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.

I'm really in need of some guidance on this HW. I've read and re-read the sections but still can't quite determine how to formulate the time series that are asked for. I've looked through the sample problems other posts but can't find anything that relates to this assignment, it is asked in such a different format than the reading and other samples.

To begin with I think the root of my confusions lies in the apparent inclusion of 't' in the V_t formula in A)

V_t = 100t*Y_t

Additionally, all of the examples in the forums and the readings are in forms that include additional terms ( that seem necessary for the series ) and weights/coefficients. Maybe someone could give the initial steps for part A and I could go from there, for part B) is seems that I'm being asked to multiply by a white noise process, I couldn't find even a simple example of something like this.

[NEAS: The intent of this problem is to explain why differences make a time series stationary. Stock prices are random walks, often with trends. In truth, stock prices are geometric random walks, so we take logarithms and then first differences to make them stationary, but this problem assumes they are simple random walks. if the total investment is the sum of annual stock purchases, we must take second differences to make the time series stationary. Cryer and Chan discuss this topic theoretically; the birthday gift of stocks is an ilustration. You need not form the exact ARIMA process for this assignment; just explain why we need a certain number of differences.]

I was wondering how you know that if the total investment is the sum of annual stock purchases, you must take second differences to make the time series stationary.  

[NEAS: If total investment is the sum of annual stock purchases, then the first difference of total investment is the annual stock purchase. If stock purchases have a constant trend, then the first difference of stock purchases is stationary.]