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What is Linear Regression?
Linear regression is a statistical modeling technique used in quantitative finance for various purposes, including asset pricing, risk management, portfolio management, and factor modeling to analyze and predict the relationship between two variables. It provides a way to understand the linear association between a dependent variable and one or more independent variables and allows for the estimation of a linear equation that best fits the data.
The dependent variable (also called the response variable) is the variable being predicted or explained, while the independent variables (also known as explanatory variables or predictors) are used to explain or predict the dependent variable. In finance, the dependent variable can be, for example, a stock price, an asset return, or a financial indicator, and the independent variables can be factors like interest rates, economic indicators, or company-specific variables. Linear regression assumes a linear relationship between the dependent and independent variables. It assumes that changes in the dependent variable are directly proportional to changes in the independent variables, with a constant slope. The goal is to estimate the slope and intercept of the linear equation that best fits the observed data.
Linear regression estimates the parameters of the linear equation using a method called ordinary least squares (OLS). OLS minimizes the sum of the squared differences between the observed values and the predicted values based on the linear equation. The estimated parameters represent the coefficients that determine the impact of the independent variables on the dependent variable. The coefficients in linear regression provide insights into the relationship between the independent variables and the dependent variable. They indicate the direction and magnitude of the impact of the independent variables on the dependent variable. Positive coefficients suggest a positive relationship, while negative coefficients suggest a negative relationship.
Linear regression can be used for prediction by plugging in values for the independent variables into the estimated linear equation to estimate the corresponding value of the dependent variable. Additionally, various statistical measures, such as R-squared, adjusted R-squared, and significance tests, are used to evaluate the model's goodness of fit and the statistical significance of the estimated coefficients. Linear regression allows for the identification of relationships between financial variables, the prediction of future values, and the analysis of the impact of independent variables on the dependent variable.
Linear regression is covered in more detail in module 4 of the CQF program.