ECON 266: Bivariate OLS Model

Consider a bivariate Ordinary Least Squares (OLS) model expressed as $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$. If you are presented with a dataset and told it perfectly explains the data, what specific characteristic defines the OLS estimates of the $\beta$ parameters in this scenario?

In the context of deriving the OLS estimator for $\beta_1$ in a bivariate regression model ($Y_i = \beta_1X_i + \epsilon_i$), what is the most critical initial step after setting up the sum of squared residuals?

Given the final OLS estimator for $\beta_1$ in a simplified bivariate model (assuming no intercept) as $\hat{\beta}1 = \frac{\sum{i=1}^{N} Y_i X_i}{\sum_{i=1}^{N} X_i^2}$, under what specific condition does this estimator become inconsistent?

In the context of Ordinary Least Squares, what is the theoretical justification for the approximate normality of the $b_1$ estimates in large samples, and under what specific condition might this approximation fail?

Consider a scenario where you're estimating a bivariate regression model, and you suspect that the error term's variance is not constant. How would you assess and address the potential bias and efficiency issues in the OLS estimator of $b_1$?

Suppose you estimate a bivariate regression model and observe that the residuals exhibit a non-random pattern, suggesting autocorrelation. Evaluate the implications of this autocorrelation for the OLS estimator of $b_1$ and the validity of standard inference procedures.

In the context of estimating the bivariate OLS model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, what are the implications if the independent variable $X_i$ is endogenous, and what econometric techniques can be used to address this endogeneity?

Imagine you are analyzing a large dataset, and after running an OLS regression, you suspect that the functional form of the relationship between $X$ and $Y$ is misspecified. What diagnostic tests and remedies would you employ to address this issue?

Suppose an econometrician is using OLS to estimate the effect of education ($X$) on income ($Y$) but is concerned about omitted variable bias due to unobserved ability. What specific econometric technique could be used to mitigate this bias, and under what assumptions would this technique be valid?

Which of the following statements precisely describes the implications of sampling randomness and modeled randomness on the Ordinary Least Squares (OLS) estimator $b_1$ in a bivariate regression?

In the context of distributions of $b$ estimates, how does an increase in the sample size affect the distribution, and what specific theorem underpins this change?

In the context of econometrics, differentiate between a discrete and a continuous random variable, providing examples of their application in regression analysis.

Consider a scenario where multiple linear regression has confirmed multicollinearity among several independent variables. Evaluate the most effective strategy for mitigating the adverse effects of multicollinearity on the Ordinary Least Squares (OLS) estimates.

Suppose you are estimating a regression model and suspect that your residuals are not normally distributed. Assess the consequences of non-normal residuals on the properties of the OLS estimator and outline strategies to address this issue.

You are analyzing a dataset with a binary dependent variable and considering whether to use a linear probability model (LPM) or a probit model for estimation. Evaluate the key differences between these models, particularly regarding the interpretation of coefficients and the potential drawbacks of each.

An econometrician is using time series data to model the relationship between two variables and suspects the presence of a unit root in one or more series. How does the presence of a unit root affect the properties of OLS estimators, and what are the appropriate steps to address this issue?

Considering a complex econometric model with several endogenous variables, which estimation technique is most appropriate for simultaneously addressing endogeneity and identifying causal effects, and what identifying assumptions must be satisfied?

What are the key advantages and disadvantages of using panel data estimation techniques compared to cross-sectional or time-series methods, particularly concerning controlling for unobserved heterogeneity and addressing endogeneity?