Why OLS Regression Residuals Are Uncorrelated with the Market
Summary
The document explains why a portfolio’s residual return is uncorrelated with the market return when beta is estimated by ordinary least squares on the same return observations. It writes portfolio returns as an intercept, a market component, and a residual, then uses the OLS beta formula, covariance divided by market variance, to show that the residual’s covariance with the market is zero.
Geometrically, OLS projects the portfolio return series onto the space spanned by the market series; the residual is orthogonal to that projection. This result follows from the regression fit rather than from a separate empirical guarantee. It applies to the sample used to estimate the regression, with an intercept included as described. It does not establish that residuals will remain uncorrelated out of sample, nor does it by itself confirm the assumptions needed for every variance decomposition or asset-pricing interpretation.
Key ideas
- OLS beta is the covariance of portfolio and market returns divided by market-return variance.
- The regression residual has zero sample covariance with the market return when beta is estimated by OLS.
- The residual is orthogonal to the fitted market component by construction.
- The property applies to the estimation sample and does not ensure zero correlation in future data.
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# how can we know the residual return will be uncorrelated with the market return # how can we know the residual return will be uncorrelated with the market return I was reading that if we know a portfolios beta we can break the excess return on that portfolio into a market component and a residual component. ``` r_p = beta_p * r_m + e_p r_p - portfolio excess return r_m - market return e_p - residual return beta_p - portfolio beta ``` It then goes on to so say, the residual return (e_p) will be uncorrelated with the market return (r_m) and so the variance of the portfolio is ``` var_p = beta_p^2 * var_m + var_p_residual var_p - variance of portfolio beta_p^2 - beta of portfolio squared var_m - variance of market var_p_residual - variance of portfolio residual ``` So my question is how can we know the residual return will be uncorrelated with the market return? I've found this web page which near the top it has a section titled The Key Assumption. Consider, for example, a case in which the residual return is correlated with factor 1. By adjusting the factor exposure (bi1) appropriately, the correlation of the residual with the factor can be made to equal zero. I'm not sure if this is linked to my question or not but I still don't follow it either ## Answer by Richi Wa (score 5, accepted) https://quant.stackexchange.com/a/25689 Let us ignore the riskless rate for simplicity of the presentation. If you have (historical or simulated) return series $r_i$ for the portfolio and $r_i^M$ for the market, then the beta is the OLS regression beta: $$ \beta = cov(r_i,r_i^M)/var(r_i^M). $$ Then if you write $r_i = \alpha + \beta r_i^M + \epsilon_i$ on the other hand $$ \epsilon_i = r_i - ( \alpha + \beta r_i^M). $$ Then the covariance of these erros with the market are given as follows: $$ cov(\epsilon_i, r_i^M) = cov(r_i - ( \alpha + \beta r_i^M),r_i^M) $$ and as $cov(\alpha,r_i^M) = 0$ ($\alpha$ is a constant) we get $$ cov(r_i - ( \alpha + \beta r_i^M),r_i^M) = \\ cov(r_i,r_i^M) - \beta cov(r_i^M,r_i^M) = cov(r_i,r_i^M) - cov(r_i,r_i^M)/var(r_i^M) * var(r_i^M) =0, $$ as $cov(r_i^M,r_i^M) = var(r_i^M)$. In a geometric sense Beta tells you the projection on the space spanned by the market. The residual is orthogonal by construction.
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