Week_2_PPT

Week 2: Ordinary Least Squares (OLS) in Applied Econometrics

Overview

• Course: ECO440/ECO640

• Institution: Niagara University

Lecture Outline

• Goal of OLS Regression: Estimating the empirical regression equation.

• Mechanics of OLS Estimator: Parameter estimates in both univariate and multivariate regression.

• OLS Regression: Intuition and interpretation of results.

• Importance of Fit: Understanding the decomposition of variance and R².

From Theory to Empirics

• Purpose of Regression Analysis: Transition from a theoretical equation to an estimated empirical regression equation.

• Key Questions:

  • What empirical regression line should be used?

  • How to choose among several alternative versions of the regression equation?

Selecting an Empirical Regression Equation

• Aim for a regression line that resembles the theoretical model, e.g., a linear function connecting schooling to wages.

• The model should provide the best fit for the data, specifically assessing the parameters.

• Example: How does an additional year of schooling affect wage levels?

Possible Empirical Regression Candidates

• Analyze and assess the fit of candidate regression equations.

• Evaluate prediction errors (residuals) using actual vs predicted wage values.

• Consider slope implications regarding education's impact on wages.

Preferred Empirical Regression Equation

• The best-fitting regression line is derived using OLS methods.

• Discuss reasons why this regression equation is favored over others.

Ordinary Least Squares Regression

• OLS as the primary method for obtaining regression estimates.

• Goal: Minimize the squared errors (residuals).

• Rationale: Minimizing the square of errors rather than their sum improves accuracy.

Distinguishing Estimators vs. Estimates

• Estimator: A technique applied to sample data to estimate population regression coefficients.

• Estimate: The computed value of a regression coefficient.

• Key components: Parameters (unknown betas, β) vs variables (Y, X).

Why Use OLS?

• Advantages of OLS:

  • Ease of use.

  • Conceptual appeal of minimizing squared errors.

  • Properties:

    • Residuals sum to zero.

    • Under certain conditions, OLS is the BLUE (Best Linear Unbiased Estimator).

Ordinary Least Squares Mechanics

• OLS minimizes the squared error:

  • Mathematically, the method involves computing values that minimize the expression ∑(Y - Ŷ)².

OLS Estimate Solutions for Single Variable Regression

• Derivation of OLS estimates is based on theoretical equations.

• Fundamental equations for computation: (2.1), (2.4), (2.5).

Intuition of OLS Slope Coefficient Estimate

• The numerator represents covariance between X and Y, indicating the relationship strength.

• The denominator indicates the variance of X, showing data dispersion.

• Interpretation: The slope (β₁) indicates how a unit change in X affects Y, weighted by X's variance.

Mechanically Calculating OLS Estimates

• Steps:

  1. Calculate means for X and Y.

  2. Compute residuals and sums of products.

  3. Derive estimates using manual calculations or software.

Examples of Mechanically Calculating OLS Estimates

• Data and intermediate calculations are crucial for estimating coefficients.

• Illustration using height/weight data to compute regression coefficients.

Estimating Multivariate Regression Models with OLS

• Multivariate models needed as many Y variables cannot be explained by a single X.

• General form of multivariate regression with K independent variables:

  • Y = β₀ + β₁X₁ + β₂X₂ + ... + βₖXₖ + ε.

Interpreting OLS Estimates from Multivariate Regression Models

• Coefficients in multivariate models indicate the change in Y with a one-unit increase in X, holding other variables constant.

Interpreting OLS Examples

• Example 1: Demand for beef in the U.S.

  • Consumption relation to income, controlling for price.

• Example 2: Financial aid effects based on parents’ contribution and student GPA.

  • Insights drawn from the coefficients and their implications on aid calculations.

The Fit of a Regression

• Importance of understanding fit, focusing on how well the model predicts Y based on X.

  • Total sum of squares (TSS) measures variation in Y.

Decomposing Total Sum of Squares

• TSS comprises two components:

  • Explained sum of squares (ESS): Variation explained by regression.

  • Residual sum of squares (RSS): Variation not explained.

Measuring the Overall Fit of an Estimated OLS Regression

• ESS must represent a large portion of TSS for a successful model fit.

• Introduces the coefficient of determination, R².

R² and Overall Fit of Estimated OLS Regression

• R² is the ratio of ESS to TSS, indicating how well the regression explains the data.

• Values vary between 0 and 1, with higher indicating better fit.

Adding Variables and R² Limitations

• Adding variables can artificially inflate R², potentially without meaningful impact on the model.

• New variables require estimation and affect degrees of freedom, which should be considered.

Using R² to Describe Overall Fit

• Adjusted R² accounts for degrees of freedom, allowing for more meaningful comparisons when adding variables.

Appropriate and Inappropriate Uses of R²

• R² is useful for comparing equations with the same dependent variable but not for different ones.

• Warning against optimizing R² at the expense of meaningful theory behind model choice.

Example of Misusing Fit and R²

• Example with mozzarella cheese consumption illustrates the danger of adding nonsensical adjustments to the model, leading to misleading conclusions about R².