Econometrics: Multivariate OLS

In the context of multivariate OLS, under what specific condition does including additional control variables fail to mitigate endogeneity, subsequently leading to potentially spurious causal inferences, assuming all OLS assumptions generally hold?

Suppose a researcher omits a salient variable, $Z$, from a regression model predicting $Y$ based on $X$. Under what precise circumstances would the omission of $Z$ not induce omitted variable bias in the estimated coefficient for $X$?

Consider a structural equation where $Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \epsilon_i$, but a researcher mistakenly estimates $Y_i = \beta_0^{OX2} + \beta_1^{OX2} X_{1i} + \epsilon_i$. If $X_{2i} = \delta_0 + \delta_1 X_{1i} + \tau_i$, and (\tau_i) is uncorrelated with (\nu_i) and (X_1), what is the precise mathematical expression for the omitted variable bias in the estimator (\beta_1^{OX2})?

In a scenario where (\text{Corr}(X_1, X_2) < 0) and (\beta_2 > 0), what is the most likely direction of the bias introduced when (X_2) is omitted from a regression model attempting to estimate the effect of (X_1) on (Y)?

Assume a regression analysis aims to estimate the effect of mileage (mpg) on the price of a car. In a bivariate regression, the coefficient on mpg is -238.9 (significant at p<0.01). After adding 'weight' and 'foreign' (whether the car is foreign-made) as controls in a multivariate regression, the coefficient on mpg becomes 21.85 (insignificant). What is the most rigorous interpretation of this coefficient change, assuming that the multivariate model is correctly specified?

In the context of econometric modeling, under what precise condition will measurement error in the dependent variable lead to biased OLS estimates, assuming standard OLS assumptions (besides the absence of measurement error) are met?

Consider a scenario where the 'true' model is given by $Y_i = \beta_1 X_i^* + \epsilon_i$, but $X_i^$ is unobserved. Instead, we observe $X_i = X_i^ + \nu_i$, where (\nu_i) is a random error term with mean zero and is uncorrelated with both (X_i^*) and (\epsilon_i). What is the technical term for the type of bias that OLS estimation of (Y_i = b_1 X_i + \epsilon_i) will inevitably produce?

Given the model $Y_i = \beta_1 X_i^* + \epsilon_i$ and the observation equation $X_i = X_i^* + \nu_i$, where (\nu_i) is uncorrelated with (X_i^*) and (\epsilon_i), what is the probabilistic limit (plim) of the OLS estimator (b_1) in the regression (Y_i = b_1 X_i + \epsilon_i)?

Assume X and Y are independent random variables. Which of the following statements accurately describes their covariance?

Given random variables A, B, W, Z and constants a, b, e, d, which of the following represents the correct expansion of $cov(aA + bB, eW + dZ)$?

Let X and Y be random variables. Under what specific condition is $var(X + Y) = var(X) + var(Y)$?

A researcher posits the model $Y_i = \beta_1 X_i^* + \epsilon_i$, where $X_i^$ represents the 'true' value of an independent variable and (\epsilon_i) is an error term. However, $X_i^$ is measured with error, such that the observed value is $X_i = X_i^* + \nu_i$, where (\nu_i) is a purely random measurement error, independent of (X_i^) and (\epsilon_i). The researcher estimates the misspecified model, $Y_i = b_1 X_i + e_i$. If (\text{var}(X^)= 5) and (\text{var}(\nu) = 2 ), by what approximate percentage will the OLS estimate (b_1) be attenuated compared to the 'true' (\beta_1)?

Consider a scenario where data on individual incomes are collected using a survey. Respondents are asked to self-report their annual income, but some individuals intentionally overstate or understate their income due to privacy concerns or social desirability bias. If this misreporting is correlated with individuals' education levels (i.e., more educated individuals tend to underreport income to avoid appearing boastful), what is the likely consequence for a regression analysis attempting to estimate the relationship between education and income?

A researcher is using OLS regression to estimate the returns to schooling. However, the researcher suspects that individuals may systematically misreport their years of education completed. Specifically, individuals with lower cognitive abilities tend to overstate their educational attainment, while those with higher cognitive abilities tend to report it accurately. This misreporting is also correlated with their future earnings. What is the most likely consequence of this non-random measurement error on the estimated return to schooling?

A researcher is modeling the relationship between neighborhood socioeconomic status (SES) and student test scores. However, SES is difficult to measure directly, so the researcher uses median household income as a proxy. Suppose that median household income is an imperfect measure of SES because it doesn't capture wealth inequality within the neighborhood. Furthermore, neighborhoods with high wealth inequality tend to have lower average test scores due to social stratification. What is the nature of the endogeneity that this measurement error introduces?

Researchers are analyzing the effect of air pollution ($X$) on respiratory health outcomes ($Y$) using observational data. Ideally, they would measure individual-level exposure to pollutants using personal air quality monitors. However, due to cost constraints, they rely on publicly available data from regional air quality monitoring stations. These stations provide aggregate pollution levels for large geographic areas, leading to measurement error as they don't capture individual-level variation in exposure. If individuals living closer to major roadways experience systematically higher pollution exposure than captured by the regional monitors while also experiencing worse respiratory health due to other confounding factors, what is the most likely consequence for the estimated effect of air pollution on respiratory health?

Suppose you are estimating the effect of parental income on children's educational attainment. However, parental income is often underreported in surveys, particularly by high-income individuals seeking to avoid scrutiny. What type of bias does this measurement error most directly introduce, and how would you expect this bias to affect your estimated coefficient on parental income?

A researcher wants to study the effect of education on wages, but finds that educational attainment in their survey is top-coded (e.g., all individuals with 16 or more years of education are coded as '16'). If individuals with top-coded education levels also tend to have significantly higher unobserved skills that are correlated with wages, what specific problem does this generate, and how will it affect estimates?

A researcher wants to estimate the causal effect of exercise ($X$) on weight loss ($Y$). However, individuals self-report their exercise habits, leading to potential measurement error. Specifically, individuals who are more self-conscious about their weight might overestimate their exercise levels (social desirability bias). If these same individuals also tend to weigh more, how does this non-random measurement error affect the estimated causal effect of exercise on weight loss?

In a randomized controlled trial (RCT) designed to measure the impact of a new educational intervention on student test scores, researchers discover that some students randomly assigned to the treatment group did not fully comply with the intervention protocol (e.g., they attended only a fraction of the tutoring sessions). How does this imperfect compliance affect the estimated intent-to-treat (ITT) effect, and how does it relate to attenuation bias?

A researcher is studying the impact of financial literacy training on household savings. To measure financial literacy, they use a multiple-choice quiz. However, some quiz questions are poorly worded, leading to random guessing and measurement error in the financial literacy scores. If high levels of random guessing are especially prevalent among individuals with lower cognitive abilities, even after the training, what specific consequence is most likely?

A study aims to assess the relationship between neighborhood walkability (X) and residents' physical activity levels (Y). However, walkability is difficult to measure directly, so the researchers use an index based on factors like street connectivity. Suppose residents who are more health-conscious select into more walkable neighborhoods, and these same residents tend to overreport their physical activity levels on surveys (social desirability bias). What issue does this correlation create, and how it does alter estimations?