ECON 266: Bivariate OLS

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Questions and Answers

In the context of econometric modeling, what is the MOST precise interpretation of the parameters $\beta_0$ and $\beta_1$ within a bivariate model?

They represent merely descriptive statistics devoid of any causal inference capabilities, useful only for summarizing observed data.
They are simply numerical artifacts of the Ordinary Least Squares (OLS) estimation procedure, lacking any inherent meaning or interpretability outside of the sample.
They serve as unbiased estimators of the population parameters under any conditions, providing straightforward policy implications.
They quantify the magnitude and direction of the _potential_ causal effect of the independent variable on the dependent variable, contingent upon satisfying rigorous identification assumptions. (correct)

Under what specific condition is the Ordinary Least Squares (OLS) estimator for a bivariate regression model considered the Best Linear Unbiased Estimator (BLUE)?

When the sample size is sufficiently large, irrespective of the underlying data distribution.
When the error term exhibits homoscedasticity and is serially correlated.
When the error term exhibits heteroscedasticity and zero covariance.
When the error term exhibits homoscedasticity, zero covariance, and follows a normal distribution. (correct)

In the context of bivariate regression, what is the fundamental difference between the error term $\epsilon_i$ and the residual $\hat{\epsilon}_i$?

They are conceptually identical; both represent the unexplained variation in the dependent variable, with any differences arising solely from computational approximations.
The error term $\epsilon_i$ is observable directly from the data, while the residual $\hat{\epsilon}_i$ is unobservable.
The error term $\epsilon_i$ is unobservable and represents the true deviation of an observation from the population regression line, while the residual $\hat{\epsilon}_i$ is the estimated deviation from the sample regression line. (correct)
The error term $\epsilon_i$ represents the difference between the observed and predicted values in the sample, ignoring the population, while the residual $\hat{\epsilon}_i$ captures the difference between the observed values in the sample and the true population regression line.

Consider a scenario where you run a bivariate regression model and observe that the sum of squared residuals (SSR) is zero. What implications can be unequivocally derived from this?

This implies that all observations lie perfectly on the estimated regression line, but it does not guarantee a perfect fit in the population. (B) Signup and view all the answers

Ordinary Least Squares (OLS) estimation aims to minimize the sum of squared residuals. Why is the sum of squared residuals minimized, as opposed to, for instance, the sum of absolute values of residuals?

The square function is differentiable, enabling the use of calculus (derivatives) to find the parameter values that minimize the sum of squared residuals. (D) Signup and view all the answers

How does the interpretation of the R-squared statistic change, if at all, when comparing a bivariate regression model to a multivariate regression model?

In a bivariate model, R-squared represents the percentage of variance in the dependent variable explained by the single independent variable, but in a multivariate model, it represents the percentage of variance explained by all independent variables collectively. (B) Signup and view all the answers

Consider an econometrician estimating a bivariate regression model. Under what circumstance would the estimated coefficient on the independent variable, $b_1$, be exactly zero?

When the independent and dependent variables are perfectly uncorrelated in the sample. (C) Signup and view all the answers

What is the MOST accurate interpretation of the phrase 'Ordinary Least Squares (OLS) is the Best Linear Unbiased Estimator (BLUE)'?

Among all linear and unbiased estimators, OLS has the minimum variance, conditional on the OLS assumptions being satisfied. (B) Signup and view all the answers

In a bivariate regression model, the slope coefficient is primarily influenced by:

The covariance between the independent and dependent variables, scaled by the variance of the independent variable. (D) Signup and view all the answers

What is the KEY distinction between using Ordinary Least Squares (OLS) for prediction versus using it for causal inference?

For causal inference, one must justify strong assumptions (e.g., exogeneity) to ensure the estimated coefficients reflect causal effects, not just correlations. (C) Signup and view all the answers

Which of the following statements offer the MOST precise explanation of why Ordinary Least Squares (OLS) estimation aims to minimize the sum of squared residuals, rather than simply the sum of residuals?

Minimizing the sum of squared residuals enables the use of differential calculus to derive closed-form solutions for the parameter estimates. (A) Signup and view all the answers

Concerning bivariate regression, if the variance of the independent variable is zero, what is the implication?

The regression slope is undefined. (A) Signup and view all the answers

How might heteroscedasticity impact the validity of inferences drawn from a bivariate Ordinary Least Squares (OLS) regression model?

Heteroscedasticity does not affect the coefficient estimates themselves, but it renders the standard errors unreliable. (B) Signup and view all the answers

In the context of Ordinary Least Squares (OLS) estimation, what does it mean for an estimator to be 'unbiased'?

On average, across many repeated samples, the estimator's expected value equals the true population parameter. (D) Signup and view all the answers

When would applying Ordinary Least Squares (OLS) to a non-linear relationship be most likely to yield misleading or invalid results?

When visualizing the residuals produces a clear indication and discernible pattern of non-linearity. (A) Signup and view all the answers

Suppose you estimate a bivariate regression and observe that the R-squared is exceptionally high (e.g., 0.99). What potential problem should you be MOST concerned about?

Spurious Regression. (D) Signup and view all the answers

In instrumental variables regression, under what circumstances would the use of a 'weak' instrumental variable lead to biased estimates?

A weak instrument exacerbates the bias from endogeneity, potentially leading to estimates that are even more biased than OLS. (D) Signup and view all the answers

How will the interpretation of the $b_1$ coefficient value differ in a bivariate OLS regression if the independent variable is changed from level-form to log-form?

The slope coefficient represents the percentage change in Y for a one-unit increase in X. (A) Signup and view all the answers

In the context of bivariate regression, when is it most appropriate to use the 'regression through the origin' (i.e., forcing the intercept to be zero)?

Whenever there is strong theoretical justification that the dependent variable must be zero when the independent variable is zero. (D) Signup and view all the answers

What is the MOST significant limitation of relying solely on R-squared to compare the fit of two different bivariate regression models?

R-squared will always increase as variables are added to the equation, even if there is no relationship. (D) Signup and view all the answers

Flashcards

What is a parameter?

A number describing a characteristic of a population or relationship between variables.

Parameters β0 and β1

Summarize how X is related to Y; quantify the degree to which two variables move together.