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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?

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  • $\begingroup$ 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. $\endgroup$ – user2763361 Feb 26 '14 at 7:17
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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.

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