lecture6

Multiple Regression Analysis: Inference

Introduction to Econometric Methods

• Presenter: Abhimanyu Gupta

• Date: November 11, 2024

Outline of Lecture

• Sampling Distribution of the OLS Estimator

  • Normality assumption on the error term

  • Result on the distribution of the OLS estimators

• Hypothesis Testing: t-test

  • Introduction to hypothesis testing

  • One-sided alternatives vs Two-sided alternatives

  • p-values and Confidence Intervals

• Reading: Wooldridge (2018), Introductory Econometrics, Chapter 4

Sampling Distributions of the OLS Estimators

Objective of the Lecture

• To understand how to test hypotheses about the parameters in the population regression model.

Review Topics from Previous Lectures

1. Statistical inference (Lecture 2)

2. Mean and variance of the OLS estimators (Lecture 5)

Key Focus

• Determining the sampling distribution of the OLS estimators

• Conditional distribution of y given x's determines the distribution of (βˆ0, βˆ1, …, βˆk)

Assumption of Normality

Assumption MLR.6

• The population error u must:

  • Be independent of explanatory variables (x1, x2, ..., xk)

  • Be normally distributed with zero mean and variance σ2: u ∼ Normal(0, σ2)

Justification

• This assumption is practical but needs to be examined in specific examples.

Implications of MLR.6 on Variance and Expectation

1. Expectation: E(u|x1, ..., xk) = E(u) = 0, thus MLR.4 is satisfied.

2. Variance: Var(u|x1, ..., xk) = Var(u) = σ2, thus MLR.5 is satisfied.

Classical Linear Model (CLM) Assumptions

Overview of MLR.1–MLR.6

• The classical linear model consists of Gauss-Markov assumptions plus the distribution assumption of the error term.

• OLS estimators are minimum variance unbiased estimators under these assumptions, leading to stronger efficiency properties than just the Gauss-Markov conditions.

Conditional Mean and Distribution

Summary of Population Assumptions of the CLM

• Distribution: y|x1, …, xk ∼ N(β0 + β1x1 + β2x2 + … + βk xk, σ2)

  • Linear combination of x's as conditional mean

  • Constant variance σ2

Normality in Practical Applications

• Economic theory may offer limited guidance for normality assumptions.

• Examples where normality might fail:

  • Hourly wages with a minimum wage floor: y ≥ ymin

  • Number of children born (y ≥ 0)

• Empirical investigation of residuals can assess how well normality holds.

Handling Non-normality

• Log transformations can help improve normality assumptions:

  • Example: log(wages) typically exhibits normality, unlike raw wages.

Normal Sampling Distributions of OLS Estimators

Theorem 4.1 (Normal Sampling Distributions)

• If CLM assumptions (MLR.1-MLR.6) hold, then:

  • βˆj ∼ Normal(βj, Var(βˆj))

  • Thus: βˆj - βj / sd(βˆj) ∼ Normal(0, 1)

Understanding the Conditional Mean Function

• The conditional mean function E(y|x) shows how the distribution of y surrounds its mean with a normally distributed error term.

Hypothesis Testing: t-test Overview

Testing a Single Parameter

• Population model: y = β0 + β1x1 + … + βkxk + u

• Use OLS estimators to test parameters.

Theorem 4.2 (t Distribution for Standardized Estimators)

• Under CLM assumptions:

  • βˆj - βj / se(βˆj) ∼ tn−k−1

• Requires estimation of σ2, leading to a t-distribution due to additional estimation steps.

One-Sided Test Procedure

• Null hypothesis (H0): βj = 0

• t-statistic: tβˆj ≡ βˆj / se(βˆj) measures distance from zero.

• Significant levels guide rejection of H0 based on generated t-statistic.

bExample: Wage Effects of Education

Steps to Perform One-Sided Test

1. Estimate βˆj and se(βˆj)

2. Compute t-statistic

3. Determine degrees of freedom

4. Use t-table for critical value

5. Apply rejection rule.

Two-Sided Alternative Testing

Framework

• Null hypothesis: H0: βj = 0

• Rejection rule: H0 is rejected if |tβˆj| > c

• Interpretation when H0 is rejected or not.

Computation of p-Values for t Tests

• P(|T| > |t|) provides the p-value to assess significance vs null hypothesis.

• p-value indicates probability of observing as extreme statistic under the null hypothesis.

Caveats on Statistical Testing

1. Failure to reject H0 doesn’t mean acceptance.

2. Distinction between statistical significance (size of tβˆj) vs economic significance (magnitude of βˆj).

Confidence Intervals for Population Parameters

• 95% CI for βj using t-distribution.

• Common mistakes include unnecessary use of sample size adjustments in CI formulation.

Conclusion for Example with Wage Effects

1. CI and critical value application to assess statistical significance.

2. p-values and CI yield insights to reject or fail to reject hypotheses.

Next Lecture

• Continue with Multiple Regression Analysis: Inference.