Consider the simple transfer function model

(1 − ????)????1???? = ????1???? − ????????1,????−1 ????2???? = ????????1???? + ????2????

where ????1???? and ????2???? are independent white noise processes.

(a) Determine the univariate ARIMA model for ????2????, and note that ????2???? is nonstationary.

(b) Express the bivariate model for ???????? = (????1????, ????2????)

′ in the general form of a ‘‘generalized’’ ARMA(1, 1) model, (???? − ????1????)???????? = (???? − ????1????)????????, and determine that one of the eigenvalues of ????1 is equal to one.

(c) Determine the bivariate model for the first differences (1 − ????)????????, and show that it has the form of a bivariate IMA(1, 1) model, (1 − ????)???????? = (???? − ????∗????)????????,

where the MA operator(???? − ????∗????)is not invertible. Hence, this model represents an ‘‘overdifferencing’’ of the bivariate series ????????.