TS Module 8: Non-stationary time series basics HW

(The attached PDF file has better formatting.)

Homework assignment: Stationarity through differencing and logarithms

Automobile liability claim severities have a geometric trend of +8% per annum.

The average claim severity in year t is the average claim severity in year t-1 adjusted for the geometric trend, plus or minus a random error term.

Assume the error term is added to the logarithm of the average claim severities.

The average claim severities are multiplied by a random error term.

　

Is the time series of average claim severities stationary?

Is the first difference of this time series stationary?

Is the second difference of this time series stationary?

Is the logarithm of this time series stationary?

What transformation makes the time series stationary?

Jacob:

What is the form of this time series?

Rachel:

Actuaries write: Y_t = 1.08 Y_t-1. The error term is multiplicative: Y_t = 1.08 Y_t-1 × (1 +

å).

A separate discussion forum posting shows the solution.

"....Jacob: What is the form of this time series?

Rachel: Actuaries write: Yt = 1.08 Yt-1. The error term is multiplicative: Yt = 1.08 Yt-1 × (1 + å)...."

What does mean the sentences above in the question?
We use Yt = 1.08 Yt-1 × (1 + å)as the times series in this question? but, I don't think it's right.
Even if the time series we should use is like that, both the first difference and the second difference are nonstationary, right? because:
D1(Yt)=0.08Y(t-1)+sigma(t) its mean is not constant. And,
D2(Yt+1)=(Yt+1-2*Yt+Yt-1), its mean is not constant also.

But from the text part 5.1, we can see the derivation results of first and second differences are stationary, so I can't be sure the results above are true.

Please help anybody.