Testing the Random Walk Model

The Efficient Market Hypothesis (EMH)

EMH states that markets are efficient in the sense that investors take into account all available information when making an investment decision. Therefore, the only reason prices change is due to randomness. In other words, randomness of prices is a sign of markets operating efficiently.

The random walk model

The random walk model is similar to the EMH and states that asset prices are due to randomness and can be modelled using the equation

r_t = μ + σz_t

where μ and σ are constants. 

Hence, if we observe, measure, and estimate μ and σ, they should be the same within any time period we look at (as long as the duration of the time periods are the same), to within sampling error and statistical estimates. 

How to test the random walk model

The following section describes the approach used by Andrew Lo and Craig Mackinlay to test the random walk model in their 1987 paper “Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test”. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=346975 

To test the model, they tested the property that the variance of a random walk model scales with time. The approach was to look at the variance over random walks of different lengths taken over the period 1962-1985. 

Steps

They started by computing weekly log returns, where

r_t = log(P_t/P_t-1)

Next, they computed returns for varying periods (e.g. 2-weeks returns, 4-weeks returns, q-weeks returns etc).

For instance, for 2-period (i.e. 2-weeks) returns:

r_t⁽²⁾
= r_t + r_t-1
= log(P_t/P_t-1) + log(P_t-1/P_t-2)
= log(P_t/P_t-2)

For q-period returns:

r_t^(q)
= log(P_t/P_t-q)

If the returns are uncorrelated, variance computed from each series should be proportional to its length. In other words, Var(r_t^(q)) should equal qVar(r_t).

Therefore, Lo and Mackinlay tested whether the ratio Var(r_t^(q)) / qVar(r_t) equals 1. This ratio is known as the variance ratio and denoted as VR(q). 

The variance ratio will not be exactly 1. The question is, how far away from 1 is it acceptable?

To answer this question, Lo and Mackinlay scaled the variance ratio to a value called z(q) and tested if z(q) follows a normal distribution with mean 0 and variance 1. 

z(q) is given by the formula

z(q) = T/(2(q-1)(VR(q) – 1) 

where T = number of r_t in the time series.

Their conclusion was asset prices do not follow a random walk model.