Journal of Population Research

Spatial Weighting Improves the Accuracy of Small-Area Demographic Forecasts ofUrban Populations--Manuscript Draft--

Manuscript Number: JPOR-D-13-00051R2

Full Title: Spatial Weighting Improves the Accuracy of Small-Area Demographic Forecasts ofUrban Populations

Article Type: Original Research

Keywords: small-area estimation, demography, forecasts, urban, census tract

Corresponding Author: Jack Baker, Ph.D.University of New MexicoAlbuquerque, NM UNITED STATES

Corresponding Author SecondaryInformation:

Corresponding Author's Institution: University of New Mexico

Corresponding Author's SecondaryInstitution:

First Author: Jack Baker, Ph.D.

First Author Secondary Information:

Order of Authors: Jack Baker, Ph.D.

Adelamar Alcantara, PhD

Xiaomin Ruan, MA, MS

Kendra Watkins, MACRP

Order of Authors Secondary Information:

Abstract: Existing research in small-area demographic forecasting suffers from two importantlimitations: (1) a paucity of studies that quantify patterns of error in either total orage/sex-specific estimates and (2) limited methodological innovation aimed specificallyat improving the accuracy of such forecasts. This paper attempts to fill, in part, thesegaps in existing research by presenting a comparative evaluation of the accuracy ofstandard and spatially-weighted Hamilton-Perry forecasts for urbanized census tractswithin incorporated New Mexico municipalities. These comparative forecasts areconstructed for a ten-year horizon (base = April 1, 2000 and target = April 1, 2010),then compared to the results of the 2010 Census in an ex post facto evaluation.Results are presented for the standard Hamilton-Perry forecasts as well as two setsthat incorporate two common variants of spatial weights to improve forecast accuracy.Findings are discussed in the context of what is currently known about error in small-area demographic forecasts and with an eye toward continued innovations.

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Dear Dr. Charles-Edwards,

We appreciate the editorial comments you made and have incorporated them as is in this revision.

The issue of Queen vs Rook contiguity is corrected—essentially a textual mix-up. There is no

significance that we infer from the modest differences between the two models of spatial dependency

so we have simply switched the text to conform to the accurate description. We have made no changes

in the remainder of the text in presenting or discussing results.

Thanks for your hard work on our manuscript. The only other change is an acknowledgement of our

reviewers and of your inputs.

Response to reviewer's comments

Spatial Weighting Improves Accuracy in Small-Area Demographic Forecasts of

Urban Census Tract Populations

Jack Baker, PhD Senior Research Scientist Geospatial and Population Studies, 1 University of New Mexico MSC 063510 Albuquerque, NM, 87131 USA kali@unm.edu (505) 504-1020 Adélamar Alcántara, PhD Director Geospatial and Population Studies, 1 University of New Mexico MSC 063510 Albuquerque, NM, 87131 USA Xiaomin Ruan Research Scientist Geospatial and Population Studies, 1 University of New Mexico MSC 063510 Albuquerque, NM, 87131 USA Kendra Watkins Socio-economic Program Manager Mid-Region Council of Governments 809 Copper NW Albuquerque, NM 87102

Acknowledgements: This manuscript has been greatly improved by the suggestions made by

two anonymous referees as well as Dr. Elin Charles-Edwards, editor for the Journal of

Population Research. This research was supported by an annual appropriation to Geospatial

and Population Studies by the Legislature of the State of New Mexico to support the Census

Data Dissemination and Demographic Analysis project. While we wish to acknowledge these

contributions, any errors or omissions in either logic or content remain the responsibility of the

authors.

Page Contaning Author's Details

Introduction

Small-area demographic estimation has a long history. Within the United States, the

need for small-area forecasts of population appears to have grown out of a need to plan for

public education demand in the post World War II era (Armstrong and Harris, 1949; Zitter,

1954; Schmid and Shanley, 1952). While this interest has continued (Fabricant and Weinman,

1972; Legare, 1972; Mckibben, 2006), the utility of generating small-area demographic data in

between censuses—and beyond them—has reached the fields of public planning (Cavanaugh,

1981; Smith et. al., 2001; Landis and Zhang, 1998; George, 2004), market analysis (Le Sage and

Pace, 2004; Pace and O-Gilley, 1997; Harris et. al., 2005; Swanson et. al., 2010), public health

surveillance (CDC, 1999; Kuldorff, 1997, 1999), biostatistics (Hund et. al., 2012; Best and

Wakefield, 1999), and GIScience/Geography (Herold et. al., 2003; Ward et. al., 2000; Dietzel

and Clarke, 2007). With such an expanded audience in recent years, it is surprising that so few

studies have attempted to quantify the magnitude and direction of errors associated with

small-area demographic models or to introduce new methodological innovations aimed at

enhancing their accuracy. In this paper, we attempt to fill gaps both in existing knowledge of

the structure of errors for small-area demographic models and in innovations to improve their

accuracy. While we focus on the problem of forecasting: predicting future values of total

population and population within five-year age/sex-specific intervals, it should be noted that

methodological and mathematical equivalency exists with studies focused on the task of

“estimation” (modeling past population values—see Swanson and Tayman, 2012; Smith et. al.,

2001; Baker et. al., 2013). Thus, although we focus on forecasts in the current paper—the

results are relevant to demographers making demographic estimates using equivalent

methods.

Forecast accuracy is typically measured using ex post facto evaluations that compare

forecasts against decennial census counts and compute distributions of “error”—or

discrepancies with the two sets with the census considered as a gold-standard (Tayman, 1999;

Tayman et. al., 1998; Smith, 1987). While it is clear that the census itself is not a gold-standard

(Hogan, 1992, 1993; Fellegi, 1968; Cavanaugh, 1981), it is likely true that a census is always

more accurate than a demographic forecast and it is largely agreed upon that such evaluations

Manuscript (Please exclude authors names from the file)

provide useful information on forecast accuracy. In such evaluation studies, it is a general

finding that existing methods tend to be less accurate in modeling rapidly-changing areas or

those with smaller populations or population groupings (Smith et. al., 2001; Swanson and

Tayman, 2012; Keyfitz, 1981; Isserman, 1977; Smith, 1987; Hoque, 2010), only a handful of

studies have specifically attempted to quantify forecast errors for small-areas. These studies

(Smith and Shahidullah, 1995; Baker et. al., 2012, 2013; Voss and Kale, 1985; Hoque, 2010) have

suggested that errors in total population forecasts over a ten-year forecast horizons should

reach between 15.00 and 25.00 percent. When estimates are made for specific population

age/sex categories, these studies suggest that errors may climb significantly—sometimes as

high as 100.00 percent or greater within a given age/sex interval—though they typically fall in

the range of 20.00 to 40.00 percent. Clearly, demographers should use caution when making

small-area population forecasts—large amounts of error may be present and both Hoque

(2010) and Baker et. al. (2013) report the existence of large-scale outliers that are clearly not

reliable enough for release in a demographic dataset.

With so few studies to validate existing methods for small-area population forecasting,

demographers would benefit from more evaluative studies across a greater range of study

areas. They would also benefit from the introduction of innovative extensions of existing

methods—or entirely new ones—to enhance the accuracy of small-area demographic forecasts.

Perhaps one obstacle to the development of innovations in demography has been a lack of

input data on demographic components of change (births, deaths, migration, residential

construction, etc.) at appropriately small levels to permit implementation of more traditional

methods (Voss and Kale, 1985; Smith et. al., 2001; Cai, 2007; Baker et. al., 2008; Swanson et.

al., 2010). While geocoding (mapping to specific street segments—see Drummond, 1995;

Karimi and Durcik, 2004 for general explanations and Baker et. al., 2012 for demographic

examples) has opened the possibility of such straightforward implementation of demographic

methods at small geographic levels, the known incompleteness of geocoding has been

observed to significantly down-bias demographic models even after significant efforts at data

quality control and remediation and spatial imputation methods have been employed (Baker

et. al., 2012, 2013). Since geocoded data are also known to be biased with respect to important

population characteristics such as race and ethnicity (Gilboa et. al., 2005; Oliver et. al., 2005;

Goldberg et. al., 2007; Zandbergen, 2009), demographers wishing to implement such

procedures also have run the risk of differentially impacting forecast accuracy for specific

population groupings. The findings of Baker et. al. (2013) suggest that methods of trend

extrapolation might perform similarly to the best-performing estimates made using geocoded

data, even within specific age-sex categories. This raises the question of whether such

intensive remediation is worth the effort involved. A conclusion offered by Baker et. al. (2014)

is that in cases where information on population dynamics is necessary—such efforts have a

benefit in estimating components of change; however, existing evidence suggests that

additional complexity in small-area demographic models is not the magic-bullet that

demographers seek to improve forecast accuracy (see also Kale et. al., 1981; Long, 1995; Smith

and Sincich, 1992; Stoto, 1983; White, 1954; Leach, 1981; Pflaumer, 1992).

Chi and Zhu (2008) and Chi and Voss (2011) have suggested an alternative: that the

incorporation of spatial dependency into small-area demographic forecasting may provide

improvements in accuracy that applied demographers have sought. Spatial dependency refers

to the tendency of data physically closer in space to be more similar—this is Tobler’s first law of

geography (Tobler, 1979; Voss, 2004; Haining, 2003). This suggestion is compelling for a

number of reasons. First, it has long been established that the grouping of small-area data into

larger areal units distorts important locally-specific effects and spatial statisticians and

GIScientists make use of spatial autocorrelation models in order to capture such small-area

heterogeneity (Getis and Ord, 2004; Getis, 2009; Getis and Alstadt, 2004; Frotheringham et.

al., 2002). Chi and Zhu (2008) , Chi and Voss (2011), and Baker et. al. (2012) have presented

analyses suggesting that demographic dynamics may be similarly spatially-autocorrelated.

Census data—which form the basis of demographic forecasts—are known to be spatially

autocorrelated (Hogan and Tchernis, 2004; Pace and O’Gilley, 1997; Pattachini and Zenou,

2007) and significant spatial dependency has been observed in socioeconomic status (Duncan

and Duncan, 1955), ethnicity and education (Duncan and Duncan, 1955; Legare, 1972; Alba et.

al., 2000; Vasan et. al., 2014), housing prices (Pace and O’Gilley, 1997; se Can and Meglolugbe,

1997) and other important variables that might also impact models of small-area population

dynamics (Myer, 1954; Massey and Denton, 1985; Legare, 1972). The potential for leveraging

such patterns of spatial-dependency for improving estimation is well-known in other fields

(Pace and O’Gilley, 1997; Le Sage and Pace, 2004; Frotheringham et. al., 2002)—its full

potential in small-area demographic forecasts remains largely untapped (Chi and Zhu, 2008;

Chi and Voss, 2011).

This paper attempts to leverage spatial-dependency within the framework of a well-

known existing component method of demographic forecasting: the Hamilton-Perry or

Component III method (Hamilton-Perry, 1962; Smith and Shahidullah, 1995; Smith et. al.,

2001; George, 2004; Romaniuk, 2009; Swanson et. al., 2010; Baker et. al., 2013). This method

utilizes historical intercensal shifts in population as an extrapolative basis for forecasting

population—and it does so by modeling change for each specific geography to which the model

will be applied (in the current study: US census tracts—polygons containing 1,500 to 8,000

persons, but targeted to 4,000). In this sense, the model is inherently local to the extreme and

one might be tempted to view this model as an extreme example of leveraging spatial

dependency; however, upon careful consideration it is clear why this is not so from a

conceptual basis. In related fields interested in small-area analysis (such as geography or real

estate economics), it has been observed that what is happening within a single neighborhood is

not usually happening in isolation. In fact, it may impact and interact with surrounding

neighborhoods in shaping future growth or decline (Le Sage and Pace, 2004; Dietzel and Clarke,

2007; Landis and Zhang, 1998). Cellular automata models attempting to model urban

expansion have utilized the fact that limits to growth in one neighborhood (build-out given

suitable terrain for example) will shift growth to surrounding, appropriate areas (Herold et. al.,

2003; Ward et. al., 2000). Likewise, Baker et. al. (2008) have found that incorporating carrying

capacity into models of housing unit growth in urban census tracts may provide more accurate

forecasts for similar reasons. Overly-local application of the Hamilton-Perry method might tend

to obscure such dynamics while incorporating them may provide a “smoothing” effect that

allows both a better reflection of the real patterns of demographic change that population

forecasts seek to capture while also leveraging existing patterns of spatial dependency to

improve their accuracy (Chi and Voss, 2011; Chi and Zhu, 2008). Incorporating spatial

dependency may provide an important innovation in small-area demographic forecasting that

when combined with a method such as the Hamilton-Perry procedure—which is capable of

forecasting both total population and age/sex composition—comprises a more adequate

solution to many of the issues associated with small-area demographic forecasting as well as

the needs of data users in disparate fields of study.

From this contextual perspective, this paper attempts to achieve two goals. First it

seeks to add to a growing body of literature that reports the ex post facto errors associated

with demographic forecasts. It accomplishes this goal by reporting the results of an April 1,

2010-based evaluation of errors associated with 1990-2000 based Hamilton-Perry forecasts

when compared against the 2010 census counts (n = 221). Next, it compares these results to

identical evaluations of error for two sets of spatially-weighted Hamilton-Perry forecasts. The

results of the sets are compared in terms of percentage-point improvement in accuracy for

both total population forecasts as well as those made for population within five-year age/sex-

specific intervals. Comparisons are made only for urban census tracts—those within the

municipal boundaries of New Mexico’s fastest-growing and most urbanized cities:

Albuquerque, Las Cruces, Rio Rancho, and Santa Fe. The average size of these census tracts is

approximately 4,000 persons and their standard deviation is approximately 1,250 persons. To

avoid biases associated with size-differences between census tracts (Cavanaugh, 1981; Tayman

et. al., 1999), all comparisons are made using percentage errors. To avoid biases in

comparisons of central tendencies given the well-known non-normality of such error

distributions (Smith and Sincich, 1992; Tayman et. al., 1999), we utilize the median as a

comparison measure as suggested by Baker and coworkers (2013, 2014). The results of are

compared not only across methods, but also by method across strata of the percentage

population change observed in each census tract between 2000 and 2010. These results are

discussed in light of the existing literature on small-area demographic forecasts and further

potential refinements of the method are reviewed.

Materials and Methods

Study Area and Database Formulation

This study focuses on census tracts within urbanized incorporated cities in New Mexico

including: Albuquerque, Las Cruces, Rio Rancho, and Santa Fe. The study area comprises a large

segment of the New Mexico population and provides the only urbanized census tracts within

the State that are comparable to other communities in the Western United States characterized

by rapid growth and booms of residential construction. As such, they are ideally suited to the

purposes of the research reported here. The suitability of similar datasets for the purposes of

research on demographic modeling is evidenced in several previously-published articles in peer-

reviewed journals (Baker et. al, 2008, 2012, 2013a,b). Census 2010 population counts were

extracted for each census tract in the study area (n = 221) to form the basis of the evaluation.

Census 1990 and 2000 data were extracted at the block level and re-aggregated to Census 2010

boundaries to arrive at both a base 2000 population and a 1990 population necessary to

execute the Hamilton-Perry model.

Hamilton-Perry Forecasts: Standard and Spatially-Weighted Variants The Hamilton-Perry method (Hamilton and Perry, 1962; Smith et. al., 2001; Swanson et.

al., 2010; Baker et.al., 2013) relies upon cohort-change ratios that capture change between

census in population counts within corresponding age-intervals as in this example:

CCR 0-4, year0|10-14, year10 = Population 10-14 (2000)/Population0-4 (1990)

Cohort change ratios capture the effects of both deaths and migration within each age/sex

grouping and Smith et. al. (2001), Swanson et. al. (2010), and Swanson and Tayman (2012) have

argued that it represents a true “component” method of population dynamics with the

qualification that the effects of each component are composed into a single change-factor.

Since no data on those aged 0 to 4 or 5 to 9 ten years earlier exists (as these individuals were

not born prior to the previous decennial census), these populations are modeled using

dependency ratios (Hamilton and Perry, 1962; Smith et. al., 2001; Swanson and Tayman, 2012)

that proportionally relate the number of individuals within each age/sex group to the number

of reproductive-aged women (typically between ages 15 and 49), as in this example:

DR males 0 to 4 = Population of males 0 to 4 (2000)/Population of Women 15 to 49 (2000)

The Hamilton-Perry forecast is accomplished by multiplying the previous census population

against the appropriate cohort change ratio to arrive at an estimate of the target population

then multiplying the forecasted population of reproductive-aged women against the

appropriate dependency ratio.

The Hamilton-Perry forecasts were implemented by computing cohort change ratios

based on observed 1990 to 2000 intercensal population change, then multiplication of the

Census 2000 population counts by age/sex in five-year groupings against these ratios to arrive

at estimates of population in each age/sex grouping for the years 10-14 through 80 plus. Both

the cohort-change ratios and the resulting ten-year forecasts were computed for each census

tract individually (the standard Hamilton-Perry). Forecasts were made in the standard method

without spatial weighting, then subjected to spatial-weighting algorithms that average the

values of groups of census tracts depending on specific geographic relationships. These

relationships are captured in spatial weights matrices (Getis and Aldstadt, 2004; Getis, 2009; )

that represent weighted sum of spatially-proximate census tracts based on an apriori

conceptualization of spatial relationships based on contiguity or proximity. In this study, two

commonly-used spatial weight matrices were utilized including those based on rook contiguity

(shared sides) and queen contiguity (shared sides and/or corners). Arithmetically, if a given

census tract had two neighbors j and k (as defined by the contiguity models rook or queen),

then its spatially-weighted forecast would be determined as:

Pop Census Tract i = (Pop tract h*.333) (Pop tract i original*.333) + (Pop tract j *.333)

Rook and Queen contiguity matrices were constructed using ESRI’s Arc-GIS software

package (version 10.1).

Populations aged 0 to 4 and 5 to 9 for each sex were forecasted by computing Census

2000 dependency ratios and holding them constant throughout the forecast horizon. Different

formulations of dependency ratios such as those associated children ages 0 to 4 divided by

women 15 to 44 or children ages 5 to 9 by women aged 20 to 49 were implemented, but made

little difference in the results (less than a percent in any specific instance) and are not reported

or discussed in this paper.

Statistical Characterization of Accuracy

Typically, applied demographers use average errors to capture information on model

accuracy. Mean absolute percentage errors are typically utilized to measure “accuracy” while

mean algebraic percentage errors provide information on “bias”—the tendency of a set of

estimates to be higher or lower than the census count (Tayman et. al., 1998; Smith et. al., 2001;

Smith and Sincich, 1992). Here, we use percentage errors to account for the large variation

observed in population size across the census tracts included in this study since it is well-known

that using simple numeric differences can tend to exaggerate the contribution of tracts with

larger populations to summary measures of error (Tayman, 1999; Tayman et. al., 1998; Smith

et. al., 2001; Swanson and Tayman, 2012; Cavanaugh, 1981). In the current study, nearly all of

the average errors reported were negative, indicating an overall negative bias in both weighted

and unweighted Hamilton-Perry forecasts. Presentation of both sets of errors would create a

very large number of complicated tables to review. Given that nearly all average errors were

negative and, therefore, presentation of absolute errors would provide redundant information,

here we focus on presenting signed, percentage errors to summarize the central tendency of

error distributions associated with each set. Average percentage errors are presented for each

five year age/sex-specific category and across a variety of categories of percentage change

ranging from tracts that lost over 20 percent of their population between 2000 and 2010 and

those that gained over 20 percent during the equivalent time interval. Finally, here we focus on

median errors to better capture central tendency, as mean errors are extremely sensitive to

both non-normality (Tayman et. al., 1999) and outlying values present in small-area

demographic datasets (Witmer and Samuels, 1998; Baker et. al., 2012, 2013).

Results

Table 1 reports errors associated with standard and spatially-weighted Hamilton-Perry

forecasts of total populations. The improvements in accuracy associated with the spatially-

weighted forecasts are clear. Though all sets showed negative bias, non-spatially weighted

forecasts are nearly two times less accurate than spatially-weighted variants: -10.98 percent vs

-5.30 and – 5.19 percent for forecasts based on rook and queen contiguity, respectively. It is

noteworthy that neither spatial weighting model out-performed in forecasts of total

population.

Tables 2 and 3 consider age-specific errors for each model, stratified by sex (males in

Table 2 and females in Table 3). Again, the spatially-weighted variants significantly reduced

error in the forecasts; this result holds for both sexes. Rook-weighted models out-performed

the standard method in 14 out of 18 age groups, with errors ranging between a low of 0.30

percent (35 to 39 year old men) and a high of 23.48 percent (85 plus). Queen-weighted

variants (which are less inclusive since only shared sides qualify them as a “neighbor”) out-

performed the standard Hamilton-Perry forecasts in 16 of 18 age-groups, ranging from a high of

20.39 percent (85 Plus) and a low of .10 percent (40 to 44 year old men). In 15 out of 18 age-

categories, the queen weighed variant was more accurate than either standard or rook-

weighted alternatives. The improvements associated with the queen-weighted forecasts (when

compared to the standard method) ranged from a high of 9.10 percentage points to a low of

only 0.10 percentage point. Most improvements were between 3 and 7 percentage points.

Among females the rook-weighted forecasts out-performed the standard method in 11

of 18 age-groups, with errors ranging between a low of -1.60 percent (85 plus year old women)

and a high of -19.95 percent (50 to 54 year olds). Queen contiguity based forecasts again out-

performed, with improvements over the standard methods observed in 14 of 18 age intervals.

Errors for these forecasts varied between a low of .70 percent (again 85 plus women) and a

high of -20.40 (also again in 50 to 54 year-old women. In 13 of 18 age-groups, the queen-

weighted forecasts were the most accurate. Improvements ranged between only 0.04 percent

(70 to 74 year olds) and 23.97 percentage points (85 plus). Most improvements among females

were more modest, ranging from 1 to 4 percentage points.

Errors in total population estimates across the methods varied widely in relation to the

percentage population change between 2000 and 2010 (Table 4, Figure 1). Again, forecasts of

total population were rather insensitive to the weighting scheme used, with very similar results

between the two spatially-weighted variants across all percentage change groupings. In nearly

all cases, the spatially-weighted variants were more accurate in general and tended to less

grossly mis-estimate the largest growth and decline categories (greater than 20 percent change

in either direction). Within these two groupings, spatially weighted estimates were nearly

twice as accurate (88.47 and 31.16 percent for standard methods compared to 46.26 and 14.93

percent with the queen-weighted forecasts). One notable exception, however, occurred within

those census tracts growing between 0 and 10 percent between 2000 and 2010; among these

census tracts errors exceeded -18.00 percent for the spatially-weighted forecasts while the

standard method displayed an error of only -10.03 percent. These trends are well-captured in

Figure 1. While there is clearly a relationship between percentage change and the accuracy of

forecasts that holds across both spatially-weighted and standard forecasts, it is clear that within

modestly-growing census tracts the standard method out-performed.

Discussion

As with other extrapolation-based methods, the performance of Hamilton-Perry

forecasts depends upon the correspondence between historical and future trends: the

method will perform best when rates and direction of change within any age-cohort are similar

between the two periods and fertility is constant or growing at a constant rate (Smith et. al.,

2001; Swanson and Tayman, 2012). In cases where either of these trends are incongruent

(change in direction) between the historical and horizon-specific periods one should anticipate

relatively poorer performance. Previous studies have uniformly suggested that demographic

forecasts will perform most poorly in areas where rapid change has occurred in either direction

(Smith and Shahidullah, 1995; Isserman, 1977; Keyfitz, 1981; Smith, 1987; White, 1954; Smith

et. al., 2001). In this study, visual inspection of errors revealed that the largest errors observed

were in those census tracts where such incongruent trends have occurred. While there is not a

simple solution to this issue in the context of the Hamilton-Perry method, the results of the

current study are encouraging for its utility and, especially, for the utility of spatially-weighted

versions of it.

Since the Hamilton-Perry method (or any component method) is mathematically

equivalent whether it is applied for making estimates or forecasts (Swanson and Tayman, 2012;

Baker et. al., 2013), the relevance of these results are reviewed in the context of both sets of

studies. From this perspective, the average errors for forecasts of total population using

spatially-weighted variants is at the low end of those reported in previous research, while the

non-weighted standard variant falls well within the range of errors reported in these efforts.

Hoque (2010) presents size-stratified results in his study of vintage 2000 estimates for Texas

counties made using two variants of the Component method, one of which was the Hamilton-

Perry (he refers to it as the Component III method in the sense of Smith et. al., 2001 and

Swanson and Tayman, 2012). He also presents results for a well-known alternative method

based on regression—the ratio-correlation technique (Schmitt and Crossetti, 1954; Swanson

and Tayman, 2012). Hoque reports an average error or -18.93% for a component-based

estimate based on births, deaths, and school-enrollment and an average error or only 10.98%

when using the Hamilton-Perry approach. These results are comparable those obtained using

the standard Hamilton-Perry method in the current study (-10.98 percent), and clearly larger

than the spatially-weighted total population forecasts presented here (-5.30 for Rook-weighted

and – 5.19 for Queen-weighted forecasts). Hoque reports average errors ranging between 4.70

percent and 17.74 percent over similar size categories using the ratio-correlation method.

Smith (1987) reported errors between 18.00 and 25.00 percent using a variety of trend-

extrapolation procedures for Florida census tracts and Voss and Kale (1985) report errors of a

similar magnitude (15.00 to 25.00 percent) in their study of census tract extrapolation-based

forecasts. Smith and Shahidullah (1995) report errors between 17.30 and 27.61 percent for a

variety of methods. Cai (2007) reports a very small average error (0.5 percent) in her study of a

select group of Oregon census tracts (n=147) for which building permit data were available to

track net-migration through housing within a component-based framework that might be

utilized for either estimation or projection.

To our knowledge, only two studies have attempted to address errors in forecasts of

age/sex composition for census tracts or equivalently-small populations: Smith and Shahidullah

(1995), Baker et. al. (2013a) and Baker et. al. (2013b). Smith and Shahidullah (1995) report

their results in ten year age intervals. Errors in that study ranged between -7.3 percent and

11.4 % (0 to 9 year olds and 65 plus with no sex-specific analysis reported). We would expect

that errors would have been somewhat larger in that study if they had been grouped in five-

year age/sex intervals—perhaps similar to the magnitude of errors reported here. However,

perhaps the most important study to compare to the current one is Baker et. al. (2013a). In

that study, census tract level estimates were compared to the 2010 census results for all census

tracts in four representative counties that broadly reflect the statewide characteristics of the

New Mexico population (Bernalillo, Dona Ana, San Juan, and Curry). Errors in this study were

also consistently negative. Among males, these errors ranged between a high of -33.42 percent

(40 to 44 year olds) and a low of only 4.28 percent (0 to 4 year olds). Among females,

magnitudes of error were similar, ranging between a low of 3.68 percent (0 to 4 year olds) and

a high of 24.02 percent (30 to 34 year olds). It is noteworthy that the lowest errors were

observed in intervals where child/woman dependency ratios were utilized to make forecasts,

rather than cohort change ratios. Unlike in the current study—no clear relationship between

percentage change and accuracy of estimates was observed in that research. While the study

area in that research overlapped with the current one, it is important to note that some

important differences exist between the two studies. First, and perhaps foremost, in the

current study only urban census tracts situated within city boundaries were included, while

census tracts within two largely rural counties (Curry and San Juan) were included in the Baker

et. al. (2013) study. This should tend to enhance accuracy even in the absence of spatial

weighting. This limits the comparability of the two studies in spite of some overlap in study

population (Bernalillo and Dona Ana counties)—and it makes the greater accuracy observed in

this study much more comprehensible. The results of this study apply only to urbanized

population dynamics—they are likely generalizable to other urbanized settings of the American

West, but only additional studies in a variety of settings will confirm the conclusions reached

here.

The significantly lower error patterns observed here suggest that the spatial weighting

of demographic forecasts may be a promising avenue for future research within applied

demography. While limitations associated with generalizability are apparent, the results

suggest that further research on the utility of spatial weighting of demographic forecasts is

merited. A variety of spatial-weighting algorithms exist—some base the definition of a

“neighborhood” on a specific number of k closest neighbors while others (such as Delaunay

triangulation) conceptualize proximity in terms of centroids (Getis, 2009; Ord and Getis, 2009;

Getis and Aldstadt, 2004). Statistical methods such as “spatial-scanning” (Kuldorff, 1997, 1999;

Kuldorff and Nagarwala,1995; Kuldorff et. al., 2005) apply a rough equivalent to a

neighborhood matrix in applying a variably-sized circle-based scanning window to define

groupings of census tract population data. Such a flexible scanning window based on a

stochastic neighborhood matrix might provide an avenue for extending the concept of

“neighborhood” to provide refinements to the simple method presented here. Likewise,

though it would be computationally more intensive, it is also possible that spatial weightings

could be adjusted to optimize fit between forecasts and census data at one point in time, then

carried forward within forecasts. This would be analogous to the optimization of portfolio

weightings within an investment model except that the objective function here would be

minimization of error rather than an maximization of return on investment given a specific risk

appetite (Markowitz, 1952; De Miguel et. al., 2009).

The most significant challenge to successful implementation of the Hamilton-Perry

method for demographic forecasts observed in this study, was in zones where there was an

incongruency of trends between the previous intercensal period and the forecast period.

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