12/09/02Urban Economics 7800Tina Kleehaupt Some Tests of Alternative Urban Population Density...

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)


12/09/02 Urban Economics 7800

1

2

3

Introduction

Tests and Results

Conclusion

Introduction


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


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


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



1

2

3

Introduction

Tests and Results

Conclusion

Tests and Results


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


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


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


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


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

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1

2

3

Introduction

Tests and Results

Conclusion

Conclusion


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

12/09/02Urban Economics 7800Tina Kleehaupt Some Tests of Alternative Urban Population Density...

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