Review of Basic Econometric Concepts: OLS and Regression

In behavioral economics and experimental design, understanding fundamental econometric tools is crucial for analyzing data and drawing valid conclusions. This module revisits the core concepts of Ordinary Least Squares (OLS) regression, a cornerstone for estimating relationships between variables.

What is Regression Analysis?

Regression analysis is a statistical method used to estimate the relationship between a dependent variable (the outcome you're interested in) and one or more independent variables (the factors you believe influence the outcome). It helps us understand how changes in independent variables are associated with changes in the dependent variable.

Regression quantifies the relationship between variables.

Regression analysis aims to find a line (or curve) that best fits the observed data points, allowing us to predict the value of the dependent variable based on the values of the independent variables.

The core idea is to model the expected value of the dependent variable, Y, as a function of the independent variable(s), X. For a simple linear regression with one independent variable, the model is typically expressed as: $Y = \beta_0 + \beta_1 X + \epsilon$ . Here, $\beta_0$ is the intercept (the expected value of Y when X is 0), $\beta_1$ is the slope coefficient (representing the change in Y for a one-unit change in X), and $\epsilon$ is the error term, capturing all other unobserved factors affecting Y.

Ordinary Least Squares (OLS)

Ordinary Least Squares (OLS) is the most common method for estimating the coefficients ( $\beta_0$ , $\beta_1$ ) in a regression model. It works by minimizing the sum of the squared differences between the observed values of the dependent variable and the values predicted by the regression line.

OLS minimizes the sum of squared errors.

OLS finds the 'best-fitting' line by ensuring that the sum of the vertical distances (errors) between each data point and the line, when squared, is as small as possible. This prevents positive and negative errors from canceling each other out.

The objective function that OLS seeks to minimize is: $SSE = \sum_{i=1}^{n} (Y_i - \hat{Y}_i)^2$ , where $Y_i$ is the observed value of the dependent variable for observation $i$ , and $\hat{Y}_i$ is the predicted value of the dependent variable for observation $i$ (calculated as $\hat{Y}_i = \hat{\beta}_0 + \hat{\beta}_1 X_i$ ). The resulting estimators, $\hat{\beta}_0$ and $\hat{\beta}_1$ , are known as the OLS estimators.

Imagine a scatter plot of data points. The regression line is the line that passes through these points in a way that minimizes the total vertical distance from each point to the line, after squaring those distances. This process is visualized by showing the data points and a line that appears to be the 'closest' fit, with vertical lines representing the errors (residuals) from each point to the line. The squaring of these errors is a mathematical step to ensure that deviations above and below the line are treated equally and to penalize larger errors more heavily.

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Assumptions of OLS

For OLS estimators to be the best linear unbiased estimators (BLUE), several key assumptions must hold. Violations of these assumptions can lead to biased or inefficient estimates and unreliable statistical inference.

Assumption	Description	Implication of Violation
Linearity	The relationship between the dependent and independent variables is linear.	Model misspecification, biased coefficients.
No Perfect Multicollinearity	Independent variables are not perfectly linearly related to each other.	Inability to estimate coefficients, inflated standard errors.
Exogeneity (Zero Conditional Mean)	The expected value of the error term is zero, conditional on the independent variables.	Endogeneity, biased and inconsistent coefficients.
Homoskedasticity	The variance of the error term is constant across all levels of the independent variables.	Inefficient estimates, incorrect standard errors (though coefficients remain unbiased).
No Autocorrelation	Error terms are uncorrelated with each other.	Inefficient estimates, incorrect standard errors.

In behavioral research, understanding these assumptions is vital. For instance, endogeneity (a violation of exogeneity) is common when unobserved factors influence both the treatment and the outcome, requiring advanced techniques like instrumental variables or experimental designs to address.

Interpreting Regression Output

Interpreting the output of a regression analysis is key to understanding the findings. Key components include coefficients, standard errors, p-values, and R-squared.

What does the coefficient of an independent variable in a regression represent?

It represents the estimated change in the dependent variable for a one-unit increase in the independent variable, holding all other independent variables constant.

What does a low p-value (typically < 0.05) for a coefficient suggest?

It suggests that the independent variable is statistically significant, meaning it has a relationship with the dependent variable that is unlikely to be due to random chance.

What does R-squared measure?

R-squared measures the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

Learning Resources

Introduction to Regression Analysis | Coursera(video)

A foundational video explaining the basic concepts of regression analysis and its purpose in understanding relationships between variables.

Ordinary Least Squares (OLS) Regression | Khan Academy(video)

Khan Academy provides a clear, step-by-step explanation of linear regression and the OLS method, suitable for beginners.

Econometrics: The Basics | The Economist(blog)

An accessible overview of econometrics, explaining its role in analyzing economic data and testing theories.

OLS Regression Assumptions | Stata Blog(blog)

This blog post details the key assumptions of OLS regression and the consequences of violating them, with practical implications.

Interpreting Regression Coefficients | UCLA Statistical Consulting(documentation)

A comprehensive guide on how to correctly interpret the coefficients, p-values, and R-squared values from regression output.

Introduction to Econometrics with R | Jeffrey Wooldridge(paper)

While a book, this is a seminal text in econometrics. Chapter 3 specifically covers OLS, its properties, and assumptions, often considered a standard reference.

What is Econometrics? | Wikipedia(wikipedia)

Provides a broad overview of econometrics, its history, methods, and applications, including regression analysis.

Regression Analysis Explained | Towards Data Science(blog)

A practical explanation of regression analysis, covering the intuition behind OLS and how to interpret results in a data science context.

OLS Regression: Assumptions and Violations | DataCamp Community(tutorial)

A tutorial that breaks down the assumptions of OLS regression and discusses common violations and how to detect them.

The Classical Linear Regression Model | MIT OpenCourseware(documentation)

Lecture notes from MIT covering the foundational aspects of the Classical Linear Regression Model, including OLS and its assumptions.

Review of Basic Econometric Concepts: OLS, Regression