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Population ForecastingPopulation Forecasting

Time Series Forecasting TechniquesTime Series Forecasting Techniques

Wayne Foss, MBA, MAI

Wayne Foss Appraisals, Inc.Email: wfoss@fossconsult.com

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Extrapolation TechniquesExtrapolation Techniques

Real Estate Analysts - faced with a difficult taskReal Estate Analysts - faced with a difficult task long-term projections for small areas such aslong-term projections for small areas such as

» CountiesCounties

» Cities and/orCities and/or

» NeighborhoodsNeighborhoods

Reliable short-term projections for small areasReliable short-term projections for small areas Reliable long-term projections for regions countriesReliable long-term 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 of Forecasts 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 past is 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 would are calculations of future conditions that would

exist as a result of adopting a set of underlying exist as a result of adopting a set of underlying assumptions.”assumptions.”

Forecast:Forecast: ““is a judgmental statement of what the analyst is 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 and The distinction between projections and forecasts are important because:forecasts are important because: Analysts often use projections when they should be Analysts 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 be Analysts prepare projections that they know will be

accepted as forecasts without evaluating the accepted 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 project Using 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 the total 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 without deals only with aggregate trends from the past without attempting to account for the underlying demographic and attempting to account for the underlying demographic and economic processes that caused the trends.economic processes that caused the trends.

Less appealing than the cohort-component Less appealing than the cohort-component techniques or economic analysis techniques that techniques 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 data Disaggregated 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 Fitting -Curve Fitting - Analyzes past data to identify overall trends of Analyzes past data to identify overall trends of

growth or declinegrowth or decline

Curve Extrapolation -Curve 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) is This 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 in aggregate Population change reflects the change in aggregate of three factors:of three factors: birthsbirths deathsdeaths migrationmigration

These factors are time related and are caused by These factors are time related and are caused by other time related factors:other time related factors: health levelshealth levels economic conditionseconomic conditions

Time is a proxy that reflects the net effect of a large Time is a proxy that reflects the net effect of a large number of unmeasured events.number of unmeasured events.

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CaveatsCaveats

The extrapolation technique should never be used to The extrapolation technique should never be used to blindly assume that past trends of growth or decline will blindly assume that past trends of growth or decline will continue into the future.continue into the future. Past trends observed, not because they will always continue, Past trends observed, not because they will always continue,

but because they generally provide the best available but because they generally provide the best available information about the future.information about the future.

Must carefully analyze:Must carefully analyze: Determine whether past trends can be expected to continue, Determine whether past trends can be expected to continue,

oror If continuation seems unlikely, alternatives must be consideredIf continuation seems unlikely, alternatives must be considered

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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 geometric Difference 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 variable increasesgrowth increments increase or decrease as the x variable increases

Caution should be exercised when using for long Caution should be exercised when using for long range projections.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 will This 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 patterns very 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 / Yc Yc = 1 / Yc-1-1 where Yc where Yc-1-1 = c + ab = c + abX X c = Upper limitc = Upper limit b = ratio of successive growthb = ratio of successive growth a = constanta = constant

Identical to the Modified Exponential and Identical to the Modified Exponential and Gompertz curves, except:Gompertz curves, except: observed values of the modified exponential curve and the observed values of the modified exponential curve and the

logarithms of observed values of the Gompertz curve are replaced by logarithms of observed values of the Gompertz curve are replaced by the reciprocals of the observed values.the reciprocals of the observed values.

Result: the ratio of successive growth increments of the reciprocals Result: the ratio of successive growth increments of the reciprocals of the Yc values are equal to a constantof 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 Extrapolation Selecting 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 increments - or the first Linear: if growth increments - or the first

differences for the observation data are differences for the observation data are approximately equal - approximately equal -

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

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Selecting Appropriate Extrapolation Selecting Appropriate Extrapolation Projections, con’tProjections, con’t

Curve AssumptionsCurve Assumptions Parabolic: Characterized by constant 2nd Parabolic: Characterized by constant 2nd

differences (differences between the first difference differences (differences between the first difference and the dependent variable) if the 2nd differences and the dependent variable) if the 2nd differences are approximately equal are approximately equal

Modified Exponential: characterized by first Modified Exponential: characterized by first differences that decline or increase by a constant differences that decline or increase by a constant percentage; ratios of successive first differences are percentage; ratios of successive first differences are approximately equalapproximately equal

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Selecting Appropriate Extrapolation Selecting Appropriate Extrapolation Projections, con’tProjections, con’t

Curve AssumptionsCurve Assumptions Gompertz: Characterized by first differences in the Gompertz: Characterized by first differences in the

logarithms of the dependent variable that decline logarithms of the dependent variable that decline by a constant percentageby a constant percentage

Logistic: characterized by first differences in the Logistic: characterized by first differences in the reciprocals of the observation value that decline by reciprocals of the observation value that decline by a constant percentagea constant percentage

Observation data rarely correspond to any Observation data rarely correspond to any assumption underlying the extrapolation curvesassumption underlying the extrapolation curves

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Selecting Appropriate Extrapolation Selecting Appropriate Extrapolation Projections, con’tProjections, con’t

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,ME and MAPE In General: Curve with the lowest CRV,ME and MAPE should be considered the best fit for the observation should be considered the best fit for the observation datadata

JudgementJudgement is required is required Select the Curve that produces results consistent with Select the Curve that produces results consistent with

the most likely futurethe most likely future

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Selecting Appropriate Extrapolation Selecting Appropriate Extrapolation Projections, con’tProjections, con’t

Year Linear Linear Geometric Parabolic ModifiedOdd Even Exponential

1960 98,640 101,114 98,956 94,683 101,0171965 109,050 109,669 108,263 111,029 109,9791970 119,460 118,223 118,444 123,417 118,6721975 129,870 126,777 129,583 131,849 127,1051980 140,280 135,331 141,770 136,323 135,2851985 150,690 143,886 155,102 136,840 143,2201990 161,100 152,440 169,688 133,400 150,9171995 171,510 160,994 185,647 126,003 158,3842000 181,920 169,549 203,106 114,649 165,626

Alternate Estimates and Projections

Curve CRV ME MAPE Upper LimitLinear (Odd) 0.01 0.00 2.82% noneLinear (Even) 0.01 -1,030.95 1.61% noneGeometric 0.01 56.87 3.17% noneParabolic 351.27 0.00 1.41% noneModified 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)-D+HU = ((BP*N)-D+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 Average Number of Bldg. Permits

issued per year since most recent censusissued per year since most 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)-D+HU = ((BP*N)-D+HUaa)*OCC)*OCC

» HHHHgg = ((193*5)-0+0)*95.1% = ((193*5)-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 That’sSo That’s Population Forecasting Population Forecasting

Wayne Foss, MBA, MAI, Fullerton, CA USA Email: waynefoss@usa.net