Tim Olson
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61 lines
3.6 KiB
Markdown
61 lines
3.6 KiB
Markdown
---
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description: "Extends cluster mean-reversion to a general loadings matrix with non-binary (continuous) risk factor exposures and regression weights, enabling neutrality to arbitrary factor sets."
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tags: [stocks, mean-reversion, weighted-regression, statistical-arbitrage]
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---
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# Mean-Reversion — Weighted Regression
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**Section**: 3.10 | **Asset Class**: Stocks | **Type**: Mean-Reversion / Statistical Arbitrage
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## Overview
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This strategy generalizes the cluster mean-reversion approach (Section 3.9) by replacing the binary loadings matrix with a general (possibly non-binary) loadings matrix `Omega_{iA}`. The resulting demeaned returns are orthogonal to all K loadings vectors, providing neutrality to the corresponding risk factors. Non-binary columns can represent industry-neutral, style factor, or principal component exposures.
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## Construction / Signal
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The orthogonality condition for the twiddled (demeaned) returns `R_tilde_i` to a general loadings matrix `Omega_{iA}`:
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```
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sum_{i=1}^{N} R_tilde_i Omega_{iA} = 0, A = 1,...,K (313)
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```
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The twiddled returns are the residuals `epsilon_i` of the regression of `R_i` on `Omega_{iA}` with regression weights `z_i`:
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```
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R_tilde = Z epsilon (314)
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epsilon = R - Omega Q^{-1} Omega^T Z R (315)
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Z = diag(z_i) (316)
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Q = Omega^T Z Omega (317)
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```
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When the intercept is included in `Omega_{iA}` (i.e., a linear combination of columns equals the unit N-vector nu), then automatically:
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```
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sum_{i=1}^{N} R_tilde_i = 0 (318)
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```
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(dollar-neutrality is automatic).
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Weights `z_i` can be taken as `z_i = 1/sigma_i^2` where `sigma_i` are historical volatilities (inverse-variance weighting).
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## Entry / Exit Rules
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- **Entry**: Compute residuals from the weighted regression; enter positions proportional to `-R_tilde_i` (buy underperformers relative to factor model, short outperformers).
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- **Exit**: Close when residuals converge; or at a fixed holding horizon.
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- **Dollar-neutrality**: Automatically satisfied if intercept is included in Omega.
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## Key Parameters
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- **Loadings matrix Omega_{iA}**: Binary (industry/sector) or non-binary (continuous risk factors, PCA components)
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- **Regression weights z_i**: Often `1/sigma_i^2` (inverse variance) or uniform
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- **Number of factors K**: At least 1; more factors remove more systematic risk exposures
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- **Holding period**: Short-term, matching the mean-reversion horizon
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## Variations
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- **Binary Omega (reduces to Section 3.9)**: When Omega is a binary cluster membership matrix, recovers the single-cluster or multi-cluster mean-reversion formula exactly
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- **PCA-based Omega**: Use principal components of the return covariance matrix as non-binary columns
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- **Style factor neutrality**: Add style factor exposures (value, momentum, size, liquidity, volatility) as columns in Omega
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## Notes
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- This is the most general form of the cluster mean-reversion strategy family.
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- Non-binary columns in Omega (e.g., industry-based continuous risk factors, or PCA-derived factors) allow neutralization of more complex systematic risks.
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- In the zero specific-risk limit (all variance is factor-driven), optimization reduces to weighted regression.
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- The choice of Omega and z_i is the key design decision; binary industry/sector classifications are stable out-of-sample; continuous factor exposures require more frequent recalibration.
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- Building a reliable loadings matrix Omega is closely related to constructing a risk model (see Kakushadze and Yu references).
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