# How to use factor models for prediction?

I was looking at this thread here, reading about how to run regressions and thereby construct factor models.

Assuming these factor models are properly specified, I am trying to better understand how they facilitate prediction of future returns. How do they exactly help with the estimation of a future return vector's mean vector and covariance matrix? These regressions estimate the conditional distribution of returns, given contemporaneous information about factors.

These are my questions:

1. how are future factors forecasted from old ones?
2. Are there economic arguments that justify the avoidance of this issue? I suspect that some people will assume factors have conditional (on previous factors) mean zero and constant conditional covariance matrix.
3. Are there any references where people run these regressions on lagged factors?
4. Why aren't the dynamics of the factors talked about more? Perhaps they are, though, and I just need to read more.

Ideally your answers will provide highly cited papers and other resources. I'm looking for sources and specifics here.

# Edit:

Thanks to @Will Gu for pointing out the difference between "risk" models and "alpha" models. I'm not totally convinced that modelling should be done separately for these two tasks, though. In either case, one wants the probability distribution of the future returns $r_{t+k}$, given the past returns $r_{1:t}$ and past factor information $f_{1:t}$ (of course $f_{1:t}$ may be some deterministic transformation of the returns data $r_{1:t}$.

I suspect for these "risk" models, one takes concurrent factors and is implicitly using the fact that $$p(r_{t+k}|r_{1:t},f_{1:t}) = \int p(r_{t+k}|f_{t+k})p(f_{t+k}|r_{1:t},f_{1:t})df_{t+k}$$ if one can assume conditional independence in certain areas. These regressions that people run emphasize estimation of $p(r_{t+k}|f_{t+k})$ and neglect the other part: $p(f_{t+k}|r_{1:t},f_{1:t})$.

I suspect that when one is fitting an "alpha" model one is using the decomposition above OR just trying to estimate

$$p(r_{t+k}|f_{1:t},r_{1:t})$$

directly.

• Separating "risk" and "alpha" is in general a hack and a bad idea. The only real separation in my view is that between prediction (has a clear simple loss function) and taking actions (which involves optimizing some other objective). The research side of this is how, under limited model capacity, you balance between allocating model resources to prediction in general vs prediction in areas more relevant to decisions in your action space. For example, you don't care about accuracy in domains where you can not find opportunities. Apr 1, 2020 at 13:08

You kinda mentioned in your questions, but the predictive model is essentially a lagged version of the "factor model". Part of the problem comes from the subscripts, the model itself doesn't really constrain when you observe the predictors. For now, we can assume the subscript $t$ means information at or up to time $t$, then if you change your response vector to $t+1$, this is effectively a predictive model.
Of course, in reality you might find a lot of factors have strong explanatory power but not much predictive power, e.g. regressing individual stock return on SP500 index return of the same day might get you a decent $R^2$. It explains the variance of your stock return, but can't be used for prediction. You would need to study the residuals for unexplanatory variances.