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2. Consider two least-squares regressions y = Xıı + è

2. Consider two least-squares regressions y = Xıı + è

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Least-Squares Regressions Analysis **Problem Statement:** Consider two least-squares regressions: 1. \( y = X_1\hat{\beta}_1 + \bar{e} \) 2. \( y = X_1\hat{\beta}_1 + X_2\hat{\beta}_2 + \hat{e} \) Let \( R_1^2 \) and \( R_2^2 \) be the R-squared values from the two regressions. Show that \( R_2^2 \geq R_1^2 \). **Explanation:** This problem involves understanding how adding additional explanatory variables in a regression model affects the R-squared value. The R-squared value (\( R^2 \)) provides a measure of how well the independent variables explain the variability in the dependent variable. Increasing the number of independent variables generally cannot decrease the R-squared value, and it often increases it. This is because adding more variables either explains additional variance or results in no change at all if the variables are not significant. **Details:** 1. **First Regression:** \( y = X_1\hat{\beta}_1 + \bar{e} \) Here, the dependent variable \( y \) is regressed on a single independent variable \( X_1 \) with estimated coefficient \( \hat{\beta}_1 \) and error term \( \bar{e} \). The R-squared value for this regression is denoted as \( R_1^2 \). 2. **Second Regression:** \( y = X_1\hat{\beta}_1 + X_2\hat{\beta}_2 + \hat{e} \) In this case, the dependent variable \( y \) is regressed on two independent variables \( X_1 \) and \( X_2 \) with estimated coefficients \( \hat{\beta}_1 \) and \( \hat{\beta}_2 \) respectively, and error term \( \hat{e} \). The R-squared value for this regression is denoted as \( R_2^2 \). **Objective:** To demonstrate that adding an additional variable \( X_2 \) in the regression model will result in an R-squared value (\( R_2^2 \)) that is at least as large as the R-squared value (\(
Transcribed Image Text:### Least-Squares Regressions Analysis **Problem Statement:** Consider two least-squares regressions: 1. \( y = X_1\hat{\beta}_1 + \bar{e} \) 2. \( y = X_1\hat{\beta}_1 + X_2\hat{\beta}_2 + \hat{e} \) Let \( R_1^2 \) and \( R_2^2 \) be the R-squared values from the two regressions. Show that \( R_2^2 \geq R_1^2 \). **Explanation:** This problem involves understanding how adding additional explanatory variables in a regression model affects the R-squared value. The R-squared value (\( R^2 \)) provides a measure of how well the independent variables explain the variability in the dependent variable. Increasing the number of independent variables generally cannot decrease the R-squared value, and it often increases it. This is because adding more variables either explains additional variance or results in no change at all if the variables are not significant. **Details:** 1. **First Regression:** \( y = X_1\hat{\beta}_1 + \bar{e} \) Here, the dependent variable \( y \) is regressed on a single independent variable \( X_1 \) with estimated coefficient \( \hat{\beta}_1 \) and error term \( \bar{e} \). The R-squared value for this regression is denoted as \( R_1^2 \). 2. **Second Regression:** \( y = X_1\hat{\beta}_1 + X_2\hat{\beta}_2 + \hat{e} \) In this case, the dependent variable \( y \) is regressed on two independent variables \( X_1 \) and \( X_2 \) with estimated coefficients \( \hat{\beta}_1 \) and \( \hat{\beta}_2 \) respectively, and error term \( \hat{e} \). The R-squared value for this regression is denoted as \( R_2^2 \). **Objective:** To demonstrate that adding an additional variable \( X_2 \) in the regression model will result in an R-squared value (\( R_2^2 \)) that is at least as large as the R-squared value (\(
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