Simple regression

Regression fits a straight line through points and reports how well it fits. In finance it is how beta is measured, how fund performance is attributed, and how most empirical claims are made.

The model

$y_i = \alpha + \beta x_i + \varepsilon_i$

• $y$ — the dependent variable, what you are explaining
• $x$ — the independent variable, what you are explaining it with
• $\alpha$ — intercept: $y$ when $x$ is zero
• $\beta$ — slope: how much $y$ moves per unit of $x$
• $\varepsilon$ — the residual, what the line does not explain

Least squares

The line is chosen to minimise the sum of squared residuals, $\sum \varepsilon_i^2$. Squaring rather than taking absolute values makes the solution a closed form rather than an iterative search — and has the side effect of weighting large errors heavily, so a few outliers can move the line a long way.

The slope comes out as:

$\hat{\beta} = \frac{\text{Cov}(x, y)}{\text{Var}(x)}$

Worth pausing on. Beta is a covariance divided by a variance — exactly the quantities from chapter 5. The intercept then follows from the line passing through the means:

$\hat{\alpha} = \bar{y} - \hat{\beta}\bar{x}$

The finance application

Regress a stock's excess returns on the index's excess returns:

$r_{\text{stock}} - r_f = \alpha + \beta (r_{\text{market}} - r_f) + \varepsilon$

$\beta$ is sensitivity to the market. A beta of 1.35 says that historically, when the index moved 1%, this stock moved about 1.35%. Above 1 is more volatile than the market; below 1, less; negative would mean it moved against it.

$\alpha$ is return unexplained by market exposure. A positive, statistically significant alpha is the empirical definition of skill — and chapter 8's standard errors are why so few managers can demonstrate one.

$\varepsilon$ is the firm-specific part, the piece diversification can remove because it is uncorrelated across companies.

The index matters here. The Nifty 50 is free-float market-capitalisation weighted, so a beta against it is a beta against a particular, concentrated definition of "the market" — not against the Indian economy.

R-squared

$R^2 = \frac{\text{explained variance}}{\text{total variance}} \in [0, 1]$

The share of $y$'s variation the line accounts for. An $R^2$ of 0.41 means 41% of the stock's movement tracks the index and 59% is its own.

Three things it does not mean:

It is not a measure of correctness. A high $R^2$ on a nonsense relationship is still nonsense. Two series both trending upward over time will regress beautifully against each other with no connection whatever — chapter 13 names this spurious regression.

It does not say the slope is reliable. $R^2$ and the standard error of $\beta$ are different questions; a low $R^2$ with many observations can still pin the slope down tightly.

Low is not bad. For a single stock against an index, 0.3 to 0.5 is typical and expected — most of a company's movement should be about the company. A fund with an $R^2$ of 0.98 against its benchmark is telling you something useful and unflattering: it is an index fund with active fees.

Is the slope different from 1?

The interesting null for a beta is usually $\beta = 1$, not $\beta = 0$. From chapter 10:

$t = \frac{1.35 - 1.00}{0.22} = 1.59$

Against a critical value near 2, that does not reject $\beta = 1$ at 95%. The point estimate says the stock is a third more volatile than the market; the data does not establish that it differs from the market at all.

This is the chapter's main practical warning. A beta quoted to two decimals on a screen carries a standard error nobody prints.

What regression assumes

Least squares is only unbiased under conditions that market data strains:

• Linearity. An option's relationship to its underlying is not linear, so beta is the wrong description of it.
• Independent residuals. Violated by the volatility clustering of chapter 13.
• Constant residual variance. Violated for the same reason; calm periods and crises have different error sizes.
• No reverse causation. Regression fits association. It cannot tell you which variable moved first.

The point

Regression fits a line by minimising squared residuals, and its slope is a covariance over a variance — which is exactly what beta is. Alpha is the part of return the market exposure does not explain, and $R^2$ is the share of variation the line captures, not a verdict on whether the relationship is real. A beta is an estimate with a standard error, and a stock whose beta reads 1.35 may not be statistically distinguishable from the market.