Econometrics Concepts and Assumptions

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Questions and Answers

What does the validity of estimates in econometrics depend on?

The number of observations in the dataset

The underlying assumptions related to the estimator (correct)

The complexity of the model used

The presence of measurement errors in the data

Which assumption states that the expected value of the error term is zero?

A4: Independence of Errors

A3: Constant Variance

A1: Linearity

A2: Mean Zero (correct)

What does Assumption A3 concerning errors imply?

The sum of errors must equal zero

The variance of errors must be constant across observations (correct)

Errors must be normally distributed

Errors must be correlated with the independent variable

Which of the following is true about Assumption A5?

It indicates that the error is unrelated to the independent variable Signup and view all the answers

Assumption A6 is necessary for which aspect of econometric modeling?

Inference and hypothesis testing Signup and view all the answers

What is the consequence if assumptions do not hold in the data-collection process?

Estimates may not reflect the true population parameters Signup and view all the answers

In the Classical Linear Regression Model, what do α and β represent?

Populations parameters Signup and view all the answers

Which of the following reflects randomness and unobserved factors in the Classical Linear Regression Model?

Idiosyncratic error term, εi Signup and view all the answers

What is the primary estimation required for conducting a Wald test?

Estimation of only the unrestricted model Signup and view all the answers

What does a larger decrease in the likelihood function when imposing restrictions indicate?

The restrictions are likely invalid. Signup and view all the answers

What is the form of the test statistic for the likelihood ratio test?

$LR = 2[LUR - LR]$ Signup and view all the answers

In limited dependent variable models, what estimation is used when the dependent variable is binary?

Probit, logit, LPM Signup and view all the answers

When dealing with a count model, what type of regression can be used for estimation?

Poisson and negative binomial regression Signup and view all the answers

What characteristic defines a qualitative response model?

Values correspond to choices with no natural ordering. Signup and view all the answers

What is typically necessary for maximizing the likelihood function in MLE?

Numerical methods due to messy analytical derivatives Signup and view all the answers

Which of the following correctly describes the conditions for a censored model?

Dependent variable ranges within known bounds, a and b. Signup and view all the answers

What is the significance of taking the natural logarithm of the likelihood in maximum likelihood estimation?

It simplifies the calculation by turning products into sums. Signup and view all the answers

Which statement accurately describes the role of θ̂ML in maximum likelihood estimation?

It represents the value of θ that maximizes the likelihood function. Signup and view all the answers

Which equation is crucial in identifying the maximum likelihood estimator for θ?

∂ln[L(θ)]/∂θ = 0 Signup and view all the answers

What form does the probability Pr(yi |xi , θ) take in the context of the example provided?

Normal Probability Density Function Signup and view all the answers

What does the likelihood function lead to when it is maximized in this context?

Minimization of the sum of squared residuals. Signup and view all the answers

Which of the following properties is NOT associated with maximum likelihood estimators?

Unbiased Signup and view all the answers

In the context of the provided equations, what does the εi term represent?

The error term associated with the model. Signup and view all the answers

What role does the parameter σ² play in the likelihood equation?

It represents the variance of the error term. Signup and view all the answers

What is a characteristic of an ordered QR model?

The 'distance' between values can vary. Signup and view all the answers

In which of the following situations is a binary model applicable?

Determining if someone defaults on a loan. Signup and view all the answers

What is a limitation of the Linear Probability Model (LPM)?

Predictions can exceed the range of 0 to 1. Signup and view all the answers

Which functional form models the probability of a binary outcome using a proper bounded approach?

Pr(yi = 1|xi) = F(xi β) Signup and view all the answers

Which of the following correctly describes a feature of the Probit model?

It uses a cumulative distribution function to map outcomes. Signup and view all the answers

What does the Logit model estimate using the logistic function?

It provides a probability estimate of outcomes ranging from 0 to 1. Signup and view all the answers

What is one significant limitation of using an Ordinary Least Squares (OLS) method with LPM?

It may yield unrealistic predictions. Signup and view all the answers

How does a cumulative distribution function (CDF) function in binary models?

It maps real numbers to the unit interval. Signup and view all the answers

What is typically the format of the dependent variable in qualitative response models?

Positive integers corresponding to specific choices Signup and view all the answers

In the multinomial logit model, what is the significance of the parameters βj?

They measure the % change in odds ratio from a unit change in x. Signup and view all the answers

What does the functional form of F in the multinomial logit model describe?

The relationship between choices and their probabilities. Signup and view all the answers

Which of the following methods is considered more complex than the multinomial logit model?

Multinomial probit Signup and view all the answers

In the context of multinomial logit, what can the log odds ratio be used to compare?

The probability of a choice relative to a base choice. Signup and view all the answers

What type of response models analyze choices among ordered alternatives?

Ordered response models Signup and view all the answers

Which of the following is an example of a situation where qualitative response models would be applicable?

Selecting a mode of transportation Signup and view all the answers

What does the marginal effect represent in a probit or logit model?

The change in probability of an event occurring for a unit change in a predictor Signup and view all the answers

Which of the following is NOT a common way to report marginal effects?

Marginal effects for every individual in the sample Signup and view all the answers

In the latent variable framework, what does $y_i^*$ represent?

The unobserved latent variable Signup and view all the answers

What is the correct interpretation of $y_i$ in the model $y_i = 1$ if $y_i^* > 0$ and $y_i = 0$ if $y_i^* ≤ 0$?

It indicates the occurrence of an event Signup and view all the answers

Which of the following is true about marginal effects in regression analysis?

They provide insights into the conditional probability Signup and view all the answers

In the context of probit/logit models, what does MLE stand for?

Maximum Likelihood Estimation Signup and view all the answers

What is the purpose of the error term $ε_i$ in the model $y_i^* = x_i β + ε_i$?

To represent unobserved factors affecting the outcome Signup and view all the answers

Study Notes

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.

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Description

Test your understanding of key concepts and assumptions in econometrics. This quiz covers the Classical Linear Regression Model, estimation techniques, and important assumptions that affect the validity of econometric models. Perfect for students who want to strengthen their grasp of econometric principles.