ARCH Volatility Clustering and Uncorrelated Returns
Summary
The document explains why autoregressive conditional heteroskedasticity can be consistent with a version of the random walk hypothesis in which returns are uncorrelated but not statistically independent. In an ARCH(1) model, the return residual is white noise scaled by a volatility term that depends on the previous squared residual. This creates serial dependence in conditional variance even when the return innovations have zero autocorrelation.
The answer distinguishes three formulations: constant-distribution returns, independent returns with changing distributions, and uncorrelated returns whose volatility depends on recent history. ARCH and GARCH models represent this final kind of volatility clustering. The explanation is conceptual and confirms the main distinction between linear correlation and broader dependence. Its phrasing of the random-walk variants is informal, and the setup assumes a stationary ARCH process; it does not explore estimation, forecasting performance, or the conditions needed for uncorrelated returns in every model specification.
Key ideas
- ARCH models let current volatility depend on past squared return residuals.
- Changing conditional variance can create dependence in squared returns while raw returns remain uncorrelated.
- Uncorrelated returns are not necessarily statistically independent.
- ARCH and GARCH models describe autoregressive patterns in volatility.
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# How come the existence of ARCH effect is not a violation of Random Walk Hypothesis 3?
# How come the existence of ARCH effect is not a violation of Random Walk Hypothesis 3?
An ARCH (autoregressive conditional heteroscedastic) (1) model is:
$r_t=\mu +a_t$, where $a_t=$return residual, and $\mu$ is the drift of the stock return
$a_t=\sigma_t\epsilon_t$, where $\sigma_t=$standard deviation at time $t$ and $\epsilon_t=$ white noise
$\sigma_t^2=\alpha_0+\alpha_1a_{t-1}^2$, where $\alpha_1<1$ so that the process is stationary
Random walk 3 states that returns are dependent but uncorrelated, such that
$Cov(\epsilon_t,\epsilon_{t-k})=0$
$Cov(\epsilon_t^2,\epsilon_{t-k}^2)\neq0$
If we take the square root of $\sigma^2$, then $\sigma_t=\sqrt{\alpha_0+\alpha_1a_{t-1}^2}$ so $a_t=\sqrt{\alpha_0+\alpha_1a_{t-1}^2}\epsilon_t$.
Therefore the dependence of $a_t$ and $a_{t-1}$ is nonlinear, therefore they are uncorrelated but dependent, and satisfies RW3.
Can someone confirm if this looks correct?
## Answer by user7056 (score 1)
https://quant.stackexchange.com/a/10469
I would confirm it.
For time series forecasting, one can use 3 versions of random walk:
RW model 1 (basic geometric random walk): stock returns in different periods are statistically independent (uncorrelated) and identically distributed (constant volatility)
RW model 2: stock returns in different periods are statistically independent bot not identically distributed: volatility might change deterministically over time or depend on the current price level.
RW model 3: stock returns in different periods are statistically independent (uncorrelated) but not otherwise independent, so the volatility in one period might depend on the volatility in recent periods. (G)ARCH models give a particular behaviour for such volatility dependence: it follows an autoregressive process.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.