Why Squared Lagged Shocks Can Break Return Independence

Summary

The document poses a question about whether a return process qualifies as a random walk under a definition requiring zero covariance between linear transformations of returns at distinct times. The process includes the square of the previous period’s innovation alongside the current innovation, with innovations assumed independent and identically distributed with zero mean and constant variance.

This setup invites checking both ordinary return autocovariance and dependence in transformed returns. Although the lagged squared shock can be uncorrelated with the current return under symmetric or suitable innovation distributions, squared returns can carry information about later returns because the conditional distribution depends on the prior shock’s magnitude. Thus zero linear autocorrelation alone would not establish the stronger condition stated in the question. The document provides no answer, derivation, or empirical evidence, so the issue remains an analytical prompt; conclusions can also depend on the innovation distribution and the exact interpretation of the definition’s allowed functions.

Key ideas

• The proposed return process depends on both the current innovation and the previous innovation’s squared value.
• Zero autocorrelation does not by itself establish independence across time.
• Transformed returns, such as squared returns, may reveal dependence that linear correlation misses.
• The document asks the classification question but does not include a solution.

Tags

# Is it random walk?


# Is it random walk?












I would like to ask a question about random walk. Campbell, Lo & Mackinlay defined the random walk, in the following way (RW3):

$$ cov[f(r_{t}),g(r_{t+k})]=0,\qquad k\neq0 $$

for all $f(\cdot)$ and $g(\cdot)$, where $f(\cdot)$ and $g(\cdot)$ are linear functions, and $\{r_{t}\}$ is a series of returns. So, the question is about the following equation: $$ r_{t}=\alpha\varepsilon_{t-1}^{2}+\varepsilon_{t},\qquad\varepsilon_{t}\sim IID(0,\sigma^{2}). $$

Is it random walk or not? And why? (I have an idea, but i don't know, if it's true.)

Thanks

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.