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

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The error term in a regression model captures the unobserved factors that affect the dependent variable but are not explicitly included in the model.

It accounts for the discrepancies between the observed values of the dependent variable and the values predicted by the regression equation. The inclusion of the error term acknowledges that there are other factors influencing the dependent variable that are not explicitly accounted for in the model.

Assumptions about the Error Term:

  1. Zero Mean: The error term has a population mean of zero, indicating that, on average, the model is correctly specified.
  2. Independence: The errors are independent of each other, meaning that the occurrence of an error for one observation does not provide information about the occurrence of an error for another observation.
  3. Homoscedasticity: The variance of the error term is constant across all levels of the independent variables. This assumption ensures that the spread of errors is consistent throughout the range of the independent variable.
  4. Normality: The errors are normally distributed. While this assumption is not crucial for large sample sizes due to the Central Limit Theorem, it facilitates the use of statistical inference techniques.
  5. Linearity: The relationship between the independent variables and the dependent variable is linear.

Implications of Assumptions:

  1. Efficiency of Estimators: When the assumptions are met, the Ordinary Least Squares (OLS) estimators of the regression coefficients are unbiased, efficient, and have the minimum variance among the class of linear unbiased estimators.
  2. Validity of Inference: Assumptions about the error term are crucial for valid statistical inference, including hypothesis testing and confidence intervals.
  3. Reliability of Predictions: Meeting the assumptions enhances the reliability of predictions made by the regression model.

Consequences of Violating Assumptions:

  1. Biased Estimates: If the assumptions are violated, OLS estimators may become biased, leading to inaccurate estimates of the population parameters.
  2. Inefficient Estimation: Violations can result in inefficient estimates, meaning that the estimators may have larger variances than they would if the assumptions were met.
  3. Invalid Inference: Hypothesis tests and confidence intervals may become invalid, undermining the reliability of statistical inferences.
  4. Model Misinterpretation: Violations of assumptions may indicate that the specified model is not an accurate representation of the underlying data-generating process.
  5. Heteroscedasticity: Violation of the homoscedasticity assumption can lead to inefficiency in estimating standard errors and affect the validity of statistical tests.
  6. Serial Correlation: If errors are correlated across observations, it can lead to inefficient estimates and affect the precision of hypothesis tests.

Handling Violations:

  1. Transformations: Nonlinear transformations of variables may help address violations of linearity.
  2. Weighted Least Squares: If heteroscedasticity is present, weighted least squares can be used to account for varying variances.
  3. Robust Standard Errors: Robust standard errors can be employed to mitigate the impact of heteroscedasticity on inference.
  4. Residual Analysis: Diagnostic checks, such as residual analysis, can help identify and address violations of assumptions.

It is essential to conduct thorough diagnostics and consider alternative modeling approaches if assumptions are violated, as failure to do so may compromise the validity and reliability of the regression analysis.