Explain why an error term is added to the regression model. What assumptions are made about the error term? What are the implications of such assumptions? What will happen to the estimators of the parameters of the regression model, if these assumptions are violated

The error term ((u_i)) in a regression model is introduced to account for unobserved factors that affect the dependent variable ((Y_i)) but are not included in the model.

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It represents the difference between the observed values and the values predicted by the model. The inclusion of the error term acknowledges that there are factors beyond the control of the model that influence the dependent variable.

Assumptions about the error term in the classical linear regression model include:

1. Linearity: The relationship between the independent variable ((X_i)) and the expected value of the dependent variable ((E(Y_i))) is linear.
2. Independence: The errors are independent of each other, meaning the error in one observation does not influence the error in another.
3. Homoscedasticity: The variance of the errors is constant across all levels of the independent variable.
4. Normality: The errors are normally distributed.

Violation of these assumptions can have several implications:

• Bias in Parameter Estimates: If the assumptions are violated, the estimators of the regression model parameters ((\alpha) and (\beta)) may be biased, meaning they may systematically deviate from the true population values.
• Inefficiency: Violation of assumptions may lead to less precise (inefficient) estimates, reducing the reliability of the statistical inferences drawn from the model.
• Incorrect Inference: Inference tests, such as hypothesis tests and confidence intervals, rely on these assumptions. If violated, the results of these tests may be invalid.
• Inconsistent Results: In the presence of serious violations, the estimators may become inconsistent, meaning they may not converge to the true values as the sample size increases.

To mitigate these issues, it’s crucial to check the assumptions of the regression model and, if necessary, consider alternative modeling approaches or transformations to address the violations. Various diagnostic tests and methods, such as residual analysis, can be employed to assess the validity of these assumptions.