Hidden Group Time Profiles: Heterogeneous Drawdown Behaviours in Retirement
Igor Balnozan, Denzil G. Fiebig, Anthony Asher, Robert Kohn, Scott A. Sisson
HHidden Group Time Profiles:Heterogeneous Drawdown Behaviours in Retirement
Igor Balnozan ∗† , Denzil G. Fiebig ∗ , Anthony Asher ∗ , Robert Kohn ∗ , Scott A. Sisson ∗ Version: September 2020
Abstract
This article investigates retirement decumulation behaviours using the Grouped Fixed-Effects (GFE) estimator applied to Australian panel data on drawdowns from phasedwithdrawal retirement income products. Behaviours exhibited by the distinct latent groupsidentified suggest that retirees may adopt simple heuristics determining how they drawdown their accumulated wealth. Two extensions to the original GFE methodology areproposed: a latent group label-matching procedure which broadens bootstrap inference toinclude the time profile estimates, and a modified estimation procedure for models withtime-invariant additive fixed effects estimated using unbalanced data.
Keywords
Panel data, discrete heterogeneity, microeconomics, retirement
JEL codes
C51, D14, G40
Acknowledgements
Balnozan is grateful for the support provided by the Commonwealth Government of Australia through theprovision of an Australian Government Research Training Program Scholarship. We thank Plan For Life, Ac-tuaries & Researchers, who collected, cleaned and allowed us to analyse the data used in this research. Thisdata capture forms one part of a broader survey into retirement incomes, commissioned by the Institute of Ac-tuaries of Australia. This research includes computations using the computational cluster Katana supportedby Research Technology Services at UNSW Sydney. Balnozan, Kohn and Sisson are partially supported bythe Australian Research Council through the Australian Centre of Excellence in Mathematical and StatisticalFrontiers (ACEMS; CE140100049).
Conflict of interest statement
The authors have no conflict of interest to declare. ∗ University of New South Wales, UNSW Sydney, NSW 2052, Australia † Correspondence: [email protected] / West Lobby Level 4, UNSW Business School Building, UNSWSydney, NSW 2052, Australia. a r X i v : . [ ec on . E M ] S e p Introduction
The importance and prevalence of preference heterogeneity is a pervasive feature of microe-conomic modelling. Such heterogeneity is usually unobserved, remaining unexplained aftercontrolling for the observable characteristics of individuals. Where the effects of unobserv-ables have clear economic interpretations, methods of identifying latent groups of individualsthat share common behaviours can provide insights into economic phenomena.In the retirement incomes literature, there remains a need to understand retiree behavioursin countries where retirees primarily generate income by drawing down savings accumulatedin personal Defined Contribution accounts. Appropriate policy design and product develop-ment depends on understanding these behaviours. While theoretical work in this area is wellestablished, existing empirical studies lack the statistical methodology required to draw clearconclusions regarding the heterogeneity in retiree behaviours observed in the data. Work byBateman and Thorp (2008) and Balnozan (2018) gives reason to expect distinct behaviouralgroups, where group behaviours correspond to strategies that individuals employ when ac-cessing their retirement savings using phased withdrawal products. One of the expected be-haviours is to draw constant dollar amounts, while another is to follow closely the legislatedminimum drawdown rates.This article investigates drawdown behaviours in phased withdrawal retirement income prod-ucts by studying latent group effects estimated using the Grouped Fixed-Effects (GFE) esti-mator of Bonhomme and Manresa (2015). In this application, using the GFE estimator alsoallows testing whether retirees were heterogeneous in their responses to the Global FinancialCrisis and retirement incomes policy changes. This analysis demonstrates the usefulness ofthe GFE estimator applied to such event studies.In the presence of group-level time-varying unobservable heterogeneity, using the standardtwo-way fixed-effects (2WFE) model, defined in Section 4, generally gives biased covariateeffect estimates and unrepresentative time effect estimates. This is what the present studyfinds: estimates for the time effects in the standard 2WFE model, which allows for only oneset of time effects, obscure the distinct latent group-level effects which suggest retirees mayadopt simple heuristics determining how they draw down their accumulated wealth duringretirement. The importance of this comparison with the 2WFE model results motivates ourmodification to the GFE estimator, described in Section 4.A key novelty in this application of the GFE estimator is the focus on performing statisticalinference on the estimated effects of the latent heterogeneity. To robustify the GFE estimatorin this scenario, this article proposes an extension to the pairs bootstrap method outlined inthe supplement to Bonhomme and Manresa (2015). While these authors describe a procedurefor using the bootstrap to obtain standard errors for covariate effects, a label-switching prob-lem prevents finding standard errors for the effects of group-level time-varying unobservableheterogeneity. A method to match labels across independent GFE estimations is required so2hat the bootstrap can also explore uncertainty in the latent heterogeneity effect estimates.Related to this, the fixed- T variance estimate formula given in the supplement to Bonhommeand Manresa (2015) provides an alternative approach to performing inference on the effectsof the latent heterogeneity, where T is the maximum number of observations on each unit inthe sample. The proposed label-matching method also allows studying the properties of thestandard errors for the latent group effects derived from this analytical formula, by observingtheir distribution across a large number of simulated datasets. This is useful in determiningthe potential applicability of the analytical formula in obtaining standard errors for the latentgroup effects in finite samples.The rest of the paper is organised as follows. Section 2 summarises the literature to whichthis article contributes. Section 3 describes the available data for our application to retire-ment decumulation behaviours. Section 4 outlines the GFE estimator from Bonhomme andManresa (2015), and explains our proposed extensions. Section 5 presents the main resultsfrom applying the GFE estimator to the available data. Section 6 examines the implicationsfor retirement incomes research and policy, considers limitations and discusses areas for futureresearch. Section 7 concludes. The supporting material provides robustness checks, a simula-tion exercise, and descriptive analysis of the data. Recent work studying drawdown behaviours builds on Horneff, Maurer, Mitchell, and Dus(2008). These authors compare three rate-based drawdown strategies against a level guar-anteed lifetime annuity, using a model of retiree utility that incorporates stochastic interestreturns and retiree lifetimes. The first strategy draws a fixed proportion of the account bal-ance each year; the second determines a terminal time horizon T and draws a proportion 1 /T of the account balance in the first year, 1 / ( T −
1) in the second year, and so on, continuinguntil the T th year of the plan where all the remaining funds are drawn down; the third drawsa proportion of the balance that updates each year based on the surviving retiree’s expectedremaining lifetime.Bateman and Thorp (2008) extend Horneff et al. (2008), motivated by newly-legislated mini-mum annual drawdown rates for a phased withdrawal retirement income product in Australiaknown as the account-based pension. Table 1 details these age-based rates, which start at 4%of the account balance for retirees under age 65, and increase as a step function of age to amaximum of 14% for retirees aged 95 or above; these are specified in Schedule 7 of the Su-perannuation Industry (Supervision) Regulations 1994. The rates, when applied to a retiree’saccount balance at the start of the relevant financial year, give the dollar amount the retiree3ust draw down from their account during that financial year.Bateman and Thorp (2008) use a stochastic lifecycle model to compare five drawdown strate-gies. Horneff et al. (2008) examine three of these; the remaining two rules are based on leg-islated minimum drawdown rates in Australia: always drawing at the newly-legislated mini-mum drawdown rates effective from 1 July 2007 ; and drawing at the minima previously ineffect. They also use their model to derive the implied optimal drawdown plan, and examinethe sensitivity of these results to changes in the model assumptions. The authors find thatwhile across many scenarios the newly-legislated minima follow relatively closely the optimalbehavioural pattern, in many cases a fixed-rate drawdown strategy generates higher simulatedutilities. The fixed-rate strategy considered by Bateman and Thorp (2008) draws a fixed pro-portion of the remaining account balance each year, where this fixed proportion is the annuitypayout rate the retiree would receive in the market at the time of writing by purchasing aguaranteed lifetime annuity using their account balance at retirement.Based on this theoretical literature alone, researchers can a priori expect that analysing draw-downs data will reveal groups of retirees exhibiting distinct drawdown behaviours based onsimple rules or heuristics. Following from this theoretical work, empirical research has madesome progress in understanding how closely actual behaviours follow those predicted by themodels.Balnozan (2018) gives the first analysis of drawdown behaviours in phased withdrawal retire-ment income products using a panel dataset. His method involves traditional panel regres-sion models and a combination of handwritten filters and cluster analysis to find patternsbased on the drawdowns. The analysis finds descriptive evidence for two simple behaviours:a) drawing constant dollar amounts; and b) following closely the legislated minimum draw-down rates. However, the work is limited because the panel regression models associatingavailable covariates with the observed drawdowns do not control for unobserved time-varyingheterogeneity from behavioural effects, and the methodology for identifying distinct draw-down behaviours lacks a formal foundation.This article uses recent advances in panel data methods in an attempt to overcome both ofthese shortcomings. Section 2.2 summarises recent methodological progress in panel dataeconometrics. Bonhomme and Manresa (2015) present the Grouped Fixed-Effects (GFE) estimator and ap-ply it to a linear panel data model that allows for group-level time-varying unobserved het-erogeneity and unknown group membership. Importantly, the latent group effects have unre-stricted correlation with the observed regressors. Being able to control for and estimate the Superannuation Industry (Supervision) Amendment Regulations 2007 (No. 1) Schedule 3 T grows, and can have correspondingly better finite-sample performance (Bonhomme &Manresa, 2015). This makes the GFE estimator more suitable when analysing a typical mi-croeconomic panel, which may have a large number of units N , but rarely more than a mod-erate length T .A related research area concerns finite mixture models, which can incorporate unobservableheterogeneity when the heterogeneity is constant over time. Finite mixture models are parsi-monious solutions to modelling unobserved heterogeneity in populations, wherein data fromlatent subpopulations can be drawn from different distributions, and the true allocation ofspecific individuals to subpopulations is unobserved. Deb and Trivedi (2013) develop finitemixture models with time-constant fixed effects. Critically, these models cannot specify thelikelihood of being in a particular group as a function of the covariates. By contrast, one ofthe primary advantages of using the GFE estimator over a standard 2WFE model estimatoris the ability to control for correlation between latent group effects and the included covari-ates. Mixture-of-experts models (e.g., Jacobs, Jordan, Nowlan, & Hinton, 1991; Jordan &Jacobs, 1994) generalise mixture models by parametrically specifying a link between groupidentity and covariate values; however, these models do not control for unit-level fixed effectsin panel data.The discussion above suggests that the GFE estimator is the most appropriate for the presentapplication, because: the available dataset only has a moderate panel length T ; a review ofthe theoretical drawdowns literature suggests that there may be a finite number of distinctbehavioural groups in the population, consistent with the GFE assumption that there are afinite number of latent group time profiles; the time profiles recovered by the GFE estimatorare likely to have economically meaningful interpretations.5 Data on superannuation drawdowns
The available superannuation dataset contains information on drawdowns from account-basedpensions (ABPs), a phased withdrawal retirement income product in Australia. Using ABPs,retirees generate personal income streams or receive lump-sum payments by drawing downtheir accumulated savings during retirement. Throughout, the balance remains invested infinancial markets based on each retiree’s chosen combination of safe and risky exposures.Following the Global Financial Crisis (GFC), the Australian government introduced tempo-rary minimum drawdown rate reductions as per Schedule 7 of the Superannuation Indus-try (Supervision) Regulations 1994. These resulted in ‘concessional’ minimum drawdownrates, which came into effect for the financial year ended 30 June 2009. For the financial years2009–2011 inclusive, the reduction was 50%, so that any retiree’s minimum drawdown rateover this period was halved relative to the standard schedule in Table 1. For financial years2012 and 2013, the reduction decreased to 25%, meaning the minimum drawdown rates werethree-quarters of those given in the table. For subsequent financial years, the rates returnedto their nominal values.To generate an income stream from an ABP, retirees can elect a payment amount and a fre-quency, for example $2000 monthly, which the superannuation fund follows in paying the re-tiree from their account balance. This type of drawdown, which is specified by the beginningof a given financial year, is referred to here as a ‘regular’ drawdown, and is the primary ob-ject of interest in this study. These regular drawdowns are distinct from ‘ad-hoc’ drawdowns,referring to any additional lump-sum payments requested by the retiree during the financialyear. A rigorous analysis of ad-hoc drawdowns is crucial in understanding retirees’ needs forflexibility and insurance, but is not the focus of this paper.The dependent variable studied here is the log of the regular drawdown rate as a proportionof account balance. Qualitatively, one interpretation of this rate is the speed at which re-tirees deplete their accumulated savings throughout retirement. In ABPs, retirees are exposedto the risk of outliving their savings and their retirement income becoming reliant on non-superannuation assets or taxpayer-funded transfer payments known as the age pension. Policyand product design should support retiree needs for regular income, and so understanding thedrawdown behaviours that manifest in this flexible withdrawal product has the potential toinform policymakers, financial advisors and superannuation fund trustees.The superannuation dataset contains N = 9516 individuals, combining source data from mul-tiple large industry and retail superannuation funds. Due to the small number of funds in thesample whose data permitted determining the regular drawdown rate, the economic results inthis paper may not be representative of the Australian superannuation system as a whole.The data capture window spans the financial years 2004 to 2015 inclusive, for T = 12 annualobservations. The main results use a balanced subset of each superannuation fund’s data, re-moving missing observations. However, as the data capture window is not the same for each6und, when combining these balanced subsets, the resulting dataset is unbalanced. The sup-porting material shows that the results are robust to using a fully balanced subsample of allthe available data.Individuals recorded as dying during the sample observation period do not appear in the dataanalysed. For some records, obtaining regular and ad-hoc drawdown amounts requires imput-ing these from the observed total drawdown amount using a method which, in periods wherethe retiree makes an ad-hoc drawdown, errs on the side of understating the amount of the ad-hoc drawdown; this affects fewer than 2.4% of the records in the sample.The superannuation dataset is derived from administrative data, and so has limited demo-graphic information on members. The two covariates considered are the minimum drawdownrate for each individual at each time point, and their account balance at the beginning of therespective financial year, both on the log scale. Thus the model, with regular drawdown rateon the log scale as the dependent variable, estimates the elasticity of the regular drawdownrate with respect to the minima and account balances. The remainder of the variation in logregular drawdown rates is attributed to the latent group effects estimated by the GFE proce-dure, and residual noise.The lack of demographic information, particularly health and marital status, is also a data-based motivation for using GFE as an estimation procedure. These unobserved characteris-tics are a potential source of omitted variable bias; they are possibly correlated with the de-pendent variable as well as the included covariates. Because relevant characteristics such ashealth and marital status may vary over time, controlling for time-constant unobserved het-erogeneity by using a standard fixed-effects model may not remove the biasing effect of all rel-evant unobservables. For this reason, a model allowing for sources of time-varying unobservedheterogeneity, such as the linear panel model to which the GFE estimator is applied, may bemore appropriate; the results in Section 5.1 provide evidence for this proposition. The GFE estimator introduced by Bonhomme and Manresa (2015) considers a linear modelof the form y it = x it θ + α g i t + v it ; (1)in our application i ∈ { , , ..., N } indexes individual retirees; t ∈ { , , ..., T } indexes fi-nancial years; y it is the dependent variable; x it is a vector of covariates; θ are the partial ef-fects of covariates on the dependent variable after controlling for group-level time profiles; g i ∈ { , , ..., G } identifies the group membership for unit i , where G is the chosen numberof latent groups in the sample; α g i t is a term representing time-varying, group-specific unob-7erved heterogeneity; v it represents the residual effect of all other unobserved determinants ofthe dependent variable.The linear model in (1) can be extended to include time-constant, individual-specific fixedeffects c i , so that y it = x it θ + c i + α g i t + v it , (2)because applying the within transformation (centering the variables around individual-specificmeans) reduces (2) to the form of (1). To see this, for any variable z it , define ˙ z it := z it − ¯ z i ,where ¯ z i := T − P Tt =1 z it ; then ˙ y it = ˙ x it θ + ˙ α g i t + ˙ v it , (3)which has the same form as (1). Note that the ˙ α g i t in (3) have a different economic interpre-tation to the α g i t from (1). Section 4.2 explains the treatment and interpretation of the grouptime profile estimators in the transformed model (3).Throughout, “GFE model” refers to the model in (2). A useful way to consider the GFEmodel is as a generalisation of the 2WFE model y it = x it θ + c i + α t + v it , (4)where the GFE model allows distinct time profiles for G groups, with group membership un-observed.Bonhomme and Manresa (2015) state the assumptions needed for large- N , large- T consis-tency of their GFE estimator, in models both with, and without, time-invariant fixed effects.Moreover, the asymptotics show that their estimator converges in distribution even when T grows substantially more slowly than N . To examine the finite- T properties of the estima-tor, Bonhomme and Manresa (2015) use a Monte Carlo exercise with simulated datasets thatmatch the N = 90, T = 7 panel used in their empirical application. In the present study, thesuperannuation dataset has large N = 9516 but moderate T = 12; the supporting materialfinds that the estimator performs well on simulated datasets of this size.When using the standard 2WFE model, if there is group-level time-varying unobservable het-erogeneity α g i t correlated with observable characteristics x it , two problems arise: first, esti-mates of the covariate effects θ may suffer from omitted variable bias; second, the group timeprofiles, which in many applications have interesting economic interpretations, remain un-covered. The 2WFE model is a special case of the GFE model with G = 1; hence, the GFEmodel presents an appealing alternative to the 2WFE model in applications where there isthe possibility of unobservable time-varying heterogeneity in addition to time-invariant un-observable heterogeneity. While determining the precise number of groups G lacks a generalsolution, estimating the GFE model can provide evidence for whether only controlling for oneset of time effects, as in the 2WFE model, is adequate; Section 5.1 discusses this issue.8he GFE estimator for (1) minimises the sum of squared residuals, giving( b θ, b α, b γ ) = arg min ( θ,α,γ ) ∈ Θ ×A GT × Γ G N X i =1 T X t =1 ( y it − x it θ − α g i t ) , (5)where g i ∈ { , , ..., G } ; the vector γ = { g , g , ..., g N } defines the grouping of each of the N units into one of the G groups; Γ G is the set of all possible groupings of N units into G or fewer groups; Θ is a subset of R k , the k -dimensional real space, where k is the number ofcovariates or equivalently the dimension of x it ; A is a subset of R .Bonhomme and Manresa (2015) present an iterative algorithm for estimating (5). The algo-rithm initialises values for the grouping vector γ and the model parameters ( θ, α ); it then al-ternates between the following two steps until convergence: a grouping update step, whichallocates each unit to the group minimising the sum of squared residuals given the most re-cent estimate of the model parameters θ and α ; a parameter update step, which estimates theparameters ( θ, α ) conditioning on the most recent estimate of the grouping vector γ .Estimating the GFE model does not require a balanced panel; the algorithm can be adjustedto run even when the sample contains unit–period combinations with missing data. However,when considering the relationship between the 2WFE model (4) and the GFE model (2) with G = 1, the unbalanced data case presents subtleties requiring further discussion.The original GFE estimator for (2) with G = 1 gives the same estimates as applying thewithin transformation to the data and running a least-squares regression on the time-demeaneddata, including T time dummy variables and no constant term. In balanced samples, this re-turns the same numerical results as standard implementations of the 2WFE model: for ex-ample Stata ’s xtreg command with the fe option and time dummy variables as covariates.However, in unbalanced panels, the results differ.With unbalanced data, obtaining the results from standard implementations of the 2WFEmodel requires time-demeaning the time dummy variables at the unit level. Hence, this arti-cle proposes and utilises a modified GFE estimation procedure having the following property:when G = 1, it recovers precisely the same estimates as a standard implementation of the2WFE model even when the data is unbalanced and the model includes time-invariant unit-level unobservable heterogeneity. Section 4.5 describes this alternative estimation procedure.When the panel is balanced, or when there are no time-invariant unit-level unobservables,the results from this method are identical to the unmodified GFE estimation procedure re-sults. However, panel data models in microeconomic applications generally control for time-invariant unit-level unobservable heterogeneity. Hence, this is a useful extension when apply-ing the GFE estimator to microeconomic applications more broadly. To examine the sensi-tivity of the present results to using the modified algorithm, the supporting material providesa robustness check using the unmodified GFE estimation procedure in place of the proposedmodified algorithm. 9onhomme and Manresa (2015) explain that in the absence of covariates, their algorithm out-lined above reduces to k -means clustering. Similarly to standard implementations of k -meansclustering algorithms, the results depend on the starting values. Running the algorithm multi-ple times with randomly generated starting values increases the likelihood of finding solutionscorresponding to smaller values of the objective function in (5).In the present study, the algorithm is initialised by drawing starting values for each covari-ate effect from independent Gaussian distributions, centred at the coefficient estimates froma 2WFE regression fit to the data, and with standard deviations equal to the magnitude ofthese coefficient estimates. Results for the application take the most optimal solution across1000 independent runs of the algorithm using randomly drawn starting values to initialise theparameters. The supporting material also tests the sensitivity of the results to the number ofstarting values used.To facilitate inference on the group time profiles, this study implements the fixed- T vari-ance estimate formula in the supplement to Bonhomme and Manresa (2015), as well as anextension to their non-parametric bootstrap method. Notably, approximating the variabil-ity in time profile estimates using the non-parametric bootstrap method described by Bon-homme and Manresa (2015) requires solving a label-switching problem. Section 4.4 explainsthis problem and proposes an extension to the bootstrap procedure which matches group la-bels across replications to allow for bootstrap inference on the time profiles. Within-transforming the data to remove the effect of any time-constant unobservable hetero-geneity results in time-demeaned group time profiles, ˙ α gt . As the ˙ α gt contain only informa-tion about changes in the group effects over time relative to their mean, and no informationabout the absolute level of the effect, the estimates of the ˙ α gt are modified in order to ob-tain the desired economic interpretations. All estimated time profiles are shifted to begin at avalue of 0 on the y -axis, so that the interpretations of the values are as changes relative to thefirst time period. This is analogous to estimating a set of T − α gt := ˙ α gt − ˙ α g, , for all g and t . First, estimates for V ar ( ˙ α gt ), V ar ( ˙ α g, ) and Cov ( ˙ α gt , ˙ α g, ) are obtained by applying the fixed- T variance estimate formulaafter estimating the GFE model on the within-transformed data, and Normal-approximation95% confidence intervals are constructed. These are compared to the corresponding 95% con-fidence intervals obtained from the bootstrap, using the proposed extension for matchinggroup labels across bootstrap replications.Here, a comparison of the time profile standard errors derived from the fixed- T variance es-10imate formula versus simulated standard errors from a Monte Carlo exercise shows that thefixed- T variance estimate formula performs well on simulated data calibrated to the super-annuation dataset; the supporting material provides details of this comparison. Moreover,in the present study, confidence intervals constructed from the bootstrap procedure resultsare broadly similar to those derived from the fixed- T variance estimate formula, resulting inequivalent inference on the estimated parameters. For smaller datasets, where asymptotic re-sults are less likely to provide accurate standard errors, and where the computational costof running the bootstrap is lower, it may be preferable to use the bootstrap. However, theresults suggest that in larger datasets, using the fixed- T variance estimate formula may besufficiently accurate while keeping the computational burden low compared to using the boot-strap. G Bonhomme and Manresa (2015) propose an information criterion that correctly selects thenumber of groups when N and T are both large and similar in magnitude; however, for large- N moderate- T panels, like the superannuation dataset, this criterion may overestimate thetrue value of G (Bonhomme & Manresa, 2015). An alternative method to choose the numberof groups, which Bonhomme and Manresa (2015) employ in their empirical application, usesthe fact that underestimating G leads to a type of omitted-variable bias in the θ parameters,to the extent that the unobserved effects are correlated with the included covariates. Con-versely, overestimating G does not bias the θ estimates, although it does increase the numberof parameters to estimate and the parameter space of possible groupings, resulting in noisierestimates for all model parameters.Taken together, these properties suggest that observing a plot of how the coefficient estimateschange as G increases can reveal the point at which the coefficients have been de-biased rela-tive to the G = 1 case. Visually, the coefficients might vary significantly between G = 1 andsome higher value G ? of G , after which point they may appear to stabilise, albeit becomingnoisier as G continues to increase. One can then select G ? as the number of groups suggestedby the data. This is how Bonhomme and Manresa (2015) decide on G = 4 in their applica-tion, rather than G = 10 as suggested by the information criterion—which, given the dimen-sions of their panel, may overestimate the true number of groups.For applications that focus only on estimating unbiased effects of the covariates, the abovesuggests choosing the number of groups by selecting the smallest value of G that places thecovariate effect estimates close to their stable values. However, the present application takes aunique interest in the time profiles for all latent groups in the sample. It is possible that stop-ping at the first value of G that seems to successfully de-bias the regression coefficients willresult in at least one estimated group whose time profile is an average over distinct time pro-files of constituent subgroups. For the purposes of obtaining unbiased regression coefficientsalone, separating out these mixed groups by further increasing G may unnecessarily increase11he complexity of the model. However, identifying all distinct behavioural groups with eco-nomically meaningful interpretations may require searching beyond the smallest G that de-biases the regression coefficients.On the other hand, overestimating G can result in the separation of a particular time profileinto multiple biased representations of itself, where the difference between the representationsis spuriously driven by noise (Bonhomme & Manresa, 2015). This means that the more sim-ilar a pair of time profiles appear, the more sceptical the researcher should be in regardingthem as distinct effects.Hence, this study posits a selection rule that picks the greater of two values of G : the small-est G that appears to result in unbiased estimates of the regression coefficients; or the lastvalue of G for which the marginal change from G − G groups finds an economically im-portant behaviour, sufficiently distinct from all others and exhibited by a non-trivial portionof the sample. Section 5 discusses this selection process alongside the main results. Bonhomme and Manresa (2015) address obtaining bootstrap standard errors for the covari-ate effect estimates but not the time profile estimates. As the covariate effect estimates areunambiguously labelled across multiple runs of the GFE procedure, estimating their standarderrors by comparing estimates across a large number of bootstrap replications requires no spe-cial consideration. By contrast, group-level results from the procedure are unique only up toa relabelling of the groups, and the group labels are determined at random during estimation.This means that the same economic group may receive different labels across multiple inde-pendent estimation runs, e.g., when performing the bootstrap. Hence, estimating the stan-dard errors of group-level time effects by comparing time profile estimates across bootstrapreplications requires a method to match labels across runs.The bootstrap procedure outlined in the online supplement to Bonhomme and Manresa (2015)involves running the GFE procedure B times, each time on a different bootstrap replicatedataset. To create the bootstrap replicate datasets, all N units in the original data are sam-pled with replacement N times, taking all observations corresponding to a given unit into thereplicate dataset when that unit is sampled. Each of the B model runs produces a set of coef-ficient estimates for the model covariates, as well as an estimated grouping and a set of timeprofiles for each group.To match group labels across replications, this article proposes the following label-matchingprocedure, extending the solution Hofmans, Ceulemans, Steinley, and Van Mechelen (2015)use for the analogous problem in k -means clustering.1. Fix the labelling generated by the GFE model estimation completed on the originalsample. These labels act as the reference labels to which all subsequent estimates will12lign their group labels. Define the initial time profile estimates labelled using the refer-ence labels as b α rg := (cid:16)b α rg, , b α rg, , ..., b α rg,T (cid:17) for g ∈ { , , ..., G } .2. For b = 1 , , ..., B , find the permutation of group labels for the b th bootstrap run whichminimises the sum of Euclidean distances, aggregated across all G time profiles, betweenthe time profiles estimated in the b th bootstrap run and the original sample estimates asidentified by the reference labels; i.e., for b = 1 , , ..., B , do:2.1. Using the unmodified labels generated by the b th bootstrap run, define the esti-mated time profiles from the b th bootstrap run with these labels as the length- T vectors b α b, g for g ∈ { , , ..., G } .2.2. Index the G ! permutations of the label sequence (1 , , ..., G ) by p ∈ { , , ..., G ! } .Each permutation p for the b th bootstrap run defines a permuted set of time pro-files b α b,pg for g ∈ { , , ..., G } and a corresponding mapping function m p : { , , ..., G } → { , , ..., G } such that m p ( g ) is the g th element of the p th permuted label sequence and b α b,pg = b α b, m p ( g ) .2.3. For p = 1 , , ..., G !, do:2.3.1. For g = 1 , , ..., G , compute the Euclidean distance between the length- T vec-tors b α b,pg and b α rg , given by k b α b,pg − b α rg k .2.3.2. Sum all G Euclidean distances to compute an aggregate distance metric forthat permutation, P Gg =1 k b α b,pg − b α rg k .2.4. Select the permutation p ? with the smallest aggregate distance metric: p ? = arg min p ∈{ , ,...,G ! } G X g =1 k b α b,pg − b α rg k . (6)Relabel the G groups in the b th bootstrap run using the labelling given by the se-lected permutation.This method matches time profiles across different bootstrap replications in the main resultspresented in Section 5, as well as across different simulated datasets and bootstrap replica-tions in the simulation exercise reported in the supporting material.Deriving standard errors for the shifted time-demeaned group time profiles from the fixed- T variance estimate formula requires first estimating variances for the time-demeaned time pro-file estimators ˙ α g i t , then using these to estimate standard errors for the shifted time profilesto construct Normal-approximation confidence intervals. By contrast, the bootstrap resultsdirectly estimate the variability of the shifted time profile estimators, by matching shiftedtime profiles across the B bootstrap replications, then using the empirical distribution of thematched time profile estimates to obtain bootstrap intervals.13 .5 Extension 2: An alternative estimation method for unbalanced data Consider the model in (4), which, after time-demeaning, has the form˙ y it = ˙ x it θ + ˙ α t + ˙ v it . (7)The primary estimands of interest are the T − α t = ˙ α t − ˙ α for t = 2 , , ..., T .Since ˙ α t := α t − ¯ α , where ¯ α = T − Σ Tt =1 ˙ α t , the set of T values of ˙ α t satisfy Σ Tt =1 ˙ α t = 0.In balanced panels, the following three methods obtain identical estimates for the ˜˙ α t :1. regress ˙ y it on ˙ x it and T time dummy variables with no constant term to estimate all T values for ˙ α t as the coefficients of the time dummies; then compute ˜˙ α t = ˙ α t − ˙ α for t = 2 , , ..., T ;2. regress ˙ y it on ˙ x it and T − T − α t are the estimated coefficients of the included T − , , ..., T ; then,regress ˙ y it on ˙ x it and the T − T − α t are the estimated coeffi-cients of the included T − α t instead of ˙ α t , the values of ˙ α t for t = 1 , , ..., T are recovered using ˙ α = − T − Σ Tt =2 ˜˙ α t , followed by ˙ α t = ˜˙ α t + ˙ α for t = 2 , , ..., T . Showingthis involves expanding the left-hand side of the identity Σ Tt =1 ˙ α t = 0 to obtain Σ Tt =2 ˙ α t = − ˙ α . By definition, Σ Tt =2 ˜˙ α t = Σ Tt =2 ( ˙ α t − ˙ α ), for which the right-hand side simplifies to − ˙ α − ( T −
1) ˙ α , so that ˙ α = − T − Σ Tt =2 ˜˙ α t as required. This reduces to the same trans-formation used in Suits (1984, p. 178), but applied to a model with no constant term. Whilethe economic interest in applications like ours will generally lie in the T − α t ,this method to convert between ˜˙ α t and ˙ α t is required in the discussion below.When the panel is unbalanced, the choice of estimation strategy requires further consider-ation. With unbalanced data, methods 1 and 2 described above for obtaining estimates of˜˙ α t no longer guarantee that the corresponding values of ˙ α t sum to zero; a nonzero sum con-tradicts a property of the model given by (7). Conversely, method 3 obtains the desired esti-mates while ensuring that the T estimates of ˙ α t , generated by converting the T − α t using the conversion method outlined above, sum to 0, consistent with the model.Using the GFE model implementation provided by Bonhomme and Manresa (2015) with G = 1 on unbalanced data gives estimates for the time effects in line with those producedby methods 1 and 2, whereas standard implementations of the 2WFE model give the same es-timates as method 3. This motivates our modification to the GFE methodology, which alignsthe G = 1 results with the standard 2WFE model estimates regardless of whether the panel isbalanced or not. 14he unmodified GFE model estimation procedure obtains the time profiles that would arisefrom interacting group identity with a set of T time dummy variables, analogous to usingmethod 1 in the G = 1 case. In the modified procedure’s parameter update step, the algo-rithm interacts the group identity with T − G × ( T −
1) relative effect estimates ˜˙ α gt := ˙ α gt − ˙ α g, ; this is analogous to using method3 for G = 1. As the GFE algorithm requires G × T estimates of ˙ α gt in computing the sum ofsquared residuals across all i and t , it is necessary to perform a conversion step, identical tothe conversion method outline above but performed for each of the G time profiles.Implementing these modifications gives the algorithm the desired behaviour for G = 1. Thesupporting material provides results using the unmodified estimation procedure to comparewith the main results in the paper, which are obtained with the modified GFE model esti-mation procedure. Comparison of the two methods shows that while the numerical resultsdepend on the procedure used, the economic interpretations of the results are similar. Figure 1 plots the covariate effect estimates versus G for the GFE model fitted to the su-perannuation drawdowns dataset. The values of G on the horizontal axis correspond to thenumber of latent groups specified; the vertical axis denotes partial effect values for the twoincluded covariates: log minimum drawdown rate and log account balance. The lines connectcovariate effect estimates, while the shaded regions around these correspond to 95% confi-dence intervals constructed using standard errors derived from the fixed- T variance estimateformula in the supplement to Bonhomme and Manresa (2015).As the dependent variable and covariates are on the log scale, the covariate estimates repre-sent the elasticity of regular drawdown rates to changes in account balances and the mini-mum drawdown rates. Section 5.2 provides the economic interpretation for the correspondingestimates after choosing the value of G .A critical feature of Figure 1 is that as G increases, changes in the log account balance covari-ate effect estimates are large relative to the confidence intervals for the first few values of G .After G = 7, the point estimates become more stable and successive 95% confidence intervalsshow significant overlap. This suggests that from G = 7, the GFE procedure removes the biasarising from correlation between group-level latent time effects and the included covariates.Using the proposed modified GFE model estimation procedure, the point estimates corre-sponding to G = 1 are identical to the results from a standard 2WFE estimation on the draw-downs dataset. Hence, Figure 1 implies that the estimators using a more traditional analysisincluding only one time profile are biased compared to the estimators in the GFE model for15 ≥
7. The supporting material shows that this result holds in the fully balanced case, whereboth the modified and unmodified GFE procedures give results identical to the 2WFE modelwhen G = 1. G Figure 2 shows plots of the time profiles estimated for G = 4 , , ...,
9, all shifted to begin ata y -axis value of 0, and omitting confidence interval bounds for clarity. This figure suggeststhat while increases in G initially produce new and economically distinct time profiles, at G = 7, incremental moves to a larger value of G split existing time profiles into highly sim-ilar representations of the same trajectory. This is consistent with a theoretical result fromBonhomme and Manresa (2015) that implies overestimating G can create spurious copies ofthe true time profiles, differing by random noise. Hence, the number of latent groups to use inthe model is set to G = 7. Section 5.3 explains in detail the economic interpretations of theresulting time profile estimates.Selecting G = 7 groups corresponds to estimated covariate effects of 0.144 and − e . − .
5% on average, hold-ing account balance equal and controlling for group-level time-varying heterogeneity. A sim-ilar proportional increase in a retiree’s account balance is expected to effect a proportionaldecrease of 14 .
6% (1 − e − . ) in their regular drawdown rate on average, holding the mini-mum drawdown rate constant and controlling for group effects. Figure 3 shows the time profiles estimated using the selected value of G = 7. Figure 4 presentsthe equivalent plot with confidence interval bounds computed from the empirical 2.5 and 97.5percentiles of the time profile estimates from 1000 bootstrap replications. While qualitativelysimilar overall to the intervals constructed using standard errors derived from the fixed- T variance estimate formula, a comparison highlights some differences. In particular, the inter-vals are wider around financial years 2011–2013 for group 6; the interval is wider for financialyear 2010 for group 4; the intervals are tighter for group 5 throughout; the interval is wider inthe terminal financial year for group 3; and the point estimates from the original sample tendtowards the lower end of the 95% bootstrap confidence intervals for groups 1, 2 and 7. How-16ver, due to the estimated effect magnitudes, these numerical differences in the sets of confi-dence intervals do not meaningfully change inferential conclusions regarding the estimates ortheir economic interpretations.The values of the time profiles represent behavioural effects on the proportional changes in aretiree’s regular drawdown rate, relative to their own rate of drawdown in 2004. As the de-pendent variable is on the log scale, a value of, say, 0 . y -axis represents a proportionalchange in the regular drawdown rate of e . ≈ .
65, or roughly a 65% increase in the draw-down rate, relative to 2004 levels.Figure 5 shows the time profile estimates for the G = 1 model estimated using the modi-fied GFE procedure, with the y -axis scale set to the same scale used for the G = 7 modelgroup time profile plot (Figure 3). Comparing these shows that a model controlling for onlyone set of time effects fails to capture the shape and magnitude of any of the time profiles inthe seven-group model. Visually, the resulting single time profile appears to ‘average over’the seven markedly distinct time profiles. Hence, not only are the covariate effect estimatesbiased when G = 1, but the single time profile is entirely unrepresentative of the latent group-level time profiles. The following refers to the group time profiles in Figure 3. All the time profiles exhibit sim-ilar, slowly rising trajectories until the financial year ended 30 June 2008, after which theydiverge in statistically and economically significant ways.Consider group 5, whose time profile describes a rapid reduction in drawdown rates betweenfinancial years 2008 and 2010, and then a gradual return towards pre-reduction levels. Thetiming of these movements follows closely the progression of concessional minimum drawdownrates (see Section 3). This implies that members of group 5 were the most responsive to thechanging minimum drawdown rates over this period, compared to the rest of the sample. Fig-ure 6 confirms that many individuals in group 5 show a step-like pattern in their log regulardrawdown rate series during the second half of the observation period. This pattern is sugges-tive of the new step-function schedule for minimum drawdown rates which came into effect 1July 2007.Groups 1 and 2 display time profiles rising steadily following financial year 2008 for the re-mainder of the sample. Group 7 follows a similar rising trend with a reduced magnitude forthe first few financial years after 2008, before stabilising for the remainder of the observationwindow. The panel plots in Figure 7 suggest a tendency for individuals in these groups todraw constant dollar amounts, while their account balances gradually decline over time. Thecorresponding plots for group 3 show a similar preference for constant dollar amounts, butafter financial year 2013, individuals in this group suffer a rapid decline in both the amountsdrawn down and account balances. Many members of group 6 undergo a similar evolution,17haracterised by mostly constant drawdowns initially and a subsequent reduction in the levelin later years. The magnitudes of the downwards revisions appear smaller for this group, andoccur earlier for many retirees. Moreover, while group 3 members continue to revise downover successive financial years, it appears that many individuals in group 6 revise down onceand then continue to draw at the new, reduced level. The group 7 plots show gradual down-ward revisions in the amounts year on year.Thus, groups 1, 2, 3, 6 and 7 appear to be similar in that members draw mostly constant dol-lar amounts over time; however, these groups seem to differ in how many members make adownwards revision in the level of their drawdown amounts, and whether this revision is once-off or the beginning of a downward trend. Crucially, the timing of downwards revisions of-ten appears to align with periods where account balances begin falling at an accelerated rate.Hence, the heuristic of drawing constant amounts over time may be adversely impacting re-tiree financial security by contributing to a premature exhaustion of account balances.Figure 3 shows that the time profile of group 4 follows closely the zero line before financialyear 2011. A time profile with all values equal to zero suggests that the correct model for thisgroup’s data is y it = x it θ + c i + v it . Correspondingly, one possible interpretation for group4’s behaviour is that before financial year 2011, they were constantly tuning their drawdownrate as they progressed to higher minimum drawdown rates and as their account balanceschanged. This could potentially represent a group of ‘engaged’ retirees, who regularly occupythemselves with determining their desired rate of drawdown; however, it is not possible to fur-ther investigate this hypothesis using this dataset.Group 4’s time profile drops significantly below 0 after financial year 2010. These movements,while of a smaller magnitude compared to the time profile values of other groups, are stilleconomically significant; they suggest a behavioural response, after netting out the effect ofcovariates, of reducing drawdown rates relative to earlier levels from 2011 onwards.The supporting material provides summary statistics and descriptive plots for all seven groups.Based on these, some of the ways in which the groups differ in terms of observable charac-teristics are: the proportion of males in groups 1 to 3 are 62%, 62% and 69% respectively,while the sample on aggregate is 56% male; the members of groups 1 and 3 tend to be older,with median retirees aged 81 at 31 Dec 2015; members in groups 4, 5 and 6 have the youngestmedian retirees, aged 78 at 31 Dec 2015; group 5 members have the highest median risk ap-petite, a variable defined in the supporting material as a summary measure of the relativesizes of derived equity returns within individual accounts compared to the S&P/ASX 200market index; group 7 members have the lowest median risk appetite; group 7 members makead-hoc drawdowns least frequently, in roughly 3% of person–years observed, followed by group1, for 5% of person–years observed—for all other groups this frequency was 9–11%; in yearswhere they make ad-hoc drawdowns, group 3 members tend to draw down the largest propor-tions of their account balance using ad-hoc drawdowns over the course of a year—on average,these ad-hoc drawdowns over the year amount to 35% of their account balance at the start ofthe respective financial year. 18 .5 Prior expectations Section 1 presents this study’s prior expectations for finding at least two types of strategies inthe data: drawing constant dollar amounts; following closely the minimum drawdown rates.Most of the group behaviours found under the seven-group assumption can be interpreted ascases of these two hypothesised behaviours. Moreover, a restricted model assuming only twogroups already begins to show evidence for the existence of both of these groups. Figure 8shows the time profiles for the two-group model. As in Figure 5, the y -axis scale allows for di-rect comparison of the estimated magnitudes with the G = 7 model results. The time profileof the first group appears similar to those for the groups in the seven-group model that seemto target constant dollar amounts for the regular drawdowns; the second profile shows a dipcomparable to the group that seems to follow the minimum drawdown rate.The supporting material also provides descriptive panel plots for these two groups. Theseplots are broadly consistent with the interpretation that the first group often attempts tohold drawdown amounts constant, although with downward revisions in the amount com-mon, as well as rapid declines in account balance towards the end of the period; the secondgroup mainly makes decisions regarding their drawdown rate. This analysis, alongside the ob-servation that moving to a two-group model already removes much of the bias in the covari-ate effect estimates, suggests that accounting for both rate-based and amount-based strate-gies seems to be the most important step in controlling for unobservable heterogeneity in thedata. The key implication for retirement incomes research and policy is that there is now a statis-tical basis for empirical results that indicate a behavioural stance explains much of the varia-tion in the drawdowns in phased withdrawal retirement income products.Some groups identified by the GFE model reveal retirees whose preference for generating levelincome streams using their account-based pensions may be partly responsible for the prema-ture depletion of their account balances. Hence, retirees with this preference may be an at-risk group. Practitioners and policymakers may wish to consider this when giving advice ordesigning retirement income products to align with the Comprehensive Income Products forRetirement framework, which places an emphasis on generating income streams without com-promising the ability to retain superannuation savings until late into retirement (The Aus-tralian Government the Treasury, 2016). In particular, financial advisors may benefit retireesby predicting their individual needs for income and precautionary savings later in life, andmonitoring their actual experience relative to this forecast.19he present study makes clear how a robust behavioural analysis of drawdowns data is vitalfor informing the various stakeholders in the retirement incomes space, including retirees, thegovernment, and the financial services industry. However, Section 3 notes that despite havingdata from multiple superannuation funds, due to the small number of funds in the estimationsample, the results may not describe the population of Australian retirees in general. In par-ticular, using the current dataset may overlook more meaningful behavioural patterns thatare present in funds not included in the sample.
The present study asserts the value of utilising the GFE model not only in applications wheretime-varying latent group effects are unwanted sources of endogeneity, but also in cases wherethe time profiles are the primary estimands of interest. This article provides a case study forhow researchers may utilise the GFE model in these scenarios. This includes proposing a so-lution to a label-matching problem to align group labels across independent GFE model es-timations. Importantly, this allows standard error estimation for the time profiles, insteadof only the covariate effects, when using the bootstrap method described in the supplementto Bonhomme and Manresa (2015). It also enables simulation testing of the accuracy of stan-dard errors derived from the fixed- T variance estimate formula for various sizes of panel datasets.In applications where the need to capture time-varying latent group effects is unclear, theGFE model provides a valuable diagnostic tool; it is able to both test for the presence oftime-varying unobservable heterogeneity, and show the impact of failing to account for it.The GFE model achieves this by observing how regression coefficient estimates evolve as thenumber of assumed groups increases from an initial value of one, with the one-group modelequation corresponding to the standard 2WFE model specification. In applications wherethe model includes time-constant unobservable heterogeneity and the available data is un-balanced, our proposed modification to the estimation procedure maintains the equivalencebetween the 2WFE estimation results and the GFE estimation results when G = 1.We believe that the GFE model is valuable for research in behavioural microeconomics andevent studies. While the present paper’s focus is on the behavioural interpretations of the la-tent group effects, the superannuation data covers a period in which multiple common macroe-conomic and policy shocks affect all members in the sample. Hence, the superannuation ap-plication also invites an event-study interpretation, although it is impossible to disentan-gle the effects of several common shocks which occur in close proximity. These include theGlobal Financial Crisis, the introduction of a new schedule of minimum drawdown rates foraccount-based pensions, and temporary tweaks made to the minimum drawdown rates. Takentogether, however, the time profile plots presented in Figure 3 clearly show how otherwisehomogeneous-looking trends diverge radically, plausibly driven by one or more of these shocks.Hence, the results broadly show how researchers can utilise the same methodology in an event-study framework. 20hile the theory derived in Bonhomme and Manresa (2015) describes the asymptotic per-formance of the GFE estimator when T can grow substantially more slowly than N , our un-derstanding is that the theory does not provide precise guarantees for how well the procedureworks with any given values N and T . Hence, the supporting material reports a simulationexercise to see if the dimensions of the superannuation dataset pose a barrier to the analysis.This simulation exercise finds that the procedure works well on the simulated datasets, whichuse G = 7, N = 9516 and T = 12.The GFE model assumption of a group structure to the time-varying unobservable hetero-geneity also deserves discussion. In the superannuation application, a finite support for thetime-varying unobservable heterogeneity is tenable, as it was a priori expected that an impor-tant source of heterogeneity in the drawdowns arises from the latent motivations that driveindividuals to choose one of a finite number of drawdown strategies. In general, many be-havioural economics studies with panel data may have analogous motivations to assume afinite support on the time-varying heterogeneity. This prior assumption may be importantas it is not known precisely how the GFE estimator behaves when a finite support only ap-proximates, rather than accurately captures, the nature of the unobservable heterogeneity.For a discussion on approximating more general forms of heterogeneity using a finite support,see e.g. the distinct, but conceptually related, work of Bonhomme, Lamadon, and Manresa(2019). The set of possible solutions to the optimisation problem that produces the GFE model es-timates grows quickly with N and G , and solutions are in general sensitive to the startingvalues used to initialise the algorithm (Bonhomme & Manresa, 2015). For the superannuationdataset, a robustness check provided in the supporting material shows that the results ap-pear relatively insensitive to changing the number of starting values. Hence, the present pa-per does not further address this general problem. However, Bonhomme and Manresa (2015)and authors they cite develop exact and heuristic methods for finding solutions to the inher-ent difficulty in optimising functions determining the grouping of individual units. Develop-ing better solutions to this problem in the context of GFE model estimation will increase thesuitability of the GFE model applied to large- N datasets, which in general may not prove asrobust to increasing the number of starting values as the superannuation dataset.The issue of estimating G when the true number of latent groups is unknown only has a clearsolution when N and T tend to infinity at the same rate, as described in the supplement toBonhomme and Manresa (2015); developing a solution under weaker assumptions would bevaluable to applied research. An alternative approach to allow for a grouped data structurewithout specifying the number of groups exists using non-parametric Bayesian methods. Kimand Wang (2019) construct a Bayesian estimation procedure for (1), placing a Dirichlet Pro-cess prior on the time profiles to induce flexible data-driven clustering. Future research could21nvestigate the suitability of using a similar method in typical microeconomic applications,especially when interest lies in performing inference on the time profiles.Another possible methodological contribution relates to GFE estimators for nonlinear paneldata models, discussed by Bonhomme and Manresa (2015) as a potential extension to themain methodology. A nonlinear model removes the constraint of expressing the dependentvariable as a linear combination of the covariates, and thus broadens the applicability of GFE-style approaches, for example to binary dependent variable models. To our knowledge, thereis not yet an algorithm that can jointly estimate all the parameters of such a model, nor isthere a statistical treatment allowing for inference on the parameter estimates. This article motivated an application of the GFE estimator from Bonhomme and Manresa(2015) that shows how to perform inference on the group time profiles when these are the pri-mary estimands of interest, rather than only nuisance parameters. This was achieved by ap-plying a GFE estimator to a linear panel model using data on drawdowns from phased with-drawal retirement income products in Australia. The application also contributes to the em-pirical retirement incomes literature, justified by a prior lack of statistical treatment of thebehavioural dimension to the observed drawdowns from phased withdrawal retirement incomeproducts.Broadly, the results show that capturing latent group behaviours in account-based pensiondrawdowns explains much of the variation observed in the data. Two behaviours—drawing aconstant dollar amount and following the minimum drawdown rate—reveal themselves in theestimation results without explicitly being searched for. Furthermore, some groups of retireesin the sample appear to be at risk of exhausting their account balances quickly during retire-ment. These tend to be a subset of those individuals whose drawdowns target level incomestreams.Interest in the group time profile estimates motivates the proposed solution to a label-matchingissue in the existing bootstrap procedure for obtaining standard errors. This allows compar-ing bootstrap standard errors for the time profiles with those derived from an asymptoticallyvalid analytical formula. In addition, this permits studying the finite-sample performance ofthe formula using simulated data. Furthermore, this article proposes a modified GFE estima-tion procedure which has the following property: even when the data is unbalanced and themodel includes time-invariant unit-level unobservable heterogeneity, when G = 1 it recoversprecisely the same estimates as a standard implementation of the two-way fixed-effects model.22 eferences Bai, J. (2009). Panel data models with interactive fixed effects.
Econometrica , , 1229–1279.doi:10.3982/ECTA6135Balnozan, I. (2018). Slow and steady: Drawdown behaviours in phased withdrawal retirementincome products.
Unpublished manuscript, UNSW Business School, University of NewSouth Wales, UNSW Sydney, Australia. Retrieved from https://sites.google.com/view/igorbalnozan/research
Bateman, H., & Thorp, S. (2008). Choices and constraints over retirement income streams:Comparing rules and regulations.
Economic Record , , S17–S31. doi:10.1111/j.1475-4932.2008.00480.xBonhomme, S., Lamadon, T., & Manresa, E. (2019). Discretizing unobserved heterogeneity.
Manuscript submitted for publication. Retrieved from https://sites.google.com/site/stephanebonhommeresearch/
Bonhomme, S., & Manresa, E. (2015). Grouped patterns of heterogeneity in panel data.
Econometrica , , 1147–1184. doi:10.3982/ECTA11319Deb, P., & Trivedi, P. K. (2013). Finite mixture for panels with fixed effects. Journal ofEconometric Methods , , 35–51. doi:10.1515/jem-2012-0018Hofmans, J., Ceulemans, E., Steinley, D., & Van Mechelen, I. (2015). On the added value ofbootstrap analysis for k-means clustering. Journal of Classification , , 268–284. doi:10.1007/s00357-015-9178-yHorneff, W. J., Maurer, R. H., Mitchell, O. S., & Dus, I. (2008). Following the rules: Inte-grating asset allocation and annuitization in retirement portfolios. Insurance: Mathe-matics and Economics , , 396–408. doi:10.1016/j.insmatheco.2007.04.004Jacobs, R. A., Jordan, M. I., Nowlan, S. J., & Hinton, G. E. (1991). Adaptive mixtures oflocal experts. Neural Computation , , 79–87. doi:10.1162/neco.1991.3.1.79Jordan, M. I., & Jacobs, R. A. (1994). Hierarchical mixtures of experts and the EM algo-rithm. Neural Computation , , 181–214. doi:10.1162/neco.1994.6.2.181Kim, J., & Wang, L. (2019). Hidden group patterns in democracy developments: Bayesianinference for grouped heterogeneity. Journal of Applied Econometrics , , 1016–1028.doi:10.1002/jae.2734Su, L., & Ju, G. (2018). Identifying latent grouped patterns in panel data mod-els with interactive fixed effects. Journal of Econometrics , , 554–573. doi:10.1016/j.jeconom.2018.06.014Suits, D. B. (1984). Dummy variables: Mechanics v. interpretation. The Review of Economicsand Statistics , , 177–180. doi:10.2307/1924713The Australian Government the Treasury. (2016). Development of the framework for compre-hensive income products for retirement (Discussion Paper). Retrieved from https://consult.treasury.gov.au/retirement-income-policy-division/comprehensive-income-products-for-retirement/supporting documents/CIPRs DiscussionPaper 1702.pdf
Minimum drawdown rates by age for account-based pensions, effective 1 July 2007.
Age at financial year start <
65 65–74 75–79 80–84 85–89 90–94 95+Minimum drawdown rate 0.04 0.05 0.06 0.07 0.09 0.11 0.14Figure 1:
Point estimates and 95% confidence intervals for partial effects of log minimum drawdownrate and log account balance covariates on log regular drawdown rate, controlling for group-level time-varying unobservable heterogeneity for a range of values of G . Note: Shaded regions denote 95% con-fidence intervals constructed using standard errors derived from fixed- T variance estimate formula inthe supplement to Bonhomme and Manresa (2015) −0.20.00.2 1 3 5 7 9 11 G C o v a r i a t e E ff e c t Covariate
Log Account Balance Log Minimum Drawdown Rate
Point estimates for effects of group-level time-varying unobservable heterogeneity on logregular drawdown rates assuming G = 4 , , ...,
9. Note: Estimated time-demeaned group time profilesshifted to begin at 0 on the vertical axis. −101 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=1447)2 (n=2368)3 (n=1358)4 (n=4343)
G = 4 Time Profiles −101 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=1227)2 (n=2320)3 (n=369)4 (n=4159)5 (n=1441)
G = 5 Time Profiles −101 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=4154)2 (n=358)3 (n=1309)4 (n=1987)5 (n=527)6 (n=1181)
G = 6 Time Profiles −101 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=2057)2 (n=865)3 (n=331)4 (n=1854)5 (n=1298)6 (n=508)7 (n=2603)
G = 7 Time Profiles −101 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=428)2 (n=828)3 (n=1909)4 (n=2721)5 (n=1807)6 (n=332)7 (n=487)8 (n=1004)
G = 8 Time Profiles −101 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=974)2 (n=301)3 (n=416)4 (n=2543)5 (n=226)6 (n=1595)7 (n=2047)8 (n=661)9 (n=753)
G = 9 Time Profiles
Point estimates and 95% confidence intervals from analytical formula for effects of group-level time-varying unobservable heterogeneity on log regular drawdown rates assuming G = 7. Note:Shaded regions denote 95% confidence intervals constructed using standard errors derived from fixed- T variance estimate formula in the supplement to Bonhomme and Manresa (2015). Time-demeanedgroup time profiles shifted to begin at 0 on the vertical axis. −101 2004 2006 2008 2010 2012 2014 Financial Year T i m e P r o f il e V a l ue Group
Figure 4:
Point estimates and 95% bootstrap confidence intervals for effects of group-level time-varying unobservable heterogeneity on log regular drawdown rates assuming G = 7. Note: Shadedregions denote 95% confidence intervals computed from empirical percentiles across 1000 bootstrapreplications. Time-demeaned group time profiles shifted to begin at 0 on the vertical axis. −101 2004 2006 2008 2010 2012 2014 Financial Year T i m e P r o f il e V a l ue Group
Point estimates and 95% confidence intervals for effects of group-level time-varying unob-servable heterogeneity on log regular drawdown rates assuming G = 1. Note: Shaded regions (indis-tinguishable from point estimates in plot) denote 95% confidence intervals constructed using standarderrors derived from fixed- T variance estimate formula in the supplement to Bonhomme and Manresa(2015). Time-demeaned time profile shifted to begin at 0 on the vertical axis. Owing to the modifica-tion of the GFE model estimation algorithm, point estimates here are identical to those recovered fromestimating a standard 2WFE model on the data. The supporting material gives the corresponding re-sults using the unmodified GFE model estimation procedure as a comparison. −101 2004 2006 2008 2010 2012 2014 Financial Year T i m e P r o f il e V a l ue Group
Figure 6:
Panel plot showing all individual time series for group 5 of the time-demeaned (TD) logregular drawdown rate variable vs. financial year. −101 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n R a t e Panel plots showing all individual time series for groups 1, 2, 3, 6 and 7 (top to bottom)of the time-demeaned (TD) log regular drawdown amount (left column) and TD log account balance(right column) variables on the vertical axes vs. financial year on the horizontal axes. Note: Plots ineach column maintain same y -axis scale for comparability. −4−20 2004 2006 2008 2010 2012 2014 G r oup TD Log Regular Drawdown Amount −6−4−202 2004 2006 2008 2010 2012 2014
TD Log Account Balance −4−20 2004 2006 2008 2010 2012 2014 G r oup −6−4−202 2004 2006 2008 2010 2012 2014−4−20 2004 2006 2008 2010 2012 2014 G r oup −6−4−202 2004 2006 2008 2010 2012 2014−4−20 2004 2006 2008 2010 2012 2014 G r oup −6−4−202 2004 2006 2008 2010 2012 2014−4−20 2004 2006 2008 2010 2012 2014 Financial Year G r oup −6−4−202 2004 2006 2008 2010 2012 2014 Financial Year
Point estimates and 95% confidence intervals from analytical formula for effects of group-level time-varying unobservable heterogeneity on log regular drawdown rates assuming G = 2. Note:Shaded regions (indistinguishable from point estimates in plot) denote 95% confidence intervals con-structed using standard errors derived from fixed- T variance estimate formula in the supplement toBonhomme and Manresa (2015). Time-demeaned group time profiles shifted to begin at 0 on the verti-cal axis. −101 2004 2006 2008 2010 2012 2014 Financial Year T i m e P r o f il e V a l ue Group upporting Material for‘Hidden Group Time Profiles: Heterogeneous DrawdownBehaviours in Retirement’ Igor Balnozan ∗† , Denzil G. Fiebig ∗ , Anthony Asher ∗ , Robert Kohn ∗ , Scott A. Sisson ∗ Version: September 2020
Abstract
This material supplements the paper with: robustness checks to examine the sensi-tivity of the main results to changes in the dataset and estimation strategy; a simulationexercise to compare simulated standard errors with those estimated from the analyticalformula and the bootstrap; additional details on the superannuation drawdowns dataset;additional descriptive results for the seven-group model; and panel plots for the two-groupmodel.
Keywords
Panel data, discrete heterogeneity, microeconomics, retirement
JEL codes
C51, D14, G40
Acknowledgements
Balnozan is grateful for the support provided by the Commonwealth Government of Australia through theprovision of an Australian Government Research Training Program Scholarship. We thank Plan For Life, Ac-tuaries & Researchers, who collected, cleaned and allowed us to analyse the data used in this research. Thisdata capture forms one part of a broader survey into retirement incomes, commissioned by the Institute of Ac-tuaries of Australia. This research includes computations using the computational cluster Katana supportedby Research Technology Services at UNSW Sydney. Balnozan, Kohn and Sisson are partially supported bythe Australian Research Council through the Australian Centre of Excellence in Mathematical and StatisticalFrontiers (ACEMS; CE140100049).
Conflict of interest statement
The authors have no conflict of interest to declare. ∗ University of New South Wales, UNSW Sydney, NSW 2052, Australia † Correspondence: [email protected] / West Lobby Level 4, UNSW Business School Building, UNSWSydney, NSW 2052, Australia. a r X i v : . [ ec on . E M ] S e p Robustness checks
This section explores the sensitivity of the main results to changes in data composition andestimation methodology.
The first robustness check determines whether the results depend materially on using a bal-anced subsample of each fund’s data. It uses the same model as the main results, but thedata is filtered down to retain a fully balanced sample, leaving N = 8274 units in the sam-ple. Covariate effects
Figure 1 plots covariate effect estimates for different values of G using the fully balanced sub-sample. Like the dataset used for the main results in the paper, the fully balanced subsamplesupports a seven-group model using the selection method in the paper. Time profiles
Figure 2 plots the time profiles for G = 4 , , ..., G = 7, with confidence intervals computed using standard errorestimates derived from the fixed- T variance estimate formula. While numerical values differ,the economic interpretation of these results is similar to the interpretation of the main resultsin the paper. The number of possible allocations of N units into G groups is G N , and solutions found bythe GFE estimation procedure depend on starting values for the algorithm (Bonhomme &Manresa, 2015); hence, as N and G increase, the GFE procedure is more likely to find a local,rather than global, optimum. This may require an increasing number of randomly selectedstarting values for the algorithm to adequately explore the solution space, if individual runsbecome trapped in regions around local optima.To test the sensitivity of the main results to the choice of 1000 starting values, Figure 4 pro-vides the estimated group time profiles from the equivalent estimation using 1 million starting2alues. With 1000 starting values, the objective function value is 2673.732, while with 1 mil-lion starting values the value is 2673.716.The time profiles are nearly identical to those from the main results in the paper. The largestabsolute difference in any pair of corresponding time profile value estimates is approximately1 . × − . With 1 million starting values, using the group labels as in the main results inthe paper, group 4 contains four more people, group 7 has one person less and group 5 hasthree fewer people, compared to the run with 1000 starting values.Table 1 also provides the covariate effect estimates for the estimation with 1 million startingvalues. The estimates and standard errors are identical to those in the main results to threedecimal places. The largest difference in estimated effect magnitudes is approximately 2 . × − , for the coefficient on the log minimum drawdown rate variable. These results suggestthat the economic interpretation of the main results in the paper are robust to the number ofstarting values used; however, it is still possible that there exists a more optimal solution withmaterially different results for the covariate effects or time profiles that was not found using 1million starting values. The paper uses a modified estimation method for unbalanced data, having the property thatfor G = 1, the results align precisely with those obtained by running a standard two-wayfixed-effects regression model. Comparison output is presented here, using the unmodifiedalgorithm to test the sensitivity of the main results in the paper to this alternative procedure.Figure 5 shows how the covariate effect estimates evolve with G . Overall, the results for thelog account balance variable appear almost identical to those in the main results. Althoughthe results for the log minimum drawdown rate differ more significantly in magnitude to thosein the main results, their economic implications are similar.Figure 6 presents time profile point estimates for G = 4 , , ...,
9. Figure 7 shows the timeprofile plot for G = 7, including 95% confidence intervals constructed using standard errorsderived from the fixed- T variance estimate formula. The plots show that the time profiles ob-tained from both estimation strategies are similar.Figure 8 shows the time profile plot for G = 1, with axes identical to the corresponding timeprofile plot in the main results section of the paper. Comparing the plots reveals that thepoint estimates differ depending on the algorithm used, and the economic interpretations oftime effects around financial year 2008 are different. Using the unmodified procedure suggestsa small downward effect in 2008 followed by a gradual rise, while this initial drop is absent inthe corresponding plot created using the modified algorithm in the paper.3 Simulation exercise
This section uses simulation evidence to argue for the validity of the GFE procedure in appli-cations with data resembling the superannuation dataset. It also uses the proposed methodfor matching labels between estimations to investigate how standard errors derived from thefixed- T variance estimate formula and the bootstrap compare to the simulated standard er-rors estimated across simulations. The creation of the simulated datasets and a framework for analysing the results are now de-scribed.
Consider the data generating process (DGP)˙ y ?it := ˙ x ? it b θ + b ˙ α g i t + ˙ v ?it , (1)where: • ˙ x ?it = ( ˙ x ? ,it , ˙ x ? ,it ) is a column vector of simulated covariate values for unit i at time t ; • b θ and b ˙ α g i t are the GFE estimates for the covariate effects and time-demeaned grouptime profiles from the main results in the paper, respectively; • the simulated covariates have mean zero and there is no time-constant individual-specificfixed effect – that is, the generated data resembles the true data after time-demeaning; • ˙ x ?k,it ∼ N (0 , b σ x k ), where b σ x k is the sample variance of all values of the ˙ x k,it , for k = 1 , • ˙ v ?it ∼ N (0 , b σ b ˙ v ), where b σ b ˙ v is the sample variance of the empirical residuals b ˙ v it := ˙ y it − ˙ x it b θ − b ˙ α g i t ; • ˙ x ?k,it are generated using a method that induces correlation between the time profile val-ues b ˙ α g i t and the covariates ˙ x ?k,it , for k = 1 ,
2; this approximates the correlation observedin the data. The details of this are given below.A simulation exercise using data simulated from (1) with the ˙ x ?k,it uncorrelated with b ˙ α g i t would be unfaithful to the challenges involved in estimating the model on the original data.Recall that the GFE method allows for arbitrary correlation between the covariates and theunobserved grouped fixed effects. Moreover, in the absence of correlation between ˙ x ?k,it and b ˙ α g i t , a standard two-way fixed-effects regression of ˙ y on the ˙ x -es directly obtains unbiased es-timates b θ . Thus, recovering accurate estimates using the simulated data is unrealistically easyif the covariates are uncorrelated with the time profiles.A realistic exercise simulates correlation between the covariates and the time profiles to match4hat observed in the data. Using the data to estimate the correlation statistics b ρ k,g , for all( k, g ), allows inducing a flexible correlation structure. The b ρ k,g values are the correlations be-tween values of ˙ x ?k,it and values of b ˙ α g i t ; i.e., b ρ k,g i is the correlation between the value of co-variate k and the value contributed to the dependent variable by individual i ’s group timeprofile. The b ρ k,g are estimated for all ( k, g ) by:1. filtering the observed data to keep only records where g i = g ;2. computing the sample correlation statistic between the observed values of ˙ x ?k,it and cor-responding estimated values of b ˙ α g i t ; call this value b ρ k,g .Having estimated the correlation statistics, the aim is to generate Gaussian random variables˙ x ?k,it which have correlation b ρ k,g i with the b ˙ α g i t . The following procedure induces this correla-tion structure, treating the model estimates of b ˙ α gt for ( g, t ) ∈ { , , ..., G } × { , , ..., T } as ifthey had been drawn from a Gaussian distribution. For all ( k, i, t ) ∈ { , } × { , , ..., N } ×{ , , ..., T } :1. set W ,kit = b ˙ α g i t / b σ α , where b σ α is estimated using the sample standard deviation of theset of G × T estimated values b ˙ α gt ;2. draw W ,kit ∼ N (0 , W ,kit = b ρ k,g i W ,kit + q − b ρ k,g i W ,kit ;4. set ˙ x ?k,it = b σ ˙ x k W ,kit , where b σ ˙ x k is the sample standard deviation of all observed valuesof covariate ˙ x k .The error term ˙ v ?it ∼ N (0 , b σ b ˙ v ), and (1) gives the simulated values of the dependent variable˙ y ? . The GFE procedure is then run on a large number of simulated datasets. Comparing theGFE estimation results to the DGP values checks the validity of the GFE procedure appliedto this setting and the code implementing the method. The simulation exercise investigates the following properties of the GFE estimator:1. whether the GFE procedure applied to a known DGP, constructed from the results ofapplying the GFE procedure to the superannuation dataset, estimates the DGP accu-rately in simulated datasets whose dimensions are the same as in the superannuationdataset;2. the closeness of the confidence intervals derived from the fixed- T variance estimate for-mula to the ‘simulated’ confidence intervals—the intervals obtained by matching timeprofile estimates across a large number of simulated datasets and observing the empiri-cal spread of estimates;3. the closeness of the simulated confidence intervals to the bootstrap confidence intervals—the intervals obtained by matching time profile estimates across a large number of boot-strap samples drawn from the first simulated dataset and observing the empirical spread5f the estimates.The following steps are used to generate the simulation results:1. M = 1000 datasets are generated independently from the DGP, each with N = 9516units and covering T = 12 time periods. The GFE procedure is then run on each ofthese using 1000 random starting values for each estimation, and the standard errors areobtained from the fixed- T variance estimate formula;2. using the resulting set of M estimates of b θ and b ˙ α := { b ˙ α gt } ( g,t ) ∈{ , ,...,G }×{ , ,...,T } as sam-ples { b θ ( m ) } Mm =1 and { b ˙ α ( m ) } Mm =1 from the sampling distributions of the estimators, simu-lated standard errors and 95% confidence intervals are estimated;3. all bootstrap results are obtained by creating B = 1000 bootstrap replicate datasets us-ing the method outlined in the methodology section of the paper—except that here thefirst simulated dataset is treated as the source dataset for the bootstrap sampling. TheGFE procedure is then run on each of the resulting bootstrap replicate datasets. Thebootstrap results that follow are different to the results obtained in the main resultssection of the paper, which use the observed data as the source dataset for bootstrapsampling;4. estimated time profiles are compared to the DGP time profiles after shifting all timeprofiles to begin at 0. For the first property listed in Section 2.1.2, the input values used for the DGP are comparedto those estimated using the GFE procedure on the first simulated dataset. Figure 9 showsthis comparison for the time profiles; Table 2 compares the covariate effects numerically. Theresults are close and suggest that the GFE procedure recovers the parameters of the DGPwith a high degree of accuracy.Figure 10 summarises the distribution of time profile estimates across all 1000 simulated datasets.The 95% confidence interval bounds represent the empirical 2 . . T variance estimate formula, where the input data is the first simu-lated dataset. The plots are nearly indistinguishable up to group relabelling, suggesting thatthe fixed- T variance estimate formula applied to the first simulated dataset estimates the truestandard errors with reasonable precision. Table 4 gives a similar comparison for the covariateeffects, and has the same interpretation.As with the previous comparison, all 1000 simulated datasets are considered next, by com-paring the empirical distribution of estimated standard errors—derived from the results ofapplying the fixed- T variance formula to 1000 simulated datasets—to the simulated standarderrors—computed by calculating the sample standard deviation of parameter estimates acrossthe 1000 simulated datasets. Table 5 provides the figure references for the standard error dis-tribution plots by group of the time-demeaned group time profile values, shifted to begin atzero in the first time period, corresponding to the financial year ended 30 June 2004. For eachgroup, the plots show the empirical distribution of the standard errors for estimates corre-sponding to financial years 2005 to 2015, inclusive. The group labels follow Figure 10, whichshows the group time profiles with confidence intervals derived from the simulated standarderrors. These are the same group labels presented in the figures for the main results in thepaper. In general, the simulated standard error value is in an area of nontrivial density in thecorresponding empirical standard error distribution.Figure 19 provides the corresponding plots for standard errors of the covariate estimates. Thefixed- T variance estimate formula tends to overestimate the true standard error for the firstcovariate; for the second covariate, the true standard error is more centrally located in thedistribution of empirical standard errors.For the third property listed in Section 2.1.2, Figures 10 and 20 are compared. Figure 20gives time profile estimates and 95% confidence intervals constructed from the empirical dis-tribution of estimates across 1000 bootstrap replications, where the input data is the firstsimulated dataset. The results are again nearly indistinguishable up to group relabelling.This suggests that inference conducted using the bootstrap procedure gives almost identicalresults to that using the fixed- T variance estimate formula, which approximates the true stan-dard errors well. Ideally, the bootstrap procedure would be performed on all 1000 simulateddatasets, and the resulting distributions of bootstrap standard errors compared to the simu-lated standard errors, as done for the fixed- T variance estimate formula. However, computa-tional constraints prevent this.Considering the simulation evidence, the fixed- T variance estimate formula performs well indatasets simulated from the generating process implied by the GFE estimation results for thesuperannuation dataset. This suggests that if the GFE assumptions are satisfied, the analyt-ical formula may be adequate in datasets of similar size to the superannuation dataset, notrequiring the bootstrap procedure for inference on the parameter estimates. This is impor-tant as using our code to perform the bootstrap on the superannuation dataset is currentlytoo computationally intensive to run on a standard machine; it requires access to a high-7erformance computing cluster. As for the bootstrap results on the simulated data, these sug-gest that the bootstrap may also perform comparably well; however, due to computationalconstraints, our implementation is unable to test this as rigorously as for the analytical for-mula. Possible covariates for the analysis included in the dataset are limited to age, account bal-ance and gender. From the available information, two derived covariates are also constructed:the minimum drawdown rate, and a crude estimate of an individual’s risk appetite over theobservation period. For each person–year observation, the age of the member maps to the rel-evant minimum drawdown rate the retiree is constrained by, with concessional reductions inthe rates applying to the 2009–2013 financial years.The construction of the risk appetite variable is the same as in Balnozan (2018). As this vari-able is only used descriptively, and does not enter into the model estimation, its precisenessdoes not affect the results. Its construction involves observing movements in account balancesand comparing these with the amounts drawn down and contributed to the funds by retirees.From this, ignoring administrative fees on the accounts, one can roughly estimate the returnon assets. As the source data is at a monthly frequency, this return is computed monthly andthen annualised; comparing it to the S&P/ASX 200 market index over matching time periodsgives an approximate measure of sensitivity to market returns. Taking the magnitude of theaverage of these sensitivities then serves as a proxy for risk appetite.Applying the within transformation—centering all variables around their time averages foreach individual—prevents estimating the effect of any time-invariant covariates, which do notshow within-individual variation; this includes gender as well as the constructed risk appetitevariable. Thus, gender and risk appetite do not enter in the GFE estimation, although theyare used to qualitatively characterise the groups that the GFE method finds in the data.Furthermore, age is not included as a covariate because the focus is on estimating group ef-fects for each time period; these time effects cannot easily be separated from the effect of age-ing after within-transforming the data.
The remainder of this section presents a preliminary descriptive analysis of the dataset usedto generate the main results. 8 .2.1 Summary statistics
Table 6 summarises characteristics that vary across individuals but not time. The age at 31December 2015 represents the individual’s cohort, equivalent to measuring a year-of-birthvariable. The median retiree in the sample was born in 1936, with more than 50% of the sam-ple born in an interval capturing four years on either side.The age at account opening is the age when the retiree initiates a phased withdrawal prod-uct and begins drawing down from the account. In the superannuation dataset, the medianretiree was aged 64 when opening their account. In general, opening an account before age 65requires an individual to cease an employment arrangement.The sex indicator variable equals 1 if the the retiree is male, and 0 otherwise. The mean valueof 0.56 represents the proportion of retirees in the sample that are male.The risk appetite variable is a proxy for the returns in the account relative to the referenceS&P/ASX 200 index. The median retiree earned approximately 46% of the index returnswhile under observation, with 75% of the sample earning less than 55% of the index returns.This proxy variable suggests that most retirees have asset mixes that are conservative or bal-anced, with few retirees seeking aggressive returns in these accounts.Table 7 summarises the time-varying variables in the dataset. The regular drawdown rate isthe dependent variable of interest. The median drawdown rate in the sample is 9% of the ac-count balances annually, while the mean is 12%. In absolute terms, the median regular draw-down amount is $4800 while the average is $6436.The ad-hoc drawdown indicator variable equals 1 if the retiree made an ad-hoc withdrawalfrom their account balance during a given financial year; its mean value of 0.07 indicates that7% of the observations recorded contain an ad-hoc drawdown. An interpretation is that theaverage retiree in the sample makes an ad-hoc drawdown roughly once every 14 years. Con-ditional on making an ad-hoc drawdown, the median ad-hoc drawdown rate is 7% of the ac-count balance at the start of the year, while the mean is 15%. In dollars, the median ad-hocdrawdown is $4656 and the average is $10,217.Median account balances, as measured at the beginning of each financial year, are $52,063,with roughly 50% of balances lying in the interval ($30,000, $87,000).
Figure 21 plots the histograms of the time-invariant covariates. Most notable is the spike inthe age at account open distribution around age 65, corresponding to the age at which indi-viduals can open a phased withdrawal account without satisfying any other conditions. Alsoinstructive are the multiple peaks in the risk appetite distribution, suggesting a bunching ofretirees into distinct asset mix options. 9igure 22 plots the histograms for the time-varying variables. The plots show some evidenceof ad-hoc drawdown rates bunching around several modes for the larger values.
This section provides summary statistics and histograms created for each of the estimatedgroups from the main results. This allows comparison of each of the groups’ characteristicsagainst other groups, using the following set of tables and plots. Alternatively, it is possibleto compare group-level characteristics against the aggregate sample, through comparison withthe results in Section 3.2. This section also presents panel plots to supplement those given inthe paper. Throughout this section, group labels follow the main results section in the paper.
Table 8 lists table references for the summary statistics of the time-invariant variables bygroup.
Table 16 lists table references for the summary statistics of the time-varying variables bygroup.
Table 24 lists figure references for the histograms of the time-invariant variables by group.
Table 25 lists figure references for the histograms of the time-varying variables by group.
Table 26 lists figure references for time-demeaned panel plots by group. Each figure showsfour variables after time-demeaning by unit: the log regular drawdown rate; the log regu-10ar drawdown dollar amount; the log account balance at the financial year start; compositeresiduals from the estimation. Composite residuals are the estimated group time profiles plusthe model residual, which arise from subtracting the estimated effect of covariates from thedependent variable: b ˙ α g i t + b ˙ v it := ˙ y it − ˙ x it b θ . The black line represents the estimated time-demeaned group time profile values b ˙ α gt . Figures 44 and 45 show panel plots of time-demeaned variables for the two groups in the two-group model. Group labels in these plots follow the two-group model results in the paper.
References
Balnozan, I. (2018).
Slow and steady: Drawdown behaviours in phased withdrawal retirementincome products.
Unpublished manuscript, UNSW Business School, University of NewSouth Wales, UNSW Sydney, Australia. Retrieved from https://sites.google.com/view/igorbalnozan/research
Bonhomme, S., & Manresa, E. (2015). Grouped patterns of heterogeneity in panel data.
Econometrica , , 1147–1184. doi: 10.3982/ECTA1131911able 1: One million starting values – Point estimates and 95% confidence intervals for par-tial effects of log minimum drawdown rate and log account balance covariates on log regulardrawdown rate, controlling for group-level time-varying unobservable heterogeneity assuming G = 6. Covariate Estimate(Standard Error)Log Minimum Drawdown Rate 0.144(0.0248)Log Account Balance − Standard errors derived from fixed- T variance estimate formula. Table 2: DGP covariate effects vs. first simulated dataset estimates.
Log Minimum Drawdown Rate Log Account Balance
DGP 0.1436 − − Log Minimum Drawdown Rate Log Account Balance
DGP value 0.1436 − − − Simulated 95% CI bounds represent empirical 2 . . Table 4: Covariate effect estimates from first simulated dataset, CIs from formula.
Log Minimum Drawdown Rate Log Account Balance
Covariate Effect Estimate 0.1445 − − −
95% CIs derived from fixed- T variance estimate formula. Table 5: Lookup table – Standard error distributions for time profile estimates. Note: Groupslabelled as per Figure 10.Group Label Figure1 122 133 144 155 166 177 18 12able 6: Summary statistics – Time-invariant variables.
Age at 31 December 2015 Age at Account Open Sex: Male Risk Appetitemean 79.4 63.57 0.56 0.41SD 5.22 4.17 0.5 0.21median 79.78 64.23 1 0.46Q1 76.37 60.9 0 0.25Q3 83.03 65.39 1 0.54min 60.66 48.48 0 0max 101.46 85.44 1 1.88count 9516 9516 9516 9507
Table 7: Summary statistics – Time-varying variables.
Regular Drawdown Regular Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Account BalanceRate Amount Indicator Rate Amountmean 0.12 6435.91 0.07 0.15 10,216.56 72,686.55SD 0.12 6121.76 0.25 0.21 24,672.68 78,721.39median 0.09 4800 0 0.07 4655.9 52,063Q1 0.07 2992 0 0.02 1132.33 30,532.5Q3 0.12 7728 0 0.18 10,000 87,427min 0 1 0 0 1 1max 2 166,695 1 0.9 600,000 2,427,083count 107,935 107,975 108,717 7450 7454 108,635
Table 8: Lookup table – Summary statistics for time-invariant variables by group. Note:Group labels follow the main results section in the paper.Group Table1 92 103 114 125 136 147 15 Table 9: Group 1 summary statistics – Time-invariant variables.
Age at 31 Dec 2015 Age at Account Open Sex: Male Risk Appetitemean 81.53 64.67 0.62 0.43SD 4.22 3.84 0.49 0.18median 81.76 65 1 0.46Q1 78.87 62.69 0 0.34Q3 84.57 66.67 1 0.54min 67.67 50.45 0 0max 94.7 80.22 1 0.92count 2057 2057 2057 2055
Age at 31 December 2015 Age at Account Open Sex: Male Risk Appetitemean 80.03 63.36 0.62 0.41SD 5.23 5 0.48 0.2median 79.41 63.67 1 0.46Q1 76.42 59.82 0 0.24Q3 83.54 65.57 1 0.54min 67.53 50.26 0 0.01max 96.23 81.78 1 1.02count 865 865 865 863
Table 11: Group 3 summary statistics – Time-invariant variables. age at 31DEC15 age at account open sex male risk appetitemean 81.75 64.66 0.69 0.35SD 4.43 4.29 0.46 0.22median 81.39 64.94 1 0.4Q1 79.24 61.93 0 0.17Q3 83.88 65.74 1 0.5min 65.82 55.92 0 0max 99.87 85.44 1 1.52count 331 331 331 331
Table 12: Group 4 summary statistics – Time-invariant variables.
Age at 31 December 2015 Age at Account Open Sex: Male Risk Appetitemean 78.17 63.54 0.51 0.44SD 6.09 4.19 0.5 0.21median 78.93 64.1 1 0.47Q1 74.25 60.79 0 0.31Q3 82.57 65.46 1 0.57min 60.66 51.3 0 0max 97.06 79.79 1 1.59count 1854 1854 1854 1852
Table 13: Group 5 summary statistics – Time-invariant variables.
Age at 31 December 2015 Age at Account Open Sex: Male Risk Appetitemean 78.12 63.92 0.49 0.47SD 6.17 4.25 0.5 0.21median 78.8 64.35 0 0.49Q1 73.35 61.48 0 0.35Q3 82.77 65.93 1 0.6min 60.81 51.58 0 0max 101.46 81.91 1 1.88count 1298 1298 1298 1298
Age at 31 December 2015 Age at Account Open Sex: Male Risk Appetitemean 78.19 63.16 0.53 0.45SD 5.7 4.16 0.5 0.21median 78.99 63.78 1 0.48Q1 74.56 60.5 0 0.34Q3 82.34 65.29 1 0.58min 61.74 52.68 0 0max 94.21 76.01 1 1.09count 508 508 508 508
Table 15: Group 7 summary statistics – Time-invariant variables.
Age at 31 December 2015 Age at Account Open Sex: Male Risk Appetitemean 78.97 62.56 0.54 0.35SD 3.93 3.77 0.5 0.2median 79.08 63.06 1 0.39Q1 76.49 60.15 0 0.22Q3 81.83 65.02 1 0.5min 65.27 48.48 0 0max 92.5 78.32 1 0.88count 2603 2603 2603 2600
Table 16: Lookup table – Summary statistics for time-varying variables by group. Note:Group labels follow the main results section in the paper.Group Table1 172 183 194 205 216 227 23 Table 17: Group 1 summary statistics – Time-varying variables.
Regular Drawdown Regular Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Account BalanceRate Amount Indicator Rate Amountmean 0.13 6687.54 0.05 0.14 10434.4 60246.23SD 0.07 5615.68 0.21 0.16 20751.13 56052.49median 0.11 5172 0 0.09 6000 44855Q1 0.09 3360 0 0.05 3432 27314Q3 0.15 7944 0 0.15 10013 73574.5min 0.01 1 0 0 1 1max 1.22 70800 1 0.9 500000 812479count 24638 24644 24684 1144 1145 24675
Regular Drawdown Regular Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Account BalanceRate Amount Indicator Rate Amountmean 0.25 6933.94 0.1 0.18 10224.05 46450.67SD 0.23 5594.9 0.3 0.17 19495.22 47647.52median 0.15 5460 0 0.12 5000 33775Q1 0.11 3372 0 0.06 2000 15689.5Q3 0.3 8928 0 0.25 10075 62153.75min 0.01 2 0 0 1 1max 2 42000 1 0.9 277831 543370count 10344 10345 10372 1114 1115 10368
Table 19: Group 3 summary statistics – Time-varying variables.
Regular Drawdown Regular Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Account BalanceRate Amount Indicator Rate Amountmean 0.34 6076.35 0.1 0.35 9789.94 33754.28SD 0.3 6305.35 0.3 0.3 19162.35 46781.99median 0.2 4400 0 0.23 4013 19960Q1 0.14 2291 0 0.09 42 5732.25Q3 0.46 7481.25 0 0.6 10000 42861.5min 0 1 0 0 1 1max 1.38 48000 1 0.9 233305 632325count 3716 3716 3944 461 461 3914
Table 20: Group 4 summary statistics – Time-varying variables.
Regular Drawdown Regular Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Account BalanceRate Amount Indicator Rate Amountmean 0.07 6412.78 0.09 0.11 8635.62 90976.64SD 0.03 6524.65 0.29 0.2 25313.44 94553.22median 0.07 4588.91 0 0.02 2253.33 65456Q1 0.06 2712 0 0.01 732.5 39864Q3 0.08 7560 0 0.09 7500 104543min 0 30 0 0 1 15max 0.94 99768 1 0.9 430000 1573153count 19460 19472 19609 1831 1833 19597
Table 21: Group 5 summary statistics – Time-varying variables.
Regular Drawdown Regular Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Account BalanceRate Amount Indicator Rate Amountmean 0.06 5898.44 0.11 0.11 9308.75 99415.18SD 0.04 6219.52 0.31 0.19 26711.86 103532.05median 0.06 4201.49 0 0.03 2565.83 71760Q1 0.05 2310 0 0 622.93 43445.75Q3 0.07 7094.75 0 0.09 8705.42 115423.25min 0 10 0 0 3 53max 0.75 109320 1 0.9 600000 1514586.47count 13184 13192 13294 1403 1403 13286
Table 22: Group 6 summary statistics – Time-varying variables.
Regular Drawdown Regular Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Account BalanceRate Amount Indicator Rate Amountmean 0.1 6377.06 0.11 0.21 13530.6 73469.07SD 0.08 5853.41 0.31 0.27 32379.36 71795.16median 0.08 4800 0 0.09 5000 53776Q1 0.07 2670 0 0.02 1159.84 30503Q3 0.11 8060 0 0.31 13318.59 89727min 0 1 0 0 1 1max 1.57 86688.13 1 0.9 455548 781753.8count 5437 5438 5592 606 606 5585
Regular Drawdown Regular Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Ad-hoc Drawdown Account BalanceRate Amount Indicator Rate Amountmean 0.09 6366.74 0.03 0.16 12575.79 73117.21SD 0.03 6367.71 0.17 0.2 26520.99 76839.21median 0.09 4764 0 0.09 6000 53267.5Q1 0.07 3036 0 0.04 3000 33820Q3 0.11 7416 0 0.18 12000 87103.25min 0.02 12 0 0 31 13max 0.92 166695 1 0.9 520802 2427083count 31156 31168 31222 891 891 31210
Table 24: Lookup table – Histograms of time-invariant variables by group. Note: Group la-bels follow the main results section in the paper.Group Figure1 232 243 254 265 276 287 29Table 25: Lookup table – Histograms of time-varying variables by group. Note: Group labelsfollow the main results section in the paper.Group Figure1 302 313 324 335 346 357 36Table 26: Lookup table – Time-demeaned (TD) panel plots by group. Note: Group labelsfollow the main results section in the paper.Group Figure1 372 383 394 405 416 427 43 17igure 1: Fully balanced subsample – Point estimates and 95% confidence intervals for par-tial effects of log minimum drawdown rate and log account balance covariates on log regulardrawdown rate, controlling for group-level time-varying unobservable heterogeneity assuming G = 1 , , ...,
16. Note: Shaded regions denote confidence intervals constructed using standarderrors derived from fixed- T variance estimate formula. −0.4−0.20.00.2 1 3 5 7 9 11 13 15 G C o v a r i a t e E ff e c t Covariate
Log Account Balance Log Minimum Drawdown Rate G = 4 , , ...,
9. Note:Estimated time-demeaned group time profiles shifted to begin at 0 on the vertical axis. −1.0−0.50.00.51.01.5 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=1802)2 (n=4201)3 (n=1054)4 (n=1217)
G = 4 Time Profiles −1.0−0.50.00.51.01.5 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=1062)2 (n=1199)3 (n=3985)4 (n=246)5 (n=1782)
G = 5 Time Profiles −1.0−0.50.00.51.01.5 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=973)2 (n=1292)3 (n=1186)4 (n=623)5 (n=3962)6 (n=238)
G = 6 Time Profiles −1.0−0.50.00.51.01.5 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=913)2 (n=2514)3 (n=2034)4 (n=599)5 (n=1194)6 (n=202)7 (n=818)
G = 7 Time Profiles −1.0−0.50.00.51.01.5 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=809)2 (n=1998)3 (n=363)4 (n=1146)5 (n=2493)6 (n=670)7 (n=585)8 (n=210)
G = 8 Time Profiles −1.0−0.50.00.51.01.5 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=117)2 (n=669)3 (n=351)4 (n=736)5 (n=220)6 (n=1128)7 (n=584)8 (n=2016)9 (n=2453)
G = 9 Time Profiles G = 7. Note: Shaded regions denote 95% confidence inter-vals constructed using standard errors derived from fixed- T variance estimate formula. Time-demeaned group time profiles shifted to begin at 0 on the vertical axis. −101 2004 2006 2008 2010 2012 2014 Financial Year T i m e P r o f il e V a l ue Group
Figure 4: One million starting values – Point estimates and 95% confidence intervals from an-alytical formula for effects of group-level time-varying unobservable heterogeneity on log reg-ular drawdown rates assuming G = 7. Note: Shaded regions denote 95% confidence intervalsderived from fixed- T variance estimate formula. Time-demeaned group time profiles shifted tobegin at 0 on the vertical axis. −101 2004 2006 2008 2010 2012 2014 Financial Year T i m e P r o f il e V a l ue Group G = 1 , , ...,
16. Note: Shaded regions denote confidence intervals constructed usingstandard errors derived from fixed- T variance estimate formula. −0.20.00.20.4 1 3 5 7 9 11 13 15 G C o v a r i a t e E ff e c t Covariate
Log Account Balance Log Minimum Drawdown Rate G = 4 , , ...,
01 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=1363)2 (n=4320)3 (n=2377)4 (n=1456)
G = 4 Time Profiles
01 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=372)2 (n=4131)3 (n=1222)4 (n=2340)5 (n=1451)
G = 5 Time Profiles
01 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=349)2 (n=530)3 (n=2004)4 (n=4125)5 (n=1189)6 (n=1319)
G = 6 Time Profiles
01 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=1293)2 (n=504)3 (n=847)4 (n=334)5 (n=1970)6 (n=1900)7 (n=2668)
G = 7 Time Profiles
01 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=898)2 (n=491)3 (n=1494)4 (n=2593)5 (n=2048)6 (n=332)7 (n=799)8 (n=861)
G = 8 Time Profiles
01 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group1 (n=229)2 (n=689)3 (n=295)4 (n=2026)5 (n=957)6 (n=1600)7 (n=752)8 (n=441)9 (n=2527)
G = 9 Time Profiles G = 7. Note: Shaded regions denote 95% confidenceintervals constructed using standard errors derived from fixed- T variance estimate formula.Time-demeaned group time profiles shifted to begin at 0 on the vertical axis. −101 2004 2006 2008 2010 2012 2014 Financial Year T i m e P r o f il e V a l ue Group
Figure 8: Unmodified estimation procedure – Point estimates and 95% confidence intervalsfrom analytical formula for effects of group-level time-varying unobservable heterogeneity onlog regular drawdown rates assuming G = 1. Note: Shaded regions denote 95% confidenceintervals constructed using standard errors derived from fixed- T variance estimate formula.Time-demeaned group time profiles shifted to begin at 0 on the vertical axis. −101 2004 2006 2008 2010 2012 2014 Financial Year T i m e P r o f il e V a l ue Group −101 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Source
DGPFirst Simulated Dataset
Figure 10: DGP time profiles and simulated 95% CIs. Note: Time-demeaned group timeprofile values are the inputs to the DGP and here shifted to begin at 0 on the vertical axis.Shaded regions denote 95% confidence intervals compute from empirical percentiles of theshifted, time-demeaned group time profile estimates across 1000 simulated datasets. −101 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group T variance estimate formula. −101 2004 2006 2008 2010 2012 2014 Financial Year T i m e P r o f il e V a l ue Group T vari-ance estimate formula after estimating the GFE model on 1000 simulated datasets. Red verti-cal lines represent the value of the simulated standard error. Time Profile Value D en s i t y Group 1, FY 2005
Time Profile Value D en s i t y Group 1, FY 2006
Time Profile Value D en s i t y Group 1, FY 2007
Time Profile Value D en s i t y Group 1, FY 2008
Time Profile Value D en s i t y Group 1, FY 2009
Time Profile Value D en s i t y Group 1, FY 2010
Time Profile Value D en s i t y Group 1, FY 2011
Time Profile Value D en s i t y Group 1, FY 2012
Time Profile Value D en s i t y Group 1, FY 2013
Time Profile Value D en s i t y Group 1, FY 2014
Time Profile Value D en s i t y Group 1, FY 2015 T vari-ance estimate formula after estimating the GFE model on 1000 simulated datasets. Red verti-cal lines represent the value of the simulated standard error. Time Profile Value D en s i t y Group 2, FY 2005
Time Profile Value D en s i t y Group 2, FY 2006
Time Profile Value D en s i t y Group 2, FY 2007
Time Profile Value D en s i t y Group 2, FY 2008
Time Profile Value D en s i t y Group 2, FY 2009
Time Profile Value D en s i t y Group 2, FY 2010
Time Profile Value D en s i t y Group 2, FY 2011
Time Profile Value D en s i t y Group 2, FY 2012
Time Profile Value D en s i t y Group 2, FY 2013
Time Profile Value D en s i t y Group 2, FY 2014
Time Profile Value D en s i t y Group 2, FY 2015 T vari-ance estimate formula after estimating the GFE model on 1000 simulated datasets. Red verti-cal lines represent the value of the simulated standard error. Time Profile Value D en s i t y Group 3, FY 2005
Time Profile Value D en s i t y Group 3, FY 2006
Time Profile Value D en s i t y Group 3, FY 2007
Time Profile Value D en s i t y Group 3, FY 2008
Time Profile Value D en s i t y Group 3, FY 2009
Time Profile Value D en s i t y Group 3, FY 2010
Time Profile Value D en s i t y Group 3, FY 2011
Time Profile Value D en s i t y Group 3, FY 2012
Time Profile Value D en s i t y Group 3, FY 2013
Time Profile Value D en s i t y Group 3, FY 2014
Time Profile Value D en s i t y Group 3, FY 2015 T vari-ance estimate formula after estimating the GFE model on 1000 simulated datasets. Red verti-cal lines represent the value of the simulated standard error. Time Profile Value D en s i t y Group 4, FY 2005
Time Profile Value D en s i t y Group 4, FY 2006
Time Profile Value D en s i t y Group 4, FY 2007
Time Profile Value D en s i t y Group 4, FY 2008
Time Profile Value D en s i t y Group 4, FY 2009
Time Profile Value D en s i t y Group 4, FY 2010
Time Profile Value D en s i t y Group 4, FY 2011
Time Profile Value D en s i t y Group 4, FY 2012
Time Profile Value D en s i t y Group 4, FY 2013
Time Profile Value D en s i t y Group 4, FY 2014
Time Profile Value D en s i t y Group 4, FY 2015 T vari-ance estimate formula after estimating the GFE model on 1000 simulated datasets. Red verti-cal lines represent the value of the simulated standard error. Time Profile Value D en s i t y Group 5, FY 2005
Time Profile Value D en s i t y Group 5, FY 2006
Time Profile Value D en s i t y Group 5, FY 2007
Time Profile Value D en s i t y Group 5, FY 2008
Time Profile Value D en s i t y Group 5, FY 2009
Time Profile Value D en s i t y Group 5, FY 2010
Time Profile Value D en s i t y Group 5, FY 2011
Time Profile Value D en s i t y Group 5, FY 2012
Time Profile Value D en s i t y Group 5, FY 2013
Time Profile Value D en s i t y Group 5, FY 2014
Time Profile Value D en s i t y Group 5, FY 2015 T vari-ance estimate formula after estimating the GFE model on 1000 simulated datasets. Red verti-cal lines represent the value of the simulated standard error. Time Profile Value D en s i t y Group 6, FY 2005
Time Profile Value D en s i t y Group 6, FY 2006
Time Profile Value D en s i t y Group 6, FY 2007
Time Profile Value D en s i t y Group 6, FY 2008
Time Profile Value D en s i t y Group 6, FY 2009
Time Profile Value D en s i t y Group 6, FY 2010
Time Profile Value D en s i t y Group 6, FY 2011
Time Profile Value D en s i t y Group 6, FY 2012
Time Profile Value D en s i t y Group 6, FY 2013
Time Profile Value D en s i t y Group 6, FY 2014
Time Profile Value D en s i t y Group 6, FY 2015 T vari-ance estimate formula after estimating the GFE model on 1000 simulated datasets. Red verti-cal lines represent the value of the simulated standard error. Time Profile Value D en s i t y Group 7, FY 2005
Time Profile Value D en s i t y Group 7, FY 2006
Time Profile Value D en s i t y Group 7, FY 2007
Time Profile Value D en s i t y Group 7, FY 2008
Time Profile Value D en s i t y Group 7, FY 2009
Time Profile Value D en s i t y Group 7, FY 2010
Time Profile Value D en s i t y Group 7, FY 2011
Time Profile Value D en s i t y Group 7, FY 2012
Time Profile Value D en s i t y Group 7, FY 2013
Time Profile Value D en s i t y Group 7, FY 2014
Time Profile Value D en s i t y Group 7, FY 2015 T varianceestimate formula after estimating the GFE model on 1000 simulated datasets. Red verticallines represent the value of the simulated standard error. Log Minimum Drawdown Rate D en s i t y Log Minimum Drawdown Rate
Log Account Balance D en s i t y Log Account Balance
Figure 20: Time profile estimates from first simulated dataset, CIs from bootstrap. Note: Re-sults from point estimates aggregated over 1000 bootstrap replications using the first sim-ulated dataset to generate bootstrap replicate datasets. Time-demeaned group time profilevalues from GFE estimation on first simulated dataset and shifted to begin at 0 on the verti-cal axis. Shaded regions denote 95% confidence intervals computed from empirical percentilesacross 1000 bootstrap replications. −101 2004 2006 2008 2010 2012 2014
Financial Year T i m e P r o f il e V a l ue Group
Age at 31 Dec 2015 c oun t Age at 31 Dec 2015
Age at Account Open c oun t Age at Account Open
Sex: Male c oun t Sex: Male
Risk Appetite c oun t Risk Appetite
Regular Drawdown Rate c oun t Regular Drawdown Rate
Regular Drawdown Amount c oun t Regular Drawdown Amount
Made Ad−hoc Drawdown c oun t Made Ad−hoc Drawdown
Ad−hoc Drawdown Rate c oun t Ad−hoc Drawdown Rate
Ad−hoc Drawdown Amount c oun t Ad−hoc Drawdown Amount
Account Balance c oun t Account Balance
Age at 31 Dec 2015 c oun t Age at 31 Dec 2015
Age at Account Open c oun t Age at Account Open
Sex: Male c oun t Sex: Male
Risk Appetite c oun t Risk Appetite
Age at 31 Dec 2015 c oun t Age at 31 Dec 2015
Age at Account Open c oun t Age at Account Open
Sex: Male c oun t Sex: Male
Risk Appetite c oun t Risk Appetite
Age at 31 Dec 2015 c oun t Age at 31 Dec 2015
Age at Account Open c oun t Age at Account Open
Sex: Male c oun t Sex: Male
Risk Appetite c oun t Risk Appetite
Age at 31 Dec 2015 c oun t Age at 31 Dec 2015
Age at Account Open c oun t Age at Account Open
Sex: Male c oun t Sex: Male
Risk Appetite c oun t Risk Appetite
Age at 31 Dec 2015 c oun t Age at 31 Dec 2015
Age at Account Open c oun t Age at Account Open
Sex: Male c oun t Sex: Male
Risk Appetite c oun t Risk Appetite
Age at 31 Dec 2015 c oun t Age at 31 Dec 2015
Age at Account Open c oun t Age at Account Open
Sex: Male c oun t Sex: Male
Risk Appetite c oun t Risk Appetite
Age at 31 Dec 2015 c oun t Age at 31 Dec 2015
Age at Account Open c oun t Age at Account Open
Sex: Male c oun t Sex: Male
Risk Appetite c oun t Risk Appetite
Regular Drawdown Rate c oun t Regular Drawdown Rate
Regular Drawdown Amount c oun t Regular Drawdown Amount
Made Ad−hoc Drawdown c oun t Made Ad−hoc Drawdown
Ad−hoc Drawdown Rate c oun t Ad−hoc Drawdown Rate
Ad−hoc Drawdown Amount c oun t Ad−hoc Drawdown Amount
Account Balance c oun t Account Balance
Regular Drawdown Rate c oun t Regular Drawdown Rate
Regular Drawdown Amount c oun t Regular Drawdown Amount
Made Ad−hoc Drawdown c oun t Made Ad−hoc Drawdown
Ad−hoc Drawdown Rate c oun t Ad−hoc Drawdown Rate
Ad−hoc Drawdown Amount c oun t Ad−hoc Drawdown Amount
Account Balance c oun t Account Balance
Regular Drawdown Rate c oun t Regular Drawdown Rate
Regular Drawdown Amount c oun t Regular Drawdown Amount
Made Ad−hoc Drawdown c oun t Made Ad−hoc Drawdown
Ad−hoc Drawdown Rate c oun t Ad−hoc Drawdown Rate
Ad−hoc Drawdown Amount c oun t Ad−hoc Drawdown Amount
Account Balance c oun t Account Balance
Regular Drawdown Rate c oun t Regular Drawdown Rate
Regular Drawdown Amount c oun t Regular Drawdown Amount
Made Ad−hoc Drawdown c oun t Made Ad−hoc Drawdown
Ad−hoc Drawdown Rate c oun t Ad−hoc Drawdown Rate
Ad−hoc Drawdown Amount c oun t Ad−hoc Drawdown Amount
Account Balance c oun t Account Balance
Regular Drawdown Rate c oun t Regular Drawdown Rate
Regular Drawdown Amount c oun t Regular Drawdown Amount
Made Ad−hoc Drawdown c oun t Made Ad−hoc Drawdown
Ad−hoc Drawdown Rate c oun t Ad−hoc Drawdown Rate
Ad−hoc Drawdown Amount c oun t Ad−hoc Drawdown Amount
Account Balance c oun t Account Balance
Regular Drawdown Rate c oun t Regular Drawdown Rate
Regular Drawdown Amount c oun t Regular Drawdown Amount
Made Ad−hoc Drawdown c oun t Made Ad−hoc Drawdown
Ad−hoc Drawdown Rate c oun t Ad−hoc Drawdown Rate
Ad−hoc Drawdown Amount c oun t Ad−hoc Drawdown Amount
Account Balance c oun t Account Balance
Regular Drawdown Rate c oun t Regular Drawdown Rate
Regular Drawdown Amount c oun t Regular Drawdown Amount
Made Ad−hoc Drawdown c oun t Made Ad−hoc Drawdown
Ad−hoc Drawdown Rate c oun t Ad−hoc Drawdown Rate
Ad−hoc Drawdown Amount c oun t Ad−hoc Drawdown Amount
Account Balance c oun t Account Balance G = 7 model – Group 1 time-demeaned (TD) panel plots. Note: Account bal-ances as at financial year start. The black series in the bottom-right panel represents esti-mated time-demeaned group time profile values. −2−1012 2004 2006 2008 2010 2012 2014 Financial Year T D Log R egu l a r D r a w do w n R a t e TD Log Regular Drawdown Rate −7.5−5.0−2.50.02.5 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n A m oun t TD Log Regular Drawdown Amount −9−6−303 2004 2006 2008 2010 2012 2014
Financial Year T D Log A cc oun t B a l an c e TD Log Account Balance −2−1012 2004 2006 2008 2010 2012 2014
Financial Year T D C o m po s i t e R e s i dua l s TD Composite Residuals G = 7 model – Group 2 time-demeaned (TD) panel plots. Note: Account bal-ances as at financial year start. The black series in the bottom-right panel represents esti-mated time-demeaned group time profile values. −2−1012 2004 2006 2008 2010 2012 2014 Financial Year T D Log R egu l a r D r a w do w n R a t e TD Log Regular Drawdown Rate −6−3036 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n A m oun t TD Log Regular Drawdown Amount −505 2004 2006 2008 2010 2012 2014
Financial Year T D Log A cc oun t B a l an c e TD Log Account Balance −2−1012 2004 2006 2008 2010 2012 2014
Financial Year T D C o m po s i t e R e s i dua l s TD Composite Residuals G = 7 model – Group 3 time-demeaned (TD) panel plots. Note: Account bal-ances as at financial year start. The black series in the bottom-right panel represents esti-mated time-demeaned group time profile values. −1012 2004 2006 2008 2010 2012 2014 Financial Year T D Log R egu l a r D r a w do w n R a t e TD Log Regular Drawdown Rate −7.5−5.0−2.50.0 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n A m oun t TD Log Regular Drawdown Amount −10−50 2004 2006 2008 2010 2012 2014
Financial Year T D Log A cc oun t B a l an c e TD Log Account Balance −2−1012 2004 2006 2008 2010 2012 2014
Financial Year T D C o m po s i t e R e s i dua l s TD Composite Residuals G = 7 model – Group 4 time-demeaned (TD) panel plots. Note: Account bal-ances as at financial year start. The black series in the bottom-right panel represents esti-mated time-demeaned group time profile values. −2−1012 2004 2006 2008 2010 2012 2014 Financial Year T D Log R egu l a r D r a w do w n R a t e TD Log Regular Drawdown Rate −4−202 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n A m oun t TD Log Regular Drawdown Amount −6−4−202 2004 2006 2008 2010 2012 2014
Financial Year T D Log A cc oun t B a l an c e TD Log Account Balance −2−101 2004 2006 2008 2010 2012 2014
Financial Year T D C o m po s i t e R e s i dua l s TD Composite Residuals G = 7 model – Group 5 time-demeaned (TD) panel plots. Note: Account bal-ances as at financial year start. The black series in the bottom-right panel represents esti-mated time-demeaned group time profile values. −202 2004 2006 2008 2010 2012 2014 Financial Year T D Log R egu l a r D r a w do w n R a t e TD Log Regular Drawdown Rate −4−202 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n A m oun t TD Log Regular Drawdown Amount −6−4−202 2004 2006 2008 2010 2012 2014
Financial Year T D Log A cc oun t B a l an c e TD Log Account Balance −202 2004 2006 2008 2010 2012 2014
Financial Year T D C o m po s i t e R e s i dua l s TD Composite Residuals G = 7 model – Group 6 time-demeaned (TD) panel plots. Note: Account bal-ances as at financial year start. The black series in the bottom-right panel represents esti-mated time-demeaned group time profile values. −2024 2004 2006 2008 2010 2012 2014 Financial Year T D Log R egu l a r D r a w do w n R a t e TD Log Regular Drawdown Rate −7.5−5.0−2.50.02.55.0 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n A m oun t TD Log Regular Drawdown Amount −50 2004 2006 2008 2010 2012 2014
Financial Year T D Log A cc oun t B a l an c e TD Log Account Balance −2024 2004 2006 2008 2010 2012 2014
Financial Year T D C o m po s i t e R e s i dua l s TD Composite Residuals G = 7 model – Group 7 time-demeaned (TD) panel plots. Note: Account bal-ances as at financial year start. The black series in the bottom-right panel represents esti-mated time-demeaned group time profile values. −1012 2004 2006 2008 2010 2012 2014 Financial Year T D Log R egu l a r D r a w do w n R a t e TD Log Regular Drawdown Rate −6−4−202 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n A m oun t TD Log Regular Drawdown Amount −7.5−5.0−2.50.02.5 2004 2006 2008 2010 2012 2014
Financial Year T D Log A cc oun t B a l an c e TD Log Account Balance −1012 2004 2006 2008 2010 2012 2014
Financial Year T D C o m po s i t e R e s i dua l s TD Composite Residuals G = 2 model – Group 1 time-demeaned (TD) panel plots. Note: Account bal-ances as at financial year start. The black series in the bottom-right panel represents esti-mated time-demeaned group time profile values. −2−10123 2004 2006 2008 2010 2012 2014 Financial Year T D Log R egu l a r D r a w do w n R a t e TD Log Regular Drawdown Rate −8−404 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n A m oun t TD Log Regular Drawdown Amount −10−505 2004 2006 2008 2010 2012 2014
Financial Year T D Log A cc oun t B a l an c e TD Log Account Balance −2−10123 2004 2006 2008 2010 2012 2014
Financial Year T D C o m po s i t e R e s i dua l s TD Composite Residuals G = 2 model – Group 2 time-demeaned (TD) panel plots. Note: Account bal-ances as at financial year start. The black series in the bottom-right panel represents esti-mated time-demeaned group time profile values. −2024 2004 2006 2008 2010 2012 2014 Financial Year T D Log R egu l a r D r a w do w n R a t e TD Log Regular Drawdown Rate −7.5−5.0−2.50.02.55.0 2004 2006 2008 2010 2012 2014
Financial Year T D Log R egu l a r D r a w do w n A m oun t TD Log Regular Drawdown Amount −8−404 2004 2006 2008 2010 2012 2014
Financial Year T D Log A cc oun t B a l an c e TD Log Account Balance −2024 2004 2006 2008 2010 2012 2014