HIAS-E-103

Fiscal Adjustments and Debt-Dependent Multipliers:
Evidence from the U.S. Time Series
Yasuharu Iwata(a), Hirokuni Iiboshi(b)(a) Cabinet Secretariat, Government of Japan(b) Tokyo Metropolitan University

December, 2020

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
2-1, Naka, Kunitachi, Tokyo 186-8601, Japan
tel:+81 42 580 8668 http://hias.hit-u.ac.jp/
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Fiscal Adjustments and Debt-Dependent Multipliers: Evidence
from the U.S. Time Series
Yasuharu Iwatay

Hirokuni Iiboshiz

Cabinet Secretariat, Government of JapanTokyo Metropolitan University

December 22, 2020

Abstract
Using sign restrictions within a time-varying parameter vector autoregressive (TVPVAR) framework, we provide new time-series evidence of debt-dependent multipliers for the
U.S. while simultaneously obtaining larger multipliers during recessions in line with previous
studies. The Ricardian channel where households reduce consumption expecting larger …scal
adjustments is shown to be relevant for the debt-dependent multipliers. The TVP-VAR
framework also allows us to observe changes in the magnitude of …scal adjustments. We…nd that the larger …scal adjustments in the presence of rising indebtedness is the major
driving force behind the smaller multipliers in the post-Volcker period rather than debt
accumulation itself.
JEL classi…cation: E32, E62, H60.
Keywords: Bayesian VARs; Time-varying parameters; Fiscal multipliers; Fiscal policy.

We would like to thank Torben Andersen, Alan Auerbach, Harris Dellas, James Hamilton, Eiji Kurozumi,
Fabio Milani, Hiroshi Morita, Jouchi Nakajima, Tatsuyoshi Okimoto, Mototsugu Shintani, Etsuro Shioji, Ricardo
M. Sousa, Toshiaki Watanabe, Yohei Yamamoto, Andreas Zervas, and participants at the 5th Hitotsubashi
Summer Institute (Tokyo), the International Symposium in Computational Economics and Finance (Paris),International Conference on Applied Theory, Macro and Empirical Finance (Thessaloniki), and International
Conference on Computing in Economics and Finance (Bordeaux) for helpful comments and suggestions. The
views expressed in this paper are solely the responsibility of the authors and should not be interpreted asre‡ecting the views of the Japanese government.
yCabinet Secretariat, Government of Japan, 1-6-1 Nagata-cho, Chiyoda-ku, 100-8968 Tokyo, Japan (E-mail:yiwata.econ@gmail.com).
z
Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-shi, 192-0397 Tokyo, Japan (Email: iiboshi@tmu.ac.jp).

11

Introduction

The past decade has witnessed increased attention to the size of government spending multipliers
and their heterogeneities over time and across countries. Investigating sources of heterogeneity
in multipliers across countries, the literature has provided ample evidence that government
spending multipliers are large in countries with low public debt (e.g., Ilzetzki, Mendoza andVégh (2013); Nickel and Tudyka (2014); Huidrom et al. (2020)). The role of public debt ina¤ecting the size of multipliers has also become a very relevant issue for the United States.
As illustrated in Figure 1, the public debt-to-GDP ratio in the U.S. has been on an upward
trajectory since the 1980s. After the Global Financial Crisis, it has soared to the level above
the thresholds used to de…ne high-debt countries in previous studies.

Debt- to- GDP ratio in %

120
100

Huidrom et al. (2020): 92%8060

Ilzetzki et al. (2013): 60%

40
20
0
1950

1960

1970

1980

1990

2000

2010

FIG. 1. U.S. debt-to-GDP ratio. Notes: The red solid line and the blue dashed line represent the debt-toGDP ratio of general government and that of federal government, respectively. The horizontal lines indicate thethreshold debt-to-GDP ratios used to de…ne high-debt countries in Huidrom et al. (2020) and Ilzetzki, Mendoza
and Végh (2013). The thresholds used in Huidrom et al. (2020) and Ilzetzki, Mendoza and Végh (2013) are thosefor general government and federal government, respectively.

However, public debt dependency of government spending multipliers in the U.S. time-series
data has been somewhat neglected in the literature. Although time variation in the U.S. multipliers is an area of active research, existing studies have focused on its state-dependent natureacross business cycles relying on a regime switching framework. The growing body of empiricalevidence suggests that multipliers are larger in recessions than in expansions (e.g., Auerbach
2and Gorodnichenko (2012); Bachmann and Sims (2012); Candelon and Lieb (2013); Caggiano
et al. (2015)).1 In a similar vein, Bernardini and Peersman (2018) …nd larger multipliers inperiods of private debt overhang while considering public debt as a control variable.
Whereas theoretical literature highlights the importance of policy regimes in a¤ecting thesize of multipliers, there has been relatively limited empirical evidence from the U.S. time series
with a few exceptions. Bilbiie, Meier and Müller (2008) …nd smaller multipliers in the post1980 period than in the preceding period and attribute the cause to changes in the conduct ofmonetary policy after Volcker’s appointment as Fed Chairman.2 Ramey and Zubairy (2018)
report rather mixed results on the size of multipliers when monetary policy is constrained by thezero lower bound.3 Leeper, Traum and Walker (2017) demonstrate that monetary-…scal policy
regime is the dominant factor in determining the size of multipliers, but they do not provide
evidence of time variation in multipliers across di¤erent regimes.4
The nexus between public debt and …scal policy e¤ects has been studied since Giavazzi andPagano (1990) found cases of expansionary …scal adjustments from Danish and Irish experiences in the 1980s.5 The transmission where households reduce consumption in anticipation of
future …scal adjustments has been examined in the following literature (e.g., Blanchard (1990);
Sutherland (1997); Perotti (1999)). Huidrom et al. (2020) call the transmission a Ricardianchannel and consider the channel as the underlying cause of the debt-dependent multipliers.
1
Studies that consider data from a panel of countries also …nd business cycle dependency of multipliers (e.g.,
Auerbach and Gorodnichenko (2013); Riera-Crichton, Vegh and Vuletin (2015)). For its theoretical account, seeMichaillat (2014), Canzoneri et al. (2016), and Shen and Yang (2018). In contrast, Ramey and Zubairy (2018)…nd little evidence of business cycle-dependent multipliers from their U.S. historical data.
2
Bilbiie, Meier and Müller (2008) also suggest that smaller multipliers in the post-Volcker period can be
attributed to increased asset market participation as well as the more active monetary policy of the period. In
theory, asset market participation allows households to save or borrow to smooth their consumption in anticipationof future …scal adjustments. Therefore, its increase could also represent strengthening of the Ricardian channel
where households reduce consumption expecting larger …scal adjustments.
3
Theoretical literature predicts large multipliers when interest rates are at the zero lower bound (e.g., Woodford (2011); Christiano, Eichenbaum and Rebelo (2011)).
4
Leeper, Traum and Walker (2017) calculate multipliers based on estimated DSGE models under di¤erent
policy regimes. Although multipliers are shown to be larger in regime F (active …scal policy coupled with passive
monetary policy) than in regime M (active monetary policy coupled with passive …scal policy) in general, they
do not …nd signi…cant di¤erence between the log marginal data densities for the two regimes. They only reportmodest time variation in multipliers in regime M. In contrast, Traum and Yang (2011) report that regime F isnever favored by the post-war U.S. data.
5
The literature on the expansionary e¤ects of …scal adjustments (‘expansionary austerity hypothesis’) remains
divided (e.g., Alesina, Favero and Giavazzi (2019); House, Proebsting and Tesar (2020)).

3

On the other hand, the empirical literature investigating the size of U.S. multiplier documents
the importance of capturing dynamics of …scal adjustments (e.g., Chung and Leeper (2007);
Favero and Giavazzi (2012); Corsetti, Meier and Müller (2012)). Bohn (1998) …nds a positive
correlation between the magnitude of …scal adjustments and the debt-to-GDP ratio in providing
evidence of the government’s reaction to debt accumulation. In search of theoretical grounds fordebt-dependent multipliers, Bi, Shen and Yang (2016) show that larger magnitude of …scal adjustments induces stronger negative e¤ects on consumption, thus leading to smaller multipliers
when debt levels are high.
Against this background, this paper aims to provide time-series evidence of debt-dependent
multipliers from the U.S. data and to investigate the transmission paying particular attention to the role of …scal adjustments. For these purposes, we employ a time-varying parameter vector autoregressive (TVP-VAR) model with stochastic volatility developed by Primiceri(2005), in which time-varying contemporaneous relations among variables are assumed. Unlike
regime-switching models widely used in the previous literature, the TVP-VAR model allows
the parameters to vary continuously over time in a stochastic manner and, hence, is suitablefor capturing permanent and gradual changes in the transmission mechanism.6 Therefore, themodel may well describe possible changes in household behavior and those in the magnitude
of …scal adjustments. Although rapid changes in the economic state are di¢ cult to capture
within the model, we consider them with the assistance of sign restrictions following Canova
and Pappa (2011).7 Together with the assumption of time-varying contemporaneous relations
among variables, we identify government spending shocks during recessions and expansions
by imposing additional sign restrictions in accordance with the phases of the business cycle
on a period-by-period basis. The method allows us to simultaneously …nd both cyclical and6
Primiceri (2005) provides a succinct discussion of the advantages and disadvantages of TVP-VAR models
over regime-switching models.
7
Auerbach and Gorodnichenko (2012) discuss the disadvantage of a TVP-VAR model in capturing the stateof the business cycle.

4structural variations in multipliers within a single TVP-VAR framework.8
The analysis provides evidence of two heterogeneities in multipliers: larger multipliers inrecessions than in expansions and smaller ones in the post-Volcker period than in the preceding
period. A negative correlation between government spending multipliers and public debt is alsopresented. We then examine the underlying cause of the decline in multipliers in the postVolcker period by augmenting the baseline model with interest rate and private consumption.
Comparing the results with those of the baseline model con…rms the relevance of the Ricardianchannel. By applying our TVP-VAR framework to the bivariate VAR methodology of Canzoneri,
Cumby and Diba (2001) and Canzoneri, Cumby and Diba (2011), we further show that themagnitude of …scal adjustments was increased during most of the post-Volcker period. The
Granger causality test among the debt-to-GDP ratio, the magnitude of …scal adjustments, andthe size of multiplier suggests that the increased magnitude of …scal adjustments in the presenceof rising indebtedness was the major driving force for the decline in multipliers.
The paper most closely related to ours is Kirchner, Cimadomo and Hauptmeier (2010),
which is the only study that we are aware of which explore the debt dependency of government spending multipliers based on time-series data.9 By conducting regression analysis on
the multipliers calculated from their estimated TVP-VAR model for the Euro area and possible explanatory factors, they conclude that an increase in debt-to-GDP ratio has a negative
impact on the multipliers. Our study di¤ers from them in that we consider the business cycle
dependency of multipliers aside from the debt dependency with the assistance of sign restriction
identi…cation. Furthermore, while Kirchner, Cimadomo and Hauptmeier (2010) focus on therelationship between multipliers and debt-to-GDP ratio, this paper addresses that the Ricardianchannel operates in response to the increased magnitude of …scal adjustments rather than debt
8
Bi, Shen and Yang (2016) address the di¢ culty in isolating the debt-dependent government spending e¤ects
based on structural VAR estimations because of various interrelated state variables.
9
Application of TVP-VAR framework to …scal policy analysis has been relatively limited when comparing itwith that to monetary policy. Although there has been a growing interest in applying the TVP-VAR framework
to …scal policy analysis, those studies, such as Ra…q (2012), Pereira and Lopes (2014), and Glocker, Sestieri and
Towbin (2019), focus on capturing time-varying e¤ects of …scal policy rather than investigating their transmission
using data from Japan, the U.S., and the U.K., respectively.

5

accumulation itself.
The remainder of this paper is organized as follows. Section 2 discusses the empiricalmethodology. Section 3 reports the results. Section 4 investigates the underlying transmission
mechanism of the debt-dependent multipliers. Section 5 concludes.

2

Empirical methodology

2.1

A VAR model with time-varying parameters and stochastic volatility

We consider the following VAR (p) model with time-varying parameters and stochastic volatility:

yt = B1;t yt 1 +for t = p + 1; :::; T , where yt is a k+ Bp;t yt p + ut ,(2.1)

1 vector of observed variables and Bi;t ; i = 1; :::p; are k

matrices of time-varying coe¢ cients. The ut is a kk

1 vector of heteroskedastic shocks that are

assumed to be normally distributed with a zero mean and a time-varying covariance matrix,
t . Following established practice, we decompose ut as ut = At

2

6 1
6
6
6a
6 21;tAt = 6
6 .
6 .
6 .
6
4
ak1;t
2

6 1;t
6
6
6 0
6
6
t =6
6 ..
6 .
6
4
0

..

.

..

.

..

.
.

akk 1;t

..

.

..

.

..

.
.

0

6

07
7
.. 7
.7
7
7;
7
7
07
7
5
1

(2.2)

3

0
..

t "t ; where

3

0
..

1

0 7
7
.. 7
. 7
7
7;
7
7
0 7
7
5
k;t

(2.3)

and "tN (0; Ik ). It follows that At

0t At =t

0 : Lett2
t be a stacked k p

1 vector of theelements in the rows of the B1;t ; :::Bp;t , and at be the vector of non-zero and non-one elements of
the At . Following Primiceri (2005), we assume that these vectors follow a random walk process:

t+1 =

2

3

t + u ;t ;(2.4)

at+1 = at + ua;t ;(2.5)

ht+1 = ht + uh;t ;

(2.6)

0 2

6 "t 76
7
6
7
6u 7
6 ;t 767
6
7
6
7
6 ua;t 7
6
7
4
5
uh;t

B 6I
B 6
B 6
B 6O
B 6
6
NB
B0; 6
B 6
B 6O
B 6
@ 4
O

where ht = [h1;t ; : : : ; hk;t ]0 with hj;t = ln

O

O
O

O

a

O

O

31

O 7C
7C
7C
C
O7
7C
7C ;
7C
7C
O 7C
7C
5A
h

2 for j = 1; :::; k, and I is a k-dimensional identityj;tmatrix. The prior distributions for the initial values are given by
N ( a0 ;a0 ), and hp+1

N ( h0 ;

(2.7)

p+1

N(

0

;

0

), ap+1

h0 ). Observe that the model allows both the parameters thatgovern contemporaneous relations among variables and the log of the variance for the shocks to
evolve over time as a random walk.
The stochastic volatility assumption makes the likelihood function of the model di¢ cult toconstruct and requires Bayesian inference via Markov Chain Monte Carlo (MCMC) methods.
To estimate a model that contains a relatively large number of variables, we rely on the e¢ cientalgorithm proposed by Nakajima, Kasuya and Watanabe (2011), which is developed by modifying Primiceri (2005)’s original algorithm. Following Nakajima (2011a), we further assume
for simplicity that

;

a;

h;

0

;

10
h0 are all diagonal matrices.

a0 ; andRegarding the10
Although the assumption is not essential, it greatly simpli…es the inference procedures for at and ht , thereby
contributing to increase the e¢ ciency of the algorithm (e.g., Primiceri (2005); Nakajima, Kasuya and Watanabe(2011)). Moreover, we do not expect a signi…cant di¤erence in results from allowing for correlations among
elements of at , t , and ht , as in Primiceri (2005), Nakajima (2011a), and Nakajima (2011b), respectively.

7

sampling oft and at , we use the simulation smoother of de Jong and Shephard (1995) because

the model can be written as a linear Gaussian state space form conditional on the rest of theparameters.11 In contrast, in sampling ht ; we employ the multi-move sampler of Shephard andPitt (1997) and Watanabe and Omori (2004) for non-linear and non-Gaussian state space models. The multi-move sampler is more e¢ cient than the single-move sampler of Jacquier, Polson
and Rossi (1994).12 Furthermore, it enables us to draw a sample from the exact conditional
posterior density of the stochastic volatility, unlike the mixture sampler of Kim, Shephard andChib (1998). Appendix A provides a more detailed outline of the MCMC algorithm used inthis study.
We use U.S. quarterly data for the period from 1952:Q1 to 2013:Q4.13 The observed variables
include government spending, gross domestic product (GDP), debt-to-GDP ratio, and the GDPde‡ator. The government spending and GDP are expressed in real per capita terms. We use thelogarithm for all variables except the debt-to-GDP ratio. All variables are seasonally adjusted
and detrended with a linear and quadratic trend. The lag length is set to p = 4; following
Blanchard and Perotti (2002). See Appendix B for a detailed description of the data sources.

2.2

Identi…cation strategies

The TVP-VAR framework allows parameters to vary continuously over time in a stochastic
manner and, hence, it is not suitable for capturing rapid changes in the economic state. Nevertheless, we can consider the e¤ects of such changes by implementing a shock identi…cation
11
We employ the simulation smoother of de Jong and Shephard (1995) instead of the multi-state sampler ofCarter and Kohn (1994), which is widely used in previous TVP-VAR studies. The multi-state sampler generates
the entire state vector at once and therefore converges more quickly than the single-state sampler that yields astrong correlation among the samples. However, the method is prone to the problem of degeneracies because theentire state vector is constructed recursively. The simulation smoother of de Jong and Shephard (1995) avoids
the problem by drawing disturbances rather than states.
12
The shortcoming of using the single-move sampler is that it leads to slow convergence when state variables
are highly autocorrelated. The multi-move sampler reduces the ine¢ ciency by generating randomly selected
blocks of disturbances rather than each state variable at a time.
13
The sample period starts at the time when quarterly data series on public debt is available. Because we
examine the role of monetary policy in a¤ecting the size of multipliers in the next section, the sample excludes theperiod of monetary policy normalization, which began with the tapering of quantitative easing in January 2014.
Our sample, on the other hand, covers the zero interest-rate policy period in light of the …ndings of Nakajima(2011a). Based on the Japanese experience, Nakajima (2011a) documents that a zero lower bound on nominalinterest rates has negligible e¤ects on impulse responses in a TVP-VAR model with stochastic volatility.

8

through sign restrictions in each period. As Canova and Pappa (2011) suggest, the sign restriction approach enables us to study the e¤ectiveness of …scal policy under a certain economic
state by imposing additional sign restrictions. Together with the assumption that the parameters governing contemporaneous relations among variables are time variant, we can imposedi¤erent sets of sign restrictions on a period-by-period basis, considering the economic state ofeach period. To implement the sign restriction approach within the TVP-VAR framework, we
exploit the algorithm proposed by Rubio-Ramírez, Waggoner and Zha (2010) (RWZ algorithm,
hereafter), as in Benati (2008). The algorithm allows us to identify several shocks in a highly
parameterized TVP-VAR model for each period with great e¢ ciency. Thus, it is possible toreplicate the impact of government spending shocks that re‡ect the e¤ects of rapid changes inthe economic state in addition to permanent and gradual changes in the transmission mechanism.
The RWZ algorithm proceeds as follows. We draw an independent standard normal kkmatrix Zs for period s. The QR decomposition of Zs gives an orthogonal matrix Qs that satis…es
Qs Q0s = I and an upper triangular matrix Rs : Using As 1

s Qs , we generate impulse responses

for each MCMC replication. If the impulse response satis…es the restrictions, we keep the Qs ;otherwise, we discard it. The combination of Q0s and "s , "s = Q0s "s is now regarded as a new
set of structural shocks with the same covariance matrix as the original shock "s . Because Qs is
orthogonal, the new shocks are orthogonal to each other by design. Because contemporaneous
relations among variables are assumed to be time varying in our TVP-VAR model, the algorithm
is particularly appealing for identifying shocks on a period-by-period basis.14
Table 1 reports the sign restrictions that we employed in calculating impulse responses.15
We identify two structural shocks: an expansionary government spending shock and a positive
demand shock. Thus, orthogonality to a demand shock is imposed in identifying a government
14

Although we cannot give an economic interpretation of the orthogonal matrix Qs as described by Baumeisterand Hamilton (2015), we rely on the algorithm to preserve its computational e¢ ciency. As argued by Arias,
Rubio-Ramírez and Waggoner (2018), alternative approach could become computationally ine¢ cient.
15
To compare the results with those of other studies, we restrict our focus in this study to a traditional
unanticipated government spending shock.

9

spending shock in the spirit of Mountford and Uhlig (2009). We impose a minimum set ofcontemporaneous restrictions to make our identi…cation as agnostic as possible.16 In particular,
we leave the response of output to a government spending shock unrestricted following Mountford and Uhlig (2009). A government spending shock is assumed to increase the debt-to-GDP
ratio, which is the key identifying restriction that distinguishes the shock from other shocks.17
To distinguish the e¤ects of a government spending shock during recession from that during
expansion, we use a di¤erent set of restrictions for that shock. Following Canova and Pappa(2011), we assume that a government spending shock during recession is accompanied by asimultaneous fall in the GDP de‡ator while being agnostic on the response of the GDP de‡ator
during expansion. On the other hand, a demand shock is assumed to increase GDP and the
GDP de‡ator and to decrease the debt-to-GDP ratio.
TABLE 1Sign restrictions

ShocksVariables

Gov. Spending (Expansion)

Gov. Spending (Recession)

Government spending

+

+

+

+

+

GDP
Debt-to-GDP ratio

Demand+GDP de‡ator

Notes: The table shows the signs imposed on the impulse responses of the variables to an expansionary
government spending shock and a positive demand shock. A blank indicates that the variable’s response isunrestricted. A positive [negative] sign indicates that the variable’s response is restricted to being positive[negative] on impact.

To de…ne the state of the business cycle, we use our detrended GDP data as the indicatorof economic slack.18 Okun’s Law suggests that a one-percentage point decrease in GDP fromits potential causes a half-percentage point increase in unemployment rate. On the other hand,conventional wisdom suggests that a recession is typically accompanied by a two-percentage

16

The choice of the period during which to restrict the responses does not change the basic results. It is also
computationally burdensome to estimate impulse responses from a TVP-VAR model that imposes sign restrictions
for several periods.
17
The restriction shares similarities with those in previous studies (e.g., Canova and Pappa (2011); Enders,Müller and Scholl (2011); Bouakez, Chihi and Normandin (2014)).
18
Some kind of indicator is necessary to de…ne the state of the business cycle. Auerbach and Gorodnichenko(2012) use a seven-quarter moving average of the output growth rate as an index that changes the probability of
economic state. Ramey and Zubairy (2018) use a 6.5 percent unemployment rate as the threshold value to de…nehigh and low unemployment states.

10

points increase in unemployment rate within a year, which can be reinterpreted as a halfpercentage point increase within a quarter. Applying the Okun’s Law to the conventional
wisdom on the relationship between unemployment and recession, we de…ne recessions as theperiods where a more than one-percentage point decrease in detrended quarterly GDP data isobserved.

3

Results

3.1

Two types of heterogeneities in multipliers

Figure 2 presents the stochastic volatilities of the reduced-form innovations. The time variationin the volatility estimates are largely consistent with those reported in previous studies onmonetary policy analysis. The volatility of the output shocks declined sharply in the early
1980s as in Canova and Gambetti (2009) and Mumtaz and Zanetti (2013).19 As reported in
Primiceri (2005), the volatility of price shocks reached its highest peak during the Great In‡ationof the mid-1970s. A reduction in the volatility of government spending shocks can also be foundin Justiniano and Primiceri (2008). Since the estimation results here are largely consistent withthose reported in previous studies, we can conclude that the time-varying volatilities are well
captured in our model. The inclusion of stochastic volatility in the TVP-VAR model appears
to be essential to appropriately detecting structural changes in the transmission of government
spending shocks.

19

The volatility of the unemployment innovation reported in Cogley and Sargent (2005) and Primiceri (2005)
shares similar time variation.

11

Government spending

GDP

5

3

2.5

1.5

0

0

1960 1970 1980 1990 2000 2010

Debt-to-GDP ratio

GDP deflator

5

0.18

2.5

0.12

0

1960 1970 1980 1990 2000 2010

0.06

1960 1970 1980 1990 2000 2010

1960 1970 1980 1990 2000 2010

FIG. 2. Stochastic volatilities. Notes: The solid lines represent posterior mean with the shaded areas
representing the 16th-84th percentile ranges.

Figure 3 presents the impulse responses of output to government spending shocks at selectedperiods of recessions and expansions. We picked up the responses of the period when the
maximum impact on output (peak multiplier) takes the largest and smallest values among
those during recessions and expansions. The impulse response at time t is computed for each
MCMC replication on the basis of the estimated time-varying parameters at time t.20 We
convert the impulse responses to the government spending multipliers using the sample average
ratio of output to government spending following Auerbach and Gorodnichenko (2012).21

20

Koop, Pesaran and Potter (1996) propose a method to calculate impulse responses considering the history ofobservations that a¤ects impulse responses in non-linear models. However, because we expect a slight di¤erence
from using the computationally demanding method as argued in Koop, Leon-Gonzalez and Strachan (2009),
we follow the simple computational procedure used in Primiceri (2005) and Koop, Leon-Gonzalez and Strachan(2009).
21
Ramey and Zubairy (2018) point out a potential problem arising from the use of the sample average ratio tocalculate multipliers by considering the large variation found in their long samples of historical data. Nevertheless,we use the average ratio not only because it is relatively stable in our post-war sample, but also because we intend
to highlight the time variation in multipliers without interference from changes in the ratio.

12

Recession
1975:Q1

Recession
2008:Q1

10

10

5

5

0

0

-5

-5

-10

5

10

15

-10

20

5

Expansion
1975:Q2
10

5

5

0

0

-5

-5

5

10

15

20

15

20

Expansion
2011:Q4

10

-10

10

15

-10

20

5

10

FIG. 3. Impulse responses during recessions and expansions. Notes: The solid lines represent posterior mean
impulse responses of output to a one-dollar increase in government spending with the shaded areas representing
the 16th-84th percentile ranges.

The upper panels show the impulse responses to government spending shocks during theperiods of recessions. The shapes of the impulse responses are very similar to those estimatedby Blanchard and Perotti (2002). After remaining unchanged or declining for about one year,
the government spending multipliers become larger and reach their highest peak around a threeyear horizon. Comparison of impulse responses at the two representative periods indicates thatthe expansionary output e¤ects become smaller over time, while the peak multipliers are still
positive even in the late 2000s. Time variation in the impulse response is more pronounced
when comparing those during the periods of expansions. The lower panels show the impulse
responses to government spending shocks at the two representative periods of expansions. The
expansionary e¤ects become smaller over time as those observed in recessions, however, negative
multipliers can be observed during the 2010s expansion.

13

Cumulative multipliers
4
2
0
-2
-4
-6

1960

1970

1980

1990

2000

2010

FIG. 4. Evolution of cumulative multipliers. Notes: The cumulative multipliers are calculated as the
cumulative changes in output over the cumulative changes in government spending evaluated at a …ve-year
horizon using the posterior mean impulse responses after a one-dollar increase in government spending. The
shaded areas represent recessions as de…ned by the NBER.

To illustrate the time variation, we compute cumulative multipliers using the posterior mean
impulse responses. Figure 4 presents the cumulative multipliers calculated as the cumulativechanges in output over the cumulative changes in government spending evaluated at a …ve-year
horizon after government spending shocks. Our identi…cation strategies allow us to observeboth cyclical and structural variations. The cumulative multipliers in recessions are larger thanthose in expansions and continue to be positive even in the 2000s while showing moderate
downward trend since the 1980s. Although we de…ne the state of the business cycle relying on
rules of thumb, larger multipliers are mostly observed during periods of recessions de…ned by
the NBER. On the other hand, the cumulative multipliers in expansions exhibit a substantial
declining trend since the 1980s and falls into negative territory in the 1990s. The negative
multipliers indicate that …scal adjustments during these periods can have expansionary e¤ects
on output, providing support for the expansionary austerity hypothesis introduced by Giavazzi
and Pagano (1990).
Table 2 reports the averages of the peak and cumulative multipliers over either the entire
sample or subsample periods, such as recession, expansion, pre-Volcker, and post-Volcker periods. For comparative purposes, we put the corresponding multipliers reported in Auerbach and
Gorodnichenko (2012). In line with the …ndings of previous studies, we obtain larger multipliers

14

in recession than in expansion. The di¤erence between the sizes of multipliers in recession andexpansion is comparable to that reported in Auerbach and Gorodnichenko (2012). The smaller
multipliers observed in the post-Volcker period than those in the preceding period corroborate
the …ndings of Bilbiie, Meier and Müller (2008). The analysis provides evidence of two heterogeneities in multipliers: larger multipliers in recessions than in expansions and smaller ones in
the post-Volcker period than in the preceding period.
TABLE 2Multipliers
Average
Peak

Range
Cumulative

Peak

Cumulative

This paper

AG

This paper

AG

Full sample

1.14

1.00

-0.25

0.57

-1.19 – 4.26

-4.47 – 2.67

Recession

2.28

2.48

0.91

2.24

-0.80 – 4.26

-3.44 – 2.67

Expansion

0.87

0.57

-0.52

-0.33

-1.19 – 4.25

-4.47 – 2.63

Pre-Volcker period

1.97

-

0.93

0.20 – 4.26

-1.19 – 2.67

Post-Volcker period

0.50

-

-1.15

-1.19 – 4.10

-4.47 – 2.53

Notes: The table shows the averages and ranges of the peak and cumulative multipliers calculated over either
the entire sample or the subsample periods (recession, expansion, pre-Volcker, and post-Volcker). The periods ofrecessions and expansions are those de…ned by the NBER. The peak and cumulative multipliers are evaluated at a…ve-year horizon using the posterior mean impulse responses after a one-dollar increase in government spending.
The cumulative multipliers are calculated as the cumulative changes in output over the cumulative changes ingovernment spending. The columns labeled AG report the corresponding results of Auerbach and Gorodnichenko(2012).

3.2

Extended experiments

The results from the baseline model show that the government spending multipliers in the
post-war U.S. are large in recessions and small in the post-Volcker period. Because businesscycle-dependent nature of U.S. multipliers has already been studied extensively, we henceforth
focus our analysis on the change in the size of multipliers between the pre- and post-Volcker
periods. One explanation for the decline is that more active monetary policy during the postVolcker period o¤sets the stimulative e¤ects of government spending strongly (e.g., Bilbiie, Meierand Müller (2008)). On the other hand, it should be recalled that the debt-to-GDP ratio starts
showing an upward trend around the same time as Paul Volcker was appointed as Fed Chairman.
Figure 5 displays a scatter plot of the point estimates of cumulative multipliers during expansions
15

against historical data on debt-to-GDP ratio of corresponding periods. The observed negative
correlation suggests debt-dependent nature of the U.S. multipliers. Therefore, we investigate
whether the decline in multipliers during the post-Volcker period can be attributed to more
active monetary policy or debt accumulation of the period.

Cumulative multipliers

4.0
2.0
0.0
-2.0
-4.0
-6.0
20

40

60

80

100

120

Debt-to-GDP ratio in %FIG. 5. Correlation between multipliers and debt. Notes: The …gure plots the estimated cumulative
multipliers and historical data on debt-to-GDP ratio. The cumulative multipliers are calculated as the cumulativechanges in output over the cumulative changes in government spending evaluated at a …ve-year horizon using
the posterior mean impulse responses after a one-dollar increase in government spending during expansions as
de…ned by the NBER. R-squared: 0.659.

We begin by examining the role of monetary policy by augmenting the baseline model withinterest rate. Like other variables, interest rate is detrended with a linear and quadratic trend.
See Appendix B for a detailed description of the data source. The responses of interest rate toa government spending shock and a demand shock are both left unrestricted. The estimatedvolatility of interest rate shocks depicted in Figure 6 is consistent with those reported in previous
studies (e.g., Cogley and Sargent (2005); Primiceri (2005); Canova and Gambetti (2009); Mumtaz and Zanetti (2013)). The volatility of interest rate shocks increased substantially aroundthe time of Volcker’s appointment and showed a large decline in the early 1980s.

16Interest rate
6

3

0

1960

1970

1980

1990

2000

2010

FIG. 6. Stochastic volatility. Notes: The solid line represents posterior mean with the shaded area representing the 16th-84th percentile range.

As shown in Figure 7, the inclusion of interest rate to the baseline model does not change
the time variation of cumulative multipliers. Cumulative response of interest rate, on theother hand, shows little time variation. Although monetary policy responses during recessions
appear to have become more active throughout the post-Volcker period, the responses during
expansions do not show a clear trend. The negative responses of interest rate seem puzzling
but the counterintuitive results can be found in previous studies, such as Mountford and Uhlig(2009).22 The results do not suggest stronger o¤setting monetary policy response during thepost-Volcker period, leading us to conclude that the decline in multipliers cannot be attributed
to the change in the conduct of monetary policy.

22
Mountford and Uhlig (2009) do not impose a sign restriction on the response of interest rate to a government
spending shock as we did in this paper. Enders, Müller and Scholl (2011) obtain a positive response of interest
rate to a government spending shock while restricting the response to be positive.

17

Cumulative multipliers
4
2
0
-2
-4
-6

1960

1970

1980

1990

2000

2010

2000

2010

Cumulative responses of interest rate
0.5
0.25
0
-0.25
-0.5
-0.75

1960

1970

1980

1990

FIG. 7. Evolution of cumulative responses of output and interest rate. Notes: The cumulative responses arecalculated as the cumulative changes in output [interest rate] over the cumulative changes in government spending
evaluated at a …ve-year horizon using the posterior mean impulse responses after a one-dollar [one-percentage
point] increase in government spending. The shaded areas represent recessions as de…ned by the NBER.

We now proceed to examine the role of debt accumulation. Because the Ricardian channel
has been considered as the primary cause of debt-dependent multipliers (e.g., Huidrom et al.
(2020)), we examine its relevance by augmenting the baseline model with private consumption.
Private consumption is seasonally adjusted, expressed in logarithm and real per capita term,and detrended with a linear and quadratic trend. See Appendix B for a detailed description of
the data source. We impose the same set of restrictions on the response of private consumption
as that on output response. The impulse response of consumption is scaled by the sample
average ratio of consumption to government spending so that the magnitude of the response
is comparable to that of government spending multipliers. Figure 8 illustrates the similarity
between the time variation patterns in the cumulative multipliers and cumulative responses ofconsumption. Their co-movement indicates that the decline in multipliers during post-Volcker
period is mostly led by the consumption responses to government spending shocks.
18

Cumulative multipliers
4
2
0
-2
-4
-6

1960

1970

1980

1990

2000

2010

Cumulative responses of consumption
2
1
0
-1
-2
-3

1960

1970

1980

1990

2000

2010

FIG. 8. Evolution of cumulative responses of output and consumption. Notes: The cumulative responses
are calculated as the cumulative changes in output [consumption] over the cumulative changes in government
spending evaluated at a …ve-year horizon using the posterior mean impulse responses after a one-dollar increasein government spending. The shaded areas represent recessions as de…ned by the NBER.

4

Explaining the debt-dependent multipliers

4.1

Time variation in the magnitude of …scal adjustments

The results from the extended experiments in previous section suggest the relevance of the
Ricardian channel. When public debt is high, households reduce consumption expecting alarger magnitude of …scal adjustments thereby making the e¤ects of government spending lessstimulative. In this section, we turn our attention to the role of …scal adjustments in the debtdependent multipliers. In providing historical evidence of …scal adjustments, Bohn (1998) …nds
a positive correlation between the magnitude of …scal adjustments and the debt-to-GDP ratio.
Therefore, smaller multipliers during times of high debt could be attributed to larger magnitude
of …scal adjustments as assumed in Bi, Shen and Yang (2016).

19

Canzoneri, Cumby and Diba (2001) and Canzoneri, Cumby and Diba (2011) present a VARbased methodology to test whether …scal policy follows a Ricardian regime, i.e., …scal policy
adjusts the path of primary surpluses to stabilize the debt-to-GDP ratio. Their methodology
is attractive because we can easily extend it to our TVP-VAR framework. Following the speci…cation employed in Canzoneri, Cumby and Diba (2001) and Canzoneri, Cumby and Diba
(2011), we estimate a bivariate TVP-VAR model with stochastic volatility in Surplus/GDP and
Liabilities/GDP with two lags. See Appendix B for a detailed description of the data sources.
Figure 9 presents the estimated stochastic volatilities. The overall results for the volatilityof Surplus/GDP shocks shows a pattern similar to the volatility of tax shocks estimated by
Gonzalez-Astudillo (2013), indicating that the stochastic volatility assumption e¤ectively captures the …scal events.23
Liabilities/GDP

Surplus/GDP

12

3

10

2.5

8

2

6

1.5

4

1

2

0.5

0

0

1960 1970 1980 1990 2000 2010

1960 1970 1980 1990 2000 2010

FIG. 9. Stochastic volatilities. Notes: The solid lines represent posterior mean with the shaded areas
representing the 16th-84th percentile ranges.

Figure 10 presents the impulse responses of Surplus/GDP and Liabilities/GDP to an increasein Surplus/GDP. The Surplus/GDP and Liabilities/GDP are ordered …rst in the left and theright panels, respectively. The former is consistent with a non-Ricardian regime and the latter
makes more sense in a Ricardian regime. Because the point estimate of Liabilities/GDP declined
the least and most in 1972:Q1 and 2007:Q3, respectively, when Liabilities/GDP is ordered…rst, Figure 10 compares the impulse responses in these periods. Regardless of the ordering,
Liabilities/GDP declined for several years in response to a Surplus/GDP shock across di¤erent
23
The stochastic volatility of Surplus/GDP shocks increased the most around the time of the Tax Reduction
Act of 1975, and increased during tax reforms and measures such as the Reagan Tax Reform of 1981 and 1986,
the Bush Tax Cuts of 2001 and 2003, and the American Recovery and Reinvestment Act of 2009.

20

sample dates. The results are very similar to those obtained by Canzoneri, Cumby and Diba
(2001) and Canzoneri, Cumby and Diba (2011), suggesting that the U.S. government followed
Ricardian regime throughout the post-war period. As Canzoneri, Cumby and Diba (2001)
and Canzoneri, Cumby and Diba (2011) discuss, a non-Ricardian interpretation is implausible
because it requires a negative correlation between present and future surpluses, which we cannot
observe. Furthermore, the TVP-VAR framework reveals that one unit of deterioration in surplus
leads to a larger decline in Liabilities/GDP in 2007:Q3 than in 1972:Q1. Note that the degreeof the decline in Liabilities/GDP, which measures the magnitude of …scal adjustments, shows awidening trend until 2007:Q3. The observed timing of the largest …scal adjustment is consistent
with D’Erasmo, Mendoza and Zhang (2016) who …nd a evidence of structural change leading
to smaller …scal adjustments in the post-2008 U.S. data.
The increase in the magnitude of …scal adjustments occurred around the same time as thepassage of the Congressional Budget and Impoundment Act of 1974, which established theCongressional Budget O¢ ce. Since then, Congress introduced a variety of budget rules in an
attempt to impose …scal discipline on the budgetary process. By examining the e¤ects of budget
rules, Auerbach (2008) concludes that these rules appear to have had some success with de…cit
control. Such policy change in the presence of debt accumulation might have contributed toraising expectations of the future …scal adjustments, thereby leading to smaller multipliers.

21

Surplus/GDP[ Surplus/GDP; Liabilities/GDP ]Surplus/GDP[ Liabilities/GDP; Surplus/GDP ]

1.2

1.2
1972:Q1
2007:Q3

1
0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

-0.2

5

10

15

-0.2

20

Liabilities/GDP[ Surplus/GDP; Liabilities/GDP ]

5

10

15

20

Liabilities/GDP[ Liabilities/GDP; Surplus/GDP ]00
1972:Q1
2007:Q3

-0.4

-0.4

-0.8

-0.8

-1.2

-1.2

-1.6

-1.6

-2

1972:Q1
2007:Q3

1

5

10

15

-2

20

1972:Q1
2007:Q3
5

10

15

20

FIG. 10. Evolution of surplus and debt dynamics. Notes: The …gure shows impulse responses of Surplus/GDP
and Liabilities/GDP to a one-percentage point increase in Surplus/GDP. The solid lines represent posterior mean
impulse responses for 2007:Q3 with the shaded areas representing the 16th-84th percentile ranges. The solid lines
with circles represent posterior mean impulse responses for 1972:Q1. Surplus/GDP is ordered …rst in the left
column and is reversed in the right column.

4.2

Investigating the debt-multiplier nexus

Our next question is whether the observed increase in the magnitude of …scal adjustments has
contributed to the decline in multipliers during the post-Volcker period. In order to see that, we
examine the Granger-causality relations among the debt-to-GDP ratio, the size of cumulative
multiplier, and the magnitude of …scal adjustments proxied by the cumulative response of
liabilities to a surplus shock estimated in the bivariate TVP-VAR model in Surplus/GDP and
Liabilities/GDP. The cumulative response of liabilities is calculated as the cumulative change in
Liabilities/GDP over the cumulative change in Surplus/GDP using the posterior mean impulse
responses. As the magnitude of …scal adjustments does not depend on the state of the businesscycle, we calculate cumulative multipliers without taking it into consideration.
To cope with possible non-stationarity of the variables, we employ the procedure of Toda
22and Yamamoto (1995) to test for their causal relationship. The …rst step of the procedure is toselect the optimal lag length (k) of the VAR model in levels. The Akaike information criterion(AIC) suggests k = 5. As a second step, we conduct the Augmented Dickey-Fuller (ADF) unit
root test to determine the maximum order of integration (dmax ) that might occur in the model.
Toda and Yamamoto (1995) show that we can test restrictions on the …rst k coe¢ cient matricesof a (k + dmax )th-order VAR model in levels using the standard asymptotic theory, even if
the variables are integrated or cointegrated. We then test the null hypothesis of no Granger
causality using a standard Wald statistic for the …rst k coe¢ cient matrices of the (k + dmax )thorder VAR model in levels. In fact, the ADF test results show that cumulative multipliers anddebt-to-GDP ratios are processes integrated of order 1 (i.e., dmax = 1) as reported in Table 3.
TABLE 3Unit-root and causality test results
Test statistics and p-values

Null hypothesist-statistic
ADF

t-statistic

at levelat …rst di¤erence

CM has a unit root

-2.6573(0.2555)-26.1752(0.0000)**

FA has a unit root

-3.2081(0.0853)*-4.2524(0.0044)**-1.6291(0.7788)-10.9165(0.0000)**

DR has a unit root
Granger (Toda-Yamamoto procedure)

Wald chi-square test statistic

FA does not Granger-cause CM

9.8405(0.0799)*

CM does not Granger-cause FA

9.2073(0.1011)

DR does not Granger-cause FA

16.5662(0.0054)**

FA does not Granger-cause DR

18.8549(0.0020)**

DR does not Granger-cause CM

8.5981(0.1262)

CM does not Granger-cause DR

6.3292(0.2755)

Notes: CM stands for the cumulative multiplier calculated as the cumulative change in output over the
cumulative change in government spending using the posterior mean impulse responses after a one-dollar increasein government spending. FA stands for the magnitude of …scal adjustments measured as the cumulative responseof liabilities to a surplus shock. FA is calculated as the cumulative change in Liabilities/GDP over the cumulativechange in Surplus/GDP using the posterior mean impulse responses after a one-percentage point increase in
Surplus/GDP. In calculating the FA, Liabilities/GDP is placed …rst in the ordering. Both CM and FA are
evaluated at a …ve-year horizon. DR stands for debt-to-GDP ratio. Figures between parentheses are p-values.
A double asterisk (**) denotes signi…cant at the one percent level; a single asterisk (*) denotes signi…cant at theten percent level.

The results suggest that the magnitude of …scal adjustments (FA) appear to Granger-cause
the cumulative multiplier (CM), while the null hypothesis of no-Granger-causality in the opposite direction cannot be rejected. In addition to the unidirectional causality running from themagnitudes of …scal adjustments to the multiplier, we also …nd a bidirectional causality between
23

the debt-to-GDP ratio (DR) and the magnitude of …scal adjustments. Interestingly, the results
do not support direct causal relationships between the debt-to-GDP ratio and the cumulative
multiplier.
Although we provide evidence that multipliers are negatively correlated with debt-to-GDP
ratio, Granger causality analysis tells us that the larger magnitude of …scal adjustments in thepresence of higher indebtedness was the major driving force for the decline in multipliers rather
than debt accumulation itself.

5Conclusions

This study provides new time-series evidence of government spending multipliers during thepost-war period in the United States. We obtain two types of heterogeneities in multipliers:
large multipliers in recessions and small ones in the post-Volcker period. Both phenomena havebeen shown in previous studies, but this study di¤ers from them in that we address the twoheterogeneities in multipliers simultaneously within a single TVP-VAR framework using theassistance of sign restriction identi…cation. In particular, we provide the empirical evidence forthe negative correlation between debt and multipliers by analyzing U.S. time-series data, which
has been neglected in the literature.
Another contribution of the study is to investigate the underlying transmission mechanismof U.S. debt-dependent multipliers. After examining the relevance of the Ricardian channel,
we …nd that the magnitude of …scal adjustments was increased during most of the post-Volcker
period by applying TVP-VAR technique to test for changes in a …scal policy regime. The larger
magnitude of …scal adjustments is shown to be the major driving force for the decline in multipliers rather than debt accumulation itself. The results could have major policy implications
for …scal adjustment strategies during when …scal stimulus is necessary.
Nevertheless, there remains much work ahead. Although our atheoretical approach is a
‡exible way to model the evolution of time-series data, it has limitations in explaining the24underlying mechanism. Therefore the development of a theoretical model that accounts for thetime variation in multipliers and its underlying transmission reported in this paper. Moreover,while we do not consider the relevance of a sovereign risk channel as the U.S. economy has
supposedly not yet reached the …scal limit, it would become worth exploring to consider thechannel as the concerns over the U.S. debt sustainability increase in the future (e.g., Corsetti
et al. (2013); Huidrom et al. (2020)).

A

Markov Chain Monte Carlo Methods

This appendix outlines the MCMC algorithm used to estimate the TVP-VAR models presented
in this paper. Given the data, the algorithm allows us to sample parameters and hyperparameters from their posterior density. In what follows, x denotes the entire history of xt to the endof the sample period. Letting f (x j z) denote the conditional density of x given z, the MCMC
algorithm repeats the following steps:
1. Initialize ; a; h;;a;2. Draw

from f ( j a; h;

3. Draw

from f (a ; y):

a from f ( a j a; y).

6. Draw h from f (h j ; a;
7. Draw; y).

j ; y).

4. Draw a from f (a j ; h;
5. Drawh.

h ; y):

h from f ( h j h; y).

8. Go to 2.
For the …rst step, we set the initial states of the parameters as
N (0; 10I), and hp+1

p+1

N (0; 50I). We postulate an inverse-Gamma distribution for the m-th

diagonal elements of the covariance matrices. The priors are speci…ed as (
(

2
a )m

N (0; 10I), ap+1

IG 5; 10 3 , and (

2
h )m

)2m

IG 10; 10 6 ,IG 5; 10 3 . We execute 30,000 MCMC replications and25

discard the …rst 5,000 draws to estimate the TVP-VAR models. To reduce autocorrelation
among the draws, we save only every …fth draw.

A.1

Drawing

For notational convenience, we rewrite equation (2.1) as

yt = Xt t + At 1

where Xt = Ik

yt0 1 ; :::; yt0 k andt "t ,(A.1)

denotes the Kronecker product. Because the observation

equation (A.1) and state equation (2.4) constitute a linear Gaussian state-space representation
for the dynamic behavior of yt ; we can apply the simulation smoother of de Jong and Shephard(1995) to draw samples of

A.2

from its posterior density conditioned on a; h;; and y:Drawing a

We can write equation (A.1) as
At (ytLet ybt = yt

bt be the k
Xt t and X
2

6
6
6
6
6
6
6
b
Xt = 6
6
6
6
6
6
6
4

Xt t ) =t "t .

k(k 1)
matrix de…ned by
2

0
yb1;t
0
..
.

(A.2)

0
0

0

yb[1;2];t
..

.

0

..

.

..
.

..

.

0

0

yb[1;:::;k 1];t

3

7
7
7
7
7
7
7
7;
7
7
7
7
7
7
5

where yb[1;:::;q];t represents the row vector [by1;t ; yb2;t ; :::; ybq;t ], where qk

1. We can express

equation (A.2) as

bt at +
ybt = X
26

t "t :(A.3)

Because the observation equation (A.3) and state equation (2.5) can be treated as a linear
Gaussian state-space model, assuming thata anda0 are diagonal matrices, we can apply thesimulation smoother of de Jong and Shephard (1995) to draw samples of a from its posterior
density conditioned on ; h;A.3

a ; and y:

Drawing h

For the stochastic volatilities, we apply the multi-move sampler developed by Shephard and Pitt(1997) and Watanabe and Omori (2004) because the system of equations consists of (A.1), and(2.6) is not linear in h. The diagonality assumptions ofh andh0 allow us to make inferenceson fhj;t gTt=p+1 separately for j = 1; :::; k. Let yj;t be the j-th element of At ybt . Now, consider
the following system of equations:hj;t2yj;t = exp

hj;t+1 = hj;t +

"j;t
j;t

0 0

B B 1
B
sNB
@0; @
0

"j;t ;

(A.4)

j;t ;

(A.5)
11

0 CC
CC ;
AA
2
j

where "j;t andj;t are the j-th elements of "t and uh;t , respectively, andelement of

The prior distribution for the initial value is given byh.

2 is the j-th diagonal element of
j;o

j;p

is the j-th diagonal
N (0;2 ), where
j;o

h0 .

Drawing samples of h conditional on ; a;h ; and y is di¢ cult because of an analytically in-tractable form of its posterior density. One way is to draw each sample of ht conditional on hnt ;; a;

24
h ; and y; however, the method tends to produce a highly correlated sample sequence.

Therefore, we divide the state variables fhj;t gTt=p+1 into K +1 blocks and draw each block conditional on the elements of the other blocks and parameters. Let the end elements of the blocks be

24

See Shephard and Pitt (1997) and Kim, Shephard and Chib (1998).

27

hj;kn for n = 1; :::; K. The end conditions of blocks kn , called “stochastic knots,”are determined
randomly over iterations. To cope with possible degeneracies, we drawj;p+1 ; :::; j;T 1 insteadof hj;p+2 ; :::; hj;T ; which can be constructed using (A.5) given the sampled hj;p+1 . Suppose we
draw samples from a typical block hj;r ; :::; hj;r+d , where r

p + 1; d1; and r + d

T . ByBayes’ theorem, the posterior conditional density of a block of disturbances can be expressedasf ( j;r 1 ; :::; j;r+d 1 j hj;r 1 ; hj;r+d+1 ; yj;r ; :::; yj;r+d ;
_

r+d
Y

1
ehj;t =2

t=r

yj;t2

exp

2ehj;t

where
f

j;t

f (hj;r+d ) =To draw

=

!

8
>
>
>
< exp

f

(A.6)

f (hj;r+d ) ;

j;t

t=r 1

2
j;p
2 2j;o

>
>
>
: exp

8
>
>
< exp

r+d
Y1

j ; j;o )2j;t(t = p);(t

2 2j

p + 1);
2(hj;r+d+1 hj;r+d )
2 2j

>
>
: 1

(r + d < T );(r + d = T ):j;r 1 ; :::; j;r+d 1 from the density (A.6), we consider a proposal density expressedin logarithmic form by taking the logarithm of (A.6) and applying the second-order Taylor
approximation tohj;t2g (hj;t )yj;t22ehj;taround a certain point hj;t = b
hj;t , which we choose to be near the mode of the posterior density.
We can sample from the proposal density by de…ning arti…cial variables

hj;t =

8
>
>
<

j;t

g (b
hj;t )>>: b
hj;t +

0

0

b

00
h
g (b
hj;t )b
hj;t + j;t+1
2(t = r + d < T );jj;t g (hj;t )28(t = r; :::; r + d

1 and t = r + d = T );

where
j;t =

8
>
>
<
>
>
:

2
j
00 b

(t = r + d < T );1 g (hj;t ) 2j
1
g 00 (bhj;t )(t = r; :::; r + d

1 and t = r + d = T );and then considering the following equation

hj;t = hj;t +

where

j;t

N (0;

j;t ;

(A.7)

j;t ): Using (A.5) and (A.7), we can formulate a linear Gaussian state-spacemodel and draw samples applying the simulation smoother of de Jong and Shephard (1995).
The sampling is the same as that for the proposal density. Hence, we use the Accept-Reject
Metropolis-Hastings algorithm in Tierney (1994) to produce draws from the correct density(A.6) as the sampling process is iterated.

A.4

Drawing hyperparametersLet= ( ; a; h); m-th element ofcovariance matrix oft be

be given by (m;t ; and the prior for the m-th diagonal element of the)2m

IG(s 0 =2; S 0 =2): Because we assume thatis adiagonal matrix, the m-th diagonals of covariance matrix ()2m can be sampled independently.

The posterior density conditioned on)2m j

is then given by (sb m = s 0 + T

Sb m = S 0 +

T
X1

p

( m;t+1

t=p+1

29

1;
2
m;t ) :

IG(bs m =2; Sb m =2); whereB

Description of Data Sources

We obtain all quarterly data from the Federal Reserve Bank of St. Louis’s FRED database.
Seasonally adjusted series for real government spending, real gross domestic product, and real
private consumption are Real Government Consumption Expenditures and Gross Investment(GCEC96), Real Gross Domestic Product (GDPC1), and Real Personal Consumption Expenditures (PCECC96), respectively. To convert the series into per-capita terms, we divide them by
the seasonally adjusted Civilian Labor Force (CLF16OV). The ratios of output and consumption
to government spending used to calculate the multipliers are constructed from seasonally adjusted series for Gross Domestic Product (GDP), Personal Consumption Expenditures (PCEC),
and Government Consumption Expenditures and Gross Investment (GCE), respectively. We
use the seasonally adjusted GDP Chain-type Price Index (GDPCTPI) as the price. We use
the 3-Month Treasury Bill Secondary Market Rate (TB3MS) as the nominal interest rate. The
debt-to-output ratio is calculated by dividing the sum of federal, state, and local government
liabilities by the seasonally adjusted Gross Domestic Product (GDP). We use the seasonally adjusted liabilities of the Federal government (FGDSLAQ027S) and those of the State and local
governments (SLGLIAQ027S) in the calculation. The Surplus/GDP is calculated by dividing
the seasonally adjusted primary surplus by the Gross Domestic Product (GDP). The primary
surplus is de…ned as Net Government Saving (TGDEF) minus the di¤erence between income
receipts on assets (W059RC1Q027SBEA) and interest payments (A180RC1Q027SBEA). The
Liabilities/GDP is calculated in the same way as the debt-to-output ratio.

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