Bivariate OLS in Econometrics

In the context of Ordinary Least Squares (OLS) estimation, if the long-run average of a random variable is consistently greater than its realized values, what econometric property is most directly challenged?

Consider a bivariate regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$. If $E[\epsilon_i | X_i] \neq 0$, which of the following statements most accurately describes the consequence for the OLS estimator $\hat{\beta}_1$?

Suppose that in a bivariate regression, the true parameter $\beta_1$ is known to be 0, but the estimated coefficient $b_1$ from an OLS regression is consistently non-zero across repeated samples. What inferential pitfall is MOST likely contaminating the analysis?

Given a bivariate OLS regression $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, and assuming that $Cov(X, \epsilon) \neq 0$, what adjustment can be implemented to address the bias in the OLS estimator while maintaining efficiency, if the source of endogeneity is known and measurable?

Consider a scenario where $Cov(X, \epsilon) \neq 0$ in a bivariate regression. What is the fundamental statistical implication regarding the error term and its influence on the OLS estimates?

In the equation $E(\beta) = \frac{1}{N}\sum_{i=1}^{N}\beta = \beta$, what econometric principle does this relationship primarily illustrate?

If $E(XY) = E(X)E(Y)$, what fundamental statistical condition MUST hold between the random variables X and Y?

In the context of bivariate OLS regression, what is the implication of the statement: 'OLS does not automatically produce unbiased coefficient estimates'?

Given that $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, which condition must the error term fulfill for OLS to yield unbiased estimates of $\beta_1$?

In a bivariate regression, if a researcher concludes that b₁ ≠ β₁, what can be definitively inferred about b₁?

Within the context of expectation in econometrics, what is the fundamental relevance of expectation in establishing causality using regression models?

How would you interpret the econometric implication if it's determined that exogeneity condition has been violated?

If it can be proven or found that the error term is correlated with the independent variables, what best describes the effect on the nature of the association between the two?

Given the formula for the estimated coefficient $b_1 = \frac{\sum_{i=1}^{N}(X_i - \bar{X})(Y_i - \bar{Y})}{\sum_{i=1}^{N}(X_i - \bar{X})^2}$, what is the necessary condition for ensuring that $b_1$ accurately reflects the causal effect of $X$ on $Y$?

If we start with the standard bivariate regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, and derive that $\bar{Y} = \beta_0 + \beta_1\bar{X} + \bar{\epsilon}$, which assumption MUST be valid to ensure that $\bar{\epsilon}$ converges to zero as N increases?

Given the derived expression $b_1 = \beta_1 + \frac{\sum_i (X_i - \bar{X})(\epsilon_i - \bar{\epsilon})}{\sum_i (X_i - \bar{X})^2}$, under what specific condition will $b_1$ be an unbiased estimator of $\beta_1$?

If a regression yields $E[b_1] = \beta_1 + \frac{Cov(X, \epsilon)}{Var(X)}$, what action can be taken to eliminate bias in the estimate of $\beta_1$ in a real-world scenario where $Cov(X, \epsilon) \neq 0$ and a valid instrument is unavailable.

Consider a bivariate OLS model. What is the most accurate interpretation if it's found where $Cov(X, ε)$ is nonzero?

In the context of mitigating endogeneity bias, if you find that $Cov(X, \epsilon)$ isn't zero, what technique could you employ if IV regression isn't a valid or ideal choice?

Let's say we find that b₁ estimates of β₁ can be normally distributed when the sample is sufficiently large. What is assumed?

When analyzing the biasedness and unbiasedness of our estimated coefficient ($b_1$), what type of models is most crucial?

What is the formula for determining the Y; of a dataset given what you know about the bivariate model?