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Small area population forecasting. (Features).

Small area population forecasts are used by real estate appraisers and analysts as the basis for evaluating the demand for residential and commercial real estate. Most analysts utilize population forecasts prepared by demographic forecasting companies without any knowledge of the methods used to develop the forecasts. This paper will analyze two methods, the housing unit method and the extrapolation technique, that real estate analysts can easily employ to check the forecasts received from demographic forecasting companies.

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Population projections are something every analyst considers when preparing an appraisal or completing a consulting assignment. Real estate appraisers and consultants consider the growth or decline of populations when forecasting the demand for any existing or proposed development. Projections of population growth or decline come from many different sources. The most prolific sources are federal, state, and local government agencies. Population projections also can be obtained from many third-party sources, such as demographic data marketing companies or local universities. Appraisers and analysts also can prepare their own projections from readily available data. This article focuses on two techniques that real estate analysts can use to project or forecast population changes in small areas. The forecast techniques discussed here are the housing unit method, as explained by Smith, (1) and the extrapolation techniques, as discussed by Klosterman. (2) With readily available data, real estate analysts can apply both of these techniques.

Forecasts Uses

Small area population forecasts focus on areas that are smaller than a county, generally the size of a city, census tract, or market trade area. Small area population forecasts are in great demand by public and private sector market researchers.

As part of an analysis of a proposed subdivision, the real estate analyst will consider the future growth of the population. As a first step in the demand analysis, the analyst will try to obtain a measurement of the growth in the population. This measurement can take the form of a projection or a forecast. A projection is defined as the product of a mathematical model. Assumptions are made to provide a range of outcomes, but once the assumptions are established, the calculations proceed without modifications. For example, a study is made of the population changes over the past 20 years and a projection is made based on the average rate of change observed. The assumption made here is that the average rate of change observed in the past will be constant into the future. A forecast requires judgment. Several projections may be made based on specific assumptions about future conditions, and then a forecast is made as to a probable outcome. Many users and makers of population projections use the terms projection and forecast interchangeably; however, a distinction should be made regarding the difference between these two terms.

Planners, in the private and public sectors, and market researchers use population forecasts extensively. Their use generally requires information about areas that are smaller than state or county jurisdictions. There are several distinct problems in small area population forecasting. The first problem is data availability and quality. Accurate data for small areas are only available once every ten years. Market researchers need interim forecasts of population and their characteristics for their studies for demand for consumer goods, housing, and retail purchases. Another problem with small area population projections is with shifting geographic boundaries. If one is forecasting population growth for a city, when the city grows by annexing additional land, the census data may not be consistent from one census to the next. In the public sector, decisions about future demand, need, and funding for roads, housing, schools, and other infrastructure rely on information about the expected change in the population o f small areas. Private sector market researchers study the demand for all types of housing and retail services based on the forecast change in the population of small areas.

Small Area Population Forecasting Models

The types of small area population forecasting models can be classified into five broad categories: (1) cohort-component, (2) mathematical extrapolations, (3) ratio, (4) economic base, and (5) housing unit. The cohort-component model is a complex forecasting method. With this model the researcher studies the changes in the population by age cohort, taking into consideration birth, death, and migration. This model is time consuming to construct and requires a large database from which the model develops projections of the future population.

The ratio (share) method is based on the ratio to a larger area, such as a region or state. The historical trends of the ratios are determined, projected into the future, and multiplied by the projection for the larger area population.

The economic base method is based on the assumption that changes in population are a function of changes in area employment (i.e., people migrate to where the jobs are).

Two simpler techniques are a class of models known as the extrapolation techniques and the housing unit method. Extrapolation techniques, because of their modest data requirements and ease of use, are widely used for small area population projections. Generally the extrapolation techniques are based on six curves, as summarized further in this article. Klosterman (3) discusses the six extrapolation curves in detail. He examines each curve by analyzing its assumptions, showing the curve in a graphic form, and using the same population database for comparison purposes.

The housing unit method is a type of land use model. It focuses on a certain pattern of land use, as specified by a communities general plan and/or zoning ordinances. How many housing units can be accommodated assumes full build out. With the assumption of the average number of persons per housing unit, a projection of the total population can be obtained by simple multiplication. An example of a large-scale land use modeling system is the projective land use model (PLUM) as discussed by Tayman. (4) The PLUM model has been used to prepare small area population projections for San Diego County, California.

The housing unit method (5) utilizes the last recorded census figures, then modifies the census count by adding the population growth that is the product of the new housing units, less the demolished housing units, times the persons per household in the study area. The housing unit method of population forecasting is very simple to use and the data required for its use is readily available, either from the census bureau, the city and/or the county under study.

Review of Literature

There have been many articles and sections of books written about small area population forecasting, but few articles have addressed the issue of the accuracy of the small area forecast. Little is known about the magnitude or the characteristics of the error generated in small area forecasts. Most research on the accuracy of population forecasting focuses on the nation, states, or counties. Recently, researchers have begun to determine the accuracy of sub-county forecasts. Isserman (6) studied the accuracy level of three commonly used extrapolative methods of projecting populations in subcounty areas. Smith (7) and Murdock and Hamm (8) followed up this study. Their studies addressed the accuracy and bias of forecasts made for county areas and developed factors that should be considered when evaluating small area population forecasts. Smith and Shaidullah (9) set out to test four extrapolation techniques and Tayman (10) studied the accuracy of two proprietary forecasting models.

The significance of the sub-county forecasting models is that they are most applicable to public and private sector planners and to market researchers. As I have noted above, public and private sector planners need accurate small area population forecasts for a variety of reasons; most notably the demand for infrastructure development or redevelopment and for forecasting the change in school populations. Market researchers utilize small area population forecasts for their studies of the demand for all types of housing and retail services utilized by the local population.

Smith, Isserman, and Klosterman have developed the main body of literature. All of these authors have contributed significantly to the study of population forecasting for small areas. The remaining literature referenced at the conclusion of this article can serve to inform the reader of the many tools available to analysts seeking to compile their own forecasts, and the errors that the tools can be expected to contain.

Housing Unit Method

The housing unit method of population estimation is the most commonly used method for the estimation of population for small areas. The theoretical rational behind the method is that the population for an area (as reported in the most recent census), plus the remainder of newly constructed housing units, less those demolished, times the persons per household, plus the change in the number of people residing in group quarters should equal the current population. This rational will produce a valid estimation of the current population, provided that the constructed housing units are actually occupied and that the number of people residing in group quarters can be reliability forecast. The basic formula (Formula 1) is shown below:

Population = (HH * PPH) + GQ (1)

where:

HH = Number of occupied households;

PPH = Number of people per household; and

GQ = Number of people in group quarters.

Data from the 1990 Census was used to verify the validity of the formula. The data in Table 1 shows that the error in the calculated population versus the actual census counts is generally less than 0.2%. This small error supports the proposition that the population can be calculated using the household formula.

It can be argued that disaggregating the number of households into three components and analyzing each unit separately will produce a more accurate forecast. By doing so, the forecast of the population is more focused and is a better estimate of the population. The problem with the disaggregation of the number of households into components is that the three components are never known exactly.

The procedure suggested by Smith and Lewis (11) is to disaggregate the number of households into three major categories (single family, multi-family, and mobile home households) and analyze each separately. Each of the components has its own development and occupancy rates, and by disaggregating the housing unit count into its components, Smith and Lewis argue that the estimate of the number of households will be more accurate. The use of an overall occupancy rate may lead to significant errors in the estimate, although Smith and Lewis did not test this hypothesis. For example, the 1990 census in the city of Fullerton, CA, reported a vacancy rate of 0.9% on owner occupied residences, whereas the vacancy rate of tenant occupied residences was 7.4%. The overall vacancy rate was 4.9%. Based on this small sample, the overall vacancy rate was fairly stable at around 5.0%. The exception to this is Palm Desert, where a large number of second homes are located. This community is in the desert resort area of southern California and a large number of the community residents are "snowbirds" (people who occupy their residences on a seasonal basis) or temporary residents who occupy their homes on weekends.

The estimate of the population is actually developed through a series of formulas. The first is to estimate the number of households in the study area.

This can be completed through the use of Formula 2:

HH = (HU + BP - D)* OCC (2)

where:

HH = Number of occupied households;

HU = Housing units in the most recent census;

BP = Building permits issued since the most recent census;

D = Number of demolished units; and

OCC = Occupancy rate.

The number of housing units projected using Formula 2 can then be substituted into Formula 1 and a projection of the population can then be made.

Housing Unit Methodology

Formulas 1 and 2 have several distinct components that need to be discussed. Both formulas include a households component. There are a number of different types of data that can be used to estimate the number of households. This data consists of building permits, certificates of occupancy, and property tax records, just to name a few. In this study I focus on building permits. Building permit records are widely available from many sources both public and private. The census data collects this data from cities, counties, and states. Historical building permit data is also generally available on request at most city and county building and planning departments.

Formula 2 includes a component for demolitions. To accurately forecast the number of additional households, the number of households lost through demolition needs to be considered. In most communities this component is very small. The demolition of a housing unit can occur for a variety of reasons. The main factors are changes in land use, redevelopment, fire damage replacement, and street widening. For example, in the 10-year period under study, the city of Tustin issued 33 demolition permits for residential units and 5,569 permits for new residential construction. The demolition rate amounted to less than 0.6% of the new permits issued. Other cities studied reported very low numbers of demolition permits; (12) in the case of Fullerton, the demolition records for the past 10 years were incomplete (there were eight demolition permits issued for residential housing units over the past 18 months). The impact on the projection by omitting the demolition of residential housing units is very small.

Building permit data does not directly correspond to population growth. The building permit data must be transformed into an estimate of the number of occupied households using an occupancy rate. The rate of occupancy will change over time as the composition of the population within the community changes. The most common procedure is to use the occupancy rate from the most recent census as the multiplier. The product of the building permit (new building permits less demolitions) projection and the occupancy rate will yield the number of occupied households.

Persons per household is another component of the housing unit method. The number of persons per household will change over time just as the occupancy rate changes. In practice, the number of persons per household used in the formula is that recorded in the most recent census.

Group quarters population is the last component in the formula for estimating small area population. Persons living in group quarters are those persons living in prisons, college dormitories, military bases, and long-term health care facilities. Group quarters data generally is taken from the most recent census unless more current research data is available. For small area forecasts, this component of the population is usually small.

To analyze the accuracy of these formulas as a population forecasting tool, in this study I applied the formulas to the six cities that have been used as the example. Table 3 shows the application of the formulas using 1990 census data as the base. The assumption that needs to be made about the number of building permits is the time period from which the building permit was issued until the household was occupied. For the purposes of this study, and because data is generally available on an annual basis rather than a monthly basis, it was assumed that building permits issued for residential construction during 1989 were occupied by the 1990 census and hence should not be counted. The 10-year forecasting period includes building permits issued during 1990 through 1999. To be consistent, it was assumed that building permits issued during 1999 were occupied and the occupants counted in the 2000 census of the population. A criticism that can be made of this assumption is that it takes different amounts of time fo r certain types of residential structures to be constructed. While this criticism has merit, for the purposes of short-term projections, any forecasting error will, in all likelihood, be very small.

The error in the population forecast for the first three cities is relatively small, whereas the error grows for the last three cities. The increasing error in the forecast indicates another potential source of error when using the housing unit method: municipal annexations. In the case of Palm Desert, between 1990 and 1995 the city extended its municipal boundaries and annexed an area of the county that was already developed with housing. Formula 2 does not consider the possibility of annexed areas that have existing housing units. Because of the potential for cities to annex already developed land, Formula 2 needs to be modified to take into account annexed housing units as shown in Formula 3:

HH = (HU + BP - D + [HU.sub.a])*OCC (3)

where:

[HU.sub.a] = Number of housing units in annexed area

To make a forecast of future population using the housing unit method, the only items needed are a projection period and the number of housing units that will likely be constructed during the projection period. A place to start with the projection is to plot the number of permits issued by year for the past 10-15 years. From this plot, a trend may emerge or the randomness of the permit issuances will be noted. If the data shows some sign of a trend, one of the extrapolation techniques discussed further on in this paper can be utilized for the projection. Table 4 is a tabulation of the building permits, both single and multiple family units that were issued over the past 13 years by the city of Fullerton. As Table 4 shows, there is no trend or consistency in the issuance of permits from year to year. Another method that could be utilized for this projection, if the data shows no trend at all, would be to average the past 10-15 years of building permit records to develop an indication of the average number of n ew permits issued in any typical year. A criticism of using an average is that there is no consistency in the issuance of permits for single family detached or multiple family residential construction from year to year. In fact, in plotting the number of permits issued by type and year for the cities utilized in this study, I could find no common trend. Another problem with using an average, especially when a community is mature and nearly fully developed is the potential to overestimate the population forecast if the remaining development potential is not considered.

Table 5 demonstrates an application of the housing unit method to project future population. The development of the projected population is through the use of a series of formulas, beginning with a variance of Formula 2 previously noted. The projected population is then developed by the following formulas:

[HH.sub.g] = ((BP * N) - D + [HU.sub.a])) * OCC (4)

[POP.sub.g] = [HH.sub.g] * PPH (5)

[POP.sub.f] = [POP.sub.c] + [POP.sub.g] (6)

where:

[HH.sub.g] = Growth in number of households;

BP = Building permits issued per year since the most recent census;

N = Forecast period (in years);

D = Number of demolished units;

[HU.sub.a] = Number of housing units in annexed area;

[POP.sub.g] = Population growth;

[POP.sub.c] = Population at last census; and

[POP.sub.f] = Population forecast.

One method for developing an estimate of the potential for development in a community is to examine the city's transportation analysis zone (TAZ) map and associated records. Every city should have a TAZ map that divides the city into development blocks for the purpose of analyzing the road system based upon the community's general plan. The records that will accompany the TAZ map will indicate the maximum number of residential units and/or the number of square feet of commercial and industrial development that the general plan allows for a particular block. In addition, the records should show the number of existing units developed by block and the number of units that remain to be developed under the general plan. This type of data is very helpful to the analyst when making forecasts of future population, when the ability of a city to annex additional land is limited, and the community is nearing full build out.

One might argue after reading the above description that housing development drives population growth. This argument is generally false. It is the other way around. Population growth is the driver for housing development. However, in most cases the population will not grow without a strong local and/or regional economic base and the growth of jobs. Stated in economic terms, without the presence of and the growth of jobs and the income they produce, consumers (the population) have limited purchasing power and population growth is not likely to happen. The exception to this general rule are retirement communities, where population grows first, and this growth is followed by the creation of new jobs. If the assumption holds true that the growth of jobs will continue in the same pattern as was observed in the past over your projection period, then the housing unit method of population forecasting is a simple and accurate method to use.

Extrapolation Techniques

The extrapolation technique uses aggregate data from the past to project the future. This process involves the application of two steps. The first step is the identification of the curve that best fits the data, and the second step is the extension of the best fit curve to project the future. Among the alternative curves that are available to the analyst are the linear, geometric, parabolic, modified exponential, and logistic. All of these curves are based on different growth assumptions. Depending upon the depth of the data, one curve may be more applicable to the projection than another.

One thing in common with the linear, geometric, and parabolic curves is that there is no upper limit to growth. The growth (or decline) in a population is assumed to go on indefinitely. While the basic trend may go on for some time, in reality it is unlikely that it will continue forever. The linear curve is actually straight because it plots the data in a straight line. The slope of the line is a constant rate that can be either positive, negative, or zero. The geometric curve, while similar to the linear curve, plots the data in a constant ratio. Real estate appraisers and analysts will recognize this curve as the basis for compound interest. The parabolic curve is similar to the geometric curve, except that the curve has a constantly changing slope and one bend. The direction of the curve can be either positive, negative, or zero.

The final two curves are the modified exponential and logistic curves. Both of these curves are applicable to population forecasting in that they have assumed growth limits because of their asymptotic features. An asymptotic curve recognizes that growth or decline will approach some upper or lower limit. In the case of population forecasting, the upper limit may be a trade area, city, or county jurisdictional boundary. The upper limit could be increased or decreased by a change in the land use element of a communities general plan.

To test the projections made by utilizing these five extrapolation curves, I have used the census data from 1940-1990 for the city of Fullerton. In the case where the formulas called for odd numbered data, I utilized the data from 1950-1990 and, where even number data were required, I used the entire data set.

The alternate estimates and projections are shown in Table 6. When looking at the year 2000 projection, the size of the population ranges from 111,336 as projected by the parabolic curve, to 253,971 as projected by the geometric curve. This range in projections is over 125% and the question then becomes which curve best firs the data.

The next step in this analysis of the extrapolation curves is to employ some input and output statistics. The two most common output statistics utilized by researchers are the mean error (ME) and the mean absolute percentage error (MAPE), as shown in Table 7. The mean error is computed by summing the differences between the actual and computed values and dividing the total difference by the number of observations (in this case either five or six). A problem with the mean error is that large positive and negative errors can cancel each other our, thereby making this statistic a poor measure of the total deviation between the actual and estimated values.

The second output statistic is the mean absolute percentage error. As the name implies, this statistic measures the mean variance between the observed value and the estimated value on absolute terms without the influence of the direction of the variance. This output statistic is more useful than the mean error because of the absolute value feature and it can be used to analyze data developed from databases of differing sizes.

Judgment is required to select the extrapolation curve that best fits the data. The output statistics are tools that the analyst can use to assist in the selection of the appropriate curve. The curve with the lowest mean error and mean absolute percentage error may yield the extrapolation curve that best fits the data. If that is the case, then the parabolic curve meets the test. However, if the estimation in Table 6 is examined one will note that the population projection from 1990 to 2000 has fallen, and if the table is extended out to include the years 2010 and 2020, the population continues to fall. If the estimates and projections of the population using the parabolic curve are plotted, note that the population peaks in 1990 then begins to fall. This is highly unlikely for a community in a major metropolitan area that has a widely diverse employment base.

The last two extrapolation curves, the modified exponential and logistic, have the same characteristic of an upper limit (asymptote) for population growth. This limit is based on the total land area zoned for housing within the boundaries of the city. The modified exponential curve shows a very small mean error, but a large, mean absolute percentage error, and the logistic curve shows just the opposite. The projected population for the year 2000 indicated by these two curves ranges from 122,857-136,815. The actual population for the city as reported in the 2000 census was 126,003.

Conclusion

The small area population forecasting methods used in this study provide three projections for the population of Fullerton as of the year 2000. The housing unit method indicated 118,722, the modified exponential curve indicated 122,857, and the logistic curve indicated 136,815. The housing unit method and the modified exponential curve slightly underestimated the actual population by 6.1% and 2.5%, whereas the logistic curve overestimated the population by 7.9%. Will the under or overestimation make a substantial difference in the conclusions developed in an appraisal or market study? The overestimation of the population would cause the analyst to underestimate the absorption time required for a new development, whereas the opposite is true of underestimating the population.

The housing unit method is an accurate method of population estimation. The method is simple to apply with data that is readily available to the analyst. It is easy to explain to people with very little background in planning or demography. The technique can be applied to almost any level of geography, from regions down to counties, cities, and neighborhoods. Because of this flexibility the application of testing the forecasts obtained from outside consultants or demographic forecasting firms against your own forecast can easily be done. And finally, the forecasts produced by this method are at least as accurate as the other more complex methods discussed in this paper.

The extrapolation curves offer a bit more complexity. The linear and geometric curves are simple to explain to people without a background in mathematics, however, the other curves are not. Depending on the depth of the data and the analyst's view of the area's most likely future, one curve may be more applicable to the forecasting problem than another. Complex extrapolation procedures cannot replace the need for an understanding of the past trends and whether these trends will continue into the future. What these methods have demonstrated is that simple extrapolation methods or the application of the housing unit method can be at least as accurate for short-term projections as the more rigorous, complex models.
Table 1

Calclulation of Calculated Population to Actual Census Counts

 Fullerton Irvine Tustin Anaheim San Marcos
Census: 1990 1990 1990 1990 1990

Occupied households 40,872 40,257 18,332 87,588 13,617
PPH 2.74 2.69 2.66 2.99 2.85
GQ 1,961 2,174 1,876 4,184 135
Calculated population 113,950 110,465 50,639 266,072 38,943
Actual population 1990 114,144 110,330 50,689 266,406 38,974
Error -0.17% 0.12% -0.10% -0.13% -0.08%

 Palm Desert
Census: 1990

Occupied households 10,595
PPH 2.18
GQ 118
Calculated population 23,215
Actual population 1990 23,252
Error -0.16%
Table 2

Housing Unit and Vacancy Calculations

 Fullerton Irvine Tustin Anaheim San Marcos

Housing units 42,956 42,221 19,300 93,177 14,476
Owner occupied 52.4% 59.6% 38.9% 46.3% 58.6%
Renter occupied 42.7% 35.8% 56.1% 47.7% 35.5%
Vacant units 4.9% 4.7% 5.0% 6.0% 5.9%
Owner vacancy rate 0.9% 1.6% 1.5% 1.5% 2.5%
Renter vacancy rate 7.4% 6.9% 5.6% 8.1% 8.3%

 Palm Desert

Housing units 18,248
Owner occupied 37.1%
Renter occupied 21.0%
Vacant units 41.9%
Owner vacancy rate 6.2%
Renter vacancy rate 18.6%

Source: 1990 Census.
Table 3

Comparison of 2000 Calculated Population Projection to Actual Census
Counts

 Fullerton Irvine Tustin
Change in Population 1990-2000 1990-2000 1990-2000

Building permits (1990-1999) 1,206 12,133 5,736
Demolition permits (1990-1999) 8 4 33
Annexed housing units 0 0 0
Average occupancy 95.1% 95.3% 95.0%
Occupied housing units 1,139 11,559 5,418
PPH 2.74 2.69 2.66
Total change in population 3,122 31,094 14,411
Actual population 1990 114,144 110,330 50,689
Projected population 2000 117,266 141,424 65,100
Actual population 126,003 143,072 67,504
Error -7.45% -1.17% -3.69%

 Anaheim San Marcos Palm Desert
Change In Population 1990-2000 1990-2000 1990-2000

Building permits (1990-1999) 6,220 3,112 4,491
Demolition permits (1990-1999) 0 12 55
Annexed housing units 0 0 6,143
Average occupancy 97.2% 96.0% 58.1%
Occupied housing units 6,046 2,976 6,146
PPH 2.99 2.85 2.18
Total change in population 18,077 8,482 13,399
Actual population 1990 266,406 38,974 23,252
Projected population 2000 284,483 47,456 36,651
Actual population 328,014 54,997 41,155
Error -15.30% -15.89% -12.29%
Table 4

Total Residential Building Permits Issued City of Fullerton

Year Single Multi Total

1987 282 58 340
1988 17 188 205
1989 147 612 759
1990 23 107 130
1991 21 38 59
1992 13 32 45
1993 30 65 95
1994 89 0 89
1995 165 168 333
1996 66 31 97
1997 67 3 70
1998 77 11 88
1999 200 0 200
Total 1,197 1,313 2,510
Average 92 101 193
Table 5

Population Projection 2005 City of Fullerton

Change In Population 2000-2005

Building permits (per year) 193
Times forecast period 5
Less demolition permits 0
Plus annexed housing units 0
Average occupancy 95.1%
Occupied housing units 918
Persons per household 2.74
Total change in population 2,515
Actual population 2000 126,003
Projected population 2005 128,518
Table 6

Alternate Estimates and Projections

 Linear Linear Modified
Year Odd Even Geometric Parabolic Exponential Logistic

1940 1,255 -9,159 10,602
1950 25,215 27,854 23,021 14,616 31,477 22,076
1960 49,859 54,453 37,210 55,159 61,738 42,206
1970 74,503 81,052 60,143 85,102 84,273 70,567
1980 99,147 107,650 97,212 104,447 101,054 100,260
1990 123,791 134,249 157,128 113,191 113,551 123,086
2000 148,434 160,848 253,971 111,336 122,857 136,815
Table 7

Input and Output Evaluation Statistics

Curve ME MAPE Upper Limit

Linear (odd) 0 23.35% none
Linear (even) -3,926 16.52% none
Geometric -440 34.27% none
Parabolic 0 2.11% none
Modified exponential 4 54.47% 150,000
Logistic 2,360 18.71% 150,000


(1.) Stanley K. Smith and Bart B. Lewis. "Some New Techniques for Applying the Housing Unit Method of Local Population Estimation," Demography (17: 3, 1980): 323-339. Stanley K. Smith "A Review and Evaluation of the Housing Unit Method of Population Estimation," Journal of the American Statistical Association (81, 1986): 287-296

(2.) Richard E. Klosterman, Community Analysis and Planning Techniques. (Maryland: Rowan and Littlefield Publishers, Inc., 1990).

(3.) Ibid.

(4.) Jeff Tayman, "The Accuracy of Small-Area Population Forecasts Based on a Spatial interaction Land-Use Modeling System," Journal of the American Planning Association (62:1 1996): 85-99.

(5.) Smith and Lewis, Ibid. Smith (1986), Ibid.

(6.) Andrew M. Isserman, "The Accuracy of Population Projections for Subcounty Areas," Journal of the American Institute of Planners (43, 1977): 247-259.

(7.) Stanley K. Smith, "Tests of Forecast Accuracy and Bias for County Population Projections," Journal of the American Statistical Association (82, 1987): 991-1003.

(8.) Steve H. Murdock and Rita R. Hamm, "Evaluating Small-Area Population Projections," Journal of the American Planning Association (57:4 1991): 432-444.

(9.) Stanley K. Smith and Mohammed Shaidullah, "An Evaluation of Population Projection Errors for Census Tracts," Journal of the American Statistical Association (90:429, 1995): 64-72.

(10.) Jeff Tayman, "The Accuracy at Small-Area Population Forecasts Based on a Spatial Interaction Land-Use Modeling System," Journal of the American Planning Association (62:1, 1996): 85-99.

(11.) Stanley K. Smith and Bart B. Lewis. "Some New Techniques for Applying the Housing Unit Method of Local Population Estimation," Demography (17:3, 1980): 323-339.

(12.) Richard Irwin, "Guide for Local Area Population Projections," U.S. Bureau of the Census, Technical Paper 39. Washington, D.C., 1977.

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Wayne Foss, MAI, is the president of Wayne Foss Appraisals, Inc., in Fullerton, California. His practice involves real estate valuations, market and feasibility studies. In addition he teaches real estate valuation and market analysis courses at California State University, Fullerton and Course 520 for the Appraisal Institute. Mr. Foss has an MBA degree from the University of California, Irvine, and is a candidate for the DBA degree from Argosy University. Contact: wfoss@fossconsult.com.
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