QUESTION FOUR Given a simple model: \[ y=\beta_{0}+\beta_{1} x+u \] a) Write the OLS estimators for \( \beta_{0} \) and \( \beta_{1} \). ( 8 marks) b) What is \( u \) and what does it include? ( 10 marks) c) If you are to estimate the model above, what would be the required OL.S assumption about the expectation of the error term \( u \) and variable \( x \)

Below are detailed answers to each part of the question: ────────────────────────────── a) OLS Estimators for β₀ and β₁ For the simple linear regression model   y = β₀ + β₁x + u the Ordinary Least Squares (OLS) estimators are obtained by minimizing the sum of squared residuals. They are given by:   β₁̂ = (Σ(xᵢ – x̄)(yᵢ – ȳ)) / (Σ(xᵢ – x̄)²)   β₀̂ = ȳ – β₁̂ · x̄ where: • x̄ = (1/n) Σ xᵢ is the mean of x, • ȳ = (1/n) Σ yᵢ is the mean of y, and • the summations run over all n observations. ────────────────────────────── b) Definition and Components of u In the model, u denotes the error term. It represents the difference between the observed value of y and the value predicted by the linear relationship (β₀ + β₁x). In other words, u includes:  • The effect of all other factors influencing y that are not captured by the explanatory variable x.  • Omitted variables that affect y.  • Random shocks or disturbances.  • Measurement errors in the dependent variable (if present). Thus, u embodies all influences on y that are omitted from the model and any randomness that prevents the model from perfectly predicting y. ────────────────────────────── c) The Required OLS Assumption on u and x A key OLS assumption needed for unbiased estimation of β₀ and β₁ is that the error term has a zero conditional mean given x. Formally, this assumption is written as:   E(u | x) = 0 This assumption implies that:  • On average, the error term does not depend on the value of x.  • There is no systematic relationship between x and u.  • All omitted factors or random shocks (captured by u) are uncorrelated with the explanatory variable x. This condition is crucial because if u and x were correlated, the OLS estimators would be biased, leading to incorrect inference about the impact of x on y. ────────────────────────────── Summary a) β₁̂ = Σ(xᵢ – x̄)(yᵢ – ȳ) / Σ(xᵢ – x̄)² and β₀̂ = ȳ – β₁̂ · x̄. b) u is the error term capturing all factors affecting y other than x, including omitted variables, measurement error, and random disturbances. c) The required OLS assumption is E(u | x) = 0, meaning the error term has zero conditional mean and is uncorrelated with the explanatory variable x.