FIN454 Population Forecasting

8/2/2019 FIN454 Population Forecasting

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Population ForecastingPopulation Forecasting

Time Series Forecasting TechniquesTime Series Forecasting Techniques

Wayne Foss, MBA, MAI

Wayne Foss Appraisals, Inc.Email: [email protected]


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Extrapolation TechniquesExtrapolation Techniques

Real Estate AnalystsReal Estate Analysts -- faced with a difficult taskfaced with a difficult task

longlong--term projections for small areas such asterm projections for small areas such as

CountiesCounties

Cities and/orCities and/or

NeighborhoodsNeighborhoods

Reliable shortReliable short--term projections for small areasterm projections for small areas

Reliable longReliable long--term projections for regions countriesterm projections for regions countries

Forecasting task complicated by:Forecasting task complicated by:

Reliable, Timely and Consistent informationReliable, Timely and Consistent information


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Sources of ForecastsSources of Forecasts

Public and Private Sector ForecastsPublic and Private Sector Forecasts

Public: California Department of FinancePublic: California Department of Finance

Private: CACIPrivate: CACI

Forecasts may be based on large quantities ofForecasts may be based on large quantities of

current and historical datacurrent and historical data


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Projections are ImportantProjections are Important

Comprehensive plans for the futureComprehensive plans for the future

Community General Plans forCommunity General Plans for

Residential Land UsesResidential Land Uses

Commercial Land UsesCommercial Land Uses

Related Land UsesRelated Land Uses

Transportation SystemsTransportation Systems

Sewage SystemsSewage Systems SchoolsSchools


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DefinitionsDefinitions

Estimate:Estimate:

is an indirect measure of a present or pastis an indirect measure of a present or past

condition that can be directly measured.condition that can be directly measured.

Projection (or Prediction):Projection (or Prediction):

are calculations of future conditions that wouldare calculations of future conditions that would

exist as a result of adopting a set of underlyingexist as a result of adopting a set of underlying

assumptions.assumptions. Forecast:Forecast:

is a judgmental statement of what the analystis a judgmental statement of what the analyst

believes to be the most likely future.believes to be the most likely future.


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Projections vs. ForecastsProjections vs. Forecasts

The distinction between projections andThe distinction between projections and

forecasts are important because:forecasts are important because:

Analysts often use projections when they should beAnalysts often use projections when they should be

using forecasts.using forecasts.

Projections are mislabeled as forecastsProjections are mislabeled as forecasts

Analysts prepare projections that they know will beAnalysts prepare projections that they know will be

accepted as forecasts without evaluating theaccepted as forecasts without evaluating the

assumptions implicit in their analytic results.assumptions implicit in their analytic results.


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ProcedureProcedure

Using Aggregate data from the past to projectUsing Aggregate data from the past to project

the future.the future.

Data Aggregated in two ways:Data Aggregated in two ways:

total populations or employment without identifying thetotal populations or employment without identifying the

subcomponents of local populations or the economysubcomponents of local populations or the economy

I.e.: age or occupational makeupI.e.: age or occupational makeup

deals only with aggregate trends from the past withoutdeals only with aggregate trends from the past without

attempting to account for the underlying demographic andattempting to account for the underlying demographic andeconomic processes that caused the trends.economic processes that caused the trends.

Less appealing than the cohortLess appealing than the cohort--componentcomponent

techniques or economic analysis techniques thattechniques or economic analysis techniques that

consider the underlying components of change.consider the underlying components of change.


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Why Use Aggregate Data?Why Use Aggregate Data?

Easier to obtain and analyzeEasier to obtain and analyze

Conserves time and costsConserves time and costs

Disaggregated population or employment dataDisaggregated population or employment data

often is unavailable for small areasoften is unavailable for small areas


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Extrapolation: A Two Stage ProcessExtrapolation: A Two Stage Process

Curve FittingCurve Fitting --

Analyzes past data to identify overall trends ofAnalyzes past data to identify overall trends of

growth or declinegrowth or decline

Curve ExtrapolationCurve Extrapolation --

Extends the identified trend to project the futureExtends the identified trend to project the future


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Assumptions and ConventionsAssumptions and Conventions

Graphic conventions Assume:Graphic conventions Assume:

Independent variable:Independent variable: x axisx axis

Dependent variable:Dependent variable: y axisy axis

This suggests that population change (y axis) isThis suggests that population change (y axis) is

dependent on (caused by) the passage of time!dependent on (caused by) the passage of time!

Is this true or false?Is this true or false?


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Assumptions and ConventionsAssumptions and Conventions

Population change reflects the change inPopulation change reflects the change in

aggregate of three factors:aggregate of three factors:

birthsbirths

deathsdeaths

migrationmigration

These factors are time related and are causedThese factors are time related and are caused

by other time related factors:by other time related factors: health levelshealth levels

economic conditionseconomic conditions

Time is a proxy that reflects the net effect of aTime is a proxy that reflects the net effect of a

large number of unmeasured events.large number of unmeasured events.


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Alternative Extrapolation CurvesAlternative Extrapolation Curves

LinearLinear

GeometricGeometric

ParabolicParabolic

Modified ExponentialModified Exponential

GompertzGompertz

LogisticLogistic


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Linear CurveLinear Curve

Formula:Formula: Yc = a + bxYc = a + bx

a = constant or intercepta = constant or intercept

b = slopeb = slope

Substituting values of x yields YcSubstituting values of x yields Yc

Conventions of the formula:Conventions of the formula:

curve increases without limit if the b value > 0curve increases without limit if the b value > 0

curve is flat if the b value = 0curve is flat if the b value = 0

curve decreases without limit if the b value < 0curve decreases without limit if the b value < 0


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Linear CurveLinear Curve


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Geometric CurveGeometric Curve

Formula:Formula: Yc = abYc = abxx

a = constant (intercept)a = constant (intercept)

b = 1 plus growth rate (slope)b = 1 plus growth rate (slope)

Difference between linear and geometricDifference between linear and geometric

curves:curves: Linear:Linear: constant incremental growthconstant incremental growth

Geometric:Geometric: constant growth rateconstant growth rate

Conventions of the formula:Conventions of the formula:

if b value > 1 curve increases without limitif b value > 1 curve increases without limit

b value = 1, then the curve is equal to ab value = 1, then the curve is equal to a

if b value < 1 curve approaches 0 as x increasesif b value < 1 curve approaches 0 as x increases


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Geometric CurveGeometric Curve


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Parabolic CurveParabolic Curve

Formula:Formula: Yc = a + bx + cxYc = a + bx + cx22

a = constant (intercept)a = constant (intercept)

b = equal to the slopeb = equal to the slope c = when positive: curve is concave upwardc = when positive: curve is concave upward

when = 0, curve is linearwhen = 0, curve is linear

when negative, curve is concave downwardwhen negative, curve is concave downward

growth increments increase or decrease as the x variablegrowth increments increase or decrease as the x variable

increasesincreases

Caution should be exercised when using forCaution should be exercised when using for

long range projections.long range projections.

Assumes growth or decline has no limitsAssumes growth or decline has no limits


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Parabolic CurveParabolic Curve


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Modified Exponential CurveModified Exponential Curve

Formula:Formula: Yc = c + abYc = c + abxx

c = Upper limitc = Upper limit

b = ratio of successive growthb = ratio of successive growth a = constanta = constant

This curve recognizes that growth willThis curve recognizes that growth will

approach a limitapproach a limit Most municipal areas have defined areasMost municipal areas have defined areas

i.e.: boundaries of cities or countiesi.e.: boundaries of cities or counties


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Modified Exponential CurveModified Exponential Curve


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Gompertz CurveGompertz Curve

Formula:Formula: Log Yc = log c + log a(bLog Yc = log c + log a(bxx)) c = Upper limitc = Upper limit

b = ratio of successive growthb = ratio of successive growth

a = constanta = constant

Very similar to the Modified Exponential CurveVery similar to the Modified Exponential Curve

Curve describes:Curve describes:

initially quite slow growthinitially quite slow growth

increases for a period, thenincreases for a period, then growth tapers offgrowth tapers off

very similar to neighborhood and/or city growth patternsvery similar to neighborhood and/or city growth patterns

over the long termover the long term


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Gompertz CurveGompertz Curve


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Logistic CurveLogistic Curve

Formula:Formula: Yc = 1 / YcYc = 1 / Yc--11 where Ycwhere Yc--11 = c + ab= c + abXX

c = Upper limitc = Upper limit

b = ratio of successive growthb = ratio of successive growth

a = constanta = constant

Identical to the Modified Exponential andIdentical to the Modified Exponential and

Gompertz curves, except:Gompertz curves, except: observed values of the modified exponential curve and theobserved values of the modified exponential curve and the

logarithms of observed values of the Gompertz curve are replacedlogarithms of observed values of the Gompertz curve are replacedby the reciprocals of the observed values.by the reciprocals of the observed values.

Result: the ratio of successive growth increments of theResult: the ratio of successive growth increments of the

reciprocals of the Yc values are equal to a constantreciprocals of the Yc values are equal to a constant

Appeal: Same as the Gompertz CurveAppeal: Same as the Gompertz Curve


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Logistic CurveLogistic Curve


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Selecting Appropriate ExtrapolationSelecting Appropriate Extrapolation

ProjectionsProjections

First: Plot the DataFirst: Plot the Data

What does the trend look like?What does the trend look like?

Does it take the shape of any of the six curvesDoes it take the shape of any of the six curves

Curve AssumptionsCurve Assumptions

Linear: if growth incrementsLinear: if growth increments -- or the firstor the first

differences for the observation data aredifferences for the observation data areapproximately equalapproximately equal --

Geometric: growth increments are equal to aGeometric: growth increments are equal to a

constantconstant


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Projections, contProjections, cont


Parabolic: Characterized by constant 2ndParabolic: Characterized by constant 2nd

differences (differences between the first differencedifferences (differences between the first differenceand the dependent variable) if the 2nd differencesand the dependent variable) if the 2nd differences

are approximately equalare approximately equal

Modified Exponential: characterized by firstModified Exponential: characterized by first

differences that decline or increase by a constantdifferences that decline or increase by a constantpercentage; ratios of successive first differences arepercentage; ratios of successive first differences are

approximately equalapproximately equal


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Gompertz: Characterized by first differences in theGompertz: Characterized by first differences in the

logarithms of the dependent variable that declinelogarithms of the dependent variable that declineby a constant percentageby a constant percentage

Logistic: characterized by first differences in theLogistic: characterized by first differences in the

reciprocals of the observation value that decline byreciprocals of the observation value that decline by

a constant percentagea constant percentage

Observation data rarely correspond to anyObservation data rarely correspond to any

assumption underlying the extrapolationassumption underlying the extrapolation

curvescurves


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Test Results using measures of dispersionTest Results using measures of dispersion

CRV (Coefficient of relative variation)CRV (Coefficient of relative variation)

ME (Mean Error)ME (Mean Error)

MAPE (Mean Absolute Percentage Error)MAPE (Mean Absolute Percentage Error)

In General: Curve with the lowest CRV,MEIn General: Curve with the lowest CRV,ME

and MAPE should be considered the best fitand MAPE should be considered the best fit

for the observation datafor the observation data JudgementJudgementis requiredis required

Select the Curve that produces resultsSelect the Curve that produces results

consistent with the most likely futureconsistent with the most likely future


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Year Linear Linear Geometric Parabolic Modified

Odd Even Exponential

1960 98,640 101,114 98,956 94,683 101,017

1965 109,050 109,669 108,263 111,029 109,979

1970 119,460 118,223 118,444 123,417 118,6721975 129,870 126,777 129,583 131,849 127,105

1980 140,280 135,331 141,770 136,323 135,285

1985 150,690 143,886 155,102 136,840 143,220

1990 161,100 152,440 169,688 133,400 150,917

1995 171,510 160,994 185,647 126,003 158,384

2000 181,920 169,549 203,106 114,649 165,626

Alternate Estimates and Projections

Curve CRV ME MAPE Upper Limit

Linear (Odd) 0.01 0.00 2.82% none

Linear (Even) 0.01 -1,030.95 1.61% none

Geometric 0.01 56.87 3.17% none

Parabolic 351.27 0.00 1.41% none

Modified Exponential 73.29 -46.53 3.55% 400,000

Input and Output Evaluation Statistics


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Housing Unit MethodHousing Unit Method

Formulas:Formulas:

1) HH1) HHgg = ((BP*N)= ((BP*N)--D+HUD+HUaa)*OCC)*OCC

2) POP2) POPgg = HH= HHgg * PHH* PHH

3) POP3) POPff= POP= POPcc + POP+ POPgg Where:Where: HHHHgg Growth In Number of HouseholdsGrowth In Number of Households

BPBP Average Number of Bldg. Permits issued per year sinceAverage Number of Bldg. Permits issued per year since

most recent censusmost recent census

NN Forecast period in YearsForecast period in Years

HUHUaa No. of Housing Units in Annexed AreaNo. of Housing Units in Annexed Area

OCCOCC Occupancy RateOccupancy Rate

POPPOPgg Population GrowthPopulation Growth

PHHPHH Persons per HouseholdPersons per Household

POPPOPcc Population at last censusPopulation at last census

POPPOPff

Population ForecastPopulation Forecast


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Housing Unit Method ExampleHousing Unit Method Example

Forecast Growth in Number of Housing UnitsForecast Growth in Number of Housing Units

1)1) HHHHgg = ((BP*N)= ((BP*N)--D+HUD+HUaa)*OCC)*OCC

HHHHgg = ((193*5)= ((193*5)--0+0)*95.1%0+0)*95.1%

HHHHgg = 918= 918

Forecast Growth in PopulationForecast Growth in Population 2) POP2) POPgg = HH= HHgg * PHH* PHH

POPPOPgg = 918 * 2.74= 918 * 2.74

POPPOPgg = 2,515= 2,515

Forecast Total PopulationForecast Total Population

3)3) POPPOPff= POP= POPcc + POP+ POPgg POPPOPff= 126,003 + 2,515= 126,003 + 2,515

POPPOPff= 128,518= 128,518


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So ThatsSo Thats

Population ForecastingPopulation Forecasting

Wayne Foss, MBA, MAI, Fullerton, CA USA

Email: [email protected]

Documents

FIN454 Population Forecasting