Some clarifications:

1. What is a Random Walk?
By repeated substitution we have
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2. Do stock market prices follow a Random Walk?

That wouldn't make any sense at all since nobody would invest in something that has an expected value of zero.

3. What process do stock market prices follow under the Random Walk Hypothesis?

Under the Random Walk Hypothesis stock market returns are comprised of a constant part (the risk free return?) plus a random part, the process for stock prices takes the following form:One shouldn't mix continuous time with discrete time but if one keeps in mind that (X_t - X_t-1)/X_t-1 is approximately Δln(X_t) it is quite clear what is meant by saying that the log-price follows a random walk with drift (r is the drift).

4. What is the easiest way to reject the random walk hypothesis?

Well, one important property of the random walk Y_t is that the variance of its increments ε_t is linear in the observation interval. That is, the variance of Y_t - Y_t-2 is twice the variance of Y_t - Y_t-1. There are quite a lot of statistical test around for checking this property.

5. Wow, is this exciting?

No. Nowadays everybody knows that that the variance of stock returns isn't constant over time (= nowadays everybody knows that log-prices don't follow a random walk). So you could go on an say: Hey, maybe the variance isn't constant over time but it could still be the case that the increments are independent (under the assumption that the increments are normally distributed this would be the same as saying that increments are uncorrelated). Then somebody could claim that the log-price process is pretty close to a random walk. Your task would be find a test that is robust to heteroscedasticity (= unqueal variances), i.e. a test that would ignore the non-constant variance but can tell you whether the increments are independent. See MacKinley and Lo for further information.

6. What has the Random Walk Hypothesis to do with the Efficient Market Hypothesis?

I couldn't care less. But according to MacKinley and Lo:

Under very special circumstances, e.g. risk neutrality, the two are equivalent. However, LeRoy (1973), Lucas (1978), and many others have shown in many ways and in many contexts that the Random Walk Hypothesis is neither a necessary nor sufficient condition for rationally determined security prices. In other words, unforecastable prices need not imply a well-functioning financial market with rational investors, and forecastable prices need not imply the opposite.

These conclusions seem sharply at odds with Samuelson’s “proof” that properly anticipated prices fluctuate randomly, an argument so compelling that it is reminiscent of the role that uncertainty plays in quantum mechanics. Just as Heisenberg’s uncertainty principle places a limit on what we can know about an electron’s position and momentum if quantum mechanics holds, Samuelson’s version of the Efficient Markets Hypothesis places a limit on what we can know about future price changes if the forces of economic self-interest hold. Nevertheless, one of the central insights of modern financial economics is the necessity of some trade-off between risk and expected return, and although Samuelson’s version of the Efficient Markets Hypothesis places a restriction on expected returns, it does not account for risk in any way. In particular, if a security’s expected price change is positive, it may be just the reward needed to attract investors to hold the asset and bear the associated risks. Indeed, if an investor is sufficiently risk averse, he might gladly pay to avoid holding a security that has unforecastable returns. In such a world, the Random Walk Hypothesis—a purely statistical model of returns—need not be satisfied even if prices do fully reflect all available information. This was demonstrated conclusively by LeRoy (1973) and Lucas (1978), who construct explicit examples of informationally efficient markets in which the Efficient Markets Hypothesis holds but where prices do not follow random walks.

Grossman (1976) and Grossman and Stiglitz (1980) go even further. They argue that perfectly informationally efficient markets are an impossibility, for if markets are perfectly efficient, the return to gathering information is nil, in which case there would be little reason to trade and markets would eventually collapse. Alternatively, the degree of market inefficiency determines the effort investors are willing to expend to gather and trade on information, hence a non-degenerate market equilibrium will arise only when there are sufficient profit opportunities, i.e., inefficiencies, to compensate investors for the costs of trading and information-gathering. The profits <continue reading>