Clearing Up the Fiscal Multiplier Morass

Abstract

We quantify government spending multipliers in US data using Bayesian prior and posterior analysis of a monetary model with fiscal details and two distinct monetary-fiscal policy regimes. The combination of model specification, observable data, and relatively diffuse priors for some parameters lands posterior estimates in regions of the parameter space that yield fresh perspectives on the transmission mechanisms that underlie government spending multipliers. Short-run output multipliers are comparable across regimes-posterior means around 1.3 on impact-but much larger after 10 years under passive money/active fiscal than under active money/passive fiscal-90 percent credible sets of [1.5, 1.9] versus [0.1, 0.4 ] in present value, when estimated from 1955 to 2016.

Full text

Clearing Up the Fiscal Multiplier Morass∗
Eric M. Leeper†Nora Traum‡Todd B. Walker§
March 19, 2017
Abstract
We quantify government spending multipliers in U.S. data using Bayesian prior and pos-
terior analysis of a monetary model with fiscal details and two distinct monetary-fiscal policy
regimes. The combination of model specification, observable data, and relatively diffuse pri-
ors for some parameters lands posterior estimates in regions of the parameter space that yield
fresh perspectives on the transmission mechanisms that underlie government spending multipli-
ers. Short-run output multipliers are comparable across regimes—posterior means around 1.3
on impact—but much larger after 10 years under passive money/active fiscal than under active
money/passive fiscal—90-percent credible sets of [1.5,1.9] versus [0.1,0.4] in present value, when
estimated from 1955 to 2007.
Keywords: government spending; monetary-fiscal interactions; prior predictive analysis; Bayesian
estimation
JEL Codes: C11, E62, E63
∗We would like to thank seminar participants at the Bank of Canada, the 2011 Bundesbank Spring Conference, the
Federal Reserve Bank of Dallas, the 2011 Konstanz Seminar on Monetary Theory and Policy, the 2011 SED annual
meeting, the 2016 National Tax Association Conference, the Federal Reserve Bank of Kansas City, the Federal
Reserve Bank of St. Louis and Henning Bohn, Hess Chung, G¨unter Coenen, Marco Del Negro, Berthold Herrendorf,
Campbell Leith, Giorgio Primiceri, Morten Ravn, Harald Uhlig, Tao Zha, Marty Eichenbaum, and anonymous referees
for helpful comments.
†Indiana University and NBER; eleeper@indiana.edu.
‡North Carolina State University; nora traum@ncsu.edu
§Indiana University; walkertb@indiana.edu.

The global recession of 2008 and the resulting fiscal stimulus packages in many countries
reignited academic interest in government spending multipliers to spawn a new and growing theo-
retical and empirical literature. Despite intense professional attention, no consensus has emerged
on the dynamic impacts of government spending on macroeconomic aggregates. Because the fiscal
multiplier depends on nearly every detail of private and policy behavior, different model specifica-
tions or identifying assumptions can produce wildly different quantitative predictions of multipliers.
Sharply different conclusions from similar models and data constitute a morass.1
This paper uses Bayesian prior and posterior analyses to trace differences in estimates of multi-
pliers to different model specifications. We augment a monetary DSGE model from the class that
Christiano et al. (2005) and Smets and Wouters (2007) develop with a rich set of fiscal details:
government spending that may be valued as a public good, explicit rules for fiscal instruments,
a maturity structure of government debt, and distorting steady-state taxes. We also go beyond
existing empirical analyses of multipliers to consider alternative monetary-fiscal regimes: either
active monetary policy coupled with passive fiscal policy (regime M) or active fiscal policy together
with passive monetary policy (regime F).2
Prior predictive analysis reports the probability distribution of multiplier values that a particular
specification can produce before confronting data. That analysis speaks to the models the literature
employs: for example, it is impossible for standard real business cycle models to produce large
multipliers, while new Keynesian models with a substantial fraction of rule-of-thumb agents are
unlikely to generate small multipliers, regardless of the information data contain about multipliers.
Implications drawn from prior predictive analysis guide our choice of model to take to the data.
We seek a specification that a priori is consistent with either small or large multipliers, depending
on estimated parameter values. The prior analysis suggests that a model that permits government
spending to complement or substitute for private consumption and conditions on either regime M
or regime F supports the widest ranges for multipliers.
We maintain the agnostic spirit of the prior predictive analysis when we estimate by using
relatively diffuse prior distributions over some model parameters and by considering distinct priors
that place the economy in one of the two monetary-fiscal regimes. The fiscal details in our model and
the data set, both of which rarely appear in estimated DSGE models, permit the posterior to land
in regions of the parameter space that produce fresh perspectives on the transmission mechanisms
that underlie government spending multipliers. Starting with diffuse a priori views about the sizes
and the dynamics of multipliers permits any messages in data to come through clearly. U.S. data
are highly informative: they narrow the posterior range of multipliers substantially. Combining
a general model specification and diffuse a priori views with data delivers posterior estimates of
multipliers that help clear up the morass.
1Gechert and Will (2012, p. 28) examine 89 multiplier studies spanning many methodologies to conclude that
“reported multipliers very much depend on the setting and method chosen.”
2An active authority is not constrained by current budgetary conditions and freely chooses the decision rule it
wants. A passive authority is constrained by the consumers’ and firms’ optimizations and by the actions of the active
authority, so the passive authority must stabilize debt. See Leeper (1991), Sims (1994), and Woodford (1995).

Leeper, Traum & Walker: Fiscal Multiplier Morass
Over our baseline sample period, 1955q1–2007q4, and across both policy regimes, the posterior
estimates entail high degrees of nominal rigidities, strong habit formation and complementarity
between government and private consumption in both policy regimes. These estimates produce
comparable short-run output multipliers across regimes—mean impact multipliers are about 1.3—
but substantially larger multipliers in regime F than in regime M at long horizons—after 10 years the
90-percent credible set for present-value multipliers is [1.5,1.9] in F, but [0.1,0.4] in M. Consumption
effects are positive in both regimes, with mean short-term multipliers that hover around 0.1 to 0.2
in present value. Investment multipliers are decidedly negative in regime M but more likely to be
positive in regime F: 90-percent credible sets at 10 years are [−1.6,−1.0] in M and [−0.4,0.2] in F.
All these estimated multipliers are marginally larger when the sample extends through 2014q2 to
include the years when the federal funds rate was near its effective lower bound.
Although private parameter estimates are quite similar across policy regimes, the two monetary-
fiscal mixes imply different fiscal financing schemes that transmit government spending through the
economy in different ways. Posterior estimates for the full sample yield somewhat unusual passive
fiscal behavior in regime M: higher government debt modestly raises future lump-sum transfers,
so the full brunt of debt stabilization is borne by government spending reversals of the kind that
Corsetti et al. (2012) emphasize. In regime F, stabilization occurs from revaluations of debt through
surprise changes in inflation and bond prices. Steady-state distorting tax rates ensure that revenues
endogenously respond to economic conditions in both regimes, even though the constant tax rates
cannot stabilize debt. Endogenous revenues attenuate the necessary revaluation effects.
Data do not exhibit strong preference for one monetary-fiscal regime over the other. Nearly
equivalent fits of the two different policy mixes, with their associated very different implications for
transmission mechanisms and policy effects, lead to the paper’s broader message that estimated pol-
icy models should routinely consider monetary-fiscal policy specifications beyond the conventional
mix that regime M embodies.
At the risk of some oversimplification, we can succinctly describe the transmission mechanisms.
Three aspects of behavior lie behind government spending impacts in regime M: strong rigidities—
price and wage stickiness and habit formation—complementarity of government spending to private
consumption, and fiscal financing through spending reversals. Complementarity ensures that higher
spending initially raises consumption even though long-run real interest rates also rise. Anticipated
cuts in future government spending, coupled with higher transfers, raise household wealth and
temper long-run real rate increases to support consumers’ strong desire to smooth consumption at
a level above steady state for many years after the initial spending impulse. Because the output
boost is short-lived, higher consumption in the long run comes out of reduced investment.
This estimated transmission mechanism differs from convention—as in, for example, Gal´ı et al.
(2007), Woodford (2011) or Corsetti et al. (2012)—along several dimensions. First, most studies
do not permit government spending to interact directly with consumption through preferences.
Second, high estimated nominal rigidities dampen inflationary and real-interest rate effects. Third,
estimated fiscal financing produces positive, rather than the usual negative, wealth effects. These
2

Leeper, Traum & Walker: Fiscal Multiplier Morass
differences account for the persistently positive consumption multipliers.
Based on previous work on government spending multipliers when monetary policy is passive, it
may be surprising that our reported multipliers are not many times larger in regime F than in M.3
Although very large fiscal effects are possible when our model resides in regime F, the moderate
impacts that the posterior estimates produce stem primarily from three factors: high nominal
rigidities, the existence of a maturity structure for nominal government debt, and the presence of
steady-state taxes on labor and capital income.
Higher government spending financed by nominal bond sales raises household wealth when
fiscal policy is active and future surpluses are not expected to adjust to stabilize debt. Rigid prices
convert higher nominal debt into sustained increases in real debt and household wealth. Higher
wealth boosts consumption demand, which price stickiness translates into higher labor demand,
rather than higher goods prices. Because the real value of debt cannot fall significantly through
a higher price level, it declines instead through lower bond prices and revaluation occurs through
higher future inflation. With inflation rising only modestly, long-run real interest rates rise even
under passive monetary policy, just as they do when monetary policy is active.
Long-run output multipliers are substantially larger in regime F because real wages and employ-
ment increase strongly and persistently to increase human wealth and sustain consumption demand.
Consumption multipliers remain positive many years after the government spending increase has
dissipated without crowding out investment, as occurs in regime M. Multipliers are not implausi-
bly large in regime F, as previous research may suggest, because steady-state taxes levied against
factor incomes raise aggregate tax revenues along with the expansion in real economic activity to
temper the wealth effects that active fiscal policy engenders. Steady-state tax rates capture the
reality that even if a government does not systematically adjust tax schedules when government
debt rises, revenues nonetheless rise with incomes because existing tax rates remain in place.
As in regime M, the posterior estimates in regime F deliver a very different transmission mech-
anism for government spending than appears elsewhere in the literature. Sizeable multipliers for
output and consumption arise despite higher long-run real interest rates. Dupor and Li (2015)
argue that passive monetary policy gives government spending expansions unreasonably large in-
flationary consequences that are inconsistent with empirical evidence. This does not occur in our
estimates because the model includes fiscal details that most analyses neglect.
1 The Models
The models we use for prior predictive analysis share several details with the class of models used
to evaluate the size of fiscal multipliers: (1) forward-looking, optimizing agents; (2) households who
receive utility from consumption and leisure and additionally may value government consumption;
(3) a distinction between households who can save (“savers”) and who are constrained to consume
their income each period (“non-savers”); (4) production sectors that use capital and labor inputs;
3Kim (2003), Christiano et al. (2011), Davig and Leeper (2011), and Dupor and Li (2015) find that in regime F
or at the lower bound for nominal interest rates, output multipliers can exceed 2, with falling real interest rates.
3

Leeper, Traum & Walker: Fiscal Multiplier Morass
(5) monopolistic competition in the goods and labor sectors; (6) empirically relevant nominal and
real frictions; (7) fiscal and monetary authorities who set their instruments using feedback rules;
and (8) the economy at its cashless limit.
Our model structure nests frameworks that researchers use to quantify fiscal multipliers, but
expands on those frameworks by filling in details of the fiscal side of the model. Those details
include allowing for public goods that may be valued in utility, explicit rules for several fiscal
instruments, a maturity structure for nominal government debt, and steady-state distorting taxes.
1.1 Firms and Price Setting
The production sector consists of firms that produce intermediate and final goods. A perfectly
competitive final goods producer uses a continuum of intermediate goods Yt(i), i∈[0,1], to produce
the final good, Yt, with the constant-return-to-scale technology (R1
0Yt(i)
1
1+ηp
tdi)1+ηp
t≥Yt,where ηp
t
denotes an exogenous, time-varying markup to intermediate goods’ prices.
The price of intermediate good iis ¯
Pt(i) and the price of final goods Ytis ¯
Pt. The final goods
producing firm chooses Ytand Yt(i) to maximize profits subject to the constant-return-to-scale
technology. Dixit-Stiglitz aggregation yields the demand Yt(i) = Yt¯
Pt(i)/¯
Pt−(1+ηp
t)/ηp
t.
Intermediate goods producers are monopolistic competitors in their product market. Firm ihas
access to the technology Yt(i) = Kt(i)α(AtLt(i))1−α−AtΩ, where α∈[0,1] and Ω >0 represents
fixed costs to production that grow at the rate of technological progress. Atis a permanent shock
to technology. The logarithm of its growth rate, ua
t= ln At−ln At−1, follows the stationary AR(1)
process ua
t= (1 −ρa)γ+ρaua
t−1+ǫa
t, ǫa
t∼N(0, σ2
a), where γdefines the logarithm of the steady-
state gross growth rate of technology. Firms face perfectly competitive factor markets for capital
and labor. Cost minimization implies that the firms have identical nominal marginal costs per unit
of output, M Ct= (1 −α)α−1α−α(Rk
t)αW1−α
tA−1+α
t.
Prices evolve by a Calvo (1983) mechanism. An intermediate firm faces probability (1−ωp) each
period that it may reoptimize its price. Firms that cannot reoptimize partially index their prices
to past inflation according to the rule Pt(i) = (πt−1)χp(π)1−χpPt−1(i), where πt−1≡Pt−1/Pt−2is
the inflation rate, πis the steady state inflation rate, and χp∈[0,1].
Firms that reoptimize their price in period tmaximize expected discounted nominal profits
subject to the demand for Yt(i). Given the production function, average and marginal costs coincide,
which allows expected discounted nominal profits to be written as
Et
∞
X
s=0
(βωp)sλt+s
λt" s
Y
k=1
πχp
t+k−1π1−χp!Pt(i)Yt+s(i)−MCt+sYt+s(i)#(1)
where λis the marginal utility of wealth of saver households, defined below.
1.1.1 Labor Agency Each household supplies a continuum of differentiated labor services
indexed by l. These differentiated labor services are supplied by both savers and non-savers,
and demand is uniformly allocated among households. A competitive labor agency combines the
4

Leeper, Traum & Walker: Fiscal Multiplier Morass
differentiated labor services into a homogeneous labor input that is sold to intermediate firms,
according to the technology Lt= (R1
0Lt(l)
1
1+ηw
tdl)1+ηw
t, where ηw
tdenotes a time-varying exogenous
markup to wages. The competitive labor agency’s demand function comes from solving its profit
maximization problem, which yields Lt(l) = Ld
t(Wt(l)/Wt)−(1+ηw
t)/ηw
t, where Ld
tis the demand for
composite labor services, which is given by intermediate firms, and Wtis the aggregate nominal
wage that satisfies Wt= (R1
0Wt(l)
1
ηw
tdl)ηw
t.
1.2 Households
The economy is populated by a continuum of households on the interval [0,1], of which a fraction
µare non-savers and a fraction 1 −µare savers. Superscript Sindicates a variable associated with
savers and Nwith non-savers.
1.2.1 Savers An optimizing saver household jderives utility from composite consumption,
C∗S(j), consisting of private, CS
t(j), and public, Gt, consumption goods, C∗S(j)≡CS
t(j) + αGGt.
Parameter αGgoverns the degree of substitutability of the consumption goods: when αG<0, pri-
vate and public consumption are complements; when αG>0, the goods are substitutes. The house-
hold values consumption relative to a habit stock defined in terms of lagged aggregate consumption
of savers (θ˜
C∗S
t−1where θ∈[0,1)). Each household jsupplies a continuum of differentiated labor
inputs, LS
t(j, l), l∈[0,1]. The aggregate quantity of these labor services is LS
t(j)≡R1
0LS
t(j, l)dl.
Households maximize lifetime utility EtP∞
t=0 βtub
t(ln (C∗S
t(j)−θ˜
C∗S
t−1)−(LS
t(j)1+ξ)/(1+ ξ)), where
βis the discount rate, ξis the inverse of the Frisch labor elasticity and ub
tis an exogenous shock
to preferences.
Savers have access to one-period nominal private bonds, Bs,t, that pay 1 unit of currency in t+1,
sell at price R−1
tin t, and are in zero net supply. They also have access to a portfolio of long-term
nominal government bonds, Bt, which sells at price PB
tin t. Maturity of these zero-coupon bonds
decays at the constant rate ρ∈[0,1] to yield the duration (1 −βρ)−1.
Savers receive after-tax wage and rental income, lump-sum transfers from the government, ZS,
and profits from firms, D. Savers spend income on consumption, investment in future capital, IS,
and on government bonds. The nominal flow budget constraint for saver jis
Pt(1 + τC
t)CS
t(j) + PtIS
t(j) + PB
tBt(j) + R−1
tBs,t(j) = (1 + ρP B
t)Bt−1(j) + Bs,t−1(j)
+ (1 −τL
t)Z1
0
Wt(l)LS
t(j, l)dl + (1 −τK
t)Rk
tvt(j)¯
KS
t−1(j)−ψ(vt)¯
KS
t−1+PtZS
t(j) + Dt(j)
Nominal consumption, PCC, is subject to a sales tax τC.Wt(l) is the nominal wage rate for labor
input l, and R1
0Wt(l)LS
t(j, l)dl is the total nominal labor income for household j, which is taxed at
the rate τL. Each saver-type household supplies all differentiated labor inputs in the economy, so
all saver households have the same total after-tax labor income in equilibrium.
Effective capital is related to the physical capital stock ¯
Kby Ks
t(j) = vt(j)¯
KS
t−1(j), where vt(j)
is the utilization rate of capital. Utilization incurs a cost of Ψ(vt) per unit of physical capital. In
steady state, v= 1 and Ψ(1) = 0. Define parameter ψ∈[0,1) such that Ψ′′(1)
Ψ′(1) ≡ψ
1−ψ, as in Smets
5

Leeper, Traum & Walker: Fiscal Multiplier Morass
and Wouters (2003). As ψ→1, utilization costs become infinite, and the capital utilization rate
becomes constant. Rental income on effective capital is taxed at the rate τK. Capital evolves as
¯
KS
t(j) = (1 −δ)¯
KS
t−1(j) + ui
t"1−s IS
t(j)
IS
t−1(j)!#IS
t(j)
where s(·)IS
tis an investment adjustment cost, as in Smets and Wouters (2003) and Christiano et
al. (2005) and satisfies s′(eγ) = 0, and s′′ (eγ)≡s > 0. Investment costs decrease as sdeclines and
are subject to an investment-specific efficiency shock ui
t.
Saver households reset their nominal wages for each differentiated labor service with probability
(1 −ωw) each period. Wages that cannot be reoptimized are partially indexed to past inflation
according to the rule Wt(l) = Wt−1(l) (πt−1eua
t−1)χw(πeγ)1−χw, where χw∈[0,1] measures the
degree of indexation. When wages are reset, households choose the nominal wage rate Wt(l) to
maximize their utility.
1.2.2 Non-savers Non-savers have the same preferences as savers. Non-savers are rule-of-
thumb agents who consume their entire disposable income each period, which consists of after-tax
labor income and lump-sum transfers from the government, ZN. Like savers, non-savers supply all
differentiated labor services. The budget constraint for a non-saver j∈(µ, 1] is
(1 + τC
t)PtCN
t(j) = (1 −τl
t)Z1
0
Wt(l)LN
t(j, l)dl +PtZN
t(j) (2)
We assume that savers optimally set wage rates, while non-savers follow a rule-of-thumb to set
their wage rates to be the average wage rates chosen by savers, as in Erceg et al. (2006) and Forni
et al. (2009). Since non-savers face the same labor demand schedule as savers, they work the same
number of hours as the average for savers.
Non-savers’ nominal consumption, PCCN, is taxed at the same rate as savers, τC, and their
nominal wage income is taxed at the same rate as savers, τL. Because non-savers elastically meet
the demand for their labor and set their nominal wages according to savers’ optimization, budget
constraint (2) determines non-savers’ consumption.
1.3 Monetary & Fiscal Policy
The monetary authority follows a Taylor-type rule, in which the nominal interest rate, Rt, responds
to its lagged value, the current inflation rate, and current output relative to trend technology,
yt=Yt/At. We denote a variable in percentage deviations from steady state by a hat. The interest
rate obeys ˆ
Rt=ρrˆ
Rt−1+ (1 −ρr) [φπˆπt+φyˆyt] + um
t,where umis a monetary policy shock, defined
by the process um
t=ρemum
t−1+ǫm
t, ǫm
t∼N(0, σ2
m) .
The government collects tax revenues from capital, labor, and consumption taxes, and sells the
nominal bond portfolio, Bt, to finance its interest payments and expenditures, Gt, Z S
t, ZN
t. Fiscal
choices satisfy the identity PB
tBt+τK
tRK
tKt+τL
tWtLt+PtτC
tCt= (1 + ρP B
t)Bt−1+PtGt+PtZt.
6

Leeper, Traum & Walker: Fiscal Multiplier Morass
Parameter Restrictions
Model 1: RBC Real Frictions ωw=ωp=ηw=ηp=χw=χp=φπ=φy=ρr=µ=αG= 0
Model 2: New Keynesian µ=αG= 0
Model 3: New Keynesian Nonsavers αG= 0
Model 4: New Keynesian G in Util µ= 0
Table 1: Parameter restrictions on the general prior predictive model that deliver nested models.
Lump-sum transfers are identical across households, so Zt=R1
0Zt(j)dj =ZS
t=ZN
t.
Fiscal rules include a response of fiscal instruments to the market value of the debt-to-GDP
ratio and an autoregressive term to allow for serial correlation. Fiscal instruments follow the rules
ˆgt=ρGˆgt−1−(1 −ρG)γGˆsb
t−1+uG
t,ˆzt=ρZˆzt−1−(1 −ρZ)γZˆsb
t−1+uZ
t,ˆτJ
t=ρJˆτJ
t−1+ (1 −ρJ)γJˆsb
t−1
where J=K, L,gt=Gt/At,zt=Zt/At,sb
t−1≡PB
t−1Bt−1
Pt−1Yt−1,us
t=ρesus
t−1+ǫs
tand ǫs
t∼N(0, σ2
s) for
s={G, Z}. Consumption taxes are restricted to a constant, steady state value.4
1.4 Aggregation
Aggregate consumption is Ct=R1
0Ct(j)dj = (1 −µ)CS
t+µCN
t. Because only savers have access
to the asset and capital markets, aggregate bonds, private capital, investment, and dividends are
Υt=R1−µ
0Υt(j)dj for Υ={B, K, I , D}. Goods market clearing is Yt=Ct+It+Gt+ψ(vt)¯
Kt−1.
1.5 Nested Models
This model nests models commonly used to examine the size of the fiscal multiplier. Table 1lists
the restrictions that deliver each of the five nested models. Model 1 eliminates all nominal frictions
(ωw=ωp=ηw=ηp=χw=χp= 0) and monetary policy (φπ=φy=ρr= 0) to reduce to an RBC
model. Model 2 is a basic new Keynesian model with sticky prices and wages, which introduces a
role for monetary-fiscal policy interactions. Model 3 adds non-savers to the new Keynesian model.
Model 4 eliminates non-savers and allows instead for government spending to be non-separable in
the utility function.5
2 Prior Predictive Analysis
Models that permit analytical calculations of the multiplier are important for building economic
intuition [Uhlig (2010), Woodford (2011)], but they tend to be too simple to take to data. Models
that include real and nominal frictions, which fit data well, do not yield clean analytics. We echo
Geweke (2010) in arguing for the use of prior predictive analysis to shed light on the black-box
4We do not allow consumption taxes to respond to debt. In U.S. federal government data, consumption taxes
consist of excise taxes and custom duties, which average one percent of GDP. The online appendix documents that
adding consumption tax financing has little quantitative affect on multipliers.
5We restrict attention to closed economies. Leeper et al. (2011) and the appendix explore open economies.
7

Leeper, Traum & Walker: Fiscal Multiplier Morass
nature of empirically validated DSGE models. Prior predictive analysis pinpoints precisely which
elements of a model are critical to determine fiscal multipliers and it delivers the range of multipliers
that a model can produce. We use the results of the prior predictive analysis to determine which
models to take to the data. We also show that many of the DSGE models that have played a role
in the fiscal policy debate impose tight ranges on fiscal multipliers a priori.
This section lays out the prior predictive technique and the priors that we employ. After
defining the government spending multipliers that we report throughout the paper, the section
reports statistics that summarize the prior predictive distributions of multipliers across a wide
variety of model specifications.
2.1 Prior Predictive Technique
Prior predictive analysis entails four steps:6
i. Given a DSGE model, Aj, and associated model parameters, θAj, for j= 1,2,...,n, we
posit a prior density function p(θAj|Aj), which specifies the range of values and the proba-
bilities that the parameters take those values. Calibration is an example of a degenerate or
dogmatic prior density. We assume that the parameters are drawn independently, and let
˜p(θAj|Aj) be the product of the marginal parameter distributions. We restrict analysis to
the parameter subspace that delivers a unique rational expectations equilibrium and denote
this subspace as ΘDj. Let I{θAj∈ΘDj}be an indicator function that is one if θAjis in
the determinacy region and zero otherwise. Then the joint prior distribution is defined as
p(θAj|Aj) = 1
c˜p(θAj|Aj)I{θAj∈ΘDj}, where c=RθAj∈ΘDj
˜p(θAj|Aj)dθAj.
ii. The log-linearized DSGE model that section 1describes and the nested models in table 1
constitute the set of models under consideration. Those models generate ex-ante predictive
distributions for the models’ observables, yT, from p(yT|Aj) = RΘAj
p(θAj|Aj)p(yT|θAjAj)dθAj.
iii. We specify a vector of interest, ω, with corresponding distribution p(ωT|yT, θAj, Aj). Our vector
of interest consists of various measures of the fiscal multiplier, which we define in section 2.4.
As the conditional distribution makes explicit, the fiscal multiplier depends on the choice of
model (Aj), model-implied observables (yT), and parameters (θAj).
iv. To generate prior predictive distributions for fiscal multipliers, the algorithm draws from θ(m)
Aj∼
p(θAj|Aj), and y(m)
T∼p(yT|θ(m)
Aj, Aj). Drawing sequentially from these distributions delivers
p(yT|Aj) and any function of yTincluding the vector of interest, ω(m). A model specification
and a prior distribution produce prior distributions for fiscal multipliers.
The distribution p(yT|Aj) gives the prior distribution of observables, which implies the distri-
bution of fiscal multipliers, p(ωT|yT, θAj, Aj). Given a prior density over model parameters, prior
predictive analysis produces the entire range of a model’s possible multipliers, to shed light on a
6These steps follow Geweke (2010, Chapter 3).
8

Leeper, Traum & Walker: Fiscal Multiplier Morass
model’s predictions before confronting data. This narrows the set of models to estimate. For ex-
ample, if prior predictive analysis suggests that it is nearly impossible for a model to produce large
multipliers, then conclusions drawn from estimates of that model need to be tempered by the fact
that regardless of the information in actual data, the model will imply that multipliers are small.
Prior predictive analysis helps to gauge whether a model is appropriate to study fiscal multipliers.
2.2 Prior Distributions
In all model specifications we fix a few parameters. The discount factor is set to β= 0.99. The
capital income share of total output is set to α= 0.33. The quarterly depreciation rate for private
capital, δ, is set to 0.025 so that the annual depreciation rate is 10 percent. Steady state inflation
is π= 1. Because the price and wage markups cannot be separately identified in the estimation,
we calibrate them as ηw=ηp= 0.14.
Steady-state fiscal variables are calibrated to the mean values from U.S. data over the period
1955q1–2014q2. Federal government consumption as a share of model output—GDP excluding net
exports—is 0.11, the ratio of federal debt to model output is 1.47, the average federal labor tax
rate is 0.186, the capital tax rate is 0.218, and the consumption tax rate is 0.023. See the online
appendix for details of the data construction.
Table 2lists the priors. The prior distributions cover a broad range of parameter values and
are similar to those employed for Bayesian estimation of models closely related to ours [Coenen
and Straub (2005), Forni et al. (2009), Leeper et al. (2010), Drautzburg and Uhlig (2015), F`eve
et al. (2013), Zubairy (2014), and Traum and Yang (2015)]. An important difference is that we
adopt priors over nominal rigidities and habit formation—parameters ωp,ωw, and θin the table—
with somewhat broader support. The prior predictive analysis tells us which model/parameter
specification permits a wide range of fiscal multipliers. An agnostic a priori view of the signs
and sizes of multipliers is an essential step toward clearing up the multiplier morass. To avoid pre-
judging the estimation results, we use an equally agnostic prior to obtain the posterior distribution.
2.3 Policy Regimes
In versions of our model that integrate monetary and fiscal policies, two distinct regions of the pa-
rameter space deliver unique bounded rational expectations equilibria—an active monetary/passive
fiscal policy regime (regime M) or a passive monetary/active fiscal policy regime (regime F). To
reflect these two policy regimes, we consider two sets of policy parameter priors: the first places
nearly all probability mass on regions of the parameter space consistent with regime M and the
second does the same for regime F. In regime M the monetary authority raises the interest rate
aggressively in response to inflation while the fiscal authority adjusts expenditures and tax rates
to stabilize debt. Regime F has monetary policy respond only weakly to inflation, while fiscal
instruments adjust weakly to government debt. The two regimes appear in table 2as different
9

Leeper, Traum & Walker: Fiscal Multiplier Morass
Parameter Prior
distribution mean std. 90% int.
Preference and HHs
100γ, ss ln growth rate N 0.4 0.05 [0.42, 0.58]
ξ, inverse Frisch labor elast. G 2 0.5 [1.18, 2.80]
θ, habit formation B 0.5 0.2 [0.17, 0.83]
µ, fraction of non-savers B 0.3 0.1 [0.14, 0.46]
αG, substitutability of private/public cons. U 0 1.01 [-1.58, 1.58]
Frictions and Productio n
ψ, capital utilization B 0.6 0.15 [0.36, 0.85]
s, investment adj. cost N 6 1.5 [3.54, 8.47]
ωp, price stickiness B 0.5 0.2 [0.17, 0.83]
ωw, wage stickiness B 0.5 0.2 [0.17, 0.83]
χp, price partial indexation B 0.5 0.2 [0.17, 0.83]
χw, wage partial indexation B 0.5 0.2 [0.17, 0.83]
Monetary policy
φπ, interest rate resp. to inflation, Regime M N 1.5 0.2 [1.17, 1.83]
φπ, interest rate resp. to inflation, Regime F B 0.5 0.15 [0.25, 0.75]
φy, interest rate resp. to output N 0.125 0.05 [0.04, 0.21]
ρr, resp. to lagged interest rate B 0.5 0.2 [0.17, 0.83]
Fiscal policy
γi, debt responses for i=G, K, L, Z, Regime M N 0.15 0.1 [-0.015, 0.31]
γi, debt responses for i=G, K, L, Z, Regime F N 0 0.001 [-0.0016, 0.0016]
ρi, lagged resp. for i=G, K, L, Z B 0.5 0.2 [0.17, 0.83]
Shocks
ρa, technology B 0.5 0.2 [0.17, 0.83]
ρb, preference B 0.5 0.2 [0.17, 0.83]
ρi, investment B 0.5 0.2 [0.17, 0.83]
ρp, price markup B 0.5 0.2 [0.17, 0.83]
ρw, wage markup B 0.5 0.2 [0.17, 0.83]
ρem, monetary p olicy B 0.5 0.15 [0.25, 0.75]
ρeg, govt cons B 0.5 0.15 [0.25, 0.75]
ρez, transfers B 0.5 0.15 [0.25, 0.75]
100σa, technology Inv. Gamma 0.1 1 [0.01, 0.19]
100σb, preference Inv. Gamma 0.1 1 [0.01, 0.19]
100σm, monetary policy Inv. Gamma 0.1 1 [0.01, 0.19]
100σi, investment Inv. Gamma 0.1 1 [0.01, 0.19]
100σp, price markup Inv. Gamma 0.1 1 [0.01, 0.19]
100σw, wage markup Inv. Gamma 0.1 1 [0.01, 0.19]
100σG, govt cons Inv. Gamma 0.1 1 [0.01, 0.19]
100σZ, transfers Inv. Gamma 0.1 1 [0.01, 0.19]
Table 2: Prior distributions.
priors on φπin the monetary policy rule and on the γi’s in the fiscal rules. The priors assign a
small, non-zero density outside the determinacy regions of the parameter space, so we restrict the
parameter space to the subspaces in which the log-linearized model has a unique bounded rational
expectations solution by discarding draws from the indeterminacy region.
Cochrane (2001), Sims (2013) and Leeper and Leith (2017) show that in regime F, long-term
nominal government debt can have important effects on inflation dynamics. When prices and
wages are sticky, dynamics of real variables will also be affected by the presence of long debt, so we
examine specifications with one-period debt—the typical assumption in the literature—and with a
fixed duration of five years. Maturity structure is irrelevant in regime M when all fiscal financing
is lump sum and Ricardian equivalence holds; otherwise, maturity structure can matter even in
regime M.
2.4 Multiplier Definition
Present-value multipliers, which embody the full dynamics associated with exogenous fiscal actions
and properly discount future macroeconomic effects, constitute our vector of interest. The present
value of additional output, Yt+k, over a k-period horizon produced by an exogenous change in the
10

Leeper, Traum & Walker: Fiscal Multiplier Morass
present value of government spending is
Present Value Multiplier(k) =
EtPk
j=0 Qk
i=0(1 + rt+i)−1∆Yt+j
EtPk
j=0 Qk
i=0(1 + rt+i)−1∆Gt+j
(3)
where rt+iis the model-implied real interest rate. Private consumption and investment multipliers
are defined analogously. At k= 0 the present-value multiplier equals the impact multiplier. Because
a present-value multiplier is cumulative, its value at t+kreports the total effect over kperiods of
a change in spending at time t.
2.5 Likelihood of Large Multipliers
To compare multipliers across models, we focus on prior predictive p-values, which report the
probability of observing a multiplier greater than a particular value in repeated sampling from the
model and prior. Tables 3and 4compare output, consumption and investment multiplier p-values
at various horizons across the four model specifications and variants within those specifications.
Table 3reports the probability that present-value multipliers for output exceed unity at various
horizons. The top panel of table 4records the probability that multipliers for consumption exceed
0, while the bottom panel reports the same information about investment multipliers.
We examine four broad model specifications: a real business cycle model with frictions; a basic
new Keynesian model with sticky prices and wages; an extension of the new Keynesian model
that adds non-saving rule-of-thumb agents; and a new Keynesian model that eliminates rule-of-
thumb agents, but permits government purchases to enter utility directly, either as a substitute
or a complement. The three monetary models include both regime M and regime F monetary-
fiscal policy regimes. Because the presence of long-maturity debt matters in regime F for inflation
and output dynamics, those specifications are subdivided between short debt and long debt.7To
shed light on the estimation results that appear in section 3.2, we consider several fiscal variants
on model 4 in which government purchases enter utility: all fiscal instruments—capital and labor
tax rates, government purchases, and government transfers—respond to government debt; only
purchases and transfers respond to debt but tax distortions enter the steady state (labeled “ss tax
only” in the tables); purchases and transfers respond to debt but steady state taxes rates are set
to zero (labeled “no tax”). The last fiscal variant often appears in multiplier studies and implies
negative transfers or lump-sum taxation.
Consider the real business cycle model with flexible prices and real frictions that include habit
formation, investment adjustment costs, and capacity utilization (model 1 in tables 3and 4).8It
is impossible for this model to generate output multipliers greater than one or to produce positive
7See the online appendix for a comparison of multipliers in regime M with short and long debt. At longer horizons,
a longer maturity often implies higher multipliers, but the differences are small compared to regime F.
8An earlier draft reports these probabilities for the basic RBC model without frictions [Leeper et al. (2011)]. That
basic model is similar to Baxter and King (1993), Monacelli and Perotti (2008), Uhlig (2010) and Woodford (2011),
with the addition of distortionary fiscal financing, as in Leeper et al. (2010), so it has been extensively studied.
11

Leeper, Traum & Walker: Fiscal Multiplier Morass
consumption multipliers at any horizon. A persistent increase in government spending creates a
negative wealth effect, as taxes are expected to increase in the future to finance the new spending.
Agents decrease consumption and work more. These wealth effects are reinforced by negative
substitution effects. Real wages decrease with the increase in work effort and the rental price
of capital increases with the rising marginal product of capital. Consumption and investment
are likely to decrease, though their declines are tempered by real frictions in the model.9Habit
formation makes agents less willing to decrease consumption quickly as changes in consumption are
costly. Investment adjustment costs and capacity utilization costs deter large swings in investment,
offsetting some of the potential crowding out of investment. Despite these tempering forces, declines
in private demand offset most of the increased public demand, causing output to increase by less
than the increase in government consumption.
ProbP V ∆Y
∆G>1
Impact 4 qtrs 10 qtrs 25 qtrs 10 years
Model 1: RBC Real Frictions 0.00 0.00 0.00 0.00 0.00
Model 2: New Keynesian Sticky Prices & Wages
Regime M 0.11 0.01 0.00 0.00 0.01
Regime F, short debt 1.00 0.98 0.94 0.92 0.92
Regime F, long debt 0.96 0.79 0.67 0.68 0.68
Model 3: New Keynesian Nonsavers
Regime M 0.58 0.22 0.06 0.04 0.03
Regime F, short debt 1.00 1.00 0.97 0.95 0.94
Regime F, long debt 1.00 0.94 0.81 0.77 0.76
Model 4: New Keynesian G-in-Utility
Regime M, substitutes 0.00 0.00 0.00 0.00 0.00
Regime M, complements 0.84 0.69 0.49 0.33 0.29
Regime M, complements, ss tax only 0.83 0.68 0.50 0.40 0.39
Regime M, complements, no tax 0.86 0.71 0.53 0.45 0.45
Regime F, substitutes, short debt 0.43 0.48 0.65 0.78 0.80
Regime F, substitutes, long debt 0.23 0.18 0.21 0.38 0.44
Regime F, complements, short debt 1.00 1.00 0.99 0.97 0.97
Regime F, complements, long debt 1.00 0.98 0.93 0.88 0.86
Regime F, complements, short debt, ss tax only 1.00 1.00 0.99 0.97 0.97
Regime F, complements, long debt, ss tax only 1.00 0.98 0.93 0.88 0.86
Regime F, complements, short debt, no tax 1.00 1.00 0.99 0.99 0.98
Regime F, complements, long debt, no tax 1.00 0.99 0.95 0.92 0.91
Table 3: Government spending output multiplier probabilities implied by prior predictive analysis
based on 20,000 draws from the prior distribution. Short debt is all one-period; long debt has 5-year
duration. Substitutes (complements) restricts αG>0 (<0). In all cases except “ss tax only” and
“no tax,” government purchases, transfers, and distorting taxes on capital and labor may respond
to government debt. “ss tax only” shuts down the distorting tax responses, but maintains positive
steady state capital and labor taxes; “no tax” eliminates distorting taxes from the model—both
dynamic responses and in steady state.
9See Monacelli and Perotti (2008) for a more detailed examination of the effect of habit formation and investment
adjustment costs on multipliers in an RBC model. Bilbiie (2009) shows non-separable preferences can give positive
consumption multipliers but require consumption to be an inferior good, while F`eve et al. (2011) show that a model
with a labor externality can give positive consumption multipliers. Finn (1998) discusses how private and public
consumption complementarity affect consumption in an RBC model.
12

Leeper, Traum & Walker: Fiscal Multiplier Morass
There is only a small probability that investment will increase at any horizon. This result is
consistent across all regime M specifications, except in the short run when government purchases
enter utility as substitutes for private consumption, as in model 4 in the bottom panel of table 4.
Apart from that exception, any possibility of higher investment stems from a subset of very high
draws for ρG, the serial correlation of government spending. As ρGapproaches one, agents view
an exogenous change in government spending as approximately permanent. Permanent increases
in government consumption encourage households to save more, which can raise investment. This
difference between permanent and temporary changes to public expenditures echoes earlier work
[Aiyagari et al. (1992) and Baxter and King (1993)]. In the absence of a near-unity value of ρGor
sufficiently strong substitution of purchases for consumption, investment would never rise in regime
M.
Model 2 introduces sticky prices and sticky wages, which increase output multipliers at all
horizons, as Woodford (2011) shows analytically. Greater price stickiness means that more firms
respond to higher government spending by increasing production rather than prices, so markups
respond more strongly. Although the likelihood of large multipliers tapers off over time, in the long
run there continues to be some small probability of sizeable multipliers in regime M. RBC models
cannot produce these positive long-run multipliers; nominal rigidities are necessary.
Non-savers (model 3) raise fiscal multipliers substantially, a point that Gal´ı et al. (2007), Furlan-
etto (2011), and Colciago (2011) emphasize. In this model, the fraction of non-savers is the most
influential parameter for the output multiplier, as variations in this parameter are necessary to get
mean impact output multipliers greater than one in regime M. Unlike savers, non-savers ignore the
wealth effects of future taxes and consume their entire income each period. If wages are sticky,
then real wages rise with government spending, increasing non-savers’ consumption. With enough
non-savers in the economy, the increase in non-saver consumption can be large enough to cause
total consumption to increase on impact.10 Both output and consumption effects in regime M are
short-lived, with most of the increase in multipliers disappearing after two years.
In regime M, permitting government spending to enter utility can consistently generate large
multipliers, even in the long run (model 4). The effect is direct: when government purchases
substitute for private consumption, higher purchases raise output, crowd out consumption and
increase investment; when purchases complement consumption, output and consumption multipli-
ers are likely to be large and fairly persistent.11 Higher consumption comes at the cost of lower
investment. The preference parameter that determines the elasticity of substitution between gov-
ernment and private consumption, αG, is by far the most important parameter for determining the
magnitude of multipliers within a given policy regime.
Across all model specifications, the monetary-fiscal policy regime is the dominant factor in
10Alternatively, Bilbiie (2011) and Monacelli and Perotti (2008) suggest non-separability in preferences over con-
sumption and leisure also can produce positive consumption multipliers, as can deep habits, as Ravn et al. (2006)
show. Devereux et al. (1996) show an externality in production also can give large output responses.
11Models with public spending in the utility function have a long history, see for example Barro (1981), Aschauer
(1985), and Christiano and Eichenbaum (1992).
13

Leeper, Traum & Walker: Fiscal Multiplier Morass
ProbP V ∆C
∆G>0
Impact 4 qtrs 10 qtrs 25 qtrs 10 years
Model 1: RBC Real Frictions 0.00 0.00 0.00 0.00 0.00
Model 2: New Keynesian Sticky Prices & Wages
Regime M 0.00 0.00 0.00 0.00 0.01
Regime F, short debt 0.98 0.96 0.92 0.90 0.89
Regime F, long debt 0.77 0.56 0.49 0.55 0.56
Model 3: New Keynesian Nonsavers
Regime M 0.48 0.21 0.09 0.05 0.04
Regime F, short debt 1.00 0.99 0.98 0.96 0.95
Regime F, long debt 0.99 0.92 0.82 0.77 0.74
Model 4: New Keynesian G-in-Utility
Regime M, substitutes 0.00 0.00 0.00 0.00 0.00
Regime M, complements 0.83 0.75 0.70 0.61 0.51
Regime M, complements, ss tax only 0.82 0.74 0.70 0.65 0.58
Regime M, complements, no tax 0.84 0.77 0.75 0.71 0.66
Regime F, substitutes, short debt 0.30 0.27 0.29 0.49 0.59
Regime F, substitutes, long debt 0.13 0.07 0.06 0.13 0.19
Regime F, complements, short debt 1.00 1.00 0.99 0.99 0.98
Regime F, complements, long debt 0.99 0.97 0.94 0.92 0.90
Regime F, complements, short debt, ss tax only 1.00 1.00 0.99 0.99 0.98
Regime F, complements, long debt, ss tax only 0.99 0.97 0.94 0.93 0.90
Regime F, complements, short debt, no tax 1.00 1.00 1.00 0.99 0.99
Regime F, complements, long debt, no tax 1.00 0.98 0.97 0.96 0.94
ProbP V ∆I
∆G>0
Impact 4 qtrs 10 qtrs 25 qtrs 10 years
Model 1: RBC Real Frictions 0.00 0.00 0.00 0.00 0.01
Model 2: New Keynesian Sticky Prices & Wages
Regime M 0.00 0.00 0.00 0.00 0.00
Regime F, short debt 0.96 0.89 0.82 0.81 0.81
Regime F, long debt 0.62 0.50 0.50 0.55 0.57
Model 3: New Keynesian Nonsavers
Regime M 0.00 0.00 0.00 0.00 0.01
Regime F, short debt 0.91 0.81 0.74 0.74 0.74
Regime F, long debt 0.43 0.32 0.34 0.43 0.46
Model 4: New Keynesian G-in-Utility
Regime M, substitutes 0.35 0.31 0.20 0.07 0.04
Regime M, complements 0.00 0.00 0.00 0.00 0.01
Regime M, complements, ss tax only 0.00 0.00 0.00 0.01 0.01
Regime M, complements, no tax 0.00 0.00 0.00 0.01 0.02
Regime F, substitutes, short debt 1.00 0.98 0.96 0.96 0.96
Regime F, substitutes, long debt 0.94 0.91 0.89 0.89 0.89
Regime F, complements, short debt 0.67 0.55 0.53 0.58 0.59
Regime F, complements, long debt 0.19 0.13 0.14 0.22 0.26
Regime F, complements, short debt, ss tax only 0.67 0.55 0.54 0.58 0.59
Regime F, complements, long debt, ss tax only 0.20 0.13 0.14 0.22 0.25
Regime F, complements, short debt, no tax 0.92 0.83 0.76 0.75 0.75
Regime F, complements, long debt, no tax 0.43 0.35 0.36 0.43 0.44
Table 4: Government spending consumption and investment multiplier probabilities implied by
prior predictive analysis based on 20,000 draws from the prior distribution. Short debt is all one-
period; long debt has 5-year duration. Substitutes (complements) restricts αG>0 (<0). In all
cases except “ss tax only” and “no tax,” government purchases, transfers, and distorting taxes
on capital and labor may respond to government debt. “ss tax only” shuts down the distorting
tax responses, but maintains positive steady state capital and labor taxes; “no tax” eliminates
distorting taxes from the model—both dynamic responses and in steady state.
14

Leeper, Traum & Walker: Fiscal Multiplier Morass
determining government spending impacts: output, consumption and investment multipliers are
far more likely to be large in regime F than in regime M. Long-term debt reduces the probability
of large multipliers in regime F, compared to when all debt is one-period. For example, even when
αGis restricted to being positive, so government spending substitutes for private consumption,
there is a substantial probability of sizeable output and consumption multipliers in regime F; those
probabilities are 0 in regime M. Long-term debt cuts those probabilities in regime F by factors of
between two and five.
Similar patterns emerge in models 2 (sticky prices and wages) and 3 (rule-of-thumb agents).
Moving from regime M to regime F dramatically increases output and consumption multipliers at
all horizons. While the likelihood of large multipliers with non-saving agents in regime M tapers off
sharply beyond horizons of four quarters, in regime F the tapering off is barely discernible. Once
again, though, long debt systematically reduces the probability of realizing large multipliers.
In regime F, large consumption multipliers do not come at the expense of lower investment, as is
true in regime M. All regime F specifications produce a high probability of positive investment mul-
tipliers along with positive consumption effects. The least likely specifications to generate positive
investment impacts combine two factors: government and private consumption are complements
and distorting taxes are present, either in steady state or dynamically responding to increases in
debt. Even in those cases, positive investment effects occur in about 20 percent of the parameter
draws in regime F. Eliminating steady-state taxes increases the likelihood of large multipliers for
all three variables.
2.6 Prior Predictive for Model Selection
Rule-of-thumb agents are prevalent in models of government spending multipliers. Models that
include a sufficiently large fraction of such agents are likely to produce sizeable output and con-
sumption multipliers in the short run in both policy regimes, as tables 3and 4show. In contrast,
when government spending enters utility, both a broader range and a larger persistence of mul-
tipliers are possible, depending on whether the spending substitutes for or complements private
consumption. This information gleaned from the prior predictive helps to select a model specifica-
tion with which to confront data.
Figure 1plots prior means and 90-percent probability bands for multipliers in regime M for
models with rule-of-thumb agents (dashed and dotted-dashed lines) and government-spending-in-
utility (solid lines); figure 2repeats the results for regime F. The prior in table 2over the fraction
of rule-of-thumb agents, µ, is centered at 0.30 and puts 90-percent of the probability on fractions
between 0.14 and 0.48. The preference parameter for government spending, αG, obeys a uniform
prior centered at 0, and places equal probability on spending being a substitute or a complement,
with the 90-percent interval covering [−1.58,1.58].
Rule-of-thumb models deliver much tighter prior distributions for multipliers in both policy
regimes. In regime M, when all fiscal instruments respond to stabilize debt (dashed lines, figure
1), output, consumption and investment multipliers are uniformly smaller than when there are
15

Leeper, Traum & Walker: Fiscal Multiplier Morass
0 10 20 30 40
−0.5
0
0.5
1
1.5
2
(a) Output Multiplier
0 10 20 30 40
−1.5
−1
−0.5
0
0.5
1
(b) Consumption Multiplier
0 10 20 30 40
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
(c) Investment Multiplier
Figure 1: Present-value government spending multipliers in regime M for output, consumption
and investment at various horizons with 90-percent probability bands. Government spending in
utility unrestricted, steady-state taxes only, long debt (solid lines); rule-of-thumb agents, everything
responds to debt, long debt (dashed lines); rule-of-thumb agents, steady-state taxes only, long debt
(dotted-dashed lines).
steady-state tax distortions, but only lump-sum transfers and government spending adjust to debt
(dotted-dashed lines). Regardless of the fiscal adjustments, regime M rule-of-thumb models leave
no possibility of positive investment multipliers.
A uniform prior over αGpermits both large positive and large negative consumption multipliers
(solid lines), which rule-of-thumb agents preclude. Although most probability mass is on negative
investment effects, this specification does offer some chance for small positive investment multipliers.
Government spending in utility can also generate more persistence in multipliers.
Differences between the two specifications are less stark in regime F (figure 2). Although rule-
of-thumb agents can produce large short- and long-run output multipliers, the range remains more
tightly circumscribed than when government spending yields utility in an unrestricted manner.12
As the figure makes clear, government-spending-in-utility supports far wider ranges for all three
multipliers, offering a more agnostic model with which to examine data.
This prior predictive analysis leads us to choose to take the government-spending-in-utility
model to data, rather than the rule-of-thumb specification. Existing empirical work reports mul-
tipliers that vary substantially both in their magnitudes and in their persistence. A government-
spending-in-utility model, together with a uniform prior over αG, which is centered at 0, can cover
that reported range of multipliers, while it also admits the possibility of positive investment multi-
pliers in either policy regime. We do not have strong prior beliefs about whether in the aggregate
the elasticity of substitution between government and private consumption is positive or negative.
Our estimates will permit data to determine that elasticity.
12A uniform prior for µbetween [0.2,0.5] raises output and consumption multipliers in the rule-of-thumb models
at all horizons, but the differences are not large.
16

Leeper, Traum & Walker: Fiscal Multiplier Morass
0 10 20 30 40
0
0.5
1
1.5
2
2.5
(a) Output Multiplier
0 10 20 30 40
−1
−0.5
0
0.5
1
1.5
(b) Consumption Multiplier
0 10 20 30 40
−0.4
−0.2
0
0.2
0.4
(c) Investment Multiplier
Figure 2: Present-value government spending multipliers in regime F for output, consumption
and investment at various horizons with 90-percent probability bands. Government spending in
utility unrestricted, steady-state taxes only, long debt (solid lines); rule-of-thumb agents, everything
responds to debt, long debt (dashed lines); rule-of-thumb agents, steady-state taxes only, long debt
(dotted-dashed lines).
3 Data and Estimates
We estimate a variant of model 4 from section 1using quarterly U.S. data. There are eight
observables: log differences of aggregate consumption, investment, real wages, real government
consumption, the real market-value of government debt, and the GDP deflator; log hours worked;
the federal funds rate. Data are neither detrended nor demeaned. Details of the data construction
and linkage to observables appear in the online appendix. Our full sample period is 1955q1 to
2014q2, but we also estimate over three sub-samples: pre-financial crisis, 1955q1 to 2007q4; the
pre-Volcker era, 1955q1 to 1979q4; and the Great Moderation, 1982q1 to 2007q4. We treat the
pre-crisis sample as the baseline because it ends before the Federal Reserve fixed the federal funds
rate near its effective lower bound. To further investigate the sensitivity of results to specific sub-
samples, we conduct rolling window estimation. The first rolling window sample consists of 100
quarters from 1955q1 to 1979q4. Then we consecutively increase the start and end dates by four
quarters until the end of our data, making the last sample estimated from 1989q1 to 2013q4.
Our dataset differs from the conventional ones used to estimate new Keynesian models [for
example, Christiano et al. (2005) or Smets and Wouters (2007)] because it includes government
debt and government consumption. These are natural additions for the question at hand, but they
change the structure of the data in important ways. Fiscal data, particularly the market value of
government debt, are more persistent than other macro aggregates. The addition of government
debt means our data have more prominent lower frequency variation, so stronger-than-usual model
frictions are likely to improve the model’s fit.
3.1 Methodology
We use Bayesian methods to construct the parameters’ posterior distribution, which combines our
priors with the likelihood function, calculated using the Kalman filter. Drawing on the information
17

Leeper, Traum & Walker: Fiscal Multiplier Morass
from the prior predictive analysis, we eliminate rule-of-thumb agents, which restricts µ= 0. We
also do not include tax revenues or tax rates in the observables and restrict the model so that
only public consumption and transfers potentially respond to debt. Tax distortions enter only the
steady state, which restricts γK=γL=ρK=ρL= 0.13 The remaining parameters have either the
priors listed in table 2or the dogmatic priors discussed in section 2.2. As in the prior predictive
with long debt, we assume a five-year duration for government bonds. We estimate sub ject to a
monetary-fiscal regime prior. For regime M, we further restrict the parameters ρZand ρez. Since
transfers are non-distortionary in regime M, ρZ,ρez , and σZcannot be separately identified. We
restrict ρZ= 0.98 and ρez = 0.8.14 Finally, the investment-specific, price and wage markup shocks
are normalized to enter with a unit coefficient in the investment, price and wage inflation equations
respectively.
We take 1.5 million draws from the posterior distribution using the random walk Metropolis-
Hastings algorithm. For purposes of inference, we discard the first 500,000 draws and keep one out
of 50 draws to remove some correlation among draws and to obtain a sample from the posterior
equal to our prior sample of 20,000.15
3.2 Posterior Estimates
Table 5reports the posterior estimates for the entire sample—1955q1 to 2014q2—and the pre-crisis
sample—1955q1 to 2007q4—for regimes M and F. The online appendix contains parameter esti-
mates for other sub-periods and parameter estimates for the structural shock processes in all sample
periods. Three aspects of the estimates are critical for inferences. First, despite the diffuse priors,
the credible sets indicate tight posteriors for nearly all parameters and across both regimes. Diffuse
priors preserve agnosticism with respect to the multipliers. But data are sufficiently informative to
push the posterior distributions into much smaller regions of the parameter space to deliver tightly
estimated multipliers.
Second, the posterior means and credible sets are roughly in line with the values reported in
the literature. Estimates imply public and private consumption are complements, as in Bouakez
and Rebei (2007) and F`eve et al. (2013). Parameters governing nominal rigidities are consistent
with values reported in Del Negro and Schorfheide (2008) and Herbst and Schorfheide (2014), who
note that higher values of wage and price stickiness parameters arise from more diffuse priors.
Relatively high degrees of stickiness make the inflation and wage Phillips curves quite flat. Our
estimates of habit formation are high, but they are within the 90-percent bands for external habits
that Havranek, Rusnak, and Sokolova’s (2017) meta study reports.
13We do not include tax rates as observables because quarterly measures of marginal tax rates are problematic.
14In regime M, combinations of high (low) AR(1) coefficients and low (high) standard deviations are similar. The
calibration for ρZand ρez was based on estimates from regime F and estimates in regime M with the high AR(1)
coefficient and low standard deviation combination. AR coefficients in both policy rules and policy shocks are essential
in regime F to match features of the data.
15We set the step size to target an acceptance rate in the range of 20 to 40 percent across all cases. Diagnostics
to determine chain convergence include cumulative sum of the draws (CUMSUM) statistics and Geweke’s Separated
Partial Means (GSPM) test. See the online appendix for details.
18

Leeper, Traum & Walker: Fiscal Multiplier Morass
Finally, table 6reports the log marginal data densities for both regimes, and for the entire
sample and subsamples. Log marginal data densities are calculated using Geweke’s (1999) modified
harmonic mean estimator with a truncation parameter of 0.5. The data do not systematically
prefer one regime over the other across the sub-periods, so our analysis gives equal weight to the
two regimes.16
4 Multipliers
Government spending multipliers depend on every aspect of a model’s specification. Our estimates
reveal some obvious aspects: the degree of nominal and real rigidities; the role that government
spending plays as a complement or substitute for private consumption; the stance of monetary
and fiscal policies, which encompasses the sources of fiscal financing and the prevailing monetary-
fiscal policy regime. But more subtle aspects of the model specification also emerge as important
for determining multipliers: the absence or presence of steady-state distorting taxes; the level of
steady-state government debt; and the maturity structure of outstanding debt. All these elements
affect the transmission mechanism of government spending.
To understand the economic mechanisms that underlie the estimated multipliers, we present re-
sults in several parts. We discuss similarities and differences in estimated responses to a government
spending increase across the two policy regimes and then explain the transmission mechanisms. Be-
cause differences in labor market behavior account for much of the variation in government spending
effects in the two regimes, we discuss these differences in detail. Finally, fiscal financing of gov-
ernment spending differs markedly between regimes, so we end with an analysis of the sources of
financing. In all results, government spending initially rises by 1 percent of steady-state government
purchases.
4.1 Overview of Multipliers Across Policy Regimes
Tables 7and 8summarize present-value multipliers for output, consumption and investment from
the prior predictive and posterior estimates over four sample periods: the full sample, 1955q1–
2014q2; the pre-crisis sample, 1955q1–2007q4; the pre-Volcker period, 1955q1–1979q4; the post-
Volcker pre-crisis period, 1982q1–2007q4. The tables report mean values and 90-percent credible
sets for multipliers at selected horizons. Prior predictive analysis produces very wide ranges for
possible multipliers, suggesting that a priori the model is agnostic about both the magnitudes
and signs of government spending effects. Data are highly informative about multipliers: posterior
credible sets are substantially narrower than the prior sets and in many cases leave little ambiguity
about government spending impacts.
Table 7reports that in regime M posterior mean estimates of output multipliers are positive
at all horizons and quite likely to be greater than 1 in the short run, but well below 1 over longer
16In contrast to Tan (2014) and Traum and Yang (2011), we find regime F is preferred by the data over some
periods, particularly 1955q1–2014q2. This difference stems from our inclusion of long-term debt and steady-state tax
rates. See section 4.3 for more discussion.
19

Leeper, Traum & Walker: Fiscal Multiplier Morass
1955q1:2014q2
Parameter Regime M Regime F
mean 90% C.S. mean 90% C.S.
Preference and HHs
ξ, inverse Frisch labor elast. 1.77 [1.11, 2.47] 2.33 [1.49, 3.18]
θ, habit formation 1.00 [0.99, 1.00] 0.99 [0.99, 1.00]
αg,Gin utility -0.24 [-0.41, -0.07] -0.20 [-0.38, -0.01]
Frictions & Production
100γ, SS tech growth 0.25 [0.18, 0.31] 0.25 [0.18, 0.31]
ψ, capital utilization 0.16 [0.09, 0.23] 0.15 [0.07, 0.23]
s, inv adj cost 5.46 [3.75, 7.06] 4.80 [3.24, 6.35]
ωp, price stickiness 0.92 [0.90, 0.94] 0.95 [0.94, 0.96]
ωw, wage stickiness 0.91 [0.89, 0.94] 0.87 [0.84, 0.90]
χp, price indexation 0.06 [0.01, 0.11] 0.06 [0.01, 0.11]
χw, wage indexation 0.18 [0.10, 0.26] 0.18 [0.10, 0.25]
Monetary Policy
φπ, interest rate resp. to inflation 0.90 [0.74, 1.06] 0.15 [0.08, 0.23]
φy, interest rate resp. to output 0.10 [0.08, 0.12] 0.14 [0.12, 0.16]
ρr, lagged interest rate resp. 0.71 [0.64, 0.77] 0.15 [0.06, 0.23]
Fiscal Policy
γG, govt cons. resp. to debt 0.26 [0.17, 0.34] 0.0001 [-0.0016, 0.0016]
γZ, transfer resp. to debt -0.11 [-0.20, -0.02] 0.0000 [-0.0016, 0.0017]
ρG, lagged govt cons resp. 0.98 [0.98, 0.99] 0.99 [0.98, 0.99]
ρZ, lagged transfer resp. n.e. 0.98 [0.97, 0.99]
1955q1:2007q4
Parameter Regime M Regime F
mean 90% C.S. mean 90% C.S.
Preference and HHs
ξ, inverse Frisch labor elast. 1.54 [0.92, 2.14] 2.32 [1.49, 3.20]
θ, habit formation 0.99 [0.98, 1.00] 0.99 [0.98, 1.00]
αg,Gin utility -0.19 [-0.36, -0.02] -0.16 [-0.34, 0.02]
Frictions & Production
100γ, SS tech growth 0.24 [0.18, 0.31] 0.27 [0.20, 0.33]
ψ, capital utilization 0.13 [0.08, 0.17] 0.13 [0.06, 0.19]
s, inv adj cost 5.21 [3.68, 6.71] 3.97 [2.47, 5.36]
ωp, price stickiness 0.89 [0.86, 0.91] 0.95 [0.94, 0.96]
ωw, wage stickiness 0.87 [0.83, 0.92] 0.85 [0.81, 0.89]
χp, price indexation 0.06 [0.01, 0.11] 0.06 [0.01, 0.11]
χw, wage indexation 0.09 [0.03, 0.15] 0.09 [0.03, 0.15]
Monetary Policy
φπ, interest rate resp. to inflation 1.14 [0.98, 1.31] 0.15 [0.08, 0.23]
φy, interest rate resp. to output 0.18 [0.13, 0.22] 0.17 [0.14, 0.20]
ρr, lagged interest rate resp. 0.76 [0.71, 0.81] 0.15 [0.06, 0.23]
Fiscal Policy
γG, govt cons. resp. to debt 0.21 [0.13, 0.30] 0.0000 [-0.0017, 0.0016]
γZ, transfer resp. to debt -0.03 [-0.13, 0.08] 0.0000 [-0.0017, 0.0016]
ρG, lagged govt cons resp. 0.98 [0.98, 0.99] 0.98 [0.98, 0.99]
ρZ, lagged transfer resp. n.e. 0.98 [0.97, 0.99]
Table 5: Posterior distributions for estimated parameters: means and 90-percent credible sets; n.e.
denotes not estimated
Log Marginal Data Density
1955q1 1955q1 1955q1 1982q1
–2014q2 –2007q4 –1979q4 –2007q4
Regime M -2557 -2211 -1121 -957
Regime F -2549 -2222 -1125 -969
Table 6: Log marginal data densities
20

Leeper, Traum & Walker: Fiscal Multiplier Morass
periods. These multipliers are larger over the full sample, which includes the financial crisis, than
over shorter samples.17 This pattern carries over to consumption multipliers: positive on impact
and the first few years in all samples, but zero or even negative after ten years in all cases but the
full sample. Higher government spending unambiguously crowds out private investment in regime
M: at all horizons and in all sub-periods, the 90-percent credible sets for investment multipliers are
strongly negative even though the prior predictive places some probability on positive investment
multipliers.
Output Multiplier: PV ∆Y
∆G
Impact 4 qtrs 10 qtrs 25 qtrs 10 years
Prior 0.80 0.67 0.55 0.48 0.46
[-0.57,2.12] [-0.46,1.77] [-0.41,1.49] [-0.39,1.46] [-0.42,1.49]
Posterior
1955q1-2014q2 1.36 1.16 0.90 0.69 0.70
[1.17,1.55] [0.99,1.34] [0.71,1.07] [0.48,0.91] [0.45,0.94]
1955q1-2007q4 1.21 0.93 0.57 0.30 0.24
[1.04,1.40] [0.78,1.09] [0.45,0.71] [0.17,0.41] [0.10,0.38]
1955q1-1979q4 1.41 1.06 0.65 0.40 0.34
[1.15,1.68] [0.83,1.27] [0.48,0.81] [0.27,0.51] [0.22,0.46]
1982q1-2007q4 1.25 1.01 0.67 0.37 0.31
[1.02,1.47] [0.80,1.20] [0.48,0.84] [0.19,0.55] [0.09,0.52]
Consumption Multiplier: P V ∆C
∆G
Impact 4 qtrs 10 qtrs 25 qtrs 10 years
Prior -0.22 -0.30 -0.33 -0.37 -0.41
[-1.51,1.04] [-1.47,0.81] [-1.43,0.72] [-1.40,0.68] [-1.43,0.62]
Posterior
1955q1-2014q2 0.23 0.23 0.23 0.22 0.23
[0.07,0.41] [0.07,0.41] [0.07,0.40] [0.05,0.39] [0.05,0.41]
1955q1-2007q4 0.17 0.15 0.11 0.05 -0.01
[-0.00,0.34] [-0.02,0.32] [-0.05,0.28] [-0.12,0.21] [-0.19,0.18]
1955q1-1979q4 0.45 0.36 0.22 0.02 -0.11
[0.20,0.71] [0.12,0.60] [-0.00,0.44] [-0.19,0.23] [-0.31,0.09]
1982q1-2007q4 0.19 0.14 0.04 -0.10 -0.20
[-0.01,0.41] [-0.08,0.35] [-0.20,0.28] [-0.36,0.15] [-0.45,0.05]
Investment Multiplier: P V ∆I
∆G
Impact 4 qtrs 10 qtrs 25 qtrs 10 years
Prior -0.06 -0.11 -0.19 -0.25 -0.26
[-0.17,0.05] [-0.31,0.07] [-0.53,0.13] [-0.72,0.18] [-0.76,0.20]
Posterior
1955q1-2014q2 -0.15 -0.31 -0.56 -0.91 -1.12
[-0.19,-0.10] [-0.40,-0.22] [-0.72,-0.42] [-1.16,-0.68] [-1.43,-0.80]
1955q1-2007q4 -0.20 -0.42 -0.73 -1.10 -1.33
[-0.25,-0.15] [-0.51,-0.33] [-0.87,-0.58] [-1.33,-0.88] [-1.64,-1.03]
1955q1-1979q4 -0.31 -0.51 -0.76 -0.94 -1.02
[-0.42,-0.20] [-0.65,-0.37] [-0.94,-0.56] [-1.21,-0.67] [-1.35,-0.67]
1982q1-2007q4 -0.15 -0.30 -0.53 -0.76 -0.85
[-0.20,-0.10] [-0.41,-0.20] [-0.71,-0.34] [-1.07,-0.44] [-1.27,-0.44]
Table 7: Prior versus posterior multipliers for Regime M. 90% probability intervals in brackets.
17Ramey (2011) finds the output multiplier is between 0.8 and 1.5, close to our impact estimates in both regimes.
21

Leeper, Traum & Walker: Fiscal Multiplier Morass
Regime F multipliers appear in table 8. Unlike regime M, now the prior predictive suggests
that positive output multipliers are nearly certain over longer horizons. For the 1955–2014 sample,
mean estimates of output multipliers are more than double those in regime M; at 10-year horizons,
the 90-percent credible set in F is [1.66,2.08], whereas it is [0.45,0.94] in M. Starker differences
emerge in estimates from the shorter sub-periods, where the longer-run output multipliers range
from four to seven times larger in F. Consumption multipliers are comparable across the regimes,
but somewhat more likely to be positive in F. Investment impacts also display regime differences:
whereas those multipliers are strongly negative in M, there is significant probability mass on positive
investment impacts in F, particularly at longer horizons.
Figure 3displays the impacts—posterior means and 90-percent credible intervals—of an exoge-
nous increase in government spending in both regimes, estimated from 1955q1 to 2007q4. Mean
responses in regime M appear as dashed lines, while those in regime F are solid lines. Consump-
tion multipliers in both regimes are positive and about 0.2 for the first two years, before rising to
roughly 0.4 in F at longer horizons.18 Output and investment multipliers are substantially larger
in regime F than in regime M. Average impact output multipliers are similar across regimes—1.21
in M and 1.42 in F—but multiplier estimates diverge over longer horizons: after 20 years, well
below 1 in M and above 2 in F. Striking differences appear in the effects of government spending
on investment: in regime M, investment is strongly crowded out, while in regime F investment is
virtually unchanged. At the end of the horizon in the figure, the posterior mean of the present
value multiplier for investment is $1.92 lower in M and is 1 cent higher in F.
In neither regime do higher multipliers arise from lower real interest rates, a finding that differs
from existing literature. Monetary policy reduces the one-period real rate in regimes M and F
only on impact. Long-run real interest rates, which are the relative prices that directly affect
consumption decisions, are higher in both regimes, but about twice as high in M as in F.19 Higher
real rates are associated with both higher nominal rates and higher inflation rates, which rise
more in regime F than in M. In neither case, though, are these increases large—the mean inflation
increase in M after 20 years is about 8 basis points and 12 basis points in F. Long-run inflation,
however, does rise a fair amount in both regimes: a little over 1.5 percentage points in regime M
and about 2.2 percentage points in F.
Substantial differences across regimes appear in labor market responses. Real wages remain
unchanged in regime M, but rise strongly and persistently in F. Although short-run increases in
labor are similar in the two regimes, in regime M the increase is not sustained, while in F hours
worked remain high over the 20-year period in the figure. These differences, which section 4.4
dissects, highlight how the transmission mechanisms vary across regimes.
18These present-value multipliers mean that if government spending rises in present value by $1 over the 20-year
horizon, then the present value of consumption is 40 cents higher over that horizon.
19Long-run real rates are derived from combining the consumption Euler equation with the term structure relation
to define the long-run real rate, ˆrL
t, recursively as ˆrL
t=−ˆ
PB
t−Etˆπt+1 +βρ
eγEtˆrL
t+1 +ˆ
PB
t+1. Long-run inflation,
ˆπL
t, is defined as ˆπL
t=−ˆrL
t−ˆ
PB
t. Because long-run real interest and inflation rates are discounted sums over the
infinite future, they have a similar flavor to multipliers by reporting the discounted present value of rates.
22

Leeper, Traum & Walker: Fiscal Multiplier Morass
Output Multiplier: PV ∆Y
∆G
Impact 4 qtrs 10 qtrs 25 qtrs 10 years
Prior 1.34 1.22 1.16 1.25 1.30
[-0.13,2.75] [0.03,2.38] [0.23,2.12] [0.28,2.15] [0.30,2.24]
Posterior
1955q1-2014q2 1.51 1.53 1.58 1.73 1.87
[1.31,1.69] [1.35,1.70] [1.40,1.77] [1.52,1.94] [1.66,2.08]
1955q1-2007q4 1.42 1.39 1.40 1.52 1.66
[1.22,1.61] [1.22,1.57] [1.21,1.56] [1.31,1.70] [1.46,1.86]
1955q1-1979q4 1.42 1.26 1.12 1.18 1.34
[1.15,1.70] [1.02,1.50] [0.91,1.34] [0.96,1.37] [1.14,1.55]
1982q1-2007q4 1.24 1.20 1.18 1.28 1.43
[0.99,1.49] [0.96,1.43] [0.95,1.40] [1.05,1.50] [1.22,1.65]
Consumption Multiplier: P V ∆C
∆G
Impact 4 qtrs 10 qtrs 25 qtrs 10 years
Prior 0.17 0.08 0.04 0.09 0.12
[-1.14,1.52] [-1.08,1.23] [-0.99,1.06] [-0.90,0.99] [-0.83,1.00]
Posterior
1955q1-2014q2 0.20 0.21 0.22 0.26 0.31
[0.03,0.38] [0.04,0.38] [0.06,0.39] [0.12,0.43] [0.15,0.46]
1955q1-2007q4 0.16 0.16 0.17 0.20 0.24
[-0.02,0.34] [-0.01,0.34] [0.00,0.34] [0.04,0.36] [0.08,0.40]
1955q1-1979q4 0.34 0.31 0.25 0.17 0.14
[0.08,0.62] [0.05,0.57] [0.00,0.49] [-0.05,0.40] [-0.07,0.36]
1982q1-2007q4 0.05 0.02 -0.02 -0.04 0.00
[-0.18,0.27] [-0.20,0.23] [-0.21,0.18] [-0.22,0.15] [-0.18,0.19]
Investment Multiplier: P V ∆I
∆G
Impact 4 qtrs 10 qtrs 25 qtrs 10 years
Prior 0.01 0.00 -0.01 0.02 0.04
[-0.11,0.11] [-0.20,0.19] [-0.37,0.34] [-0.46,0.51] [-0.46,0.52]
Posterior
1955q1-2014q2 -0.01 -0.01 -0.01 0.03 0.08
[-0.06,0.04] [-0.11,0.08] [-0.17,0.14] [-0.20,0.25] [-0.18,0.33]
1955q1-2007q4 -0.04 -0.08 -0.12 -0.13 -0.10
[-0.09,0.01] [-0.18,0.02] [-0.28,0.03] [-0.35,0.09] [-0.36,0.15]
1955q1-1979q4 -0.18 -0.29 -0.40 -0.39 -0.30
[-0.28,-0.07] [-0.45,-0.13] [-0.62,-0.18] [-0.69,-0.09] [-0.63,0.04]
1982q1-2007q4 -0.02 -0.04 -0.04 0.02 0.10
[-0.07,0.03] [-0.13,0.06] [-0.20,0.11] [-0.19,0.26] [-0.13,0.35]
Table 8: Prior versus posterior multipliers for Regime F. 90% probability intervals in brackets.
23

Leeper, Traum & Walker: Fiscal Multiplier Morass
10 20 30 40 50 60 70 80
0
0.5
1
1.5
2
Output Multiplier
10 20 30 40 50 60 70 80
−0.2
0
0.2
0.4
Consumption Multiplier
10 20 30 40 50 60 70 80
−2
−1
0
Investment Multiplier
10 20 30 40 50 60 70 80
5
10
15 Nominal Interest Rate
10 20 30 40 50 60 70 80
4
6
8
10
12
14
Inflation
10 20 30 40 50 60 70 80
−2
0
2
4
Real Interest Rate
10 20 30 40 50 60 70 80
0
2
4
6Mkt Value Debt/Output
10 20 30 40 50 60 70 80
100
150
200
250
Long−Run Inflation
10 20 30 40 50 60 70 80
10
20
30
40
Long−Run Real Rate
10 20 30 40 50 60 70 80
0
0.05
0.1
0.15
Real Wage
10 20 30 40 50 60 70 80
0.05
0.1
0.15
Labor
10 20 30 40 50 60 70 80
−8
−6
−4
−2
0
2Primary Surplus
Figure 3: Responses over 80 quarters to a government spending increase in estimated regime M
(dashed lines) and regime F (solid lines) over period 1955q1–2007q4. Responses displayed are for the
posterior mean parameters and the 90-percent impulse response credible intervals. Top panels are
present value multipliers for output, consumption and investment, interest rates and inflation rates
are converted to annualized basis points, and the remaining variables are in percentage deviations
from steady state.
In both regimes, the fiscal expansion initially lowers the market value of debt as a share of
output because output rises and bond prices fall. The very different sources of fiscal financing
in the two regimes appear in figure 4, which reports fiscal responses over 1000 periods to reveal
the model’s low-frequency dynamics. The return to steady state is extremely slow. In regime M
estimates, fiscal policy raises transfers and reduces government purchases in response to higher
debt, while in regime F those responses are muted. Both regimes have tax revenues rise passively
as capital, labor and consumption tax bases increase. A critical difference in financing comes from
the spending reversals that regime M triggers. These reversals raise surpluses and the value of
debt more than in F, but then cause debt to overshoot the steady state to generate low-frequency
damped oscillations around steady state. Oscillating government spending produces oscillations
in other variables that are absent from regime F, where long-run convergence to steady state is
monotonic.
We now dig more deeply into the transmission mechanisms in the two regimes to better under-
stand the differences that appear in figure 3: multipliers are larger in regime F than in M, especially
over the long run; variables are substantially more persistent in F than in M.
4.2 Transmission Mechanism in Regime M
Regime M combines active monetary policy with passive fiscal policy. Estimates of fiscal behavior,
however, differ somewhat from canonical new Keynesian models that assume lump-sum taxes are
24

Leeper, Traum & Walker: Fiscal Multiplier Morass
200 400 600 800 1000
(a) Market Value of Debt / Output
0
1
2
3
4
5
200 400 600 800 1000
(b) Primary Surplus
-6
-4
-2
0
2
4
6
Figure 4: Responses of debt and primary surplus to a government spending increase in estimated
regime M (dashed lines) and regime F (solid lines) over period 1955q1–2007q4. Responses displayed
are for the posterior mean parameters and variables are in percentage deviations from steady state.
the passive instrument that stabilizes debt. In the 1955q1–2007q4 estimates, government spending
is the stabilizing instrument: estimates have spending fall as the debt-output ratio rises, while
lump-sum transfers systematically are unresponsive (90-percent bands for γZencompass zero).
The estimated spending reversals, in Corsetti et al.’s (2012) terminology, play a key role in regime
M’s transmission mechanism.
4.2.1 Important Parameters To shed light on the transmission mechanisms that under-
lie the estimated multipliers, we calculate a measure of root mean square deviation (RMSD)
for each parameter. For each draw of the posterior parameters, ˜
θ= [˜
θ1... ˜
θn]′from the
posterior distribution p(θ), we calculate multipliers ˜ω(˜
θ). Denote the new parameter vector by
˜
θi= [˜
θ1... E[θi]... ˜
θn]′, where E[θi] fixes the ith parameter at its posterior mean, and calcu-
late the multipliers, ˜ωi(˜
θi). Repeat this for each i= 1,2,...n. The RMSD is the root mean square
deviation between the two multipliers ˜ω(˜
θ) and ˜ωi(˜
θi): it measures how much the multiplier varies
on average due to parameter i. The RMSD is largest for the parameters that are most influential
for the multiplier.
A table in the appendix reports RMSD results for output and consumption present-value mul-
tipliers on impact and after 25 quarters following a government spending shock. Most of the
parameters that RMSD calculations identify as important for output and consumption multipliers
fall into three categories: preferences (αG, the coefficient on government consumption in utility; θ,
the degree of habit formation), nominal rigidities (ωp, the Calvo parameter for price setting; ωw,
the Calvo parameter for wage setting; χp, the degree of inflation indexation), and policy parameters
(φπand φy, the responses of monetary policy to inflation and output; γZand γG, the responses
of transfers and government spending to debt in regime M; ρGand ρeg , the persistence of the
government spending shock).
4.2.2 Counterfactuals RMSD calculations inform our counterfactual exercises. Table 9re-
ports posterior means and 90-percent credible sets for impact and 25-quarter multipliers under a
25

Leeper, Traum & Walker: Fiscal Multiplier Morass
variety of counterfactual parameter settings in regime M. For comparison, the table also displays
multipliers in the estimated model. For each counterfactual, we fix the parameters indicated in the
table, and let the remaining parameters vary over the posterior.20
The persistently positive consumption multipliers in regime M that figure 3depicts come from
a combination of two estimated parameters: the complementarity of government spending (αG<
0) and strong external habit formation (large θ). With complementarity, the initial increase in
government spending raises private consumption despite higher real interest rates. Strong habits
increase the desire for smooth consumption paths that rise only gradually over time, even as
government spending decays back to steady state. Reducing habit formation, θ= 0.8, or removing
government spending’s complementarity to private consumption, αG= 0, reduces output impacts
and shifts the estimated consumption multipliers from positive to negative or zero. These preference
parameters interact: it is the combination of the two counterfactuals that moves credible sets into
negative territory for consumption and reduces the negative impacts on investment, confirming the
source of persistent positive consumption impacts in the baseline estimates.
A higher capital utilization rate, ψ= 0.3, weakens the increase in utilization. For effective
capital to expand and boost production, the capital stock must decline less, tempering the strongly
negative investment multipliers in the baseline estimates. Reducing nominal rigidities by setting
ωp=ωw= 0.7 does not significantly alter the message of the estimates in regime M. Less rigid
prices and wages soften real interest rate increases to raise output multipliers and attenuate the
sharply negative investment multipliers. More hawkish monetary policy (φπ= 1.5) raises the real
interest rate and reduces private demand and the output multiplier, while a less aggressive response
to output (φy= 0.05) raises output responses and dampens investment.
The remaining counterfactuals that table 9reports have minor effects on multipliers in regime
M. Those counterfactuals include raising the speed at which transfers and spending adjust to
stabilize debt (γZ=γG= 0.5), making all government debt one period (ρ= 0), and setting to zero
steady-state tax rates as well as the response of transfers to debt (τK=τL=τC=γZ= 0), which
forces all fiscal adjustments to occur through future government spending changes. Among these,
eliminating steady-state taxes and transfer responses to debt (τK=τL=τC=γZ= 0) has the
largest effect. In this case, output multipliers are higher as the elimination of distortionary taxes
raises disposable income and demand.
Broader consequences of three counterfactuals appear in the posterior mean responses in figure 5.
Baseline estimates (solid lines) and three sets of counterfactuals appear in the figure. Intervening on
preferences to eliminate government spending’s estimated complementarity to consumption and to
reduce the intensity of habits, converts the baseline positive consumption multipliers into strongly
negative multipliers over the full 20-year horizon (dashed lines). This also reduces the output
multiplier and, by crowding out investment less, softens the decline in investment. Strong habits and
complementary of government spending are essential to deliver the sustained positive consumption
multipliers in the baseline estimates.
20Results come from the same set of 20,000 draws in all cases. We discard draws that lead to indeterminacy.
26

Leeper, Traum & Walker: Fiscal Multiplier Morass
Counterfactuals in Regime M
Posterior (Impact) Posterior (25 qtrs)
P V ∆Y
∆GP V ∆C
∆GP V ∆I
∆GP V ∆Y
∆GP V ∆C
∆GP V ∆I
∆G
Estimated Model 1.21 0.17 -0.20 0.30 0.05 -1.10
[1.04,1.40] [-0.00,0.34] [-0.25,-0.15] [0.17,0.41] [-0.12,0.21] [-1.33,-0.88]
C1: θ= 0.8 & αG= 0 0.93 -0.15 -0.11 0.18 -0.45 -0.55
[0.88,0.98] [-0.19,-0.11] [-0.13,-0.08] [0.11,0.25] [-0.57,-0.33] [-0.71,-0.41]
A: θ= 0.8 1.11 0.01 -0.13 0.21 -0.34 -0.66
[0.94,1.27] [-0.14,0.15] [-0.16,-0.09] [0.12,0.29] [-0.51,-0.18] [-0.85,-0.46]
B: αG= 0 1.02 -0.01 -0.17 0.25 -0.12 -0.93
[0.98,1.07] [-0.02,-0.00] [-0.20,-0.14] [0.16,0.35] [-0.20,-0.04] [-1.07,-0.78]
C2: ψ= 0.3 1.19 0.17 -0.18 0.33 0.00 -0.93
[1.02,1.37] [0.00,0.34] [-0.23,-0.13] [0.17,0.48] [-0.16,0.18] [-1.14,-0.71]
C3: ωp=ωw= 0.7 1.26 0.17 -0.17 0.44 0.03 -0.91
[1.05,1.46] [-0.00,0.34] [-0.25,-0.09] [0.25,0.62] [-0.13,0.19] [-1.18,-0.69]
C4a: φπ= 1.5 1.18 0.17 -0.22 0.19 0.02 -1.17
[1.01,1.37] [-0.00,0.33] [-0.27,-0.17] [0.07,0.31] [-0.15,0.18] [-1.43,-0.90]
C4b: φy= 0.05 1.33 0.18 -0.12 0.54 0.08 -0.88
[1.13,1.52] [0.01,0.35] [-0.16,-0.08] [0.33,0.76] [-0.09,0.25] [-1.13,-0.64]
C5: γG=γZ= 0.5 1.20 0.17 -0.22 0.34 0.04 -1.05
[1.02,1.38] [-0.00,0.34] [-0.26,-0.17] [0.26,0.42] [-0.11,0.20] [-1.23,-0.87]
C6: ρ= 0 1.22 0.17 -0.20 0.30 0.06 -1.12
[1.04,1.40] [0.01,0.35] [-0.25,-0.15] [0.17,0.42] [-0.11,0.23] [-1.36,-0.90]
C7: τK=τL=τC= 0, γZ= 0 1.27 0.17 -0.23 0.28 0.08 -1.18
[1.08,1.46] [0.00,0.34] [-0.28,-0.18] [0.18,0.38] [-0.09,0.24] [-1.40,-0.97]
Table 9: Counterfactual multipliers for regime M estimated over 1955q1–2007q4. Posterior means
and 90-percent credible intervals (in brackets).
More aggressive monetary policy raises real interest rates after the fiscal expansion, tempering
the increase in inflation for several years (dotted-dashed lines). Sharply higher real rates reduce
consumption multipliers relative to baseline estimates and lower investment. A reduced capital
stock, together with substantially lower wages and short-lived labor increases, drive the present-
value output multiplier to near zero at longer horizons.
The last counterfactual makes regime M Ricardian in the sense that only non-distorting trans-
fers respond to debt; spending reversals that arise through the rule for government purchases are
eliminated (dotted lines).21 Ricardian fiscal financing coincides with analyses in Christiano et al.
(2011) and Cogan et al. (2010) and other multiplier studies. It produces smaller consumption mul-
tipliers that turn negative twice as fast as the baseline estimates. These effects stem from higher
real interest rates and much lower wages.
21“Ricardian” refers only to the sources of financing that respond to the state of government debt. Some revenue
is raised through (constant) steady-state tax rates on capital, labor and consumption.
27

Leeper, Traum & Walker: Fiscal Multiplier Morass
20 40 60 80
0
0.5
1
Output Multiplier
20 40 60 80
-0.4
-0.2
0
Consumption Multiplier
20 40 60 80
-2
-1.5
-1
-0.5
Investment Multiplier
20 40 60 80
0
2
4
6
8
Nominal Interest Rate
20 40 60 80
0
2
4
6
8
Inflation
20 40 60 80
0
2
4
Real Interest Rate
20 40 60 80
0
2
4
Mkt Value Debt
20 40 60 80
0
50
100
150
Long-Run Inflation
20 40 60 80
10
20
30
40
Long-Run Real Rate
20 40 60 80
-0.03
-0.02
-0.01
Real Wage
20 40 60 80
0.02
0.04
0.06
0.08
0.1
0.12 Labor
20 40 60 80
-6
-4
-2
0
Primary Surplus
Figure 5: Counterfactual posterior mean responses to a government spending increase in estimated
regime M, 1955q1–2007q4. Baseline estimates (solid lines); lower habits, θ= 0.8 and no government
spending in utility, αG= 0 (dashed lines); more aggressive monetary policy, φπ= 1.5 and φy= 0.2
(dotted-dashed lines); Ricardian model, γG= 0, γZ= 0.2 (dotted lines). Top panels are present
value multipliers for output, consumption and investment, interest rates and inflation rates are
converted to annualized basis points, and the remaining variables are in percentage deviations from
steady state.
4.3 Transmission Mechanism in Regime F
Regime F couples passive monetary policy with active fiscal policy, a policy mix that breaks Ricar-
dian equivalence. Debt-financed government spending does not trigger expectations of sufficiently
high surpluses to stabilize debt. Instead, changes in bond prices and the price level ensure that
the market value of debt is aligned with the expected present value of surpluses. Unlike simple
expositions of this policy regime, the estimated model includes constant tax rates levied against
capital and labor income and consumption, so a fiscal expansion does generate expectations of
somewhat higher surpluses; those surpluses, though, cannot stabilize debt.
4.3.1 Important Parameters RMSD calculations imply that important parameters in regime
F include price and wage stickiness (ωpand ωw), preferences over government spending and habits
(αGand θ), monetary policy reactions to inflation and output (φπand φy), and the persistence of
government spending (ρG). Missing from the RMSD analysis is whether changes in steady-state
variables matter for multipliers. As table 9’s C6 and C7 counterfactuals suggest, steady-state
changes in average maturity or tax rates have small effects in regime M. This is not the case in
regime F.
28

Leeper, Traum & Walker: Fiscal Multiplier Morass
Counterfactuals in Regime F
Posterior (Impact) Posterior (25 qtrs)
P V ∆Y
∆GP V ∆C
∆GP V ∆I
∆GP V ∆Y
∆GP V ∆C
∆GP V ∆I
∆G
Estimated Model 1.42 0.16 -0.04 1.52 0.20 -0.13
[1.22,1.61] [-0.02,0.34] [-0.09,0.01] [1.31,1.70] [0.04,0.36] [-0.35,0.09]
C1: θ= 0.8 & αG= 0 1.26 0.01 -0.02 1.44 0.11 -0.07
[1.19,1.33] [-0.03,0.05] [-0.05,0.01] [1.25,1.65] [-0.05,0.27] [-0.24,0.09]
A: θ= 0.8 1.39 0.14 -0.04 1.46 0.18 -0.13
[1.22,1.57] [-0.02,0.29] [-0.08,0.00] [1.26,1.66] [0.00,0.36] [-0.33,0.05]
B: αG= 0 1.27 0.01 -0.01 1.48 0.07 0.00
[1.21,1.32] [-0.00,0.01] [-0.05,0.03] [1.29,1.67] [0.02,0.13] [-0.16,0.16]
C2: ψ= 0.3 1.40 0.16 -0.01 1.58 0.18 0.06
[1.21,1.59] [-0.02,0.33] [-0.06,0.04] [1.38,1.77] [0.02,0.34] [-0.13,0.24]
C3: ωp=ωw= 0.7 2.40 0.21 0.68 3.53 0.41 1.95
[2.02,2.75] [0.02,0.38] [0.45,0.91] [2.93,4.11] [0.21,0.59] [1.34,2.50]
C4a: φπ= 0.225, ρr= 0.71 1.48 0.17 0.00 1.52 0.20 -0.12
[1.29,1.67] [-0.01,0.35] [-0.05,0.06] [1.32,1.71] [0.05,0.37] [-0.34,0.11]
C4b: φy= 0.0 2.65 0.21 0.85 6.21 0.62 4.27
[2.25,3.05] [0.03,0.39] [0.61,1.09] [5.13,7.22] [0.40,0.87] [3.23,5.27]
C6: ρ= 0 1.48 0.17 -0.01 1.76 0.26 0.03
[1.28,1.67] [-0.01,0.35] [-0.06,0.05] [1.53,1.98] [0.10,0.43] [-0.20,0.27]
C7: τK=τL=τC= 0 2.96 0.27 0.82 7.38 1.18 3.43
[2.44,3.45] [0.08,0.45] [0.55,1.10] [5.65,9.02] [0.75,1.59] [2.38,4.45]
Table 10: Counterfactual multipliers for regime F estimated over 1955q1–2007q4. Posterior means
and 90-percent credible intervals (in brackets).
4.3.2 Counterfactuals Table 10 repeats for regime F many of the counterfactuals conducted
in table 9for regime M. Reducing habit intensity (θ= 0.8) and removing government spending’s
complementarity (αG= 0) have much less effect on consumption multipliers in F than in regime
M. The impact multiplier falls from 0.16 to zero, but the 25-quarter multiplier continues to be
positive. Merely setting αG= 0 still reduces the impact consumption multiplier, but it raises the
longer-term investment multiplier from being negative to zero.
Multipliers are uniformly higher when prices and wages are more flexible, an outcome that is no
mystery in regime F. Less stickiness permits inflation to rise more after the increase in government
spending which, when monetary policy is passive, reduces real interest rates in the short run. Lower
real rates raise consumption and investment demand at the same time that they raise supply of
labor.
Monetary policy’s reaction to output, φy, becomes quite powerful when nominal rigidities are
strong, as in the baseline estimates. Making monetary policy unresponsive to output (φy= 0) raises
impact multipliers for output and investment, but the largest effects occur at longer horizons: the
output multiplier exceeds six and the investment multiplier is over four at 25 quarters.
Very large multipliers arise when steady-state tax rates are zero. Impact multipliers for output
and investment rise substantially, but the biggest increases appear in longer-run multipliers, which
29

Leeper, Traum & Walker: Fiscal Multiplier Morass
are several times larger than in baseline estimates. Eliminating steady-state taxes as a source of
revenue produces very large wealth effects in regime F: an expansion in debt is now completely
unbacked by changes in future surpluses, sharply increasing current and future demand. This
counterfactual brings the model closest to the canonical fiscal theory setting with exogenous primary
surpluses, which Dupor and Li (2015) study. It also explains the source of a critical misperception
in the received wisdom that regime F policies necessarily generate high and volatile inflation.
Figure 6reports dynamic impacts of a public spending expansion on the baseline posterior mean
estimates (solid lines) and three counterfactuals. Reducing nominal rigidities (dashed lines) makes
the real interest rate the dominant force in the transmission mechanism because inflation, which
rises dramatically on impact, is transformed into sharply lower real rates by the passive monetary
policy in regime F (dashed lines). Lower real rates trigger the typical reactions: households raise
consumption and investment demand, firms with sticky prices increase labor demand to satisfy
production, increasing equilibrium labor. All three multiplier measures are significantly higher
than in the baseline estimates. Enhanced price and wage flexibility raises the slopes of the inflation
and wage Phillips curves to transmit increased real activity into still higher inflation and wages,
with larger adjustments in both variables in the short run. Bond prices drop precipitously and drive
the market value of debt-output ratio below steady state over the 20-year horizon in the figure.
These large initial impacts are more fleeting because diminished stickiness reduces persistence in
many responses, most notably nominal interest rates, inflation and labor-market variables.
Dramatic effects come from intervening on preference parameters to reduce habit intensity and
eliminate government spending’s complementarity, while also reducing the persistence of govern-
ment spending (dotted-dashed lines). Without complementarity, the short-run consumption mul-
tiplier can turn negative as real interest rates rise modestly, while weaker habits permit long-run
consumption multipliers to rise to the baseline’s levels, even as the government spending injection
dissipates. Output and investment multipliers also fall below their baseline levels. With a less
sustained increase in demand due to reduced serial correlation in government purchases, labor and
wages rise only tepidly.
The third counterfactual combines the first two to show that complementarity of government
and private consumption and extremely persistent government spending—which emerge from the
baseline estimates—are not necessary to generate persistently positive consumption multipliers in
regime F (dotted lines). This scenario generates small, transitory increases in inflation, nominal
interest rates and labor, yet multipliers that are larger than in the baseline.
Persistence in regime F—in contrast to regime M—comes in large part from slowly evolving
government debt. Analytical models of the two regimes make clear that government debt is an
important state variable in F, but disappears in equilibrium in purely Ricardian versions of M.
This role of debt is difficult to glean from the counterfactuals in figure 6, so we now turn to
interventions on features of the steady state that have a direct bearing on the state of government
debt.
Figure 7reports the baseline responses (solid lines) and responses from interventions on three
30

Leeper, Traum & Walker: Fiscal Multiplier Morass
20 40 60 80
1
2
3
Output Multiplier
20 40 60 80
0
0.2
0.4
0.6
Consumption Multiplier
20 40 60 80
0
0.5
1
1.5
Investment Multiplier
20 40 60 80
10
20
30
Nominal Interest Rate
20 40 60 80
10
20
30
40
50
Inflation
20 40 60 80
-30
-20
-10
0
Real Interest Rate
20 40 60 80
-1
0
1
Mkt Value Debt
20 40 60 80
200
400
600
Long-Run Inflation
20 40 60 80
-100
-50
0
Long-Run Real Rate
20 40 60 80
0.02
0.04
0.06
0.08
0.1
0.12
Real Wage
20 40 60 80
0
0.2
0.4 Labor
20 40 60 80
-6
-4
-2
0
Primary Surplus
Figure 6: Counterfactual posterior mean responses to a government spending increase in estimated
Regime F, 1955q1–2007q4. Baseline estimates (solid lines); lower nominal rigidities, ωp=ωw= 0.7
(dashed lines); lower habits, θ= 0.8, no government spending in utility, αG= 0, and less persistent
spending shock, ρG= 0.9 (dotted-dashed lines); lower nominal rigidities, ωp=ωw= 0.7, lower
habits, θ= 0.8, no government spending in utility, αG= 0, and less persistent spending shock,
ρG= 0.9 (dotted lines). Top panels are present value multipliers for output, consumption and
investment, interest rates and inflation rates are converted to annualized basis points, and the
remaining variables are in percentage deviations from steady state.
aspects of the steady state that directly impact government debt. As tables 9and 10 show, making
all debt only one period or eliminating steady-state taxes on capital, labor and consumption has
minor effects on regime M multipliers, but substantial impacts on regime F multipliers. Eliminating
longer-term debt by setting ρ= 0 prevents bond prices from absorbing the higher government
spending and brings more inflation into the present (dashed lines). One-period inflation rates are
a bit higher than baseline, but long-term inflation and higher long-term real interest rates all but
disappear. Because all debt revaluations must occur through contemporaneous inflation, the market
value of debt is uniformly higher, increasing wealth effects from higher spending and shifting up
demand for consumption. This higher demand induces firms to demand more labor and produce
more goods, driving up wages.
Raising the annualized steady-state debt-output ratio from the baseline of 36.8 percent to 150
percent reduces all the multipliers (dotted-dashed lines), an outcome that at first blush might seem
counterintuitive. But a larger stock of debt presents a larger “nominal tax base” against which
surprise inflation and bond prices operate. With more nominal debt outstanding, the market value
of debt can adjust to a given reduction in the present value of surpluses with a smaller decline in
bond prices and a smaller jump in inflation. Bond prices fall and inflation rises by less than in
the baseline estimates.22 One-period and long-term real interest rates rise throughout the horizon
22Additional evidence that revaluation through bond prices is crucial comes from noticing that if all debt is one
31

Leeper, Traum & Walker: Fiscal Multiplier Morass
20 40 60 80
1.5
2
2.5
Output Multiplier
20 40 60 80
0.2
0.4
0.6
Consumption Multiplier
20 40 60 80
-0.2
0
0.2
0.4
Investment Multiplier
20 40 60 80
10
12
14
16
18
Nominal Interest Rate
20 40 60 80
8
10
12
14
16
18
Inflation
20 40 60 80
-4
-2
0
2
Real Interest Rate
20 40 60 80
0
1
2
Mkt Value Debt
20 40 60 80
100
200
300
Long-Run Inflation
20 40 60 80
0
10
20
Long-Run Real Rate
20 40 60 80
0.05
0.1
0.15
Real Wage
20 40 60 80
0.08
0.1
0.12
0.14
0.16
0.18 Labor
20 40 60 80
-6
-4
-2
0
Primary Surplus
Figure 7: Counterfactual posterior mean responses to a government spending increase in estimated
Regime F, 1955q1–2007q4. Baseline estimates (solid lines); only one-period debt, ρ= 0 (dashed
lines); higher steady state debt-GDP ratio, sb= 150 percent (dotted-dashed lines); steady-state
tax rates reduced by 40 percent (dotted lines). Top panels are present value multipliers for output,
consumption and investment, interest rates and inflation rates are converted to annualized basis
points, and the remaining variables are in percentage deviations from steady state.
in the figure to dampen demand, wages and employment. Mean investment multipliers shift from
being mildly positive to mildly negative.
Table 10 shows that eliminating steady-state tax rates significantly raises multipliers. Figure 7
reduces capital, labor and consumption tax rates by 40 percent (dotted lines). All else equal, lower
steady-state tax rates reduce expected future endogenous revenues, raise wealth, and increase con-
sumption demand, labor demand, hours worked and output, all of which would increase multipliers
dramatically. But more robust economic activity increases the tax bases against which the tax
rates apply to generate revenue that partially offsets the lower tax rates and weaken wealth effects
and demand. This attenuates but does not eliminate the boost to multipliers.
4.4 Labor Market Behavior in the Two Regimes
Both the baseline estimates in figure 3and the counterfactuals make clear that an essential difference
in the transmission of government spending in the two policy regimes stems from labor market
behavior: labor responses are more persistent in F, and real wages rise sharply in regime F but
remain flat in M. We now explore that aspect of the transmission mechanism in detail.
In the baseline model with wages that are both sticky and indexed to past wages, the real wage
period (ρ= 0), then higher steady-state debt is irrelevant for inflation, interest rates and real economic activity.
32

Leeper, Traum & Walker: Fiscal Multiplier Morass
satisfies23
ˆwt=1
1 + βˆwt−1+β
1 + βEtˆwt+1 −κwˆ̟t+χw
1 + βˆπt−1
−1 + βχw
1 + βˆπt+β
1 + βEtˆπt+1 +χw
1 + βˆua
t−1−1 + βχw−ρaβ
1 + βˆua
t
(4)
where ˆ̟tdenotes deviations of the economy’s average wage markup, defined as
ˆ̟t≡ˆwt−hξˆ
Lt+ ˆub
t−ˆ
λti+κ−1
wˆuw
t(5)
ˆ
λtdenotes the marginal utility of wealth, ˆ
Ltis labor, ˆub
tis the preference shock and ˆuw
tis the wage
markup shock.24 When shocks to the economy cause the wage markup to be below its desired
level, households increase their nominal wages. With sticky prices, this can raise the real wage, as
equation (4) suggests. When nominal wages are flexible—ωw= 0—the wage markup is constant,
ˆ̟t= 0, and the real wage equals the marginal rate of substitution between consumption and
leisure, given by the terms in brackets in equation (5).25 In this case, sticky prices can produce
higher real wages after government demand rises: some firms hire more labor to raise production,
which drives up real wages.26 When nominal wages are sticky, labor supply is forced to adjust more
to increases in labor demand.
Figure 8explores the wage mechanisms triggered by higher government spending by using
several counterfactual experiments in regimes M (top row) and F (bottom row). For comparability
across the policy specifications, we calibrate both models to the posterior mean estimates from
regime F estimates over 1955q1–2007q4 (solid lines). To produce regime M, we use this calibration
but replace φπand γGwith their posterior mean estimates in M (φπ= 1.14 and γG= 0.21), so
differences across rows in the figure stem from distinct policy behavior. To this reference case that
employs parameters estimated in regime F (solid lines), we add two counterfactuals: lower habit
formation and no complementarity of government spending, θ= 0.8 and αG= 0 (dotted-dashed
lines) and a smaller degree of price rigidity, ωp= 0.7 (dashed lines).
In regime M, the real wage falls on impact, despite the initial decline in the wage markup,
as the first row of the figure depicts. High estimates of nominal rigidities reduce the real wage’s
responsiveness to current wage markups, given by κwin equation (4). Instead, expected higher
future wage markups drive the real wage down on impact. Higher labor demand raises the marginal
rate of substitution, which reduces wage markups initially. But the marginal utility of wealth also
23The Calvo parameter that determines wage stickiness, ωw, is embedded in κw≡[(1 −βωw) (1 −
ωw)]/[ωw(1 + β)1 + (1+ηw)ξ
ηw].
24See the online appendix for derivations of these expressions.
25Under flexible prices, this yields the standard RBC model’s labor demand equation, modified to include
government spending in the utility function. Further restricting αG= 0 delivers the familiar form: ˆwt=
ξˆ
Lt+eγ
eγ−θˆct−θ
eγˆct−1.
26This result differs from RBC models where real wages decrease with the increase in work effort [Monacelli and
Perotti (2008)]. Several empirical studies, starting with Rotemberg and Woodford (1992), find that an increase in
government spending raises real wages and labor [Fatas and Mihov (2001), Gal´ı et al. (2007), and Pappa (2009)].
33

Leeper, Traum & Walker: Fiscal Multiplier Morass
rises to more than offset the effect on the marginal rate of substitution over time: gradually wage
markups rise and labor declines.
When prices are more flexible (dashed lines), current inflation rises more, depressing the real
wage. At longer horizons, firms demand more labor relative to the reference case, given its much
lower cost. Counterfactuals on consumption preferences (dotted-dashed lines) lower consumption
demand, muting the responses of labor and the real wage relative to the reference case.
Regime F, which appears in the second row of figure 8, produces much stronger negative wage
markups. Positive wealth effects from a higher market value of government debt encourage con-
sumption in regime F, and as more goods are demanded, labor demand expands. Increases in
consumption and labor both increase the marginal rate of substitution. Larger, sustained devia-
tions of wage markups from their desired level lead households to raise their nominal wages, causing
real wages to rise. Consumption remains positive even without government spending complemen-
tarity because in regime F positive wealth effects from government debt increase consumption and
investment demand (dotted-dashed lines). Higher goods and labor demand confirm the importance
of positive wealth effects for wage markups in regime F.
Despite these large deviations in markups, it is possible for the real wage to decline on impact
in regime F, as the counterfactual with less price stickiness shows (dashed lines). Larger initial
increases in prices lower the real wage, but strong negative markups make this effect short-lived.
Higher inflation devalues a larger share of government debt, and with more flexible prices, the real
interest rates falls; both effects fuel consumption and further depress wage markups, leading the
real wage to increase over time.
Returning to our baseline posterior estimates in the two regimes that appear in figure 3, dif-
ferences in the effects of government spending on real wages lie at the heart of the differences in
multiplier estimates. Regime F produces larger declines in wage markups, which lead to higher
nominal and real wage adjustments. Coupled with the slightly higher estimated degree of price
stickiness in regime F, which further raises real wages on impact, it is not surprising that real
wages are far more expansive in regime F.
4.5 Fiscal Financing in the Two Regimes
Further insights into the observed differences in multipliers come from accounting for the financing
of government spending increases, which differs across policy regimes and sample periods. Letting
ˆrB
t≡(βρ/eγ)ˆ
PB
t−ˆ
PB
t−1−ˆπtdenote the ex-post real return on government bonds and ˆ
Stbe the
primary surplus, the intertemporal equilibrium condition is27
ˆ
bt−1=−βρ
eγˆ
PB
t+ˆ
PB
t−1+ ˆπt+ (1 −β)Et
∞
X
j=0
βjˆ
St+j−Et
∞
X
j=1
βjˆrB
t+j
27The primary surplus consists of the sum of revenues from capital, labor and consumption taxes less lump-sum
transfers and government purchases: ˆ
St=TK
SˆτK
t+ ˆrK
t+ˆ
Kt+TL
SˆτL
t+ ˆwt+ˆ
Lt+TC
SˆτC
t+ˆ
Ct−Z
Sˆ
Zt−G
Sˆ
Gt.
To focus solely on financing of government purchases, we set all disturbances to zero other than government spending.
34

Leeper, Traum & Walker: Fiscal Multiplier Morass
20 40 60 80
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
M: PV Output Multiplier
20 40 60 80
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
M: PV Consumption Multiplier
20 40 60 80
−0.03
−0.025
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
M: Real Wage
20 40 60 80
−0.14
−0.12
−0.1
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
M: Wage Markup
20 40 60 80
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
M: Labor
20 40 60 80
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
F: PV Output Multiplier
20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
F: PV Consumption Multiplier
20 40 60 80
0
0.02
0.04
0.06
0.08
0.1
0.12
F: Real Wage
20 40 60 80
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
F: Wage Markup
20 40 60 80
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
F: Labor
Figure 8: Counterfactual labor market responses to a government spending increase in regime M
(top row) and F (bottom row) calibrating both regime using regime F estimates over 1955–2007.
Regime M replaces monetary and fiscal parameters with their regime M estimates (φπ= 1.14
and γG= 0.21). Reference case (solid lines); reduced habits and eliminating complementarity of
government spending, θ= 0.8 and αG= 0 (dotted-dashed lines); reduced price rigidities, ωp= 0.7
(dashed lines).
The model’s rule for government spending at tincludes both the exogenous disturbance to
spending, uG
t, and an endogenous response of spending to the debt-output ratio, ˆsb
t−1,ˆ
Gt=
ρGˆ
Gt−1−γG(1 −ρG)ˆsb
t−1+uG
t.
We separate government spending into its exogenous, ˆ
Gx
t, and endogenous, ˆ
Ge
tcomponents,
so ˆ
Gt=ˆ
Gx
t+ˆ
Ge
t. Define the effect on the present value of surpluses of an exogenous change in
spending at tby ξt≡ −(1 −β)G
SEtP∞
j=0 βjˆ
Gx
t+j. We can now split the present value of primary
surpluses following a shock to government spending into exogenous and endogenous parts
(1 −β)Et
∞
X
j=0
βjˆ
St+j= (1 −β)Et
∞
X
j=0
βj[ˆ
Sx
t+j+ˆ
Se
t+j] = ξt+ (1 −β)Et
∞
X
j=0
βjˆ
Se
t+j
where ˆ
Se
tis the surplus exclusive of exogenous government spending, ˆ
Gx
t−ρGˆ
Gx
t−1+uG
t.
Combining these two expressions yields
ξt=ˆ
bt−1+βρ
eγˆ
PB
t−ˆ
PB
t−1−ˆπt−uG
t+Et
∞
X
j=1
βjˆrB
t+j−(1 −β)Et
∞
X
j=0
βjˆ
Se
t+j(6)
Table 11 reports the fraction of ξtaccounted for by each element in (6) dated tand later,
with endogenous surpluses broken into their component parts. The table includes posterior means
and 90-percent credible sets for the six estimated models, as well as mean predictions for select
35

Leeper, Traum & Walker: Fiscal Multiplier Morass
counterfactuals. Positive entries in the table mean that the component supports financing of higher
government spending, while negative entries counter financing.
In both regimes, the baseline estimates (1955–2007) imply that a drop in bond prices at the time
of the fiscal shock supports the financing of spending by reducing the market value of debt ( ˆ
PB
t
column). Lower ˆ
PB
talone accounts for 10 percent of the financing in regime M and over 17 percent
in F. By spreading inflation into the future, lower bond prices coincide with contemporaneous
inflation that accounts for less than 1 percent in each regime (ˆπtcolumn). Because the fiscal
expansion raises real interest rates in both M and F, the higher ex-post return on bonds counters
fiscal financing (P V (ˆrB) column), and counters more strongly in M, where real rates rise more.
Important financing differences between regimes emerge from the components of the primary
surplus. Higher tax revenues—the sum of columns P V (TK), P V (TL) and P V (TC)—provide nearly
5 percent of financing in regime M, but 84 percent in F. Under regime M policies, higher government
spending leaves wages unchanged, raises labor moderately, reduces the capital stock, and raises the
return to capital to produce offsetting effects that net out to a modest increase in tax revenues.
Regime F, in contrast, permits higher spending to raise wages and hours worked dramatically and,
if anything, increase the capital stock. This passive, but large, increase in tax revenues in regime F
moderates the wealth effects of fiscal expansions that would otherwise produce huge multipliers.28
Of course, if passive tax revenues are not stabilizing debt in regime M, then fiscal adjustments
must be occurring on the expenditures side. Posterior estimates of the response of transfers to debt
in M suggest that transfers actually rise with higher debt [see estimate of γZin table 5]. In the
accounting exercise, this policy implies that transfers counter the financing of a fiscal expansion.
Financing’s heavy lifting comes from endogenous government spending reversals that rise in present
value by over 105 percent to compensate for the contrary movement in transfers [P V (ˆ
Z) and
P V (ˆ
Ge) columns]. This underscores the centrality of government spending reversals in baseline
estimates of regime M. Regime F’s prior is tightly centered on no response of expenditures to debt,
so P V (Z) and P V (Ge) account for little of the financing.
This general pattern of financing continues in the whole sample inclusive of the recent financial
crisis, 1955–2014 and is largely robust across the 1955–1979 and 1982–2007 sub-periods. One
notable difference is that in the earlier sub-period contemporaneous bond prices play a bigger role
in financing in regime F, accounting for nearly a quarter of the expansion in spending. In the earlier
sub-period, both transfers and spending help to stabilize debt in regime M, so spending reversals
are less pronounced and consumption multipliers are smaller.
The counterfactual exercises whose dynamic impacts appear in figures 5through 7shift sources
of financing substantially. A Ricardian environment in regime M reduces the role of debt revaluation
through bond prices and current inflation to push nearly all financing into lower future lump-
sum transfers. Reduced habits and removing government spending from utility produces stronger
spending reversals, with P V (ˆ
Ge) = 108 percent. More aggressive monetary policy shifts more
28Tables 9and 10 show that eliminating these passive revenue adjustments by setting steady-state tax rates to zero
has little impact on multipliers in regime M, but raises them significantly in regime F.
36

Leeper, Traum & Walker: Fiscal Multiplier Morass
financing into declines in current bond prices and, by increasing ex-post real returns on debt, raises
debt service to produce a higher path for debt that generates still larger spending reversals. In
these last two cases, large ultimate declines in government purchases make consumption multipliers
higher than they would be in the absence of reversals.
More dramatic reshuffling of fiscal financing appears in regime F counterfactuals. Reduced
nominal stickiness enhances the role of debt revaluations to concentrate more than a third of
financing in current bond prices and inflation. Because less rigidity reduces real interest rates,
ex-post returns on bonds now support the financing of spending. But less expansive real wages and
less persistent labor increases conspire to make endogenous tax revenues less important, cutting
P V (ˆ
Se) in half.
Removing the maturity structure on government debt also removes any role for drops in bond
prices to support financing. This produces larger multipliers and pushes most financing into en-
dogenous tax revenues that account for 100 percent of the present-value increase in government
spending. Higher steady-state debt-output makes drops in bond prices and surprise inflation more
potent, accounting for 55.5 percent of financing. The lower associated multipliers reduce the fi-
nancing role of steady-state tax rates. Of course, the role of endogenous revenues is also diminished
when steady-state tax rates are reduced 40 percent. With lower future tax revenues, bond prices
and inflation take on larger revaluation roles, as in conventional fiscal theory exercises.
37

Leeper, Traum & Walker: Fiscal Multiplier Morass
Percentage of Government Spending Financing, ξt, Due to
Specification ˆ
PB
tˆπtP V (ˆrB)P V (ˆ
TK)P V (ˆ
TL)P V (ˆ
TC)P V (ˆ
Z)P V (ˆ
Ge)P V (ˆ
Se)
Posterior Estimates
1955q1–2014q2
Regime M 11.0 0.4 -6.2 8.2 14.1 2.0 -42.9 113.3 94.8
[8.3, 13.5] [0.3, 0.6] [-8.6, -3.8] [4.8, 11.5] [8.3, 19.9] [1.1, 3.0] [-77.8, -6.8] [77.3, 147.4] [91.8, 97.7]
Regime F 14.3 0.7 -1.7 30.8 53.4 2.5 0.0 0.0 86.8
[11.7, 16.8] [0.5, 0.8] [-3.2, -0.2] [29.7, 32.0] [51.4, 55.4] [1.8, 3.2] [-0.2, 0.2] [-0.2, 0.2] [84.0, 89.6]
1955q1–2007q4
Regime M 10.1 0.3 -6.8 1.5 2.6 0.6 -13.6 105.3 96.3
[6.8, 13.4] [0.2, 0.5] [-8.7, -4.8] [-0.8, 3.8] [-1.4, 6.7] [-0.4, 1.5] [-60.0, 32.3] [61.7, 149.9] [93.4, 99.4]
Regime F 17.1 0.8 -2.3 30.0 52.1 2.3 0.0 0.0 84.4
[14.0, 20.3] [0.6, 0.9] [-3.8, -0.8] [28.7, 31.5] [49.8, 54.6] [1.5, 3.1] [-0.2, 0.3] [-0.3, 0.3] [81.1, 87.7]
1955q1–1979q4
Regime M 6.7 0.0 -6.7 1.0 1.7 -0.6 51.5 46.4 100.0
[0.5, 12.6] [-0.3, 0.3] [-8.9, -4.4] [-0.8, 2.8] [-1.3, 4.9] [-0.9, -0.3] [29.4, 73.4] [24.6, 66.7] [94.1, 105.8]
Regime F 24.0 0.9 -9.7 30.4 52.7 1.7 0.0 0.0 84.8
[18.2, 29.5] [0.6, 1.2] [-14.2, -4.7] [28.7, 32.3] [49.6, 55.9] [1.0, 2.4] [-0.3, 0.3] [-0.4, 0.5] [80.1, 89.7]
1982q1–2007q4
Regime M 10.7 0.3 -9.0 4.0 7.0 0.5 -21.5 108.1 98.0
[7.1, 14.0] [0.1, 0.4] [-12.1, -6.0] [0.1, 8.2] [0.2, 14.2] [-0.2, 1.2] [-56.0, 17.6] [69.8, 145.6] [95.2, 100.8]
Regime F 17.7 0.5 -4.8 31.0 53.6 2.0 0.0 0.0 86.6
[12.9, 22.3] [0.1, 0.9] [-7.4, -2.1] [29.1, 32.7] [50.4, 56.6] [1.1, 2.7] [-0.3, 0.3] [-0.4, 0.4] [82.1, 91.3]
Counterfactuals Based on 1955q1–2007q4 Estimates
Regime M
Ricardian 1.4−0.1−6.5 1.3 2.3−1.5 103.1 0 105.2
θ= 0.8, αG= 0 5.9 0.2−4.0 0.8 1.3 0.4−12.3 107.6 97.9
φπ= 1.5, φy= 0.28.5 0.2−7.9−1.7−3.0 0.2−13.3 117.1 99.2
Regime F
Parameters
ωp=ωw= 0.7 35.9 4.3 14.2 16.2 28.0 1.5 0.0 0.0 45.6
θ= 0.8, αG= 0, ρG= 0.923.1 0.8−9.8 30.5 52.8 2.6 0.0 0.0 85.9
ωp=ωw= 0.7 &
θ= 0.8, αG= 0, ρG= 0.9 43.6 4.7 4.1 16.8 29.1 1.8 0.0 0.0 47.6
Regime F
Steady State
ρ= 0 0 1.0−1.5 35.7 61.8 3.0 0.0 0.0 100.5
sb= 4 ∗150% 53.3 2.2−14.5 21.2 36.7 1.1 0.0 0.0 59.0
Lower τ24.7 1.2−0.7 26.5 46.0 2.3 0.0 0.0 74.8
Table 11: Reports the percentage of ξtaccounted for by each element on the right side of equation (6); means and 90-percent credible intervals (in brackets) of priors
and posteriors displayed. Positive entries support financing and negative entries counter financing of higher government spending. 0.0 entries represent values <0.05. Mean
rows may not sum to 100 percent due to rounding. Regime M (Ricardian) sets γG= 0 and γZ= 0.2; regime M (θ= 0.8, αG= 0) reduces habit formation and removes
government spending from utility; regime M (απ= 1.5, αy= 0.2) raises monetary policy reactions to inflation and output; regime F (ωp=ωw= 0.7) reduces price and wage
stickiness; regime F (θ= 0.8, αG= 0, ρG= 0.9) reduces habit formation, removes government spending from utility, and makes government spending less persistent; regime F
(ωp=ωw= 0.7, θ = 0.8, αG= 0, ρG= 0.9) combines the two previous counterfactuals; regime F (sb= 150%) sets the annualized steady-state ratio of the market value of debt
to output at 150 percent; regime F (ρ= 0) makes all debt one period; regime F (lower τ) reduces steady-state tax rates on capital, labor and consumption by 40 percent.
38

Leeper, Traum & Walker: Fiscal Multiplier Morass
5 Multipliers at the Effective Lower Bound
This section explores how multipliers vary when the monetary authority is constrained by a lower
bound on nominal interest rates. We use the estimates from the model conditional on regimes M
and F prior to the financial crisis, 1955q1–2007q4.29 Conditional on the estimates, we calculate
multipliers for a range of counterfactual scenarios where the lower bound on the nominal interest
rate binds. We raise the level of government spending by 1 percent for two years, accompanied by
two years in which taxes and spending do not respond to the growing government debt [Coenen et
al. (2013) consider a similar scenario]. Although the shock is unanticipated, its future time profile
is known. Government spending evolves as
ˆgt=


0.01,for t= 1,2,...,8
ρGˆgt−1−(1 −ρG)γGˆsb
t−1,for t > 8
Transfers follow an analogous rule: ˆzt= 0 for t= 1,2,...,8 and ˆzt=ρZˆzt−1−(1 −ρZ)γZˆsb
t−1
for t > 8. In addition to the fiscal accommodation, the monetary authority accommodates the
expansionary policy for Jperiods, following the rule: ˆ
Rt= 0 for t= 1,2,...,J and ˆ
Rt=ˆ
Rt−1+
(1 −ρr) [φπˆπt+φyˆyt] for t > J .30
Table 12 reports output, consumption, and investment multipliers in regimes M and F for
various lengths of the lower-bound state. The first row for each regime reports multipliers without
the lower-bound constraint. The implied multipliers for regime M support several results in the
literature. First, multipliers always increase with the length of the lower-bound state [Woodford
(2011), Christiano et al. (2011)]. The longer the lower bound lasts, the more expected inflation
lowers real interest rates, further stimulating the economy. Second, multipliers increase with the
degree of wage and price flexibility in the lower-bound state [Coenen et al. (2012), Christiano et
al. (2011)]. Greater price and wage flexibility produces larger adjustments in both variables in the
short run, which enhance the expected inflation effect on real interest rates. Third, expectations
about future policy matter for the size of the multiplier [Eggertsson (2010), Denes et al. (2013),
Erceg and Lind´e (2014)]. To see this, we consider two counterfactuals that vary the expected form
of fiscal financing: either all adjustments to higher debt come from lower lump-sum transfers or
the size of expected spending reversals doubles relative to the benchmark scenario. Lower expected
lump-sum transfers generate a negative wealth effect that lowers multipliers. Multipliers decrease
in the response of spending to debt, γg, because faster public spending reversals imply quicker
declines in demand. The length of the lower-bound state enhances these effects.
The bottom half of table 12 repeats the lower-bound multipliers in regime F. Multipliers are in-
creasing with the length of the lower-bound state, enhanced price and wage flexibility, and a smaller
29The procedure follows Christiano et al. (2015).
30Erceg and Lind´e (2014) and Bianchi and Melosi (2016) show that future policy expectations can endogenously
affect the length of the lower bound and the effects of government spending. To account for the endogenous length
of the lower bound, we also performed the exercises using the Occbin toolkit provided by Guerrieri and Iacoviello
(2015). Similar results hold in the two environments.
39

Leeper, Traum & Walker: Fiscal Multiplier Morass
monetary response to inflation after exiting the lower bound (see the φπ= 0 cases). Multipliers
in regime F are larger than multipliers in regime M, as expansions in government spending are
not expected to be financed with spending reversals or future lump-sum transfer decreases. As we
saw in table 10, in regime F monetary policy’s response to output, φy, becomes quite powerful, as
demonstrated by the counterfactual setting φy= 0. Eliminating this response raises all multipliers,
but this effect is unrelated to hitting the lower bound.
Although we follow the literature in calling this an analysis of the “lower bound,” the results
make clear that it is less about the level of the interest rate than it is about pegging the interest
rate. A pegged rate lies outside normal monetary policy behavior in regime M, which is why its
impacts are large in that regime. At 25 quarters, output and consumption multipliers are three
times higher when the lower bound binds for 12 quarters. But a pegged rate is merely a special
case of regime F monetary policy. Its effects on multipliers, accordingly, are more modest. Output
and consumption multipliers at 25 quarters are only 1.5 times larger when the bound holds for 12
quarters.
6 Sequential Estimation
We briefly address evidence of time variation in the fiscal multiplier. Several studies have argued
that multipliers vary over the business cycle [Auerbach and Gorodnichenko (2012)] and depend upon
the time spent at the lower bound [Coenen et al. (2012), Christiano et al. (2011)]. Others have
demonstrated that parameter estimates in standard DSGE models tend to exhibit time variation
[Fern´andez-Villaverde and Rubio-Ram´ırez (2008)]. To assess time variation in multipliers and
parameter estimates, we sequentially estimate model 4 of section 1using a 25-year rolling window
with annual steps. That is, we estimate the model using data from 1955:1 through 1979:4, and then
repeat the estimation for 1956:1 through 1980:4, 1957:1 through 1981:4, and so on, up to 1989:1
through 2013:4.
To preserve space, we report only a select few parameters and multipliers. Figure 9plots the
impact and present-value (25-quarter) multiplier for output (top row) and consumption (middle
row) in regime M. Mean values are solid lines, while dashed lines represent 5th and 95th percentiles.
The figure shows modest time variation in the multipliers. Impact consumption and output
multipliers follow a similar trend, peaking in the early part of the sample and reaching a minimum
around the 1970 to 1994 data set. The mean impact output (consumption) multiplier never dips
below one (zero) and reaches a maximum of 1.5 (0.55). As noted above, the estimate of αG, which
determines the complementarity of government consumption, is a critical parameter for multipliers
in regime M. A negative (positive) value of αGimplies private and public consumption are com-
plements (substitutes). Trends in output and consumption impact multipliers mirror movements
in αG. The early time periods yield a large negative value for αG, where impact multipliers are
largest. Estimates of αGincrease over time, reaching a mean value of zero for the 1971 to 1995
data set. This time period coincides with the smallest impact multipliers.
Longer-horizon multipliers exhibit less variation than the impact multipliers. The mean present-
40

Leeper, Traum & Walker: Fiscal Multiplier Morass
Multipliers and the Effective Lower Bound
Posterior (Impact) Posterior (25 qtrs)
P V ∆Y
∆GP V ∆C
∆GP V ∆I
∆GP V ∆Y
∆GP V ∆C
∆GP V ∆I
∆G
Not binding 1.23 0.17 -0.19 0.29 0.05 -1.11
[1.05,1.41] [0.01,0.35] [-0.24,-0.15] [0.16,0.40] [-0.12,0.22] [-1.33,-0.88]
ωp=ωw= 0.7 1.27 0.17 -0.15 0.44 0.03 -0.91
[1.06,1.47] [0.00,0.34] [-0.23,-0.08] [0.24,0.62] [-0.13,0.19] [-1.18,-0.68]
γz= 0.15, γg= 0 1.21 0.16 -0.20 0.34 -0.04 -0.93
[1.03,1.39] [-0.00,0.34] [-0.24,-0.16] [0.26,0.43] [-0.20,0.11] [-1.10,-0.74]
2γg1.22 0.17 -0.20 0.30 0.08 -1.15
[1.04,1.40] [0.01,0.35] [-0.24,-0.16] [0.20,0.41] [-0.08,0.25] [-1.36,-0.94]
ZLB binding, 8 qrtrs 1.48 0.18 -0.01 0.62 0.10 -0.78
[1.28,1.68] [0.01,0.35] [-0.06,0.04] [0.48,0.75] [-0.07,0.26] [-0.97,-0.60]
ωp=ωw= 0.7 1.65 0.19 0.12 0.88 0.11 -0.48
[1.14,2.20] [0.02,0.36] [-0.22,0.53] [0.38,1.48] [-0.09,0.29] [-0.99,0.09]
γz= 0.15, γg= 0 1.27 0.16 -0.15 0.41 -0.03 -0.86
[1.08,1.47] [-0.00,0.34] [-0.20,-0.10] [0.30,0.53] [-0.19,0.13] [-1.01,-0.70]
2γg1.44 0.18 -0.04 0.59 0.13 -0.87
[1.23,1.64] [0.01,0.36] [-0.08,-0.00] [0.48,0.69] [-0.03,0.30] [-1.05,-0.69]
ZLB binding, 12 qrtrs 1.63 0.19 0.10 0.99 0.15 -0.42
[1.41,1.85] [0.02,0.36] [0.02,0.19] [0.78,1.20] [-0.01,0.31] [-0.64,-0.21]
ωp=ωw= 0.7 1.97 0.20 0.35 1.48 0.19 0.10
[1.14,2.88] [0.04,0.38] [-0.22,1.01] [0.32,2.69] [-0.05,0.43] [-0.99,1.30]
γz= 0.15, γg= 0 1.27 0.16 -0.15 0.41 -0.03 -0.86
[1.06,1.49] [-0.01,0.33] [-0.23,-0.08] [0.21,0.62] [-0.19,0.13] [-1.06,-0.66]
2γg1.56 0.19 0.05 0.89 0.17 -0.57
[1.34,1.79] [0.02,0.36] [-0.01,0.11] [0.74,1.03] [0.01,0.33] [-0.76,-0.38]
Regime F
Not binding 1.43 0.16 -0.04 1.52 0.20 -0.13
[1.24,1.63] [-0.02,0.34] [-0.08,0.01] [1.32,1.71] [0.05,0.37] [-0.35,0.09]
ωp=ωw= 0.7 2.39 0.21 0.67 3.55 0.41 1.97
[2.04,2.73] [0.02,0.38] [0.45,0.88] [2.95,4.13] [0.23,0.60] [1.36,2.52]
φπ= 0 1.47 0.16 -0.01 1.68 0.23 0.01
[1.27,1.66] [-0.01,0.35] [-0.06,0.05] [1.45,1.90] [0.06,0.38] [-0.23,0.25]
φy= 0 2.63 0.21 0.84 6.25 0.63 4.31
[2.24,3.01] [0.03,0.39] [0.60,1.06] [5.17,7.26] [0.40,0.87] [3.27,5.32]
ZLB binding, 8 qrtrs 1.79 0.18 0.23 1.81 0.26 0.16
[1.57,2.05] [-0.00,0.36] [0.13,0.33] [1.60,2.01] [0.10,0.42] [-0.06,0.38]
ωp=ωw= 0.7 3.35 0.25 1.37 4.00 0.54 2.42
[2.74,3.92] [0.07,0.44] [0.94,1.79] [3.37,4.64] [0.31,0.76] [1.75,3.05]
φπ= 0 1.81 0.18 0.24 1.95 0.28 0.28
[1.57,2.06] [0.00,0.36] [0.14,0.35] [1.72,2.20] [0.11,0.43] [0.04,0.52]
φy= 0 2.66 0.21 0.86 6.27 0.63 4.33
[2.28,3.07] [0.03,0.39] [0.62,1.09] [5.19,7.28] [0.40,0.88] [3.27,5.33]
ZLB binding, 12 qrtrs 2.08 0.19 0.44 2.33 0.32 0.70
[1.77,2.38] [0.00,0.36] [0.27,0.61] [2.04,2.63] [0.15,0.47] [0.38,1.02]
ωp=ωw= 0.7 3.88 0.27 1.76 4.61 0.64 3.09
[3.13,4.62] [0.08,0.46] [1.19,2.28] [3.89,5.37] [0.38,0.90] [2.29,3.90]
φπ= 0 2.08 0.19 0.45 2.45 0.34 0.79
[1.78,2.39] [0.00,0.36] [0.28,0.61] [2.13,2.75] [0.16,0.49] [0.45,1.11]
φy= 0 2.69 0.21 0.88 6.30 0.63 4.36
[2.30,3.10] [0.03,0.39] [0.63,1.11] [5.25,7.35] [0.40,0.87] [3.29,5.36]
Table 12: Counterfactual multipliers with the effective lower bound for regimes M and F estimated
over 1955q1–2007q4. Posterior means and 90-percent credible intervals (in brackets).
41

Leeper, Traum & Walker: Fiscal Multiplier Morass
55-79 60-84 65-89 70-94 75-99 80-04 85-09
0.5
1
1.5
2Impact Output Multiplier
55-79 60-84 65-89 70-94 75-99 80-04 85-09
0
0.2
0.4
0.6
0.8 25-Quarter Output Multiplier
55-79 60-84 65-89 70-94 75-99 80-04 85-09
-0.5
0
0.5
1Impact Consumption Multiplier
55-79 60-84 65-89 70-94 75-99 80-04 85-09
-0.5
0
0.5 25-Quarter Consumption Multiplier
55-79 60-84 65-89 70-94 75-99 80-04 85-09
-1.5
-1
-0.5
0
0.5
αg: Government Spending in Utility
55-79 60-84 65-89 70-94 75-99 80-04 85-09
0
0.1
0.2
0.3
0.4
γg: Government Spending Response to Debt
Figure 9: Sequential estimation of the impact multipliers for output and consumption (top row),
25-quarter present value multipliers (middle row), and parameters αGand γG(bottom row) for
regime M using data 1955:1 to 1979:4 through 1989:1 to 2013:4. The solid lines are mean values
and dashed lines are 90-percent credible sets.
value output multiplier begins around 0.4, dips to roughly 0.3 mid-sample before returning to 0.5
by the end of the sample. The 25-quarter consumption multiplier trends slightly lower for several
periods, and then stabilizes around -0.2 beginning with the 1965 to 1989 sample. Movements in
the 25-quarter multipliers connect to time variation in γG, the strength of spending reversals. This
parameter increases over the same horizon that the consumption multiplier is falling, and both
stabilize around the same time period. Increases in γGbring forth faster spending reversals. At
longer horizons, this makes more goods available to the private sector and increases consumption
multipliers. But in the short run, faster spending reversals lower expected government demand,
muting inflation responses and raising the real interest rate. The higher real interest rate depresses
consumption, explaining the negative relationship between γGand consumption multipliers at the
25-quarter horizon.
Figure 9underscores the tightly estimated multipliers and parameters for all time periods.
Recall that the prior predictive range for impact multipliers extends well beyond two (one) and
below zero (−0.5) for output (consumption). The 90-percent posterior credible sets are much
narrower than the prior predictive analysis, indicating that the data are informative.
7 Concluding Remarks
This paper differs from the bulk of research on government spending multipliers in several ways: (1)
expands the set of observables used in estimation; (2) fills out details on the fiscal side of the model,
including explicit rules for fiscal instruments, maturity structure of government debt, government
42

Leeper, Traum & Walker: Fiscal Multiplier Morass
spending that may complement or substitute for private consumption, distorting steady-state taxes;
(3) adopts more diffuse priors over nominal rigidities and habit formation; (4) permits the posterior
to land in regions of the parameter space that uncover new transmission mechanisms for govern-
ment spending; (5) finds that monetary-fiscal regime is important for the size and persistence of
multipliers: they are larger and more persistent in regime F, but even regime M estimates produce
larger and longer lasting multipliers than most studies.
Our more general analysis spans a vast set of existing model-based estimates of government
spending multipliers. Although a priori our specification can produce a morass-like range of multi-
pliers, as the prior predictive analysis reveals, confronting the specification with data dramatically
narrows that range. Conditional on monetary-fiscal regime, data are sufficiently informative to
drain the swamp and help clear up the morass.
The paper highlights an issue that transcends multipliers: scrutinizing the prevailing monetary-
fiscal policy regime is the first order of business for understanding policy impacts. For determining
the magnitude and dynamics of multipliers, we find that the monetary-fiscal mix overshadows
the many other factors on which existing research dwells. The importance of monetary-fiscal
interactions for estimates of the impacts of other macro policy actions remains to be explored.
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