3

In demonstrating that the unconditional mean of $a_t=\sigma_t \epsilon_t$ is $0$, my professor uses the tower property of conditional expectation:

$\mathbb E [a_t] = \mathbb E\mathbb E[a_t|F_{t-1}]=\mathbb E[ \sigma_t\mathbb E(\epsilon_t) ]=0$

From my understanding, $\sigma_t$ and $\epsilon_t$ are independent. So we should be able to do:

$\mathbb E [a_t] = \mathbb E[\sigma_t]\mathbb E[\epsilon_t]=\mathbb E[\sigma_t]\centerdot0=0$

What is the reason that he doesn't just do this instead?

Twilight Sparkle
  • 217
  • 1
  • 7
  • 1
    Good point. From the definitions of ARCH(1) process (which you neglected to state) we have that $\sigma_t\sim F_{t-1}$ and $\varepsilon_t$ is independent from $F_{t-1}$ so we can surmise that $\sigma_t$ and $\varepsilon_t$ are independent. However the tower property is used extensively when dealing with ARCH processes, so the reason might be purely pedagogical one. Also all the time series textbooks I've seen prove this property using tower properties, so another reason might be that this is tradition. – mpiktas Mar 19 '14 at 07:58
  • 1
    You've introduced the mistake again. The first equality is false as it stands, that is why I added the second expectation sign. – mpiktas Mar 19 '14 at 08:22
  • Hi, thank you for your interesting comments. Apologies for accidentally editing it wrongly. I didn't notice that you edited, and I'd forgotten that the second Expectation was needed so I thought that I had made an editing error. I've reverted back. Haha I blame the lack of sleep. Thank you! – Twilight Sparkle Mar 19 '14 at 08:51
  • @mpiktas: Do you mind putting your comment as an answer so that I can accept it? – Twilight Sparkle Apr 13 '14 at 10:49

0 Answers0