Discuss the assumptions underlying linear regression, and elaborate on their relevance for accurately interpreting model output.

Key Assumptions of Linear Regression:

1. Linearity:Assumption: The relationship between the independent variables and the dependent variable is linear. Relevance: If this assumption is violated (e.g., the relationship is quadratic or logarithmic), the model may provide biased estimates and poor predictions. Non-linear relationships can lead to systematic errors.
2. Independence:Assumption: The residuals (errors) are independent of each other. Relevance: If the residuals are correlated (e.g., in time series data), it can lead to underestimated standard errors and misleading significance tests. This violates the assumption of independence and can produce unreliable predictions.
3. Homoscedasticity:Assumption: The variance of the residuals is constant across all levels of the independent variables. Relevance: If heteroscedasticity (non-constant variance) is present, it can lead to inefficient estimates and affect the reliability of hypothesis tests. It can indicate that the model may not adequately capture the relationship or that there are omitted variables.
4. Normality of Residuals:Assumption: The residuals are normally distributed. Relevance: While normality is not strictly necessary for estimating the coefficients, it is important for conducting hypothesis tests (like t-tests for coefficients). Non-normal residuals can lead to incorrect conclusions about the significance of the predictors.
5. No Multicollinearity:Assumption: Independent variables are not too highly correlated with each other. Relevance: Multicollinearity can inflate the standard errors of the coefficients, making it difficult to determine the individual effect of each predictor. This can lead to unreliable estimates and difficulty in interpreting the model.
6. No Autocorrelation:Assumption: The residuals are not correlated with each other, especially in time series data. Relevance: Autocorrelation can indicate that important variables or time effects are missing from the model, leading to biased estimates and inefficient predictions.