What are the assumptions underlying linear regression? What occurs if any of these assumptions are not met?

In this question, the interviewer is seeking insights into two separate aspects that you need to cover. First, you should discuss the fundamental assumptions that form the basis of a linear regression model and, consequently, how they affect the model's functioning. Next, you need to outline the implications of breaching these assumptions, specifically what occurs when real-world data does not conform to them. Some important points to include in your response might be: Identifying and clarifying the primary assumptions of linear regression. Explaining the effects or issues that may arise if any of these assumptions are not met.

Linear regression relies on five key assumptions: Linearity : A linear relationship exists between the independent and dependent variables. Independence : The residuals are independent, meaning there is no correlation among consecutive residuals in time series data. Homoscedasticity : The residuals maintain a constant variance. Normality : The model's residuals follow a normal distribution. No multicollinearity : The independent variables are not excessively correlated with one another. When any of these assumptions are violated, it can result in inaccurate predictions or misinterpretations. For example, if the assumption of linearity is not met, using a linear model on the data will yield misleading results, as it does not accurately represent the relationship. A breach of the independence assumption can lead to erroneous standard errors and unreliable estimates. Heteroscedasticity, which violates homoscedasticity, may result in the regression model producing biased and inefficient estimates, thereby impacting hypothesis testing. Additionally, violating the normality assumption can undermine the reliability of certain tests and confidence intervals. Multicollinearity can complicate the estimation of the individual relationships between each independent variable and the dependent variable.

Linear regression is based on the following assumptions: Linear relationship : There exists a linear connection between the predictors and the response variable. Independence of errors : The errors produced by the model are not correlated with one another. Equal Variances : The error variances remain consistent across all levels of the independent variables. Normality of errors : The errors of the model are normally distributed. Lack of perfect multicollinearity : The predictors are not perfectly correlated with one another. If these assumptions are violated, it can lead to inefficiencies, biases, and inconsistencies in the estimation of regression coefficients. A non-linear relationship may cause either an underestimation or overestimation of the predictors' effects. Correlated errors can distort our standard errors, potentially resulting in inaccurate inferences. Variance differences in errors can produce unreliable confidence intervals, while non-normal errors mainly impact hypothesis testing. The presence of multicollinearity makes it challenging to discern the individual effects of each predictor.