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Estimating the Effects of
Demographics on Interest Rates:
A Robust Bayesian Perspective
WP 20-14
Paul Ho
Federal Reserve Bank of Richmond
Estimating the Effects of
Demographics on Interest Rates:
A Robust Bayesian Perspective∗
Paul Ho†
Federal Reserve Bank of Richmond
paul.ho@rich.frb.org
October 7, 2020
Abstract
There are a vast range of estimates for the effect of demographics on interest rates.
I show that these magnitudes are not well-identified without data on capital and lifecycle consumption. However, these data are often omitted. Using nonparametric prior
sensitivity analysis for an overlapping generations model estimated through Bayesian
methods, I show that without these data, small changes in the prior for the discount
rate, intertemporal elasticity of substitution, and capital depreciation rate can shift
the posterior quantiles for the effects of demographics by up to 1.5 percentage points.
Data on the capital-output ratio and life-cycle consumption tighten identification.
∗
Download the latest version of the paper here.
I am indebted to Jaroslav Borovička, Chris Sims, and Mark Watson for their guidance. I thank SeHyoun
Ahn, Carlos Viana de Carvalho, Jesús Fernández-Villaverde, Federico Huneeus, Nobuhiro Kiyotaki, Ezra
Oberfield, Mikkel Plagborg-Møller, and numerous seminar participants for comments and suggestions. The
views expressed herein are those of the author and are not necessarily the views of the Federal Reserve Bank
of Richmond or the Federal Reserve System.
†
1
Introduction
Secular changes in demographics and interest rates in developed countries over the past
thirty years have made quantifying the effects of an aging population on the interest rate
crucial for forecasting and policy analysis. The literature has sought estimates using various
overlapping generations (OLG) models but disagrees on the magnitude of these effects. For
instance, Attanasio et al. (2006), Carvalho et al. (2016), and Fujita and Fukiwara (2016) find
that between the early-1980s and mid-2010s, demographic changes contribute to declines in
the real interest rate of 2.5, 1.5, and 0.9 percentage points, respectively. For the period
between 1980 and 2020, Ikeda and Saito (2014) predict a 0.3 percentage point decline due
to demographic changes.1 One key driver of these differences is the choice of structural parameters (see Figure D.1 for the range of values of the discount rate, intertemporal elasticity
of substitution, and capital depreciation rate used in the literature).
This paper shows that the effect of demographics on interest rates is not well-identified
with the data typically used in the literature. Instead, data on capital and household consumption over the life cycle are important for accurate estimates. Without these data, the
estimated effects of demographics on interest rates in an OLG model vary substantially with
the prior for the discount rate, intertemporal elasticity of substitution, and capital depreciation rate even though other parameters are well-identified.2 We establish these insights
using robust Bayesian analysis. First, we use Bayesian methods to estimate the structural
parameters of a parsimonious OLG model and show that the discount rate, intertemporal
elasticity of substitution, and capital depreciation rate are not well-identified by the data.
We then use nonparametric prior sensitivity analysis techniques from Ho (2020) to show how
the prior for these parameters influences the estimated effects of demographics on interest
rates. Finally, we show that including data on the capital-output ratio and consumption over
the life cycle can tighten the likelihood and substantially change posterior mean estimates.
These results show that calibrating or estimating OLG models without these data, as much
of literature has done, can lead to misleading conclusions about the quantitative effect of
demographics.
Our econometric framework is a vector autoregression (VAR) with a structural break
that captures the secular changes in demographics, interest rates, and other macroeconomic
1
In comparison, the Federal Reserve Bank of New York estimates a 3 percentage point decline in the
natural interest rate for the U.S. between 1980 and 2020.
2
While the choice of model is also potentially important, these parameters are present in any model
studying demographics and interest rates. Section 5.3 discusses how our results extend to OLG models more
generally. Cross-country differences can also lead to diverging estimates. However, if the model parameters
are not well-identified, then it is hard to determine if the estimates actually arise from cross-country variation
or simply the way the model is being fitted to data.
1
aggregates. The long-run averages of the VAR are determined by the steady state of an OLG
model that captures the economic effects of demographics on the natural interest rate. We
use the OLG model from Gertler (1999), which captures the main effects of demographics
on macroeconomic variables but remains tractable and transparent. Given the structural
parameters, the model implies an interest rate that we refer to as the natural interest rate,
as in Laubach and Williams (2003). To measure the historical effect of demographics, we
take the difference between the estimated natural interest rate and the counterfactual natural
interest rate that would have arisen if specific demographic parameters had not experienced
a structural break. In particular, we study the effects of population growth, life expectancy,
and the relative productivity of retirees. No further restrictions are placed on the dynamics
of the data, allowing flexibility in the high-frequency variation.
We use Bayesian methods to fit the model to eight macroeconomic and demographic time
series from 1980-2013 for Japan, where secular macroeconomic and demographic changes
have been especially pronounced. Bayesian estimation utilizes the full likelihood and avoids
having to choose how to weight a potentially large number of overidentifying moments.
Although the starkness of these changes improves identification, there remains substantial
posterior uncertainty about both the effects of demographics and the underlying parameters.
Our estimates of the realized and counterfactual natural interest rates have relatively wide
68% credible intervals of up to 0.8 percentage points. While the time-varying parameters
and structural break date are tightly identified by the data, the likelihood for the timeinvariant parameters is dispersed. In particular, the posterior standard deviations of these
time-invariant parameters are at most 35% smaller than the prior standard deviations. Since
the effects of demographics in the model are determined by these underlying parameters, the
results suggest that the estimated effects are strongly influenced by the prior and that more
data are required to pin down the effects.
To understand how the data inform the estimated effects of demographics, we use the
relative entropy prior sensitivity (REPS) methodology from Ho (2020) to determine how
much the estimated effects of demographics on interest rates depend on the prior for the three
fixed parameters—the discount rate, intertemporal elasticity of substitution, and capital
depreciation rate. REPS considers a nonparametric set of priors that are close to the original
prior in relative entropy and seeks a worst-case prior that changes the posterior results the
most. The worst-case priors yield bounds that contain the posterior credible intervals for
any prior in that set and identify parts of the prior that are important for the posterior
estimates. REPS does not limit one to parametric or infinitesimal changes in the prior,
thus allowing one to check across an uncountable set of priors. We are therefore able to
determine how informative the data are without having to estimate the model for every
2
plausible prior (i.e., changing the means, variances, correlations, parametric families, etc.
used in the original prior). In addition, the worst-case posterior is derived from the same
likelihood as the original estimation, ensuring that the data discipline the bounds we obtain.
The methodology is thus a flexible but systematic way to measure how much the effects of
demographics depend on the fixed parameters and how informative the likelihood is in the
relevant directions.
The REPS analysis reveals that a small change in the prior can lead the posterior quantiles for the effects to shift by up to 1.5 percentage points. Many features of the worst-case
prior are plausible ex ante, emphasizing the concern that the estimated effects are not wellidentified by the data. In Appendix D, we show that these results are not dependent on the
nonparametric nature of the exercise—reestimating the model under an alternative parametric prior that is consistent with existing calibrations in the literature can lead the posterior
mean for the effects of demographics to change by over one posterior standard deviation, or
up to 0.6 percentage points.
The large role of the prior for the posterior in our baseline estimation highlights the
need to include additional data to discipline the parameters. Intuitively, the effects of
demographics on interest rates are determined by the capital-demand and savings-supply
functions. Capital and savings data provide measures of quantities, while the interest rate
provides a measure of prices. By measuring savings and investment responses to changes
in macroeconomic conditions, aggregate data on capital improves estimates of the intertemporal elasticity of substitution and depreciation rate. In addition, consumption and savings
over the life cycle reveal how households trade off current and future consumption, which is
informative about the discount rate and intertemporal elasticity of substitution.
We incorporate the capital-output ratio into the original set of time series to show how
data on capital sharpens our estimates of the effects of demographics by providing information on the underlying parameters, especially the depreciation rate. The additional data
reduces the standard deviation of the intertemporal elasticity of substitution and depreciation rate by two and five times, respectively, but does not change the precision of the discount
rate estimate. We obtain a more precise estimate for the effect of the relative productivity of
retirees on interest rates, corroborating the result from REPS analysis that the depreciation
rate is especially important for determining this effect. In addition, the new data shifts the
posterior means of the effects of demographics by up to 1.5 percentage points.
As the appropriate data on life-cycle patterns of consumption are not available for Japan,
we show indirectly that such data would be able to improve identification. In particular, we
use Monte Carlo draws from the posterior for the estimation with the capital-output ratio and
run quadratic regressions of the model-implied effects of demographics on the correspond3
ing steady-state life-cycle consumption. The steady-state consumption levels of workers and
retirees (relative to total output) account for between 35% and 80% of the posterior uncertainty in the effects of demographics, and the ratio of average retiree consumption to average
worker consumption can account for between 4% and 74% of the posterior uncertainty in the
effects of demographics. We find suggestive evidence that the life-cycle consumption data is
able to account for the effects of demographics by informing the posterior estimates for the
discount rate and intertemporal elasticity of substitution.
Related Literature. Our results suggest that the numerical results from calibrated OLG
models (e.g., Ikeda and Saito (2014); Sudo and Takizuka (2018) for Japan and Carvalho et al.
(2016); Gagnon et al. (2016); Aksoy et al. (2019); Eggertsson et al. (2019) for the rest of the
world) may be sensitive to their calibration strategies, as there is a wide range of possible
parameter values consistent with the time series in our baseline estimation. Including the
appropriate data on capital and consumption over the life cycle can help pin down the
discount rate, intertemporal elasticity of substitution, and capital depreciation rate, which
determine the effects of demographics on interest rates in OLG models. In Section 5.3, we
discuss how these conclusions extend to questions about other effects of demographics and
models with additional channels driving interest rates.
The prior sensitivity analysis extends the literature on identification in representative
agent models (e.g., Canova and Sala (2009); Iskrev (2010); Komunjer and Ng (2011)) to
OLG models. In related work, Janssens (2020) finds that the labor share, the capital depreciation rate and the intertemporal elasticity of substitution3 are not well-identified in the
Aiyagari model estimated using indirect inference on aggregate data. In our OLG model, we
find a similar conclusion that aggregate data is insufficient for identifying the intertemporal
elasticity of substitution. On the firm side, while Janssens (2020) highlights that the labor
share and capital depreciation rate are jointly identified, we find that the capital depreciation
rate is poorly identified even with direct observations of the labor share.
In terms of methodology, the REPS analysis nonparametrically establishes the lack of
identification given the data. The Bayesian approach acknowledges the relatively short
length of the available time series and does not rely on asymptotics. REPS checks across a
wider range of priors than the local method of Müller (2012), which only considers a limited
parametric class of infinitesimal changes in the prior. In addition, the structural break
framework contributes to the literature on estimating models using specific frequencies of
the data, which we discuss in Section 2.2.
3
The Aiyagari model in Janssens (2020) uses a constant relative risk aversion utility function that,
unlike the recursive preferences in this paper, does not allow us to disentangle the intertemporal elasticity
of substitution from household risk aversion.
4
Outline. The organization of the paper is as follows. I introduce the econometric model in
Section 2 and outline the OLG model in Section 3. Section 4 describes the Bayesian estimation and Section 5 describes the prior sensitivity analysis. In Section 6, I show how additional
data can improve the estimates for the effects of demographics. Section 7 concludes.
2
Econometric Framework
We now present the econometric framework that we use to disentangle secular changes in
the data from the high-frequency variation. The secular change is modeled as a change
in the steady state of an OLG model that occurs in response to a structural break in the
macroeconomic and demographic parameters. The data is annual and the high-frequency
variation is modeled as a mean-zero VAR(1) process appended to a constant term whose
values are determined by the OLG model’s steady states. The empirical analysis will examine
how the data inform estimates of both the time-varying and fixed parameters, as well as how
the effects of demographics on interest rates in the OLG model depend on these parameters.
2.1
Setup
We observe annual data yt that follows the model:
yt = µ (st ) + vt
(2.1)
vt = Φ (st ) vt−1 + ut
(2.2)
ut ∼ N (0, Σ (st )) ,
(2.3)
where st = 1 {t ≥ t∗ } is an indicator for the structural break.4 The data yt has a mean µ and
dynamics vt , which follow a mean zero VAR(1) process. The mean and VAR process change
after the structural break in period t∗ . The structural change in µ captures the secular
changes in levels observed in the data. The structural break in the VAR process in period
t∗ allows for differences in the dynamics, including changes in the volatility, persistence, or
comovement of the data after the structural break.
We model the secular changes in the data by assuming that µ is determined by the steady
state of the OLG model with parameters that depend on st . These cross-equation restrictions
on µ account for economic forces that determine the secular comovements in yt . The OLG
model also allows us to compute counterfactuals for µ. On the other hand, we allow for
flexibility in the dynamics by modeling vt as a reduced-form VAR. Since the OLG model
4
Including a second break does not materially affect results.
5
plays no role in the high-frequency variation of the data, we avoid modeling the frictions
necessary to match business cycle dynamics, which keeps the OLG model tractable both for
estimation and interpretation. Instead, we isolate features of the data that the simple OLG
model is best suited to explain.5
Estimating the econometric framework (2.1)-(2.3) is analogous to fitting the steady state
of the OLG model to data from the start and end of the sample. However, instead of using
data from a specific year, we estimate these steady states based on the full time series of data.
By estimating the break date t∗ , we allow the data to determine which periods correspond
to the steady state before and after the structural break. The VAR flexibly accounts for
correlation across time and variables.
2.2
Relation to Other Approaches
The model (2.1)-(2.3) decomposes the data into a long-run mean component µ (st ) and a
high-frequency component vt . The long-run mean captures low-frequency changes in the
data. We now discuss alternative approaches to extract low-frequency variation from data.
By using the OLG model to discipline only the long-run averages instead the full variation
of the data, the approach described above is similar to the literature on Bayesian limited
information estimation of DSGE models. For example, Christiano et al. (2010) estimate
a DSGE model using only impulse responses to a set of identified structural shocks, thus
focusing on the effect of these shocks while ignoring predictions of the model that are of
less interest to the researcher. Since we are interested in secular changes in the economy
here, we only use the steady state of the model to discipline the means µ. Sala (2015)
estimates a DSGE model in the frequency domain, restricting the estimation to components
of the likelihood corresponding to particular frequencies. In (2.1)-(2.3), the VAR captures
the high-frequency variation, and the structural estimates are informed by the low-frequency
component consisting of two long-run means, with a structural break date that is estimated.
In related work, Del Negro and Schorfheide (2004) use information from a DSGE model
to inform the priors of a VAR. In our econometric framework, instead of using the OLG
model to construct the prior, we directly model the mean of the VAR as coming from the
steady state of the OLG model. While the object of interest for Del Negro and Schorfheide
(2004) is the VAR estimates, in this paper we are interested in the estimates of the OLG
model’s structural parameters and implied counterfactuals.
5
In the absence of additional frictions, the transition path for the OLG model we consider involves jumps
at the break dates for non-demographic variables. On the break date, these variables either overshoot the new
steady state or go in the opposite direction from the change in steady state. As a result, the model-implied
transition paths are unrealistic, leading to implausible estimates.
6
If we had fitted both low and high-frequency variation to the OLG model, we would
have required a rich set of frictions in the OLG model to produce realistic dynamics, then
incorporate these dynamics in the estimation. Smets and Wouters (2007) do this in the
context of a representative agent DSGE model. Log-linearizing such a model would yield
the system (2.1)-(2.3) without the regimes st , but would impose restrictions on Φ and Σ
arising from the equilibrium conditions in the structural model. For a given set of structural
parameters, our approach yields the same µ if the frictions affect the dynamics but not the
steady state. However, we remain agnostic about the modeling of frictions. Moreover, the
approach here ensures that the structural parameter estimates are primarily driven by the
low-frequency variation in the data, which is the object of interest. Sala (2015) shows that
the posterior estimates in a representative agent DSGE model depend on the frequencies
used in the estimation, suggesting that the distinction between µ and vt is important.
One could also filter out the low-frequency fluctuations in the data using weighted averages without reference to any model (e.g., Hodrick and Prescott (1997), Müller and Watson
(2018)), then use these low-frequency fluctuations as input to estimate the parameters of the
OLG model. However, the high-frequency observations are informative for estimating the
time-varying parameters as well as the structural break date.
3
Overlapping Generations Model
The OLG model is similar to the one in Gertler (1999), which is frequently used in the literature (e.g., Fujiwara and Teranishi (2008); Carvalho et al. (2016); Kara and von Thadden
(2016)). It is a neoclassical growth model with endogenous labor supply, stochastic retirement and death, and recursive preferences. The model is a parsimonious way to capture the
main economic forces linking demographics to interest rates. In this section, we describe
the setup of the model and focus on key equilibrium conditions for the relationship between
demographics and interest rates. The full set of steady-state equilibrium conditions is listed
in Appendix A.
3.1
3.1.1
Households
Life Cycle and Population Growth
Each individual is born as a worker. At the end of each period, a worker has probability
1 − ω of retiring and a retiree has probability 1 − γ of dying. Denoting the stock of workers
by Nt , we assume (1 − ω + n) Nt new workers are born each period, so that the workforce
7
grows at a constant rate n. The ratio of retirees to workers ψt satisfies:
ψt+1 Nt+1 = γψt Nt + (1 − ω) Nt .
3.1.2
(3.1)
Retirees
Retirees have preferences:
Vtr,i,j
=
ν
Ctr,i,j
1−
ρ
r,i,j 1−ν
Lt
+ βγ
r,i,j ρ
Vt+1
1
ρ
(3.2)
,
where i and j indicate the birth and retirement cohort, respectively. Ctr,i,j and Lr,i,j
are
t
consumption and labor of retirees, respectively. These preferences imply risk neutrality with
respect to wealth, which allows for aggregation across cohorts. The intertemporal elasticity
1
of substitution, σ ≡ 1−ρ
, controls the desire to smooth consumption over time, which is a
key force for determining households’ propensity to save and hence the interest rate. The
survival rate γ augments the discount factor β, so that retirees have an effective discount
rate of βγ.
We assume perfect annuity markets (Yaari (1965); Blanchard (1985)) that insure against
the risk arising from the uncertain time of death. Each retiree places her wealth in a mutual
fund that invests its proceeds. The surviving fraction of retirees γ receive all the returns,
while those who die receive nothing. Retirees are therefore subject to the budget constraint:
Ar,i,j
t+1 =
Rt r,i,j
A
+ Wt ξLr,i,j
− Ctr,i,j ,
t
γ t
(3.3)
where Ar,i,j
is the level of assets, Wt is the wage per effective unit of labor, ξ ∈ [0, 1] is the
t
productivity of a retiree relative to a worker, and Rt /γ is the return to a surviving retiree.
3.1.3
Workers
Workers have preferences:
Vtw,i
=
ν
Ctw,i
1−
ρ
w,i 1−ν
Lt
+β
h
w,i
ωVt+1
+ (1 −
i
r,i,t+1 ρ
ω) Vt+1
1
ρ
(3.4)
and are subject to the budget constraint:
w,i
w,i
w,i
Aw,i
t+1 = Rt At + Wt Lt − Ct .
8
(3.5)
r,i,t+1
The presence of the retirees’ value function Vt+1
in (3.4) implies that workers take retirement into account when making savings decisions. Therefore, the value of γ and ξ implicitly
enter into the workers’ decisions.
3.1.4
Aggregation
Retirees. Retiree consumption is linear in the sum of assets and discounted expected value
of labor income, with marginal propensity to consume t πt that does not depend on cohort.
This allows us to write aggregate retiree consumption as:
!
Ctr
= t πt
where human wealth:
Htr = Wt ξLrt +
Rt r
At + Htr ,
γ
(3.6)
r
Ht+1
ψt
ψt+1 (1 + n) Rt+1
(3.7)
is defined as the present discounted value of expected labor income for the entire population
t
of retirees. The first term is the labor income earned in period t, while the ψt+1ψ(1+n)
term
accounts for the population growth of retirees.
Workers. Similarly, workers have a common marginal propensity to consume πt . Aggregate worker consumption can be written as:
w
Ctw = πt (Rt Aw
t + Ht ) ,
(3.8)
where human wealth is defined as:
Htw = Wt Lw
t +ω
w
r
1
Ht+1
Ht+1
1
1
1−σ
+ (1 − ω) ξ ν−1 t+1
.
1 + n Rt+1 Ωt+1
ψt+1 (1 + n) Rt+1 Ωt+1
(3.9)
The first term is the labor income of workers. The next two terms are the expected present
discounted value of human wealth in period t + 1. The variable Ωt+1 augments Rt+1 to
account for the possibility of retiring at the end of the period and is defined as:
1
1−σ
Ωt+1 = ω + (1 − ω) ξ ν−1 t+1
.
(3.10)
Workers adjust their valuation of future labor income in response to two changes that happen when they retire—they become less productive by an exogenous factor of ξ, and their
marginal propensity to consume increases by an endogenously determined factor of t+1 .
9
State Variables. The linearity of the consumption decisions implies that we do not need
to keep track of individual cohorts when solving for the aggregate steady state or dynamics.
Instead, we can aggregate across workers and retirees, respectively, which reduces the number
of states and makes Bayesian estimation feasible.
3.2
Production and Market Clearing
Aggregate output follows the Cobb-Douglas production function:
Yt = (Xt Lt )α Kt1−α ,
(3.11)
r
where Lt ≡ Lw
t + ξLt and α determine the labor share. The labor-augmenting productivity
Xt grows at a constant rate x, and capital depreciates at rate δ. Market clearing for the
capital and goods markets implies:
3.3
A t = Kt
(3.12)
Yt = Kt+1 − (1 − δ) Kt + Ctw + Ctr .
(3.13)
Structural Break
We assume that the long-run mean µ (st ) in equation (2.1) is determined by the steady state
of the OLG model above. When the structural break occurs in period t∗ , a subset of the
structural parameters are redrawn from the same distribution that generated the parameters
before t∗ . This new set of parameters yields a new steady-state and long-run mean. The
demographic parameters affected by the structural break are the working population growth
n, the survival rate of retirees γ, and the relative productivity of retirees ξ. The nondemographic parameters affected are the productivity growth x, the labor share α, and the
parameter ν controlling the disutility of labor.
3.4
Demographics and Interest Rates
The model captures several channels through which an aging demographic affects the steadystate interest rate. The strength of each channel depends nonlinearly on the structural
parameters. In what follows, we drop the t subscript to denote steady-state variables.
Firstly, as the share of retirees increases, the supply of savings decreases, thus raising
the interest rate. This savings composition channel arises because retirees have a higher
10
marginal propensity to consume, as can be seen from the steady-state expressions:
R
π = 1 −
(1 + x)1−ν
!σ−1
RΩ
π =1−
(1 + x)1−ν
!σ−1
βσγ
(3.14)
βσ
(3.15)
for retiree and worker marginal propensities to consume π and π, respectively. Retirees
have an effective discount rate of βγ < β, which induces them to consume more from their
wealth. Moreover, workers save to smooth consumption into retirement, since retirement
leads to lower productivity and hence lower expected wealth. This force is captured by the
Ω adjustment defined in (3.10) to the interest rate R in equation (3.15).
Secondly, increases in the survival rate γ or decreases in the relative productivity ξ cause
decreases in the marginal propensity to consume within groups, which decreases interest
rates through an increase in the supply of savings. We refer to this as the within-group
savings channel. When retirees have a higher probability of survival, their effective discount
rate βγ increases, which increases the incentive to save in order to smooth consumption.
Workers anticipate the reduced propensity to consume as a retiree, reflected by the Vtr,i,t+1
term in their continuation value (3.4), and respond by decreasing their marginal propensity
to consume as well. A lower relative productivity of retirees ξ implies a greater drop in human
wealth upon retirement, leading to increased saving by workers to smooth consumption.
Changes in demographics also affect interest rates through a capital demand channel,
which is captured by the equilibrium condition for capital in steady state:
R = (1 − α) k −1 + (1 − δ) ,
(3.16)
where k is the steady-state capital-output ratio Kt /Yt . In particular, when the share of
retirees increases or the relative productivity of retirees decreases, the average household in
the economy becomes less productive. In the absence of adjustments to labor supply, this
lowers the marginal product of capital (1 − α) k −1 , which leads to a fall in the demand for
capital that pushes interest rates downward. On the other hand, when workers expect to
have a longer or less productive retirement, they increase their labor supply to accumulate
wealth for retirement. This can dampen or even reverse the capital demand effect.
Finally, there is a general equilibrium channel, as defined in Carvalho et al. (2016). In
response to a decrease in interest rates from the savings composition, within-group savings, or
capital demand channels, households decrease savings, and firms increase their use of capital.
These forces amplify the direct effect of the savings composition, within-group savings, and
11
capital demand channels.
In section 5, we analyze the sensitivity of our results to the prior for the discount factor
β, intertemporal elasticity of substitution σ, and capital depreciation rate δ, because these
three parameters influence the effects described above. A higher discount rate reduces the
incentive to smooth over time, since households place a lower weight on future utility. This
greater discounting weakens the savings composition and within-group savings channels.
Since the two channels produce opposite effects, the net effect of changing β depends on
the parameters of the model. A higher intertemporal elasticity of substitution σ strengthens
the effect of the discount factor β but decreases the sensitivity of the workers’ marginal
propensity to consume π to the interest rate adjustment Ω. In particular, when σ = 1, Ω no
longer shows up in (3.15), and the incentive to smooth consumption into retirement vanishes,
reducing the savings composition and within-group savings effects. A larger σ also decreases
households’ response to changes in the interest rate, thus reducing the general equilibrium
effect. A higher capital depreciation rate δ increases the amount of investment necessary to
maintain the steady-state level of capital, hence amplifying the capital demand effect.
Any demographic change influences the interest rate through a combination of the channels described above. The relative importance of these channels determines the equilibrium
response of interest rates. To quantify the effects of demographics, one needs to use data to
discipline the parameters in the model that control the strength of each channel. Without
more formal analysis, it is hard to establish which combinations of parameter values are supported by the data and how these change the quantitative effects of demographics. These
challenges remain or are exacerbated in larger OLG models.
4
Bayesian Estimation
We use Bayesian methods to estimate the model (2.1)-(2.3) with µ determined by the steady
state of the OLG model in Section 3. Given the prior, the posterior distribution concentrates
around parameter values supported by the data. The posterior credible intervals indicate
the uncertainty in the estimates given the prior and data. Section 5 uses prior sensitivity
analysis to distinguish the contribution of the data from that of the prior.
4.1
4.1.1
Estimation
Data
We use annual data from 1980-2013 for Japan. We focus on Japan because the macroeconomic and demographic changes there have been especially pronounced, giving the data the
12
Type
100 (β − 1) discount rate
Gamma
σ
intertemporal elasticity of substitution Normal
δ
depreciation rate
Beta
−1
Mean
3.50
0.35
0.08
Std Dev
0.50
0.15
0.02
Table 4.1: Prior for fixed structural parameters (β, σ, δ).
best chance of informing the parameter estimates.
The time series we observe are GDP growth, working population growth, the share of
workers in the population, the share of workers among the employed, the employmentpopulation ratio, real wage growth, real interest rate, and labor share. In the data, we take
the working population to be individuals from 15 to 64 years old, and take the retirement
population to be individuals age 65 and above. To match our choice of 15 to 64-year-olds as
workers, ω is calibrated to 0.98 so that workers in the model have an average working life of
50 years. The data are similar to what have been used in the literature to calibrate models
quantifying the effect of demographics on interest rates. We also include data that directly
inform us of the path of the time-varying parameters. Our results will show that without
additional data on capital and life-cycle consumption, the calibrations in the literature can
produce misleading results because the effects of demographics are not well-identified.
Several papers (e.g., Attanasio et al. (2007); Kitao (2017); Sudo and Takizuka (2018))
include the capital-output ratio in their calibration but do not directly target the interest
rate. The results in these papers suggest that omitting interest rates may also lead to
misleading results. The calibrations in Attanasio et al. (2007) and Kitao (2017) produce
interest rates that are roughly 5 percentage points higher than the real return on government
bonds,6 while Sudo and Takizuka (2018) find a real interest rate that is up to 2.5 percentage
points higher than the natural interest rate estimated using the methodology of Laubach and
Williams (2003). Since we wish to estimate the effects of demographics on interest rates,
it is natural to include interest rates in the baseline estimation. Indeed, numerous papers
calibrate the discount rate to match a given interest rate (e.g., Ikeda and Saito (2014);
Carvalho et al. (2016); Eggertsson et al. (2019)). The results in Section 6 show that even
after including data on both interest rates and the capital-output ratio, we still require
life-cycle consumption data to identify the effects of demographics.
4.1.2
Prior
We focus on the prior for (β, σ, δ), which we report in Table 4.1, and describe the rest of the
prior in Appendix B. The prior for (β, σ, δ) is of particular interest because our results will
6
Attanasio et al. (2007) argue that such an interest rate is comparable to the return on equity.
13
show that the data are especially uninformative about these three parameters, and the prior
sensitivity analysis in Section 5 will show that the prior for these parameters is important
for the estimated effects of demographics.
Our baseline prior is based on values used in existing calibrations of the Gertler (1999)
OLG model. The prior for β implies a mean discount rate of 3.5%, which is close to existing
calibrations of the same model. For σ, we choose a prior with mean 0.35 and standard
deviation 0.15 to match the calibration of similar models used in papers based on the same
model (e.g., Gertler (1999) and Fujiwara and Teranishi (2008) set σ = 0.25, while Carvalho
et al. (2016) and Ferrero et al. (2019) set σ = 0.50). Finally, the prior for δ has a mean of 0.08
and standard deviation of 0.02, which allows for the range of calibrations in the literature
(e.g., Gertler (1999), Fujiwara and Teranishi (2008), and Carvalho et al. (2016) set δ = 0.10,
while Kara and von Thadden (2016) set δ = 0.05). We keep the prior independent across
parameters, as is often done in the estimation of structural models. Figure B.1 in Appendix
B compares the prior to these calibrations from the literature.
One could have picked other equally plausible priors. For example, the empirical literature has found a wide range of estimates for the intertemporal elasticity of substitution σ
ranging from 0 to 2. Similarly, the measured depreciation rate depends on subjective choices
about the measurement process, such as how much to aggregate across different types of
capital (see, e.g., Fraumeni (1997) and Feenstra et al. (2015)). In Section 5, we show that
the prior does impact posterior inference, which highlights the need for additional data.
4.1.3
Markov Chain Monte Carlo
To sample from the posterior, we use a Metropolis-within-Gibbs algorithm described in
Appendix B. We take 2 × 105 burn-in draws, which we use to calibrate the proposal density.
We then take 2.5 × 106 draws, keeping every 25th draw to save memory. To check for
convergence, we partition the draws into four blocks and ensure that the posterior moments
and marginals are similar across blocks.
4.2
4.2.1
Results
Structural Break and Long-run Means
Figure 4.1 plots the estimated long-run means µ (st ) with 68% error bands. The structural
break date t∗ is estimated to be 1991. The long-run means before and after the break are
distinct even after accounting for the error bands. Intuitively, µ is identified by 11 periods
of data before t∗ and 33 periods of data after t∗ . In addition, the OLG model places crossequation restrictions on the comovement of µ. Since variables such as the interest rate
14
0.05
0
long-run mean
data
0
-0.02
1980 1985 1990 1995 2000 2005 2010
15-64 emp / total emp
15-64 pop / total pop
-0.05
1980 1985 1990 1995 2000 2005 2010
1
0.8
0.6
0.4
1980 1985 1990 1995 2000 2005 2010
0.65
0.6
0.95
0.9
0.85
1980 1985 1990 1995 2000 2005 2010
0.55
1980 1985 1990 1995 2000 2005 2010
0
-0.05
1980 1985 1990 1995 2000 2005 2010
0.7
labor share
0.05
interest rate
1
0.05
wages
emp-pop ratio
0.02
15-64 pop
GDP
0.1
0
-0.05
1980 1985 1990 1995 2000 2005 2010
0.65
0.6
0.55
1980 1985 1990 1995 2000 2005 2010
Figure 4.1: Estimated long-run means. Blue lines: median (solid) and 68% error bands
(dashed) of long-run means; Red dashed lines: data.
and employment-population ratio are endogenous objects in the OLG model, the long-run
means µ are therefore jointly identified by the data and the equilibrium conditions of the
OLG model. The fact that the long-run means track the data indicates that parameter
combinations exist that allow the model to fit the data well.
4.2.2
Natural Interest Rate and Counterfactuals
Natural Interest Rate. We define the natural interest rate to be the interest rate implied
by the OLG model for given parameter values. In particular, define θ∗ ≡ (β, σ, δ)0 and
ζ ≡ (x, ν, α)0 . For any set of parameters (θ∗ , ζ, n, γ, ξ), we can compute the steady-state
interest rate implied by the structural model, which we denote by R (θ∗ , ζ, n, γ, ξ). For each
15
200
200
1991-2013
1980-1990
150
1991-2013
counterfactual: n
counterfactual:
counterfactual:
150
100
100
50
50
0
0
0
0.01
0.02
0.03
0.04
0
0.01
0.02
0.03
0.04
Figure 4.2: Posterior of natural interest rate and counterfactuals. Left: natural interest rate
before and after structural break; Right: natural interest rate after structural break and
counterfactuals.
period t, define the natural interest rate:
Rt ≡ R (θ∗ , ζ (st ) , n (st ) , γ (st ) , ξ (st )) .
This is similar to the existing literature extracting the natural interest rate (e.g., Laubach and
Williams (2003); Del Negro et al. (2017); Holston et al. (2017)) using equilibrium conditions
of a DSGE model. Here, we focus on the long-run average interest rates since we are
concerned with the long-run trend in interest rates. The focus on the long run allows the OLG
model to exclude frictions that are normally included in DSGE models used for extracting
the natural interest rate at business cycle frequencies.
The left panel of Figure 4.2 shows that the posterior mean of the natural interest rate
decreased from 2.85% to 0.60% after the structural break. The posteriors of the two interest
rates are distinct, providing statistical evidence that the real interest rate has declined since
the 1980s. The widths of the 68% credible intervals are 0.84 percentage points before the
break and 0.49 percentage points after the break, widths that are comparable to the error
bands that Del Negro et al. (2017) find for the low-frequency component of the natural interest rate in the United States. These credible intervals are wide enough to imply substantial
uncertainty about the effects of a given path of interest rates.
Counterfactuals. We use the counterfactual natural interest rate to quantify the contribution of the population growth rate n, survival rate γ, and relative productivity of retirees
16
100(
1
-1
-1)
4
0.8
30
prior
posterior
3
20
0.6
2
0.4
10
1
0.2
0
0
0
1
2
3
4
5
0
0
0.5
1
0
0.05
0.1
0.15
0.2
Figure 4.3: Marginal priors and posteriors of fixed parameters. Solid blue line: prior;
Dashed red line: posterior.
ξ. In particular, we consider the counterfactual interest rates:
b n ≡ R (θ ∗ , ζ (s ) , n̂, γ (s ) , ξ (s ))
R
t
t
t
t
b γ ≡ R (θ ∗ , ζ (s ) , n (s ) , γ̂, ξ (s ))
R
t
t
t
t
b ξ ≡ R(θ ∗ , ζ (s ) , n (s ) , γ (s ) , ξ),
ˆ
R
t
t
t
t
where we pick the counterfactual parameter values ˆ· to be the median estimate for the parameter in 1980. These counterfactuals change one of the demographic parameters while keeping
all other parameters identical. Given the Monte Carlo draws for each of the parameters, we
b γ , and R
bξ.
b n, R
can construct the posterior distributions for R
t
t
t
b γ , and R
b ξ , i.e., the
bn , R
The right panel of Figure 4.2 shows the posterior distribution of R
T
T
T
counterfactual natural interest rate at the end of the sample had one of the three demographic
parameters remained at its median value from before the structural break. They have means
of 1.47%, 1.27% and 1.80% respectively, thus explaining between one-third and one-half of
the decline in interest rates.7 The widths of the 68% credible intervals range are 0.58, 0.64,
and 0.83 percentage points respectively.
4.2.3
Structural Parameters
The above estimates reveal substantial posterior uncertainty in the estimates for both the
natural interest rate and the counterfactuals. To better understand the sources of uncertainty, we now turn to the estimates of the underlying structural parameters.
7
The effects are not additive—the effect from changing more than one parameter is not the sum of
the effects of changing each of those parameters individually. In addition, variation in the macroeconomic
parameters also influence the change in the natural interest rate. For instance, the decline in labor share
puts upward pressure on the interest rate, while the decline in productivity contributes to the decline in the
interest rate.
17
150 prior
1980-1990
1991-2013
100
50
0
0
0.02
0.04
300
150
200
100
100
50
0
-0.01
productivity growth
0
0.01
0.02
0.03
0
0.88
population growth
20
0.92
0.94
0.96
survival rate
50
80
40
15
0.9
60
30
10
40
20
5
20
10
0
0
0.1
0.2
0.3
0.4
relative productivity of retirees
0
0.2
0.25
0.3
0.35
disutility of labor
0.4
0
0.6
0.65
0.7
labor share
Figure 4.4: Marginal priors and posteriors of time-varying parameters. Solid blue line:
prior; Dotted red line: posterior for parameter before structural break; Dashed green
line: posterior for parameter after structural break.
Fixed Parameters. Figure 4.3 shows that the marginal posteriors for the fixed parameters (β, σ, δ) are relatively close to their priors, suggesting a dispersed likelihood. This is
especially true for the discount rate β and capital depreciation rate δ. Intuitively, (β, σ, δ)
are identified from two steady-state observations. The dispersed posteriors arise from the
OLG model not placing substantial restrictions on the possible values of (β, σ, δ) individually
given the estimated long-run means. Nevertheless, the likelihood is informative about the
joint distribution of (β, σ, δ), producing a posterior correlation between σ and δ of −0.66.
One implication of these results is that one of the three fixed parameters could be wellidentified given the other two parameters. However, fitting these parameters jointly to the
data could yield a much wider range of possible values. Section 5 shows that the parameter
values supported by the data can imply varied effects of demographics in the OLG model.
The dispersed marginal posteriors are not unique to the model and data here. For
example, Smets and Wouters (2007) state that they calibrate δ because it is difficult to
estimate with the data they use. In addition, they obtain relatively diffuse estimates for β
and σ even though they use a longer time series of quarterly data and estimate the model
using all frequencies in the data. While we lose information from business cycle fluctuations,
we also have a more parsimonious model that has fewer parameters to be estimated.
18
Time-varying Parameters. In contrast to the fixed parameters, the time-varying parameters have marginal posteriors that are substantially tighter than their priors, as shown
in Figure 4.4. Moreover, the posteriors for each of the parameters before and after the structural break are distinct from each other. The tightness of the posterior reflects the fact that
the long-run averages of the data are tightly estimated relative to the prior.
Each time-varying parameter is closely connected to one of the time series. Working
population growth is directly observed, while the survival rate can be inferred from the fraction of workers in the population given population growth. In the OLG model, productivity
growth is equal to real wage growth as well as per capita GDP growth. The relative productivity of retirees is closely related to the fraction of workers among the employed, and the
disutility of labor is similarly connected to the employment-population ratio.
In general, the estimates match the historical narrative of Japan’s economy from 1980 to
2013. The decline in productivity growth was a symptom of the lost decade. The decrease in
population growth corresponds to the declining birth rates since the early 1970s, while the
increase in survival rate matches the growth in life expectancy. The relative productivity of
retirees is estimated to be lower after the break date due to the decrease in the fraction of
the workforce below the age of 65. Even though the aggregate employment rate decreased,
the employment rate by age group has increased since the 1980s, indicating a decrease in
the disutility of labor. Finally the decline in the labor share has been documented by
Karabarbounis and Neiman (2014) and others.
5
Prior Sensitivity
To formally establish that the data do not inform the estimated effects of demographics
due to a lack of identification for (β, σ, δ), we now analyze the sensitivity of the estimated
counterfactual interest rates to changes in the prior of (β, σ, δ) using the relative entropy prior
sensitivity (REPS) methodology introduced by Ho (2020). REPS explores a nonparametric
set of priors that are close to the original prior in terms of relative entropy and finds bounds
for the posterior results. In particular, we compute 68% robust credible intervals, defined as
bounds that contain the equal-tailed 68% posterior credible interval for any prior in a given
set of priors. In particular, we find the upper (lower) bound for the posterior 84% (16%)
quantile given a set of priors that is close to the original prior. A wider robust credible
interval relative to the original posterior credible interval indicates a greater dependence on
the prior, which corresponds to the data being uninformative about underlying parameters
and those parameters being important for the effect of demographics.
Our analysis is motivated by Section 3 showing theoretically that the three parameters
19
(β, σ, δ) have a role in determining the effect of demographics on interest rates and by Section
4 showing that these parameters are not well-identified by the data. Because the interest
rate is determined jointly by all the equilibrium conditions, it is difficult to predict how these
estimated effects change if we jointly change the prior of (β, σ, δ). Changing the prior, as
opposed to exogenously specifying new values of (β, σ, δ), ensures that the new posterior
continues to be disciplined by the data, respecting the joint likelihood of both (β, σ, δ) and
the time-varying parameters. While it is impossible to reestimate the model for all possible
priors on (β, σ, δ), REPS provides bounds on our results for a nonparametric set of priors
close to the original prior and identifies features of the prior that are important for the
estimated effects.
5.1
Methodology
Denote the full vector of parameters by θ, the prior by π, and the posterior by p. As before,
define θ∗ ≡ (β, σ, δ)0 , and let the marginal prior and posterior of θ∗ be π ∗ and p∗ , respectively.
We are interested in how much the qth quantile of a function ϕ (θ) can change as we change
the marginal prior π ∗ . In our setting, ϕ is the difference between the counterfactual and
realized natural interest rate:
n
bξ − R
bn − R , R
bγ − R , R
ϕ (θ) ∈ R
T
T
T
T
T
T
o
(5.1)
These three definitions of ϕ measure the effects of population growth n, the survival rate γ,
and the relative productivity of retirees ξ on the interest rate. Each choice of ϕ depends on
the prior in a distinct way because n, γ, and ξ influence the interest rate in different ways.
We solve for an alternative prior π̃ ∗ for θ∗ that minimizes the qth quantile:
"
min
ϕ̃ s.t. R ≥ Eπ̃∗
∗
π̃
π̃ ∗ (θ∗ )
π̃ ∗ (θ∗ )
log
π ∗ (θ∗ )
π ∗ (θ∗ )
q = Ep̃ [1 {ϕ (θ) ≤ ϕ̃}] ,
!#
(5.2)
(5.3)
where p̃ is the posterior arising from the worst-case marginal prior π̃ ∗ , keeping the conditional prior of the remaining parameters and the likelihood the same. Solving (5.2)-(5.3)
involves searching for a worst-case prior that minimizes the qth quantile of ϕ. Replacing
the minimization with maximization instead yields the upper bound for the quantile. The
constraint (5.3) states that ϕ̃ is the qth posterior quantile of ϕ. The constraint in (5.2)
limits the relative entropy or Kullback–Leibler divergence of π̃ ∗ relative to π ∗ to be less than
some constant R ≥ 0, restricting us to choose among priors that are statistically difficult
20
to distinguish from the original prior π ∗ . Problem (5.2)-(5.3) does not place parametric
restrictions on the alternative prior. In particular, the feasible set of priors includes nonparametric distributions that could introduce correlations across parameters even though we
started with a parametric and independent prior on θ∗ . In our application here, we seek
bounds for the 68% equal-tailed credible interval for effects of n, γ, and ξ. We therefore take
q ∈ {0.16, 0.84} in (5.3) with minimization replaced with maximization for q = 0.84.
To gauge the size of R, Ho (2020) recommends the rule:
R=
v
u
u
d π (µp ) t |Σ` |
1.6 r
π (µπ )
−1
Σ` ≡ Σ−1
p − Σπ
(5.4)
|Σπ |
−1
(5.5)
,
where µp and µπ are the posterior and prior means for θ∗ , respectively; Σp and Σπ are the
posterior and prior variances for θ∗ , respectively; and d is the dimensionality of θ∗ . Σ` is a
measure of how dispersed the likelihood is. The choice of r determines how large the relative
entropy is, with r < 0.05 corresponding to small levels of relative entropy.8 We shall pick r
to correspond to a one to two standard deviation change in the quantiles on average.
We implement the above computations using the sequential Monte Carlo and numerical
approximations described in Ho (2020). See Appendix C for details.
5.2
5.2.1
Results
Robust Credible Intervals
Figure 5.1 shows that the robust credible intervals are wide, indicating that a small change
in the prior for (β, σ, δ) can produce a large change in the posterior estimates for the effects
of demographics. In particular, with r = 0.005, the robust credible intervals are between
3.7 and 5.5 posterior standard deviations wider than the corresponding original credible
intervals. The quantiles shift by an economically significant amount of up to 1.5 percentage
points.9 Relative to the r = 0.05 benchmark, r = 0.005 corresponds to a very small set
of priors, which shows that the estimated effects of demographics are very sensitive to the
prior.
0
Suppose ψ is a linear combination of θ∗ , and we estimate θ∗ ≡ (β, σ, δ) from a large number of
observations θ∗ + εt , with εt ∼ N (0, Ω) where Ω is known. Then r = 0.05 and r = 0.005 would correspond
to changes in the quantile of approximately 0.4 and 0.1 posterior standard deviations, respectively. Ho (2020)
shows that if the variance of εt is equal to the prior variance of θ∗ , then ten observations are sufficient for
this to be a good approximation.
9
Smets and Wouters (2007) estimate that the standard deviation of the monetary policy shock is 0.2
percentage points and a one standard deviation monetary policy shock leads to a 3% decline in output.
8
21
population growth
300
250
survival rate
200
original
robust
100
relative productivity of retirees
80
150
200
60
150
100
40
100
50
20
50
0
0
0
0.005
0.01
0.015
0
0
0.01
0.02
-0.01
0
0.01
0.02
0.03
Figure 5.1: Robust credible intervals for difference between counterfactual and realized natural interest rate with r = 0.005. Dashed blue line: original 68% credible interval; Dotted
red line: robust 68% credible interval; Solid black line: posterior density.
There are two reasons for the high degree of prior sensitivity. Firstly, as suggested in
Figure
4.3, the likelihood for (β, σ, δ) is dispersed. Defining Σπ and Σ` as in (5.5), we find
q
|Σ` | / |Σπ | = 2.2, which indicates that the likelihood is more dispersed than the prior.
bn − R , R
b γ − R , and
Secondly, (β, σ, δ) are strong predictors of the demographic effects R
T
T
T
T
ξ
γ
ξ
−1
n
b
b
b
b
RT − RT . Quadratic regressions of RT − RT , RT − RT , and RT − RT on (β , σ, δ) yield
R-squares of 0.67, 0.74, and 0.93, respectively. Changing the prior for (β, σ, δ) thus results
in a large change in the posterior for both (β, σ, δ) and the effects of demographics.
These results show that the data are uninformative about structural parameters that
are important for determining the effects of demographics on interest rates in the OLG
model. Since the dispersion of the likelihood is a feature of the model and data only, any
other function of the model parameters that is well-predicted by (β, σ, δ) would also depend
heavily on the prior.
In Appendix D, we also reestimate the model using an alternative parametric prior that
is motivated by the results here. The posterior estimates change substantially under the new
prior, corroborating the results here.
5.2.2
Worst-case Posteriors
To understand how the estimates depend on each part of the prior, we compare the original
and worst-case posteriors for the 16% and 84% quantiles, shown in Figures 5.2 and 5.3,
respectively. The worst-case distortions for the 16% (84%) quantiles place greater weight
on parameter values that imply smaller (larger) effects of demographics. These distortions
indicate the relative importance of the savings composition, within-group savings, capital
demand, and general equilibrium effects, as well as their sensitivity to changes in (β, σ, δ).10
10
While these are not necessarily the only distortions that shift the effects in the desired direction, each
worst-case prior provides one example of an alternative prior that has a large impact on the estimated effects
22
population growth
1
survival rate
1
relative productivity of retirees
1
original
worst case
0.5
0.5
0
0.5
0
2
4
6
0
8
2
100( -1 - 1)
4
6
8
2
100( -1 - 1)
6
6
6
4
4
4
2
2
2
0
0
0
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0
40
40
40
20
20
20
0
0.05
0.1
0.15
0.2
0.25
6
8
0
0
0
4
100( -1 - 1)
0.2
0.4
0.6
0.1
0.15
0.8
1
0
0.05
0.1
0.15
0.2
0.25
0.05
0.2
0.25
Figure 5.2: Original and worst-case posteriors generating a one posterior standard deviation
bξ − R .
b γ − R ; Right: R
b n − R ; Center: R
decrease in the 16% quantile. Left: R
T
T
T
T
T
T
The distortions are asymmetric and involve joint changes in the prior for (β, σ, δ). These
features emphasize that it is difficult to predict how one’s choice of prior may be affecting
one’s estimates. Moreover, the nonlinear dependence of the estimated effects on the prior
imply that reestimating the model with an ad hoc alternative prior may not give a complete
picture of the sensitivity of estimates to the prior. Similarly, our results show that even
though a wide range of calibrations could be consistent with the data, it is hard to know ex
ante how the choice among these calibrations may affect the model’s quantitative predictions.
Population Growth. The worst-case posteriors corresponding to the effect of population
growth involve especially large distortions to the marginal of the intertemporal elasticity
of substitution σ, placing more weight on large (small) values of σ to decrease (increase)
the estimated effect of population. A change in the working population growth n affects the
interest rate through the savings composition, capital demand, and general equilibrium channels. Increasing σ reduces the worker’s incentive to save for retirement, which decreases the
savings composition effect, thus strengthening the net effect of population growth. However,
a large σ also dampens the effect of population growth by decreasing the general equilibrium
of demographics.
23
population growth
1
survival rate
1
relative productivity of retirees
1
original
worst case
0.5
0.5
0
0.5
0
2
4
6
0
8
2
100( -1 - 1)
4
6
8
2
100( -1 - 1)
6
6
6
4
4
4
2
2
2
0
0
0
0.2
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0
40
40
40
20
20
20
0
0.05
0.1
0.15
0.2
0.25
6
8
0
0
0
4
100( -1 - 1)
0.2
0.4
0.6
0.1
0.15
0.8
1
0
0.05
0.1
0.15
0.2
0.25
0.05
0.2
0.25
Figure 5.3: Original and worst-case posteriors generating a one posterior standard deviation
bξ − R .
b γ − R ; Right: R
b n − R ; Center: R
increase in the 84% quantile. Left: R
T
T
T
T
T
T
effect. That the worst-case posterior concentrates around large values of σ to decrease the
16% quantile suggests that the general equilibrium effect is especially sensitive to σ.
The distortions for β −1 are asymmetric. To decrease the 16% quantile, the worst-case
distortion places more weight on large values of β −1 . In contrast, the marginal for β −1 is
relatively unchanged for the 84% quantile. The asymmetry highlights the nonlinearity in
the mapping from (β, σ, δ) to the effect of population growth. For example, the effect of β
on the marginal propensities to consume (3.14) and (3.15) is amplified by a larger value of
σ. As a result, changing β has a smaller effect when accompanied by a decrease in σ for the
84% quantile worst-case distortion.
For both the 16% and 84% quantiles, the distortion of δ is relatively small, suggesting
that the capital demand channel is less important than the effects originating from household
savings decisions. The small increase in δ for the 84% quantile arises due to the negative
correlation of −0.66 between σ and δ.
Survival Rate. The worst-case posterior for the 16% quantile for the effect of the survival
rate is similar to that of population growth, but the distortions for the 84% quantile differ,
placing greater weight on small values of β −1 , large values of σ, and small values of δ.
24
Decreasing β −1 increases the within-group saving effect. Firstly, decreasing β −1 amplifies
the change in the effective discount rate βγ arising from a change in the survival rate γ.
Retirees thus increase savings more in response to an increase in γ. Secondly, decreasing
β −1 increases the workers’ incentive to smooth consumption into retirement, causing their
marginal propensity to consume to decline more in response to an increase in γ.
On its own, increasing σ dampens the effect of the survival rate on interest rates. However,
increasing σ also amplifies the effects of changing β. That the worst-case distortions for the
84% quantile involves an increase in σ suggests that the interaction with β is more important
than the direct effect of σ. The nonlinearity emphasizes the importance of studying the joint
distribution and effect of (β, σ, δ) rather than analyzing each parameter independently.
The worst-case distortions for the 84% quantile result in a new posterior mode for δ
around 0.05. The decrease in δ dampens the capital demand channel. This is consistent with
an increase in the effect of the survival rate γ because the increase in worker labor supply
more than offsets the increased fraction of retirees in the economy. As a result, average labor
productivity rises, which increases the marginal product of capital. Decreasing δ reduces the
resulting increase in capital demand, thus strengthening the effect of γ on interest rates.
Relative Productivity of Retirees. The worst-case posterior for the 16% quantile for
the effect of the relative productivity ξ is similar to that of the 84% quantile for the effect
of the survival rate γ. These distortions increase the effect of γ but decrease the effect of ξ.
The contrasting dependence on the prior shows that the estimates of both the absolute and
relative magnitudes of the different effects of demographics are sensitive to the prior.
The worst-case posterior indicates the importance of the capital demand channel for
determining the effect of the relative productivity of retirees. In addition, it shows that the
capital demand effect serves to decrease interest rates in response to a decrease in ξ even
though it increases interest rates in response to a decrease in γ.
Decreasing β −1 increases the incentive for workers to increase their labor supply in response to a decrease in ξ, hence decreasing the downward pressure on interest rates from
the capital demand channel. This dominates the increase in the within-group savings effect
arising from the reduction of β −1 . The decrease in δ also dampens the capital demand effect.
On the other hand, the increase in σ decreases the incentive for workers to increase their
labor supply but also decreases the within-group savings effect. The effect of σ on the latter
channel dominates. The worst-case posterior for the 84% quantile also suggests an important
role for the capital demand channel as more weight is put on large values of β −1 and δ.
The fact that the marginal distortions push the various effects of ξ in different directions
emphasizes the nonlinear dependence on (β, σ, δ). The analysis here accounts for this non25
linearity while remaining disciplined by the likelihood. In particular, the endogenous labor
supply response to changes in ξ and γ are disciplined by the data on the fraction of workers
among the employed and the employment-population ratio.
5.3
Implications for Estimation and Calibration
Additional Data. The importance of (β, σ, δ) informs us of the data needed to tighten our
estimates of the effects of demographics. Aggregate data on capital and investment, such as
the capital-output ratio, can help identify σ and δ by providing the response of savings and
investment to interest rate changes. Data on consumption and saving over the life cycle can
help to better identify β and σ. Such data provides information on how households respond
to their life-cycle trajectory of wages, which helps to quantify the savings composition effect.
The evolution of these patterns over time provides information on the exact response of
households’ marginal propensities to consume to changes in life expectancy or future wages,
which in turn helps to identify the strength of the within-group savings effect.
An alternative approach would be to include a longer time series or use a panel of countries. Increasing the length of the time series would increase the effective number of steadystate observations if we observe structural breaks prior to 1980. However, constructing a
sufficiently long time series to narrow estimates on (β, σ, δ) is challenging given the relative
scarcity of data before 1950. Including a larger panel of countries would similarly allow us to
observe more regimes. For such an exercise to narrow our estimates of (β, σ, δ), we require
that (β, σ, δ) are identical or relatively similar across countries.
Calibration. The results here also have implications for the calibration of OLG models.
In particular, when calibrating OLG models, one should consider a range of overidentifying
restrictions to pin down (β, σ, δ). For example, while it is possible to pin down β conditional
on σ and δ, given a particular interest rate, our analysis has shown that the data support
a range of possible interest rates. Once one acknowledges the uncertainty in σ and δ, one
has an even wider range of possible combinations. Our results show that these parameter
combinations can produce different conclusions. Additional data is required to determine
the appropriate parameter values with more precision.
Extensions and Other Questions. The dependence of our estimates on the prior for
(β, σ, δ) is not unique to the model considered here. The savings composition, within-group
savings, capital demand, and general equilibrium channels are present in many OLG models
used to quantify the effects of demographic changes on the interest rate. Additional channels
such as financial frictions (Ikeda and Saito (2014); Wong (2018)), public pension schemes
26
(Muto et al. (2016); Sudo and Takizuka (2018)), or international capital flows (Brooks (2003);
Attanasio et al. (2006)) also depend on (β, σ, δ). These extensions will only tighten the
likelihood of (β, σ, δ) if they place additional cross-equation restrictions that discipline the
range of plausible values for (β, σ, δ). In such cases, it would be important to understand
the sensitivity of these restrictions to the details of the extensions and to show empirical
support for the relevant parts of the model.
The identification of (β, σ, δ) is also important for quantifying other connections between
demographics and the economy. For example, the consequences of public pension policy (e.g.,
Imrohoroglu et al. (1995); Attanasio et al. (2007); Kitao (2017)) or the role of demographics
in the transmission of monetary policy shocks (e.g., Fujiwara and Teranishi (2008); Wong
(2018)) also depend on (β, σ, δ). The need for more data extends to quantitative analysis of
such questions.
6
6.1
The Role of Additional Data
Capital-output Ratio
We now show how data on capital improves identification. First, we reestimate the model
using the capital-output ratio along with the original eight time series with the original
prior from Section 4. We then repeat the REPS analysis. Both exercises show that the
capital-output ratio adds substantial information about the effects of demographics.
6.1.1
Bayesian Estimation
Table 6.1 shows that the inclusion of the capital-output ratio tightens the posterior estimate
of the structural parameters. The inclusion of data on the capital-output ratio decreases
the posterior standard deviation of δ by nearly five times, while the standard deviation
of the intertemporal elasticity of substitution σ is halved. The standard deviation of the
discount rate β −1 is roughly unchanged. The posterior means of δ and σ change by over two
posterior standard deviations, while the mean of the discount rate is relatively unaffected by
the inclusion of the capital-output ratio.
The changes in the posterior indicate that the capital-output ratio adds substantial information about the values of σ and δ but not β. Equation (3.16) shows that given the
interest rate R − 1 and the labor share α, the capital-output ratio pins down the capital
depreciation rate δ. Similarly, the marginal propensities to consume (3.14)-(3.15) show that
the responsiveness of savings to interest rates is determined by σ. Data on interest rates
measure the price of savings and capital, while the capital-output ratio measures the re27
mean
Original
sd
68% C.I.
With K/Y Ratio
mean
sd
68% C.I.
Structural Parameters
100 β −1 − 1
disc. rate
σ
IES
100δ
dep. rate
3.245
0.574
7.861
0.442
0.103
1.485
(2.804, 3.690)
(0.471, 0.677)
(6.393, 9.317)
3.443
0.282
11.394
0.427
0.053
0.300
(3.013, 3.873)
(0.230, 0.335)
(11.098, 11.692)
Effects
of Demographics
n
b
pop. growth
100 RT − RT
0.875
0.159
(0.725, 1.023)
1.480
0.421
(1.122, 1.817)
bγ
100 R
T
bξ
100 R
T
− RT
surv. rate
0.671
0.266
(0.429, 0.905)
2.140
0.808
(1.401, 2.858)
− RT
rel. prod.
1.206
0.412
(0.793, 1.618)
1.423
0.202
(1.240, 1.604)
Table 6.1: Posterior statistics for fixed structural parameters and effects of demographics,
including capital-output ratio in data.
sponse of the quantities. As a result, fitting the model to both of these variables gives us
more precise estimates of σ and δ.
With the additional data, the posterior standard deviation of the effect of the relative
productivity of retirees ξ is halved, but the posterior standard deviation of the effects of
population growth n and the survival rate γ are approximately tripled. The estimated effect
of ξ is more precise because of the tighter estimate of δ. In particular, Figures 5.2 and
5.3 show that the worst-case posteriors for the effect of ξ involve large distortions in the
marginal of δ, suggesting that the estimate of δ is important for the estimated effect of ξ.
The marginal of δ is distorted less in the worst-case posteriors corresponding to the effects
of n and γ. The posterior for the effects of n and γ become more dispersed because of
the new estimates for the structural parameters. The decrease in σ and increase in δ make
savings and investment more responsive to changes in the interest rate, which increases the
general equilibrium channel. The increased general equilibrium channel amplifies the effects
of demographics on interest rates, increasing the difference between the net effect of small
and large partial equilibrium effects. This amplification outweighs the effect of the increased
precision in the estimates.
Besides changing the posterior standard deviations, the introduction of the capital-output
ratio data also causes the posterior means of the effects to shift by up to 1.5 percentage points.
The difference in estimates shows that the choice of data can alter the conclusions from
quantitative exercises, such as policy analysis or forecasting that depend on the interaction
between demographics and interest rates.
28
150
population growth
survival rate
60
250
50
100
relative productivity of retirees
200
40
original
r = 0.005
r = 0.025
150
30
50
100
20
50
10
0
0.005
0
0.01
0.015
0.02
0.025
0
0.02
0.04
0
0.01
0.012 0.014 0.016 0.018
0.02
Figure 6.1: Robust credible intervals for difference between counterfactual and realized natural interest rate with r ∈ {0.005, 0.025} in estimation with capital-output ratio. Dashed
blue line: original 68% credible interval; Dotted red line: robust 68% credible interval
with r = 0.005; Dash-dot green line: robust 68% credible interval with r = 0.025; Solid
black line: posterior density.
6.1.2
Prior Sensitivity
Figure 6.1 shows that the inclusion of the capital-output ratio results in robust credible
intervals that are much closer to the original credible intervals, which indicates that the data
are more informative in this estimation. For comparison, we consider the robust credible
intervals with r = 0.005 with and without the capital-output ratio. Table 6.2 reports the
statistic:
χr (ϕ)
− 1,
ρ (ψ) ≡
χp (ϕ)
where χp and χr are the widths of the original and robust 68% credible intervals, respectively,
for each definition of ϕ in (5.1). The statistic ρ quantifies the sensitivity of the credible
interval to the prior by measuring how much wider the robust credible interval is compared
with the original credible interval. Including the capital-output ratio decreases ρ from 2.0,
3.1, and 2.7 to approximately 0.2. Even though the credible intervals for the estimated effects
of population growth n and the survival rate γ are wider than in the estimation without
the capital-output ratio, the REPS analysis shows that the prior now plays a smaller role
in determining these estimates. These results support the earlier claim that these wider
posterior credible intervals arise due to the new estimates for σ and δ.
The amount by which the robust credible intervals tighten is consistent with the worstcase distortions in Figures 5.2 and 5.3. In particular, we define ρk and ρo to be the values of ρ
for the estimation with and without the capital-output ratio, respectively, and take ρo /ρk as
a measure of how much additional information the capital-output ratio provides for a given
ϕ. This ratio is largest for the effect of the relative productivity of retirees ξ. The worst-case
posteriors for the effect of ξ involve largest distortions in the distribution of δ, suggesting that
the importance of information about δ for the estimated effects of demographics is greatest
29
Effects
of Demographics
b n − RT
100 R
T
bγ
100 R
T
bξ
100 R
T
68% C.I. (χp )
orig.
K/Y
Rob. C.I. (χr )
orig.
K/Y
Relative Change (ρ)
orig. K/Y ratio ( ρρko )
pop. growth
0.298
0.694
0.885
0.863
1.965
0.243
8.081
− RT
surv. rate
0.475
1.458
1.951
1.782
3.105
0.223
13.954
− RT
rel. prod.
0.826
0.365
3.056
0.429
2.702
0.177
15.307
Table 6.2: Widths of 68% credible intervals and robust credible intervals with r = 0.005
for estimations with and without capital-output ratio. Left panel: width of 68% credible
interval; Center panel: width of robust 68% credible interval; Right panel: difference
between widths of credible interval and robust credible interval normalized by width of 68%
credible interval.
Pop. Growth (n)
16% qtl 84% qtl
Structural Parameters
100 β −1 − 1
disc. rate
σ
IES
100δ
depr. rate
0.082
1.364
2.029
–0.039
–0.165
–0.059
Surv. Rate (γ)
16% qtl 84% qtl
0.701
0.830
0.109
–0.073
–0.182
–0.049
Rel. Prod. (ξ)
16% qtl 84% qtl
1.162
0.676
–0.203
–0.254
–0.306
0.017
Table 6.3: Difference between worst-case and original posterior mean of (β −1 , σ, δ), normalized by original posterior standard deviation. Worst-case posteriors correspond to a one-half
posterior standard deviation change in quantile.
for the effect of ξ. The change in ρ is greatest for the effect of ξ since the capital-output ratio
tightens the estimate for δ more than β or σ. On the other hand, the worst-case posteriors
for the effect of population growth n involve minimal distortions to the distribution of δ,
consistent with ρo /ρk being the smallest for this effect.
Nevertheless, the robust credible intervals remain wide relative to the rule of thumb in
Ho (2020). Figure 6.1 shows that with r = 0.025, which is half the r = 0.05 benchmark, the
robust credible intervals are between 1.1 and 1.4 posterior standard deviations wider than
the corresponding original credible intervals. The change is about 1.5 times as large as what
one would expect from the r = 0.05 benchmark (see footnote 8 for details). These results
show that even though the capital-output ratio greatly improves estimates, the augmented
data remain relatively uninformative about the effects of demographics.
Table 6.3 shows that in the estimation with the capital-output ratio, the worst-case
30
distortions are concentrated in the prior of (β, σ).11 In contrast, Figures 5.2 and 5.3 show
that without the capital-output ratio, the worst-case distortions in δ are relatively more
important, especially for the estimated effect of the relative productivity ξ. The worst-case
distortions reflect that the capital-output ratio data informs the estimate of δ, making it
more difficult to change the estimated effects by changing the prior of δ. It is therefore
important to incorporate additional data that can further discipline β and σ.
6.2
Consumption over the Life Cycle
One natural way to obtain information on β and σ is to use data on life-cycle patterns
in consumption and savings. However, direct observations of such data are unavailable.12
Instead of reestimating the model with missing data, we indirectly study how data on lifecycle consumption would tighten our estimates if it were available.
6.2.1
Methodology
Suppose we wish to understand the role of some data {yt∗ }, with steady states µ∗ (st ) in regime
st . For each Monte Carlo draw j from the posterior, we compute the implied model steady
states µ∗j (st ). Taking the Monte Carlo draws as observations, we run quadratic regressions
of the parameters and effects of demographics on (µ∗ (0) , µ∗ (1)) and use the R2 to measure
how much data on {yt∗ } would improve identification of the parameters and effects.
Intuitively, the exercise asks how well one can predict the objects of interest if one observed µ∗ (st ). Assuming we are able to obtain tight estimates of µ∗ (st ) from {yt∗ }, a large
R2 is suggestive evidence that the data would improve identification substantially. Conversely, an R2 that is close to zero implies that the data provides no information about the
values of the parameters or effects. We use a quadratic regression as an approximation for
the mapping between the objects of interest and the observables. Since we use the Monte
Carlo draws from the existing posterior, our results are conditional on the existing likelihood,
measuring the information contained in hypothetical data {yt∗ } on top of the information
contained in the data used for the existing estimation.
11
The only exception is the worst-case distortions corresponding to the 16% quantile for the effect of
population growth n. However, since Figure 6.1 shows that this quantile is especially robust to changes in
the prior, we focus our discussion on the other five worst-case priors.
12
To the best of my knowledge, the closest available data is from the Family Income and Expenditure
Survey, which reports average consumption per household, aggregated by head of household’s age from 2003.
Inferring worker and retiree consumption consistent with the model requires accounting for household composition in both the demographic-level and aggregate data. Estimating the model with missing observations
is computationally much costlier, as we are no longer able to analytically integrate out the VAR parameters
in (2.2) and (2.3). Given these challenges, we leave the exercise to future work.
31
(cw , cr /ψ)
ψcw /cr
Structural Parameters
100 β −1 − 1
disc. rate
σ
IES
100δ
dep. rate
0.215
0.439
0.354
0.118
0.046
0.070
Effects
of Demographics
n
b
100 RT − RT
pop. growth
0.353
0.042
surv. rate
0.588
0.280
rel. prod.
0.808
0.738
b γ − RT
100 R
T
b ξ − RT
100 R
T
Table 6.4: R2 from quadratic regression of parameters and effects on retiree and worker
consumption.
We consider two alternatives for µ∗ . First, we take µ∗ = (cw , cr /ψ), which is the steadystate average consumption of retirees and workers respectively, scaled by output. Next, we
take µ∗ = ψcw /cr , which is the steady-state ratio of average retiree consumption to average
worker consumption. The former requires consumption levels for each group, while the latter
requires only the change in consumption over the life cycle. We use Monte Carlo draws for
the posterior of the estimation that includes the capital-output ratio from Section 6.1.
6.2.2
Results
Table 6.4 shows that data on consumption levels (cw , cr /ψ) over the life cycle can provide
substantial information about the effects of demographics on interest rates, with R2 s of
0.35, 0.59, and 0.81 for population growth, the survival rate, and the relative productivity,
respectively. For comparison, the respective R2 s when we take µ∗ = k and use Monte Carlo
draws from the estimation without the capital-output ratio are 0.43, 0.49, and 0.85.
On the other hand, data on the relative change in consumption levels ψcw /cr would be
less informative about the effects of changes in population growth and the survival rate
on interest rates, with R2 s of 0.04 and 0.28 respectively. The low R2 arises from reduced
information about the intertemporal elasticity of substitution σ and depreciation rate δ,
which are reflected in reductions in R2 s from 0.44 and 0.35, respectively, to 0.05 and 0.07.
Table 6.3 shows that σ and δ are particularly important relative to β for pinning down the
effect of changes in population growth. Nevertheless, ψcw /cr is informative about the effects
of changes in relative productivity with R2 s of 0.74.
These results highlight the potential for life-cycle consumption data to identify the effects
32
of demographics on interest rates by informing estimates of the discount rate, intertemporal elasticity of substitution, and depreciation rate. Consumption levels are particularly
informative relative to only knowing changes in consumption over the life cycle.
7
Conclusion
The aging populations and falling interest rates in developed countries have led to a large
literature seeking to quantify the effects of demographics on interest rates through the lens
of OLG models. The analysis here shows that these results may be fragile. In particular,
we have shown that the estimated effects of demographics depend crucially on the discount
rate, intertemporal elasticity of substitution, and capital depreciation rate. Without the appropriate data, these parameters are not well-identified. As a result, a large set of parameter
values can be justified, leading to a wide range of possible estimated effects and explaining
the differences across estimates in the literature. Including aggregate capital and life-cycle
consumption data to discipline these parameters can help produce more accurate and precise
estimates. These insights extend to more complicated models and other related empirical
questions.
In terms of methodology, this paper makes two contributions. Firstly, it introduces an
econometric framework that disentangles secular changes from high-frequency fluctuations in
a way that uses a structural economic model to discipline only the secular changes. Secondly,
the REPS analysis illustrates how prior sensitivity analysis reveals parts of a model that
are not well-identified and matter for our objects of interest. Importantly, by avoiding
parametric restrictions on the prior, our analysis accounts for potentially large joint changes
in the parameters that could alter results without substantially worsening the fit with the
data. The analysis informs us about the data needed to sharpen the estimates and suggests
the appropriate moments for calibration.
33
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37
Appendix
A
Steady State of Overlapping Generations Model
w
Denote `t = Lt /Nt and `w
t ≡ Lt /Lt . In addition, we use lowercase letters to denote variables
normalized by y and drop the time subscripts for steady states. We need to solve for:
{ψ, cr , cw , hr , hw , `, `w , , π, Ω, λ, k, R} .
A.1
Production
Firm Capital Decision. The firm capital decision is:
Rt = (1 − α)
Yt
+ (1 − δ)
Kt
= (1 − α) kt−1 + (1 − δ) ,
which yields the steady state condition:
k=
1−α
.
R−1+δ
(A.1)
Resource Constraint. We have the aggregate resource constraint:
Yt = Kt+1 − (1 − δ) Kt + Ctw + Ctr .
Normalizing by Yt , we have, in steady state:
1 = (x + n + δ) k + cw + cr .
A.2
(A.2)
Households
Population. The population of retirees is:
ψt+1 Nt+1 = ψt+1 (1 + n) Nt = γψt Nt + (1 − ω) Nt ,
which yields the steady-state condition:
ψ=
1−ω
.
1+n−γ
38
(A.3)
Wealth Share of Retirees. We have:
w
λt+1 At+1 = λt Rt At + Wt ξLrt − Ctr + (1 − ω) [(1 − λt ) Rt At + Wt Lw
t − Ct ]
ξLrt
Lw
t
r
w
= (1 − ω + ωλt ) Rt At + αYt
− Ct + (1 − ω) αYt
− Ct .
Lt
Lt
Normalizing by Yt , we have, in steady state:
(1 + x + n) λk = (1 − ω + ωλ) Rk + α − αω`w − cr − (1 − ω) cw .
(A.4)
Consumption of Retirees. We have:
Ctr = t πt (λt Rt At + Htr ) .
Normalizing by Yt , we have, in steady state:
cr = π (λRk + hr ) .
(A.5)
Human Wealth of Retirees. If retirees choose to supply labor, we have the law of
motion:
r
Ht+1
ψt Nt
ψt+1 Nt+1 Rt+1 /γ
r
r
γ
ξL
ψt Ht+1
.
= αYt t +
Lt
1 + n ψt+1 Rt+1
Htr = ξLrt Wt +
Normalizing by Yt , we have, in steady state:
r
h
(1 + x) γ
1−
R
!
= α (1 − lw ) .
Consumption of Workers. We have:
Ctw = πt [(1 − λt ) Rt At + Htw ] .
Normalizing by Yt , we have, in steady state:
cw = π [(1 − λ) Rk + hw ] .
39
(A.6)
Human Wealth of Workers. We have the law of motion:
Htw = Lw
t Wt + ω
w
r
1
Ht+1
Ht+1
Nt
Nt
1−σ
+ (1 − ω) ξ ν−1 t+1
Nt+1 Rt+1 Ωt+1
ψt+1 Nt+1 Rt+1 Ωt+1
1
1−σ
w
r
Ht+1
Ht+1
Lw
ω
1 − ω ξ ν−1 t+1
= αYt t +
+
.
Lt
1 + n Rt+1 Ωt+1 1 + n ψt+1 Rt+1 Ωt+1
Normalizing by Yt , we have, in steady state:
w
h
(1 + x) ω
1−
RΩ
Labor Supply. Define φ ≡
is (abusing notation):
1−ν
.
ν
!
1
(1 + x) (1 − ω) ξ ν−1 1−σ hr
= αl +
.
ψRΩ
w
(A.7)
The first-order condition for an individual worker’s labor
Lw
t = 1−φ
Ctw
.
Wt
Aggregating, we have:
Lw
t = Nt −
φ Lt w
C .
α Yt t
Dividing by Lt , we have, in steady state:
`−1 = `w +
φ w
c .
α
(A.8)
The first-order condition for an individual retiree’s labor is (abusing notation):
Lrt = 1 − φ
Ctr
.
ξWt
Aggregating, we have:
Lrt = ψt Nt −
φ Lt r
C .
ξα Yt t
Dividing by Lt /ξ, we have:
ξψt
Nt
φ r
= 1 − `w
c.
t +
Lt
α t
Therefore, in steady state:
(1 + ξψ) `w = 1 +
40
φ r
(c − ξψcw ) .
α
(A.9)
Retiree Propensity to Consume. The retiree propensity to consume follows:
Wt
t πt = 1 −
Wt+1
σ−1
!1−ν
t πt
.
t+1 πt+1
βσγ
Rt+1
The firm labor decision Wt Lt = αYt implies
Wt
Yt Lt+1
1
`t+1
=
=
.
Wt+1
Yt+1 Lt
1 + x + n `t
Hence, in steady state,
R
π = 1 −
(1 + x)1−ν
!σ−1
β σ γ.
(A.10)
Worker Propensity to Consume. The worker propensity to consume follows:
Wt
πt = 1 −
Wt+1
σ−1
!1−ν
βσ
Rt+1 Ωt+1
πt
,
πt+1
where
1
1−σ
.
Ωt+1 = ω + (1 − ω) ξ ν−1 t+1
This yields steady-state conditions:
RΩ
π =1−
(1 + x)1−ν
!σ−1
βσ
1
Ω = ω + (1 − ω) ξ ν−1 1−σ .
B
B.1
(A.11)
(A.12)
Bayesian Estimation
Data
Table B.1 lists the data series that we use as observables, as well as their mapping to
the variables or parameters in the OLG model. For labor quantity variables, we focus on
employment at the extensive margin, so that labor supply in the model is taken to be the
fraction of workers and retirees respectively who are employed. For wages, we use hourly
earnings for manufacturing, which is close to the series for monthly private sector earnings.
41
Data Series
GDP growth
age 15-64 pop. growth
age 15-64 pop. / total 15+ pop.
age 15-64 emp. / total 15+ emp.
employment-population ratio
real wage growth
real interest rate
labor share
capital-output ratio
Model Counterpart
x+n
n
1/ (1 + ψ)
ξ`w / [1 − (1 − ξ) `w ]
[1 − (1 − ξ) `w ] /ξ
x
R−1
α
k
Source
FRED
OECD
OECD
OECD
OECD
FRED
Bank of Japan, IMF DSBB
Penn World Table 9.0
Penn World Table 9.0
Table B.1: Observables, model counterparts, and data sources
B.2
Prior
Structural Parameters. The structural parameters and their priors are summarized in
Table B.2. For each regime, we estimate the time-varying parameters (x, n, γ, ξ, ν, α). These
are drawn iid for each regime from a distribution whose mean and variance are estimated.
The priors for these means and variances are also reported in Table B.2. Figure B.1 compares
the prior for (β, σ, δ) to existing calibrations of the Gertler (1999) model.
VAR Parameters. For the VAR parameters {Φ (st ) , Σ (st )}, we use a normal-inverseWishart prior that shrinks toward white noise, to ensure that vt captures primarily highfrequency variation. The normal-inverse-Wishart prior makes it straightforward to integrate
out the VAR parameters. The estimation results are not significantly affected by the particular Normal-inverse Wishart prior used. The coefficients are drawn iid across regimes.
Conditional on µ (st ), we have a prior that vt∗ ∼ N (µ (0) − µ (1) , Ω) and vT ∼ N (0, Ω),
where Ω is a diagonal matrix of variances that we calibrate to be equal to the variances of
each series in {yt }. This prior on the initial and terminal condition implies that we expect
the measurement error to be large immediately after the regime change but to shrink toward
the end of the sample. Intuitively, yt starts out of steady state in t∗ , but converges toward
its steady state. The vector autoregression allows this convergence to be modeled flexibly.
The results do not change substantially if we ignore the prior for vt∗ and vT .
Structural Break. The prior for t∗ is flat for t∗ ∈ [1985, 2009] and zero otherwise. The
restriction ensures that t∗ does not lie in the initial or final periods of the sample. In the
estimation, t∗ is tightly estimated to lie around 1991.
42
Type
Mean
Std Dev
calibrated
Gamma
Normal
Beta
0.98
3.50
0.35
0.08
–
0.50
0.15
0.02
Normal
Normal
Gamma
Beta
Beta
Normal
µx
µn
µγ
µξ
µν
µα
σx
σn
σγ
σξ
σν
σα
Normal
Normal
Gamma
Beta
Beta
Normal
1.50
1.00
8.00
0.50
0.50
0.65
0.50
0.50
0.50
0.10
0.10
0.05
1.00
1.00
2.00
0.10
0.10
0.10
1.00
1.00
2.00
0.10
0.10
0.10
Structural Parameters
Fixed
ω
100 β −1 − 1
σ
δ
one minus retirement rate
discount rate
intertemporal elasticity of substitution
depreciation rate
Time-varying
100x
productivity growth
100n
population growth
−1
100 γ − 1
probability of death
ξ
relative productivity of retirees
ν
one minus disutility of labor
α
labor share
Hyperparameters
Means
µx
µn
µγ
µξ
µν
µα
productivity growth
population growth
probability of death
relative productivity of retirees
one minus disutility of labor
labor share
Standard Deviations
σx
productivity growth
σn
population growth
σγ
probability of death
σξ
relative productivity of retirees
σν
one minus disutility of labor
σα
labor share
Inv.
Inv.
Inv.
Inv.
Inv.
Inv.
Gamma
Gamma
Gamma
Gamma
Gamma
Gamma
Table B.2: Prior for structural parameters.
43
0.7
0.6
0.5
Ferrero (2010)
Carvalho et al. (2016)
0.4
Kara, Thadden (2016)
0.3
Fujiwara, Teranishi (2008)
Gertler (1999)
0.2
0.1
2
2.5
3
3.5
100(
4
-1
4.5
5
- 1)
0.12
0.11
0.1
Ferrero (2010)
Carvalho et al. (2016)
Gertler (1999)
Fujiwara, Teranishi (2008)
0.09
0.08
0.07
0.06
0.05
Kara, Thadden (2016)
2
2.5
3
3.5
100(
4
-1
4.5
5
- 1)
Figure B.1: Original prior and existing calibrations of (β, σ, δ) for the Gertler (1999) model.
Black crosses indicate calibrations; lines are level curves for the joint distribution under the
prior. Top: β and σ; Bottom: β and δ.
44
B.3
Markov Chain Monte Carlo
The MCMC algorithm to estimate the model (2.1)-(2.3) has two main blocks. First we draw
t∗ given the structural parameters. Next, we draw the structural parameters given t∗ .
Each step involves one or more Metropolis-Hastings draws. Given the structural parameters, we compute y by solving numerically solving for the steady state of the OLG model
given those parameters. Given y, we obtain vt . Since vt follows a VAR(1) with a normalinverse-Wishart prior, we can analytically integrate out the VAR parameters when evaluating
the posterior. This reduces the size of the parameter space and improves convergence.
For the burn-in draws, we break the second step into several blocks. We make draws
for the fixed and time-varying parameters in separate blocks, and break the time-varying
parameters into two blocks corresponding to their values before and after the structural
break. The blockwise draws improve convergence when our proposal density is not yet
optimized. For the main draws, we draw all the structural parameters in a single block and
scale the covariance of the burn-in draws for the proposal density.
C
Prior Sensitivity
We use the SMC algorithm from Ho (2020), which is an extension of the SMC algorithm
for Bayesian estimation in Herbst and Schorfheide (2014). We use 105 particles, 500 SMC
steps and 5 Metropolis-Hastings steps, and repeat this 20 times in parallel. This takes
approximately 15 hours to complete.13 As a reference, the main Bayesian estimation takes
approximately five hours to complete.
To produce Figures 5.1 and 6.1, we extrapolate the output from the SMC algorithm.
First, we take the median relative entropy of the 20 runs of the SMC algorithm for each
SMC step. The ith SMC step corresponds to the same worst-case quantile across runs.
Next, we run a polynomial regression of the worst-case quantile on the median relative
entropy. Finally, we use the regression to predict the worst-case quantiles for the r = 0.005
or r = 0.025.
D
An Alternative Parametric Prior
To demonstrate that the sensitivity of the estimated effects to the prior does not depend
on the nonparametric nature of REPS, we now show that the posterior results also change
substantially when we estimate the model using a different parametric prior. The new prior
13
We obtain similar results with half the SMC steps, which requires less than half the time.
45
incorporates the worst-case distortions for the 16% quantile for the effect of the relative
productivity of retirees ξ and the 84% quantile for the effect of the survival rate γ.
The alternative prior is reasonable in two dimensions. Firstly, it remains in a parametric
family, retaining potentially desirable smoothness properties that can be violated by the
worst-case prior from REPS. Secondly, the distortions put greater weight on parameter
values that are consistent with calibrations of OLG models in the literature.
D.1
Prior
We take the new prior of (β, σ) to be:
100 (β −1 − 1)
2.00 2.002 −0.315
∼ N
,
.
σ
0.50
−0.315 0.452
The new prior for δ is an independent Beta distribution that has mean 0.06 instead of 0.08,
and has the same standard deviation as original prior of 0.02.
The new prior places greater weight on regions that the REPS analysis suggests are
particularly important for the estimated effects of the survival rate γ and the relative productivity of retirees ξ. In particular, it places more weight on regions of the parameter space
with small values of β −1 and δ, as well as large values of σ. These regions coincide with the
worst-case distortions for the 16% quantile for the effect of ξ, and the 84% quantile for the
effect of γ. The REPS analysis also showed that increasing the prior mass on large values of
σ reduces the estimated effect of population growth n.
While the worst-case distortions indicate features of the prior that are important for
our estimates, some of these distortions appear implausible. For example, a discount rate
β < 0.95 or depreciation rate δ > 0.20 are unlikely. To discipline the prior, we ensure that
it is consistent with the range of values used in the OLG literature, shown in Figure D.1.
We decrease the prior mean and increase the prior variance of β −1 . In addition, the
Gaussian prior on β −1 no longer imposes the β < 1 restriction of the original Gamma
distribution prior. The new support is reasonable because the restriction that β < 1 in
representative agent models does not apply in OLG models. Instead, the stochastic death
introduces discounting on top of the time preference β. Imrohoroglu et al. (1995) argue
that a value of β > 1 matches the empirical evidence and allows their model to fit the US
wealth-income ratio. Nevertheless, we keep a mean discount rate of 2% to reflect the positive
discount rates used by a majority of the literature.
We increase the prior mean and variance of σ to reflect the wide range of values in the
literature beyond versions of the Gertler (1999) model. The increased mean is consistent
46
1
Gertler (1999)
other OLG
Muto et al. (2016)
Sudo, Takizuka (2018)
0.8
Miles (1999)
Eggertsson et al. (2019)
Imrohoroglu et al. (1995)
0.6
Ikeda, Saito (2014)
Kitao (2017)
Kitao (2018)
Brooks (2003)
Attanasio et al. (2006)
Attanasio et al. (2007) Ferrero (2010)
Fujita, Fujiwara (2016)
Wong (2018)
Brooks (2002)
Carvalho et al. (2016)
0.4
Kara, Thadden (2016)
Fujiwara, Teranishi (2008)
Gertler (1999)
0.2
-3
-2
-1
0
1
100(
2
-1
3
4
5
- 1)
0.12
Eggertsson et al. (2019)
0.11
0.1
0.09
0.08
Ferrero (2010)
Carvalho et al. (2016)
Fujiwara, Teranishi (2008)
Gertler (1999)
Kitao (2017)
Kitao (2018)
Ikeda, Saito (2014)
Imrohoroglu et al. (1995)
0.07
0.06
0.05
-3
Attanasio et al. (2006)
-2
-1
Attanasio et al. (2007)
0
Brooks (2003)
Brooks (2002)
Kara, Thadden (2016)
1
100(
2
-1
3
4
5
- 1)
Figure D.1: Existing calibrations of (β, σ, δ). Black crosses indicate calibrations of models
based on Gertler (1999); red circles indicate calibrations of other OLG models; lines are level
curves for the joint distribution under the alternative prior. Top: β and σ; Bottom: β and
δ.
47
with Figure D.1, which shows sixteen papers with σ ≥ 0.5. Furthermore, there is a wide
range of empirical estimates of σ between 0 and 2.
Under the new prior, β and σ have a correlation of −0.35, which matches the correlation
for the sample of calibrations in Figure D.1. Many papers pick σ independently and calibrate
β to match a steady-state statistic, such as the interest rate in a given year or the capitaloutput ratio. The correlation reflects the fact that β and σ jointly determine these moments.
For example, in a representative agent neoclassical growth model with constant relative risk
aversion, the household’s Euler equation implies a negative relationship between β and σ to
jointly match consumption growth and interest rates. In the model here, the dependence is
summarized by the expressions (3.14) and (3.15) for the marginal propensities to consume.
Since the REPS analysis has shown that the original assumption of independence between
β and σ is not innocuous, we introduce this correlation to our prior.
To illustrate the importance of δ for the estimated effects of the survival rate γ and relative
productivity ξ, we reduce the prior mean for δ. The prior mean of 0.06 is closer to the World
Penn Tables estimate of 0.05 for the depreciation rate in Japan. Figure D.1 shows that this
is a plausible prior mean, with values of δ ranging between 0.05 and 0.12 in the literature.
We keep δ independent from β and σ for two reasons. Firstly, the correlation between δ and
(β, σ) among the papers considered in Figure D.1 is smaller than the correlation between β
and σ. Secondly, β and σ are both preference parameters that directly influence households’
savings decisions, but δ governs the production side of the economy.
While this prior has been chosen to emphasize the results from the REPS analysis, it
is one that would be plausible without observing the data. The prior for (β, σ) was chosen
based on existing calibrations with no reference to the data. The prior for δ is also plausible
given existing empirical work that does not use the data we consider. We do not take a stand
on which prior should be preferred. Instead, we emphasize that a complete analysis of the
data requires understanding the range of results that can arise from different possible priors
and take the alternative prior as an illustrative example. In particular, although there are
numerous other plausible priors, we pick this particular one because the REPS results show
that these specific changes in the prior can lead to large changes in the posterior estimates.
D.2
Results
Table D.1 shows that changing the prior substantially changes the estimates for the effects
of demographics on the interest rate. In all three cases, the mean changes by more than
one posterior standard deviation, and the new mean is not contained in the original 68%
credible interval. The change in the posterior mean of up to 0.6 percentage points is also
48
mean
Original
sd
68% C.I.
With Alternative Prior
mean
sd
68% C.I.
Structural Parameters
100 β −1 − 1
σ
100δ
disc. rate
IES
dep. rate
Effects
of Demographics
b n − RT
100 R
T
bγ
100 R
T
bξ
100 R
T
3.245
0.574
7.861
0.442
0.103
1.485
(2.804, 3.690)
(0.471, 0.677)
(6.393, 9.317)
0.022
0.790
3.328
1.353
0.149
0.890
(–1.365, 1.425)
(0.624, 0.959)
(2.433, 4.282)
pop. growth
0.875
0.159
(0.725, 1.023)
0.696
0.136
(0.571, 0.829)
− RT
surv. rate
0.671
0.266
(0.429, 0.905)
1.110
0.225
(0.891, 1.332)
− RT
rel. prod.
1.206
0.412
(0.793, 1.618)
0.635
0.465
(0.184, 1.118)
Table D.1: Posterior statistics for effects of demographics with original and alternative parametric priors.
economically significant.
The changes in the estimated effects are driven by large changes in the posterior estimates
for (β, σ, δ). The mean for the discount rate falls from 3% to 0%; the mean for the intertemporal elasticity of substitution rises from 0.57 to 0.79; and the mean depreciation rate falls
from 8% to 3%.14 Relative to the posterior standard deviations, these are statistically large
changes. Moreover, the resulting changes in the estimated effects of demographics indicate
that these are also economically significant. In contrast to the posterior for (β, σ, δ), the
estimates for the long-run mean µ and the time-varying parameters are relatively unaffected
by the change in the prior.
The directions in which the estimates change are consistent with the worst-case posteriors
in Figures 5.2 and 5.3. The estimate for the effect of population growth decreases because it
depends negatively on σ. The new prior is chosen in line with the distortions that increase
the 84% quantile for the effect of the survival rate and thus increase the estimate of that
effect. The estimate for the effect of the relative productivity of retirees decreases as the
alternative prior places greater weight on similar regions to the worst-case distortions for the
16% quantile for that effect.
We have attained these changes in the posterior using an economically plausible alternative prior that lies in a commonly used distributional family. The prior sensitivity identified
by REPS therefore remains an issue even when we restrict ourselves to parametric alternative
14
The fact that the new posterior estimate for the depreciation rate is smaller than typical values in the
literature should not be viewed as a criticism of a prior that was deemed plausible before observing the data.
Rather, the result is further evidence that the current data are insufficient to discipline the estimate of δ.
49
priors. In addition, many of the calibrations considered in Figure D.1 fall around the high
prior density region for (β, σ, δ), suggesting that the results in the literature are also sensitive
to the exact calibration strategy. For example, while it is common to calibrate β to match
the interest rate in a given year (e.g., Ikeda and Saito (2014); Carvalho et al. (2016)), we
find a large dispersion in the likelihood for β here that translates into imprecise estimates for
the effect of demographics. Once we account for uncertainty in the natural interest rate as
well as the joint dependence of all the observables on the full parameter vector, we find that
the likelihood supports a wide range of parameter combinations that are consistent with the
data. Our analysis has shown that the estimated effects of demographics vary substantially
with parameter values in this range.
50
@FedFRASER