Regression Fits a Line and Tells You Almost Nothing About Why
The workhorse tool of quantitative finance estimates how one variable moves with another. Everything difficult about using it comes from what it does not claim.
What It Does
Linear regression finds the straight line that best fits a set of paired observations, where best means minimising the sum of squared vertical distances from the points to the line.
The output is a slope, an intercept, and a set of statistics describing how well the line fits.
In finance, the most familiar application is beta: regress a stock's returns against the market's returns, and the slope is beta. A slope of 1.3 means the stock has historically moved 1.3 percent for each 1 percent market move.
Reading the Output
| Output | Meaning |
|---|---|
| Slope, or beta | Sensitivity to the explanatory variable |
| Intercept, or alpha | Average return unexplained by the variable |
| R squared | Proportion of variation explained |
| Standard error | Uncertainty around the slope estimate |
| t statistic | Slope divided by its standard error |
R squared is the most commonly misread. An R squared of 0.4 means 40 percent of the variation in the dependent variable is associated with movement in the explanatory variable. It does not mean the model is 40 percent correct, and a low R squared does not mean the relationship is unimportant.
A regression describes an association in the sample you fed it. Every stronger claim comes from assumptions you supplied, not from the arithmetic.
The Assumptions That Get Violated
Regression rests on assumptions, and financial data violates most of them routinely.
Linearity. The relationship is assumed straight. Many financial relationships are not, particularly around extremes, which is exactly where they matter most.
Constant variance. The scatter around the line is assumed uniform. Financial data shows volatility clustering, so the scatter is much wider in some periods.
Independence. Observations are assumed unrelated to each other. Financial time series are autocorrelated.
Stability. The relationship is assumed constant across the sample. Betas and factor loadings shift over time, sometimes abruptly.
Violations mainly cause the standard errors to be understated, which means relationships look more statistically reliable than they are. That understatement is the source of a great deal of confident nonsense.
Correlation and Causation
A regression establishes that two series moved together in the sample. It does not establish direction, and it does not exclude a third variable driving both.
The financial version of this error is common. A factor found to explain returns in historical data is interpreted as a driver of returns. It may be a proxy for something else, an artefact of the sample period, or one of many relationships that appear when enough variables are tested.
The Multiple Testing Problem
Run enough regressions and some will produce impressive statistics purely by chance. At a conventional 5 percent significance threshold, one in twenty tests of pure noise will appear significant.
Modern data availability makes it possible to test thousands of relationships quickly. Published research in finance has documented a very large number of factors claimed to explain returns, and a substantial portion fail to replicate out of sample.
The correction is to demand an economic reason for the relationship before accepting the statistics, and to require the relationship to hold in data that was not used to find it.
Practical Cautions
Check whether the relationship is stable across subperiods, since a coefficient averaged over a period containing a structural break describes neither regime. Plot the data rather than trusting the summary statistics, because very different datasets can produce identical regression output. And treat any regression with a large number of explanatory variables and a small sample as fitted to noise.
The Bottom Line
Regression estimates how one variable has moved with another and provides uncertainty around that estimate. It underpins beta and factor models and is misused whenever the output is read as causation, whenever violated assumptions make significance look stronger than it is, or whenever the relationship was discovered by testing many possibilities. Demand an economic explanation and out of sample confirmation before believing any of it.