12/09/02urban economics 7800tina kleehaupt some tests of alternative urban population density...
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12/09/02 Urban 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)
Some Tests of Alternative Urban Population Density Functions
12/09/02 Urban Economics 7800
1
2
3
Introduction
Tests and Results
Conclusion
Introduction
12/09/02 Urban Economics 7800
Usefulness of some alternative density functions
2 evaluation criteria:
Maximum explanatory power in standard regression analysis
Accuracy in predicting total population
)exp()( 0 buDuD Remember Density Function:
Introduction
12/09/02 Urban Economics 7800
Name Type Analytical form
1. Clark, Muth, Mills Exponential
2. Mills Binomial
2‘. Binomial, approx. assuming
B = 0
3. Newling General normal
4. Tanner, Sherratt Standardized normal
5. Aynvarg Gamma
5‘. Gamma, assuming b = 0
6. Linear
6‘. Quadratic
7. Suggested function
)exp()( 0 buDuD
guubBAuD )()( guuDuD )()( 0
)exp()( 20 buauDuD
)exp()( 20 buDuD
)exp()( 0 buuDuD aauDuD 0)(
buDuD 0)(2
0)( cubuDuD
)/exp()( 0 ubauDuD
Introduction
12/09/02 Urban 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
Some Tests of Alternative Urban Population Density Functions
12/09/02 Urban Economics 7800
1
2
3
Introduction
Tests and Results
Conclusion
Tests and Results
12/09/02 Urban 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 R2 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
)exp()( 20 buauDuD
Tests and Results
12/09/02 Urban Economics 7800
Accuracy of Total Population Prediction
duuuDN uu )(
Example of exponential form:
)exp()( 0 buDuD
ubbu eubebubDN )1()1()/( 20
Factor out , integrate from to uu
Tests and Results
12/09/02 Urban 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
Tests and Results
12/09/02 Urban 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 R2
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 D0
substituting ln D0 into the estimation form
ubeubNbD )1(1/20
ueuNuD u222
2)1(1lnlnlnln2)(ln
)exp()( 0 buDuD
Tests and Results
12/09/02 Urban 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
2
2
Some Tests of Alternative Urban Population Density Functions
12/09/02 Urban Economics 7800
1
2
3
Introduction
Tests and Results
Conclusion
Conclusion
12/09/02 Urban 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