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

An error term (often denoted as 𝑢ᵢ) is added to a regression model for several important reasons:

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1. **Unexplained Variation**: In a real-world dataset, the relationship between the independent variable(s) and the dependent variable may not be perfectly deterministic. There are typically other unobservable factors that influence the dependent variable. The error term captures this unexplained or random variation.

• **Modeling Imperfections**: Regression models aim to capture the average or expected relationship between variables. However, they cannot perfectly account for every data point’s exact value. The error term accounts for the discrepancy between the predicted value (based on the model) and the actual observed value.

Assumptions about the error term (𝑢ᵢ) in a linear regression model include:

1. **Independence**: The errors are assumed to be independent of each other. This means that the error for one observation does not depend on the errors for other observations. Violation of this assumption can lead to biased parameter estimates and incorrect standard errors.

• **Zero Mean**: The expected value of the error term is zero, which means that on average, the model is correct. In other words, the model is not systematically over- or under-predicting the dependent variable.

• **Constant Variance (Homoscedasticity)**: The errors have constant variance for all values of the independent variable(s). If this assumption is violated (heteroscedasticity), it can affect the efficiency of parameter estimates, making some less reliable than others.

• **Normality**: The error term is assumed to follow a normal distribution. This assumption is important mainly for hypothesis testing and constructing confidence intervals for parameter estimates. Departure from normality may affect the accuracy of statistical tests but may not necessarily bias parameter estimates.

Implications of these assumptions:

• If the assumptions about the error term are met, the Ordinary Least Squares (OLS) estimators for the parameters (𝛼 and 𝛽) are the Best Linear Unbiased Estimators (BLUE). This means that OLS provides efficient and unbiased estimates of the model parameters.

– Violation of these assumptions can lead to biased parameter estimates. For example:

  – Independence violation can lead to biased and inefficient estimators.

  – Non-zero mean of the errors can lead to bias in parameter estimates.

  – Heteroscedasticity can lead to inefficient estimates, making some coefficients less reliable.

  – Departure from normality can affect hypothesis testing results but might not necessarily bias parameter estimates.

In practice, it’s essential to diagnose potential violations of these assumptions using techniques like residual analysis, and if violations are severe, consider alternative regression methods or transformations to address the issues. Failure to account for these assumptions can lead to inaccurate model results and unreliable inferences.