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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

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  • $\begingroup$ It is good practice here to share your idea, meaning: Show any work and progress that you have done/made yourself. $\endgroup$ – vonjd May 16 '13 at 15:52
  • $\begingroup$ $f(\cdot)$ and $g(\cdot)$ can be written as $f(x)=ax+b$ and $g(y)=cy+d$ . Then, $$ cov[f(r_{t});g(r_{t+k})]=ac\alpha^{2}(E[\varepsilon_{t-1}^{2}\varepsilon_{t+k-1}^{2}]-E[\varepsilon_{t-1}^{2}]E[\varepsilon_{t+k-1}^{2}])+ac\alpha(E[\varepsilon_{t-1}^{2}\varepsilon_{t+k}]-E[\varepsilon_{t-1}^{2}]E[\varepsilon_{t+k}])+ac\alpha(E[\varepsilon_{t}\varepsilon_{t+k-1}^{2}]-E[\varepsilon_{t}]E[\varepsilon_{t+k-1}^{2}]) $$ If $k=\pm1$ , then it's $ac\alpha(E[\varepsilon_{t}^{3}])$ , else it's 0. So it can be only 0 if $E[\varepsilon_{t}^{3}]=0$ . $\endgroup$ – Daniel May 16 '13 at 18:28
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    $\begingroup$ Include this not as comment, but within the body of the question... $\endgroup$ – SRKX May 17 '13 at 8:18

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