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Engle comments in "Risk and Volatility: Econometric models and Financial Practice" that

If the price of risk were constant over time, then rising conditional variances would translate linearly into rising expected returns. Thus the mean of the return equation would no longer be estimated as zero, it would depend upon the past squared returns exactly in the same way that the conditional variance depends on past squared returns. This very strong coefficient restriction can be tested and used to estimate the price of risk. It can also be used to measure the coefficient of relative risk aversion of the representative agent under the same assumptions.

What return equation is he referring to? What else is being described here?

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Can you edit the title to be more informative? –  John Feb 26 at 15:55
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up vote 4 down vote accepted

The return equation is just an econometric equation that models stock returns (or other asset returns) as a function of: (i) intercept (i.e. the average return), (ii) some independent variables/features, (iii) noise that has zero mean and time-varying variance. There are sometimes other things in the return equation too that form more advanced models. The main difference between a return equation in this kind of modelling and in standard OLS is that the variance of the errors (and hence returns) is allowed to change over time, dropping the homoscedasticity assuption .

The return equation in ARCH/GARCH type models is usually given by $R(t) = \mu(t) + \epsilon(t)$, where $R(t)$ is the evenly spaced return, $\mu(t)$ is some intercept (perhaps $\mu(t) = \mu(z) \forall t \not=z \wedge t,z\in \mathbb{R}_+$ specifying the population expected return at time step $t$, and $\varepsilon(t) := \sigma(t) z(t) \sim D(0,\sigma(t)^2)$ is some heteroscedastic disturbance, whose direction is unpredictable but whose variance is predictable within some error. Such a prediction equation is given by a variance equation (as opposed to a return equation).

The above is a simple return equation. You can have whatever you like. It is common to see $R(t) = \mu(t) + \sum_{i=1}^N \beta_i(t)f_i(X_i(t)) + \varepsilon(t)$ where $f_i$ is a transform (perhaps identity mapping), $X_i$ is a feature, $\beta_i(t)$ are time-varying or time-invariant factor loadings with their own distributions. Even more generally this equation can model a vector of returns and we would be entering the territory of MGARCH modelling.

In my mind, this is all superseded by machine learning models.

I believe what he's talking about can and has been modelled as a GARCH-IN-MEAN model (GARCH-M), where the garch term makes a fancy appearance in the mean equation (ie the return equation). I believe this is what's used to analyse the dynamics between noise and informed traders in the academic literature. The most basic return equation you have above has it that $E[R(t)] = E[\mu(t)]$ which is often set to zero a priori when modelling daily or higher frequencies, thus leaving the modelled return equation as the zero mean $R(t) := z(t)\sigma(t)$. Or if this assumption is not made, the calibrated mean would be small for stocks. By supposing that $R(t)$ depends linearly and positively on $\sigma(t)$, and since $E[\sigma(t)]>0$, we can see what he means when he says the mean will no longer be zero in the return equation. The coefficient estimate on this term in the return equation can be interpreted as the compensation that investors need for holding an asset that gives them the risk inherent in $\sigma(t)$.

If the coefficient estimate is $>0$, we would say that we have suggestive evidence that the representative agent (a mathematical object whose properties/state are supposed to model the 'average' agent) is risk seeking, and risk averse for $<0$ estimate.

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Thank you. That was very helpful. I am intrigued by your comment that machine learning models supersede modeling Engle talks about. Why would that be? Is it because of superior predictive power of machine learning models? What models would these be? I look forward to your clarifying my confusion. –  user7359 Feb 26 at 18:33
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