Regression is a powerful statistical method used across various fields such as finance and investing, aimed at determining the strength and nature of the relationships between a dependent variable (often denoted by Y) and multiple independent variables.
Essential Types of Regression
Linear Regression
Also known as simple regression or ordinary least squares (OLS), linear regression establishes a linear relationship between two variables based on a line of best fit. This is depicted using a straight line, where the slope describes how a change in one variable impacts the change in another. The y-intercept represents the value of the dependent variable when the independent variable equals zero.
Nonlinear Regression
While the nonlinear models are more complex, they can more accurately fit data that doesn’t follow a straight line. These models are more adaptable but challenging to apply and interpret.
Key Takeaways
- Interpret Nature and Strength: Regression connects a dependent variable to one or more independent variables.
- Predict Trends: The technique is instrumental in forecasting economic trends, asset valuation, and more by determining whether and how variables change together.
- Graphical Representation: It essentially fits a best-fit line to see data dispersion around this line.
- Informed Decision Making: From asset valuation to modeling economic predictions, regression provides deep insights.
- Critical Assumptions: Be aware that several assumptions about data must hold for valid results.
Understanding Regression
Regression quantifies correlations between variables, assessing whether they are statistically significant. Both simple and multiple linear regression models exist for handling varying complexities in data analysis.
Role in Finance
Professionals in finance leverage regression for predicting sales based on various factors like weather or GDP growth, and to model prices using the Capital Asset Pricing Model (CAPM).
Regression and Econometrics
Econometrics employs regression to interpret financial and economic data, helping to explain relationships such as income vs. consumption. Multiple linear regression tools help in explaining phenomena using multiple explanatory variables.
Criticisms and Requirements
It’s essential that the regression outcomes are linked to sound economic theories, avoiding over-reliance on data without appropriate theoretical backing.
Calculating Regression
Linear regression often employs a least-squares approach to ascertain the line of best fit. Minimizing the sum of squares–the squared differences between actual and predicted values–creates the regression model. The general forms are as follows:
Simple Linear Regression:
Y = a + bX + u
Multiple Linear Regression:
Y = a + b1X1 + b2X2 + ... + btXt + u
- Y: Dependent Variable
- X: Explanatory Variable(s)
- a: y-intercept
- b: Slope or Beta Coefficient
- u: Residual or Error Term
Finance in Focus: Practical Example
Regression is crucial for assessing how factors like commodity prices or industry trends influence asset prices. For instance, CAPM relies on regressing stock returns against a market index to determine a stock’s risk in relation to the market, represented as beta.
Expanded Models
Adding variables like market capitalization or valuation ratios improves predictive power for more accurate stock returns, exemplified in Fama-French models.
The Origin of the Term “Regression”
Sir Francis Galton coined the term in the 19th century, describing biological data’s propensity to regress toward a mean level, a statistical observation still significant in today’s analyses.
Purpose and Interpretation
Regression illuminates the associations between variables, quantifying both their strength and statistical significance, essential for making informed predictions based on empirical data.
Example Model Interpretation:
For a regression model written as Y = 1.0 + 3.2X1 - 2.0X2 + 0.21, every unit change in X1 results in a 3.2× change in Y, controlling for X2, which decreases Y by 2× per unit increase.
Core Assumptions
For proper analysis, these assumptions must be satisfied:
- Linear relationship
- Homoskedasticity
- Independence of explanatory variables
- Normal distribution of variables
Final Thoughts
Regression is an indispensable statistical technique that elucidates relationships between variables. While powerful, it’s imperative to couple its use with sound theory for valid and actionable insights.
Related Terms: linear regression, nonlinear regression, CAPM, beta, econometrics.
References
- Margo Bergman. “Quantitative Analysis for Business: 12. Simple Linear Regression and Correlation”. University of Washington Pressbooks, 2022.
- Margo Bergman. “Quantitative Analysis for Business: 13. Multiple Linear Regression”. University of Washington Pressbooks, 2022.
- Eugene F. Fama and Kenneth R. French, via Wiley Online Library. “The Cross-Section of Expected Stock Returns”. The Journal of Finance, Vol. 47, No. 2 (June 1992), Pages 427–465.
- Jeffrey M. Stanton, via Taylor & Francis Online. “Galton, Pearson, and the Peas: A Brief History of Linear Regression for Statistics Instructors”. Journal of Statistics Education, vol. 9, no. 3, 2001, .
- CFA Institute. “Basics of Multiple Regression and Underlying Assumptions”.
