Econometrics Quiz: Hypothesis Testing Concepts

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Questions and Answers

What does the Wald test require for hypothesis testing?

Estimation of both restricted and unrestricted models

Estimation of only the unrestricted model (correct)

Estimation only if the likelihood function is maximized

Comparison of multiple restricted models

In a likelihood ratio test, what does a larger decrease in the likelihood function indicate?

The restriction does not hold (correct)

The restricted model is preferred

The unrestricted model is better

The restriction is likely to hold

What is the distribution of the likelihood ratio test statistic under the null hypothesis?

Poisson distribution

Chi-squared distribution (correct)

Binomial distribution

Normal distribution

Which model type is applicable when the dependent variable is binary?

Probit or logit model Signup and view all the answers

What characterizes Limited Dependent Variable (LDV) models?

The dependent variable can only take on specific values Signup and view all the answers

Which estimation methods are used for a count model?

Poisson and negative binomial Signup and view all the answers

If a dependent variable y is limited to the range [a, b], what type of model is this?

Censored model Signup and view all the answers

In qualitative response models, how are the values of the dependent variable characterized?

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

What is the significance of consistency for an estimator in econometrics?

It guarantees that the estimator approaches the true parameter value as the sample size increases. Signup and view all the answers

In the context of multiple regression modeling, what does the notation $y = Xβ + ε$ represent?

The relationship between the dependent variable and independent variables with an error term. Signup and view all the answers

Which statement best describes the law of large numbers in relation to estimators?

It indicates that estimators will approach their expected value as the sample size increases. Signup and view all the answers

When discussing regressors in multiple regression models, what does $K$ represent?

The number of independent variables or covariates. Signup and view all the answers

What does the term 'unbiasedness' of an estimator signify?

The estimator's expected value equals the true parameter value. Signup and view all the answers

What does $etâ_j ightarrow eta_j$ imply regarding estimators?

As the sample size approaches infinity, the estimated parameter converges to the true parameter. Signup and view all the answers

What type of data censoring occurs when some values are not observed above a certain threshold?

Right censoring Signup and view all the answers

What role does the error term $ε_i$ serve in the regression model $y_i = x_iβ + ε_i$?

It accounts for the variability in the dependent variable not explained by the model. Signup and view all the answers

In a censored regression model, which variable is always observed?

The independent variable (xi) Signup and view all the answers

Why is consistency considered a minimal requirement for estimators in econometrics?

Without it, estimators could diverge from the true values as data increases. Signup and view all the answers

What does the parameter β represent in the latent variable framework of censored models?

The impact on the latent variable (yi*) Signup and view all the answers

What is an example of a variable that may be right-censored?

Income reported with a maximum limit Signup and view all the answers

Which of the following statements is true regarding a censored regression model?

The dependent variable can be censored at both ends Signup and view all the answers

What type of models are used for analyzing choice among unordered alternatives?

Qualitative response models Signup and view all the answers

In a multinomial logit model, what does the term 'βj' represent?

Choice-specific coefficients Signup and view all the answers

What do the values of the dependent variable typically represent in qualitative response models?

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

Which alternative estimation method is noted as more complex than multinomial logit?

Multinomial probit Signup and view all the answers

In the explained multinomial logit formula, what does the expression 'exp{xi βj}' represent?

The odds ratio for a specific alternative Signup and view all the answers

What does the log odds ratio imply about the coefficient βj?

It represents a % change in the odds ratio from unit change in x Signup and view all the answers

Which STATA command is used for multinomial logit estimation?

-mlogit- Signup and view all the answers

Ordered response models are best suited for analyzing which type of alternatives?

Ordered alternatives Signup and view all the answers

What is the primary purpose of the Tobit model?

To handle data with a limit at zero Signup and view all the answers

In the Tobit model, what does the notation $yi∗$ represent?

A latent variable Signup and view all the answers

What is the main estimation method used for the Tobit model?

Maximum Likelihood Estimation (MLE) Signup and view all the answers

Which of the following is a characteristic of count models?

Dependent variable is typically a non-negative integer Signup and view all the answers

In the Poisson count model setup, what does $E[yi |xi]$ represent?

The expected number of events given $xi$ Signup and view all the answers

What kind of distribution is assumed for $yi$ in the Poisson count model?

Poisson distribution Signup and view all the answers

What does the term $F(xi β)$ imply in the context of the Poisson count model?

$F$ is a non-negative function estimating event rates Signup and view all the answers

In the context of the Tobit model, how are marginal effects interpreted?

They're specific to individual observations Signup and view all the answers

What does the marginal effect represent in the context of probit/logit models?

The change in the predicted probability of an outcome for a one-unit change in a predictor variable Signup and view all the answers

In the equations provided, what does $y_i^*$ represent?

The unobserved latent variable Signup and view all the answers

Which of the following is NOT a common option for reporting marginal effects?

Marginal effects represented as average fixed effects Signup and view all the answers

What is the purpose of using maximum likelihood estimation (MLE) in the context of probit/logit models?

To estimate the parameter vectors that maximize the likelihood of observing the given data Signup and view all the answers

Which command would you use in STATA to compute marginal effects after running a probit model?

-margin- Signup and view all the answers

What is the significance of the error term ($ε_i$) in the latent variable model?

It captures unobservable factors affecting the dependent variable Signup and view all the answers

What is the role of the indicator function $y_i$ in the model?

To convert the continuous latent variable to a binary outcome Signup and view all the answers

Which marginal effect calculation considers only the average of marginal effects across observations?

Sample mean of the marginal effects Signup and view all the answers

Study Notes

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.

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Description

Test your knowledge on key concepts in econometrics, focusing on hypothesis testing and model types. This quiz covers the Wald test, likelihood ratio tests, and various dependent variable models. Perfect for students looking to reinforce their understanding of econometric principles.