Econometrics Concepts and Assumptions

Course Information

• Course Title: Advanced Econometrics
• Course Code: ADEC-3070
• Instructor: Dr. Manini Ojha
• Semester: Fall, 2024
• JSGP Elective

Lecture Design

• Lectures aim to expose students to various models rather than focusing on specific models in detail.
• Consistent estimation is essential, as no estimator is guaranteed to yield consistent results.
• All estimators rely on assumptions, and the validity of the estimates depends on the validity of these assumptions.
• Assumptions can change due to data collection processes (e.g., measurement error, sample selection).

Recap: Ordinary Least Squares (OLS)

• Classical Linear Regression Model (CLRM) assumes a true relationship: yi = a + βxi + εi, where i = 1,..., N.
• α, β are population parameters.
• â, β are parameter estimates.
• εi is idiosyncratic error (randomness, unobserved factors).
• Estimated residual = €i.
• Exercise: State the assumptions!

Assumptions of CLRM

• (A1) Linearity: The relationship is linear in parameters.
• (A2) Mean Zero Error: E[εi] = 0. This implies E[yixi] = a + βxi.
• (A3) Homoscedasticity: Var(εi) = σ². The variance is constant across observations.
• (A4) No Autocorrelation: Cov (εi, εj) = 0 for i ≠ j (no covariance between errors).
• (A5) Exogeneity: E[εixi] = 0. x is independent of the error.
• (A6) Normality: ε¡ ~ N(0, σ²). Errors are normally distributed. (Not required for unbiasedness or consistency, needed for inference).

OLS Estimation

• Given a random sample, OLS minimizes the sum of squared residuals.
• Solution implies formulas for â and β.

Properties of OLS

• â and Β are unbiased (finite sample property) and consistent (asymptotic property).
• â and β are efficient (smallest variance of any linear, unbiased estimate).

Consistency (Asymptotics)

• Unbiasedness is not always attainable.
• Consistency is a fundamental requirement for any estimator (essential for reliable results).

Multiple Regression Model

• True relationship: yi = x¡β + ε¡, where K = number of independent variables.
• Stacking observations: X y = xβ + ε y is N × 1; x is N × (K + 1); β is (K + 1) × 1; ε is N × 1 xβ is N × 1.
• Assumptions:
  • E[Xiki] = 0 for all k
  • x's are linearly independent(no perfect multicollinearity)

Maximum Likelihood Estimation (MLE)

• Alternative to OLS, especially useful for nonlinear models.
• Equivalent to OLS in the classical linear regression model.
• Likelihood function L(θ) captures the probability of observing the realized data.

Intuition of MLE

• Outcome variable depends on parameters (e.g. θ).
• Estimation aims to maximize the probability of observing the given data.
• MLE identifies parameter values that are most likely to have generated the observed data.

MLE Likelihood Function

• The likelihood function gives the total probability of observing realized data as a function of parameters.
• Joint density is the likelihood function.
• To maximize the likelihood function, a commonly used approach is to maximize the log-likelihood function.

Example CLRM and Probability

• yi = x¡β + ε¡, and ε; α N(0, σ²).
• The probability equation is given by the normal PDF.

Properties of MLE

• Consistent (plim @ML → θ).
• Asymptotically normal
• Asymptotically efficient.

Hypothesis testing - MLE

• Wald tests are equivalent to F-tests in OLS and only involve the unrestricted model.
• Likelihood Ratio test, which involves estimating both the restricted and unrestricted models. Intuition: Imposing restrictions/dropping variables tends to yield a smaller likelihood function, and larger decrease indicates a likely invalid restriction.

Limited Dependent Variable Models (LDV)

• Models which have a dependent variable that is not continuous include:
  • Binary models (e.g., {0, 1})-probit, logit, LPM (labor force participation, loan default).
  • Censored models (e.g., [a, b])-censored regression, tobit (income, wealth).
  • Counts models (e.g., {0, 1, 2,...})- Poisson, negative binomial (e.g., number of children).
  • Qualitative response (QR) models, Ordered QR/Ordinal models (e.g. brand choice, mode of transportation, schooling level, bond ratings).

Linear Probability Model (LPM)

• Estimated by OLS and has issues with predictions that lack appropriate boundaries.

Probit/Logit model

• These are functional forms that use the cumulative distribution functions (CDFs) of probability distributions (e.g. normal function (probit)); logistic function (logit)).
• Interpretations of B are marginal effects.

Latent variable framework

• Probit and logit models can be expressed using a latent variable framework.
• Similar models use indicator functions that produce binary or categorical outputs.

STATA commands for LDV

• probit, logit, dprobit, -margin-, -mfx-

Censored Regression Models

• Applicable to situations where the dependent variable is censored (potentially from above and below)
• Examples: income/wealth top coding, age at first birth
• Latent framework approach
• Estimation via MLE
• Tobit model (a¡ = 0, b; = ∞)

Count Models

• Applicable to situations with a non-negative integer counts.
• Examples: number of children, number of patents held by a firm, number of doctor visits, etc.

Poisson Count Model Setup

• Aims to model expected count of events conditional on a variable. Estimated via MLE, and uses a Poisson distribution assumption, dependent on a mean value.
• Alternative models are negative binomial models.

Zero-inflated Poisson Models (ZIP)

• Models count data with a mass at zero, where the decision for the zero vs a positive outcome may hinge on different factors from those which influence larger positive counted outcomes.

Application: Talley et al. (2005)

• Analyzing determinants of crew injuries in ship accidents (use of count data, mixture of discrete and continuous data over years).

Qualitative Response Model

• Applicable for analyzing choices among a set of alternatives that are unordered (e.g., choice of brand, mode of transport). Multinomial logit and probit;
• The dependent variable is typically coded with integers corresponding to alternative choices.

Multinomial Logit Setup

• Used to model the probability of a particular choice among a set of J+1 alternatives, in the presence of several covariates or other variables which may impact the choice. Uses an exponential functional structure.

Properties of Multinomial Logit Setup

• Note Beta's are choice specific outputs, the output for the Log odds relative to the base choice. The output can be interpreted as a percentage change in odds from a unit change in another parameter.

Alternative Estimation Methods

• Multinomial probit (more complex), Nested logit (sometimes used).

Ordered Response Models

• Applicable to analyses of choices with ordinal alternatives (e.g., labor force status, education levels, bond ratings).
• Data is typically coded as discrete integers; the ordering of values is relevant.

Latent Variable Framework for Ordered Models

The model setup is similar to logit/probit models; using indicator functions with thresholds for each category.

Applications and Estimations

• Specific example application examples are given for each model type
• Data and estimations are carried out using STATA commands ( e.g., -probit-, -logit-, -dprobit-, -margin-, -mfx-, -tobit-, -cnreg-, -poisson-, -nbreg-, -zip-, -zinb-, -ivreg2-, -xtreg-, -fe-, -fd-, and -areg-).

Endogenous Sample Selection

• If outcome and selection variables are mutually dependent, then standard OLS is not suitable.
• Selection based on unobserved factors leads to bias.

Inverse Mills Ratio (IMR)

• Used to correct for endogenous sample selection
• IMR estimation involves a probit model (use of the latent variable approach) and regression.

Endogenous Variables

• An endogenous variable is correlated with the error term in a model; which would cause bias.
• Potential reasons for endogeneity include omitted variable bias, reverse causation and measurement error.
• A relevant regressor is excluded from a model and is correlated with regressor, which leads to omitted variable bias.

Reverse Causation

• The outcome and the treatment variables themselves may reciprocally influence one another.

Measurement Error (ME)

• Data measurement error can lead to bias.
• If ME affects only the dependent variable, then consistent estimates may be obtained.

Classical Error-in-Variables (CEV) Model

• Assumptions for the model include μ and ε are independent from one another; μ and x are also independent from one another.
• If these assumptions do not hold, then OLS estimators will be subject to bias.

Instrumental Variables (IV)

• Used for correcting endogeneity.
• A valid instrument (z) correlates with the endogenous variable (x) but is uncorrelated with the error.

Two-Stage Least Squares (2SLS)

• A two-step process for obtaining IV estimates.
• First stage, finds the predicted values for the endogenous regressor(s).
• Second stage uses the predicted value(s) for the endogenous variable(s) when running the regression- thus using instrumental variables to improve the endogeneity issue.

Panel Data Models

• Models which utilize multiple observations for various groups (e.g., countries, individuals, firms) across different time periods.
• Models include (pooled OLS, fixed effects, random effects, and difference-in-difference (DID))

Time Trends

• The impact of a time trend on the model, which can take into account any variations or tendencies over time.

Structural Breaks

• Models which explicitly factor in structural breaks (e.g., changes in the intercepts or slopes in the presence of sudden events like a war/policy change)

Time Specific Intercepts

• A model which factors in varying time periods to account for various intercepts which may accompany any time variable or treatment effect.

Difference-in-differences (DID)

• Used in policy analysis, especially for situations involving policy implementation in a subset of groups, utilizing variation in observable factors between groups in a pre- and post- policy change to isolate the impact of the policy itself.

Summary

• This course material presents a comprehensive overview of advanced econometrics, covering various methodologies in the presence of several challenging considerations including issues with endogeneity and estimation in cases where data is limited.