Regression, beta and alpha

When a strategy makes money, the first question is whether it did anything clever or simply held the market in a rising year. Linear regression answers it.

The market model

Regress the strategy’s excess return on the market’s excess return:

$$r_t - r_f = \alpha + \beta\,(r_{m,t} - r_f) + \varepsilon_t$$

• $\beta$ measures exposure to the market. A beta of 1.2 means the strategy tends to move 1.2% for every 1% market move.
• $\alpha$ is the average return left over after paying for that exposure. This is the part you can claim as skill.
• $\varepsilon_t$ is idiosyncratic noise, uncorrelated with the market by construction.

Ordinary least squares

OLS picks $\alpha$ and $\beta$ to minimise the sum of squared residuals. For a single regressor the solution is

$$\hat\beta = \frac{\operatorname{Cov}(r, r_m)}{\operatorname{Var}(r_m)} = \rho\,\frac{\sigma_r}{\sigma_m}, \qquad \hat\alpha = \bar r - \hat\beta\,\bar r_m$$

With many regressors in a matrix $X$, it is $\hat\beta = (X^\top X)^{-1} X^\top y$.

The standard error of $\hat\beta$ shrinks with more data and more variation in the regressor. The t-statistic $\hat\beta / \operatorname{SE}(\hat\beta)$ tests whether the coefficient differs from zero; values above about 2 are conventionally “significant”, though in strategy research you should demand much more.

R-squared

$R^2$ is the fraction of variance in $y$ explained by the regressors. For a market-neutral strategy you want low $R^2$ against the market. For a hedge you want high. It says nothing about whether the relationship is causal or stable.

Multi-factor models

Beyond the market, returns load on other systematic factors: size, value, momentum, quality, low volatility, and in crypto, BTC beta and funding-rate carry. Regressing on several factors at once,

$$r_t = \alpha + \sum_k \beta_k f_{k,t} + \varepsilon_t$$

often shows that an apparently novel strategy is mostly a known factor in disguise. The residual alpha is usually much smaller than the raw return.

Assumptions that bite in finance

• Constant coefficients. Betas drift. Use rolling or exponentially weighted regressions to see how much.
• Homoskedastic, uncorrelated errors. Financial residuals are neither. Use robust (Newey–West) standard errors for t-stats on overlapping or autocorrelated returns.
• No look-ahead. Regressors must be known at the time of the trade. A factor built from end-of-month data cannot explain returns earned during that month.