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 $$ x $$

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Below are detailed answers to each part of the question:

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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.

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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.

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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.

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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.
a) The OLS estimators for β₀ and β₁ are: - β₁̂ = (Σ(xᵢ – x̄)(yᵢ – ȳ)) / (Σ(xᵢ – x̄)²) - β₀̂ = ȳ – β₁̂ · x̄ b) $$ u $$ is the error term, representing factors not included in the model that affect y, such as omitted variables and random disturbances. c) The OLS assumption is that the expected value of the error term $$ u $$ given x is zero, meaning $$ E(u | x) = 0 $$.

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