Ordered Probit Model Quiz
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Questions and Answers

What do ordered models indicate regarding the order of options?

  • The order should always be randomized
  • The order is irrelevant
  • The order only matters for visual representation
  • The order is informative about outcomes (correct)

In the ordered probit model, the marginal effects are dependent on alternative-specific variables.

False (B)

What is the effect of relabelling options in an ordered model?

It breaks up the natural ordering of the outcomes.

In a survey data example, rating very poor would be represented as _____ in an ordered model.

<p>1</p> Signup and view all the answers

Match the following elements with their correct descriptions:

<p>Yi* = Latent variable representing satisfaction Φ(·) = Cumulative distribution function of the standard normal distribution β = Coefficient indicating the effect of variables on outcomes αk = Cut-off parameter defining thresholds in ordered models</p> Signup and view all the answers

Which of the following is definitive in determining the choice probabilities in an ordered probit model?

<p>Cut-off parameters (A)</p> Signup and view all the answers

The conditional logit model allows β to vary with alternatives.

<p>False (B)</p> Signup and view all the answers

The choice probabilities for an ordered probit model are calculated using the standard normal cumulative distribution function, denoted as _____ .

<p>Φ</p> Signup and view all the answers

Which of the following is NOT a requirement for Instrumental Variables (IV) to work?

<p>The instrument must influence the outcome directly (C)</p> Signup and view all the answers

The Local Average Treatment Effect (LATE) applies to individuals who do not change treatment status regardless of the instrument.

<p>False (B)</p> Signup and view all the answers

What is the sharp RDD?

<p>It is a type of Regression Discontinuity Design where treatment switches cleanly at a cutoff.</p> Signup and view all the answers

The treatment variable for KIPP attendance is represented as ______.

<p>Di</p> Signup and view all the answers

Match the following terms with their definitions:

<p>Instrument Zi = Dummy variable for being offered a seat at KIPP Outcome Yi = Fifth-grade math scores Treatment Di = Dummy variable for attending KIPP Running variable a = Variable determining treatment according to threshold</p> Signup and view all the answers

In which scenario is the concept of Fuzzy RDD typically applied?

<p>When there are variations in treatment assignment around the cutoff (D)</p> Signup and view all the answers

The exclusion restriction assumption can be directly tested with data.

<p>False (B)</p> Signup and view all the answers

What is the significance of the first stage effect in IV analysis?

<p>It indicates that the instrument has a causal effect on the treatment variable.</p> Signup and view all the answers

The average mortality rates at age a is represented as ______ in RDD.

<p>M̄a</p> Signup and view all the answers

Which statement best describes the role of the instrument in the IV setup?

<p>It must only influence the outcome through the treatment variable (C)</p> Signup and view all the answers

In a multinomial logit model, how is the probability of an alternative k calculated?

<p>$\frac{exp(x_i \beta_k)}{1 + \sum_{h=1}^{K} exp(x_i \beta_h)}$ (C)</p> Signup and view all the answers

In the multinomial logit model, the coefficients βk are consistent across all alternatives.

<p>False (B)</p> Signup and view all the answers

What defines the marginal effect of an alternative k regarding covariate j in the multinomial model?

<p>It is given by the formula $\frac{\partial P(Y_i = k|x_i)}{\partial x_{ij}} = P(Y_i = k|x_i) \beta_{jk} - \frac{\sum_{h=1}^{K} exp(x_i \beta_h) \beta_{jh}}{1 + \sum_{h=1}^{K} exp(x_i \beta_h)}$</p> Signup and view all the answers

In a multinomial logit model, probabilities must ____ to one.

<p>sum</p> Signup and view all the answers

Match the model type with its characteristic:

<p>Multinomial Logit = Coefficients vary with alternatives Conditional Logit = Variables vary across alternatives Binary Logit = Used when there are only two outcomes Ordered Logit = Used for ordered categories</p> Signup and view all the answers

Which of the following statements regarding covariates in a multinomial logit model is true?

<p>Covariates are fixed across alternatives. (C)</p> Signup and view all the answers

In terms of probabilities, a higher value of a coefficient βjk always indicates a higher probability of choosing alternative k.

<p>False (B)</p> Signup and view all the answers

What element of the multinomial logit model normalizes the equation with regard to alternative effects?

<p>The inclusion of $\beta_0 = 0$ ensures that probabilities sum to one.</p> Signup and view all the answers

What does the variable Ti indicate in a fuzzy regression discontinuity design (RDD)?

<p>It represents the treatment assignment based on the GRE cutoff. (D)</p> Signup and view all the answers

In a sharp RDD, treatment is not deterministic based on the running variable.

<p>False (B)</p> Signup and view all the answers

What does the outcome equation Yi = 1 + Di + 1 xi + represent?

<p>It represents the earnings of student i based on their treatment status and GRE score.</p> Signup and view all the answers

In a linear probability model, P(Yi = 1|xi) is expressed as __________.

<p>0 + j xij</p> Signup and view all the answers

What is a key characteristic of fuzzy RDD compared to sharp RDD?

<p>Treatment probability can change at the cutoff. (D)</p> Signup and view all the answers

The linear probability model can produce estimated probabilities greater than 1.

<p>True (A)</p> Signup and view all the answers

What is the first stage equation in a fuzzy RDD?

<p>Di = 2 + Ti + 2 xi + i</p> Signup and view all the answers

In a probit model, the cumulative distribution function used is __________.

<p>(xi )</p> Signup and view all the answers

Match the following elements with their corresponding roles:

<p>Yi = Earnings of student i xi = GRE score of student i = Effect of treatment on earnings 1 = Effect of GRE score on earnings</p> Signup and view all the answers

Why is maximum likelihood estimation used in binary discrete choice models?

<p>It can handle probabilities constrained between 0 and 1. (A)</p> Signup and view all the answers

Fuzzy RDD requires scaling of estimates based on the treatment probability at the cutoff.

<p>True (A)</p> Signup and view all the answers

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

<p>It can produce predicted probabilities that are negative or greater than 100%.</p> Signup and view all the answers

In the logit model, the function G(xi ) is expressed as __________.

<p>(xi ) = exp(xi ) / (1 + exp(xi ))</p> Signup and view all the answers

Flashcards

Instrument Variable (IV)

A variable that is used to influence the treatment variable, but has no direct effect on the outcome variable.

First Stage Assumption of IV

The IV must have a causal impact on the treatment variable. This means that the instrument can change the probability of receiving the treatment.

Random Assignment Assumption of IV

The IV must be as good as randomly assigned. This eliminates the possibility of confounding factors affecting both the instrument and the outcome.

Exclusion Restriction Assumption of IV

The IV should only affect the outcome variable through the treatment variable. There should be no other pathways.

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Local Average Treatment Effect (LATE)

The effect of the treatment on the outcome variable, specifically for those who changed their treatment status due to the instrument.

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Running Variable in Regression Discontinuity Design (RDD)

A variable that determines whether someone receives treatment based on crossing a threshold.

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Treatment Variable in RDD

A variable that indicates whether someone is treated, based on the value of the running variable and the threshold.

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Outcome Variable in RDD

The outcome variable in an RDD, measured at different values of the running variable.

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Sharp RDD

A type of RDD where the treatment effect causes a sharp discontinuity in the outcome variable at the threshold.

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Fuzzy RDD

A type of RDD where the treatment effect causes a less distinct discontinuity in the outcome variable at the threshold.

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Multinomial Logit Model

The equation giving the probability of an outcome, given each alternative, and individual characteristics.

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βk coefficients

The coefficients in the multinomial logit model that are specific to each alternative.

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Marginal effect of an alternative

The change in the probability of choosing a specific alternative when a particular characteristic changes.

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Multinomial Logit Model

The type of model where individual characteristics remain constant for each alternative, while coefficients differ between options.

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βk coefficients

The coefficients in the multinomial logit model that DO NOT vary between alternatives.

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Multinomial Logit vs. Conditional Logit

The difference between a standard multinomial logit model and a multinomial logit model where a variable is allowed to vary between alternatives.

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Conditional Logit Model

The type of model where alternative-specific variables are included as independent variables.

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Multinomial Model

The type of model used when the outcome variable has more than two categories.

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Ordered Model Variables

Variables in an ordered model that capture the effect of individual characteristics on the probability of choosing a particular outcome. These variables are the same for all outcomes.

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Cut-off Parameters in an Ordered Model

A series of cut-off points that divide the latent variable into categories. The categories represent distinct ordered outcomes, like very poor, poor, good, and very good.

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Ordered Model

A model where the order of the outcomes is important and influences the probability of choosing each outcome. This is common in surveys where respondents choose among ordered options.

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Ordered Probit Model Probability

The probability of choosing a particular outcome in an ordered model can be calculated using the cumulative distribution function (CDF) of the standard normal distribution. It involves plugging in the cut-off parameters and the values of the independent variables.

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Marginal Effects in an Ordered Model

The effect of a change in an independent variable on the probability of choosing a specific outcome in an ordered model. These effects are calculated using the derivatives of the probability functions.

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β Coefficients in an Ordered Model

The coefficients in an ordered model that capture the effect of independent variables on the latent variable, but they do not differ across the options. They reflect the overall impact on the outcome.

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Ordered Model (General Definition)

A statistical method that analyzes categorical data where the choices are ordered. It is used to determine the influence of various factors on individuals' choices within a given ordered scale.

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Latent Variable in an Ordered Model

The underlying unobserved variable representing a person's true value on a given aspect, such as their true satisfaction with a service. This variable is not directly observed but is estimated based on the observed categorical choices.

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Treatment dummy variable (Ti)

A dummy variable that is set to 1 when the running variable (e.g., GRE score) is above the cutoff point (e.g., the minimum GRE score for Harvard admission), and 0 otherwise.

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Fuzzy treatment variable

A variable that changes the probability of a person receiving treatment, but not necessarily guaranteeing it.

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Complier proportion

The degree to which the treatment variable changes at the cutoff point, reflecting the strength of the relationship between the treatment and the cutoff.

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Regression Discontinuity (RDD)

A statistical method for estimating the causal effect of a treatment on an outcome, when treatment is not assigned randomly but based on a continuous variable (running variable) and a cutoff point.

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Sharp RDD Estimation

A statistical method that relies on a sharp RDD setup, where the treatment assignment is entirely determined by the running variable crossing the cutoff point. This method estimates the treatment effect by comparing the outcomes of individuals just above and just below the cutoff.

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Fuzzy RDD Estimation

A statistical method that relies on a fuzzy RDD setup, where the treatment assignment is not fully determined by the running variable crossing the cutoff point. This method uses an instrumental variable (the cutoff) to estimate the treatment effect.

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First-stage equation

The first-stage equation in a fuzzy RDD setup, which relates the treatment variable to the instrumental variable (the cutoff) and other covariates.

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Second-stage equation

The second stage equation in a fuzzy RDD setup, where the outcome variable is regressed on the estimated treatment effect from the first stage and other covariates.

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Linear Probability Model (LPM)

A model that estimates the probability of a binary outcome (e.g., completing high school) based on a linear combination of covariates.

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Probit and Logit Models

Models that estimate probabilities of binary outcomes using non-linear functions, ensuring that the predicted probabilities stay within the valid range of 0 to 1.

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Marginal effects

The effect of a change in a covariate on the probability of a binary outcome, calculated as the derivative of the predicted probability function with respect to the covariate.

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Maximum Likelihood Estimation (MLE)

A method for estimating the parameters of a statistical model by maximizing the likelihood of the observed data, assuming a specific probability distribution for the data.

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Log-likelihood

The likelihood function is defined by a product of probability terms, one for each data point, and the log-likelihood function is the logarithm of the likelihood function. It is used in maximum likelihood estimation (MLE) to find the best parameters for the model.

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Study Notes

Microeconometrics Weeks 6-10 Revision

  • This revision covers topics from weeks 6-10 of EC338 - Microeconometrics.

Instrumental Variables (IV)

  • IV Setup:

    • Instrument (Zᵢ): A dummy variable equal to 1 if a student was offered a seat at KIPP, and 0 otherwise.
    • Treatment (Dᵢ): A dummy variable equal to 1 if a student attends KIPP, and 0 otherwise.
    • Outcome (Yᵢ): Fifth-grade math scores for student i.
    • Causal chain reaction: Zᵢ (instrument) → Dᵢ (treatment) → Yᵢ (outcome).
    • IV uses first-stage and reduced-form effects to find the effect of interest.

  • IV Assumptions:

    • First Stage: The instrument (Zᵢ) must have a causal effect on the treatment (Dᵢ). This is equivalent to the instrument's effect on the treatment being non-zero; this can be checked using data.
    • Random Assignment: The instrument (Zᵢ) should be as good as randomly assigned. This assumption is untestable, but balance checks can offer supporting evidence.
    • Exclusion Restriction: The instrument (Zᵢ) should only affect the outcome (Yᵢ) through its effect on treatment (Dᵢ). This assumption is also untestable, especially with only one instrument.

  • Local Average Treatment Effect (LATE):

    • Applies to cases with heterogeneous treatment effects.
    • The effect of interest is the LATE.
    • Measures the treatment effect for the compliers.
    • Compliers are individuals who change treatment status because of the instrument.
    • The subsample of compliers depends on the instrument and setting.
    • LATE for binary instrument and treatment: λ = E[Yᵢ|Zᵢ = 1] – E[Yᵢ|Zᵢ = 0] / E[Dᵢ|Zᵢ = 1] – E[Dᵢ|Zᵢ = 0]

Regression Discontinuity Designs (RDD)

  • RDD Setup:

    • Running variable (a): A variable determining treatment according to a threshold (e.g., age).
    • Treatment variable (Dₐ): A dummy variable, equal to 1 if eligible for treatment (above the threshold), 0 otherwise.
    • Outcome variable (Mₐ): E.g., average mortality rates at age a.
    • Key feature: sharp switch on treatment at a cutoff point.

  • Fuzzy RDD:

    • Exploits discontinuities in the probability of treatment at a cutoff point.
    • Treatment probability (or intensity) changes at the cutoff.
      • Useful when treatment isn't fully deterministic.
    • The treatment doesn't switch from 0 to 1, but its probability changes due to some condition.

  • Difference between Sharp and Fuzzy RDD:

    • Sharp RDD: Treatment is a deterministic function; everyone above the threshold gets the treatment; everyone below does not.
    • Fuzzy RDD: Treatment is not deterministic; some individuals below the threshold might receive the treatment, and some above the threshold might not.

  • Fuzzy RDD as IV:

    • Intuitively, treatment becomes more likely to the right of the cutoff.
    • Fuzzy RDD provides a way to estimate that effect; it scales the estimates by the fraction of individuals who get treated due to crossing the cutoff.
    • Outcome equation: Yᵢ = α₁ + xDᵢ + γ₁xᵢ + εᵢ
    • First stage equation: Dᵢ = α₂ + φTᵢ + γ₂xᵢ + ξᵢ
    • 2SLS second stage: Yᵢ= α₁ + λ₂Dᵢ + γ₁xᵢ + εᵢ

Binary Outcomes

  • Linear Probability Model (LPM):

    • In an LPM, P(Yᵢ = 1|xᵢ) = E(Yᵢ|xᵢ) = β₀ + Σβⱼxᵢⱼ
    • Leads to a linear regression model: Yᵢ = β₀ + Σβⱼxᵢⱼ + εᵢ
    • Fitted values estimate the probability, but probabilities can be < 0 or > 1.

  • Probit and Logit Models:

    • Non-linear functions: G(xᵢβ) ∈ [0, 1] ⇒ P(Yᵢ = 1|xᵢ) ∈ [0, 1].
    • Probit: Uses the cumulative distribution function (CDF) of the standard normal distribution.
    • Logit: Uses the CDF of the logistic distribution.
    • Used to get probabilities bounded within [0,1].

  • Marginal Effects:

    • With non-linear models, marginal effects are not equal to the coefficients.
    • Marginal effects depend on xᵢ.

  • Maximum Likelihood Estimation:

    • The likelihood function and its log are used to find the best parameter estimates that match observed data.
    • Econometric statistical packages do this calculation for us in practice.

Multinomial and Ordered Models

  • Multinomial Logit:

    • Used for models with multiple unordered categories.
    • P(Yᵢ = k|xᵢ) = exp(xᵢβₖ) / [1 + Σₖ₋₁exp(xᵢβₙ)].
    • Marginal effects now depend on multiple covariates (compared to just single X variable).

  • Ordered Models:

    • Outcome is categorical but ordered.
    • Latent (unobserved) variable model is assumed (e.g., measure of satisfaction).
    • Choice probabilities are based on a latent variable and cutoff values to determine observed values.
    • Marginal effects of ordered models are calculated similarly to other models, based on specific conditions of the model.

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EC338 Microeconometrics Revision Weeks 6-10 PDF

Description

Test your knowledge on ordered probit models and their characteristics. This quiz covers topics like choice probabilities, marginal effects, and the implications of relabelling options. Dive into the specific details of how these models function within survey data and treatment variables.

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