Lecture 4

Consider an econometric model where schooling is suspected to be endogenous. A researcher proposes using a change in student loan policies as an instrument. Under what stringent condition would this instrument be considered valid, assuming it is relevant?

In the context of Angrist's (1990) study on the effects of Vietnam-era military service on earnings, which of the following scenarios would most severely threaten the validity of draft-eligibility as an instrument for veteran status?

Suppose a researcher aims to estimate the causal effect of an additional year of schooling on wages using the Wald estimator. The reduced form estimate shows that individuals exposed to a policy change completed 0.2 years more schooling and earned 2% higher wages. However, the first-stage effect (policy change on schooling) is found to be statistically insignificant. Which of the following is the MOST appropriate conclusion?

In the context of Angrist and Krueger's (1991) study using quarter of birth as an instrument for education, which unobserved factor would MOST severely bias the Wald estimate of the return to schooling?

Consider a scenario where a researcher attempts to use proximity to a new community college as an instrument for college attendance. What specific threat to the exclusion restriction should the researcher be MOST concerned about?

In the context of the 'constant effect case' model, if a researcher posits that an individual's wage ($y_i$) is determined by $\alpha$, the return to schooling ($p s_i$), and a composite error term ($\eta_i$), and they further decompose $\eta_i$ into observed abilities ($A_i' \gamma$) and a random error term ($\upsilon_i$), under what specific condition does ordinary least squares (OLS) estimation of the 'short' regression ($y_i = \alpha + p s_i + \eta_i$) yield biased estimates of $p$?

Given the model $y_i = \alpha + p s_i + A_i' \gamma + \upsilon_i$, where $y_i$ represents wages, $s_i$ represents schooling, $A_i$ represents observed abilities, $\gamma$ represents the effect of abilities, and $\upsilon_i$ is a random error term, what key identifying assumption must hold true to ensure that the population regression of $y_i$ on $s_i$ and $A_i$ yields unbiased estimates of the causal effect of schooling ($p$) on wages?

In the context of instrumental variables (IV) estimation, which of the following scenarios would unequivocally violate the exclusion restriction, rendering the instrument invalid for consistently estimating the causal effect of schooling ($s_i$) on wages ($y_i$)?

Consider a scenario where an instrumental variable $z_i$ is proposed to address the endogeneity of schooling ($s_i$) in a wage regression. While $z_i$ exhibits a statistically significant correlation with $s_i$, a formal overidentification test using multiple instruments fails to reject the null hypothesis that all instruments are valid. Which of the following inferences is most warranted?

In an instrumental variables (IV) framework designed to estimate the causal effect of schooling ($s_i$) on wages ($y_i$), where the error term ($\eta_i$) is potentially correlated with schooling, what specific assumption regarding the instrument ($z_i$) ensures that it breaks the endogenous variation in $s_i$ into components that are both correlated and uncorrelated with $\eta_i$ , allowing for consistent estimation of the causal effect?

Suppose you are employing an instrumental variable ($z_i$) strategy to estimate the causal effect of years of schooling ($s_i$) on individual earnings ($y_i$). You have reason to suspect that even though $z_i$ is ostensibly exogenous, its effect on earnings may be heterogeneous across different subgroups of the population. Which econometric technique would be most appropriate to assess the potential bias arising from this heterogeneous treatment effect and obtain a more precise estimate of the average treatment effect?

Consider a scenario where researchers are using proximity to a college ($z_i$) as an instrument for educational attainment ($s_i$) in a wage regression. However, they discover that individuals who grow up near colleges also tend to have better access to healthcare and other resources that directly impact their future earnings, irrespective of their educational choices. This situation exemplifies which of the following threats to the validity of the instrumental variable?

In the context of instrumental variables (IV) analysis, what fundamental threat does the exclusion restriction aim to mitigate, and why is its untestability a critical consideration for econometricians?

Consider an instrumental variable $z_i$ that purportedly affects a causal variable of interest $s_i$. What specific condition must be met to ensure the validity of $z_i$ as an instrument, and why is directly testing this condition inherently problematic?

In an instrumental variable framework, why is the first-stage requirement—that the instrument must have a demonstrable effect on the causal variable—considered essential, and how does a failure in this stage undermine the entire analysis?

Consider the scenario where an instrument $z_i$ (e.g., quarter of birth) is proposed to address endogeneity in a causal variable $s_i$ (e.g., years of schooling) when estimating its impact on an outcome $y_i$ (e.g., earnings). If $z_i$ is found to directly influence unobserved determinants of $y_i$ or ability, what specific IV assumption is violated, and what is the implication for the validity of the IV estimates?

Random assignment in experiments is often likened to the exclusion restriction in instrumental variables. In what critical sense does a valid instrument, satisfying the exclusion restriction, emulate random assignment, and why is this equivalence fundamental for causal inference?

Suppose you are employing an instrumental variable $z_i$ to estimate the causal effect of $s_i$ on $y_i$. How does the strength of the correlation between $z_i$ and $s_i$ (the first stage) specifically affect the properties of the resulting instrumental variable estimator, particularly in terms of bias and efficiency?

In the framework of instrumental variables, the exclusion restriction is pivotal. Enumerate the potential consequences of violating this restriction, elucidating how such a violation undermines the integrity and interpretability of the causal inferences drawn from the analysis.

Consider the application of quarter of birth as an instrument for educational attainment. What are the primary, theoretically-grounded concerns regarding the validity of this instrument, especially concerning potential violations of the exclusion restriction in contemporary, developed economies?

In the context of the exclusion restriction in instrumental variables, why is it inherently impossible to empirically 'test' the validity of this assumption, and what implications does this untestability have for the interpretation of IV estimates?

Critically evaluate the assertion that a valid instrument provides variation in a causal variable 'as good as random.' In what precise sense does this hold, and what are the key limitations or caveats to this analogy when dealing with real-world instrumental variable applications?

In the context of Instrumental Variables (IV) estimation, if the exclusion restriction is demonstrably violated, what inferential challenge is most likely to arise, assuming all other IV assumptions hold?

Consider an IV model analyzing the effect of educational attainment ($s_i$) on earnings ($y_i$), where $z_i$ represents proximity to a college. What specific threat to validity is most directly addressed by including a rich set of geographic and socioeconomic controls ($X_i'$) in both the first-stage and reduced-form equations?

In the context of the Wald estimator, assume you are analyzing the impact of military service ($s_i$), a binary variable, on future civilian earnings ($y_i$). You use draft lottery number ($z_i$) as an instrument, also a binary variable. If $E[y_i | z_i = 1] = $50,000$, $E[y_i | z_i = 0] = $40,000$, $p = P(z_i = 1) = 0.3$, and $Cov(s_i, z_i) = 0.0525$, what is the Wald estimate of the effect of military service on earnings?

Consider an instrumental variable (IV) regression where both the endogenous variable ($s_i$) and the instrument ($z_i$) are binary. Which of the following statements accurately describes the conditions under which the IV estimator will produce a consistent estimate of the local average treatment effect (LATE)?

Suppose an econometrician uses quarter of birth ($z_i$) as an instrument for years of schooling ($s_i$) in a regression examining the effect of education on earnings ($y_i$). Assuming a linear model as described, what specific concern arises if individuals strategically choose their birth quarter to maximize their schooling opportunities?

In IV estimation, what is the consequence of using an instrument ($z_i$) that is only weakly correlated with the endogenous variable ($s_i$) in the first-stage equation, even if the instrument satisfies the exclusion restriction?

Consider a scenario where $s_i$ represents years of schooling and $y_i$ represents log earnings. Using the described IV framework, which of the following best describes limitations of interpreting $$ as the causal effect of an additional year of schooling on log earnings for all individuals?

In the two-equation IV system provided, how does the inclusion of the covariates $X_i'$ affect the interpretation of the coefficients ${11}$ and ${21}$?

In a scenario where $z_i$ (the instrument) is a valid instrument, $s_i$ is the endogenous variable, and $y_i$ is the outcome, what is the most accurate interpretation of $Cov(y_i, z_i)$?

Assuming that $z_i$ is a binary instrument. Under what condition does $Cov(y_i, z_i) = {E[y_i | z_i = 1] - E[y_i | z_i = 0]}p(1 - p)$ accurately compute covariance even if the sample data exhibits substantial heteroskedasticity?

In the context of Angrist and Krueger's (1991) study on the causal effect of education on earnings, how does the quarter of birth instrument leverage institutional knowledge and compulsory schooling laws to address omitted variable bias, specifically in relation to unobserved ability?

Within the framework of instrumental variables (IV) estimation, if an instrument $z_i$ is correlated with the error term $\eta_i$, what is the most likely consequence for the estimation of the parameter $\pi_1$ representing the causal effect of schooling ($s$) on earnings?

In the context of instrumental variables, what critical assumption must hold true for an instrument $z_i$ to validly isolate the 'unproblematic' variation in a variable $s$ (schooling) that is uncorrelated with the error term $\eta_i$ (unobserved factors), and how is this assumption typically assessed in empirical research?

Considering Angrist and Krueger's study utilizing quarter of birth as an instrument, what potential threat to the validity of their instrument could arise if individuals born in different quarters of the year systematically differ in unobservable characteristics that also affect earnings, and how might this compromise the identification of the causal effect of education?

Suppose a researcher aims to replicate Angrist and Krueger's (1991) study in a different cultural context where seasonal employment patterns significantly influence family income. How might these seasonal employment patterns affect the validity of quarter of birth as an instrument for educational attainment, and what specific econometric strategies could be employed to address this potential confounding?

Consider a scenario where the compulsory schooling laws exhibit significant heterogeneity across different states within the US during the period studied by Angrist and Krueger (1991). How would this heterogeneity impact the interpretation of their findings, and what methodological adjustments would be necessary to account for this variation in compulsory schooling requirements?

Suppose that advancements in fertility treatments have led to a higher incidence of multiple births (twins, triplets, etc.). How might the presence of multiple births within a family introduce a potential bias in Angrist and Krueger's instrumental variable approach, and what analytic strategies could be implemented to address this bias?

In the context of instrumental variables estimation, define and differentiate between the concepts of 'relevance' and 'exclusion restriction' concerning the validity of an instrument $z_i$ and explain how a failure of either condition would compromise the identification of a causal effect.

In the context of Angrist and Krueger’s study, consider the potential for differential migration patterns based on quarter of birth. If individuals born in certain quarters are more likely to migrate to regions with different labor market conditions and educational opportunities, how could this migration bias affect the validity of their instrumental variable estimates, and what statistical techniques could be employed to mitigate this potential bias?