Quiz on OLS estimator interpretation and validity

📚 Quick Study Guide

• 📈 OLS Estimator: The Ordinary Least Squares (OLS) estimator minimizes the sum of the squared errors between the observed values and the values predicted by the linear regression model.
• 📏 Linear Regression Model: The basic form is $Y = X\beta + \epsilon$, where $Y$ is the dependent variable, $X$ is the independent variable(s), $\beta$ are the coefficients, and $\epsilon$ is the error term.
• ✅ Assumptions of OLS:
  • Linearity: The relationship between $X$ and $Y$ is linear.
  • Randomness: The error term $\epsilon$ is a random variable with a mean of zero.
  • Exogeneity: $E[\epsilon|X] = 0$. The error term is uncorrelated with the independent variables.
  • Homoskedasticity: The error term has constant variance. $Var(\epsilon|X) = \sigma^2$.
  • No Autocorrelation: The error terms are uncorrelated with each other. $Cov(\epsilon_i, \epsilon_j) = 0$ for $i \neq j$.
  • No perfect multicollinearity: The independent variables are not perfectly correlated.
• 🎯 Interpretation of Coefficients: The coefficient $\beta_i$ represents the change in $Y$ for a one-unit change in $X_i$, holding all other variables constant.
• 📊 Validity: The validity of the OLS estimator depends on the assumptions being met. Violations of these assumptions can lead to biased and inconsistent estimates.
• 🤔 Bias: Occurs when the expected value of the estimator does not equal the true parameter value.
• 🌱 Consistency: An estimator is consistent if it converges in probability to the true parameter value as the sample size increases.

Practice Quiz

1. What does the OLS estimator aim to minimize?
   1. The sum of absolute errors
   2. The sum of squared errors
   3. The mean of the errors
   4. The variance of the errors
2. Which of the following is NOT a key assumption of OLS regression?
   1. Homoskedasticity
   2. Autocorrelation of error terms
   3. Exogeneity
   4. Linearity
3. What does the assumption of exogeneity imply?
   1. The error term has a non-zero mean.
   2. The error term is correlated with the independent variables.
   3. The error term is uncorrelated with the independent variables.
   4. The error term has constant variance.
4. What does homoskedasticity mean?
   1. The error term has a mean of zero.
   2. The error term has constant variance across all levels of the independent variables.
   3. The error term is correlated with the independent variables.
   4. The error term follows a normal distribution.
5. In the linear regression model $Y = X\beta + \epsilon$, what does $\beta$ represent?
   1. The error term
   2. The dependent variable
   3. The coefficients
   4. The independent variable
6. What is the consequence of violating the exogeneity assumption?
   1. The OLS estimator is unbiased.
   2. The OLS estimator is inconsistent.
   3. The OLS estimator is efficient.
   4. The OLS estimator is BLUE (Best Linear Unbiased Estimator).
7. What does it mean for an estimator to be consistent?
   1. It always equals the true parameter value.
   2. It converges in probability to the true parameter value as the sample size increases.
   3. It is unbiased.
   4. It has the smallest possible variance.