# How can I test for the unknown parameter of this model?

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.

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