In simple linear regression, what are the common residual assumptions?

In simple linear regression, what are the common residual assumptions?

• A. Linearity, independence, homoscedasticity, and normality of residuals.
• B. Linearity, independence, heteroscedasticity, and normality of residuals.
• C. Nonlinearity, dependence, homoscedasticity, and non-normal residuals.
• D. Linearity, autocorrelation, unequal variance, and non-normal residuals.

Answer: (A)

Residuals in simple linear regression are expected to be random around zero with no systematic pattern and with constant spread across all levels of X. The standard assumptions reflect this: the relationship between X and Y is linear, the residuals are independent from one observation to the next, their variance does not change with the level of X (homoscedasticity), and they are approximately normally distributed. This combination ensures the ordinary least squares estimates are unbiased and efficient, and it underpins the validity of usual inference procedures like t-tests and F-tests in small samples. If any of these fail—nonlinearity suggests the model is mis-specified, dependence or autocorrelation indicates correlated residuals, heteroscedasticity means unequal variance, or non-normal residuals—the usual inference can be distorted. So the best description of common residual assumptions is linearity, independence, homoscedasticity, and normality of residuals.

Residuals in simple linear regression are expected to be random around zero with no systematic pattern and with constant spread across all levels of X. The standard assumptions reflect this: the relationship between X and Y is linear, the residuals are independent from one observation to the next, their variance does not change with the level of X (homoscedasticity), and they are approximately normally distributed. This combination ensures the ordinary least squares estimates are unbiased and efficient, and it underpins the validity of usual inference procedures like t-tests and F-tests in small samples. If any of these fail—nonlinearity suggests the model is mis-specified, dependence or autocorrelation indicates correlated residuals, heteroscedasticity means unequal variance, or non-normal residuals—the usual inference can be distorted. So the best description of common residual assumptions is linearity, independence, homoscedasticity, and normality of residuals.