. Briefly comment on their relationships.
2. (1) Verify whether the following process is stationary or not. If so, calculate
its autocovariance. (2) Verify whether the following process is Ergodic
or not.
yt = (1+2.4L + 0.8L2) t t ∼ WN(0, 1)
3. Consider the following bilinear time series model,
yt = t + β t−1 t−2
where ∼ SWN(0, σ2). Explain whether yt is stationary or not. (SWN
stands for the strict WN.)
4. Consider the following bivariate probability distribution
X=0 X=1
Y=0 0.2 0.5
Y=1 0.1 0.2
(Note: The table can be read as prob(X=0, Y=0)=0.2 ,etc.)
(i) Find the marginal distribution of X and Y; Prove that X and Y are not
statistically independent.
(ii) Find E(X|Y = 0) and V ar(X|Y = 1);
(iii) Find Cov(X, Y) and Corr (X, Y).
5. Plot the Autocorrelation Function (ACF) for the ARMA (2,1) process
yt = φ1yt−1 + φ2yt−2 + t + θ1 t−1
for the lags j=0, 1, 2, …, 20 with the parameter values, φ1 = 0.8, φ2 = −0.6
and θ1 = 0.5. Repeat the exercise with θ1 = 0, the other two parameters
being unchanged, in order to see how the moving-average component affects
the ACF in this case.
