Instrumental Variables and probit/logit

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Questions and Answers

In instrumental variables regression, what does it mean for a variable to be exogenous?

The variable is endogenous

The variable is determined within the system

The variable is uncorrelated with the error term (correct)

The variable is correlated with the error term

What is the main purpose of instrumental variables (IV) regression?

To estimate coefficients with high precision

To test for instrument validity

To address endogeneity in regression models (correct)

To identify exogenous variables

What is the term used for a variable that is correlated with the error term in a regression model?

Exogenous variable

Endogenous variable (correct)

Inconsistent variable

Omitted variable

What is the focus of instrumental variables regression when considering the relationship between X and Z?

X is endogenous and Z is exogenous Signup and view all the answers

In the context of instrumental variable regression, when are instruments considered weak?

When all the coefficients on the instruments are either zero or nearly zero in the first-stage regression Signup and view all the answers

What does the first-stage F-statistic test in instrumental variable regression?

The hypothesis that the instruments do not enter the first-stage regression Signup and view all the answers

How is the strength of instruments measured in practice in the context of instrumental variable regression?

By computing the first-stage F-statistic Signup and view all the answers

According to the rule-of-thumb, when is a set of instruments considered weak in instrumental variable regression?

If the first stage F-statistic is less than 10 Signup and view all the answers

In TSLS with a single endogenous regressor, how many instruments are involved?

m instruments Signup and view all the answers

What is included in the first stage of the two-stage process in TSLS?

Regressing X1 on all exogenous regressors to compute predicted values Xˆ 1i Signup and view all the answers

What is regressed on in the second stage of the TSLS process?

Y is regressed on Xˆ 1i, W1,…, Wr, and an intercept Signup and view all the answers

To ensure correct standard errors in TSLS, what process is recommended in regression software?

A single-step process Signup and view all the answers

What variables are included in the cigarette demand model as exogenous variables?

ln(Incomei) Signup and view all the answers

What does TSLS estimation using two instruments offer compared to using one instrument for the cigarette demand model?

Smaller standard errors and more information Signup and view all the answers

What are the general instrument validity assumptions?

Instrument exogeneity and relevance Signup and view all the answers

What do the IV regression assumptions involve?

Exogeneity, independence, and finite moments for the variables and instruments Signup and view all the answers

What does checking instrument validity require?

Ensuring relevance and exogeneity, and considering the consequences if these requirements are not met Signup and view all the answers

What is the critical requirement for TSLS and its t-statistic?

The instruments be valid Signup and view all the answers

In IV regression, what statistics are reported for the combined TSLS regression?

R-squared, adjusted R-squared, standard error of regression, F-statistic, and J-statistic Signup and view all the answers

What does the concept of identification in IV regression distinguish between?

Exactly identified, overidentified, and underidentified coefficients based on the number of instrumental variables Signup and view all the answers

What does the general IV regression model extend IV regression to?

Multiple endogenous regressors, included exogenous variables, and instrumental variables Signup and view all the answers

What is cautioned about the interpretation of R-squared in TSLS regression?

It may not have the same interpretation as in classical linear regression Signup and view all the answers

What is discussed in the concept of identification in IV regression?

Distinguishing between exactly identified, overidentified, and underidentified coefficients based on the number of instrumental variables Signup and view all the answers

What is the process of solving the X equation for Z and substituting it into the Y equation to collect terms called?

Two-stage least squares (TSLS) Signup and view all the answers

In the context of the supply and demand for butter, what is highlighted as the need for IV estimation due to simultaneous causality bias in OLS regression?

The interaction of demand and supply Signup and view all the answers

What is explained in the general IV regression model jargon?

The dependent variable, endogenous regressors, included exogenous regressors, control variables, unknown regression coefficients, and instrumental variables Signup and view all the answers

What is the graphical representation of the interaction of demand and supply in a time series model called?

Demand-supply equilibrium chart Signup and view all the answers

What is mentioned in the need for more instruments in IV regression?

The need for more instruments is highlighted in the underidentified case Signup and view all the answers

What is used to isolate shifts in supply to estimate the demand curve in the context of IV estimation?

Two-stage least squares (TSLS) Signup and view all the answers

In the application of TSLS in the supply-demand example, what is highlighted as a valid instrument?

Rainfall in dairy-producing regions Signup and view all the answers

What is emphasized in the process of inference using TSLS?

The normal distribution of the TSLS estimator in large samples Signup and view all the answers

In the context of the demand for cigarettes, what is discussed as a valid instrument for IV estimation?

General sales tax per pack Signup and view all the answers

What results are presented along with correct, heteroskedasticity-robust standard errors in the combined TSLS regression for cigarette demand?

The IV estimator from the reduced form Signup and view all the answers

What is the IV estimator from the reduced form used to show the relationship between?

Changes in X and Y Signup and view all the answers

What is the need for IV estimation highlighted in the context of the supply and demand for butter?

Simultaneous causality bias in OLS regression Signup and view all the answers

What process is used to estimate the demand curve by isolating shifts in supply in the context of IV estimation?

Two-stage least squares (TSLS) Signup and view all the answers

What is the bias in OLS regression of ln(Qibutter) on ln(Pibutter) due to the interaction of demand and supply called?

Simultaneous causality bias Signup and view all the answers

What does instrumental variables (IV) regression aim to address?

All of the above Signup and view all the answers

What does the Two Stage Least Squares (TSLS) method involve?

Two regressions to estimate $\beta_1$ using an instrumental variable Signup and view all the answers

What does the TSLS estimator obtained through two-stage regression using a valid instrumental variable guarantee?

Consistency Signup and view all the answers

Which of the following is a characteristic of multinomial regression models?

The dependent variable can take several discrete values, such as 0, 1, 2, 3, 4 Signup and view all the answers

What is a key similarity between probit and logit regression?

They are both used for binary dependent variables Signup and view all the answers

What is the main focus of multinomial regression models?

Modeling scenarios where the dependent variable has multiple categories Signup and view all the answers

What is the distinguishing feature of a binary dependent variable in the context of probit and logit regression?

It takes the value of either 0 or 1 Signup and view all the answers

What is the main reason for historical preference of logit regression over probit models?

Logit regression is computationally faster and easier Signup and view all the answers

What is the critical consideration for interpreting coefficients in logit and probit models?

The level of probability Signup and view all the answers

What does EViews software allow computation of for the explanatory variables in probit models?

Fitted probability or index Signup and view all the answers

What is the key similarity between predicted probabilities from probit and logit models in the HMDA regressions?

They are based on different probability functions Signup and view all the answers

What is the marginal effect for the P/I ratio in the EViews example used for?

Forecasting Signup and view all the answers

What is the main difference between logit and probit models?

The probability functions used Signup and view all the answers

What is the impact of a unit change in an explanatory variable in logit and probit models dependent on?

The level of probability and values of other explanatory variables Signup and view all the answers

What is the key advantage of logit regression over probit models?

Faster computation Signup and view all the answers

What is the significance of the marginal effect for the P/I ratio in the EViews example?

It demonstrates the impact of a specific variable on the outcome Signup and view all the answers

What does the EViews example with HMDA data demonstrate about the predicted probabilities from logit and probit models?

They are nearly identical Signup and view all the answers

What is the method used for estimating the logit and probit models?

Maximum Likelihood (ML) Signup and view all the answers

What specialized measures are used instead of R-squared in logit and probit models?

Fraction correctly predicted and other specialized measures Signup and view all the answers

What is the main problem with the regressions in the context of potential omitted variable bias?

Potential omitted variable bias Signup and view all the answers

What does the inclusion of covariates do to the effect of race on denial probability?

Reduces the effect Signup and view all the answers

What is the focus of the remaining threats to internal validity?

Sample selection bias Signup and view all the answers

What does the conclusion state about the effect of $\Delta X$ in logit and probit models?

Depends on the initial X Signup and view all the answers

What is the main purpose of the log-likelihood function in logit and probit models?

To maximize the likelihood of observing the data Signup and view all the answers

What is the term used to describe the process of maximizing an additive function of a set of variables in logit and probit models?

Natural logarithm maximization Signup and view all the answers

What does the estimated effect of race on the probability of denial indicate?

Large effect Signup and view all the answers

What is the critical assumption made when using critical values from a normal distribution rather than a t-distribution in logit and probit models?

Normality of the errors Signup and view all the answers

What is the main advantage of the linear probability model (LPM) for modeling binary dependent variables?

It is simple and easy to interpret Signup and view all the answers

What is a limitation of the linear probability model (LPM) for modeling binary dependent variables?

It assumes a constant change in predicted probability for all values of X Signup and view all the answers

What does the coefficient for black applicants in the model indicate?

There is a significant impact on mortgage denial Signup and view all the answers

What is the critical requirement for accurate inference in the linear probability model (LPM)?

Heteroskedasticity-robust standard errors Signup and view all the answers

What is the change in the probability of denial when the P/I ratio changes from 0.3 to 0.4, based on the example from the HMDA data set?

6.1 percentage points Signup and view all the answers

What is the primary reason for the need to use more advanced models like Probit and Logit for binary dependent variables?

To handle non-constant change in predicted probability for all values of X Signup and view all the answers

What does the coefficient β1 in the linear probability model (LPM) represent?

The change in the predicted probability of Y=1 for a unit change in X Signup and view all the answers

What is the interpretation of the predicted value of Y in the linear probability model (LPM)?

It represents the predicted probability that Y=1 Signup and view all the answers

What does the coefficient for the P/I ratio in the LPM model example indicate?

An increase in the probability of denial Signup and view all the answers

What is the primary advantage of using the cumulative normal probability distribution in probit regression?

It ensures that the probability of Y=1 is between 0 and 1 for all values of X Signup and view all the answers

What does the interpretation of a positive coefficient and standard errors in probit regression indicate?

The positive coefficient indicates a higher probability of Y=1 as X increases Signup and view all the answers

What is the calculation of predicted probabilities in probit regression based on?

The values of the independent variables and their coefficients Signup and view all the answers

What is the key similarity between predicted probabilities from probit and logit models in the HMDA regressions?

Both models yield predicted probabilities between 0 and 1 Signup and view all the answers

What is the main focus of multinomial regression models?

Modeling categorical dependent variables with more than two categories Signup and view all the answers

What is the term used to describe the process of maximizing an additive function of a set of variables in logit and probit models?

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

What is the critical requirement for TSLS and its t-statistic?

The requirement for the instrumental variable to be exogenous Signup and view all the answers

Study Notes

Instrumental Variable Estimation: Examples and Applications

The text discusses the use of instrumental variable (IV) estimation in econometrics.
It presents the mathematical equations for IV estimation, including the equations for X and Y in terms of Z.
It explains the process of solving the X equation for Z and substituting it into the Y equation to collect terms.
The IV estimator from the reduced form is presented, showing the relationship between changes in X and Y.
The text provides an example of IV regression in the context of the supply and demand for butter, highlighting the need for IV estimation due to simultaneous causality bias in OLS regression.
It explains the simultaneous causality bias in OLS regression of ln(Qibutter) on ln(Pibutter) due to the interaction of demand and supply.
The text presents a graphical representation of the interaction of demand and supply in a time series model.
It discusses the need to isolate shifts in supply to estimate the demand curve and explains how two-stage least squares (TSLS) can be used for this purpose.
The application of TSLS in the supply-demand example, using rainfall in dairy-producing regions as a valid instrument, is highlighted.
The text explains the process of inference using TSLS, emphasizing the normal distribution of the TSLS estimator in large samples and the importance of correct standard errors.
It provides another example of IV estimation in the context of the demand for cigarettes, discussing the use of general sales tax per pack as a valid instrument.
The results of the combined TSLS regression for cigarette demand, along with correct, heteroskedasticity-robust standard errors, are presented.

Instrumental Variables (IV) Regression: Key Concepts and Estimation Techniques

Endogeneity in regression occurs when X is jointly determined with Y, leading to simultaneous causality bias, omitted-variable bias, and errors-in-variable bias.
The fundamental assumption of regression analysis is that the right-hand side variables are uncorrelated with the disturbance term.
Violation of this assumption leads to biased and inconsistent OLS and weighted LS estimators.
Three possible violations include simultaneous causality bias, omitted variable bias, and errors-in-variables bias.
Instrumental variables (IV) regression can address bias when the assumption of uncorrelated right-hand side variables with the disturbance term is violated.
An example of endogeneity due to simultaneous causality is seen in housing market equilibrium equations, where OLS is not suitable for estimating the price regression.
IV regression breaks X into two parts, isolating the part that is not correlated with the disturbance term using an instrumental variable.
For an instrumental variable to be valid, it must satisfy two conditions: instrument relevance and instrument exogeneity.
The Two Stage Least Squares (TSLS) method involves two regressions to estimate β1 using an instrumental variable.
The TSLS estimator is consistent and is obtained through two-stage regression using a valid instrumental variable.
An alternative explanation for the IV estimator involves a direct algebraic derivation, replacing population covariances with sample covariances.
The "reduced form" relates Y to Z and X to Z, enabling the estimation of the IV estimator from the reduced form.

Estimation and Inference in Probit and Logit: Application to Racial Discrimination in Mortgage Lending

Binary dependent variables present different modeling challenges compared to continuous variables
An example from the Boston Fed HMDA Dataset illustrates the use of independent variables such as income, wealth, employment status, and race of the applicant to predict mortgage acceptance or denial
The linear probability model (LPM) is a starting point for modeling binary dependent variables, with the predicted value of Y interpreted as the predicted probability that Y=1
In the LPM, the coefficient β1 represents the change in the predicted probability of Y=1 for a unit change in X
An example from the HMDA data set uses the LPM to model mortgage denial in relation to the ratio of debt payments to income, with a coefficient of -0.080 for P/I ratio
Including race as a regressor in the model shows that the coefficient for black applicants is -0.091, indicating a significant impact on mortgage denial
The LPM has advantages such as simplicity and ease of interpretation, but it assumes a constant change in predicted probability for all values of X, which may not always hold true
LPM predicted probabilities can reach 1, which presents a limitation of the model
The example illustrates that a change in P/I ratio from 0.3 to 0.4 increases the probability of denial by 6.1 percentage points
The significance of the coefficient on black applicants in the model suggests racial bias in mortgage lending decisions
Inference in the LPM requires heteroskedasticity-robust standard errors for accurate estimation
The LPM's limitations underscore the need for more advanced models like Probit and Logit for binary dependent variables, especially in scenarios like racial discrimination in mortgage lending

Probit Regression Model in Econometrics

Linear probability model's limitations: models the probability of Y=1 as linear
Desire for the probability of Y=1 to be increasing in X for β1>0 and 0 ≤ Pr(Y = 1|X) ≤ 1 for all X
Probit model satisfies the desired conditions for the probability of Y=1
Probit regression models the probability that Y=1 using the cumulative standard normal distribution function
Example of probit regression model's computation with specific values
Advantages of using the cumulative normal probability distribution in probit regression
EViews example using HMDA data for binary probit regression
Interpretation of positive coefficient and standard errors in probit regression
Calculation of predicted probabilities in probit regression
Probit regression with multiple regressors and its formula
EViews example with multiple regressors for binary probit regression
Calculation of predicted probit probabilities with multiple regressors using EViews command and interpretation of coefficients

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Description

Test your understanding of instrumental variable (IV) estimation with this quiz. Explore key concepts, estimation techniques, and real-world applications in econometrics. Practice identifying valid instruments and applying Two Stage Least Squares (TSLS) method to address endogeneity and bias in regression analysis.