We have used a sample of 30 observations to fit a regression...

We have used a sample of 30 observations to fit a regression model. The full model has nine regressors, the variance estimate is $\hat{\sigma}^{2}=\mathrm{MS}_{\mathrm{E}}=100,$ and $R^{2}=0.92$ (a) Calculate the $F$ -statistic for testing significance of regression. Using $\alpha=0.05,$ what would you conclude? (b) Suppose that we fit another model using only four of the original regressors and that the error sum of squares for this new model is $2200 .$ Find the estimate of $\sigma^{2}$ for this new reduced model. Would you conclude that the reduced model is superior to the old one? Why? (c) Find the value of $C_{p}$ for the reduced model in part (b). Would you conclude that the reduced model is better than the old model?

We have used a sample of 30 observations to fit a regression model. The full model has nine regressors, the variance estimate is $\hat{\sigma}^{2}=\mathrm{MS}_{\mathrm{E}}=100,$ and $R^{2}=0.92$
(a) Calculate the $F$ -statistic for testing significance of regression. Using $\alpha=0.05,$ what would you conclude?
(b) Suppose that we fit another model using only four of the original regressors and that the error sum of squares for this new model is $2200 .$ Find the estimate of $\sigma^{2}$ for this
new reduced model. Would you conclude that the reduced model is superior to the old one? Why?
(c) Find the value of $C_{p}$ for the reduced model in part (b). Would you conclude that the reduced model is better than the old model?

Key Concepts

Model Selection and Comparison

Model selection involves choosing the most appropriate set of predictors that provide a balance between complexity and predictive power. It often includes comparing the full model against reduced models to determine whether simpler models might yield comparable or superior predictive performance without unnecessary complexity. Considerations typically include statistical tests, goodness-of-fit measures, and criteria that address both fit and parsimony.

Multiple Linear Regression

This is a statistical method used to model the relationship between a dependent variable and several independent variables. The approach involves estimating the coefficients of linear regressions so as to minimize the errors between the predicted and observed values, and it forms the basis for assessing the influence and significance of each predictor within the overall model context.

F-Statistic

The F-statistic is used in regression analysis to test the overall significance of the model. It compares the model with multiple regressors to a model with no predictors (intercept only) by examining the ratio of systematic variance explained by the model to the unsystematic variance. A large F-statistic indicates that the explanatory variables as a group provide a statistically significant explanation of the variability observed in the dependent variable.

Error Variance Estimation

Error variance, often estimated by the Mean Squared Error (MSE), represents the variability of the residuals around the regression line. It provides insight into the accuracy of the predictions made by the model. A smaller error variance suggests that the model’s predictions are closer to the actual data values, enhancing the model’s reliability.

Mallows' Cp

Mallows' Cp is a criterion often used in model selection to assess the trade-off between a model's complexity and its fit to data. It helps in identifying models that are neither overfitted nor underfitted by comparing the residual sum of squares adjusted for the number of predictors. A model with a Cp value close to or below the number of predictors plus the intercept indicates a potentially good model.

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