Conditional Return Expectations in an ARCH Model

Summary

The document poses a question about the conditional expectation of stock returns in a model with a return driver, a volatility term, and an ARCH(1) variance process. The innovation is written as conditional volatility multiplied by a standard normal shock. The key calculation separates quantities known from the conditioning information from the random innovation: given the previous residual, current conditional variance is determined, while the current shock has mean zero.

This implies the conditional expectation is the predictable component of the return equation, including the model's volatility effect, rather than an expectation of the squared shock inside a square root. The question itself does not include an accepted answer, derivation, or empirical evidence, so it offers a setup for reasoning about conditional expectations rather than a fully developed treatment. Its result depends on the stated normal, zero-mean innovation assumption and on which variables are included in the conditioning set.

Key ideas

• Given the previous residual, the ARCH variance for the current period is known.
• The current innovation has conditional mean zero when its standardized shock is independent and zero-mean.
• The volatility term in the return equation remains in the conditional expectation if it is known under the conditioning information.
• Squaring the innovation does not provide a shortcut for finding the expectation of the signed innovation.
• The document gives a model setup but no full solution or empirical evidence.

Tags

# ARCH; Expectation and Variance


# ARCH; Expectation and Variance












I have got the following question that I am struggling to answer.

The stock return $S_t$ follows the following DL model, with $Z_t$ being a dependent variable explaining the stock return:

$S_t = \beta_0 + \beta_1 Z_t + \delta \sigma_t + \epsilon_t$

with $\epsilon_t=\sigma_t\zeta_t$ and $\sigma^2_t= \omega + \alpha \epsilon^2_{t-1}$.

Assuming that $\zeta $ is i.i.d. $N(0,1)$, in order to calculate $E[S_t|X_t,\epsilon_{t-1}]$, I set: $\epsilon_t^2 = \sigma_t^2\zeta_t^2 = \zeta_t^2[\omega + \alpha \epsilon^2_{t-1}]$

$E[S_t|X_t,\epsilon_{t-1}] = E[\beta_0 + \beta_1 Z_t + \delta \sigma_t + \epsilon_t|X_t,\epsilon_{t-1}] = E[\beta_0 + \beta_1 Z_t + \delta \sigma_t + \sqrt{\zeta_t^2[\omega + \alpha \epsilon^2_{t-1}]}|X_t,\epsilon_{t-1}]$

Is there an intuition that makes the calculation easier?

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.