Question

Time Series transformation

Let an annual series Yt be stationary. However, the series transformed and differentiated Dt = ln(Yt) - ln(Yt-1) is stationary. Moreover, we suppose that it obeys the following theoretical model: Dt = -0.12 + 0.75 Dt-1 + et, in which the error term and is a white noise of variance σ2 = 0.012.

How can I transform this model to get the original one before the transformation?

Answer 1

As,   we can write , $e^{D_{t}}=Y_{t}/Y_{t-1}.$

and since, the transformed and differentiated series $D_{t}$ follows model given by -

0.12 +0.75Dt-1+ t
, where  $\epsilon_{t}$ is white noise with variance 0.012, we can write the original model as -

0.12+0.75D-iterY/Y

i.e.
0.12+0.75D-1+et

0.12+0.75D-1+Et
will be the original model.