Consider the regression model

???????? = ????1????1,???? + ????2????2,???? + ????????

where ???????? is a nonstationary error term following an IMA(0, 1, 1) process ∇???????? =

???????? − ????????????−1. Show that the regression model may be rewritten in the form

???????? − ????̄

????−1 = ????1(????1,???? − ????̄

1,????−1) + ????2(????2,???? − ????̄

2,????−1) + ????????

where ????̄

????−1, ????̄

1,????−1, and ????̄

2,????−1 are exponentially weighted moving averages so that, for example,

????̄

????−1 = (1 − ????)(????????−1 + ????????????−2 + ????2????????−3 + ⋯)

It will be seen that the fitting of this regression model with nonstationary noise by maximum likelihood is equivalent to fitting the deviations of the independent and dependent variables from local updated exponentially weighted moving averages by ordinary least-squares. (Refer to Section 9.5.1 for related ideas regarding transformation of regression models with autocorrelated noise ????????.)