Uncertainty in Sociai Security Trust Fund Projections an… · Uncertainty in Sociai Security Trust...

Uncertainty in Sociai Security TrustFund Projections

Noah Meyerson &John SabelhausCongressional BudgetOffice, Washington,D.C. 20515

National Tax JournalVol. Llll, No. 3, Part 1

Abstract - This paper presents measures of uncertainty about So-cial Security Trust Fund projections based on the new Long-TermActuarial Model (LTAM) being developed at the Congressional Bud-get Office. Measuring the variance in Social Security outcomesinvolves three steps: specifying a model, characterizing uncertaintyabout model inputs, and generating useful measures of the uncer-tainty about model outputs. There are significant trade-offs to bemade at each step, which can affect measured uncertainty in im-portant ways. The UAM framework is a promising approach forreconciling differences in other studies of uncertainty about long-term Social Security finances.

INTRODUCTION

Current projections for the Old Age Survivor's and Dis-ability Insurance (OASDI) program suggest a dramatic

change in financing fundamentals starting in about ten years.As the baby boom cohort begins to retire around 2010, thecurrent excess of payroll tax revenues over benefits paid isexpected to dwindle and eventually turn negative. Theseprojections are, of course, based on a number of demographicand economic assumptions, all of which are uncertain. Thispaper considers the issue of uncertainty of OASDI projec-tions using the new Long-Term Actuarial Model (LTAM) be-ing developed at the Congressional Budget Office.'

Evaluating vmcertainty about Social Security projections isnot a new idea. The Office of the Chief Actuary (OCACT) atthe Social Security Administration (SSA) annually producesboth baseline and alternative (low cost and high cost) projec-tions when evaluating OASDI finances. Those alternativesare intended to give some sense of the uncertainty about thelong-run forecasts, although the range of outcomes is not as-sociated with any specific probabilistic interpretation. Thelack of specific probabilistic interpretations (such as confi-dence intervals and standard deviations) from OCACT forstandard financial measures has led the Social Security Ad-visory Board (1999) to advocate more effort in this direction,

' The LTAM model is being developed by the Long-Term Modeling Group atCBO, which includes Amy Rehder Harris, Michael Simpson, and Joel Smith,and the authors. We are grateful to them and other colleagues at CBO whohave supported the LTAM project in general and this paper in particular.

515

NATIONAL TAX JOURNAL

and has also led researchers outside ofOCACT (for example, Hoimer (1996;1999; 2000) and Lee and Tuljapurkar(1998a; 1998b)) to develop their own mod-els in order to measure uncertainty aboutOASDI projections.

The first principle for evaluating alter-native approaches to measuring uncer-tainty is the ability of a given measure toinform policy decisions. Expected valuesabout input assumptions (and modelstructure) are not sufficient to make goodpolicy; two policies with the same ex-pected values for system finances butdifferent variances for the outcomes areobviously not equivalent. However, vari-ance in OASDI outcomes arises for tworeasons—uncertainty about input as-sumptions and the sensitivity of theOASDI system to changes in those in-puts—and those have different implica-tions for policy. Both estimates of imcer-tainty about inputs and model specifica-tion are subject to error, so the overallcalculated variance in trust fund projec-tions will reflect both sources of error.

The first (and most widely studied)source of variance in OASDI projectionsis that which stems directly from uncer-tainty about the input assumptions. ForOASDI, there are several demographic as-sumptions (mortality improvement, fer-tility, and immigration) and economic as-sumptions (inflation, interest rates, wagegrowth, unemployment, disability inci-dence, and termination) that go into mak-ing a projection, all of which are uncer-tain. Although historical data exists forthese inputs, the extent of uncertaintyabout the future path for those inputs isstill subject to interpretation. Indeed, it canbe argued that some approaches to usinghistorical data for measuring how muchuncertainty exists about certain input as-sumptions may be suspect because secu-lar changes have occurred in the under-lying process generating that assumption.For example, some might claim (withmore certainty than historical data sug-

gest) that fertility rates will probably notdouble from the levels today to those thatexisted during the baby boom era.

The second source of variability is frommodel specification (meaning the OASDIforecasting model, as distinct from themodel(s) that generate values for the in-puts). When making projections, it seemsobvious that starting with the best fore-casting model is a good idea. However, amodel which reflects everything that isknown about a system as complex asOASDI by definition will never get built,and even feasible models (such asOCACT's) are too complex to be solvedrepeatedly, as in a Monte Carlo simula-tion. In the real world researchers buildsimplified models, and the degree of sim-plification may limit the ability to mea-sure uncertainty because some input as-sumptions are not incorporated in thestripped-down model. Thus, two differ-ent forecasting models may generate twodifferent variance estimates for outcomeseven if the variability of input assump-tions are the same.

The fact that there are two sources ofvariance in outcomes complicates esti-mates of the overall level and relativesources of variance in program projec-tions. Both are important, but the sourceof uncertainty across the various input as-sumptions is especially interesting be-cause policy choices may depend on esti-mated sensitivity. One set of policy rulesmay have "shock absorber" propertieswith respect to an input assumption,meaning the long-run financial viabilityof the system is not much affected by val-ues for that assumption. For example,under current rules, if wage growth risespayroll tax revenues will inunediately in-crease, but benefits paid will also eventu-ally rise and (in the long-run) offset theincreased revenues, leaving the overallfinances basically the same. There is asmall improvement in system financesbecause of the timing, but that is second-ary. Changes in mortality rates do not

516

Uncertainty in Social Security Trust Fund Projections

have this "shock absorber" property un-der current rules, however, because retire-ment age is not explicitly linked to ex-pected longevity.

Clearly, there is a direct relationshipbetween the estimate of uncertainty aboutan input assumption and the degree ofoutcome sensitivity with respect to thatinput. If the OASDI system will automati-cally absorb shocks from a given input,the inability to measure the uncertaintyabout that input is not important for esti-mating uncertainty about system finances,though it may be important for measur-ing outcomes like rates of retum to indi-viduals in the system.

If the overall OASDI system financesare sensitive to changes in an input, thenthe extent of (measured or assumed) vari-ance in that input will determine the vari-ance of the overall system outcomes (withrespect to that input). If one analyst in-fers there is little variance in mortalityimprovements, while a second analystfinds there is significant variance, andthey use an otherwise identical forecast-ing model for system finances, the secondanalyst will obviously find a wider vari-ance in program outcomes. This issuecould be important because it is easy toimagine policy changes being recom-mended because of how much they wouldreduce (estimated) variance in the trustftmd projections.

Distinguishing the two sources of un-certainty is a main organizing principle forthis paper. The next section discusses thespecification of a projection model, witha particular focus on the OCACT tech-niques as a reference point. The third sec-tion shows the results of comparing LTAMinput responses with those fromOCACT's model. The model specifictiondictates which input assumptions (and/or model characteristics) can be varied,but the issues of how to vary the assump-tions in a Monte Carlo simulation is ad-dressed separately in the fourth section.The fifth section discusses an interesting

aspect of stochastically-generated fore-casts; average outputs from those projec-tions are not necessarily equal to themeans from deterministic solutions of thesame model. The last section presents es-timates of OASDI system variance usingthe LTAM model and several alternativespecifications for variability in the inputassumptions.

MODELING LONG-TERM SOCIALSECURITY FINANCES

The specification of a forecasting model(given values for the input parameters)generally receives limited consideration instudies of imcertainty about long run So-cial Security finances—most of the atten-tion is devoted to estitnating the varianceof the input parameters. The premise isthat one should develop a forecasting toolwhich adequately captures how changesin assumptions will affect the OASDI out-comes of interest, then vary the assump-tions stochastically to generate ranges forthose outcomes. However, the choice ofmodel structure does involve a crucialtradeoff: the ability to accurately measurethe sensitivity of system finances with re-spect to changes in the input parametersversus the ability to build and solve themodel quickly under alternative valuesfor those parameters. In this section thattradeoff is described in some detail; thereference point is the OCACT forecast pro-cedure, and the comparison is with othermodels used for studying uncertainty,including the LTAM model used in thispaper.

Given what goes into generating abaseline OASDI forecast, OCACT cur-rently has little choice but to use the sce-nario approach for measuring imcertainty,because the number of times they cansolve their model under alternative as-sumptions is limited. To some extent it isnot even meaningful to speak of runningthe OCACT "model" over and over againwith alternative parameter assumptions—

517


the OCACT projection methodology in-volves a sequence of several modelswhich are run by individual analysts, us-ing a variety of different software appli-cations, each suited to the particularmodel being implemented. Thus, each al-ternative simulation involves a number ofsteps for a series of people, not just a se-ries of steps in one linked set of computercode.

The steps that go into generating anOASDI forecast are easy to list, but diffi-cult to describe concisely and with anyreasonable level of detail.^ OCACT firstprojects population by detailed age, sex,and marital status groups; then projectsemployment, payroll tax revenues, num-ber of beneficiaries, and average benefitsby combining economic assumptions withthe underlying demographics; then finallycomputes trust fund outcomes by sum-ming payroll tax revenues, interest re-ceived, other inflows and subtracting ben-efits paid, administrative costs, and otheroutflows. For all but the demographic pro-jections, there are also generally distinctshort and long run projection methodolo-gies.

There is nothing inherent in the OCACTdemographic modeling that rules out sto-chastic analysis, but it is easy to see whyit is complicated to introduce randomness.Incrementing the cells in the populationmatrix involves subtracting deaths by ageand sex, adding births by sex, adding im-migrants by age and sex, and distribut-ing the population within each age-sexgroup by marital status in each forecastyear. Each of the four demographic pro-cesses is itself a model which builds onthe outputs from previous models. Forexample, fertility rates are set exog-enously, but the level of fertility will de-pend on fertility rates and the femalechild-bearing population, which dependson the number of female deaths in thechild-bearing age groups. Marital status

for a given age-sex group depends on thenumber of (available) people in the age-sex cell and the pattems of marriage acrossthe age distribution. Thus, to make thisprocess stochastic, all of the models haveto be linked together.

Some parts of the OCACT projectionprocess would be even more difficult (ifnot impossible) to make stochastic with-out significant restructuring. For ex-ample, the macroeconomic and laborforce/employment sector of the model isactually managed by a time-series soft-ware package. The macro model takesthe population by age, sex, and maritalstatus as given, and generates projectionsfor the number of covered workers andaggregate earnings for the forecast period.Aggregate earnings feed into a calcula-tion of taxable payroll (based on the tax-able maximum OASDI parameter) andthose outcomes are fed directly into amodel (again, in a different software pro-gram than the macro model) which ana-lyzes OASDI program outcomes. Thecovered-worker outcomes (by age andsex) also flow into the computer programwhich determines expected benefitawards.

The covered-worker model outcomesaffect OAI and DI worker average benefitsin the OCACT projection sequence (asthey should) because benefits are esti-mated using a micro-simulation based onthe Continuous Wage History Sample(CWHS) longitudinal data set which iscalibrated to be consistent with the esti-mated group-level work histories. Themicro-simulation starts with samples ofnew OASI and DI beneficiaries in 1996and alters the micro-level data so that theindividual work histories are consistentwith the overall patterns by age and sexfor the relevant cohort, where thosegroup-level employment patterns aretaken as given from the macro/labor mar-ket model. Benefits are also affected by

For a highly technical and fairly comprehensive description of CXIACT projection techniques, see Frees (1999).

518


average wage growth in the economy,which is also solved for in the macro/la-bor market model.

The new benefit awards (by age andsex) solved for in the micro-simulationmodel then feed into other parts of themodel. Auxiliary benefit levels for the 25or so different benefit recipient programsare set relative to the insured OAI and DIworker average benefits. Then, averagebenefits for both insured and auxiliarybeneficiaries are multiplied by the num-ber of beneficiaries (independently solvedfor in another series of programs) to solvefor total benefits paid. The outputs of thevarious models are pulled together by afinal set of programs, which add othercomponents of income and cost (such asadministrative expenses and revenuefrom income taxes on benefits) and thenincrement trust fund balances.

This highly simplified description of theOCACT projection sequence does notcome close to revealing the true complex-ity of the models. For example, estimatesof mortality rates by age and sex are basedon quite complicated procedures that be-gin with estimates of disease-specificdeath rates and incidence. Also, short andlong-term forecasting techniques for eachof the sub-models (except demographics)differ, because recent trends are morelikely to influence one's view of the shortrun, while stability is a guiding principlefor the long run. In any case, it is easy tosee why synthesizing all of these modelsinto one program unit where one can re-petitively draw values for the input as-sumptions and run Monte Carlo simula-tions would be very difficult.

That difficulty has led other research-ers to develop stripped-down models forprojecting OASDI finances in order tomeasure uncertainty (in particular,Hoimer (1996; 1999; 2000) and Lee andTuljapurkar (1998a; 1998b)). The recentmodels developed by Hoimer (1999; 2000)also incorporate macroeconomic feedbackeffects (in addition to stochastic capabili-

ties for the input assumptions) which isalso made possible by simplifying the pro-jection model. The LTAM model used herefor measuring vmcertainty about OASDIprojections is another example of astripped-down model capable of makingstochastic projections, though it is muchcloser to the OCACT approach than othermodels.

The various stripped-down models usedifferent approaches to simplifying theprojection process. For example. Lee andTuljapurkar (1998a; 1998b) use their ownmodel of demographics combined withempirical age-«ex profiles of average pay-roll taxes collected and benefits received,which are aged forward through time us-ing an aggregate productivity growthvariable (which is stochastic). This limitsthe capability of the model to do policysimulations, because, for example, ben-efits are not linked to an underlying mi-cro data set as in the OCACT approach.But the authors are able to conduct somesimple policy experiments, like raising thenormal retirement age, by making ad hocadjustments to average payroll taxes andbenefits for the age groups that will beaffected. This model also limits the num-ber of assumptions which can be varied,because, for example, disability incidenceand termination are not modeled explic-itly.

The Hoimer (1999; 2000) model doesallow for stochastic analysis of all the mainOCACT input assumptions: Fertility, im-migration, mortality improvement, fe-male and male labor force participation,unemployment rate, inflation rate, pro-ductivity growth rate, wage share growthrate, hours worked growth rate, nominalinterest rate, and disability incidence andrecovery rates are all stochastic. TheHoimer (1999; 2000) model seems to fol-low OCACT projection techniques moreclosely than Lee and Tuljapurkar (1998a;1998b), but there are still simplifications.For example, OAI worker benefits arebased on group-level labor force partici-

519


pation and average wages for cohorts inthe years right before they retire, ratherthan a micro-simulation using longitudi-nal work histories.

The initial objective for the LTAMproject was to build a model which mim-ics the crucial pieces of the OCACTprocedure, so that if an input or policyvariable is changed, the model will auto-matically respond appropriately. LTAMembodies the crucial machinery for pro-jecting population, labor force, payroll taxrevenues, and benefit awards (using amicro-simulation) in a way that is consis-tent with OCACT. Discrepancies betweenLTAM and OCACT projections are re-flected in a series of calibration factors thatare applied in all simulations, so baselineprojections will match, and alternativesimulations (those with different inputassumptions or policy parameters) willyield answers that should be consistentwith OCACT's.

As in other stripped-down models,LTAM does not attempt to duplicate muchof the effort that goes into an OCACT pro-jection. Rather, the model embodies onlythe long-run projection techniques, andmuch of the model is based on exogenousratios derived from OCACT data files. Forexample, LTAM solves for the ratio of aux-iliary (spouse, widow, child) benefits tounderlying OAI or DI worker benefitsusing OCACT benefit projections, thenapplies those ratios in simulations. Aux-iliary benefits will then change if and onlyif underlying worker benefits (determinedby the micro-simulation) change. Otherparts of the model work in a similar way;these exogenous ratios will be replacedwith explicit forecasting models as LTAMevolves.

Though the initial goal of buildingLTAM is generating deviations fromOCACT baselines under alternative inputassumptions and policy parameters, themodel architecture was set up to allow

subsequent expansion in at least two di-mensions. First, the model is written as aself-contained unit in one piece of soft-ware, so it is straightforward to introduceMonte Carlo simulation as was done forthis paper. Second, the model is solvedyear by year (rather than sector by sector,as in the OCACT procedure), which willmake it feasible to extend the model toinclude macro feedbacks and even simul-taneity across sectors, which could even-tually lead to a full general equilibriumapproach.

As of this writing LTAM has basic ca-pabilities for stochastic simulation on fourinput assumptions: mortality improve-ment, fertility, immigration, and the realinterest rate. Although the current modeldoes allow users to change other assump-tions that are on the input assumption list(unemployment, inflation, real wagegrowth, labor force participation, disabil-ity incidence, and termination), those in-puts all require the benefits micro-simu-lation to be re-run, which takes abouteight minutes, and is thus prohibitive foruseful stochastic runs of 1,000 or so simu-lations. Speeding up the micro-simula-tion is a priority for near-term LTAM de-velopment work.

SENSITIVITY ANALYSIS USING LTAM

Although the stripped-down modelsuse simpler approaches to estimating howdeviations in assumptions cause differ-ences in outcomes, there is a well-estab-lished set of criteria used to evaluatewhether or not these simplified models areadequate for measuring the stochasticproperties of the OASDI forecast. The cri-teria is whether they can (1) duplicateOCACT baseline projections and (2) dupli-cate OCACT's sensitivity analysis experi-ments in which one input assumption ischanged and the impact on some summarytrust fund outcome is evaluated.' This

These sensitivity analysis results are published in the armual Trustees Reports.

520


suggests a general consensus that OCACTtechniques (setting aside possible disagree-ment about input assumptions) generatesa good reference baseline and that theirsensitivity analysis also adequately repre-sents how the OASDI system will respondto given changes in the inputs.

Table 1 compares the effect of changingseveral input assumptions in both theOCACT and LTAM models. The tableshows the 75 year actuarial balance, whichis (roughly) the present value of OASDIcosts divided by taxable payroll less the

present value of OASDI income dividedby taxable payroll.'' The baseline OCACTactuarial balance of -2.07 percent impliesthat payroll tax rates would have to beraised (or cost rates would have to be cut)by 2.07 percent of payroll to put the sys-tem in balance over the 75 year period.The estimate of -2.03 percent from LTAMdiffers only because of some approxima-tions built into the model.^

For each of the inputs in Table 1, actu-arial balance estimates are produced forthe range of that input used in the low,

TABLE 1SENSITIVITY OF 75-YEAR ACTUARIAL BALANCE TO CHANGES IN INPUT ASSUMPTIONS: SOCIALSECURITY ADMINISTRATION (OCACT) AND CONGRESSIONAL BUDGET OFFICE (LTAM) ESTIMATES

Mortality Improvement

Fertility

Immigration

Interest Rate

Wage Growth

Inflation (CPI-W)

Disability Incidence

Disability Termination

OCACTLTAM

OCACTLTAM

OCACTLTAM

OCACTLTAM

OCACTLTAM

OCACTLTAM

OCACTLTAM

OCACTLTAM

Low Cost

-1.47-1.72

-1.74-1.67

-1.90-1.90

-1.59-1.53

-1.55-1.62

-1.84-1.85

-1.77-1.36

-2.01-1.87

Intermediate Cost

-2.07-2.03

-2.07-2.03

-2.07-2.03

-2.07-2.03

-2.07-2.03

-2.07-2.03

-2.07-2.03

-2.07-2.03

High Cost

-2.75-2.51

-2.42-2.40

-2.18-2.12

-2.64-2.62

-2.57-2.46

-2.29-2.36

-2.35-2.74

-2.12-2.11

Source: OCACT values from 1999 Trustees Report. LTAM values are author's calculations using CongressionalBudget Office Long-Term Actuarial Model (LTAM).Note: All values are a percent of taxable payroll.

' The reason these are "roughly" true is that the actual measure also considers the balance currently in theOASDI trust fund and the actuarial requirement that the fund have one year's worth of outlays left at the endof the valuation period.

' Both of these values relate to the 1999 Trustees Report estimates. LTAM does not yet have 2000 Trustees Reportinput data, so the OCACT 1999 values were used to keep things comparable. The 2000 actuarial balance isslightly less negative than the 1999 value.

521


medium, and high cost scenarios pro-duced by OCACT. Most of those rangesare easily reproduced in LTAM—for ex-ample, the fertility rate ranges from 1.6(high cost) to 2.2 (low cost) per woman,and since LTAM uses the same basic fer-tility model as OCACT, that parametercan be varied directly. The same is true ofimmigration, the interest rate, wagegrowth, and inflation. Mortality improve-ment is somewhat more difficult to repro-duce, because the range is for "overall"mortality improvement, yet the rate ofimprovement in the OCACT alternativesand LTAM is also differentiated by ageand sex. To generate the table, a range ofmortality improvement for LTAM waschosen such that the average of male andfemale life expectancy matched the rangepublished by OCACT.

In general the sensitivity of LTAM com-pares well to the sensitivity of the OCACTmodel with respect to key input assump-tions. The first four inputs will be variedin the Monte Carlo simulation presentedbelow. Within those four, the results forfertility, immigration, and the interest rateare basically identical across the two mod-els, while the results for mortality are closebut show some indication that LTAM isnot allocating the improvements in mor-tality across age and sex groups appropri-ately, which indicates this part of themodel requires some attention. In general,the actuarial balance is not as sensitive tochanges in life expectancy in LTAM as itis in the OCACT model.

The other four input assumptions in thebottom half of Table 1 are not currently inthe stochastic simulations for LTAM, be-cause changing any of these will lead to achange in average benefits, which causesthe internal micro-simulation to be in-voked. It is straightforward to run acouple of simulations on each input forthe purpose of doing sensitivity analysis,but repeated draws at eight minutes persolution are currently infeasible. (With-out the micro-simulation the model solves

in a little less than one second). Improv-ing the speed of the nnicro-simulation inorder to make these inputs stochastic isan important development area for thenear-term.

Table 1 shows that LTAM does a prettygood job capturing the sensitivity of theactuarial balance with respect to changesin the non-stochastic assumptions,though the match is not as close as it waswith the demographic variables, espe-cially for disability. This makes sense, be-cause the demographic simulations effec-tively (through the calibration procedure)take all of the economic variables (for ex-ample, payroll taxes per worker, numberof workers relative to population, averagebenefits, and number of beneficiaries rela-tive to population) as given. If simulatedchanges in population by age and sexmatch the changes in the OCACT model,the overall system outcomes will match.It is much more difficult to match on in-puts like wage growth, because this feedsright into some of the more complicatedparts of the model. Wage growth affectsthe benefits micro-simulation directly, in-teracts with the taxable maximum in anon-linear way to determine total taxablepayroll, and even affects labor force par-ticipation of older workers as benefitschange.

CHARACTERIZING UNCERTAINTYABOUT MODEL INPUTS

Given a model which is able to repli-cate sensitivity with respect to changes ininput assumptions, the second step instudying variance of OASDI projectionsis characterizing uncertainty about thoseinput assumptions. The first decisionwhen characterizing uncertainty is tochoose between "scenario" and MonteCarlo approaches to measuring variabil-ity in outcomes. There is little doubt thatmore information can be gleaned usingMonte Carlo simulation, but the choicemay be constrained by the nature of the

522


projection model. The second decision iswhether to specify a single "ultimate"long-nm outcome or annual values foreach input assumption. Finally, one hasto choose ad hoc or empirical methods tomeasure variability in the inputs; if theempirical approach is taken, the methodused to recover estimates of variance (andcorrelation with other inputs) from his-torical data is crucial, because differentapproaches may lead to very differentanswers about the variance of input as-sumptions.

In order to give a sense of how far thevarious OASDI outcomes (summary in-come and cost rates, actuarial balance)might deviate from the baseline (mid-range) values, OCACT bundles what itconsiders to be a reasonable range for as-sumptions into "low cost" and "high cost"scenarios, and thus produces a total ofthree projections. This scenario approachwas critiqued by the 1999 Technical Ad-visory Panel on four grounds: it assumesthat trajectories for the inputs are alwayshigh or low (ruling out boom/bust pat-tems), it assumes correlations between theoutcomes are rigid (all assumptions highcost or all assumptions low cost), it is pos-sible that the likelihood a given assump-tion will fall within the high/low rangewill be much less than the likelihood thesummary measure will fall within thehigh/low range, and OCACT does notexplicitly assign probability distributionsfor inputs.

The first two critiques are explicitlyabout how to estimate variability in in-puts, and the last two are about why thosevariance estimates are necessary to assignprobabilistic interpretations to outcomes.It is true that one needs input probabili-ties to assign output probabilities, but itis also true that Monte Carlo simulation(or systematic evaluation of each possibleprobability-weighted "state" for the in-puts) is required (in the multivariate in-put case) in order to derive variances forthe outcomes. That is, even if we know

the joint probability distribution of theinputs, evaluating the system with justtwo points from that joint distribution isnot sufficient to draw conclusions aboutthe probability distribution for the out-come being analyzed. It is sort of a mootpoint that OCACT does not explicitly statewhat the assumed probability distributionis for each input, because the requisitesteps for making inferences about the dis-tribution of outcomes involves repeatedsolutions to the model that are infeasible.

Given the capability to run Monte Carlosimulations, one has to choose ultimate orannual draws for the inputs and be ex-plicit about correlations between thosevariables in order to run simulations. Thecurrent version of LTAM is designed tobe solved in a non-stochastic mode usingthe same input "levers" as the OCACTmodel, and thus the assumptions are allmodeled as ultimate values (with linearinterpolation between the first year andthe ultimate value in a fixed year; for ex-ample fertility hits its ultimate value in2023). Thus, there is only one stochasticdraw per input assumption per simula-tion.

The probability distributions for the in-puts are also specified in a very ad hocway. The model assumes that the rangefor low and high cost values of each in-put represents some points on a normaldistribution for that input. For example,the average of the gap between the low-medium and high-medium range for avariable is taken as one or two standarddeviations in the simulation. The correla-tions for the variables are set to either one(meaning all four move towards low costor high cost together) or zero (meaningthere is no correlation).

The ad hoc specification for the prob-ability distributions is a useful place tostart, but it is obvious that empirical in-vestigation of the input assumptions is acrucial area for LTAM development work.There is a substantial literature on mea-suring variability of input assumptions for

523

NATiONAL TAX JOURNAL

OASDL including Foster (1994), Sze(1996), Hoimer (1996), Frees, et al. (1997),and the previously mentioned studies byLee and Tuljapurkar (1998a; 1999b) andHoimer (1999; 2000). In general, theseanalyses are based on time-series model-ing of the input processes, and the find-ings raise a number of issues about thefuture work planned for LTAM. In par-ticular, one can draw different inferencesabout the stochastic properties of inputassumptions using the same historicaldata, and that will affect conclusions aboutoverall variability in OASDI outcomes aswell as the sensitivity of the baseline withrespect to particular inputs.

A good example of this issue ariseswhen modeling fertility. Attempts to fitunconstrained time-series models to fer-tility rates (see, for example. Lee andTuljapurkar (1994)) imply unrealistic pro-jections—negative fertility rates in the fu-ture in particular—because the down-ward trend following the baby boom erais so pronounced. Therefore the modelhas to be constrained in some way. Themethod chosen by Lee and Tuljapurkar(1994; 1998a; 1998b) is to fix the long-runmean for the fertility rate at the OCACTmid-point (currently 1.9 children perwoman) and interpret deviations fromthat trend as (autocorrelated) variability.

This approach to inferring the variabil-ity in fertility rates can certainly be ques-tioned. If one believes that structuralchanges in marital pattems, labor forceactivity of women, living arrangements,acceptance of birth control, or other de-terminants has dramatically reduced thechance that fertility rates will climb backto baby boom levels, then the pure time-series estimate of the variance will be bi-ased upwards. Of course, the only way totest this is to build a model of fertility thatincludes those structural determinantsand then compute the residual variance.This opens a can of worms, however, be-cause a comprehensive variance analysiswould then require consideration of un-

certainty about the structural determi-nants and even the uncertainty about thecoefficients on those determinants.

The time-series approach can also leadto conclusions which work in the other di-rection. For example. Lee and Tuljapurkar(1994; 1998a; 1998b) find that a trend-based model of mortality improvementfits the historical data quite well. Thatmodel leads them to two conclusions: first,the authors conclude that the overallOCACT rate of mortality improvement inthe mid-range assumptions is too low,because it implies a slowdown from trend,and second, the authors (implicitly) con-clude the variance of OASDI outcomeswith respect to uncertainty about mortal-ity reduction is relatively low, becausethey find that the overall variance ofOASDI outcomes when just mortality isstochastic is relatively low. OCACT, on theother hand, builds their projections ofmortality improvement by consideringdisease-specific death rates, so more thanjust trend analysis is involved.

Although it is clear that the next stepfor improving the LTAM stochastic simu-lation capability is to work with empiri-cal models of input imcertainty, these ob-servations about time-series analysis sug-gest some caution should be exercised. Ata minimum, some sensitivity analysis ofthe time-series specification is in order. Itis also likely (see Hoimer (2000) in par-ticular) that for some of the economic in-put assumptions it is important to modelthe correlations between the inputs, us-ing either a Vector Auto Regression (VAR)or explicit macro-model framework todraw inferences about variability in theinputs.

STOCHASTIC BiAS

Before presenting the estimates of un-certainty about OASDI finances usingLTAM, it is important to address one phe-nomenon that can occur when a modellike this is used in a stochastic setting. It

524


seems natural that the expected outcomeof a model should be the same in a sto-chastic and non-stochastic setting, butthat is not the case for some previous stud-ies and (to some extent) for the LTAM re-sults presented here. This section explainswhy "stochastic bias" will affect the out-comes in these types of simulations.

Consider the following: Y is the OASDIoutcome of interest, say the 75 year or an-nual actuarial balance. X̂ is the vector ofinputs in state s, ;r is probability that states will occur, and/ ( ) is the model that re-lates outputs to inputs. Therefore, Y =f(X) is the outcome for OASDI in state s.When OCACT makes its medium-costprojection, it plugs in the expected valuefor inputs, £(X) = X ;r X ,̂ and solves forthe expected outcome using E(Y) =/(E(X))=/(X ;r X )̂. In a stochastic setting, how-ever, the expected value of the OASDIoutcome is computed using £ (Y) = E n f{X), where each ;r is set equal to the in-verse of the number of stochastic draws.

The non-stochastic and stochasticmeans for the OASDI outcome will matchif the OADSI projection model (that is,/ (X) itself) is probability-weighted sym-metric in X, so that equal probability val-ues of X on either side of its expected val-ues cause the same (but opposite signed)

change in the outcome of interest. Thissymmetry occurs, for example, if/ (X) islinear with respect to inputs or the prob-ability distributions for the inputs andsensitivity of / (X) arovind the expectedvalue for X are both symmetric.

The possibility of stochastic bias israised because Hoimer (1999), for ex-ample, finds a significant change in theestimated actuarial balance when hismodel is solved stochastically and deter-ministically. His estimated 75 year actu-arial balance worsens from -2.41 percentto -2.95 percent when the inputs are madestochastic, and the only possible explana-tions are the phenomenon of asymmetrydescribed above (which Hoimer implic-itly argues for in his description) and thepossibility that some drift in the expectedvalues for the inputs is (inadvertently)introduced through his specification forthe input assumption equations (perhapsthe correlations between variables). In thatcase, it is the E (X) that has changed, whichis a separate problem.

Table 2 shows that there is a possibilityof stochastic bias in LTAM. The effect onthe 75 year actuarial balance from vary-ing each of the four input assumptions iscompared over the range moving fromlow to intermediate cost, and from inter-

TABLE 2CONDITIONAL SENSITIVITY OF 75-YEAR ACTUARIAL BALANCE IN LTAM

TO CHANGES IN INPUT ASSUMPTIONS

AssumptionVaried

Mortality

Fertility

Immigration

Interest Rate

Range Varied

Low->IntennediateIntermedia te->High

Low->IntermediateIntermediate->High

Low->IntermediateIntermediate->High

Low->IntermediateIntermedia te->High

Low Cost

0.260.39

0.280.28

0.120.07

0.370.41

Change in Actuarial BalanceWhen All Other Inputs Set To...

IntermediateCost High Cost

0.310.48

0.360.37

0.130.09

0.500.59

0.390.60

0.500.51

0.170.10

0.670.82

Source: Congressional Budget Office Long-Term Actuarial Model (LTAM).

Note: All values are a percent of taxable payroll.

525


mediate to high. If those are equal, thesystem response is basically linear. Eachof these comparisons is done with all otherassumptions set to intermediate cost(which corresponds to differences betweenthe columns in Table 1), low cost, or highcost. The differences in pairs within eachcolumn imply some non-linearity; the ef-fect of mortality improvement when allother inputs are set to intermediate is .31percent of payroll over the low to inter-mediate cost range, and .48 percent of pay-roll over the intermediate to high range.

The estimated non-linearities in Table2 are noticeable, but actually turn out tohave only a small impact on the mean es-timates of system outcomes presented inthe next section. That conclusion is en-tirely dependent on the assumption aboutinput variance. If the input variance is in-creased significantly, the non-linearitiesare more pronounced, and the level of sto-chastic bias rises.

VARIANCE ESTIMATES FOR OASDIUSING LTAM

This section presents estimates of vari-ability in OASDI projections using LTAM.

The estimates are generated using verysimple assumptions about the variabilityof inputs. Ultimate values for each of thefour inputs are first assumed to be inde-pendent and normally distributed, withmeans equal to the OCACT intermediatevalues and base case standard deviationsequal to the average of the gap betweenintermediate/low and intermediate/highranges for the assumption. Two of thevariants run involve shrinking the stan-dard deviations so the OCACT scenariorange represents two standard deviations,and making the inputs perfectly corre-lated rather than independent.

Table 3 shows several output statisticsfor three sets of model solutions: deter-ministic, and the independent draws (for1,000 simulations each) where the stan-dard deviations are set to the base case andone-half the base case. The first observa-tion is that there is a small (but statisti-cally significant amoimt) of stochastic biasin the base case—the mean estimate forthe 75 year actuarial balance falls from-2.03 in the deterministic case to -2.05.The other statistics are also affected, es-pecially the measures from the end of theprojection period in 2075.

TABLE 3OASDI OUTCOMES IN LTAM BASED ON 1000 INDEPENDENT MONTE CARLO SIMULATIONS FOR

MORTALITY IMPROVEMENT, FERTILITY, IMMIGRATION, AND THE REAL INTEREST RATE

Outcome

75-Year Actuarial Balance75-Year Cost Rate75-Year Income Rate

Actuarial Balance in 2030Cost Rate in 2030

Actuarial Balance in 2075Cost Rate in 2075

Aged-Dependency Ratio in 2030Aged-Dependency Ratio in 2075

DeterministicSolution

-2.0315.5613.52

-4.6417.71

-6.5319.88

19.8022.70

Stochastic Solution, OCACT Input Range Interpreted As...

One Standard Deviation

Mean

-2.0515.5813.53

-4.6517.72

-6.9520.32

19.8023.10

StandardDeviation

0.780.780.07

0.400.42

3.473.66

0.804.40

Two Standard Deviations

Mean

-2.0315.5613.53

-4.6417.71

-6.6620.00

19.8022.80

StandardDeviation

0.390.380.04

0.200.21

1.671.76

0.402.20

Source: Congressional Budget Office Long-Term Actuarial Model (LTAM).

Note: All values are a percent of taxable payroll, except aged-dependency ratio, which is the fraction of thepopulation aged 65+.

526


The base case standard deviation for the75 year actuarial balance is 0.78, suggest-ing a 95 percent confidence interval ofroughly -0.5 percent to -3.5 percent. Ofcourse this estimate (and all the estimatesin Table 2) are entirely dependent on thead hoc assumption about the size of thestandard deviation and the independenceof the input assumptions. The last twocolumns in Table 2 show the impact ofcutting the standard deviations in half (sothe assumption range in the Trustees Re-port is interpreted as two standard devia-tions). The overall standard deviations fallby almost exactly half, suggesting the sys-tem response is very linear over this range.Note also that tightening the standarddeviations effectively eliminates the sto-chastic bias for the summary measures.

Although the absolute level of uncer-tainty is entirely dependent on the as-sumptions about input variability, thereare some interesting observations aboutuncertainty for the various time periodsconsidered. The actuarial balance and costrates for 2030 are much more certain thanthey are for 2075. Remember, this versionof the model does not have uncertaintyabout economic assumptions (except thereal interest rate) so fertility and mortalityare the only significant influences, andboth of those take time to cause big

changes. But, the same variance assump-tions that overwhelmingly suggest the sys-tem will be in significant deficit in 2030(actuarial balance of -4.65 percent of pay-roll, standard deviation of 0.4) also sug-gest it is unclear what the system will looklike in 2075 (actuarial balance of-6.95 per-cent of payroll, standard deviation of 3.47).

Table 4 shows some other aspects ofhow the model responds to variability inthe inputs, focusing on just two outputstatistics but with different assumptionsabout which variables are stochastic orhow they are related (all standard devia-tions are set to base case values). The twostatistics considered are the 75 year actu-arial balance, and the actuarial balance in2075. Again the table shows (on the firstline) the deterministic solution and (on thesecond line) the four independent stochas-tic variables solution. The next four linesshow the effect of varying each input in-dividually—all other inputs are fixed atthe intermediate values. The rank orderof the variability estimates is generallyconsistent with OCACT univariate sensi-tivity analysis, though the problems withmortality improvement in LTAM are evi-dent because of the (relatively) weaker re-sponse than the interest rate (the standarddeviation for mortality should be a httlehigher, not lower).

TABLE 4OASDI OUTCOMES IN LTAM BASED ON 1000 MONTE CARLO SIMULATIONS UNDER ALTERNATIVE

SPECFICATIONS FOR INPUT ASSUMPTIONS

Input Assumptions

Deterministic

Independent Stochastic DrawsVarying Only;

Mortality ImprovementFertilityImmigrationReal Interest Rate

Perfectly Correlated Stochastic Draws

Output Statistic

75-Year Actuarial Balance

Mean

-2.03

-2.05

-2.01-2.05-2.04-2.06

-2.30

StandardDeviation

n.a.

0.78

0.410.370.110.52

1.53

Actuarial Balance in 2075

Mean

-6.53

-6.95

-6.45-6.99-6.56-6.53

-7.56

StandardDeviation

n.a.

3.47

1.662.990.26n.a.

5.37Source; Congressional Budget Office Long-Term Actuarial Model {LTAM).


527


TABLE 5COMPARISON OF UNIVARIATE AND SCENARIO OUTCOMES IN OCACT MODEL

75-Year Cost RateFull ScenarioSum of Univariate Sensitivity Analyses

75-Year Income RateFull ScenarioSum of Univariate Sensitivity Analyses

75-Year Actuarial BalanceFull ScenarioSum of Univariate Sensitivity Analyses

Lovif Cost

13.3713.37

13.1412.74

0.230.63

Intermediate Cost

13.4913.49

15.5615.56

-2.07-2.07

High Cost

13.6213.60

18.6018.43

-4.97-4.83

Source: 1999 Trustees Report.


The other interesting aspect of the fourindividual simulations is that the sum ofthe four individual standard errors (forthe 75 year actuarial balance, 1.41) is abouttwice the overall standard deviation. Thisis the same basic result as reported by Leeand Tuljapurkar (1998b), and reinforcesthe idea that only Monte Carlo simulationallows one to construct a cumulative den-sity function for the system outcomes^tcannot be inferred just from the univariateresponses.

The last row of Table 4 shows the LTAMsolution with perfectly correlated drawsfor the input assumptions. Two observa-tions jump out from the table. First, theevidence of stochastic bias is quite strong,as the mean 75 year actuarial balance fallsfrom -2.03 (deterministic) to -2.30, and themean actuarial balance in 2075 falls from-6.53 (deterministic) to -7.56. Second, theestimated standard deviations for the twostatistics shown are only slightly largerthan the sum of the univariate standarddeviations, suggesting that negative andpositive draws for all the variables rein-force the overall negative or positive im-pact on the system finances, but not bymuch. That possibility is confirmed inTable 5, which shows OCACT's own esti-mates for 75 year cost rates, income rates,and actuarial balances from their scenario(perfectly correlated errors) and sum ofunivariate analyses. OCACT also findsthat the system is fairly "additive" across

the inputs, because the sum of the effectson the 75 year actuarial balance estimatein the low or high cost univariate simula-tions roughly correspond to the outcomesin the corresponding scenarios. (Also,there is some variation in the scenario,which is not reflected in the univariateanalysis, so the actual values are evencloser).

CONCLUSIONS

The LTAM project at the CongressionalBudget Office is focused on developing acapability to study expected long-termOASDI finances with respect to the effectsof both baseline assumptions and policyparameters. In this paper the model wasused to analyze the variance in long-^termprojections by making simple assump-tions about the variability of four modelinputs: mortality improvement, fertility,immigration, and the real interest rate.The results show that LTAM generatesresults that are consistent with OCACTmodel outputs, and that adding uncer-tainty has predictable and stable effects onthe model results.

This paper represents the first efforts atmaking LTAM stochastic. The next stepfor the model is to make the remaininginput variables (inflation, wage growth,disability incidence and termination, un-employment) stochastic. As noted, thatwill involve speeding up parts of the

528


solution algorithm, particularly the mi-cro-simulation component, but it iscomputationally feasible. The larger taskis to decide how the variability of all theinputs should be modeled, which involvesthe use of time-series techniques to re-cover the variance estimates from histori-cal data. As argued in the paper, how-ever, this step should be taken with a gooddeal of caution, because historical datamay have only limited capacity for pre-dicting future uncertainty.

Acknowledgments

The views expressed here are those ofthe authors and should not be interpretedas those of the Congressional Budget Of-fice.

REFERENCES

Board of Trustees, Federal Old-Age andSurvivors Insurance Disability InsuranceTrust Funds.

1999 Annual Report of the Board of Trustees ofthe Federal Old-Age and Survivors InsuranceDisability Insurance Trust Funds. (HouseDocument 106^8). Washington, D.C: Gov-emment Printing Office, March, 1999.

Frees, Edward W. (Jed)."Summary of Social Security AdministrationProjections of the OASDI System." SocialSecurity Advisory Panel Working Paper,December, 1999.

Frees, Edward W. Qed), Yueh-Chuan Kung,Marjorie A. Rosenberg, Virginia R. Young,and Siu-Wai Lai.

"Forecasting Social Security Actuarial As-sumptions." North American Actuarial jour-nal 1 No. 4 (October, 1997): 49-84.

Foster, Richard S."A Stochastic Evaluation of the Short-RangeEconomic Assumptions in the 1994 OASDITrustees Report." Social Security Adminis-tration Actuarial Study No. 109. Baltimore:

MD: SSA Office of the Actuary, August,1994.

Hoimer, Martin R."Demographic Results from SSASIM, aLong-Run Stochastic Simulation Model ofSocial Security." Appendix A in Report of the1994-1995 Advisory Council on Social Secu-rity, Volume U. Washington, D.C: U.S. Gov-emment Printing Office, 1996.

Hoimer, Martin R."Stochastic Simulation of Economic GrowthEffects of Social Security Reform." In Pros-pects for Social Security Reform, edited byOlivia S. Mitchell. Philadelphia: Universityof Pennsylvania Press for the Pension Re-search Council, 1999.

Hoimer, Martin R.Introductory Guide to SSASIM. Washington,D.C: Policy Simulation Group, March, 2000.Available at www.polsim.com.

Lee, Ronald D., and Shripad Tuljapurkar."Stochastic Population Projections for theUnited States: Beyond Low, Medium,High." Journal of the American Statistical As-sociation 89 No. 428 (December, 1994): 1175-89.

Lee, Ronald D., and Shripad Tuljapurkar."Uncertain Demographic Futures and SocialSecurity Finances." American Economic Re-view 88 No. 2 (May, 1998a): 237-41.

Lee, Ronald D., and Shripad Tuljapurkar."Stochastic Forecasts of Social Security." InFrontiers in the Economics of Aging, edited byDavid A. Wise. Chicago: IJniversity of Chi-cago Press, 1998b.

Social Security Advisory Board.The 1999 Technical Panel on Assumptions andMethods. Washington, D.C: Social SecurityAdvisory Board, November, 1999.

Sze, Michael."Stochastic Simulation of the Financial Sta-tus of Social Security Trust Funds in the Next75 Years." Appendix B in Report of the 1994r-1995 Advisory Council on Social Security, Vol-ume II. Washington, D.C: U.S. GovemmentPrinting Office, 1996.

529

Uncertainty in Sociai Security Trust Fund Projections an… · Uncertainty in Sociai Security Trust...

Documents

Transcript of Uncertainty in Sociai Security Trust Fund Projections an… · Uncertainty in Sociai Security Trust...