Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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



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?

share|improve this question
Dependence being non-linear is not a sufficient or necessary condition for uncorrelation. e.g. $X = a + bX + cX^2 + e(t)$. Thus your second last paragraph is incorrect. – user2763361 Feb 26 '14 at 7:17

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.