Econometrics: OLS Estimator Variance and Assumptions Study Guide

Probability and Statistics Primer

An introduction to fundamental statistical concepts including random variables, measures of central tendency, variability, and linear association.

• Random variables
• Quantitative vs. categorical variables
• Discrete vs. continuous random variables
• Population vs. sample
• Expected value and population mean
• Sample mean
• Median and mode
• Population variance and standard deviation
• Sample variance and standard deviation
• Normal distribution empirical rule
• Covariance and correlation

Hypothesis Testing and Confidence Intervals

An introduction to the components, steps, and logic of hypothesis testing and the use of confidence intervals to estimate population parameters.

• Null Hypothesis (H0)
• Alternative Hypothesis (H1)
• Significance Level (α)
• Test Statistic
• Critical Region
• Two-Sided and One-Sided Tests
• Confidence Interval calculation
• Standard Error
• P-Value interpretation
• Relationship between Confidence Intervals and Hypothesis Testing

Introduction to Random Variables and Descriptive Statistics

This material covers the definitions of random variables, measures of central tendency, measures of dispersion, and basic concepts of probability distributions and correlation.

• random variables
• quantitative variables
• discrete random variables
• expected value
• median
• mode
• range
• sample variance
• population variance
• standard deviation
• normal distribution
• covariance
• correlation

Introduction to Hypothesis Testing

A guide to the process of rejecting or failing to reject a null hypothesis using t-tests, significance levels, and critical values.

• null hypothesis
• alternative hypothesis
• two-tailed hypothesis test
• one-tailed hypothesis test
• significance level
• rejection region
• critical values
• t statistic
• type I error
• type II error
• standard error
• degrees of freedom
• p-value

Introduction to Econometrics and Economic Models

An introduction to econometrics as an area of economics that uses statistical methods to estimate economic relationships and test theories using non-experimental data.

• estimating economic relationships
• testing economic theories
• non-experimental data
• constructing economic models
• econometric error term
• linear functional form
• cross-sectional data
• random sampling from underlying population

Types of Data and Econometric Issues

An introduction to cross-sectional, time series, pooled cross-sectional, and panel data along with common econometric challenges.

• cross-sectional data
• time series data
• pooled cross-sectional data
• panel or longitudinal data
• dummy variables
• balanced and unbalanced panel data
• sample selection
• simultaneously determined variables
• self selectivity issue
• dependency of time series data

Simple Linear Regression Econometrics

An introduction to the simple linear regression model, Ordinary Least Squares estimation, and measuring goodness of fit.

• Simple Linear Regression Model
• Dependent and Independent Variables
• Error Term components
• Ordinary Least Squares (OLS) minimization
• Sum of squared residuals
• Total Sum of Squares (SST)
• Explained Sum of Squares (SSE)
• Residual Sum of Squares (SSR)
• R-Squared (R2) interpretation
• Linearity in parameters

• Introduction to Simple Linear Regression in Econometrics

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Simple Linear Regression Model

An introduction to the simple linear regression model, including functional forms, parameter estimation, and statistical significance testing.

• simple linear regression model
• linear functional form
• error term
• population regression line
• conditional expectation
• log-linear models
• log-log models
• parameter estimates
• statistical significance
• p-value
• confidence intervals
• null hypothesis
• OLS estimators

OLS Estimator Variance and Homoskedasticity

This material discusses the variance of OLS estimators, the implications of homoskedasticity and heteroskedasticity, and the impact of omitted variables on regression models.

• variance of OLS estimator
• homoskedasticity and heteroskedasticity
• unbiased estimator
• omitted variable bias
• error term correlation
• degrees of freedom
• standard errors calculation
• conditional variance

• Understanding OLS Estimator Variance and Homoskedasticity

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Interval Estimation and Hypothesis Testing

This topic explores how to determine if an estimator $\hat{\beta}_{j}$ is close to the true population parameter $\beta_{j}$ using sampling distributions and confidence intervals.

• Assumption SLR.6 (Normality)
• Sampling Distribution of $\hat{\beta}_{j}$
• Student's t-Distribution
• Standard Error (se)
• Degrees of Freedom
• Confidence Intervals
• Hypothesis Testing

Prediction and Marginal Effects in Econometrics

This topic covers how to predict the dependent variable using OLS parameter estimates and how to calculate marginal effects for various functional forms.

• OLS parameter estimates
• Prediction of dependent variable
• Marginal effects
• Quadratic functional form
• Partial derivative
• Functional forms

OLS Estimation and Goodness of Fit

This content examines OLS model implications, functional forms, and the coefficient of determination as a measure of goodness of fit.

• OLS estimation
• Non-linear explanatory variables
• Log-log model
• Elasticity
• Total Sum of Squares (SST)
• Sum of Squared Due to Regression (SSR)
• Sum of Squared Due to Error (SSE)
• Coefficient of determination (R2)
• Goodness of fit

Prediction and Marginal Effects in Econometrics

Using estimated specific models to predict y for a given x and calculating marginal effects.

• Prediction
• Estimated Model
• Marginal Effects
• Slope
• Intercept
• Non-Linearity

The Multiple Regression Model

This topic covers the multiple regression model, OLS estimation, OLS assumptions, omitted variable bias, and goodness of fit.

• Multiple regression model
• OLS estimation
• OLS assumptions
• Omitted variable bias
• Goodness of fit
• Multicollinearity
• Zero conditional mean
• Sum of squared residuals
• Polynomials
• Marginal effect
• Elasticity
• Semi-elasticity
• Intercept
• Ceteris paribus

Gauss-Markov Assumptions and Multiple Regression

This content explores the Gauss-Markov assumptions for Multiple Linear Regression, including the consequences of omitting relevant variables or including irrelevant variables.

• MLR.1 Linear in Parameters
• MLR.2 Random Sampling
• MLR.3 No Perfect Collinearity
• MLR.4 Zero Conditional Mean
• MLR.5 Homoskedasticity
• Omitted Variable Bias
• Unbiasedness
• Efficiency
• Standard Error
• Endogeneity
• Selection Bias

Logarithms in Simple Linear Regression

This material explains how to use natural logs to transform variables for interpreting coefficients as percentages or elasticities in different regression models.

• Level-Level model interpretation
• Log-Level model semi-elasticity
• Level-Log model semi-elasticity
• Log-Log constant elasticity model
• Transforming variables using natural logs
• Interpreting coefficients as percentage changes
• Compressing large scales to reduce outliers

Hypothesis Testing and T-Statistics

This topic covers hypothesis testing procedures, including type I and type II errors, t-statistics, and p-values.

• Type I error
• Type II error
• Significance level (α)
• t-statistic
• Standard error
• Degrees of freedom
• Two-tailed test
• One-tailed test
• Critical value (tc)
• P-value
• Confidence interval

Confidence Intervals and Hypothesis Testing

This topic covers the application of t-distributions to calculate confidence intervals and perform hypothesis testing for regression coefficients.

• Confidence Interval
• t-Distribution
• Degrees of Freedom (df)
• Critical Value (tc)
• Hypothesis Testing
• Null Hypothesis (H0)
• Alternative Hypothesis (H1)
• Significance Level (alpha)
• Test Statistic

Marginal Effects in Non-Linear Models

This topic explores calculating marginal effects in linear and non-linear models using calculus rules such as the constant rule, power rule, and sum rule.

• Marginal Effects
• Linear Models
• Polynomial Models
• Derivatives
• Constant Rule
• Power Rule
• Sum Rule
• Quadratic Models
• Cubic Models

Regression Model Specification and RESET Testing

• Initial Estimation
• Determinants of Log Sales
• Omitted variables
• Null Hypothesis
• Misspecification
• Extended Model
• Adjusted R-squared
• Parameters

Gauss-Markov Assumptions for Simple Linear Regression

• Gauss-Markov Assumptions
• Linear in Parameters
• Random Sampling
• Sample Variation
• Zero Conditional Mean
• Homoskedasticity
• Heteroskedasticity
• Best Linear Unbiased Estimators
• Omitted Variable Bias
• Residual Plots

Simple Linear Regression Interpretations and Violations

This material covers interpreting log-log and log-lin regression models, evaluating goodness of fit, and identifying issues like omitted variable bias and heteroskedasticity.

• Log-log model elasticity
• Log-lin model interpretation
• R-squared and correlation versus causation
• Omitted variable bias
• Heteroskedasticity and non-constant variance
• Residual plot visualization

Hypothesis Testing for Regression Coefficients

This topic covers setting up hypotheses, determining rejection regions, and calculating the t-statistic to test the significance of regression coefficients.

• Null Hypothesis (H0)
• Alternative Hypothesis (H1)
• Two-Sided Test
• One-Sided Test
• Rejection Regions
• t-Statistic
• Standard Error
• Statistical Significance

Intuition and Application of t-Statistics

The t-statistic uses the standard error as a ruler to normalize the absolute difference between an estimate and a hypothesized value.

• t-statistic
• Standard Error
• Two-sided test
• Right-tailed test
• Null Hypothesis
• Alternative Hypothesis
• Degrees of Freedom
• Critical Value

Hypothesis Testing and P-values in Regression

This topic covers computing p-values, t-statistics, and decision rules for hypothesis testing in simple linear regression.

• p-value
• Null Hypothesis (H0)
• Alternative Hypothesis (H1)
• t-statistic
• Significance level (alpha)
• Critical value (tc)
• Rejection Region
• Two-sided test
• One-sided test
• Confidence Interval
• Standard Error (SE)

Statistical Hypothesis Testing and t-statistics

Methods for conducting hypothesis tests using t-statistics, p-values, and confidence intervals to evaluate single parameters.

• t-statistic calculation
• One-sided vs Two-sided tests
• P-value adjustment
• Confidence Interval logic
• Statistical vs Economic significance
• Null hypothesis testing

Log-Linear Prediction and Model Issues

Methods for log-linear and log-log prediction, calculating marginal effects via derivatives, and addressing impossible predictions or log issues.

• Log-Linear Prediction
• Log-Log Prediction
• Exponentiation to Level
• Marginal Effect
• Derivative of the Right-Hand Side
• Impossible Predictions
• Log Issue Approaches

Prediction and Model Comparison Issues

This topic covers prediction techniques, non-linearity, and the constraints of comparing R2 values.

• Prediction
• Non-Linearity
• Marginal Effect
• Log-Log Prediction
• Log-Linear Models
• Derivatives

Variance of OLS Estimators

The variance of the OLS estimator depends on error variance, total sample variation in x_j, and the linear relationship between x_j and other independent variables.

• Variance of OLS Estimators
• Error Variance
• Total Sample Variation
• Linear Relationship
• Multicollinearity
• Standard Error
• Precision
• Noise

Multicollinearity Detection and Remedies

High correlation between independent variables causes large standard errors and insignificant individual coefficients, which can be detected using the Variance Inflation Factor.

• Multicollinearity
• Standard errors
• t-statistics
• F-statistic
• Variance Inflation Factor (VIF)
• Regression analysis
• Joint significance
• Omitted Variable Bias
• Adjusted R-squared
• RESET Test