HIAS-E-97

Empirical Analysis on the Eﬀects of Japanese Fiscal Policy
under the Eﬀective Lower Bound
Hiroshi Morita
Institute of Comparative Economic Studies, Hosei University 4342 Aihara, Machida, Tokyo 194-0298, Japan

March, 2020

Hitotsubashi Institute for Advanced Study, Hitotsubashi University
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Empirical Analysis on the Eﬀects of Japanese Fiscal Policy under
the Eﬀective Lower Bound✩
Hiroshi Morita∗
Institute of Comparative Economic Studies, Hosei University
4342 Aihara, Machida, Tokyo 194-0298, Japan

Abstract
For the Japanese economy, we examine whether the fiscal multiplier is higher underthe eﬀective lower bound of the nominal interest rate. Using a time-varying parameter
vector autoregression model with Tobit-type nonlinearity, we calculate the fiscal multipliers under two monetary policy positions. We find that when government spending shocks
are inflationary, the fiscal multiplier under the zero interest rate policy increases steadily
as a result of the decrease in the real interest rate. This evidence is robust to diﬀerent
definitions of eﬀective lower bound, output, and government spending.
JEL classification: E62, C32, E52
Keywords: TVP-VAR model; Fiscal multiplier; Eﬀective lower bound; Implied rate✩

I am grateful to Etsuro Shioji, Toshiaki Watanabe, Takayuki Tsuruga, Junko Koeda, Ryuzo Miyao,
and the seminar participants of Frontiers in Macroeconomics and Macroeconometrics held at Hitotsubashi
University, the 11th International Conference on Computational and Financial Econometrics, and theWestern Economic Association International 14th International Conference for their helpful comments
and suggestions. This research was funded by Grant-in-Aid for Young Scientists 19K13652 and the
Nomura Foundation. Errors, if any, are entirely the author’s responsibility.
∗
E − mail: morita-h@hosei.ac.jp, T el: +81-42-783-2558.

1. Introduction
In response to the economic downturn after the global financial crisis in 2008, thecentral banks of the major industrialized economies adopted a zero interest rate policy(ZIRP). As a result, their nominal short-term interest rates have fallen to the eﬀective
lower bound (ELB). Although there remains room to implement unconventional monetary actions, such as quantitative easing and forward guidance, the central banks havelost a conventional monetary tool they would otherwise use to stimulate their economies
under a liquidity trap. On the other hand, the fiscal authorities in these economies have
implemented aggressive large-scale fiscal stimulus packages within this low interest rateenvironment (e.g., the American Recovery and Reinvestment Action in the United States,
and the Policy Package to Address the Economic Crisis in Japan). The consecutive implementation of fiscal stimulus packages has ignited debate on the eﬀectiveness of fiscal
policy and, in particular, whether the fiscal multiplier is higher under the ELB of thenominal interest rate. In the aftermath of the crisis, the literature focused on answering
this question within a theoretical framework, resulting in conflicting theoretical predictions. For example, some studies reported that the fiscal multiplier becomes much larger
when the economy is in the ELB (Braun and Waki, 2006; Eggertsson, 2010; Christiano etal., 2011), whereas others demonstrated that it does not necessarily become larger in the
ELB (Erceg and Lindé, 2014; Boneva et al., 2016). This theoretical conflict motivated us
to examine this question empirically.
Using a structural time-varying parameter vector autoregression (TVP-VAR) modelwith a censored variable, we analyze whether the Japanese government spending multiplierbecomes larger under the ELB. Long samples from the period near the zero lower boundin Japan are suitable for our study. It is important that we separate the contribution of
the ZIRP from those of all other possible factors that might aﬀect the fiscal multiplier. For
instance, Auerbach and Gorodnichenko (2012) and Fazzari et al. (2015) demonstrate that
fiscal policy is more eﬀective when the economy is in a recession. However, because thelow interest rate environment virtually coincides with a recession, it is diﬃcult to evaluate
the contribution of the ZIRP separately by comparing the multipliers calculated in the
ELB and non-ELB periods, as in Ramey and Zubairy (2018) and Miyamoto et al. (2018).
In other words, it is probable that the eﬀects of the low interest rate and the recession
are mixed in the size of the multipliers estimated after dividing the sample period. Our2

model includes three notable features to clarify the role of the ZIRP in determining thesize of the fiscal multiplier. First, the time-varying specification allows us to estimate
the policy eﬀect in each period without dividing the sample period. Second, we employ
Iwata and Wu’s (2006) Tobit specification, which treats the short-term interest rate as acensored variable. Third, the interest rate equation in our VAR model is formulated asa structural Taylor rule, not a reduced-form equation. This formulation has been usedin some empirical studies (Stock and Watson, 2001; Hayashi and Koeda, 2019), and isadopted here to accord with the theoretical model mentioned above. Exploiting the timevarying structure and the Tobit specification, we calculate two impulse responses (IRs)and fiscal multipliers at each period, one of which is derived from the lower bound valueof the interest rate (referred to as the benchmark), while the other uses the implied rate
obtained from the Tobit model (referred to as the counterfactual). Because the diﬀerence
between the two stems only from the responses of the interest rate, we can regard it
as the contribution of the monetary policy stance to the fiscal multiplier. Hence, ouranalysis diﬀers from those of Ramey and Zubairy (2018) and Miyamoto et al. (2018) inthat we eliminate the possibility of factors other than the ZIRP aﬀecting the size of themultipliers.
As mentioned above, we conduct a counterfactual simulation in which hypotheticalresponses are drawn by changing only the coeﬃcients in the interest rate equation. It is
well known that this sort of simulation is likely to conflict with Lucas’s (1976) argumentthat the coeﬃcients of other equations in the VAR system also change if the monetary
policy rule is changed. This occurs because rational agents immediately adjust their
behavior in response to the policy changes. To resolve this problem, Hayashi and Koeda(2019) estimate the reduced-form coeﬃcients for each monetary policy regime, and thenuse these to compute the counterfactual responses. This is the conventional solution to
the above problem in a VAR analysis. However, in practice, it is not clear whether the
behavior of the private sector is reversible when a policy regime switch occurs. In theJapanese economy in particular, private agents are unlikely to act as before if the central
bank lifts the zero interest rate, because the economy has been in the ELB since themid-1990s. Therefore, we conduct our counterfactual simulation following Sims and Zha(2006), who argue that the nonlinearities stemming from a regime change, as emphasizedby Lucas (1976), matter only as far as private agents change their beliefs on the policy

3

regime. Leeper and Zha (2003) also show that the hypothetical path conditioning on
invariant coeﬃcients in the nonpolicy sector could suﬃciently approximate the true eﬀects,
unless the policy intervention shifts agents’ beliefs. Based on this argument, we run ourcounterfactual simulation assuming that the private sector and the fiscal authority cannot
instantly shift their beliefs on the prevailing monetary policy rule when the monetary
authority changes this rule. Indeed, the Japanese policy rate has fluctuated only slightly
with daily transactions in the short-term money market during the ELB period. Thus, it
seems reasonable to assume that agents cannot detect a change in the monetary policy rulein the short run. On the other hand, our counterfactual is naturally subject to the Lucas
critique in the long run, because agents can adjust their beliefs accordingly. Therefore, in
the actual estimation below, we calculate the hypothetical responses for two years after
the shock, during which the agents’ beliefs on monetary policy are unlikely to be adjusted
adequately.
The main finding of this study is that when a government spending shock is inflationary, the fiscal multiplier under the zero interest rate policy increases steadily as a resultof the decrease in the real interest rate. This finding is consistent with the mechanisms
presented in Eggertsson (2010) and Christiano et al. (2011). In addition, we reveal substantial variations over time in the Taylor rule coeﬃcients and in the IRs of each variable
to a government spending shock, which supports our time-varying specification of themodel. Moreover, the estimated implied interest rate takes a negative value with active
fluctuations during the ELB period. This implies that the interest rate may have changed
in response to structural shocks had the Bank of Japan (BOJ) not successfully controlled
the short-term interest rate.
The remainder of this paper is organized as follows. In Section 2, we explain ourempirical framework. Specifically, we describe the structural TVP-VAR model with Tobit
nonlinearity, the Bayesian technique, and our data. Section 3 presents the estimatedresults for the implied rate, IRs, and fiscal multipliers. In the robustness check in Section
4, we change the definitions of the ELB period, output measure, and government spending
measure, and then re-estimate the model. Finally, Section 5 concludes the paper.

42. Empirical model
2.1. TVP-VAR model with censored variables
To examine the eﬀect of a government spending shock under the ELB of the nominal
short-term interest rate, we employ a four-variable TVP-VAR model with a censoredvariable. The VAR model incorporates the quarter-on-quarter growth rate of government
spending (gt ), inflation rate (pt ), GDP growth rate (xt ), and nominal interest rate, in thisorder. The notable feature of our VAR model is that we impose the Tobit specificationon the nominal short-term interest rate. As in Iwata and Wu (2006), we distinguishthe observed nominal interest rate (rt ) and latent rate (rt∗ ). Here, we suppose that theobserved rate is equal to the latent rate if it exceeds the lower bound of the nominalinterest rate, and is equal to the lower bound otherwise. Denoting the lower bound of thenominal interest rate as c, the relationship between rt and rt∗ is specified as⎧⎨ r∗
if rt∗ > c
t
rt =
⎩ c
if r∗ ≤ c.

(1)t

Other than adopting time-varying parameters and a censored variable, our VAR modelis essentially the same as those employed by Stock and Watson (2001) and Hayashi andKoeda (2019), in the sense that the first three equations are reduced-form equations andthe fourth equation is a structural Taylor rule.
Defining Yt = [gt , pt , xt ]′ and Ỹt = [gt , pt , xt , rt ]′ , the three reduced-form equations areformulated as
Yt = B0,t + B1,t Ỹt−1 + · · · + Bs,t Ỹt−s + A−1t Σt ε t ,εt ∼ N (0, I).

(2)

$
%
In addition, we define θt = [vec(B0,t )′ , · · · , vec(Bs,t )′ ]′ and Xt = I ⊗ 1, Ỹt−1 , · · · , Ỹt−s ,where the vec operator creates a column vector from Bi,t (i = 1, · · · , s) by stacking thecolumn vectors of Bi,t , and ⊗ denotes the Kronecker product. Exploiting these newly
defined variables, eq. (2) can be rewritten as
Yt = Xt θt + A−1t Σt εt .

(3)Here, εt = [εg,t , εp,t , εx,t ]′ is a vector of structural shocks, which are assumed to follow
mutually independent standard normal distributions; At is a matrix of time-varying coeﬃcients specifying the simultaneous relationship between the endogenous variables; and5Σt denotes the time-varying volatilities of the structural shocks in the diagonal elements.
We assume At and Σt take the following form:
⎛
⎞
⎛
⎞
1
0 0
σ
0
0
⎜
⎟
⎜ g,t
⎟
⎜
⎟
⎜
⎟
At = ⎜ agp,t
1 0 ⎟ , and Σt = ⎜ 0 σp,t 0 ⎟ ,⎝⎠⎝⎠agx,t apx,t 100 σx,t(4)

respectively, where akl,t denotes the simultaneous response of variable l to the shock εk,tat time t, and σl,t denotes the time-varying volatility of the structural shock εl,t . In
other words, the structural shocks are identified by a recursive restriction, assuming thatgovernment spending is the most exogenous variable not simultaneously aﬀected by otherstructural shocks.1
The fourth equation is the time-varying censored Taylor rule with interest rate smoothing, specified as
rt = max [c, rt∗ ] ,rt∗ = (1 − ρt ) [φ0,t + φπ,t πt + φχ,t χt ] + ρt rt−1 + εr,t ,(5)2
εr,t ∼ N (0, σr,t).

The implied interest rate is assumed to be a function of the contemporaneous valuesof year-on-year inflation (πt ≡ pt + pt−1 + pt−2 + pt−3 ), year-on-year GDP growth rate(χt ≡ xt + xt−1 + xt−2 + xt−3 ), and the lagged observed interest rate. Following theliterature on fiscal policy analysis (e.g., Blanchard and Perotti, 2002; Galı́ et al., 2007),
we employ the GDP growth rate rather than the more usual output gap (e.g., Taylor,
1993; Clarida et al., 1998) as an output measure in the Taylor rule.2 We also incorporate
inflation and the GDP growth rate into the equation as year-on-year rates, assumingthat the central bank adjusts the policy rate in response to the slow movement of the1

Perotti (2002) discusses the possibility of a simultaneous response of real government spending toprice. That is, real government spending may fluctuate in response to inflation when government spending is contracted in nominal terms. In the time-varying specification, we encounter diﬃculty when we
consider such simultaneity in our estimation of the time-varying contemporaneous matrix. Fortunately,
as noted by Kato (2003), in a constant-parameter VAR model, the qualitative eﬀect on output remains
unchanged independently of the contemporaneous elasticity of government spending to price. Although
we recognize the importance of this simultaneous relationship, we adopt a specification in which pricehas no contemporaneous eﬀect on government spending.
2
In the robustness check, we estimate the model after replacing the GDP growth rate with the GDP
gap in order to reconcile with the standard Taylor rule specification.

6

economy. In addition, the error term in eq. (5) is assumed to be orthogonal to the errors
in eq. (2). As mentioned above, the VAR system employed in this study comprises three
reduced-form equations (2) and a structural Taylor rule (5). Note that the implied interest
rate appears only on the left-hand side of the system in order to be consistent with thespecification of Iwata and Wu (2006).
Following the seminal work of Primiceri (2005) on the TVP-VAR model, the timevarying parameters in the proposed model are assumed to follow a random-walk process.
Let αt be a stacked vector of the lower triangular elements in At , ht = (hg,t , hp,t , hx,t , hr,t )′2
with hl,t = ln(σl,t), and ϕ = (ϕ0,t , ϕπ,t , ϕχ,t , ρt )′ with ϕm = (1−ρt )φm,t . Then, the dynamic

process of the model’s time-varying parameters is given by
⎛
⎞ ⎛
⎛
⎞⎞
θt+1 = θt + uθ,t ,uθ,tΣθ O O O
⎜
⎟⎟
⎜
⎟ ⎜
⎜
⎟ ⎜
⎜
⎟⎟
⎜ uα,t ⎟ ⎜⎜ O Σ α O O ⎟⎟αt+1 = αt + uα,t ,⎜⎟ ∼ ⎜ 0, ⎜
⎟⎟ ,
⎜
⎟ ⎜
⎜
⎟⎟
⎜ uϕ,t ⎟ ⎜⎜ O O Σ ϕ O ⎟⎟ϕt+1 = ϕt + uϕ,t ,
⎝
⎠ ⎝
⎝
⎠⎠
ht+1 = ht + uh,t ,uh,tO O O Σh(6)

for t = s+1, · · · , T , where θs+1 ∼ N (µθ,0 , Σθ,0 ), αs+1 ∼ N (µα,0 , Σα,0 ), ϕs+1 ∼ N (µϕ,0 , Σϕ,0 ),and hs+1 ∼ N (µh,0 , Σh,0 ). The model innovations in eq.(6) are assumed to be mutuallyuncorrelated, and the variance-covariance structure for (Σθ , Σα , Σϕ , Σh ) is assumed to bea diagonal matrix.
2.2. MCMC estimation
As widely employed in the literature, we estimate the TVP-VAR model with a censoredvariable using the Gibbs sampler of the Bayesian Markov Chain Monte Carlo (MCMC)
method. Because our model includes latent variables in addition to many parameters,
the posterior distribution is too complicated to calculate analytically. Thus, the MCMC
method is appropriate for our estimation.
Define θ = {θt }Tt=s+1 , α = {αt }Tt=s+1 , ϕ = {ϕt }Tt=s+1 , h = {ht }Tt=s+1 , r∗ = {rt∗ }Tt=s+1 ,Ω = (Σθ , Σα , Σϕ , Σh ), and y = {Ỹt }Tt=1 . Given the data y and prior density function

π(Θ), where Θ = (θ, α, ϕ, h, r∗ , Ω), the samples from the posterior distribution π(Θ|y) are
obtained as follows:
1. Set initial values of θ(0) , α(0) , ϕ(0) , h(0) , r∗ (0) , Ω(0) , and j = 0.
$%(j) ∗ (j)(j+1)(j)
2. Draw ϕfrom π ϕ | Σϕ , r , h , y .
,(j+1)
3. Draw Σϕfrom π Σϕ | ϕ(j+1) .
7

(j+1)

$

(j)

(j)

(j)
, Σβ , y%from π θ | α , h
.
,
(j+1)
5. Draw Σθfrom π Σθ | θ(j+1) .
$%(j)
6. Draw α(j+1) from π α | θ(j+1) , h(j) , Σα , y .
,(j+1)
7. Draw Σαfrom π Σα | α(j+1) .
$
%
(j)
(j+1)
(j+1)
(j+1)
(j+1)
8. Draw hfrom π h | θ,α,ϕ, Σh , y .
,(j+1)
9. Draw Σhfrom π Σh | h(j+1) .
,10. Draw r∗ (j+1) from π r∗ | ϕ(j+1) , hj+1 , y .
4. Draw θ

11. Return to step 2 until N iterations have been completed.
For the above process, N is set to 50,000. However, the first 20,000 samples are discarded
as a burn-in. Then, every 10th draw is saved in order to alleviate the autocorrelation
between the draws. Thus, the inference exhibited below is implemented based on 3,000
samples. Moreover, we only select draws in which the roots of the VAR coeﬃcients are
inside the unit circle throughout the sample period. This ensures the stationarity of the
VAR system.
The sampling of ϕ, θ and α in steps 2, 4, and 6 are conducted using the Kalmanfilter and smoother, and the stochastic volatility h in step 8 is sampled using the multimove sampler of Watanabe and Omori (2004), whose algorithm generates samples fromthe exact conditional posterior of h. The inverse of variances in Ω are generated from
Gamma distribution under conjugate priors. The implied interest rate in the ELB period
is generated from a truncated normal distribution.
2.3. Data and specification
Our empirical analysis uses quarterly data on Japan’s government spending, consumer
price index (CPI; all items, less fresh food), GDP, and nominal interest rate for the period
1985Q3 to 2019Q1. Figure 1 shows the time series for these data. The data on government
spending and GDP are obtained from the System of National Accounts, published by the
Cabinet Oﬃce, Government of Japan. We define the sum of government consumption andpublic investment as government spending. The CPI data are released by the Statistics
Bureau of Japan. This price index is eﬀectively a policy target of the BOJ. The nominal
short-term interest rate is the uncollateralized overnight call rate (end of month), which
was the policy rate in Japan before the adoption of the ZIRP. Because this series isavailable only after July 1985, our sample period starts from 1985Q3. All data other than8the short-term interest rate are seasonally adjusted, and government spending and GDP
are transformed into per capita values by dividing them by the total population.3 The lag
length in the TVP-VAR model, denoted by s in eq. (2), is set to three, following the lag
structure in the Taylor rule eq. (5), which includes the year-on-year inflation and GDP
growth rates.
[Figure 1 about here.]
The ELB periods in this study essentially adhere to the definition of Hayashi andKoeda (2019), in which a period is part of the ELB if the net policy rate (the call rateminus the rate paid on reserves) is below 50 basis points.4 Based on this definition, we
regard the periods from 1999Q1 to 2000Q1, 2001Q1 to 2006Q2, and 2008Q4 to date as
the ELB periods. The BOJ has set the rate paid on the reserve to 0.1% since November
2008; thus, the periods after 2008Q4 are also categorized as ELB periods, even though
the call rate is above 0.05%. In addition, the BOJ has adopted a negative interest rate
policy since January 2016, imposing a negative interest rate of 0.1% on part of the excess
reserve. Taking into account this transition of the BOJ’s monetary policies, the lower
bound of the nominal interest rate, denoted as c in eq. (1), is set to 0.05 for the periods
1999Q1 to 2000Q2 and 2001Q1 to 2006Q2, 0.1 for the period from 2008Q4 to 2015Q4,
and −0.1 for the period after 2016/Q1. During the ELB period, the implied interest rate
is generated from a normal distribution truncated between −∞ and c, based on eq. (5).
The priors for the inverse of the ith diagonal element of Σj (j = θ, α, ϕ, h) are assumed
to follow an independent gamma distribution, as follows:−2(Σθ )−2i ∼ Gamma(10, 0.01), (Σα )i ∼ Gamma(2, 0.01),(Σϕ )−2i ∼ Gamma(2, 0.01),3(Σh )−2i ∼ Gamma(2, 0.01).

(7)

In addition to seasonally adjusting the data, we remove the eﬀect of the increase in the consumption

tax from the CPI. Specifically, we first perform an X-12-ARIMA, with level shift dummies that takethe value one at April 1989, April 1997, and April 2014 in the original monthly CPI series. Then, we
eliminate the coeﬃcient associated with each dummy from the seasonally adjusted series, and construct
the quarterly series by taking the average over three months.
4
Iwata and Wu (2006) and Nakajima (2011) set the threshold between the ELB and non-ELB periods
to 50 basis points of the nominal short-term interest rate, without paying attention to the interest rate
payment on the bank reserve. Here, we follow the stricter definition adopted in Hayashi and Koeda(2019).

9

These priors are set based on Nakajima et al. (2011), who also estimate a four-variable
TVP-VAR model with quarterly frequency for the Japanese economy. For the initial stateof the time-varying parameters, we set the following priors:θs+1 ∼ N (0, I),

αs+1 ∼ N (0, I),

ϕs+1 ∼ N (ϕ̄, I), hs+1 ∼ N (0, I).

(8)With the exception of the initial state of the coeﬃcients in the Taylor rule, we adopt flatpriors rather than the ordinary least square estimators obtained in the pre-sample, which
is an alternative way of selecting the initial state of the time-varying parameters (e.g.,
Primiceri, 2005; Hofmann et al., 2012). We do so because we cannot exploit a suﬃcient
size of pre-sample to estimate the model, owing to the aforementioned lack of data. On
the other hand, the initial state of the parameters in the Taylor rule, denoted by ϕ̄, is set
based on the results of Clarida et al. (1998), who estimate the Japanese Taylor rule forthe period April 1974 to November 1994.5
3. Estimation results
3.1. Parameter estimatesFirst, we check the convergence of our MCMC algorithm. Following Primiceri (2005),
we show (a) the 20-th-order sample autocorrelation, (b) the ineﬃciency factor (IF), and(c) Raftery and Lewis’s (1992) total number of runs for the hyperparameters (50 freeelements of Σθ , Σα , Σϕ , and Σh ) and volatilities (four sets of volatility states for 132
periods); see Figure 2.
[Figure 2 about here.]
In Figure 2(a), most of the 20-th-order autocorrelation values are around zero, except
for the volatilities of the shocks in the Taylor rule between points 447 and 578. However,even in the volatilities of the shocks in the interest rate equation, the values are allless than 0.4. Related to the autocorrelation, Figure 2(b) plots the IF for the posterior.
estimates of the parameters, which are defined by 1 + 2 ∞τ =1 ρτ , where ρτ is the sample
autocorrelation at lag τ . In Figure 2(b), the IF estimates are approximated by summing

5

Clarida et al.

(1998) report the following estimates for the Taylor rule:[1.21, 2.04, 0.08, 0.93].

10[φ0 , φπ , φχ , ρt ] =the autocorrelations until lag 100. The IF is a measure of eﬃciency, indicating the numberof samples per MCMC sample needed to obtain the same variance of the sample meanfrom uncorrelated draws. Most of the IFs are below 40, and even the largest is far lessthan 100. Thus, we consider our MCMC algorithm to be satisfactory for inference, as
noted in Primiceri (2005). Furthermore, the Raftery and Lewis (1992) diagnostic of thetotal number of runs required to achieve a certain precision, shown in Figure 2(c), isalways below the total number of iterations in our estimation.6 In summary, the MCMC
algorithm converges suﬃciently and is reasonable for posterior inference.
Figure 3 reports the time profiles for the estimated structural parameters in the Taylor
rule throughout the sample period. Thick and thin solid lines indicate median and 68%
credible intervals, respectively, for the estimates, and the shaded areas denote ELB periods. In contrast to prior studies that estimate the Japanese Taylor rule, our estimation
includes the period after the BOJ adopted the ZIRP.7 During the ELB spells, the Tobit
specification allows us to exploit the implied rate and the implied monetary policy rulethe monetary authority would adopt if there were no ELB on the interest rate. In the
top-left chart in Figure 3, we show the variation of the time-varying intercept, denotedby ψ0,t . As originally stated in Taylor (1993), the intercept in the Taylor rule stands forthe equilibrium real interest rate, and is set to be consistent with the growth rate in thesteady state, that is, the potential growth rate of the economy. Considering the dynamics
of ψ0,t from the viewpoint of Taylor’s (1993) original formulation, our estimate can beinterpreted as reflecting the decrease in the total factor productivity growth rate documented by Hayashi and Prescott (2002). In Figures 3(b)–(d), we plot the dynamics of thecoeﬃcients for inflation and output and the speed adjustment parameters, respectively.
The ranges of the estimated parameters are essentially compatible with the estimates reported in the literature. Specifically, as summarized in Miyazawa (2011), previous studies
estimate the inflation coeﬃcient, output coeﬃcient, and inertial parameter as lying in theranges of [1.33, 3.84], [0.17, 1.001], and [0.55, 0.93], respectively; these ranges are indicated6

Following Primiceri (2005), the parameters used to specify Raftery and Lewis’s (1992) diagnostics

set are as follows: quantile = 0.025; desired accuracy = 0.025; and required probability of attaining therequired accuracy = 0.95.
7
As noted in Miyazawa (2011), the sample periods of most studies stop before the adoption of the
ZIRP. One exception is the work of Hayashi and Koeda (2019), whose sample period covers 1992 to 2012.

11

by dotted lines in Figure 3. In addition, Hayashi and Koeda (2019) find that the inflationand output coeﬃcients are 0.69 and 0.05, respectively, and the speed of adjustment isestimated to be 9.8% (i.e., ρ = 0.902) for the sample period 1992–2012.
Note that our results exhibit substantial variations in the values of the structuralparameters over time, which are not considered in previous studies. The estimation resultsfor ψπ,t , ψx,t , and ρt show trend changes around the mid-1990s, when the interest rate falls
to near zero. The inflation coeﬃcient then becomes small and stable, while the output
coeﬃcient becomes more volatile. A possible reason for this result is that the variation in
inflation is relatively smaller than that of output during the (two) lost decades in Japan.
In contrast, it is easy to interpret the dynamics of the inertial parameter. The parameter
takes the bottom value around 1995 when the BOJ frequently cut the policy rate to copewith the economic slowdown stemming from the burst of the asset price bubble in theearly 1990s. Overall, the estimation results emphasize the variations of the parameters
over time in the Taylor rule, and thereby support the time-varying formulation of ourestimation.
[Figure 3 about here.]
Figure 4 depicts the estimated series of the implied interest rate (solid line), as wellas the actual rate (dotted line). The figure shows that the policy rate takes negativevalues if there is room to cut the rate in the ELB period. In other words, this result
implies that the interest rate may fluctuate in response to structural shocks in the ELB
period if there were no lower bound. For comparison purposes, we plot the shadow rateestimated by Krippner (2015) (dot–dash line). The negative interest rate in this study isconstructed using the Taylor rule, whereas Krippner (2015) estimates the rate using theterm structure model and the information included in the yield curve.8 To highlight thediﬀerences between the estimation methods and to emphasize that our estimated rate is
the hypothetical rate set by the central bank if no lower bound exists, as in Iwata andWu (2006), we refer to our estimated negative interest rate as the implied rate rather
than the shadow rate. In spite of the methodological diﬀerences, the variations of our
implied rate are fairly close to the shadow rate estimated in Krippner (2015), until thesecond ELB period. However, the gap between the two gradually becomes wider in the
8

Ueno (2017) also estimates the shadow rate in Japan using the term structure model.

12third ELB period. Our implied rate decreases rapidly in response to the significant dropin the GDP growth rate just after the global financial crisis; however, no such decrease isobserved in the estimates of Krippner (2015). Moreover, in contrast to our estimates, theshadow rate of Krippner (2015) describes a persistent downward trend in the third ELB
period, which coincides with the BOJ’s adoption of quantitative and qualitative easing
with yield curve control, aimed at shifting the entire yield curve downward. Although
structural models are often used to estimate the shadow rate under the lower bound of thenominal interest rate (Krippner, 2015; Ueno, 2017), Iwata and Wu (2006) and Nakajima(2011) estimate the implied rate using the Tobit model and the (reduced-form) monetary
policy rule. As stated above, the primary purpose of this study is to empirically check
whether the theoretical predictions presented in Eggertsson (2010) and Christiano et al.
(2011) hold. Thus, we regard our implied rate estimated from the structural Taylor rule
as a counterpart of the policy rates discussed in Eggertsson (2010) and Christiano et al.
(2011), and use this estimate in our analysis.
[Figure 4 about here.]
3.2. Impulse responses
To examine the eﬀect of a government spending shock on output, we derive the IRs
from the estimated TVP-VAR model. Owing to the nonlinearity of the TVP-VAR modelwith a censored variable, the IRs in this study are calculated using the generalized IR
developed by Koop et al. (1996). In general, the generalized IRs in period t + j inresponse to the shock that occurred in period t are defined as the diﬀerence E[yt+j |Ωt−1 , ε̃1t ] − E[yt+j | Ωt−1 , ε̃0t ], where Ωt−1 is the information set in period t − 1, ε̃1t is
the nonzero shock of interest, and ε̃0t denotes a zero shock in period t. In our case,∗the counterpart of yt+j is Ỹt+j = [gt+j , pt+j , xt+j , rt+j]′ , the explanatory variables in the

TVP-VAR model and in the Taylor rule in period t (i.e., Xt in eq. (3), and πt , χt ,and rt−1 in eq. (5), respectively) correspond to the initial condition Ωt−1 , and we set′′

ε̃1t = [ε1t , ε1r,t ]′ = (1, 0, 0, 0)′ and ε̃0t = [ε0t , ε0r,t ]′ = (0, 0, 0, 0)′ , because we are interested
in the eﬀect of a government spending shock. For each outcome of the Gibbs sampler,
we randomly generate future shocks, ε̃t+j (j = 1, · · · , h), from N (0, I), and then calculate

1
0
yt+j
conditional on (Ωt−1 , ε̃t+j , ε̃1t ), and yt+h
conditional on (Ωt−1 , ε̃t+j , ε̃0t ). Repeating thisprocess 100 times and averaging the outcomes, we obtain estimates of the conditional

13

1
0
1
expectation, denoted by ŷt+j
and ŷt+j, where we regard the diﬀerences between ŷt+j
and
0
ŷt+j
as estimates of the generalized IRs.

Specifically, the calculation for the generalized IRs in period t + j to the shock inperiod t is implemented as follows. Given the outcome of the Gibbs sampler and the initialcondition in period t, we first obtain a pair of values for [∆gt , pt , xt ]′ under both nonzero
and zero shocks, based on the VAR system. Second, the implied interest rates (rt∗ ) undernonzero and zero shocks are derived from the Taylor rule, using χt and πt , respectively,
obtained in the first step, where χt = xt +xt−1 +xt−2 +xt−3 and πt ≡ pt +pt−1 +pt−2 +pt−3 areconstructed using xt , pt , and the initial condition. Then, we check whether the computed
implied rate is larger than the lower bound (c). The observed interest rate (rt ) is set tothe implied rate (rt∗ ) if the implied rate exceeds the lower bound, and is set to the lower
bound (c) otherwise. Subsequently, a pair of values of endogenous variables in period t+1
are computed using the values in period t obtained above. We repeat this process until
the j-period ahead to obtain the responses in period t + j.
The generalized IRs are defined as the diﬀerences between the paths of the endogenous
variables under the two situations (i.e., nonzero and zero shocks), such that the estimatedresponse of the interest rate (rt ) is zero if the observed rates in both situations are set toc, given that rt = max[c, rt∗ ]. When the implied rate is replaced by c, the responses ofother endogenous variables are considered to be those observed in an economy in which
the ELB constraint is valid (i.e., rt = c). This is because the VAR system described ineq. (3) includes the implied rate and the observed rate on the right-hand side to calculate
the dynamics of the variables. Indeed, the Japanese economy has fallen within the ELB
for some time; thus, we regard the IRs calculated under the condition rt = max[c, rt∗ ] as abenchmark, in the sense of capturing the actual dynamics of the Japanese economy. On
the other hand, as a counterfactual simulation, we calculate the hypothetical IRs using
the implied interest rate rather than the lower bound of the nominal interest rate, even if
the ELB restriction is binding. In contrast to the benchmark, the responses of the interest
rate in this case are not zero, even when the economy hits a lower bound, because the IRs
are defined as the diﬀerences between the calculated implied rates of the nonzero and zero
shocks. Comparing the economy with a government spending shock (i.e., a nonzero shock)to that without a shock (i.e., a zero shock), there may be a diﬀerence between the values of
the implied rates, both of which are replaced in c if we follow the condition rt = max[c, rt∗ ].

14

Moreover, we calculate the responses of the other variables by incorporating the impliedrate, rather than the lower bound value, on the right-hand side of the equation. Thiscounterfactual analysis reveals the eﬀect of a government spending shock on the interest
rate, and the eﬀect of the variation in the interest rate on macroeconomic variables,
which are hidden by the ZIRP. In summary, the IRs in the benchmark model represent
the actual situation of the ELB period, in which the interest rate is set to the lower
bound. In contrast, the IRs in the counterfactual simulation describe the hypothetical
dynamics of the variables in the sense that we allow the interest rate to fluctuate beyond
the lower bound. Comparing the IRs derived from the benchmark with those derived fromthe counterfactual, we extract the role of the ELB in the eﬀectiveness of a government
spending shock on the economy.
Note that our counterfactual simulation is conducted under the condition that thecoeﬃcients in the three reduced-form equations are invariant, even in the face of a changing
monetary policy stance. In the strict sense, this kind of simulation is incompatible with
the Lucas critique; however, it is also true that such simulations are widely employed in theliterature to evaluate the eﬀects of policy changes (e.g., Sims and Zha, 2006; Baumeisterand Benati, 2013; Ramey, 2013). Nevertheless, we calculate the hypothetical paths foreach variable up to at most two-years after the shock, based on the notion that this sort ofcounterfactual simulation is justified only when agents’ beliefs are unchanged, as discussed
above.
In what follows, we show the fiscal multiplier rather than the IRs in order to alignthe size of the eﬀect of the government spending shock on output in each period. The
k-period-ahead fiscal multiplier to the shock at period t is computed as.k
j=0 IR of xt+j × GDPt
fiscal multiplierk,t = .k,
j=0 IR of gt+j × gov. spendingt(9)

because the data on the GDP and government spending are included in the first diﬀerenceof the natural logarithm within the VAR model. If the multipliers derived from thebenchmark are greater than those from the counterfactual, this implies that a government
spending shock has a larger eﬀect on the economy when the ELB constraint is binding,and vice versa. In addition, the responses of the interest rate and price are normalized to
1% of the government spending shock at the impact period.
We begin by examining the contemporaneous responses of each variable to a government spending shock. Figure 5 illustrates the time profiles of the contemporaneous
15

responses of the (a) fiscal multiplier, (b) price, (c) nominal interest rate, and (d) ex post
real interest rate,9 where solid thick and thin lines denote the median and 68% credibleintervals, respectively, corresponding to the benchmark. Because we assume that the interest rate has no contemporaneous eﬀects on government spending, inflation, or GDP,
there is no diﬀerence between the benchmark and the counterfactual at the impact period. The counterfactual responses of the interest rates, for which the diﬀerences emerge
instantly, are drawn by dashed lines. First, we can see from Figure 5 that the estimatedfiscal multiplier and responses exhibit remarkable variation over time throughout the
sample period, supporting our application of the TVP-VAR model. The variation of themultiplier shows a rapid drop after the economic bubble burst around 1990, a continuing
sluggishness around the first-half of the 2000s, and a recovery after the second-half of the
2000s. Although several researchers have pointed out that the eﬀectiveness of fiscal policy
in Japan has diminished since the bubble burst, its recent revival is a noteworthy findingof this study. With regard to price, we observe that the government spending shock induces inflation (the standard theoretical view), but that exceptional periods exist, during
which the price decreases to the shock (e.g., the end of the second and third ELB periods).
The price’s negative response to a government spending shock seems contrary to the conventional view that a government spending shock is inflationary because of the demand
shock. However, empirical evidence shows a decrease of the price in response to the shock(e.g., Mountford and Uhlig, 2009; Fatas and Mihov, 2001).10 As shown in Figure 5(c) and(d), such exceptional price behavior causes the negative response of the nominal implied
rate. Furthermore, it creates a situation in which the response of the real interest ratein the counterfactual becomes lower than that derived in the benchmark. However, given
this exception, the result suggests that monetary tightening would occur in response to
an inflationary government spending shock if no ELB constraint exists. Therefore, the
ZIRP plays a significant role in suppressing the real interest rate in such circumstances.
[Figure 5 about here.]

9

The response of ex post real interest rate is defined as the diﬀerence between the response of thenominal interest rate and current inflation.
10
Jørgensen and Ravn (2018) recently proposed a theoretical model that replicates a drop in price asa result of a government spending shock by incorporating variable technology utilization in the standard
New Keynesian model.

16Figure 6(a) displays the time profiles of the fiscal multipliers for (i) one-quarter, (ii)one-year, and (iii) two-year horizons. As in Figure 5, the solid lines correspond to thebenchmark, represented by rt = max[c, rt∗ ]. The dashed lines correspond to the resultsfrom the counterfactual, represented by rt = rt∗ , and indicate ELB periods as shaded
areas. To highlight the role of the ELB in the magnitude of the fiscal multiplier, we plot
the point estimates for the fiscal multipliers under the two cases in Figure 6(a). Insteadof plotting the credible intervals, Figure 6(b) shows the posterior ratio such that the sizeof the multiplier in the benchmark is larger than that in the counterfactual simulationfor each horizon in each period. We interpret this as follows. The diﬀerence between
the fiscal multipliers in the two simulations becomes statistically more significant as theprobability increases. Accordingly, a high probability implies that it is more plausiblethat the eﬀectiveness of a government spending shock on output is enhanced by keeping
the interest rate within the lower bound. Because the same values of future shocks ε̃t+j
are shared when calculating the generalized responses in both cases, the multiplier inthe benchmark coincides perfectly with that in the counterfactual, except for the ELB
period. Thus, the posterior probabilities are calculated for the ELB period only. In
addition, Figure 7 shows the responses of the (a) nominal interest rate, (b) price, and(c) real interest rate at each horizon, where the response of the price is defined as the
cumulative response of inflation.
We first observe from Figures 6 and 7 that the estimated multipliers in the benchmark
are larger (smaller) than those in the counterfactual when the response of the real interest
rate in the former is smaller (larger) than that in the latter.11 For example, the positive
gap between the two multipliers, which is more evident after one and two years, is observedin the period just after the global financial crisis. During this period, the responses of thereal interest rate in the benchmark are considerably suppressed compared with those of
the counterfactual. This proves empirically that the theoretical mechanism of Eggertsson
11

At first glance, our finding seems to support those of Miyamoto et al. (2018), who find that themultiplier in the ELB period is larger than that in the non-ELB period. However, there is a methodologicaldiﬀerence between the work of Miyamoto el al. (2018) and this study. The former compares the multipliers
derived from the non-ELB period (1980Q1–1994Q4) and the ELB period (1995Q1–2014Q1). Here, weestimate two values of the fiscal multipliers in the same period. The advantage of our method is that wecontrol for factors other than the monetary policy stance, because the diﬀerence between the benchmark
and the counterfactual lies only in how we determine the interest rate when we calculate the IRs.

17

(2010) and Christiano et al. (2011) holds in practice. Because the dynamics of the realinterest rate stem from those of the nominal interest rate and inflation, this finding can beinterpreted as follows. The enhancement in the fiscal multiplier is driven by the absenceof monetary tightening in an ELB period, where the upward pressure on the nominalinterest rate accompanied with the positive responses of price and output is neutralized
by the ZIRP. Therefore, we conclude that, as predicted by theory, the decrease in thereal interest rate is crucial to increasing the eﬀect of a government spending shock on
output during an ELB period. In addition to the fiscal multiplier (see Figure 7(b)), theprice responses in the benchmark model are located above those in the counterfactual, as
long as the positive response of the nominal interest rate observed in the counterfactualis oﬀset by fixing the interest rate to zero. In this respect, our result agrees with thestandard theoretical prediction that an increase in the nominal interest rate mitigates anincrease in price.
Finally, note that there is a statistical vulnerability in the diﬀerence between thebenchmark and the counterfactual. The posterior probabilities shown in Figure 6(b) takevalues around 0.6 and, at most, less than 0.7, even when the median of the benchmark
exceeds that of the counterfactual. Compared with the significance level adopted in astandard inference, the ratio obtained in this study is not suﬃcient to reject the null
hypothesis that a statistical diﬀerence between them does not exist. In other words, it
is possible that the credible intervals for each IR, which are not drawn in Figure 6(a),overlap. This statistical weakness arises from our TVP-VAR specification, because the
TVP-VAR model contains many parameters that need to be estimated from a limited
sample of time-series data. Thus, the credible intervals associated with the IRs tend to be
wider. Because the variation of the economy over time, captured by the TVP-VAR model,
seems to be supported by the results thus far, we prioritize the time-varying specification
over statistical accuracy.
[Figure 6 about here.][Figure 7 about here.]18

4. Robustness check4.1. Diﬀerent definition of the ELB period
To check that our result is robust, without depending on the definition of the ELB, we
divide the sample period into non-ELB and ELB periods based on the definition adopted
by Miyamoto et al. (2018). Thus, we employ 1% as the threshold value to separate thesample period. Following this definition, we estimate the model by resetting the lower
bound to 1% (i.e., c = 1). In what follows, we refer to the results in Section 3 as thebaseline.
Figure 8 shows the implied interest rate estimated based on the new definition, andthat of the baseline. The median and 68% credible intervals of the new model are describedby dot-dashed and thin lines, respectively; the estimate for the baseline is denoted by asolid line. In this robustness check, the period after 1995Q4 is categorized as the ELB
period, as indicated by shaded area, and the implied interest rate during the ELB period
is drawn from a truncated normal distribution between −∞ and 1. Compared with thebaseline estimate, the newly estimated implied rate tends to show values that are morenegative. However, its behavior is similar to that of the baseline model.
[Figure 8 about here.]
In addition to the implied rate, Figure 9 shows the time profiles of the fiscal multiplierand the posterior probability, as calculated in the baseline. For comparison, Figure 9(b)plots the posterior probabilities obtained in the baseline and in the robustness check. Obviously, there is no quantitative or qualitative diﬀerence between these results. Therefore,
our main result, that the multiplier derived from the benchmark economy is larger thanthat derived from the counterfactual if a government spending shock is inflationary, isrobust, even if we change the definition of the ELB period following that of Miyamoto etal. (2018).
[Figure 9 about here.]
4.2. Diﬀerent measure of output
Following the literature on fiscal policy (e.g., Blanchard and Perotti, 2002), in thebaseline formulation, we use the GDP growth rate in the VAR model and, thus, the yearon-year GDP growth rate in the Taylor rule. On the other hand, the standard specification19

of the Taylor rule contains the output gap, not the GDP growth rate (Taylor, 1993; Clarida
et al., 1998; Hayashi and Koeda, 2019). Following the literature on monetary policy, we
conduct an alternative estimation using the GDP gap as the output measure to confirm
the robustness of our results. To do so, we slightly modify the specification of the Taylor
rule, as follows:rt∗ = (1 − ρt )[ψ0,t + ψπ,t πt + ψz,t zt ] + ρt rt−1 + εr,t .

(10)Here, we directly incorporate the GDP gap, denoted by zt , into the right-hand side of theequation. The GDP gap is a series of log-diﬀerentials between the actual and potential
GDP; we use the GDP gap released by the Cabinet Oﬃce, Government of Japan, as shownin Figure 10. Similarly to the year-on-year GDP growth rate, the GDP gap exhibits slowmoving variation. Therefore, we use the GDP gap directly in the Taylor rule.
[Figure 10 about here.]
Figure 11 shows the variation in the fiscal multipliers over time and the posterior probability that the value of the multiplier in the counterfactual exceeds that of the benchmark
in each period; this is the counterpart of Figure 6. The diﬀerence between the multipliersof the benchmark and the counterfactual is clearly evident compared with the results in
Figure 6. As a result, the posterior probabilities displayed in Figure 11(b) take values
greater than those in the baseline for most of the ELB period. This suggests that thediﬀerence between the benchmark and counterfactual is likely to be more reliable whenusing the GDP gap as an output measure. However, although we acknowledge the quantitative diﬀerences in the posterior probabilities, depending on the output measure, thevariations over time are qualitatively the same between the baseline and the robustness
check. Moreover, the positional relationships between the multipliers described in Figure 11(a) are similar to those in the baseline estimation. That is, the multiplier in thebenchmark is larger than that in the counterfactual when a government spending shockcauses inflation. Hence, we conclude that the main results do not depend on the output
measure.
[Figure 11 about here.]
4.3. Eﬀect of a public investment shock
In terms of an economic stimulus package, the eﬀectiveness of a public investment
shock is more relevant in practice than that of overall spending, because fiscal policy aimed
20

at an economic stimulus is implemented mainly through an increase in public investment.12
In addition, the theoretical size of the multiplier associated with public investment under
the ELB is controversial, as discussed in Albertini et al. (2014) and Bouakez et al. (2017).
Because public investment is productive spending in the sense that it contributes to theaccumulation of public capital, it has supply-side eﬀects of reducing the real marginal
cost and inflation. When the nominal interest rate is fixed to a lower bound under
the ELB, deflation causes an increase in the real interest rate, and then attenuates theeﬀect of the fiscal policy shock. Albertini et al. (2014) reports that the multiplier inthe ELB decreases steadily as the proportion of productive spending increases. Bouakez
et al. (2017) shows that the multiplier associated with public investment may becomelarge when the time-to-build delay is comparatively long. This is because the demand-side
inflationary eﬀect stemming from an increase in public investment exceeds the supply-side
deflationary pressure, which is brought about by an increase in public capital. In orderto address the above-mentioned academic and practical concerns, we replace government
spending with public investment in the following estimation.
Similarly to the baseline estimation, Figures 12, 13, and 14 illustrate the time profilesof the contemporaneous responses of each variable, the fiscal multiplier and posterior
probability, and the responses of the nominal interest rate, price, and real interest rate,
respectively. For the comparison with the baseline model, we plot the posterior probability
obtained in the baseline estimation using dashed lines in Figure 13 (b). In addition, theIRs of the price and nominal and real interest rates are normalized such that the responseof public investment becomes 1% at the impact period.
[Figure 12 about here.][Figure 13 about here.][Figure 14 about here.]
Although the overall behavior of the multiplier, as shown in Figures 12 and 13, exhibitsa U-shape, as in the baseline, two features diﬀer from the baseline: the negative valuesof the multiplier in the period from the end of 1990s to the mid-2000s, and a sharp
12

Morita (2017) documents the amounts of public investment in the recent Japanese fiscal stimulus

packages in detail.

21

increase in the multiplier during the third ELB period. These findings suggest that thesluggishness in the multiplier during the lost decades and its recent revival observedin the baseline are mainly attributed to the eﬀect of public investment shocks. Thisemphasizes the importance of a public investment shock as an economic stimulus policy.
Focusing on the diﬀerence between the benchmark (rt = max[c, rt∗ ]) and the counterfactual(rt = rt∗ ) during the ELB period, Figure 14(b) shows that a public investment shock ismore inflationary than an overall government spending shock (Figure 7(b)). This empiricalfinding supports the theoretical prediction of Bouakez et al. (2017) rather than that ofAlbertini et al. (2014). More precisely, the demand-side inflationary eﬀect of a public
investment shock seems to dominate the supply-side deflationary pressure, at least inJapan. This inflationary eﬀect generates a positive gap in the multiplier for most of the
ELB period. This includes the end of the sample period, during which a negative gap
emerges in the baseline owing to the negative response of inflation to the shock. In otherwords, the eﬀectiveness of a public investment shock is amplified under the ELB. At thesame time, the crowding-out eﬀect observed in the lost decades is mitigated, owing to its
inflationary eﬀect. Indeed, as shown in Figure 14(c), a public investment shock reduces
the real interest rate in the benchmark compared with that in the counterfactual. This
indicates that the necessary condition for enhancing the fiscal multiplier under a lowinterest rate is also satisfied if we use public investment rather than total government
spending. However, note that the credible intervals associated with the contemporaneous
fiscal multiplier shown in Figure 12 are wider than those in the baseline estimation. Thisimplies that the fiscal multiplier when using public investment entails greater uncertainty
than when using overall spending. Consequently, as shown in Figure 13(b), the posterior
probabilities, which highlight the role of the ZIRP in the eﬀectiveness of fiscal policy,
takes comparatively lower values than those of the baseline results.
5. Conclusion
This study uses a TVP-VAR model with an implied interest rate to empirically analyze
whether the eﬀect of a government spending shock on output depends on the monetary
policy in Japan. To achieve this purpose, we compute two types of IRs based on theestimation results: a benchmark, which calculates the IRs after replacing the interest
rate with the lower bound value if the implied rate hits the ELB; and a counterfactual,22which computes the IRs using the implied interest rate as an endogenous variable, even
if the rate takes a negative value. The gap between the two IRs isolates the role of the
ZIRP in the eﬀect of a fiscal policy shock, because the diﬀerence depends only on how wedetermine the interest rate under the ELB.
The results reveal that the fiscal multiplier becomes larger as a result of the decreasein the real interest rate when we eliminate the implied fluctuation of the nominal interest
rate by remaining within the lower bound. In other words, the multiplier derived from thebenchmark is greater than that derived from the counterfactual when a government spending shock is inflationary. Moreover, this main result remains qualitatively unchanged for
diﬀerent definitions of the ELB period, output, and fiscal variable. Therefore, we conclude
that the eﬀectiveness of a government spending shock is enhanced when the central bankadopts the ZIRP. Furthermore, this empirical evidence supports the theoretical findingsof Eggertsson (2010) and Christiano et al. (2011).
In concluding this paper, we mention the foreseeability of fiscal policy, as emphasized
by Ramey (2011).13 This study does not consider fiscal foresight, because its primary aim
is to examine whether the theoretical prediction of the eﬀect of fiscal policy under the
ELB is plausible. In practice, it is diﬃcult to incorporate additional variables, which cancapture a fiscal news shock, into our TVP-VAR model, given the estimation accuracy.
However, Fisher and Peters (2010) propose that industries’ excess stock returns related
to government spending can be regarded as a proxy for a fiscal news shock. Nonetheless,
it is important to check whether the results presented here are robust after considering
fiscal foresight. This topic is left to future work.
References
Albertini, J., Poirier, A., Roulleau-Pasdeloup, J., 2014. The composition of government
spending and the multiplier at the zero lower bound. Economics Letters 122(1), 31–35.
Auerbach, A. J., Gorodnichenko, Y., 2012. Measuring the output responses to fiscal policy.
American Economic Journal: Economic Policy 4(2), 1–27.
Baumeister, C., Benati, L., 2013. Unconventional monetary policy and the great recession:

13

Eggertsson (2010) also notes the importance of announcing a fiscal stimulus.

23

Estimating the macroeconomic eﬀects of a spread compression at the zero lower bound.
International Journal of Central Banking 9(2), 165–212.
Blanchard, O. J., Perotti, R., 2002. An empirical characterization of the dynamic eﬀects ofchanges in government spending and taxes on output. Quarterly Journal of Economics
117(4), 1329–1368.
Boneva, L. M., Braun, R. A., Waki, Y., 2016. Some unpleasant properties of loglinearized
solutions when the nominal rate is zero. Journal of Monetary Economics 84, 216–232.
Bouakez, H., Guillard, M., Roulleau-Pasdeloup, J., 2017. Public investment, time to build,and the zero lower bound. Review of Economic Dynamics 23, 60–79.
Braun, R. A., Waki, Y., 2006. Monetary policy during Japan’s lost decade. Japanese
Economic Review 57(2), 324–344.
Christiano, L., Eichenbaum, M., Rebelo, S., 2011. When is the government spending
multiplier large? Journal of Political Economy 119(1), 78–121.
Clarida, R., Galı́, J., Gertler, M., 1998. Monetary policy rules in practice: Some international evidence. European Economic Review 42(6), 1033–1067.
Eggertsson, G. E., 2010. What fiscal policy is eﬀective at zero interest rate? NBER
Macroeconomics Annual 25, 59–112.
Erceg, C., Lindé, J., 2014. Is there a fiscal free lunch in a liquidity trap? Journal of theEuropean Economic Association 12(1), 73–107.
Fatas, A., Mihov, I., 2001. The Eﬀects of fiscal policy on consumption and employment:
Theory and evidence. CEPR Discussion Paper 2760.
Fazzari, S. M., Morley, J., Panovska, I., 2015. State-dependent eﬀects of fiscal policy.
Studies in Nonlinear Dynamics and Econometrics 19(3), 285–315.
Fisher, J., D.M., Peters, R., 2010. Using stock returns to identify government spending
shocks. Economic Journal 120(544), 414–436.
Galı́, J., López-Salido, D., Vallés, J., 2007. Understanding the eﬀects of government spending on consumption. Journal of the European Economic Association 5(1), 227–270.
24

Hayashi, F., Koeda, J., 2019. Exiting from quantitative easing. Quantitative Economics
10(3), 1069–1107.
Hayashi, F., Prescott, E. C., 2002. Japan in the 1990s: A lost decade. Review of Economic
Dynamics 5(1), 206–235.
Hofmann, B., Peersman, G., Straub, R., 2012. Time variation in U.S. wage dynamics.
Journal of Monetary Economics 59(8), 769–783.
Iwata, S., Wu, S., 2006. Estimating monetary policy eﬀects when interest rates are close
to zero. Journal of Monetary Economics 53(7), 1395–1408.
Jørgensen, P. L., Ravn, S., 2019. The inflation response to government spending shocks:
A fiscal price puzzle? mimeo.
Kato, R. 2003. Zaisei Jousuuno Nichibei Hikaku: Kouzou VAR to Seidoteki Youinwo
Heiyousita Approach. IMES working paper series, 03-J-4. (In Japanese)
Koop, G., Pesaran, M. H., Potter, S. M., 1996. Impulse response analysis in nonlinear
multivariate model. Journal of Econometrics 74(1), 119–147.
Krippner, L., 2015. Term structure modeling at the zero lower bound: A practitioner’s
guide. Palgrave-Macmillan.
Leeper, E. M., Zha, T., 2003. Modest policy interventions. Journal of Monetary Economics
50(8), 1673–1700.
Lucas Jr., R. E., 1976. Econometric policy evaluation: a critique. In: Brunner, K., Meltzer,
A.H. (Eds.), Carnegie-Rochester Conference Series on Public Policy, Vol. 1. NorthHolland, Amsterdam, 104–130.
Miyamoto, W., Nguyen, T. L., Sergeyev, D., 2018. Government spending multipliers under
the zero lower bound: Evidence from Japan. American Economic Journal: Macroeconomics 10(3), 1–32.
Miyazawa, K., 2011. The Taylor rule in Japan. Japanese Economy 38(2), 79–104.
Morita, H., 2017. Eﬀects of anticipated fiscal policy shock on macroeconomic dynamics
in Japan. Japanese Economic Review 68(3), 364–393.
25

Mountford, A., Uhlig, H., 2009. What are the eﬀects of fiscal policy shocks? Journal ofApplied Econometrics 24(6), 960–992.
Nakajima, J., 2011. Monetary policy transmission under zero interest rates: An extended
time-varying parameter vector autoregression approach. The B.E. Journal of Macroeconomics 11(1), Article 32.
Nakajima, J., Kasuya, M., Watanabe, T., 2011. Bayesian analysis of time-varying parameter vector autoregressive model for the Japanese economy and monetary policy.
Journal of the Japanese and International Economies 25(3), 225–245.
Perotti, R., 2002. Estimating the eﬀects of fiscal policy in OECD countries. ECB working
paper series No.168.
Primiceri, G. E., 2005. Time varying structural vector autoregressions and monetary
policy. Review of Economic Studies 72(3), 821–852.
Raftery, A. E., Lewis, S., 1992. How many iterations in the Gibbs sampler? Bayesian
Statistics 763–773.
Ramey, V. A., 2011. Identifying government spending shocks: It’s all in the timing.
Quarterly Journal of Economics 126(1), 1-50.
Ramey, V. A., 2013. Government spending and private activity. In: Alesina, A., Giavazzi,
F. (Eds.), Fiscal Policy After the Financial Crisis, University of Chicago Press.
Ramey, V. A., Zubairy, S., 2018. Government spending multipliers in good times and in
bad: Evidence from US historical data. Journal of Political Economy 126(2), 850–901.
Sims, C. A., Zha, T., 2006. Were there regime switches in U.S. monetary policy? American
Economic Review 96(1), 54–81.
Stock, J. H., Watson, M. W., 2001. Vector autoregressions. Journal of Economic Perspectives 15(4), 101–115.
Taylor, J. B., 1993. Discretion versus policy rule in practice. Carnegie-Rochester Conference Series on Public Policy 39, 195–214.

26

Ueno, Y., 2017. Term structure models with negative interest rates. IMES Discussion
Paper Series No. 2017-E-1.
Watanabe, T., Omori, Y., 2004. A multi-move sampler for estimating non-Gaussian times
series models: Comments on Shephard and Pitt (1997). Biometrika 91(1), 246–248.

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Figure 1: Data description
Note: Except for the short-term interest rate, solid lines indicate the quarter-on-quarter growth rate ofeach variable. The dashed lines in Figures 1(b) and (c) present the year-on-year growth rates of the GDP
and CPI, respectively. In addition, the shaded areas correspond to the ELB period.

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100
80
60
40
20
0

100

200

300

400

500

100

200

300

400

500

5000
4000
3000
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1000
0

Figure 2: Convergence diagnostics for hyperparameters and volatilities
Notes: This figure shows (a) the 20-th-order sample autocorrelation, (b) the ineﬃciency factor, and (c)Raftery and Lewis’s (1992) total number of runs.

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Figure 3: Estimated Taylor rule parameters
Notes: This figure shows the time profiles of the structural parameters in the Taylor rule in Eq. (5).
Solid thick and thin lines correspond to the median estimates and 68% credible intervals, respectively. In
addition, dotted lines represent the ranges of the estimated parameters reported in previous studies.

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Figure 4: Estimated implied rate
Notes: The figure shows the actual short-term interest rate (dotted line), estimated implied rate (solid
thick line), and 68% credible intervals associated with the implied rate (solid thin line). For comparison
purposes, we also plot the shadow rate (dot-dashed line) estimated by Krippner (2015).

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Figure 5: Time-varying responses of fiscal multiplier, price, and nominal and real interest rates at impact
periodNotes: The figures show the time profiles of the (a) fiscal multiplier, as well as the responses of the (b)price, (c) nominal interest rate, and (d) ex post real interest rate at the impact period. The ex post
real interest rate is calculated by subtracting the response of inflation from the nominal interest rate.
The solid thick and thin lines correspond to the median and 68% credible intervals, respectively, in thebenchmark model. The dashed lines in Figure 5(c) and (d) indicate the responses in the counterfactual
simulation.

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Figure 6: Fiscal multiplier and posterior probability
Notes: Figure 6(a) shows the time profiles of the fiscal multipliers derived from ELB periods (solid lines)
and non-ELB periods (dashed lines) at (i) one quarter, (ii) one year, and (iii) two years after the shock.
Figure 6(b) shows the corresponding posterior probabilities defined as the ratio of such samples as themultiplier in an ELB period is greater than that in a non-ELB period. Because the diﬀerences between
the ELB and non-ELB multipliers only happen in the ELB periods (by construction), the time profilesof the probabilities are depicted for the ELB periods only.

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Figure 7: Time-varying responses of nominal and real interest rates and priceNotes: This figure shows the time profiles of the responses of the (a) nominal interest rate, (b) price, and(c) ex post real interest rate to a government spending shock. Here, the response of the price is definedas the accumulated response of inflation. The ex post real interest rate is calculated by subtracting the
response of inflation from that of the nominal interest rate. In each figure, the solid and dashed lines
denote responses derived from the benchmark and counterfactual, respectively.

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Figure 8: Estimated implied rate: c = 1%
Notes: This figure plots the estimated implied rate under various definitions of the ELB. The dot-dashedline is the implied rate when we regard 1% as the threshold value of the ELB.

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Figure 9: Fiscal multiplier and posterior probability: c = 1%
Notes: This figure shows the time profiles of the fiscal multipliers and the posterior probabilities for c =1%.

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Figure 10: Data for the GDP gap
Notes: The solid line shows the evolution of the GDP gap reported by the Cabinet Oﬃce, Government ofJapan. For comparison purposes, we plot the year-on-year GDP growth rate used in the baseline (dashedline). In the robustness check, we estimate the model by replacing the GDP growth rate with the GDP
gap.

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Figure 11: Fiscal multiplier and posterior probability: GDP gap
Notes: This figure shows the time profiles of the fiscal multipliers and the posterior probabilities when
we use the GDP gap rather than the GDP growth rate.

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Figure 12: Time-varying responses for the fiscal multiplier, price, and nominal and real interest rates at
the impact period: public investment
Notes: This figure shows the time-varying responses of the fiscal multiplier, price, and nominal and realinterest rates at the impact period when we use public investment rather than total government spending.

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Figure 13: Fiscal multiplier and posterior probability: public investment
Notes: This figure shows the time profiles of the fiscal multipliers and the posterior probabilities when
we use public investment rather than total government spending.

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Figure 14: Time-varying responses of nominal and real interest rates and price: public investment
Note: This figure shows the time profiles of the responses of the (a) nominal interest rate, (b) price, and(c) ex post real interest rate to a public investment shock.

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