OLS Estimator Properties Quiz

1. Under the classical linear regression assumptions, the OLS estimator is unbiased. What does unbiasedness mean?

E(β̂) = β

β̂ is always equal to β

The estimator has the smallest variance

The estimator converges to β as n → ∞

Unbiasedness in the context of OLS estimators means that the expected value of the estimated coefficients (β̂) equals the true population parameters (β). This implies that, on average, the estimator does not systematically overestimate or underestimate the true value, ensuring accurate estimation across repeated sampling.

Explanation

Unbiasedness in the context of OLS estimators means that the expected value of the estimated coefficients (β̂) equals the true population parameters (β). This implies that, on average, the estimator does not systematically overestimate or underestimate the true value, ensuring accurate estimation across repeated sampling.

2. Which assumption is required for OLS estimators to be unbiased?

Homoscedasticity

Zero conditional mean of errors: E(u|X) = 0

Normality of errors

No multicollinearity

For OLS estimators to be unbiased, the assumption of zero conditional mean of errors (E(u|X) = 0) is crucial. This means that the expected value of the error term, given any value of the independent variables, should be zero. If this condition holds, the estimators will accurately reflect the true relationship between the variables.

Explanation

For OLS estimators to be unbiased, the assumption of zero conditional mean of errors (E(u|X) = 0) is crucial. This means that the expected value of the error term, given any value of the independent variables, should be zero. If this condition holds, the estimators will accurately reflect the true relationship between the variables.

3. The Gauss-Markov theorem states that under classical assumptions, OLS is the ____.

The Gauss-Markov theorem establishes that, given certain assumptions about the error terms in a linear regression model, the Ordinary Least Squares (OLS) estimator has the lowest variance among all linear unbiased estimators. This means that OLS is the Best Linear Unbiased Estimator (BLUE), ensuring efficient and reliable parameter estimates in the model.

Explanation

The Gauss-Markov theorem establishes that, given certain assumptions about the error terms in a linear regression model, the Ordinary Least Squares (OLS) estimator has the lowest variance among all linear unbiased estimators. This means that OLS is the Best Linear Unbiased Estimator (BLUE), ensuring efficient and reliable parameter estimates in the model.

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4. True or False: OLS is consistent even if the error term is heteroscedastic.

True

False

Ordinary Least Squares (OLS) estimators remain consistent even in the presence of heteroscedasticity, which means that the variance of the error terms varies across observations. While heteroscedasticity affects the efficiency of the estimators and may lead to biased standard errors, it does not bias the estimated coefficients, allowing OLS to still provide consistent estimates.

Explanation

Ordinary Least Squares (OLS) estimators remain consistent even in the presence of heteroscedasticity, which means that the variance of the error terms varies across observations. While heteroscedasticity affects the efficiency of the estimators and may lead to biased standard errors, it does not bias the estimated coefficients, allowing OLS to still provide consistent estimates.

5. Which of the following violates the assumption of no perfect multicollinearity?

Two regressors are uncorrelated

One regressor is a perfect linear combination of others

Errors are normally distributed

Sample size is large

Perfect multicollinearity occurs when one regressor can be expressed as a linear combination of others, leading to redundancy among predictors. This violates the assumption that regressors should provide unique information, making it impossible to isolate their individual effects on the dependent variable, thus complicating the estimation of regression coefficients.

Explanation

Perfect multicollinearity occurs when one regressor can be expressed as a linear combination of others, leading to redundancy among predictors. This violates the assumption that regressors should provide unique information, making it impossible to isolate their individual effects on the dependent variable, thus complicating the estimation of regression coefficients.

6. The variance of the OLS estimator β̂ is minimized when:

Sample variance of X is high

Error variance is high

Sample variance of X is high and error variance is low

The number of observations is small

The variance of the OLS estimator β̂ is minimized when the sample variance of X is high, as it increases the precision of the estimates, and when the error variance is low, as it reduces the noise in the data. Together, these conditions enhance the reliability of the regression estimates.

Explanation

The variance of the OLS estimator β̂ is minimized when the sample variance of X is high, as it increases the precision of the estimates, and when the error variance is low, as it reduces the noise in the data. Together, these conditions enhance the reliability of the regression estimates.

7. Consistency of an estimator means that as n → ∞, the estimator ____.

Consistency of an estimator indicates that as the sample size (n) approaches infinity, the estimator will yield results that increasingly approximate the true value of the parameter being estimated. This convergence ensures that with larger samples, the estimator becomes more reliable and accurate, reflecting the underlying population characteristics more closely.

Explanation

Consistency of an estimator indicates that as the sample size (n) approaches infinity, the estimator will yield results that increasingly approximate the true value of the parameter being estimated. This convergence ensures that with larger samples, the estimator becomes more reliable and accurate, reflecting the underlying population characteristics more closely.

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8. True or False: Omitted variable bias causes OLS estimators to be biased if the omitted variable is correlated with included regressors.

True

False

Omitted variable bias occurs when a relevant variable that affects the dependent variable is not included in the regression model. If this omitted variable is correlated with the included regressors, it can distort the estimated coefficients, leading to biased Ordinary Least Squares (OLS) estimators. Thus, the statement is true.

Explanation

Omitted variable bias occurs when a relevant variable that affects the dependent variable is not included in the regression model. If this omitted variable is correlated with the included regressors, it can distort the estimated coefficients, leading to biased Ordinary Least Squares (OLS) estimators. Thus, the statement is true.

9. Under homoscedasticity, Var(β̂) = σ²(X'X)⁻¹. If heteroscedasticity is present, this formula is:

Still correct

Biased but consistent

No longer valid; standard errors are incorrect

Only valid for large samples

Under heteroscedasticity, the assumption of constant variance in the error terms is violated, leading to incorrect calculations of standard errors. As a result, the formula Var(β̂) = σ²(X'X)⁻¹ no longer holds true, making the standard errors unreliable for hypothesis testing and confidence intervals.

Explanation

Under heteroscedasticity, the assumption of constant variance in the error terms is violated, leading to incorrect calculations of standard errors. As a result, the formula Var(β̂) = σ²(X'X)⁻¹ no longer holds true, making the standard errors unreliable for hypothesis testing and confidence intervals.

10. The efficiency of an estimator refers to having the ____.

An efficient estimator is one that provides the most precise estimates with the least variability. This means it has the smallest variance among all unbiased estimators, ensuring that its estimates cluster closely around the true parameter value, leading to greater reliability and accuracy in statistical inference.

Explanation

An efficient estimator is one that provides the most precise estimates with the least variability. This means it has the smallest variance among all unbiased estimators, ensuring that its estimates cluster closely around the true parameter value, leading to greater reliability and accuracy in statistical inference.

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11. Which condition ensures that OLS estimators are asymptotically normal?

Errors must be normally distributed

Central limit theorem applies with suitable regularity conditions

Sample size must be exactly 30

Multicollinearity must be present

The Central Limit Theorem states that as the sample size increases, the distribution of the sample mean approaches a normal distribution, regardless of the population's distribution, provided certain regularity conditions are met. This principle ensures that Ordinary Least Squares (OLS) estimators become asymptotically normal as the sample size grows large, supporting valid inference.

Explanation

The Central Limit Theorem states that as the sample size increases, the distribution of the sample mean approaches a normal distribution, regardless of the population's distribution, provided certain regularity conditions are met. This principle ensures that Ordinary Least Squares (OLS) estimators become asymptotically normal as the sample size grows large, supporting valid inference.

12. True or False: If E(u|X) ≠ 0, then OLS is biased.

True

False

If the expected value of the error term conditional on the independent variable \( X \) is not zero (E(u|X) ≠ 0), it indicates that there is a systematic relationship between the error term and the predictor variable. This violates one of the key assumptions of ordinary least squares (OLS) regression, leading to biased estimates of the coefficients.

Explanation

If the expected value of the error term conditional on the independent variable \( X \) is not zero (E(u|X) ≠ 0), it indicates that there is a systematic relationship between the error term and the predictor variable. This violates one of the key assumptions of ordinary least squares (OLS) regression, leading to biased estimates of the coefficients.