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An asset price $p_t$ at any time $t = 1,2, \ldots, n$ is given by $x_t = x_{t-1}e^{\mu + \epsilon_t}$, where $\epsilon \stackrel{iid}{\sim} N(0,1)$.

I was supposed to test $H_0: \mu = 0$ against $H_1: \mu \ne 0$, determining which is the appropriated test as well as the corresponding statistic.

What I tried so far:

  1. From $x_t = x_0e^{n\mu + \sum \epsilon_t}$, and consequently $\ln x_t = \ln x_0 + n\mu + \sum \epsilon_t$, I thought the z-test would fit, since $\ln x_t$ follows normal distribution. Haven't made any progress from this point.
  2. From $\ln x_t = \ln x_{t-1} + \mu + \epsilon_i$, I thought I'd have to test for the drift, since this is the form of a geometric random walk with drift $\mu$. Also haven't made any further progress.

I have never seen this kind of test.

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  • $\begingroup$ If you subtract the $\ln x_{t-1}$ from both sides of the equation in #2, you get... $\endgroup$
    – jbowman
    Apr 27 at 2:31
  • $\begingroup$ $\ln x_t - \ln x_{t-1} = \mu + \epsilon_t$. Sorry, but I still have no clue on how to proceed with the test. $\endgroup$
    – Lenora
    Apr 27 at 3:39
  • $\begingroup$ The phrase “I was supposed to” suggests that this is homework, and if so you should say so explicitly. $\endgroup$
    – Matt F.
    Apr 27 at 4:47
  • $\begingroup$ Try substituting $y_t$ for $\ln x_t - \ln x_{t-1}$ and see how that looks. Can you see how to test $\mu=0$ now? $\endgroup$
    – jbowman
    Apr 27 at 13:54

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