8- Nonlinear Regression Functions Regressor Stedent-tcacher ratio (STR) 1.720.69

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8- Nonlinear Regression Functions Regressor Stedent-tcacher ratio (STR) 1.720.69 (0.27) 12.4 1.02..-0.67. (0.27) (14.0) STR -0.680 (0.737) STR 0011 (0.013) % English learners 0.411 (0.306) 0303(0.300) 0.437 0.434 %English learners > median? (Binary, HiEL) -12.6 (9.8) 0.80 (056) HiEL × STR % Eligible for free lunch 0.521 -0582-0.587 0.709-0.653 (0077) (0.097) (0.104) (009) (0.72) District income (logarithm) 1653* District income -387 (2.49) -3.07 3.38 (2.49) (2.31) District income 0.165 0.164 (0.085) (0.089) (0.090) (0085) 0.174 0.184 District income 0.0022-0.0023 -00023-0.0022* (0.0010) (0.0010) (0.0010) (0.0010) 7396.. (8.6) .4.. (113) 665.5.. (81.3) 7599.. (23.2) 7474.. (20.3) 744.0.. (21.3) (Table 9.2 continard F-Statistics and p-Values Testing Exclusion of Groups of Variables (1 All STR variables and interactions 0 0.038)(0.020) 0.45 (0.641) 7.75 STR, STR Income Income HiEL. HiELx STR SER 7.74 (C0001) (0001) (0.003) (0.002) 5.85 158 (0.208) 8.62 0675 14.64 8.61 8.64 0.674 These regressions were estimated using the data ce Massachusetts clementary school districts described in Appendix 9.1. Stan- 0063 0.670 0676 0.675 dard errors are given ie parentheses under the coefficicets and p-valucs are given in parentheses under the Fstatistics Individ- ual coefficients are statistically significant at the 4% level-S kel

Explanation / Answer

15.
The variables tested for curvature versus linearity are -
i) STR (STR2, STR3)
ii) Income (Income2, Income3)

The two tests are

Null Hypothesis H0: The coefficients of STR2, STR3 (in regression 4) are equal to 0.

Alternative Hypothesis H1: The coefficients of STR2, STR3 (in regression 4) are not equal to 0.

Null Hypothesis H0: The coefficients of Income2, Income3 (in regression 3,4, 5 or 6) are equal to 0.

Alternative Hypothesis H1: The coefficients of Income2, Income3 (in regression 3,4, 5 or 6) are not equal to 0.

For the first test, the p-value is 0.641, which is greater than the significance level of 0.05. So we fail to reject the null hypothesis and conclude that at significance level of 0.05, there is no significant evidence that the coefficients of STR2, STR3 (in regression 4) are not equal to 0. So, there is a linear relationship between dependent variable and STR.

For the second test, the p-value for all regressors (3,4,5 or 6), is less than the significance level of 0.05. So we reject the null hypothesis and conclude that at significance level of 0.05, there is significant evidence that the coefficients of Income2, Income3 (in regression 3,4, 5 or 6) are not equal to 0. So, there is a non-linear(curvature) relationship between dependent variable and Income.

16.

The difference between regressions 2 and 3 is the use of transformation of Income variable.

Regression 2 uses log(Income) as one of the predictor variable.

Regression 3 uses Income, Income2 and Income3 as predictor variables.

R2 for regression 2 is 0.670 whereas R2 for regression 3 is 0.676

Standard error for regression 2 is 8.69 whereas Standard error for regression 3 is 8.61

The two statistics generally used for evaluating the model are R2 and standard error of the model.

As, R2 for regression model 3 is higher than the model 2, the proportion of variation of dependent variable explained by model 3 is greater than that of model 2.

As, standard error of regression model 3 is lower than the model 2, the model 3 has fitted the data well as compared to model 2.

So, model 3 is better regression model than model 2.

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