3.6 Regression, Simulation, and Big Data

Regression, Simulation, and Big Data

This closing module turns descriptive statistics into modeling tools. The 2026 Level I curriculum treats simple linear regression in depth and adds dedicated modules on estimation, simulation, and big data techniques.

Simple linear regression

In simple linear regression, one dependent variable Y is explained by one independent variable X through Y = b0 + b1 X + error, where b0 is the intercept and b1 is the slope. Ordinary least squares (OLS) chooses the line that minimizes the sum of squared residuals, where a residual is actual Y minus predicted Y.

The slope is the expected change in Y for a one-unit change in X. If a regression of a fund's excess return on the market's excess return has a slope of 1.20, the fund is estimated to move 1.20 percentage points per 1 percentage point market move before residual effects, and in that excess-return specification the slope is interpreted as beta. The intercept is the predicted Y when X equals zero; in an excess-return performance regression it is interpreted as alpha. A positive estimated alpha is not enough on its own; the analyst must confirm statistical significance and proper model specification.

Reading regression output

Regression output reports coefficient estimates, standard errors, t-statistics, and p-values. A coefficient t-statistic is estimate / standard error; if the p-value is below the chosen significance level, the coefficient differs statistically from its hypothesized value (usually zero). R-squared is the fraction of variation in Y explained by X and, in simple regression only, equals the squared correlation between X and Y. The standard error of estimate (SEE) measures residual dispersion around the line; smaller SEE means tighter predictions.

The ANOVA table decomposes total variation: total sum of squares = regression sum of squares + sum of squared errors, and the F-statistic (regression mean square over error mean square) tests overall model significance, equaling the square of the slope t-statistic in simple regression.

Assumptions and cautions

OLS relies on a linear relationship, homoskedastic (constant-variance) and independent residuals, and approximately normally distributed residuals. Statistical significance does not prove economic importance, a high R-squared does not prove causation, and predictions outside the observed range of X are unreliable.

Simulation

Scenario analysis evaluates a handful of defined cases (base, bull, bear). Historical simulation replays the observed data set in random order. Monte Carlo simulation draws random inputs from assumed distributions and recomputes the output thousands of times, producing a full distribution rather than a point estimate. It suits path-dependent payoffs, retirement spending, credit losses, and portfolio risk, but its outputs are only as good as the assumed volatilities, correlations, and tail behavior, so always ask which distributions and dependencies were specified.

Big data and machine learning

Big data is characterized by high volume, velocity, variety, and veracity; investment examples include transactions, web traffic, satellite imagery, text, audio, and supply-chain feeds. Alternative data can create edge but raises questions of legality, privacy, data lineage, and representativeness. Machine learning is supervised (learning from labeled outcomes such as default vs. no default), unsupervised (finding structure such as clusters), or based on deep learning and reinforcement methods.

The recurring dangers are overfitting (a model that memorizes noise and fails out of sample), look-ahead bias, and survivorship bias. For Level I, keep interpretation practical: regression estimates relationships, simulation explores ranges of outcomes, and big data techniques search for patterns at scale, but none removes the need for economic logic, clean data, and independent validation.

Worked regression example

Suppose a regression of a fund's monthly excess return on the market's excess return yields an intercept of 0.2% (standard error 0.3%) and a slope of 1.15 (standard error 0.25%), with an R-squared of 0.64. The slope t-statistic is 1.15/0.25 = 4.6, comfortably significant, so the fund's beta of 1.15 is statistically distinguishable from zero (and you could separately test whether it differs from 1.0). The intercept t-statistic is 0.2/0.3 = 0.67, well below 2, so the estimated alpha of 0.2% is not statistically different from zero, meaning the apparent outperformance is indistinguishable from noise.

The R-squared of 0.64 means market movements explain 64% of the variation in the fund's excess return, and because this is a simple regression the correlation between the two return series is sqrt(0.64) = 0.80.

Exam tactics for the modeling module

Three distinctions are tested repeatedly. First, significance is not economic importance: a tiny coefficient can be statistically significant in a huge sample. Second, correlation is not causation: a high R-squared can arise from a lurking common factor. Third, R-squared equals the squared correlation only in simple (single-variable) regression. For the big-data and machine-learning material, be ready to label a technique as supervised or unsupervised, to name overfitting as the reason a model fails out of sample, and to flag survivorship and look-ahead bias as data-integrity threats.

The graders reward candidates who can read an ANOVA table, compute a t-statistic from an estimate and its standard error, and explain what a Monte Carlo output distribution does and does not tell an analyst.