Question

Box-Jenkins : For each of the non-seasonal models presented below, indicate whether or not it complies with the Box-Jenkins approach, ie is it stationary and invertible? Justify your answers.
a) Yt = et − 0,67 Yt−1
b) Yt = et + 0,43 Yt−1 − 0,37 Yt−2
c) Yt = et + 0,25 et−1
d) Yt = et + 0,9 Yt−1 + 0,3 Yt−2
e) Yt = et + 0,9 Yt−1 + 0,3 et−1

Answer 1

a) For the non-seasonal model If Ipl <1, then the system is stationary, as follows: t-1 -0.67(et-1-0.67%-2) And further expanding, we get Thus without loss of generality, we may assume, Yo0, and for large t,(-0.67)0, thus the non- seasonal model is stationary.

For the invertibility, which is not of infinite order, and hence not invertible. Thus t-1 is not invertible.

This is of the type 1 It-1 2 t-2 Such a non-seasonal model is stationary if In our case φ,-0.43, φ2--0.37 φ1 φ2-0.43-0.37 = 0.06 〈 1 lpal 0.374 1 Hence, the non-seasonal model is stationary.

c) Ytet 0.25et-1. (1) This is of the type σ2(1 +92 ) for all t" and Then E(r)-0 for all t, var(%) cor(Yt,Yt-j)_ί0 otherwise Thus () is always stationary as both variance and covariance do not depend on t Thus Yt -Et t0.25et-1... (1) is stationary 1. This is invertible also as |θ|-0.25

In our case φ1 0.3 0.9, φ2- φ1 + φ2-0.9-0.3 0.06 1 As all the conditions are satisfied, the non-seasonal model is stationary.