TS Module 9: Non-stationary time series advanced HW

(The attached PDF file has better formatting.)

Homework assignment: random walk time series

A bank firm offers a set of investments as lifetime birthday gifts. Each investment buys shares of a stock that follow a random walk. For simplicity, assume the random walk is arithmetic: the share price can be positive or negative. The share price is Y_t = Y_t-1 +

á + å_t, where á is a constant and å_t has a constant variance ó²_t.

Investment #1 buys 100 shares of the stock on each birthday. The value of Investment #1 at time t is the value of all the shares bought so far. What is the time series followed by the value of Investment 1?

Investment #2 buys X_t shares of the stock on each birthday, where X_t is a white noise process with mean of 100 and standard deviation of 10. The value of Investment #2 at time t is the value of all the shares bought so far. What is the time series followed by the value of Investment #2?

Investment #3 buys Z_t shares of the stock on each birthday, where Z_t is a random walk = X_t + X_t-1. The value of Investment #3 at time t is the value of all the shares bought so far. What is the time series followed by the value of Investment #3?

The type of time series means the number of differences to make it stationary, not the parameters or the ARIMA form. For each investment, give a brief explanation of whether one needs to take first, second, or third differences to make the time series stationary.

{This replaces the previous homework assignment, for which the textbook was unclear.}

You mention in the homework that our explanation of the number of differences to stationarity should be brief. Having worked out part A by taking the actual differences, that method is not brief, and much more complicated for parts b and c. I am assuming you are wanting us to use some sort of intuition to come to our conclusions. Would you please give us some guidance on this intuition? I attempted to read previous forums but the pdf was not working. I was hoping to find enlightment there.

Also, the section with the reading mentions we should always log processes dealing with stock prices. However, the homework makes no mention of this. Were you intending for us to log these processes? It does not seem to make checking stationarity from first principles any easier.