Let’s go through each part of the question step by step: --- ### **(a) State the Gauss-Markov assumptions.** The Gauss-Markov assumptions for the **Classical Linear Regression Model** are: 1. **Linearity**: The model is linear in parameters: \( y = X\beta + \epsilon \). 2. **Random Sampling**: The data is obtained via random sampling. 3. **No Perfect Collinearity**: The regressors (columns of \(X\)) are linearly independent; \(X'X\) is invertible. 4. **Zero Conditional Mean**: The expected value of the error, given \(X\), is zero: \( E[\epsilon|X] = 0 \). 5. **Homoscedasticity**: The errors have constant variance: \( Var(\epsilon|X) = \sigma^2 I \) (no heteroscedasticity). 6. **No Autocorrelation** (sometimes included): Errors are uncorrelated across observations. --- ### **(b) Derive the OLS estimate of β.** The OLS estimator minimizes the sum of squared residuals: \[ \hat{\beta}_{OLS} = \arg\min_\beta (y - X\beta)'(y - X\beta) \] Taking the derivative and setting to zero: \[ -2X'(y - X\beta) = 0 \implies X'X\beta = X'y \] \[ \boxed{\hat{\beta} = (X'X)^{-1} X'y} \] --- ### **(c) Show that under the Gauss-Markov assumptions, the OLS estimate of β is unbiased.** \[ E[\hat{\beta}] = E[(X'X)^{-1} X'y] \] Since \( y = X\beta + \epsilon \): \[ E[\hat{\beta}] = (X'X)^{-1} X' E[y] = (X'X)^{-1} X' E[X\beta + \epsilon] = (X'X)^{-1} X'(X\beta + E[\epsilon]) \] Given \( E[\epsilon] = 0 \): \[ E[\hat{\beta}] = (X'X)^{-1} X'X\beta = \beta \] Thus, **\(\hat{\beta}\) is unbiased**. --- ### **(d) Calculate the variance of the OLS estimate of β.** \[ Var(\hat{\beta}) = Var((X'X)^{-1} X'y) \] \[ = (X'X)^{-1} X' Var(y) X (X'X)^{-1} \] Since \(Var(y) = Var(\epsilon) = \sigma^2 I \): \[ Var(\hat{\beta}) = (X'X)^{-1} X' (\sigma^2 I) X (X'X)^{-1} \] \[ = \sigma^2 (X'X)^{-1} \] --- ### **(e) What is the consequence of omitting relevant explanatory variables in an OLS regression?** - **Omitted Variable Bias:** If you omit a variable that is correlated with both the dependent variable and included regressors, the OLS estimates are **biased and inconsistent**. - The direction and size of the bias depend on the correlation between the omitted variable and the included variables. --- ### **(f) What is the consequence of having a non-constant variance of errors (heteroscedasticity)? How to correct for it?** - **Consequence:** OLS estimates remain unbiased, but they are **no longer efficient** (not BLUE) and the standard errors are wrong, so inference (t-tests, confidence intervals) is invalid. - **Ways to correct:** 1. Use **robust standard errors** (White’s standard errors). 2. Use **Generalized Least Squares (GLS)** or transformation methods. --- ### **(g) Estimate the parameters β₁, β₂, and β₃ given the information:** Given: - \( y = (51, 44, 57, 49, 60, 48) \) - \( x_1 = (3.0, -1.0, 4.4, 4.5, 5.3) \) - \( x_2 = (1.1, 0.0, 1.1, 1.0, 0.0) \) Assume your model is: \[ y_i = \beta_1 + \beta_2 x_{1i} + \beta_3 x_{2i} + \epsilon_i \] Construct the **X matrix** (including intercept): \[ X = \begin{bmatrix} 1 & 3.0 & 1.1 \\ 1 & -1.0 & 0.0 \\ 1 & 4.4 & 1.1 \\ 1 & 4.5 & 1.0 \\ 1 & 5.3 & 0.0 \\ 1 & 0.0 & 0.0 \\ \end{bmatrix} \] \[ y = \begin{bmatrix} 51 \\ 44 \\ 57 \\ 49 \\ 60 \\ 48 \end{bmatrix} \] Find \(\hat{\beta} = (X'X)^{-1} X'y\). **You can calculate this in R or Python:** #### **R Code Example:** ```r y <- c(51,44,57,49,60,48) x1 <- c(3.0,-1.0,4.4,4.5,5.3,0.0) x2 <- c(1.1,0.0,1.1,1.0,0.0,0.0) model <- lm(y ~ x1 + x2) summary(model) ``` #### **Python Code Example:** ```python import numpy as np import statsmodels.api as sm y = np.array([51,44,57,49,60,48]) X = np.column_stack((np.ones(6), [3.0,-1.0,4.4,4.5,5.3,0.0], [1.1,0.0,1.1,1.0,0.0,0.0])) model = sm.OLS(y, X).fit() print(model.summary()) ``` This will give you the estimates for \( \beta_1, \beta_2, \beta_3 \). --- If you want to see the actual numbers, let me know which method you'd like to use (e.g., R, Python, or manual calculation), and I can walk you through it!