Regression

Regression: Definition and Purpose

Regression is a statistical method for modeling the relationship between a dependent (response) variable and one or more independent (explanatory) variables. It quantifies how changes in the independent variables are associated with changes in the dependent variable and is used for prediction, explanation, and hypothesis testing.

Key uses:
* Predicting outcomes (e.g., sales, returns).
* Estimating the strength and sign of associations between variables.
* Informing decision-making in economics, finance, social sciences, and many applied fields.

Main Types of Regression

• Simple linear regression: one independent variable. The relationship is modeled by a straight line.
• Multiple linear regression: two or more independent variables. Allows controlling for several factors simultaneously.
• Nonlinear regression: for relationships that are not well captured by a straight line; typically more complex.

How Regression Works (Conceptually)

Regression fits a model that minimizes the discrepancy between observed outcomes and model predictions. The most common fitting method is ordinary least squares (OLS), which finds the line (or surface) that minimizes the sum of squared residuals (the squared differences between observed and predicted Y values).

Analysts may use techniques such as stepwise regression to select relevant predictors or apply specialized models when data violate OLS assumptions.

Basic Formulas

Simple linear regression:
Y = a + bX + u

Multiple linear regression:
Y = a + b1 X1 + b2 X2 + … + bt Xt + u

Where:
* Y = dependent variable (outcome)
* Xi = independent (explanatory) variables
* a = intercept (value of Y when all Xi = 0)
* bi = coefficients (slope parameters showing the effect of Xi on Y, holding others constant)
* u = residual or error term (unexplained variation)

Interpreting a Regression Model

Coefficients indicate the expected change in Y for a one-unit increase in an Xi, holding other variables constant. Example:
Y = 1.0 + 3.2 X1 − 2.0 X2

• Holding X2 constant, a one-unit increase in X1 is associated with a 3.2-unit increase in Y.
• Holding X1 constant, a one-unit increase in X2 is associated with a 2.0-unit decrease in Y.
• The intercept (1.0) is the model’s predicted Y when X1 and X2 are zero.

Statistical significance and confidence intervals help assess whether observed coefficients likely reflect real relationships versus sampling noise.

Assumptions for Ordinary Least Squares (OLS)

For standard inference using OLS, several assumptions are important:
* Linearity: the relationship between Y and the predictors is linear in the parameters.
* Exogeneity: errors have zero mean conditional on predictors (E[u | X] = 0).
* No perfect multicollinearity: explanatory variables are not exact linear functions of one another.
* Homoskedasticity: error variance is constant across observations.
* Errors are independent; normality of errors is assumed for exact small-sample inference (t-tests, confidence intervals).

Violations (e.g., heteroskedasticity, autocorrelation, multicollinearity) require alternative estimation or inference methods.

Applications in Finance and Econometrics

• Asset pricing: regress a stock’s returns on market returns to estimate beta (slope) in the Capital Asset Pricing Model (CAPM).
• Factor models: extend CAPM with additional factors (e.g., Fama–French factors) using multiple regression to better explain cross-sectional returns.
• Policy and economic analysis: estimate how changes in GDP, inflation, unemployment, or income relate to consumption, investment, or prices.

Econometrics is the branch that applies regression and related techniques to economic data, combining empirical analysis with economic theory. Good practice requires linking regression findings to plausible causal mechanisms, not just statistical associations.

Limitations and Cautions

• Correlation ≠ causation: regression uncovers associations but does not by itself establish causal relationships without additional assumptions or design (experiments, instrumental variables, natural experiments).
• Model misspecification, omitted variables, measurement error, and violations of assumptions can bias estimates.
• Overreliance on statistical output without theoretical grounding can lead to misleading conclusions.

Simple Explanation

Regression finds a pattern between things (variables). If you know how one thing has changed before, regression helps you make an educated guess about what will happen to another thing in the future.

Bottom Line

Regression is a versatile and widely used tool for describing relationships, making predictions, and testing hypotheses. Its usefulness depends on appropriate model specification, valid assumptions, and careful interpretation—especially when drawing conclusions about causation.