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 OL̈S assumption about the expectation of the error term \( u \) and variable \( x \)

Below is a detailed answer addressing 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 derived by minimizing the sum of squared residuals. • The slope estimator (β₁) is given by:   β̂₁ = [Σ (xᵢ – x̄)(yᵢ – ȳ)] / [Σ (xᵢ – x̄)²], where x̄ and ȳ are the sample means of x and y, respectively. • The intercept estimator (β₀) is computed as:   β̂₀ = ȳ – β̂₁ x̄. ────────────────────────────── b) Meaning and Contents of u (Error Term) In the model y = β₀ + β₁x + u, the term u is the error term (also called the disturbance term). It represents all the factors affecting y that are not captured by the explanatory variable x. In more detail:  • u is the difference between the observed value of y and the value predicted by the regression function (i.e., y - [β₀ + β₁x]).  • u includes:   – Omitted variables that influence y,   – Measurement errors in the dependent variable or in variables not explicitly included in the regression,   – Effects of random shocks,   – Any other influences on y that are not accounted for by x. The error term is assumed to capture the collective influence of all such unobserved elements and is considered random. ────────────────────────────── c) Required OLS Assumption about the Expectation of u Given x To ensure unbiasedness of the OLS estimators, a key assumption is that the error term u has a zero conditional mean given x. This is expressed as:   E(u | x) = 0. This assumption implies that:  • The expected value of u is zero for any given value of x,  • There is no systematic relationship (or correlation) between the explanatory variable x and the error term u.  • In other words, the explanatory variable x is exogenous. This exogeneity condition is critical because if u were correlated with x, the OLS estimators would be biased and inconsistent, invalidating inference. ────────────────────────────── Summary a) The OLS estimators are:   β̂₁ = [Σ (xᵢ – x̄)(yᵢ – ȳ)] / [Σ (xᵢ – x̄)²]  and  β̂₀ = ȳ – β̂₁ x̄. b) The error term u represents all unobserved factors affecting y, such as omitted variables, measurement error, and random shocks. c) The required OLS assumption regarding the error term is that its conditional expectation given x is zero: E(u | x) = 0, ensuring that x is exogenous.