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.}

Do you want us to specify the formula representation for each time series? Or only give the number of differences needed to get to a stationary series?

[NEAS: See the NEAS post below; just explain how the differences create a stationary model.]

For the first one, I get that second differences are stationary. Specifically if Vt is the time series of the investment value at time t, I get:

Vt = 100 * t * Yt, which is not stationary.

∇Vt = 100 * t * ∇Yt + 100Yt-1, which is still not stationary.

∇²Vt = 200 * (α + ϵ), which is stationary.



For Part B, I'm running into more trouble. Intuitively, the second difference is no longer stationary. It should have constant mean, but variance that grows with t as the increasing per share value increases the variance in Xt. Modeling this out in excel seems to support this.

Taking the log of the first difference and then taking the difference of that seems to get to a time series that is close to stationary. It looks like the result has constant mean and decreasing variance, but approaches constant variance as t grows.

If I specify the number of shares owned as St = St-1 + Xt, then I get:

∇Vt = St * Yt - St-1 * Yt-1 = St-1 * Yt + Xt * Yt - St-1 * Yt-1 = St-1 * ∇Yt + Xt * Yt

log(∇Vt) = log(St-1) + log(∇Yt) + log(Xt) + log(Yt)

∇(log(∇Vt)) = log(St-1 / St-2) + log(Yt / Yt-1) + log(∇Yt / ∇Yt-1) + log(Xt / Xt-1)

The first two terms should have means and variances that decrease towards 0, the second two terms I think should be stationary.

Am I on the right track? I haven't started on part C yet.

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.

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.]