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.

If you look under the 'old textbook' section for the module 12 homework I think it helps a little bit with the reasoning for the difference equation parts.

For part A I just thought of it intuitively. If Y_0 = 1, then
E(Y_1)=1.08...
E(Y_t)= 1.08^t
Here the limit as t increases without bound is infinite, so the series is not stationary.

For part b, we want to know if the sequence of first differences is stationary. Again assuming Y_0 = 1, we have:
E(Y_1 - Y_0) = 1.08 - 1
...
E(Y_t - Y_t-1) = 1.08^t - 1.08^(t-1) = (.08)*1.08^(t-1)
as t increases without bound, this again will increase without bound, so it is not stationary.

And so on for the second difference.

Here I'm using that the expected value of the error term is zero, so as a side note you have to use the form Y_t = 1.08*Y_t-1 *(1+e)