Using PCA of Past Returns to Predict Future Returns with OLS
Summary
The note clarifies the timing and roles of variables in an OLS model built from principal component projections of asset returns. The dependent variable is the accumulated log return over the next H periods, from t+1 through t+H. The explanatory variables are accumulated projections of returns observed at or before t, so the model relates a future return outcome to historical information.
To estimate the coefficients, form training observations pairing each historical window of projected returns with the realized forward H-period return. Once the coefficients are estimated, apply the PCA transformation to returns available through the current time and use the resulting historical projections to forecast the next H-period accumulated return. This explanation resolves the apparent issue that future returns are unavailable at prediction time: they are observed later for historical training labels, while forecasts use only information available at t. The note provides no performance evidence or implementation details about PCA choices, overlapping labels, or model validation.
Key ideas
- The dependent variable is the realized accumulated return over the next H periods.
- The explanatory variables are accumulated principal component projections based on returns observed through time t.
- OLS estimation pairs historical predictor windows with subsequently realized forward returns.
- At prediction time, use only the PCA projections available from past and current returns.
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# Regression model syntax
# Regression model syntax
I'm following the methodology outlined in Developing High-Frequency Equities Trading Models. On page 27, the author outlines an OLS regression model to obtain beta coefficients. The model is defined as:
$$r_{t+1} + ... + r_{t+H} = \beta_1\sum_{i=0}^HD_{t-i,1}+...+\beta_{k}\sum_{i=0}^HD_{t-i,k}+\eta_{t+H,H}$$
Where $r_{t+1} + ... + r_{t+H}$ is the accumulated $H$-period log return, $D=R^{T,k}$ is a $Txk$ matrix of dimentionally reduced log returns (principal components) obtained after projecting de-meaned log returns on the highest $k$ eigenvectors.
The author defines the process estimating $r_{t+1} + ... + r_{t+H}$ as such:
> We calculated the accumulated future $H$-period log returns. Then we ran a regression, estimated by OLS, on the future accumulate log returns with the last sum of $H$-period dimensionally reduced returns in the principal component space.
I'm struggling a bit to understand how to implement this model. I suppose where I am most confused is in the statement "We calculated the accumulated future $H$-period log returns." I can see clearly that the independent variables are the accumulated $H$-period dimentionally reduced log returns but I do not understand what the dependent variable would be in this case as we do not have future accumulated returns.
This is likely simply a question on syntax but one that has confused me in the past as well. So stated simply, what is the independent variable in this model?
## Answer by Richi Wa (score 1)
https://quant.stackexchange.com/a/18361
I browsed through the work and this is what I see:
- the lhs $r_{t+1} + \cdots + r_{t+H}$ is the sum of log-returns after $t$.
- the rhs is indexed by $t-i, i=0, \ldots, H$ thus this has something to do with the past before (and at) $t$.
Thus the regression models the future ($r_{t+1} + \cdots + r_{t+H}$) dependent of the past where only PCA projections of past returns are used.
In the thesis they talk about historical $H-$period (dimensionally-reduced) returns.
Thus the estimation of the model is clear. Then having estimated the vector $\beta$ you do the following at time $t$:
- Do PCA on the returns before and at $t$ to get $D_{t-i,j}$.
- Predict $r_{t+1} + \cdots + r_{t+H}$ by multiplying these quantities be the betas estimated.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.