TS Module 9: Non-stationary time series advanced HW

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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.

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.