Unit 5

Unit 5: Inference

• Inference in econometrics involves quantifying uncertainty in estimates.
• Estimates are noisy, not the true value of the parameters.
• The estimated parameter (β) depends on the random sample.
• Different random samples yield different estimates (β).
• Need to assess the precision of estimates.

Motivation

• Estimation is prone to noise.
• True parameters (β) are unknown.
• Estimates are noisy approximations.
• Estimates vary with different samples.
• How precise do estimates need to be?

Example

• True model: wage = β educ + u
• β = 0.1
• Noisy sampling process: educ ~ N(0, 1) but u ~ N(0, 10)
• Results in noisy estimate.

Example: Sample 1

• n = 100
• β^ = -0.367, SE = 0.047
• Data is noisy, but shows negative relationship.

Example: Sample 2

• n = 100
• β^ = 0.570, SE = 0.050
• Data is noisy, but shows positive relationship.

Example: Four Possible Samples

• Various samples and estimations of β are shown.
• Data looks noisy, indicates a negative relationship in sample 1, and shows a positive relationship in sample 2
• Results differ across samples.

Hypothesis Testing

• Method to quantify uncertainty:
• Establish a threshold for acceptable uncertainty.
• Example: Test if β = 0, given sample data.
• Inferring from data; crucial for inference.

The Classical Linear Model (CLM)

• Additional assumption (beyond Gauss-Markov) for hypothesis testing:
• Population error (u) is independent of explanatory variables (X).
• Normally distributed (u ~ Normal(0,σ²)).
• This assumption allows for calculations of t-statistics and F-statistics.
• The CLM assumption is y|X ~ Ν(β , σ²).
• Useful in large sample sizes.

Normal Sampling Distributions

• Conditional on sample values of the independent variables, β^ is normally distributed.
• β^ ~ N(β, Var(β))
• Var(β) = σ²(X'X)⁻¹.
• Normalized estimator is distributed N(0,1) (β/sd(β)).

Inference: The t-Test

• Under CLM assumptions, t-statistic follows a t-student distribution.
• Standardized estimate is essential in hypothesis tests.
• tβ = (β - β₀)/se(β) ~ t(N-K-1)
• Estimate of population parameter (variance) is crucial: o² estimated by ô² .

Testing Parameters: A Typical Example

• Test null hypotheses (Ho : β = 0) of the significance of a variable (x).
• Reject Ho if β is statistically significant, if x has an effect on y.

One-sided Alternatives

• Hypothesis tests need an alternative hypothesis and a significance level.
• One-sided: β > 0 or β < 0.
• Two-sided: β ≠ 0
• Critical value (c) depends on significance level (α).

One-sided vs. Two-sided

• t-distribution is symmetrical, testing H₁: β < 0 is straightforward.
• Critical value is just the negative value of c
• Two-sided test: critical values are at α/2.
• A rejection rule is used to accept or reject the null hypothesis.

Testing other Hypotheses

• Test if β; differs from other values (e.g., Ho: β₁ = a₁)
• Formula for t-statistic

Initial Example Continued: Large Sample Size

• Larger sample size (n = 10000) has less noisy data.
• Estimates are close to true value (β = 0.1).

Confidence Intervals

• Alternative to hypothesis testing—confidence intervals based on critical values.
• (1 - α)% confidence interval = β ± tα/2·se(β)
• Value c is the (1- α/2) percentile of the t distribution.

Computing p-values for t-tests

• Finding the smallest significance level at which to reject a null hypothesis.
• The p-value is the probability that we would observe a t-statistic, if the null hypothesis was true.
• P-value is calculated by using a relevant t-distribution, given degrees of freedom.
• Computer packages compute p-values; divide by 2 for one-sided tests.

OLS Estimates by Stata

• Output with coefficient estimates, standard errors, t-statistics, p-values, and confidence intervals for OLS method.
• Interpret regression results, particularly significant variables.

Testing Linear Combinations

• Test if a linear combination of coefficients equals a constant (e.g., ß₁ - ß₂ = 0).
• Formula for t-statistic uses variance and covariances of estimates as a measure of variability.
• In Stata, use the "test" command for hypotheses tests.

Multiple Linear Restrictions

• Jointly test multiple hypotheses (e.g., all coefficients).
• Evaluate a group of parameters using F-test, not just evaluating each individually.
• Need to estimate both restricted and unrestricted models.

The F-Statistic

• Measure of overall increase in residual sum of squares (SSR) when moving from unrestricted to restricted model
• q = number of restrictions or numerator degrees of freedom
• N-K-1= df_unrestricted

The R² Form of the F-statistic

• Alternative formula for F statistic using R².
• Computations in relation of SSR given R²

Overall Significance

• Test hypotheses about all model coefficients (Ho :β1=β2=…=βk = 0)
• F-statistic related to R² measure of fit

General Linear Restrictions

• Basic F-statistic formula for any set of linear restrictions.
• p-values are found in the appropriate percentile of relevant F-distribution

Example

• Illustrative applications and use-cases of hypothesis testing with regression models.
• Example voting model (with campaign expenditues).
• Specific restriction on coefficient estimates tested.

Interpretation of t-test and F-test

• Example code in R
• Example exercise

Chow Test for Structural Breaks

• Method to test for structural shifts in coefficients over time or in specific data segments.
• Determine if effect of regressors differs significantly between (e.g.) different time periods or groups in data.
• Requires estimating restricted and unrestricted models.
• Evaluating F-statistics over time to establish if structural breaks are present.

Exercise: Marginal Effects and F-test

• Exercise using real world data
• Calculate and interpret marginal effects, particularly for wage determination
• F-test for linear restrictions
• Example applications in wage regression.

Appendix A: Restricted OLS Estimation

• Summary of t-tests and F-tests within the framework of restricted regressions.
• Basic formulas and notation for restricted estimators are provided

Moments of the OLS Estimator

• Calculation and interpretation of mean of restricted OLS estimators
• Exploiting normality; estimation of variance of the restricted estimator.

Moreover

• Formula for variation of the restricted estimator.
• Properties of SSR. SSRr > SSRur

The F-Test

• Test hypotheses using framework of 'restricted regression', formulas for univariate test statistic and modified statistic are provided.

Special Case: The t-test

• Special F-test case that helps with testing hypotheses about just one coefficient.
• Useful for analyzing and assessing significance of individual variables by using a univariate t-test.