We are interested in the effect on test scores of the student-teacher ratio. The table below...

  

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We are interested in the effect on test scores of the student-teacher ratio. The table below shows different regression ((1) to (5)) results with different regressors. All the five regressions use average test score as dependent variable and include the student-teacher ratio (X₁), as the variable of interest. Then different combinations of variables refer to the first column for variable names and what they measure) are used in each regression. Significance of each variable is indicated by * ( significant at the 5% level) or ** (significant at the 1% level). Dependent variable: average test score in the district. Regressor (1) Student-teacher ratio (X₁) -2.28⁰ (0.52) Percent English learners (X₂) Percent eligible for subsidized lunch (X3) Percent on public income assistance (X₂) Intercept Summary Statistics SER R² 698.9** (10.4) 11 18.58 0.049 420 (2) -1.10 (0.43) -0.650** (0.031) 686.0** (8.7) 14.46 (3) -1.00** (0.27) -0.122 (0.033) -0.547** (0.024) 700.2** (5.6) 9.08 0.773 420 (4) -1.31** (0.34) -0.488 (0.030) -0.790⁰* (0.068) 698.0** (6.9) 11.65 0.626 420 (5) -1.01. (0.27) -0.130** (0.036) -0.529** (0.038) 0.048 (0.059) 0.424 420 These regressions were estimated using the data on K-8 school districts in California, described in Appendix 4.1. Heteroskedas- ticity-robust standard errors are given in parentheses under coefficients. The individual coefficient is statistically significant at the *5% level or **1% significance level using a two-sided test. 700.4** (5.5) 9.08 0.773 420 (a) Consider (4) and (5). Note that X4 is added in both, but X3 is omitted in (4). i. Are there any bias-correction benefits of adding X3 in regression (5)? Examine the bias-correction benefit for X₁, X2, and X₁. ii. Is there a bias-variance tradeoff in estimating the coefficient of X₁ if X3 is added in regression (5)? Why or why not? (b) Consider (3) and (5). Note that X₁ is omitted in regression (3). i. Are there bias-correction benefits of adding X4 in regression (5)? Ex- amine the bias-correction benefit for X₁, X2, and X3. ii. Is there a bias-variance tradeoff in estimating the coefficients of X₁, X2, and X3 if X3 is added in regression (5)? Why or why not? We are interested in the effect on test scores of the student-teacher ratio. The table below shows different regression ((1) to (5)) results with different regressors. All the five regressions use average test score as dependent variable and include the student-teacher ratio (X₁), as the variable of interest. Then different combinations of variables refer to the first column for variable names and what they measure) are used in each regression. Significance of each variable is indicated by * ( significant at the 5% level) or ** (significant at the 1% level). Dependent variable: average test score in the district. Regressor (1) Student-teacher ratio (X₁) -2.28⁰ (0.52) Percent English learners (X₂) Percent eligible for subsidized lunch (X3) Percent on public income assistance (X₂) Intercept Summary Statistics SER R² 698.9** (10.4) 11 18.58 0.049 420 (2) -1.10 (0.43) -0.650** (0.031) 686.0** (8.7) 14.46 (3) -1.00** (0.27) -0.122 (0.033) -0.547** (0.024) 700.2** (5.6) 9.08 0.773 420 (4) -1.31** (0.34) -0.488 (0.030) -0.790⁰* (0.068) 698.0** (6.9) 11.65 0.626 420 (5) -1.01. (0.27) -0.130** (0.036) -0.529** (0.038) 0.048 (0.059) 0.424 420 These regressions were estimated using the data on K-8 school districts in California, described in Appendix 4.1. Heteroskedas- ticity-robust standard errors are given in parentheses under coefficients. The individual coefficient is statistically significant at the *5% level or **1% significance level using a two-sided test. 700.4** (5.5) 9.08 0.773 420 (a) Consider (4) and (5). Note that X4 is added in both, but X3 is omitted in (4). i. Are there any bias-correction benefits of adding X3 in regression (5)? Examine the bias-correction benefit for X₁, X2, and X₁. ii. Is there a bias-variance tradeoff in estimating the coefficient of X₁ if X3 is added in regression (5)? Why or why not? (b) Consider (3) and (5). Note that X₁ is omitted in regression (3). i. Are there bias-correction benefits of adding X4 in regression (5)? Ex- amine the bias-correction benefit for X₁, X2, and X3. ii. Is there a bias-variance tradeoff in estimating the coefficients of X₁, X2, and X3 if X3 is added in regression (5)? Why or why not?

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