Question

Discuss the following statements and explain why they are true or false.
(a) Increasing the number of predictor variables will never decrease the R2.
(b) Multicollinearity affects the interpretation of the regression coefficients.
(c) The variance inflation factor of ˆ βj depends on the R2 of the regression of the response variable y on the predictor variable xj.
(d) A high leverage point is always highly influential.
(e) All criteria for the selection of the best regression equation lead to the same set of predictor variables.

Answer 1

(a) Increasing the number of predictor variables will never decrease the R2.

True

Every time you add a predictor to a model, the R-squared increases, even if due to chance alone. It never decreases. Consequently, a model with more terms may appear to have a better fit simply because it has more terms.

(b) Multicollinearity affects the interpretation of the regression coefficients.

True

Multicollinearity makes it hard to interpret your coefficients, and it reduces the power of your model to identify independent variables that are statistically significant. Multicollinearity affects only the specific independent variables that are correlated.

(c) The variance inflation factor of ˆ βj depends on the R2 of the regression of the response variable y on the predictor variable xj.

True

The standard error of ˆβj depends on: ... SE(ˆβj) is large. 1. 1−R2 j is called the variance-inflation factor.

(d) A high leverage point is always highly influential.

False

Although an influential point will typically have high leverage, a high leverage point is not necessarily an influential point.
(e) All criteria for the selection of the best regression equation lead to the same set of predictor variables.

False

The best regression equation is dependent on the statistical significance of the involved variables.

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