# Appendix: The Regression Model and Detailed Results

The empirical results reported in this study are obtained by estimating the following baseline regression equation using school-level panel data that varies by school and time:

(Education Outcomes)sy= β01 ln?(spending per pupil)sy2 (% free or reduced lunch)sy3 ln(grade enrollment)sy + γsysy.

The variables are denoted with school and time subscripts indicating that a particular variable is for school s in year y. The main predictor variable of academic outcomes is constructed as the natural logarithm of annual school-level total spending divided by annual school-level total student enrollment. The purpose of transforming spending per pupil into natural logarithms is to make the relationship between academic outcomes and spending per pupil linear, since it is possible that the statistical relationship between these two variables is nonlinear. In addition, academic outcomes are further modeled as functions of the percentage of students eligible for a free or reduced-price lunch and the number of students enrolled in a specific grade level.

Different measures of education outcomes represent the dependent variables in the equation. These outcomes are measures of performances of for all students who took the MEAP subject tests for grades three through eight, MME and ACT subject tests in grade 11 and the high school graduation rates of all students.

β0 is the constant term, εsy is the error term, αy is a year effect and γs is a school fixed effect. These fixed-effects terms are included in the regression equation to account for the effects of predictor variables on academic outcomes that are omitted in the regression equation. For example, measures of school quality and economic conditions for the state of Michigan are not included in the regression equation, but they have an impact on academic outcomes and so are accounted for via the fixed-effect terms. The fixed-effects terms remove both the school-specific means and year-specific means from the variables in the equation.

Finally, the equation is estimated considering that the schools are grouped into districts. If the error terms for different schools within a given district are correlated, then this could lead to relatively small standard errors being estimated and, therefore, misleading statistical inferences. Therefore, to control for this statistical issue, the estimated standard errors in the regression equation are clustered at the district level.

As is standard in reporting statistical results, a figure with one asterisk signifies statistical significance at a 90 percent level of confidence, two asterisks at a 95 percent level of confidence and three asterisks at a 99 percent level of confidence.

 Academic Indicator Grade MEAP: Math 3 4 5 6 7 8 Coefficient Estimate Spending per pupil 1.297 0.505 2.514 -0.714 0.574* 0.243 -0.908 -1.596 -1.739 -0.447 -0.346 -0.432 Pct FRL 4.962 7.11 5.338 -1.841 -1.925** -1.892** -4.383 -6.416 -4.935 -1.537 -0.912 -0.882 Grade enrollment -8.949*** -1.696 -14.89*** -10.14*** -13.00*** -9.504*** -2.128 -5.23 -3.022 -2.515 -2.499 -2.465 Additional Information N 6,626 6,434 5,327 1,896 2,274 2,256 Adjusted R-squared 0.89 0.91 0.884 0.889 0.908 0.908

 Grade MEAP: Reading 3 4 5 6 7 8 Coefficient Estimate Spending per pupil 0.894 0.035 1.419 0.514 0.386 -0.249 -0.952 -1.638 -1.324 -0.4 -0.513 -0.381 Pct FRL 6.758 6.832 6.471 -0.924 -2.799** -1.972** -4.733 -6.586 -4.881 -1.349 -1.403 -0.775 Grade enrollment -8.747*** -2.336 -9.735*** -6.001*** -12.06*** -6.674*** -2.196 -5.11 -2.564 -1.848 -2.776 -1.926 Additional Information N 6,628 6,435 5,328 1,896 2,274 2,256 Adjusted R-squared 0.892 0.907 0.883 0.892 0.865 0.896

 Grade MEAP: Science 5 8 Coefficient Estimate Spending per pupil 2.013 -0.143 -1.43 -0.511 Pct FRL 4.486 -2.614*** -4.394 -0.829 Grade enrollment -13.46*** -8.318*** -2.904 -2.207 Additional Information N 5,327 2,256 Adjusted R-squared 0.9 0.909

 Subject MME Math Reading Science Social Studies Writing Coefficient Estimate Spending per pupil 0.46 -0.535 0.304 0.559 0.259 -1.221 -0.876 -1.111 -0.501 -1.131 Pct FRL -1.323 -1.966* -1.386 0.213 -1.425 -1.165 -1.179 -1.189 -0.622 -1.343 Grade enrollment -11.05*** -4.274* -7.020** -2.139 -3.943 -2.237 -2.501 -2.784 -1.733 -2.792 Additional Information N 3,096 3,102 3,101 3,101 3,108 Adjusted R-squared 0.919 0.908 0.923 0.928 0.912

 Test ACT Composite English Math Reading Science Writing Coefficient Estimate Spending per pupil 0.0027 0.0091 0.0084 -0.049 0.031 0.064 -0.074 -0.102 -0.074 -0.1 -0.081 -0.041 Pct FRL -0.105 -0.117 -0.041 -0.029 -0.170* -0.092 -0.095 -0.126 -0.076 -0.124 -0.102 -0.084 Grade enrollment -0.711*** -0.804*** -0.814*** -0.748*** -0.482* -0.221 -0.215 -0.282 -0.162 -0.274 -0.254 -0.186 Additional Information N 3,562 3,562 3,562 3,562 3,562 2,557 Adjusted R-squared 0.952 0.935 0.952 0.931 0.937 0.865

 Graduation Rate 4-year 5-year 6-year Coefficient Estimate Spending per pupil 0.58 0.042 0.494 -0.7 -0.687 -0.758 Pct FRL 1.627 1.493 0.938 -0.988 -1.103 -1.288 Grade enrollment -9.997** -3.09 3.251 -3.924 -2.751 -3.53 Additional Information N 3,880 3,364 2,802 Adjusted R-squared 0.909 0.925 0.917

The graphics below include scatterplots of regression results. The x-axis displays a school’s change in per-pupil spending and the y-axis shows a school’s change in average scale scores. If school spending were positively correlated in a statistically significant way to student achievement, the blots on these graphs would appear to follow a line slopping upwards from the bottom left to the upper right. The results, however, show much more vertical lines, suggesting that the amount of money that schools spend have little correlation to how well students perform.

Graphic 6: MEAP Math Test Results, Grades 3-5, 2008-2013

Graphic 7: MEAP Math Test Results, Grades 6-8, 2008-2013

Graphic 8: MEAP Science Test Results, Grades 5 and 8, 2008-2013

Graphic 9: MME Test Results by Subject, 11th Grade, 2008-2013

Graphic 10: ACT Test Results by Subject, 11th Grade, 2007-2013