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R 2 and model fit. Because the coefficient of
Chapter 12, Problem 22E(choose chapter or problem)
\(R^2\) and model fit. Because the coefficient of determination always increases when a new independent variable is added to the model, it is tempting to include many variables in a model to force \(R^2\) to be near 1 . However, doing so reduces the degrees of freedom available for estimating \(\sigma^2\), which adversely affects our ability to make reliable inferences. Suppose you want to use 18 economic indicators to predict next year's gross domestic product (GDP). You fit the model
\(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\cdots+\beta_{17} x_{17}+\beta_{18} x_{18}+\varepsilon\)
where y = GDP and \(x_1,\ x_2,\ .\ .\ .\ ,\ x_{18}\) are the economic indicators. Only 20 years of data (n = 20) are used to fit the model, and you obtain \(R^2\ =\ .95\). Test to see whether this impressive-looking \(R^2\) is large enough for you to infer that the model is useful-that is, that at least one term in the model is important for predicting GDP. Use \(\alpha\ =\ .05\).
Questions & Answers
QUESTION:
\(R^2\) and model fit. Because the coefficient of determination always increases when a new independent variable is added to the model, it is tempting to include many variables in a model to force \(R^2\) to be near 1 . However, doing so reduces the degrees of freedom available for estimating \(\sigma^2\), which adversely affects our ability to make reliable inferences. Suppose you want to use 18 economic indicators to predict next year's gross domestic product (GDP). You fit the model
\(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\cdots+\beta_{17} x_{17}+\beta_{18} x_{18}+\varepsilon\)
where y = GDP and \(x_1,\ x_2,\ .\ .\ .\ ,\ x_{18}\) are the economic indicators. Only 20 years of data (n = 20) are used to fit the model, and you obtain \(R^2\ =\ .95\). Test to see whether this impressive-looking \(R^2\) is large enough for you to infer that the model is useful-that is, that at least one term in the model is important for predicting GDP. Use \(\alpha\ =\ .05\).
ANSWER:Step 1 of 2
The multiple coefficient of determination, 
using the independent variables
total sample variation (measured by 