11.5 The Random Walk Hypothesis

We have seen that econometric models, at least with stock prices and exchange rates, motivate using a random walk as a statistical model. With exchange rates we saw that as a consequence of the uncovered interest rate parity and the assumption of risk neutrality of forward and future speculators the model in (10.14) follows a random walk. Assuming a geometric Brownian motion for stock price as in (10.18), then it follows from Itô's lemma that the log of stock price follows a Wiener process with a constant drift rate:

$\displaystyle d\ln S_t = \mu^* dt + \sigma dW_t$ (11.37)

where $ \mu^* = \mu - \sigma^2/2$. If one observes (10.37) in time intervals of length $ \Delta>0$, i.e., at discrete points in time $ 0, \Delta, 2 \Delta, ...$, then one obtains

$\displaystyle \ln S_{t\Delta} = \ln S_{(t-1)\Delta} + \Delta \mu^* + \sqrt{\Delta} \sigma \xi_t$ (11.38)

with independent, standard normally distributed $ \xi_t, t=1, 2,
...$. The process (10.38) is a random walk with a drift for the logged stock prices. The log returns (see Definition 10.15) over the time interval of length $ \Delta$ are also independently normally distributed with expected value $ \Delta \mu^*$ and variance $ \Delta \sigma^2$.

With long interest rate time series the random walk appears to be less plausible, since it is assumed that in the long-run there is a stationary level around which interest rates fluctuate in the short run. Let's consider once again the process for the short rate in (10.16), the Cox-Ingersoll-Ross (CIR) model. A discrete approximation is

$\displaystyle r_t -r_{t-1} = \alpha + \beta r_{t-1} + \sigma \sqrt{r_{t-1}} \xi_t
$

or

$\displaystyle r_t = \alpha + (1 + \beta) r_{t-1} + \sigma \sqrt{r_{t-1}} \xi_t.$ (11.39)

If $ \beta$ in (10.39) is negative (and larger than -2), then the process is a stationary AR(1) process with heteroscedastic error terms. In Example 10.1 we encountered such a process with heteroscedastic error terms.

There is also the interpretation that interest rates are, at least in the short-run, explained well by a random walk. It is therefore of general interest to test whether a random walk exists. In the following we show the distinguishes through the three versions of the random walk hypothesis. In general we consider a random walk with a drift

$\displaystyle P_t = \mu + P_{t-1} + \varepsilon_t.$ (11.40)

  1. The stochastic errors in (10.40) are independent and identically distributed (i.i.d.) with expectation zero and variance $ \sigma^2$. This hypothesis has already been tested on multiple data sets in the sixties and was empirically determined to be unsupported. For example, distinct volatility clusters were discovered which under the i.i.d. hypothesis are statistically not expected.

  2. The stochastic errors in (10.40) are independent but not necessarily identically distributed with an expectation of zero. This hypothesis is weaker than the first one since, for example, it allows for heteroscedasticity. Nonetheless, even here the empirical discoveries were that a dependence between the error terms must be assumed.

  3. The stochastic errors in (10.40) are uncorrelated, i.e., $ \gamma_\tau(\varepsilon_t) = 0$ for every $ \tau \ne 0$. This is the weakest and most often discussed random walk hypothesis. Empirically it is most often tested through the statistical (in)significance of the estimated autocorrelations of $ \varepsilon_t$.

The discussion of the random walk hypotheses deals with, above all, the predicability of financial time series. Another discussion deals with the question of whether the model (10.40) with independent, or as the case may be with uncorrelated, error terms is even a reasonable model for financial time series or whether it would be better to use just a model with a deterministic trend. Such a trend-stationary model has the form

$\displaystyle P_t = \nu + \mu t + \varepsilon_t$ (11.41)

with constant parameters $ \nu$ and $ \mu$. The process (10.41) is non-stationary since, for example, $ {\mathop{\text{\rm\sf E}}}[P_t]
= \nu + \mu t$, the expected value is time dependent. If the linear time trend is filtered from $ P_t$, then the stationary process $ P_t - \mu t$ is obtained.

To compare the difference stationary random walk with a drift to the trend stationary process (10.41) we write the random walk from (10.40) through recursive substitution as

$\displaystyle P_t = P_0 + \mu t + \sum_{i=1}^t \varepsilon_i,$ (11.42)

with a given initial value $ P_0$. One sees that the random walk with a drift also implies a linear time trend, but the cumulative stochastic increments ( $ \sum_{i=1}^t \varepsilon_i$) in (10.42) are not stationary, unlike the stationary increments ( $ \varepsilon_t$) in (10.41). Due to the representation (10.42) the random walk with or without a drift will be described as integrated, since the deviation from a deterministic trend is the sum of error terms. Moreover, every error term $ \varepsilon_t$ has a permanent influence on all future values of the process. For the best forecast in the sense of the mean squared error it holds for every $ k>0$,

$\displaystyle {\mathop{\text{\rm\sf E}}}[P_{t+k} \mid {\cal F}_t] = P_0 + \mu (t+k) + \sum_{i=1}^t \varepsilon_i.$

In contrast, the impact of a shock $ \varepsilon_t$ on the forecast of the trend-stationary process (10.41) could be zero, i.e.,

$\displaystyle {\mathop{\text{\rm\sf E}}}[P_{t+k} \mid {\cal F}_t] = \nu + \mu (t+k).$

It is thus of most importance to distinguish between a difference stationary and a trend-stationary process. Emphasized here is that the random walk is only a special case of a difference stationary process. If, for example, the increasing variables in (10.40) are stationary but autocorrelated, then we have a general difference stationary process. There are many statistical tests which test whether a process is difference stationary or not. Two such tests are discussed in the next section.