1. 12/09/02Urban Economics 7800Tina Kleehaupt Some Tests of Alternative Urban Population Density Functions John F. McDonald and H. Woods Bowman Journal of Urban Economics 3, pp. 242-252 (1976)

2. Some Tests of Alternative Urban Population Density Functions 12/09/02Urban Economics 7800 1 2 3 Introduction Tests and Results Conclusion

3. Introduction 12/09/02Urban Economics 7800 Usefulness of some alternative density functions 2 evaluation criteria: Maximum explanatory power in standard regression analysis Accuracy in predicting total population Remember Density Function:

4. Introduction 12/09/02Urban Economics 7800 NameTypeAnalytical form 1. Clark, Muth, MillsExponential 2. MillsBinomial 2‘.Binomial, approx. assuming B = 0 3. NewlingGeneral normal 4. Tanner, SherrattStandardized normal 5. AynvargGamma 5‘.Gamma, assuming b = 0 6.Linear 6‘.Quadratic 7.Suggested function

5. Introduction 12/09/02Urban Economics 7800 Data Sets Census tract area and population data from 1960 Census of Population Two types: 1. Sample of approx. 20 census tracts from 16 urbanized areas 2. all census tracts of the Flint and Grand Rapids SMSAs second set allows check for sampling bias

6. Some Tests of Alternative Urban Population Density Functions 12/09/02Urban Economics 7800 1 2 3 Introduction Tests and Results Conclusion

7. Tests and Results 12/09/02Urban Economics 7800 Standard Statistical Tests of Alternative Functional Forms General normal function (3), which adds a quadratic distance term exhibits slightly improved explanatory power. This has a cost in terms of population prediction accuracy. Little basis for favoring one function over others First data set: Highest R 2 for function (3), the general normal most other functions perform nearly as well (including exponential function) Second data set: similar results, but for some functions 100% samples turned out to be useful, especially for small urbanized areas

8. Tests and Results 12/09/02Urban Economics 7800 Accuracy of Total Population Prediction Example of exponential form: Factor out, integrate from to

9. Tests and Results 12/09/02Urban Economics 7800 Accuracy of Total Population Prediction adding a second distance term tends to reduce the standard error of the regression this second term is rarely statistically significant collinearity between two distance terms leads to less accurate population predictions Tendency to overprediction exponential form is not clearly surpassed

10. Tests and Results 12/09/02Urban Economics 7800 Constrained Estimation of the Exponential Function Include both objectives in the estimation technique minimize loss function which incorporates both criteria Cost of constraint is insignificant in terms of reducing R 2 The exponential function is a more useful summary measure of population distribution if it is constrained to predict population exactly Estimation procedure (example of exponential function): assume that radius of CBD (u) is zero, solve for D 0 substituting ln D 0 into the estimation form

11. Tests and Results 12/09/02Urban Economics 7800 Constrained Estimation of the Exponential Function The estimate of the slope,,and the intercept is found by solving for ln D(0), the density gradient is steeper in the constrained regression, than in the unconstrained this is expected, as unconstrained case leads to overprediction of the population constraint did not impose significant cost in terms of explanatory power

12. Some Tests of Alternative Urban Population Density Functions 12/09/02Urban Economics 7800 1 2 3 Introduction Tests and Results Conclusion

13. 12/09/02Urban Economics 7800 a constrained estimation should be considered as most descriptive exponential function no single function will best describe population distribution for all urbanized areas due to possible sampling bias large samples are advised Further research with regard to: > data for more urban areas of different sizes and vintages > more functions > more sophisticated estimation procedures and criteria Use exponential form with a little more hesitation and present estimates with implication of reasonably accurate total population estimates