Econometrics Quiz: Hypothesis Testing Concepts

Advanced Econometrics (ADEC-3070)

• Course offered by Dr. Manini Ojha
• JSGP Elective
• Fall, 2024

Lectures

• Not designed for proficiency in specific models, but rather exposure to many models.
• Key takeaway: No estimator is guaranteed to yield consistent estimates.
• Every estimator relies on a set of assumptions.
• Validity of estimates depends on the validity of underlying assumptions.
• Crucial to understand these assumptions for conducting and assessing applied work.
• Assumptions may no longer hold during the data collection process (measurement error, sample selection, etc.).

Recap: OLS

• Classical Linear Regression Model (CLRM): yᵢ = α + βxᵢ + εᵢ, i = 1, ..., N

• α, β = population parameters

• â, β = parameter estimates

• εᵢ = idiosyncratic error term (reflects randomness, unobserved factors)

• Exercise: State the OLS assumptions.

Assumptions

• (A1) Linearity: yᵢ = α + βxᵢ + εᵢ, i = 1, ..., N
• α, β = population parameters
• εᵢ = disturbance or error term (randomness, unobserved factors)
• (A2) Expected value of error: E[εᵢ] = 0
• (A3) Variance of error: E[εᵢ²] = σ²
• (A4) Covariance of errors: E[εᵢεⱼ] = 0 for i ≠ j
• (A5) Error and regressors: E[εᵢxᵢ] = 0
• (A6) Error distribution: εᵢ ~ N(0, σ²)

OLS Estimation

• Given a random sample {yᵢ, xᵢ}ᵢ₋₁ⁿ, OLS minimizes the sum of squared residuals:
  • â, β = argmin ∑ᵢ₋₁ⁿ (yᵢ – â – βxᵢ)²

OLS Solution

• Implies BOLS and AOLS
• BOLS = Cov(yᵢxᵢ) / Var(xᵢ)
• AOLS = ỹ - BOLS*x

Properties of OLS

• â, ẞ are unbiased (finite sample property) and consistent (asymptotic property).
• â, ẞ are efficient (smallest variance of any linear, unbiased estimate).
• Var(β) = σ² / Σᵢ₋₁ⁿ (xᵢ – x)²

Consistency

• As sample size grows, the distribution of β becomes tightly distributed around β.
• For a sufficient collection of data, the estimator is arbitrarily close to β₁.

Maximum Likelihood Estimation (MLE)

• Alternative estimation technique to OLS.
• Equivalent to OLS in classical linear regression models.
• Useful for nonlinear models.
• Maximizes the probability of drawing the observed data given the parameters
• To get ML estimators: need an expression for the likelihood function.
• likelihood function: total probability of observing the realized data (L(θ)).
• joint probability distribution of the data = likelihood function L(θ).
  • L(θ) = ∏ᵢ₋₁ⁿ [f(yᵢ|θ)]

Example CLRM

• yᵢ = xᵢβ + εᵢ, εᵢ ~ N(0, σ²)
• Pr(yᵢ|xᵢ, θ) = 1 / (2πσ²) exp[ – (yᵢ – xᵢβ)²/2σ²]

Properties of MLE

• Consistent: plim(θ̂ₙ) = θ
• Asymptotically normal
• Asymptotically efficient

Hypothesis Testing (MLE)

• Wald tests: Equivalent to F-test in OLS.
• Likelihood ratio test : Estimate both restricted and unrestricted model. Compute LR and Lυr
• Intuition: Restrictions/dropping variables generally lead to smaller L(θ).

Likelihood Ratio Test (contd.)

• Test statistic : LR = 2[Lυr – Lr]
• Where q: # of restrictions; LR ~ x²q .
• Maximization typically by numerical methods since analytical derivatives are messy.

Limited Dependent Variable Models (LDV)

• Models where dependent variable(s) is/are not continuous
• Common types: • Binary models (y ∈ {0, 1}) • Censored regression models (y ∈ [a, b]) • Count models (y ∈ {0, 1, 2,...}) • Qualitative response models (y ∈ {0, 1, 2,...})

Binary Models

• Applicable to problems where the dependent variable is binary.
• Examples: Labor force participation, default on a loan, belonging to an FTA.

Linear Probability Model (LPM)

• Setup: yᵢ = xᵢβ + εᵢ , where ɛ; ~ Ν(0, σ²) .
• Estimated by OLS.
• Problems: Heteroskedasticity
• Predictions are not bounded by 0 and 1 (xᵢβ ∉ [0, 1]
• Doesn't correspond to probabilities.

Probit and Logit Models

• Used when dependent variable is binary.
• Models Pr(yᵢ = 1|xᵢ) using a "proper" functional form Pr(yᵢ = 1|xᵢ) = F(xᵢβ)
• F(•) is a cumulative distribution function (CDF).
• Possible solutions include CDFs.

Interpretation of B

• Marginal Effect = dPr(yᵢ = 1) / dxᵢ = dF(xᵢβ) /dxᵢ
  • dF(xβ) / dx₁ = { φ(xβ) / dx₁ for probit ; λ(xβ)[1 – Λ(xβ)] / dx₁ for logit}.

Latent Variable Framework

• Probit logit model framed in unobserved variable framework.
• y*= xβ + ε

STATA Commands

• -probit-
• -logit-
• -dprobit-
• -margin-
• -mfx-
• -tobit
• -cnreg-

Censored Regression Models

• Applicable to problems where dependent variable is censored.
• Right censoring/top-coding
• Examples: Income, wealth, age.

Latent Variable Framework - Censored Models

• Model setup: y*= xβ + ε, where yᵢ = {bᵢ if y* ≥ bᵢ, y* if aᵢ ≤ y* ≤ bᵢ , and aᵢ if y* ≤ aᵢ.}

Tobit Model

• Special case of censored regression models.
• aᵢ=0 , bᵢ → ∞ ; y = {y* if y* > 0 and 0 if y* ≤ 0}.

Count Models

• Applicable to non-negative integer counts of events.
• Examples include number of children, patents, doctor visits.

Poisson Count Model

• Want to model the expected number of events conditional on x, E[yi|xi]= F(xiβ) .
• F(•) ≥ 0.
• Estimation via MLE using Poisson distribution (depends only on the mean).

Additional Estimation Methods for Count models

• Negative binomial
• Zero-inflated Poisson

Qualitative Response Models

• Analyzing choice by agents from many disordered alternatives.
• Examples include brand choice, mode of transportation, types of mortgage, schools

Multinomial Logit

• Model to estimate probabilities of J+1 alternatives given x, Pr(y₁ = j|xᵢ, 0) = [exp(x¡βj) / ∑k exp(x¡βk)].
• Interpreting ẞ: log odds ratio (relative to base choice)

Alternative Estimation Methods

• Multinomial probit (more complex).
• Nested logit

Ordered Response Models

• Analyzing choice by agents from many ordered alternatives.
• Examples include labor force status, schooling, bond ratings.
• Outcomes are discrete integers: 0, 1, 2... Using logit/probit framework, y* = xβ + ε.

Applications and Examples

• Studies by various authors including examples for testing different econometric models using STATA.

Panel Data

• Studying multiple draws on the same basic unit of observation (group).
• Examples: individuals in a household over time.

Pooled OLS

• Estimating relationship with multiple time periods from cross-sectional data.
• Estimation is similar to OLS with a larger sample size (NT).

Extensions of panel data

• Time trend, Quadratic time trend & Structural breaks allowing intercepts to change over time
• Time-specific intercepts, allowing intercept to vary across time periods.

Chow test

• Used to test if there is a break, or change, in the relationship between variables over time.
• H0 : α1 = α2, β₁ = β2; H1 : not all equal

Difference-in-difference estimation

• Frequently used in policy analysis
• Used to analyze the impact of a policy on the outcome given the same variables at different time periods
• Examples: Minimum wage hike, legalized abortion on crime or the death penalty on crime.

Fixed effects

• Used to account for omitted variable bias, variables that do not change across groups.
• Similar setup but allow for individual-specific intercepts

Stata Commands

• -xtreg- (for panel data).
• -fe (Fixed effects).
• -fd (first differences).
• -areg (for average effects).
• -ivreg2- (Instrumental variables regression).
• -mlogit- (for multinomial logit).
• -mprobit- (for multinomial probit).
• -reg-,-probit-, -logit-

Endogenous Variables

• Variables whose change in the model outcome are correlated to changes in other variables. This can lead to biased estimators if the change in the variable is interpreted to only depend one the variable being investigated.

Instrument Variables

• Method used to address endogeneous variables bias in a model where the variable(s) being investigated, are correlated to error terms in the model, or other dependent variables.
• This can be corrected by using an instrumental variable which is correlated with the dependent variable in the model, but not affected by error terms or other dependent variables.

Omitted Variable Bias, Reverse Causation

• Explain how these issues can lead to biased results, what conditions are necessary for this to happen, and alternative methods to analyze the models

Measurement Error

• Explain how data that are measured imprecisely can lead to biased results, how to deal with this problem in panel data, and how to interpret results.