What is the extra assumption for multiple linear regression?

What is the extra assumption for multiple linear regression?

• A. The independent variables must be normally distributed.
• B. The independent variables can be highly correlated.
• C. The independent variables do not have measurement error.
• D. The independent variables are not highly correlated.

Answer: (D)

The key thing being tested is multicollinearity. In multiple linear regression, the predictors should not be highly correlated with one another because each predictor is supposed to contribute unique information about the outcome. When two or more predictors are very similar, their effects on the outcome become hard to separate, which inflates the standard errors of the estimated coefficients and makes them unstable or unreliable to interpret. In extreme cases, perfect or near-perfect correlation can cause the design matrix to be nearly singular, making the model difficult or impossible to estimate. So the extra assumption is that the independent variables are not highly correlated. This ensures each predictor’s contribution is identifiable and the coefficient estimates are stable. By contrast, having highly correlated predictors doesn’t fit this requirement, and normal distribution of the predictors is not a needed condition for ordinary least squares; normality concerns residuals for inference, not the predictors themselves. Measurement error in predictors can bias estimates, but the classical emphasis in this context is on avoiding multicollinearity to keep estimates interpretable and precise.

The key thing being tested is multicollinearity. In multiple linear regression, the predictors should not be highly correlated with one another because each predictor is supposed to contribute unique information about the outcome. When two or more predictors are very similar, their effects on the outcome become hard to separate, which inflates the standard errors of the estimated coefficients and makes them unstable or unreliable to interpret. In extreme cases, perfect or near-perfect correlation can cause the design matrix to be nearly singular, making the model difficult or impossible to estimate.

So the extra assumption is that the independent variables are not highly correlated. This ensures each predictor’s contribution is identifiable and the coefficient estimates are stable. By contrast, having highly correlated predictors doesn’t fit this requirement, and normal distribution of the predictors is not a needed condition for ordinary least squares; normality concerns residuals for inference, not the predictors themselves. Measurement error in predictors can bias estimates, but the classical emphasis in this context is on avoiding multicollinearity to keep estimates interpretable and precise.