This memorandum addresses questions and concepts raised by Dr
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- Bu səhifədəki naviqasiya:
- SBE Staff Questions Then vs. Now Approach
- This is not an appropriate comparison in that the county model utterly fails to measure SYSTEM level fiscal capacity.
- Proportion of Observed Zero Values 2. What is the impact of the prototype variables that contain a substantial proportion of zero values
- 2.a. Do these variables accurately capture the observed values of Tennessee’s largest counties within a 95% confidence interval
- 2. b. What is the impact of 0 values on the ability of the model to accurately predict the outcomes within a 95-percent confidence interval
- 2. c. What is the impact of potential outliers on the observed weights
- Statistical Treatment of Recurring Outliers 3. a. What is an acceptable benchmark regarding the overall percentage of potential outliers within an observed model
- 3. b. How would the identified weights change if the outliers were removed
- Elimination of Res-Farm Percent and ADM Population Ratio
- Dollar Values of Coefficients Multiplied by State Average Variables County Model vs. Prototype Model, FY 2005
- Attachment: Combined Variable Methodology Combined Property Tax Base per Pupil
- Combined Sales Tax Base per Pupil
- Combined Exportability
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Responses to State Board of Education Staff Questions – Prototype System Level Fiscal Capacity Model Background This handout addresses questions and concepts raised by Dr. Gary Nixon, Executive Director, and Mr. Art Fuller, staff, of the State Board of Education during a meeting with Cliff Lippard, Director of Fiscal Affairs, TACIR, on February 2, 2005. Mr. Fuller had prepared a series of questions based upon his statistical comparison of the 95 county model and the 136 prototype system model. This handout also addresses a request by Dr. Nixon that TACIR examine the possibility of collapsing shared and unshared variables into single variables. The first and most important test for a fiscal capacity model must be whether it does what it is intended to do, which is estimate the fiscal capacity for EACH system in the state. The prototype model, by estimating capacity for all 136 systems, succeeds in this test where the county model fails. There is a structural flaw with the current county model in that it estimates the fiscal capacity for Tennessee’s 95 county areas, not the state’s 136 school systems. Nearly 1/3 of our school systems were not accounted for in the model. In fact, only the 67 single system counties are properly accounted for in the county model. The fiscal capacity indices that it produces are then used to equalize funding in the BEP, a funding system that is based upon all 136 systems. If sufficient data had been available from the onset, the fiscal capacity model would already be a system level model. It is important to keep in mind the logic of the model, which is expressed by the following eight principles:
SBE Staff Questions Then vs. Now Approach 1. Strictly from a statistical perspective, are the observed statistical indicators related to the soundness of the model improving when analyzing and comparing prototype variables. The 95 county model performs better than the prototype model in several statistical measures; it has a higher r2 and a lower standard error, for example. However, this only means that the 95 county model does a better job statistically of fitting dependent variables that describe COUNTY AREA fiscal capacity than the prototype model does with system level fiscal capacity. This is not an appropriate comparison in that the county model utterly fails to measure SYSTEM level fiscal capacity. This by no means discounts the importance of analyzing the indicators of the prototype model for fit, etc. 1.a. How should differences in the magnitude of the r2 values be interpreted when comparing the 95-county vs. prototype. As discussed in the answer to question one, it is not appropriate to directly compare the r2 for the two models as they are being used to fit different independent variables. The county model is fitting county area fiscal capacity; the prototype model is fitting system level fiscal capacity. If one insists on comparing the performance of the county model vs. the prototype model, a more relevant comparison would be to the county model when it was first implemented. The adjusted r2 for the county-level fiscal capacity model used in the Fiscal Year 1993 Basic Education Program (BEP) model was .7647, compared to an adjusted r2 of .7574 in the current version of the prototype model. 1.b. How should differences in magnitude of the observed p-values be interpreted when comparing the 95-county vs. prototype? Should the goal be observed p-values that are as close to or below 0.05 as possible? For the purposes of the Fiscal Capacity model, p, and t values are not that important. Our intent is for the model to produce a fitted value. We are less concerned with the values or biases of the individual coefficients. Our assertion is that if the model produces a good fit of the independent variables, then it accurately estimates the fiscal capacity for each system, balancing the relative strengths and weaknesses of the coefficients used to arrive at the estimate. It is also important to remember that the model is being run with a data set that includes the complete population of Tennessee’s school systems, not a random sample. Many tests that focus on sampling errors will not be appropriate when analyzing the prototype model. That said, we continue to examine approaches to address spatial autocorrelation, multicollinearity, omitted variable bias, and other potential areas for improvement within the model. Proportion of Observed Zero Values 2. What is the impact of the prototype variables that contain a substantial proportion of zero values? The three variables with a significant number of zeros are all measures of system-level attributes that are analogous to county-level attributes. Since the prototype fiscal capacity model has county, city and special school district components, these variables are necessary to fully describe the sub-county system fiscal capacity. It may be true that these variables can, to some degree, behave like dummy variables, since they are zero for ninety-three observations and they vary within a fairly small band for the other forty-three systems, and this could distort the results. But, as it turns out, combining these variables by collapsing the two property, the two sales, and the two exportability variables into one each does more harm than good. In the prototype, the coefficient on shared property tax revenue is 25% higher than on unshared. The coefficient on shared sales tax revenue is ten times that on unshared sales tax revenue. The coefficient on city exportability is higher than on county. Whatever improvements are realized by removing the dummy-like qualities of these variables is outweighed by the harm of combining and muddying effects that are clearly of different magnitudes. When these variables are collapsed, the r2 goes down, the standard error goes up, and the equation fits the outliers less than in the regular prototype model. 2.a. Do these variables accurately capture the observed values of Tennessee’s largest counties within a 95% confidence interval? Yes, because the variables represent the entire population, not a sample of school systems; the entire universe of school systems is included. Every variable is the reported value for every individual system, not an estimate based on a sample of systems. 2. b. What is the impact of 0 values on the ability of the model to accurately predict the outcomes within a 95-percent confidence interval? Whatever negative impact they may have is not as bad as the negative impact of combining the variables. The three variables that measure unshared attributes are zero for county systems by definition. The Unshared Property Tax and Unshared Tax Exportability variables each have the same ninety-three counties with a zero value, and then forty-three systems with a positive value. The Unshared Sales Tax variable has those same ninety-three, plus an additional fifteen SSD systems, with a zero value (108 zeros; 28 non-zeros). Since ninety-three county systems are perfectly correlated across these three variables, the correlation coefficients among these variables are strong. The effects of these variables will likely not be correctly and precisely defined for each, but are more likely to be correctly described by a combination of the three. It is likely that only one will attain significance, but it could be any one of the three. It is true that the confidence intervals around these variables are large, and one would not want to make specific claims about the accuracy of the coefficient on any one of these (or even the sign of the coefficient). Our model does not claim to define the effect of any one variable, however, but instead defines the effect of the combination of all variables. Because of this, the multicollinearity among these independent variables should not present any real problems. Additional problems arise from the fact that the high number of zero values sets up a nearly dichotomous relationship between those systems that have a value and those that do not. Most of the effect picked up will explain that difference. Little of the more nuanced effect of the differences in values among those systems with non-zero values will be measured. This is not necessarily improper, though. The most important aspect may well be whether or not a system has access to additional funding from local sales and property taxes. 2. c. What is the impact of potential outliers on the observed weights? The weights of the different variables at the mean are changed somewhat by the outliers. When the outliers are removed, using the residuals to determine which ones should be removed, property tax becomes less important, as does state-shared tax revenue, exportability and median household income. The sales tax, the percent of children in poverty and the intercept become more important. Statistical Treatment of Recurring Outliers 3. a. What is an acceptable benchmark regarding the overall percentage of potential outliers within an observed model? We do not have a benchmark. There are nine systems that lie more than two standard deviations from the mean. All but one of these systems is a city systems or SSD. The one county system is Sevier County, which has extremely high per pupil tax bases for both sales and property. Sevier County is going to be an outlier no matter how this model is defined. 3. b. How would the identified weights change if the outliers were removed? When the observations falling more than two standard deviations from the mean are removed, the r2 increases (and the standard error decreases). The expected fiscal capacity of the eight non-county systems that are outliers falls (by an amount ranging from $68 per pupil to $499 per pupil). Sevier County’s expected fiscal capacity increases by $209 per pupil, by far the largest increase of any system. Elimination of Res-Farm Percent and ADM Population Ratio 4. What is the magnitude of dollars that was driven by the weights observed under the 95 county model? What impact would the elimination of these dollars have particularly on county systems that are observed outliers within the newly observed prototype? First, it is important to remember that regardless of which independent variables are included or not, the total fiscal capacity will be the same in the model. The average per pupil fiscal capacity within a model will be equal to the average per pupil local revenue within the same model. The system averages vary between the two models because one portrays an average of ninety-five county areas while the other portrays the average of 136 systems. Second, clarification is in order. The ratio of residential and farm property to total property was not removed from the model. It is now represented as the ratio of commercial, industrial, utility and personal property to total property, which is exactly 1.00 minus the residential and farm property ratio. This is done to represent the tax exportability capacity of each system. Substituting the one for the other in the county model simply changes the sign on the coefficient for that factor. The intercept is also adjusted so that the overall result (the predicted value) for each system remains exactly the same. The following table shows the “dollars” driven by the coefficients. The dollars are calculated by multiplying the coefficient weight by the state average for each variable. However, it must be stated again that the model is not designed to emphasize the various coefficients, but the overall fit of the independent variables. The dollars from each coefficient cannot really be considered separately; they should be considered for their total effect, including the effect of the intercept. For comparability, the residential farm ratio is inverted to reflect exportability in the county model. Dollar Values of Coefficients Multiplied by State Average Variables County Model vs. Prototype Model, FY 2005
Note: Total for prototype model not equal to sum of variables ($1,802) due to rounding. Total values shown for each model equal the average of the independent variables (i.e., average of the 95 county areas and average of the 136 school systems). All values are per pupil. Collapsed Variables An additional topic raised at the February 2, 2005 meeting was the possibility of collapsing the various shared and unshared variables for property and sales tax bases and for tax exportability into single variables for each. This possibility was raised by Dr. Nixon at the suggestion of SBE Chairman, Fielding Rolston. These variables are split into shared and unshared amounts in the prototype model to reflect the difference in the abilities of city, county and special school districts to raise revenue from these bases. In response to the SBE request, TACIR staff ran three versions of the system-level model collapsing the six variables into three:
A more detailed explanation of these calculations is attached. The first version was a regression based on these three collapsed variables and the remaining variables. The result was a much lower r-squared (coefficient of multiple determination), which means that this modified model does not explain as much of the variation as the current prototype. The values are 0.6578 and 0.7574 for comparison. The property and sales variables and median household income were significant at p < 0.05. However, the total fiscal capacity value for the Carroll County system is negative. Because of that system’s relative negligible ADM, its tax base variables per pupil are extreme; therefore, we use its county-area values in the regression model and then subtract the results for the other systems in Carroll County to arrive at a total fiscal capacity for the Carroll County system. That is the only method we could identify that would produce a reasonable figure for that system. If we treat it as a stand-alone system in this model and enter its extremely large values per ADM, the statistical results are weaker: the r-squared falls to 0.6127 and only property and median household income are significant. The second version was identical except that it used the base ten logarithm of the two collapsed tax base variables (property and sales). The result was a slightly higher r-squared, still well below the current prototype model at 0.6805. The intercept, as well as the property and the sales variables and median household income, were significant at p < 0.05. However, the total fiscal capacity value for the Carroll County system remained negative. Again, treating Carroll County as a stand-alone system weakens the results, producing an r-squared of 0.6583. The third version was identical to the first except that statistical outliers were removed before re-running the regression. Outliers removed were those with residual values in the first model that were more than 1.96 standard deviation units away from the mean of the residuals (zero). They included
The r-squared, at 0.7167, was higher than the second version, but still below the prototype. The intercept and the sales and median household income variables were significant at p < 0.05 and the total fiscal capacity for the Carroll County system is positive. Like the other two versions, this one would be weaker (r-squared of 0.6707 and only property and median household income significant) if Carroll County were treated as a stand-alone system. The output summary for each model is attached along with a table comparing the predicted fiscal capacity for each version and the prototype. Considering all four strictly from a statistical point of view, the current prototype is superior despite the insignificance of most of the t-statistics. Considering the models from a fiscal capacity point of view, the factors used in the prototype more accurately represent the actual fiscal structure of Tennessee’s school systems. This does not mean that there might not be a better model out there. We are continuing to review the current prototype in an effort to improve on it. Attachment: Combined Variable Methodology Combined Property Tax Base per Pupil: This variable is a combination of the system’s share of the shared county area property tax base and its un-shared property tax base.
Combined Sales Tax Base per Pupil: This variable is a combination of the system’s share of the shared county area sales tax base and its un-shared sales tax base.
Combined Exportability: This variable is the ratio of commercial/industrial/utility/ personal property to the total property tax base for a combination of shared and unshared property tax bases.
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