40 Questions
What is the potential outcome framework used to estimate in the context of schooling and wage?
The causal effect of schooling on wage
What is the observed outcome in the context of schooling and wage, according to the potential outcome framework?
Yi = Y0i + (Y1i − Y0i )Di
What is the average treatment effect on the treated (ATET) in the context of schooling and wage?
E[Y1i |Di = 1] − E[Y0i |Di = 1]
What is the assumption required to give regression a causal interpretation when the data are non-experimental?
A further assumption, such as Conditional Independence Assumption
What is the problem with using regression to estimate the causal effect of schooling on wage, in the absence of additional assumptions?
Omitted variable bias, selection bias, or bad controls
What is the consequence of omitting a relevant variable from a regression model, in the context of schooling and wage?
Omitted variable bias
What is the Constant Effect Model in the context of schooling and wage?
A model that assumes the causal effect of schooling on wage is the same for all individuals
Why is it problematic to use a bad control in a regression model, in the context of schooling and wage?
It can lead to biased estimates of the causal effect of schooling on wage
What is the main characteristic of bad controls in a regression analysis?
Variables that introduce bias when controlled for, but leaving them out is fine.
What is the primary concern with using a bad control in a causal analysis?
It may not have a causal interpretation, even if the treatment is randomized.
In a DAG, what is the effect of conditioning on a collider variable?
It opens a path between two variables that were previously blocked.
What is the assumption required for a constant effect model to hold?
The treatment effect is the same for all individuals.
How can omitted variable bias arise in a regression analysis with a binary outcome?
If an omitted variable affects both the treatment and the outcome, and is not accounted for in the model.
What is the assumption that E[vi |Xi , Di ] = E[vi |Xi ] in the context of regression and causality?
Conditional Independence Assumption (CIA)
What is the purpose of the conditional independence assumption in a regression analysis?
To ensure that the treatment is independent of the error term, conditional on the controls.
What is the main difference between the traditional assumption of independence and the Conditional Independence Assumption (CIA)?
The CIA focuses on identifying one causal effect, whereas the traditional assumption assumes all regressors are independent of the error term.
In a causal analysis, what is the difference between a direct and indirect effect of a treatment?
A direct effect is the effect of the treatment on the outcome, while an indirect effect is the effect of the treatment on the outcome through a mediator.
What is the interpretation of the regression coefficients multiplying the controls in a regression model?
They have no causal interpretation.
What is the role of a DAG in identifying bad controls?
A DAG can help identify bad controls by visualizing the causal relationships between variables.
What is the problem with including 'bad controls' in a regression model?
Omitted variable bias
What is the Constant Effect Model in the context of regression and causality?
A model that assumes the causal effect is constant across different levels of the control variables.
What is the implication of the coefficient of d being almost 0 in the regression of v on x and d?
d and v are independent, conditional on x.
What is the main difference between regression and causality?
Regression focuses on estimating associations, while causality focuses on identifying causal effects.
What is the implication of including 'bad controls' in a regression model with binary outcomes?
It can lead to biased estimates of the causal effect and incorrect inference.
What is the formula for the regression coefficient in the bivariate case?
β = Cov(Yi, Xi) / Var(Xi)
What is the purpose of 'partialling out' other variables in the multivariate regression model?
To estimate the linear effect of each regressor, controlling for the other variables in the model.
What is the consequence of omitting a variable X from the regression model when the population model is Yi = β0 + τDi + γXi + εi?
Omitted variable bias, resulting in an estimator of the coefficient of D that is biased by γδXD.
What is the Conditional Independence Assumption (CIA) in the context of regression analysis?
The assumption that the error term is independent of the regressors, i.e., εi ⊥ Xki.
What is the Constant Effect Model in regression analysis?
A model that assumes the effect of a regressor on the outcome is constant across all observations.
What are 'bad controls' in the context of regression analysis?
Variables that are correlated with both the outcome and the regressor of interest, but are not causally related to the outcome.
What is the main challenge in analyzing binary outcomes in regression analysis?
The non-normality of the outcome variable, which can lead to biased estimates of the regression coefficients.
How does the omission of a variable X affect the estimator of the coefficient of D in the regression of Y on D?
The estimator is biased by γδXD, where δXD is the coefficient of D in the regression of X on D.
What is the primary assumption of the Constant Effect Model, and how is it related to regression analysis?
The primary assumption of the Constant Effect Model is that the effect of the independent variable on the dependent variable is constant across all observations. This assumption is central to regression analysis, as it allows us to estimate the average effect of the independent variable on the dependent variable.
How does the Conditional Independence Assumption (CIA) relate to the concept of causality in regression analysis?
The Conditional Independence Assumption (CIA) states that the error term is independent of the independent variable, given the control variables. This assumption is necessary for identifying causal relationships between variables in regression analysis.
What is the problem of bad controls in regression analysis, and how can it lead to biased estimates?
Bad controls refer to the inclusion of variables in a regression model that are not truly exogenous, but are instead correlated with the error term. This can lead to biased estimates, as the model may attribute the effect of the bad control to the independent variable of interest.
What is omitted variable bias, and how can it be addressed in regression analysis?
Omitted variable bias occurs when a relevant variable is omitted from a regression model, leading to biased estimates of the independent variable's effect. This can be addressed by including the omitted variable in the model or using instrumental variable techniques.
What is the difference between a binary outcome and a continuous outcome in regression analysis, and how do the assumptions differ?
Binary outcomes are discrete variables that take on only two values, whereas continuous outcomes are continuous variables that take on any value within a range. The assumptions of regression analysis differ for binary outcomes, as the dependent variable is not normally distributed, and specialized techniques such as logistic regression are required.
How can the F-statistic and R-squared values be used to evaluate the goodness of fit of a regression model?
The F-statistic tests the overall significance of the regression model, while the R-squared value measures the proportion of variance explained by the model. Together, they provide a comprehensive evaluation of the model's goodness of fit.
What is the role of the residual plot in regression analysis, and how can it be used to identify potential problems?
The residual plot is a graphical representation of the residuals against the fitted values, and it can be used to identify potential problems such as non-normality, heteroscedasticity, and non-linear relationships.
How can the Coef. column in the regression output be used to interpret the effect of the independent variable on the dependent variable?
The Coef. column represents the estimated change in the dependent variable for a one-unit change in the independent variable, while holding all other variables constant.
Study Notes
Regression and Causality
- Regression is not necessarily causal, and to give it a causal interpretation, we need an additional assumption.
- The observed outcome can be decomposed into the average treatment effect on the treated and selection bias.
- The potential outcome framework is used to state the causal effect of schooling on wage.
Bad Controls
- Bad controls are variables that introduce bias when controlled for, but leaving them out is fine.
- Bad controls are often variables that are themselves outcomes of the treatment.
- Good controls are variables that can be thought of as having been fixed before the treatment assignment.
- Example of bad control: a college (yes/no) and occupation (blue/white-collar) setting.
- Bad control means that a comparison of earnings conditional on the occupation may not have a causal interpretation, even if college is randomized.
- A DAG (directed acyclic graph) is more intuitive than a formal derivation to understand the bad control problem.
- The bad control problem is a case where a DAG is more intuitive than a formal derivation.
Regression Fundamentals
- The mechanical properties of regression are universal features of the population regression and its sample analogue that have nothing to do with a researcher's interpretation of the output.
- Regression coefficients change as covariates are added or removed from the model.
- Bivariate case: β = Cov(Yi, Xi) / Var(Xi).
- Multivariate case: βk = Cov(Yi, X̃ki) / Var(X̃ki), where X̃ki is residual from regression of Xki on all other covariates.
- Omitted variable bias: the estimator of the coefficient of D in the regression of Y on D is biased if an omitted variable X is correlated with D and affects Y.
Summarizing Comments
- The CIA (conditional independence assumption) is a weaker and more focused assumption than the traditional assumption that all regressors are independent of v.
- Focus is on identifying one causal effect, not on obtaining unbiased estimates for all right-hand side variables.
- There is a clear distinction between cause and controls on the right-hand side of the regression.
- The regression coefficients multiplying the controls have no causal interpretation.
This quiz covers the concept of regression and causality in the context of binary outcomes, using the example of schooling decisions and their effect on wages.
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