ECMT2150 Lecture 3

Lecture Information

• Course: ECMT2150 Intermediate Econometrics

• Semester: 1, 2025

• Coordinator: Felipe Pelaio

• Contact: Room 512, Social Sciences Building

• Email: felipe.queirozpelaio@sydney.edu.au

• Office Hours: Tuesdays 5-6pm

Topics Covered in Week 3

• Statistical Inference

• Inference in Multiple Linear Regressions (MLR)

• t-tests

• Confidence intervals

• p-values

• References: Chapter 4

Recap of Previous Content

• Regression Types

  • Simple and Multiple Linear Regression: Key methods for modeling relationships between variables.

  • Zero Conditional Mean Assumption: E(u|x) = E(u) = 0; essential for unbiased estimation.

• Ordinary Least Squares (OLS): Properties derived under assumptions MLR.1 to MLR.6, including:

  • Unbiasedness: Expected values hold true under MLR assumptions.

  • Variance Formulas: Defined under MLR assumptions.

  • Gauss-Markov Theorem: Establishes OLS as the Best Linear Unbiased Estimator (BLUE) under certain conditions.

• Purpose of Assumptions: Necessary for obtaining causal estimates and conducting inference.

Key Concepts in Probability and Statistics

Random Variables

• Definition: A variable that can take various numerical values determined probabilistically; not directly observed but realized through outcomes.

  • Notation: Uppercase letters for random variables; lowercase for realizations.

Continuous Random Variables

• Definition: Variables that can take any real value within a defined range; their probabilities are represented via the Cumulative Distribution Function (CDF).

Probability Calculations

• Probability Density Function (PDF):

  • P(a < X < b) = F(b) – F(a), where F represents the CDF.

  • P(X > b) = 1 - F(b).

Standardization

• Transforming a random variable to a standard normal form:

  • Z = aX + b, where a = 1/σ and b = -μ/σ.

    

  • Results in E(Z) = 0 and Var(Z) = 1.

Statistical Inference

Purpose and Definitions

• Inference: Conclusions based on evidence and reasoning from data samples.

• Population vs Sample:

  • Population: Entire group of interest (e.g., all incomes in Australia).

  • Sample: Subset of the population (e.g., 10,000 individuals).

  • Importance of sampling due to costs and feasibility.

Sample Mean and Variance

• Population Parameters: Summary measures (e.g., μ, σ²) for a population.

• Sample Statistics: Summary measures for a sample

  • Mean: [ ar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i ]

  • Variance: [ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 ]

Sampling Techniques

• Random Sampling: Sample taken such that each individual has an equal chance of selection.

• Stratified Sampling: Divides population into strata and samples from each.

• Cluster Sampling: Selects groups (clusters) and includes all individuals within the selected ones.

Inference in Multiple Linear Regression (MLR)

• Hypothesis Testing: Evaluating population parameters through statistical methods.

• Confidence Intervals: Establish ranges for population regression coefficients based on sample estimates.

Statistical Inference and OLS Estimators

• Sampling Distributions: OLS estimators are random variables influenced by the underlying error distribution.

• Normality Assumption: Errors are normally distributed; critical for valid inference.

• Testing Hypotheses: Use t-distribution for hypothesis tests of single parameters under MLR assumptions.

  • Distributions are impacted by sample size and number of parameters (n-k-1 degrees of freedom).

Hypothesis Testing Overview

One-Sided and Two-Sided Tests

• One-Sided: You test if a parameter is greater than or less than a specific value (e.g., testing if coefficient > 0).

• Two-Sided: Testing for any significant difference from a specific value.

Critical Values and t-Statistics

• Null Hypothesis: Structure of hypothesis testing, typically stating no effect or no difference.

• t-statistic: Measure of how many estimated standard deviations the coefficient is away from hypothesized value.

Important Statistical Concepts

p-values

• Definition: Minimum significance level at which the null hypothesis can be rejected.

• Small p-value: Evidence against the null hypothesis; implies stronger evidence.

• Large p-value: Favoring the null hypothesis; indicates some support.

Confidence Intervals

• Construction: Bounds that contain the population parameter a certain percentage of the time (commonly 95%).

• Interpretation: A 95% confidence interval indicates that in repeated sampling, the true parameter falls within this interval.

F-Tests for General Hypotheses

• Used for testing linear combinations of parameters or assessing overall regression significance.

• Exclusion Restrictions: Testing if multiple variables jointly have a no-effect on the dependent variable.

• F-statistic: Comparison of variances from restricted and unrestricted models; used to evaluate hypotheses.

Summary of Key Learning Points

• Understanding the role of hypothesis tests in determining statistical significance.

• Effectiveness of confidence intervals in estimating parameters.

• Importance of distinguishing between statistical significance and practical significance.

• Recognize that proper model fitting and testing for linear restrictions are essential in econometric analysis.