QUESTION SIX Clearly explain the four unbiased OLS assumptions of the simple linear regression (25 marks)

In simple linear regression, the Ordinary Least Squares (OLS) method is used to estimate the parameters of the linear model. For the OLS estimators to be considered unbiased, certain assumptions must be met. Here are the four key assumptions: 1. **Linearity**: - The relationship between the independent variable (X) and the dependent variable (Y) is linear. This means that the expected value of Y given X is a linear function of X. Mathematically, this can be expressed as: \[ E(Y|X) = \beta_0 + \beta_1 X \] - If this assumption is violated, the OLS estimates may be biased because the model does not accurately represent the relationship between the variables. 2. **Independence**: - The observations are independent of each other. This means that the value of the dependent variable for one observation does not influence the value of the dependent variable for another observation. In time series data, this assumption can be violated if there is autocorrelation, where residuals are correlated across time. - Independence is crucial because if the observations are correlated, it can lead to underestimating the standard errors, resulting in misleading statistical inferences. 3. **Homoscedasticity**: - The variance of the errors (residuals) is constant across all levels of the independent variable. This means that the spread of the residuals should be roughly the same for all predicted values of Y. If the variance of the errors changes (a condition known as heteroscedasticity), it can lead to inefficient estimates and affect the validity of hypothesis tests. - Homoscedasticity can be visually assessed using residual plots, where the residuals should show no clear pattern. 4. **No Perfect Multicollinearity**: - In the context of simple linear regression, this assumption states that the independent variable should not be a perfect linear function of another variable. In simple linear regression, this is less of a concern since there is only one independent variable. However, in multiple regression, perfect multicollinearity occurs when two or more independent variables are perfectly correlated, making it impossible to isolate the effect of each variable on the dependent variable. - This assumption ensures that the OLS estimates can be uniquely determined. In summary, these four assumptions—linearity, independence, homoscedasticity, and no perfect multicollinearity—are essential for ensuring that the OLS estimators are unbiased and that the results of the regression analysis are valid. Violations of these assumptions can lead to biased estimates, incorrect standard errors, and unreliable hypothesis tests.