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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?

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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 $t$:

  1. Do PCA on the returns before and at $t$ to get $D_{t-i,j}$.
  2. Predict $r_{t+1} + \cdots + r_{t+H}$ by multiplying these quantities be the betas estimated.
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  • $\begingroup$ Thanks for the comment but I don't think you answered my question. I understand what the lhs and rhs represents. I also understand that the regression models the future $r_{t+1}+...+r_{t+H}$ returns. What I am asking is in the actual construction/implementation of the model, what is the independent variable? We clearly do not yet have future returns, so upon what are we regressing? $\endgroup$ – strimp099 Jun 15 '15 at 11:40
  • $\begingroup$ The term $D_{t_i,k}$ are based on past returns before $t$. So one could say that they are independent and that $r_{t+1} + \cdots$ is "dependent". I would rather say that these are explanatory for the latter. Do you want to find out how that $D$ terms are calculated precisely? $\endgroup$ – Richard Jun 15 '15 at 15:46
  • $\begingroup$ The paper outlines the construction of $D$ and $\sum_{i=0}^HD_{t-i,1}$ is basically the rolling sum of $D$ for each $k$ from $t-i$ through $t$. I also agree that $r_{t+1} + ... + r_{t+H}$ is dependent on $...\sum_{i=0}^HD_{t-i,1}...$. Think about a very simple regression where $\hat{y} = \alpha + \beta x + \epsilon$. When constructing the model, there exists an existing dependent variable $y$ upon which to regress, or train the model. What is the "$y$" in this case? $\endgroup$ – strimp099 Jun 16 '15 at 8:06
  • $\begingroup$ $y$ in this case is $r_{t+1}+ \cdots +r_{t+H}$. In training the model you have to form sets of $y$ and corresponding $x$ (which are terms involving $D$). These will overlap if you increase $t$ by $1$ only - which should be ok as this usually happens in time-series regression. $\endgroup$ – Richard Jun 16 '15 at 12:06

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