Lecture 3: Ordinary Least Squares (OLS)

Consider a scenario where an econometrician posits a linear regression model to ascertain the causal impact of variable X on outcome Y. Given the foundational principles of Ordinary Least Squares (OLS) estimation, under what precise condition is the OLS estimator for the coefficient of X guaranteed to be both unbiased and consistent in elucidating this causal effect?

In the context of econometric modeling aiming for causal inference, imagine a researcher estimates a bivariate regression of earnings (Y) on years of education (X). Invoking the 'omitted variable bias' formula, under what specific circumstance would the estimated coefficient on education from this regression spuriously overestimate the true causal effect of education on earnings?

Consider a research design where randomization is employed to assign individuals to either a treatment or control group before assessing the impact of treatment (T) on an outcome (Y). Within this experimental framework, what is the most salient econometric advantage conferred by randomization concerning the application of Ordinary Least Squares (OLS) regression for causal inference?

Imagine a researcher, aiming to estimate the causal effect of class size (X) on student test scores (Y), initially omits 'student ability' from their OLS regression. Subsequently, concerned about potential omitted variable bias, they decide to include 'parental income' (Z) as a control variable. However, unbeknownst to them, parental income is largely determined after and because of their child's inherent ability, and it also independently affects test scores. What econometric problem is most acutely introduced or exacerbated by controlling for 'parental income' in this scenario?

In the context of instrumental variables (IV) regression, consider a scenario where 'distance to college' is proposed as an instrument for 'years of education' in an earnings regression. For 'distance to college' to be considered a valid instrument, which of the following conditions must be rigorously satisfied? (Assume relevance is already established)

Suppose a researcher aims to estimate the causal effect of 'smoking during pregnancy' (X) on 'birth weight' (Y) using observational data. Recognizing the potential for confounding, they consider using 'cigarette taxes' (Z) as an instrument for 'smoking during pregnancy'. Assess the validity of 'cigarette taxes' as an instrument in this context, specifically focusing on the exogeneity assumption. Which of the following poses the most significant threat to the exogeneity of 'cigarette taxes' as an instrument?

In the framework of regression discontinuity design (RDD), researchers exploit a sharp discontinuity in treatment assignment based on a threshold of a 'forcing variable'. Consider an RDD study examining the effect of a scholarship (treatment) awarded to students scoring above a certain threshold on a standardized test (forcing variable) on college enrollment (outcome). What is the most critical assumption for the validity of causal inference in this RDD setting?

When employing difference-in-differences (DID) methodology to estimate the causal effect of a policy intervention, the 'parallel trends' assumption is paramount. Assume a policy is implemented in one region (treatment group) but not in another comparable region (control group). Which of the following scenarios would most severely undermine the validity of the parallel trends assumption in a DID analysis?

Consider a scenario where a researcher is investigating the causal effect of 'job training' (X) on 'employment status' (Y). They suspect that individuals who are more motivated are both more likely to participate in job training and more likely to be employed, regardless of training. If 'motivation' is unobserved and omitted from the regression, what type of bias is most likely to arise in the OLS estimate of the effect of job training on employment?

In the context of assessing the validity of causal claims derived from observational studies using OLS regression, which of the following strategies provides the most robust approach to mitigating concerns about omitted variable bias and bolstering the credibility of causal interpretations?

In an incorrectly specified model $y_i = \alpha + \rho s_i + u_i$ where the true model is $y_i = \alpha + \rho s_i + \gamma A_i + \epsilon_i$, under what precise condition, assuming $\gamma \neq 0$, will the OLS estimator $\rho_{OLS}$ from the incorrect model converge in probability to the true $\rho$?

In the context of the omitted variable bias formula, $\rho_{OLS} = \rho + \gamma \delta_{AS}$, where $\delta_{AS}$ represents the regression coefficient of $S_i$ in a regression of $A_i$ on $S_i$, what fundamentally critical assumption must hold true for this formulation to be valid?

Consider a scenario where you are estimating a wage regression, and you suspect omitted variable bias due to unobserved ability. You initially estimate $log(wage) = 5.0455 + 0.0667 * education$. After including a proxy for ability ($IQ$), the regression becomes $log(wage) = 4.7050 + 0.0443 * education + 0.0063 * IQ$. What is the most accurate interpretation of the change in the coefficient on education?

In the standard omitted variable bias framework, let $\hat{\beta}$ represent the coefficient of interest in a short regression, $\beta$ represent the true coefficient in the long regression, and $\delta$ represent the regression coefficient from regressing the omitted variable on the included variable. What additional information must we know with certainty to compute the exact magnitude of the bias?

Suppose a researcher estimates a wage regression omitting a crucial variable: 'motivation'. They find the coefficient on education is 0.08. When they include 'motivation' (measured imperfectly), the education coefficient drops to 0.06, and the 'motivation' coefficient is 0.30. If regressing 'motivation' on 'education' yields a coefficient of 0.05, what is the estimated bias in the original education coefficient due to omitting 'motivation'?

Assume that you estimate a simple regression model and suspect that an important variable, $Z$, has been omitted. Under what conditions would the inclusion of $Z$ not alter the coefficient of interest, $\beta$, on the included variable, $X$?

In a regression framework where the true model is $Y = X\beta + Z\gamma + \epsilon$, but you estimate $Y = X\beta + u$, derive the precise mathematical condition under which the OLS estimator of $\beta$ in the short regression is unbiased, despite the omission of $Z$.

Consider a study estimating the impact of parental education ($P$) on child's education ($C$). However, 'genetic endowment' ($G$) is unobserved and correlated with both $P$ and $C$. If you could only proxy either $P$ or $C$, which one would be optimal to proxy to address OVB and why?

In the context of a wage regression, suppose you want to assess the potential bias introduced by omitting 'grit' (i.e., perseverance and passion for long-term goals). You know that individuals with higher education ($E$) tend to exhibit greater grit ($G$), but the relationship is complex and not perfectly linear. Furthermore, 'grit' is challenging to quantify reliably. What is your strategy to best assess the potential bias?

Let's say a researcher estimates the return to education but cannot observe innate ability. They apply the omitted variable bias formula and confidently conclude that the return to education is significantly biased upwards. An astute critic points out that the true relationship between ability, education, and wages might be more complex. What nuance would most effectively challenge the researcher's conclusion?

In the context of Ordinary Least Squares (OLS) regression, under what specific condition is the derivation of causality most critically dependent?

Assume a bivariate regression model where earnings ($y$) are regressed on years of education ($x$), with unobserved innate ability represented by $u$: $y = \beta_0 + \beta_1x + u$. If $E(u|x) \neq 0$, what is the most likely consequence for the OLS estimator of $\beta_1$?

In the context of the 'omitted variable formula,' which condition must be met for the OLS estimator of a coefficient to be unbiased when a relevant variable is excluded from the regression?

Consider a scenario where you aim to estimate the impact of schooling ($S_i$) on earnings ($y_i$), but you suspect that ability ($A_i$) is an omitted variable correlated with both schooling and earnings. Given the 'true model' $y_i = \alpha + \rho S_i + \gamma A_i + \epsilon_i$, what is the most likely consequence of omitting $A_i$ from your regression?

In the framework of OLS assumptions, particularly concerning the zero conditional mean, what represents the most critical threat to the validity of causal inferences drawn from regression analysis using non-experimental data?

Suppose you estimate a regression model and suspect that the zero conditional mean assumption is violated. Which of the following strategies would be the MOST appropriate for addressing this issue, assuming you have access to rich data and advanced econometric techniques?

Given the 'omitted variable formula', under what specific circumstance will the bias resulting from omitting a relevant variable, correlated with the included variables, be precisely zero?

In a scenario where an OLS regression is performed to estimate the effect of education on wages, and it is suspected that unobserved ability is correlated with both education and wages, what econometric technique could MOST effectively address the endogeneity issue, assuming suitable data is available?

Suppose a researcher estimates a model of earnings ($y_i$) as a function of schooling ($S_i$) and ability ($A_i$), represented as $y_i = \alpha + \rho S_i + \gamma A_i + \epsilon_i$. However, ability is unobserved and omitted from the regression. Assuming $S_i$ and $A_i$ are positively correlated, and $\gamma > 0$, what is the expected direction of the bias in the OLS estimate of $\rho$ if ability is omitted?

Consider a scenario where the true data generating process includes an interaction term between two explanatory variables, but this interaction is omitted from the estimated OLS model. How does the omission of this interaction term MOST directly impact the interpretation of the coefficients on the included main effects?

Suppose a researcher aims to estimate the impact of a job search program ($T_i$) on employment status ($E_i$). Due to data limitations, the researcher cannot directly observe individual ability ($A_i$). Under what specific circumstance will the estimated coefficient on $T_i$ in the short regression ($E_i = \alpha + \rho T_i + u_i$) be biased downwards, assuming $A_i$ positively influences employment?

Consider a researcher estimating the effect of education on earnings, initially finding a coefficient of 0.0667. After including a proxy for ability, the coefficient drops to 0.0443. Given that regressing the ability proxy on education yields a coefficient of 3.5388 and the coefficient on the ability proxy in the earnings equation is 0.0063, what precisely does the product of 3.5388 and 0.0063 represent in this context?

Suppose a researcher estimates the following two regressions: 1) $E_i = \alpha + \rho T_i + u_i$, and 2) $E_i = \alpha + \rho T_i + \gamma A_i + \epsilon_i$, where $E_i$ is employment status, $T_i$ is participation in a job search program, $A_i$ is ability, and $u_i$ is the error term. If $Cov(T_i, A_i) < 0$ and $\gamma > 0$, what sign will the bias term in the omitted variable formula have, and what does this imply about the estimated effect of the job search program in the first regression?

In the context of estimating the impact of education on earnings, a researcher initially omits 'innate ability' from their regression model. Critically, the researcher posits that the covariance between education and innate ability could realistically be negative. Under what specific condition would this negative covariance lead to the OLS estimate of the return to education being underestimated?

Consider the scenario where a researcher is using the Omitted Variable Bias (OVB) formula. The researcher estimates that $\hat{\beta}{short} = \beta + \delta \cdot \gamma$, where $\beta$ is the true coefficient, $\hat{\beta}{short}$ is the biased coefficient from the short regression, $\delta$ is the coefficient from regressing omitted variable $Z$ on included variable $X$, and $\gamma$ is the coefficient from regressing $Y$ on $Z$ after including $X$. If the researcher finds that$\hat{\beta}_{short}$ is statistically insignificant despite strong theoretical reasons to believe $X$ affects $Y$, What can explain this?

A researcher suspects omitted variable bias in their wage regression due to the omission of 'social capital'. To assess the potential bias, they propose a novel approach: instead of directly measuring 'social capital,' they plan to instrument it using 'participation in extracurricular activities during high school'. Which of the following conditions MOST critically undermines the validity of this instrumental variable approach?

Consider a scenario where a researcher hypothesizes that access to high-speed internet ($I$) positively affects student test scores ($S$). However, they suspect that families with higher socioeconomic status (SES) are more likely to have access to high-speed internet and to invest in other resources that enhance their children's academic performance. If the researcher estimates the following equations, where $u_i$ and $\epsilon_i$ are error terms:\ $S_i = \alpha + \beta I_i + u_i$ \$S_i = \alpha + \beta I_i + \gamma SES_i + \epsilon_i$ \ What would imply there is omitted variable bias?

In a study examining the impact of a new teaching method ($X$) on student performance ($Y$), researchers suspect that student motivation ($Z$) is an omitted variable. The estimated model is $Y = \alpha + \beta X + \epsilon$, where $\epsilon$ is the error term. For the OLS estimator of $\beta$ to be unbiased despite the omission of $Z$, which of the following conditions must hold?

Consider a researcher using the omitted variable bias formula to assess the impact of an unobserved variable on the estimated coefficient of interest in a regression model. Under what specific condition would the application of the omitted variable bias formula lead to a misleading or inaccurate assessment of the true bias?

A researcher estimates a wage regression, but omits 'cognitive ability' from the model. They acknowledge that more educated individuals tend to have higher cognitive ability. If the true data generating process involves complex interactions between education, cognitive ability, and unobserved 'grit', what is the most salient reason why applying the basic omitted variable bias (OVB) formula might provide an incomplete or misleading assessment of the true bias?