ESRI Discussion Paper Series No.376

The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules
Masao Fukui, Masayuki Yagasaki
April 2023

Economic and Social Research InstituteCabinet Office
Tokyo, JapanThe views expressed in “ESRI Discussion Papers” are those of the authors and not those of
the Economic and Social Research Institute, the Cabinet Office, or the Government of Japan.
(Contact us: https://form.cao.go.jp/esri/en_opinion-0002.html)

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules*
Masao Fukui†

Masayuki Yagasaki‡

January 2023
Abstract
We trace the impact of central bank stock purchases by exploiting the discontinuity inBank of Japan’s policy rule, which triggers purchases when the stock market index falls below a certain threshold. In normal times, a purchase of 0.01% of market capitalization (a typical size of each intervention) persistently increases the long-term interest rate by around 1.5
b.p. while leaving virtually no detectable impact on stock prices. After the introduction of
the yield curve control, which pegs the long-term interest rate to 0%, interest rates stopped
responding and stock prices rise by around 0.2% in response to the stock purchases. These
results support a theory where both stock and bond markets are substantially inelastic.

* We thank Manuel Amador, Ahnmol Bhandari, Serdar Birinci, Mai Hakamada, Kyle Herkenhoff, Kazuhiro
Hiraki, Hidehiko Ichimura, Takatoshi Ito, Illein Kondo, Hiroyuki Kubota, Tomohide Mineyama, Emi Nakamura,
Hitoshi Shigeoka, Jón Steinsson, Kjetil Storesletten, Kazuhiro Teramoto, and Kazuo Ueda for helpful comments.
The views expressed in this study are those of the authors and do not necessarily reflect the position of the Government of Japan. Fukui gratefully acknowledges the financial supports from the George-Orbis Shultz Fund.
† Boston University.
‡ ESRI Cabinet Office, Government of Japan and the University of Tokyo.

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ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

1

Introduction

As a new form of “quantitative easing,” the Bank of Japan (henceforth, the BoJ) started topurchase stocks in 2010, becoming the largest Japanese stock owner in the world in 2021. Figure
1 shows the cumulative amount of the BoJ’s stock holdings as a fraction of Japan’s market
capitalization. The BoJ continuously kept purchasing stocks, and now holds 6% of market
capitalization. No other central banks in the world have been purchasing stocks on such aregular basis. The BoJ explains that the primary goal of this extreme form of quantitative easing
is to “reduce the risk premium”.
What is the impact of the central bank’s stock purchases? Answering this question is important for the following two reasons. First, the BoJ has been always at the frontier of implementing unconventional monetary policy. Policies such as forward guidance or government
bond purchases were first implemented by the BoJ in the early 2000s, and a decade later, central banks in the U.S. and Europe have been pursing the same path that Japan has followed.
Therefore, learning from the most pioneer of unconventional policies will likely benefit policy makers worldwide in the near future. Second, it provides an ideal laboratory to test new
theories of stock market fluctuations. Recently, Gabaix and Koijen (2021) argue that shocks toflows from bonds to stocks can be a dominant source of stock price fluctuations. Assessing thecausal impact of central bank’s stock purchases simultaneously provides an empirical test of
this theory.
We trace the impact of the BoJ’s stock purchases by exploiting the discontinuity in the BoJ’s
policy rule, which triggers purchases when the stock market index falls below a certain threshold. While the BoJ has never publicly stated their practice, it is widely known that the BoJ tends
to purchase stock market index precisely on the day when the index falls below a threshold in
the morning session. This unique feature enables us to overcome the endogeneity of policy
interventions with the regression discontinuity estimator. By comparing days where a stock
market index falls slightly below the threshold and slightly above, the policy intervention canbe viewed as orthogonal to the underlying economic fundamentals.
In the entire sample, we first show that the policy had a large impact on both the stock
prices and the long-term government bond interest rates. In response to an average size ofintervention, which amounts to 0.01% of stock market capitalization, our estimates indicate thatthe stock prices rise by around 0.3-0.4% on the same day, and by around 0.1% on the following
day of the intervention. Perhaps surprisingly, the 10-year Japanese government bond (JGB)
yield also increases by around 0.5 basis point (b.p.) on the same day and the following day.
These results seem to be persistent, lasting at least several days.
We then argue that the above results mask stark underlying heterogeneity that depends on
the presence of yield curve control (YCC) by the the BoJ. In the middle of 2016, the BoJ introduced another form of unconventional monetary policy, the so-called yield curve control,20Fraction of Market Capitalization (%)
2
4

6

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

Figure 1: Cumulative ETF Purchases by the BoJ
Notes: Figure 1 plots the cumulative amount of ETF purchases by the BoJ from 2010 to 2020 as a fraction of market
capitalization.

which pegs the 10-year JGB yield to around 0%. Since then, long-term interest rates have stabilized at around 0%. Given this, we expect that the response of long-term interest rates to be
substantially different before and after the introduction of YCC. We therefore split the sample
and re-examine the effects.
We find that before the introduction of YCC, in response to the BoJ’s stock market purchases,
the long-term interest rates strongly and persistently rise after the intervention, while leaving
virtually no impact on the stock price the next day. The long-term interest rates rise by 1.5 b.p.
on the same day, which persists at least several days. In contrast, while stock prices rise byaround 0.3-0.4%% on the same day, they revert back to zero the morning of the next day andremain there. We are unable to reject the null that the BoJ’s stock purchases left no impact on
stock prices in the following days, although the standard errors are wide.
After the introduction of YCC, the long-term interest rate stopped responding, and instead,stock prices rose persistently in response to the Bank of Japan’s intervention. The effect on
the long-term interest rate is precisely estimated zero. The stock price responds by around0.3-0.4%% on the same day, and this persists for at least several days after the intervention.
We then turn to a theoretical framework to interpret our empirical findings and argue that
our empirical results support a model in which both stock and bond markets are substantiallyinelastic. Our empirical results reject a frictionless model, which predicts the neutrality withrespect to central bank stock purchases (Wallace, 1981). Our results are also inconsistent withthe model presented in Gabaix and Koijen (2021), in which the stock market is inelastic but
the bond market is perfectly elastic. In their model, the stock market is inelastic because stock
traders face various frictions in flexibly adjusting their portfolio, such as institutional mandates

3

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

or inattention. This predicts the large increase in stock prices and zero effect on interest rates
in response to the sudden inflow into the stock market. However, our empirical results do notsupport this model. Instead, we draw on our findings to propose a model in which both stockand bond markets are inelastic. Central bank stock purchases are a swap between stocks andbonds in the market. When the bond market is inelastic, the outflow of bonds reduces the priceof bonds, resulting in an increase in interest rates. While inflow into stocks tends to increase
the stock price, the increase in interest rates acts as a countervailing force. Our empirical resultsfrom the period before YCC implies that these two forces offset each other, leaving no effect on
stock prices. We then show that once the interest rate becomes fixed, through YCC, only theformer force remains, which leads to an increase in stock prices.
Our theoretical framework demonstrates that these forces are summarized by three intuitive sufficient statistics: (i) stock market inelasticity; (ii) bonds market inelasticity; and (iii) theinterest rate sensitivity of stock prices. We show that these sufficient statistics can be identified via our reduced-form estimates, providing a structural interpretation of our reduced-form
analysis. This should serve useful for future researchers disciplining or calibrating their theoretical models.
Through the lens of the model, our estimates of stock market inelasticity, which we define
as the impact on stock prices from an inflow into the stock market holding interest rate fixed, isseveral times higher than the estimates provided by Gabaix and Koijen (2021). This implies
the frictions in the stock market is large. However, our estimates indicate the frictions in thebond market is even larger, in the sense that, without an explicit constraint on the interest rateadjustment, flows between stocks and bonds will mostly end up moving bond prices rather
than stock prices. A broader message of our paper is that is crucial to take into account frictions
in both the bond and the stock market at the same time, while the literature has been typicallyconsidered one friction at a time.1

1.1

Literature

To the best of our knowledge, we are the first to uncover the aggregate causal impact of central
bank stock purchases. Since the BoJ purchased a certain stock market index rather than theanother, many studies exploit difference in weights in the BoJ purchase basket to identify the
relative price impacts (Barbon and Gianinazzi, 2019; Charoenwong et al., 2021; Harada andOkimoto, 2021; Adachi et al., 2021; Katagiri et al., 2022). In contrast, our empirical strategy
allows us to focus on the aggregate effect. Various studies (Shirota, 2018; Fukuda and Tanaka,
2022; Chung, 2020) assume selection on observables and use the unexplained policy variationthat remains after conditioning on observables for identification. However, any endogeneity
1 See Krishnamurthy and Vissing-Jorgensen (2012) for bond market frictions and Gabaix and Koijen (2021) for

stock market frictions.

4

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

due to unobservables will bias the estimates. Our identification assumptions are substantiallyweaker than any of these studies.
More broadly, we contribute to the large literature studying the effect of the central bankasset purchases, so called “quantitative easing” (e.g., Krishnamurthy and Vissing-Jorgensen,
2011; Chodorow-Reich, 2014; Gorodnichenko and Ray, 2017). These studies have focused on
central bank purchases of long-term government bonds or mortgage backed securities, which
are the swaps between one type of bonds (e.g., long-term bonds) with another (e.g., reserves).
Our focus is conceptually distinct from them because central bank stock purchases are swaps
between stocks and bonds in the economy.
Through the lens of our theoretical model, we argue that our empirical setup provides joint
estimates of two important structural parameters: stock market inelasticity and bond market
inelasticity. While growing studies (Gabaix and Koijen, 2021; Da et al., 2018; Hartzmark andSolomon, 2021; Li et al., 2021) estimate how the flows from bonds into stocks impact stock
prices, we argue it is important to jointly take into account how bond prices are impacted bysuch flows. Our empirical results suggest that such flows increase the interest rates, whichcounteract the upward pressure on stock prices. Our empirical estimates of stock market inelasticity when interest rates can respond is indistinguishable from zero. However, the estimates of stock market inelasticity when the interest rate is fixed is several times higher thanexisting estimates. In this respect, we synthesize two strands of literature: one on bond market
inelasticity (e.g., Krishnamurthy and Vissing-Jorgensen, 2012) and the other on stock market
inelasticity (Gabaix and Koijen, 2021). We argue it is critical to take into account these twoviews jointly.
While there are many studies to isolate quasi-experimental variation in monetary policy(Romer and Romer, 2004; Cochrane and Piazzesi, 2002; Angrist et al., 2018; Nakamura andSteinsson, 2018), our approach is unique in utilizing the discontinuity in policy rule. The closest
to our approach is the one in Kuersteiner, Phillips, and Villamizar-Villegas (2018), who alsoutilize a discontinuous policy rule to investigate the effectiveness of sterilized foreign exchangeinterventions in Columbia. Our approach differs not only in terms of the empirical contexts but
also in methodology, as we employ the technique of regression discontinuity with unknown
discontinuity points (Porter and Yu, 2015).

2

Data

Our primary goal is to measure the impact of central bank stock purchases on financial markets.
We focus on the time period starting in October 2010 (the start of the BoJ stock market intervention) toward the end of 2020. We obtain dates and amount of stock purchases for each the
BoJ’s interventions from the BoJ website at https://www3.boj.or.jp/market/jp/menu_etf.htm.

5

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

Figure 1 shows the amount of ETF purchases over time. The BoJ started the stock market purchases in December 2010, and the frequency and the amount of purchases grew over time. By
the end of 2020, the BoJ held over 6% of the stock market capitalization in Japan. The amountof ETF purchases is normalized by the stock market capitalization], which we obtained fromJapan Exchange Group Data Cloud.
The BoJ publishes the amount of ETF purchases the morning following the intervention.
Based on the trading volume, it is widely considered that the BoJ submits an order during
lunchtime, although the BoJ has never made this practice public.2 Therefore, investors potentially face uncertainty about whether the large inflow into the stock market reflects the BoJ’s
intervention or other factors within the day of the intervention. For this reason, we prefer totake our empirical estimates of the one-day horizon as our benchmark estimates.
To measure the response of the stock market, we use tick-by-tick data on the Tokyo Stock
Price Index (henceforth, TOPIX), which is the index of the Tokyo Stock Exchange in Japan,
tracking all domestic companies of the exchange’s first section. We obtain these data fromJapan Exchange Group Data Cloud. To measure the response of the long-term interest rate, we
use the tick-by-tick Japanese Government Bond yield data, which we obtained from Refinitiv
Japan. Since some observations are missing in the Refinitiv data, we supplement those missing
observations with the tick-by-tick data from Bloomberg.3

3

Research Design

Our primary goal is to identify the effect of the BoJ’s stock purchases on financial markets. We
consider the following econometric model∆yt+l,h = β l,h × ETFt + Γ0l,h Xt + et+l,h ,(1)where ∆yt+l,h ≡ yt+l,h − yt,0 is the change in variable y (e.g. the log of stock prices) from theend price of the morning (h = 0) session on day t to time h on l days later, ETFt is the amountof stock market purchases by the BoJ relative to the stock market capitalization of Japan, Xt isthe vector of controls, and et+l,h contains the unmodeled determinants of the outcome variable.
We are interested in estimating β l,h , which measures the impact of central bank stock purchases
at time h at l days after day t . We choose this simple linear model for expositional purposes. In
Appendix A.1, we consider a non-linear model of (1) and present more technical interpretationof the estimated parameter as in Angrist and Imbens (1995).
An obvious concern for estimating equation (1) via OLS is reverse causality. We expect that
2 See Harada and Okimoto (2021), for example.
3 The results are very similar when we only use the data either from Refinitiv or from Bloomberg.

datasets, when they overlap, are highly correlated with each other, with R2 exceeding 99.98%.

6

The two

0

-4

-2
0
Stock Price Cahnges (%)

Billion Japanese Yen
50
100
150

2

200

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

May 7, 2019

June 3, 2019

July 1, 2019

July 31, 2019

ETF Purchases by BoJ (left axis)
Stock Price Changes in during the Morning (right axis)

Figure 2: An Example of Cutoff Policy Rule
Notes: Figure 2 illustrates the cut-off policy rule by showing the percentage TOPIX changes and the BoJ (Bank ofJapan) purchase amount for each day from May 2019 to July 2019. The solid red line shows the TOPIX changes
in the morning session, and the dashed red line is the estimated cutoff of 0.25%. The bar shows the amount ofpurchases for each intervention in billions of Japanese Yen (approximately 10 million US dollars).

the central bank is more likely to intervene when the stock market performs poorly. This leads
to the downward bias of the OLS estimates of β l,h .
To solve this endogeneity problem, we propose a regression discontinuity based identification strategy that builds on the observation that the BoJ intervention appeared to follow a
cut-off rule. It has been commonly argued among the media that the BoJ appeared to interveneon the day when the value of TOPIX falls below a certain threshold in the morning. For example, Financial Times write “the central bank has tended to step in whenever the TOPIX index
has lost more than 0.5 percent in the morning session”.4 In fact, Figure 2 shows that from May
to July 2019, the BoJ indeed followed a strict rule to intervene when the stock market index
falls more than 0.5% in the morning session. The BoJ intervenes when the index falls slightly
below the 0.5% threshold, while it does not intervene when the index falls slightly above thethreshold.

Suppose for the moment that such a cut-off is known. Then, we can apply a standard
regression discontinuity design. Formally, we assume the policy rule takes the following form,

4 Financial

Times, “Bank of Japan backs away from ETF buying scheme,”
https://www.ft.com/content/a654d1c9-7126-4587-8de6-ed15f567455f.

7(March 23,2021),

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"in which the amount of ETF purchase, ETFt , is given by
ETFt = ETF−,t (∆pt )I(∆pt < ct ) + ETF+,t (∆pt )I(∆pt ≥ ct ),(2)

where ∆pt is the log-changes in the TOPIX value in the morning, ct is the cut-off, ETF−,t andETF+,t are some random functions of the ETF puchase at day t that represent different policy
rules depending on whether ∆pt is above or below the cutoff. We assume (i) E[et+l,h |∆pt , Xt ]is continuous at ∆pt = ct , (ii) lim∆p↑ct E[ ETFt |∆pt = ∆p, Xt ] and lim∆p↓ct E[ ETFt |∆pt = ∆p, Xt ]exist, and (iii) lim∆p↑ct E[ ETFt |∆pt = ∆p, Xt ] 6= lim∆p↓ct E[ ETFt |∆pt = ∆p, Xt ]. Under theseassumptions, it follows thatlim∆p↑ct E[∆yt+l,h |∆pt = ∆p, Xt ] − lim∆p↓ct E[∆yt+l,h |∆pt = ∆p, Xt ]= β l,h .
lim∆p↑ct E[ ETFt |∆pt = ∆p, Xt ] − lim∆p↓ct E[ ETFt |∆pt = ∆p, Xt ](3)

As recommended by Hahn, Todd, and Van der Klaauw (2001) and Porter (2003), we can devise
local linear regression estimators for the left-hand side to obtain an estimate of β l,h . Imbens
and Lemieux (2008) pointed out that this is numerically equivalent to a two-stage least squares
estimator with properly defined instruments and weights. The advantage of their formulationin our context is that it is easy to accommodate heteroskedasticity and auto-correlation. We use
the optimal bandwidth proposed by Calonico, Cattaneo, and Titiunik (2014) and estimate β l,husing two-stage least squares and report Newey-West standard errors.
The difficulty in implementing the above approach is that the cut-off is not necessarily
known. While it was apparently known by the public that the BoJ followed a particular cut-off
rule in some periods, it is sometimes not in other periods. In order to formally investigate thehypothesis, we aim to estimate the cut-off with the presumption that the BoJ follows a cut-off
rule, following the approach proposed by Porter and Yu (2015). Porter and Yu (2015) propose amethod to estimate the discontinuity point and show that there is no loss in efficiency with theregression discontinuity estimator using the estimated cutoff. In implementing this approach,we proceed as follows. We first split the sample period to allow time-variation in the policy
rule. We assume the cut-off is a constant within the sample split. Then, in each of the sample
splits, we consider a set of possible cutoffs, C ≡ {c̄1 , c̄2 , . . . , c̄K }. For each c̄ ∈ C, we estimate
the jump of Prt ( ETFt > 0|∆p) around c̄, which isJt (c̄) ≡ lim Prt ( ETFt > 0|∆p) − lim Prt ( ETFt > 0|∆p).
∆p↑c̄

∆p↓c̄

We select c̄ that maximizes square of the jump, Jt2 (c̄): c∗t ∈ arg maxc̄∈C Jt2 (c̄).
We implement the above approach with the following specifications. First, we consider the
split of the sample period based on the BoJ’s announcement regarding the ETF purchases. The
BoJ made six announcements that state the changes in the target amount of ETF purchases on
8

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"(B) Discontinuities in Policy Intervention-.8

Estimated Cutoff (%)
-.6
-.4

-.2

BoJ Purchases (% of market cap.)
.002
.004
.006
.008

0

.01

(A) Estimated cutoffs over time

No consecutive drops in the past 2 days

0-1

Consecutive drops in the past 2 days

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

-1

-.5
0
.5
Changes in TOPIX in the mornig (%)

1

Figure 3: Estimated Cutoffs and Discontinuities around the Cutoffs
Notes: Figure 3A plots the path of estimated cutoffs over our sample period. Figure 3B shows the discontinuity inthe amount of the BoJ stock purchases in the range of -1% to 1% around the estimated cutoff. Each dot represents
the binned scatter plot with 0.1% bin-width, and the red line represents the linear fit on each side of the cutoff.

March 4, 2013, October 31, 2014, December 18, 2015, July 29, 2016, July 31, 2018, and March
16, 2020. We further split each period between the two announcements based on whether the
TOPIX closing price falls relative to the opening price for the past two consecutive days. We
make this choice based on widely held claims in the media,5 and we indeed found this has astrong explanatory power. Second, we consider the set of potential cutoffs ranging from −1%
to 0% with 0.05% intervals. We estimate the jump of Prt ( ETFt > 0|∆p) around the potential
cutoffs using the local linear regressions with the optimal bandwidth computed from Calonico,
Cattaneo, and Titiunik (2014).

Relevance of cutoffs. Figure 3A shows the path of estimated cutoffs. The estimated cutoffsalign well with the widely held consensus. During 2010-2013, it is widely believed that the
BoJ followed a so- called “1% rule”, in which the BoJ buys ETFs whenever the TOPIX falls
more than 1% in the morning session,6 and our estimates confirm this view. Starting in April
2013, the BoJ appears to use the different cutoffs depending on whether the daily changes in
TOPIX are negative in the past two consecutive days. Since March 2018, the cutoffs appear tobe 0.5% when there is no consecutive fall in the past two days, which is again consistent withthe so-called “0.5% rule.”
5 See, for example, Bloomberg article “The BoJ’s ETF Purchase Conditions Likely to Ease if Stocks Continue toFall” (written in Japanese) (https://www.bloomberg.co.jp/news/articles/2020-07-22/-0-3-kcwteezj).
6 For example, Nikkei Asia writes “the BoJ was widely thought to be following an unwritten rule, dubbed the
1% rule: it would buy ETFs when the Topix index of all issues on the first section of the Tokyo Stock Exchange fell
more than 1% in the morning session.” (https://asia.nikkei.com/Business/Finance/BOJ-steps-up-REIT-buyingscales-back-ETF-purchases)

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ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

Figure 3B shows the binned scatter plot of the size of the BoJ intervention against thechanges in TOPIX in the morning session on the same day relative to cutoffs. We confirm
that there is a discrete jump in the size of the BoJ interventions around zero. The implied jumpin the overall sample is 0.83% of the market capitalization with the standard error of 0.0005%.
The Cragg-Donald F statistic is 1821, and the Kleibergen-Paap F statistic is 261, which sweeps
out weak identification concerns. This discontinuity comes from the discontinuity in the likelihood of an intervention with a jump in probability of intervention of 86% with a standard
error of 0.02%. Importantly, we find strong evidence of discontinuity in any single split of thesample.7
Manipulation test. A natural concern for discontinuity-based research design is the manipulation around the cutoff. While it is unlikely that investors are able to manipulate the stock
price, we formally test the presence of manipulation using the methodology proposed by Cattaneo et al. (2020) in Appendix A.3, and the density plot is shown in Figure B.2. We do not findevidence of manipulation.
(Dis)continuity in ETF purchases across days. yt+l,h is clearly affected by the BoJ’s ETF
purchases up to l days later. Therefore, if falling below the cutoff today is correlated with the
future and past purchases, our empirical estimates cannot be interpreted as the causal effect of
the BoJ’s one-time ETF purchases. In order to address this concern, in Appendix A.4 we test thediscontinuity in the amount of ETF purchases around the cutoff across days. Figure B.3 showsthe estimates of discontinuity of the amount of ETF purchases at date t + l around the cutoff at
day t. Reassuringly, we find significant discontinuity only at l = 0. Therefore, our identified
effects are the causal effects of the BoJ’s one-time ETF purchases and are not contaminated by
future or past ETF purchases.

4

Empirical Results

Armed with the estimates of cutoffs, we implement the regression discontinuity design to assess the impact of the BoJ ETF purchases on the financial market. We report the following threemain results: (i) Bank of Japan’s stock purchases increase both stock prices and long-term interest rates in the overall sample period, (ii) in periods before Bank of Japan introduces yield curve
control, there is no evidence of stock price increase, but robustly increases the long-term interest rate, and (iii) after the introduction of yield curve control, long-term interest rate stopped
responding, and stock market robustly increases.

7 We report the discontinuity for each sample split in Appendix A.2. .

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ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"(B) Discontinuity in 10 Year JGB Yield within a Day

-1

-.2

Changes in TOPIX in the afternoon (b.p.)
-.1
0
.1

Changes in 10-year yield in the afternoon (b.p.)
-.5
0

.2

(A) Discontinuity in Stock Return within a Day

-1

-.5
0
.5
Changes in TOPIX in the mornig (%)

1

-1

-.5
0
.5
Changes in TOPIX in the mornig (%)

1

Figure 4: Discontinuities in Stock Returns and Long-Term Interest Rates
Notes: Figure 4A shows the binned scatter plot of the log-changes in TOPIX in the afternoon (from 11PM to3PM) against the changes in TOPIX in the morning session relative to the cutoff. The bin width is 0.1%. The line
represents the best fit from the linear regression, with the shaded area being a 95% confidence interval. Figure 4B
is analogous to Figure 4A, with the vertical axis being the changes in the 10-year JGB Yield in basis point (b.p.) in
the afternoon (from 11PM to 3PM).

4.1

Homogenous Effect for the Entire Sample Periods

Figure 4A first assesses whether a discontinuity in policy intervention leads to a discontinuity
in stock prices changes. It reports the binned scatter plot changes in TOPIX in the afternoon(from 11AM to 3PM) against the changes in TOPIX in the morning relative to the estimatedcutoff. The figure shows that the stock prices were around 0.2% higher when the TOPIX falls
slightly below the cutoff in the morning than when it falls slightly above the cutoff. Since
the BoJ submits the order to purchase ETF during the lunch break, this suggests that the BoJ
intervention had a large impact on the stock prices within the day. The magnitude is large,
given that the BoJ purchased around 0.01% of market capitalization in each of the interventions
on average.
Figure 4B focuses on the 10-year Japanese government bond yields as an outcome variable.
Perhaps surprisingly, we see discontinuity also in the long-term interest rate. The long-term
interest rate is 4 basis points higher on the left side of the cutoff than on the right side. Later,
we argue that through the lens of the theoretical model, this evidence supports the notion thatthe bond market is inelastic. Intuitively speaking, central banks swap the bonds with stocks. As
there are more supplies of bonds in the economy, the bond prices fall if the investors’ demandfor bonds are downward sloping. The results so far concern the price changes within a day,and therefore it could be the case that the stock and bond prices revert back to the original levelon the following day. We next systematically asses the price impacts for various horizons.
Figure 5A and 5B plot the impulse response functions of stock prices and bond prices. Formally, we plot we estimate β h in equation (1) for each h, where h = 0 is 11AM of the day of the11

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"(B) Impulse Response of JGB 10 Years Yield

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2,

M

M
1P
M

11
A

M

9A

3P

1,
ay
D

M

M
1P
M

11
A

M

9A

3P

0,
ay
D

D

ay

-1

,1
1A

M
1P
M

-.5

-20

0

0

βh
20

βh

.5

40

60

1

(A) Impulse Response of Stock Prices

Figure 5: The Impact on Stock Prices and Long-Term Interest Rates
Notes: Figure 5A shows the impulse response function of stock prices by plotting coefficient β h in equation 1. The
coefficient measures the log-changes in stock prices in response to stock purchases of 1% of market capitalization.
Figure 5B is analogous to Figure 5A and shows the impulse response of the 10-year JGB yield. The coefficientmeasures the percentage point changes in the yield in response to stock purchases of 1% of market capitalization. In all figures, the shaded areas represent 90% confidence intervals, which account for heteroskedasticity andautocorrelation.

intervention with one-hour interval of market hours. In Figure 5A, we see an immediate andlarge stock price response in the afternoon of the intervention. It implies that 1 stock purchasesof 0.01% of market capitalization, a typical size of the intervention, increases the stock value by
0.4%. This coefficient is highly statistically significant. Over the next five days, the coefficient isabout halved and the standard error is wider, but it does not revert back to zero. Reassuringly,
we do not find evidence of pre-trends, which is consistent with the continuity assumption on
the error term.
Figure 5B shows that the 10-year JGB yield also sharply rises following the intervention.
The effect appears to be quite persistent, and it remains statistically significant even five days
after the intervention. The magnitude is again sizable. In response to a typical size of thepurchases (0.01% of market capitalization), the 10-year JGB yield rises by around 0.4-0.5 basis
point.
We have shown that the central bank stock purchases have quantitatively large impacts on
both the stock and bond markets. In what follows, we argue that these average effects mask an
important underlying heterogeneity.

4.2

Heterogenous Effect and Yield Curve Control

Figure 5B showed that the stock market purchases are accompanied by the rise in the longterm interest rate. In standard theoretical models, the rise in interest rate leads to a drop in
stock prices. We therefore expect that the ability of the interest rate to respond is critical in12010-year JGB Yield (p.p.)
.5
1

1.5

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"-.5

Yield curve control
2011

2012

2012

2013

2014

2015

2016

2017

2018

2019

Figure 6: 10 Year JGB Yield
Notes: Figure 6 shows the path of 10-year JGB (Japanese Government Bond) yield over time, where the red verticaldashed line (September 21, 2016) denotes the start of the yield curve control.

determining the stock price responses to the central bank stock purchases.
The BoJ’s other unconventional policy, the so-called “yield curve control,” provides an ideal
laboratory to explore this hypothesis. On September 21, 2016, the BoJ introduced an explicit
target for the 10-year Japanese government bond yield at 0%. Figure 6 indeed shows that thelong-term rate stabilized at around 0% since the introduction of the yield curve control. The
daily standard deviation of the long-term rate is 0.37% before the introduction of yield curve
control, but it falls to 0.08% after the introduction. If the BoJ does its best to stabilize the longterm interest rate at 0%, we would expect to see a much smaller response of the long-term rate
in response to the stock purchases. To test this, we split our sample periods before and after
the introduction of yield curve control and re-do our analysis.
Figures 7A- 8B show the main results of this paper. Figures 7A and 7B plot the impulse
response of stock prices before and after the introduction of yield curve control, repsectively.
We find no evidence that the stock market responds positively after the BoJ intervention before the introduction of yield curve control after one day, although the standard error is large.
In a stark contrast, stock prices persistently rise economically and in a statistically significant
manner under the yield curve control. Quantitatively, a 1% stock purchase by the BoJ causes
around a 20-30% increase in stock prices within at least several days after the intervention.
Figures 8A and 8B explain why. The long-term interest rate responds positively before theyield curve control. Quantitatively, a 1% stock purchases by the BoJ causes around a 1-2%
increase in the long-term rates, and the effect is statistically significant. However, under theyield curve control, long-term rates stopped responding and the effect is precisely estimated
zero.

13

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

50
βh
0

M
3P

M

M
1P
M

11
A

M

9A

3P
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ay

3,

M

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11
A

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ay

2,

11
A

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ay

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11
A

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ay
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ay
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-50
-1
,1
1A
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1P
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11
A

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9A

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3,

M

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11
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ay

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11
A

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1P
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M

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9A

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ay

1,

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11
A

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1P
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M

9A

3P

0,
ay

D

D

ay

-1
,1
1A
M
1P
M

-50

0

βh

50

100

(B) Stock Price Response after YCC

100(A) Stock Price Response before YCC

Figure 7: Heterogenous Stock Price Responses before and after Yield Curve Control
Notes: Figures 7A and 7B show the impulse response of stock prices separately estimated before and after the yield
curve control, which are analogous to Figure 5A.

(A) JGB 10-Year Yield Response before YCC

βh

M
3P

M
1P

M

M
11
A

9A

M

3,
ay

M
1P

3P
D

M

M
11
A

M

9A

3P

2,
ay
D

M
1P

M

M
11
A

9A

M

1,
ay

M
1P

3P
D

M

M
0,

ay

11
A

9A

M
3P

1P

D

D

ay

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-1
M
,1
1A
-1

M

M
1P
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3P
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11
A

9A

M

3,

M

3P
ay
D

1P

M
11
A

M
9A
2,

ay
D

M

M
1P
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3P
M

11
A

9A

M

1,

M

M
1P

11
A

3P
ay
D

D

ay

0,

9A

M
1P
M
3P
M

,1
1A
-1
D

ay

M

-1

0

0

1

1

βh

2

2

3

3

4

4

(B) JGB 10-Year Yield Response after YCC

Figure 8: Heterogenous Interest Rate Responses before and after Yield Curve Control
Notes: Figures 8A and 8B show the impulse response of the 10-year JGB yield separately estimated before andafter the yield curve control, which are analogous to Figure 5B. In all figures, the shaded areas represent the 90%
confidence interval, which accounts for heteroskedasticity and autocorrelation.

14

0

Estimates
2

4

6

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

1 Year

5 Years

10 Years
Before YCC

20 Years

30 Years

After YCC

Figure 9: Response of Other Maturities
Notes: Figure 9 shows the response of the JGB yield across different maturities from 11AM of the day of theintervention to 9AM of the next day. The circle dot represents the point estimates before the yield curve control,
and the diamond dot represents the point estimates after the yield curve control. The coefficient measures thepercentage point changes in the JGB Yield in response to the purchase of 1% of market capitalization. The verticallines represent the 90% confidence interval, which accounts for heteroskedasticity and autocorrelation.

4.3

Bond Yield Responses Across Different Maturities

We show that the effect on interest rate is not specific to the 10-year JGB yield, but rather iswide spread across different maturities.
Figure 9 shows the point estimates of the effect on JGB yield across maturities of 1, 2, 5,
10, 20, and 30-years. Before the yield curve control, the yield on all maturities rose, but witha larger effect on longer maturities. Our preferred interpretation is that the zero lower boundon policy rate has been binding during this period, and therefore the shorter maturity bonds
had less room to respond relative to longer maturity bonds. After the yield curve control, allinterest rates entirely stopped responding. Even though the yield curve aimed to specificallycontrol the 10-year yield, it can prevent the response of other maturities because they are interconnected through arbitrage. For example, it is the natural prediction that arises from thepreferred habitat model of term structure by Vayanos and Vila (2021).

4.4

Robustness

Table 1 conducts a battery of robustness checks and shows that our results are robust to various
alternatives to the baseline specifications. In rows 1 and 2, we show that the results are not

15

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

sensitive to changing the bandwidth of the regression discontinuity estimator.8 Row 3 uses thequadratic local polynomial regression instead of the linear. Row 4 controls the amount of the
BoJ’s purchases in the past two days. This addresses the concern that if the stock price changes
are likely to fall on one side of the cut-off over consecutive days, our estimator confounds theeffect from the past and the future interventions. Reassuringly, the results are not sensitive tothis control. This is not surprising given that there is no evidence that falling below the cutoff
today causes the future and past ETF purchases, as we formally show in the Appendix A.4.
Row 5 and row 6 control the past stock market return and changes in the long-term interest
rate over the past two days. Finally, in row 7, we drop observations one week before and after
the dates when the cutoff changed. This addresses the concern that the changes in cutoff could
be endogenous to the underlying economic fundamentals or the cutoff change may containsignals about future policy stances of the BoJ. Collectively, our results appear to be virtuallyunchanged to any of the above modifications.

4.5

Placebo Tests

We conduct placebo tests by testing the presence of discontinuity in outcome variables aroundan arbitrary cutoff for which we do not expect to find any discontinuity. One might worry that
our results are not driven by the BoJ’s policy intervention, but rather some other factors suchas investors sentiments thatdiscontinuously respond to the stock price changes in the morning
session. For example, investors might speculate that the stock market should see a stronger
rebound when stock prices fall below 0% in the morning session.
To address this concern, for each value of c ∈ {−1%, −0.95%, . . . , −0.05%, 0%}, we test
whether there is discontinuity in our outcome variables when the stock prices fall below thethreshold c. We estimate∆yt+l,h = γl,h × I(∆pt < c) + et+l,h ,(4)where ∆pt is the percentage change in TOPIX in the morning session. When estimating, we
exclude periods for which the cutoff is identical to c under consideration. We are interested in
the estimates of γl,h , and we expect that γl,h to be indistinguishable from zero for any value ofc.
Figures 10A and 10B show the estimated value of γl,h , together with the 90% confidence
interval. Reassuringly, we find that the estimates of γl,h are indistinguishable from zero in almost all cases. Even if they are significant, the estimates are the opposite sign from our baseline
estimates. Moreover, in all cases, the estimates are far enough from our baseline estimates that
use the actual cutoff. These results suggest that our results are indeed driven by the policy
intervention itself.
8 Figure B.5 in the Appendix more systematically explore robustness with respect to the choice of bandwidths.

We find that the results are virtually unchanged for a wide variation in the bandwidths.

16

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

Panel A. Stock Price Response

0. Baseline

All
Same Day Next Day
34.90
8.65
(5.89)(10.18)Before YCC
Same Day Next Day
36.31-16.95(17.50)(26.27)After YCC
Same Day Next Day
35.24
22.23
(4.68)(11.13)

1. Narrower
Bandwidth

41.65(7.16)

16.22(16.09)30.14(22.18)-21.25(39.57)47.26(7.15)

33.08(16.25)

2. WiderBandwidth

29.49(4.84)

6.07(9.36)

30.63(13.46)-2.44(20.74)29.41(4.69)

15.66(9.14)

3. Polynominal
Order 243.40
(7.47)

16.98(15.64)41.13(21.64)-22.70(42.42)47.24(7.14)

30.63(14.39)

4. Control Past
Interventions

39.50(8.09)

12.56(15.10)54.57(25.67)-16.44(34.04)37.32(6.61)

29.65(14.48)

5. Control Past
Stock Returns

34.91(5.89)

8.62(10.20)36.15(17.70)-17.33(26.21)35.18(4.77)

20.51(11.76)

6. Control Past
10-Year Yield

35.96(5.84)

6.63(9.94)

40.91(18.33)-23.41(25.42)35.06(4.67)

21.90(11.13)

7. Drop Aroundthe Cutoff Changes

34.96(6.05)

8.00(10.92)

31.21(16.52)-20.53(28.22)35.07(5.52)

24.56(12.45)

Panel B. JGB 10-Year Yield Response

0. Baseline

All
Same Day Next Day
0.47
0.37
(0.18)(0.23)

Before YCC
Same Day Next Day
1.52
1.41
(0.59)(0.68)

After YCC
Same Day Next Day
0.04
-0.00
(0.07)(0.13)

1. Half
Bandwidth

0.54(0.22)

0.43(0.30)

1.96(0.84)

1.96(0.96)-0.00(0.09)-0.13(0.20)

2. WiderBandwidth

0.40(0.16)

0.35(0.19)

1.28(0.47)

1.05(0.50)

0.01(0.06)-0.04(0.10)

3. Polynominal
Order 20.52
(0.22)

0.43(0.27)

1.81(0.77)

1.75(0.91)

0.07(0.09)

0.01(0.18)

4. Control Past
Interventions

0.47(0.23)

0.37(0.31)

1.73(0.76)

1.79(0.90)-0.06(0.11)-0.17(0.21)

5. Control Past
Stock Returns

0.47(0.18)

0.37(0.23)

1.53(0.59)

1.43(0.68)

0.03(0.08)-0.03(0.13)

6. Control Past
10-Year Yield

0.46(0.17)

0.37(0.23)

1.51(0.60)

1.42(0.69)

0.04(0.07)-0.01(0.13)

7. Drop Aroundthe Cutoff Changes

0.37(0.16)

0.28(0.21)

1.12(0.44)

0.91(0.48)

0.02(0.06)-0.05(0.12)

Table 1: Robustness
Notes: Table 1 shows robustness checks against various modifications to our benchmark specifications. Panels
A and B show responses of stock prices and the JGB 10-year yield, respectively. In each panel, row 0 shows thebaseline estimates. Row 1 considers bandwidth that is 50% less than the original one. Row 2 considers bandwidththat is 50% more than the original one. Row 3 considers a local polynomial regression of order 2 instead of 1.
Row 4 controls the BoJ’s stock purchases in the past two days. Row 5 controls stock market returns over the past
two days. Row 6 controls changes in the 10-year yield over the past two days. Row 7 drops observations before
and after one week around the cutoff changes. Same-day response indicates the changes in the outcome variablefrom 11AM to 3PM in the same day of the intervention. Next-day response indicates the changes in the outcome
variable from 11AM of the day of the intervention to 9AM in the next day. Standard errors, which account forheteroskedasticity and autocorrelation, are reported in parenthesis.

17

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"(B) Placebo Test for 10-year JGB Response-.4

Stock price change in the afternoon (%)
-.2
0
.2
.4

10-year JGB yield change in the afternoon (b.p.)
-1
-.5
0
.5

(A) Placebo Test for Stock Response

-1

-.8

-.6
-.4
Placebo Cutoff (%)

Placebo estimates

-.2

0

-1

Baseline estimates

-.8

-.6
-.4
Placebo Cutoff (%)

Placebo estimates-.20

Baseline estimates

Figure 10: Placebo Tests
Notes: Figure 10A plots the estimates of γl,h in equation (4) as the blue dots for each placebo cutoff, where theoutcome variable is the stock price. The estimates are the response within the day of intervention (changes from
11AM to 3PM). We exclude the periods where the placebo cutoff coincides with the actual cutoff. The red line
indicates our estimates using the actual cutoff. Figure 10B is analogous to Figure 10A, where the outcome variable
is now the 10-year JGB yield. The line and the shaded area represent 90% confidence interval.

4.6

Other Discussions

We discuss several other issues. First, if the BoJ is selling the long-term government bonds atthe same time as the stock purchases, then it is not surprising that the long-term interest rate
rises in response to the stock purchases. However, during our sample periods, the BoJ has soldthe government bonds only twice, March 24, 2017 and March 23, 2020, both of which are theperiods after the yield curve control. Therefore, we can forcefully rule out the concern.
Second, one may wonder whether the difference in the amount of purchases across time
periods might be driving the heterogeneity between before and after the yield curve control.
Figure B.4 in the appendix shows that, while it is true that the absolute amount of purchases
in each intervention is four times larger after the yield curve control than before, the growth ismuch less pronounced once expressed as a fraction of market capitalization. The average sizeof intervention as a fraction of market capitalization is slightly less than twice after the yield
curve control than before.
The final issue concerns the interpretation of our results. As discussed by Krishnamurthy
and Vissing-Jorgensen (2011), there are broadly two channels through which central bank asset
purchases have an effect on asset prices. The first is the signaling channel. According to thischannel, the central bank asset purchases have an effect because they send signals about the
future policy stance of the central bank. The second channel is the liquidity channel. This
channel operates through changing the aggregate demand and supply of assets. We believe our
empirical results are likely driven by the liquidity channel rather than the signaling channel.
The BoJ announces the target amount of stock purchases in each year in advance. Therefore,
18

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

whether or not the BoJ purchases stocks today should not reveal the BoJ’s future policy stance.

5

Organizing Theoretical Framework

We provide an organizing theoretical framework to provide structural interpretations of ourempirical results. Through the lens of our model, we argue that the empirical estimates support
a theory in which both stock and bond markets are substantially inelastic.

5.1

Setup

The only factor of production is capital. We assume that the supply of capital is fixed at K and
Yt = At K,
where productivity At evolves stochastically and independently over time according to lognormally distributed growth1ln( At+1 /At ) ∼ N ( g − σ2 , σ2 ),2where g is the mean growth rate and σ2 is the variance of growth rate.
Households are divided into two parts: a consumption part and an investment part. The
consumption part of the household makes consumption and saving decisions. The investment
part of the household decides the amount to invest in stocks. Each part takes the other part’s
action, as given. This structure closely follows Gabaix and Koijen (2021), which makes ourcomparison to them easier and also helps simplify the exposition.
The consumption part of the household solves the following problem.
∞

max E0 ∑ βt u(Ct exp(vb ( Bt /Yt )){Ct ,Btc }

s.t.

t =0

Ct + Bt + St = Wt − Ft
Wt+1 = Rt Bt + Rst St − Tt+1 ,where Ct is the consumption, Bt ≡ Bti + Btc is the total bond holdings with the rate of return Rt ,and Wt is the aggregate wealth at time t. Here, Btc denotes the bond-holding decisions made by
the consumption part of the household, and Bti denotes the bond-holding decisions made by theinvestor part of the household. Central bank finances the cost of purchases through the lumpsum tax Ft , and it distributes ex-post profits through Tt+1 . The stocks, St , with return Rst andthe investor’s bond holding decisions, Bti , are managed by the investor part of the household.
19

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

The preference term vb (·) captures the preference for liquidity or safety as in Krishnamurthy
and Vissing-Jorgensen (2012), with the properties v0b ≥ 0 and v00b ≤ 0.
The investor part of the household chooses the portfolio to maximize the following objectivefunction:9
max Et

St ,Bti ,Wt+1[Wt+1 exp(−vs (St /Yt − s̄))]1−γ.
1−γ

The return from stock is given by
Rst+1 =

At+1 + Pt+1,
Pt

where Pt is the stock price. The term vs is the adjustment cost that is similar in spirit to Gabaix
and Koijen (2021). This can be broadly interpreted as any factors that prevent flexible adjustment in the portfolio, for example, inattention, inertia, or institutional mandate. We assume theinvestor has to incur adjustment costs when the portfolio deviates from the equilibrium without the central bank. Since St = Pt K holds without the central bank, it follows that s̄ ≡ Pt K/Yt .
We impose v0s ≥ 0, v00s ≥ 0, and v0s (0) = 0. We assume the investor part of the household takes
the bt , the decision of the consumption part of the household as given.
The central bank’s budget constraint is
CB
StCB + BtCB = Rst−1 StCB
−1 + Rt−1 Bt−1 + Tt + Ft .

(5)

The equilibrium of this economy consists of prices { pt , Rt }, quantities {ct , at , st }, and central
bank’s policies { BtCB , StCB , Tt , Ft } such that (i) given { Pt , Rt , St }, {Ct , Bt } solve the problem ofhouseholds and mutual funds; (ii) given { Pt , Rt , Bt }, the investor part of the household optimally chooses {St }; and (ii) { BtCB , StCB , Tt } satisfies the central bank’s budget constraint (5); and(iii) markets clear as follows:

Ct = Yt
Bti + Btc + BtCB = 0
St + StCB = Pt k.

9 The assumption that the investor solves myopic portfolio problem follows Gabaix and Koijen (2021).

Thisassumption is not essential for any part of our analysis other than to make the comparison to Gabaix and Koijen(2021) easier.

20

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

5.2

Equilibrium without Central Bank

We first characterize the balanced growth path (BGP) of the equilibrium without the central
bank. Here, we delegate the detailed derivation to Appendix C. The first order condition of theconsumption part of the household combined with the market clearing conditions, Bt = 0 and
Ct = Yt , givesu 0 (Y )1 = βRt Et 0 t+1 + v0b (0),(6)
u (Yt )
which is the usual consumption Euler equation with the additional term capturing the demand
for liquidity service. In solving the problem of the investor part of the household, we workwith the following approximate portfolio problem as the time interval goes to zero, following
Campbell and Viceira (2002).
max E(ln( Rst ) − ln R)s + (1 − γ)s2 Var (ln Rst ) − vs (sPt Kt /Yt − s̄).
s

We conjecture (and verify) that the stock price is constant proportional to the productivity,
Pt = pAt . Taking the FOC of the above problem and imposing the market clearing, st = 1, we
obtain
1+ p= γσ2 + v0s (0) p.
(7)
g − ln R + lnpEquilibrium prices { p, Rt } solve (6) and (7).

5.3

Central Bank Stock Purchases

We now study the effect of central bank stock purchases. To do so, we work with the loglinearized approximation of the no central bank equilibrium described above. We consider
two cases: a case where interest rates flexibly adjusts and a case where interest rates do not
adjust in response to the central bank’s intervention.
5.3.1

Flexible Interest Rate Adjustment

In this case, we consider the following policy intervention. We assume the central bank permanently holds a fraction of stocks dsCB = dStCB /Pt K > 0 amount of stock financed by thebond issuance dBtCB = −dsCB
t Pt K. The central bank’s ex-post return is distributed through theCBlump-sum transfers: Tt = −( Rst−1 StCB
−1 + Rt−1 Bt−1 ).
The log-linearized Euler equation isd ln R = κb dsCB ,−v00 (0)(8)

where κb ≡ 1−vb0 (0) p. We refer to the parameter κb as the bond market inelasticity, and it capb

21

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

tures how a one percentage point increase in the supply of bonds as a fraction of market capitalization affects the interest rate. The log-linearized asset pricing equation isd ln p = −γr d ln R + κs dsCB
1+ p

(9)

d ln pwhere γr ≡ 1+(1+ p) pv0 (0) = d ln R is the elasticity of stock price with respect to the interest rate,s(1+ p) p2 v00 (0)and κs ≡ 1+(1+ p) pvs 0 (0) is the stock market inelasticity holding the interest rate fixed.
s

The following proposition summarizes the qualitative theoretical predictions:
Proposition 1. Consider the small amount of stock purchases by the central bank described above.
i. Elastic stock and bonds market. If vb (·) = vs (·) = 0, then the central bank stock purchases
are neutral:d ln R = d ln p = 0.
ii. Inelastic stock market and elastic bonds market. If v00s (0) > 0 and v00b (0) = 0, thend ln R = 0,d ln p > 0.

iii. Inelastic stock and bond market. If v00s (0) > 0 and v00b (0) < 0, thend ln R > 0



>0


d ln p = 0



< 0

κs > γr κbκs = γr κb .
κs < γr κb

The first result is reminiscent of the neutrality result of the central bank portfolio by Wallace(1981). In a frcitionlessenvironment where Ricardian equivalence holds and the private agentsare free to adjust their portfolios, any movements in the central bank portfolio are completely
undone by private agents.
Figure 11 graphically illustrates the frictionless benchmark. The left and the right panels
show the demand and supply curves for stocks and bonds, respectively. Since both assets
are in fixed supply, the supply curves are fixed. In the frictionless economy, the demands are
infinitely elastic, and as a result, demand curves are horizontal. As a result, a central bank stock
purchase, which absorb the supply of stocks in the market by providing additional supply ofbonds to the market, have no consequence on the stock prices or interest rates.
The second result is reminiscent of the stock market inelasticity hypothesis recently proposed by Gabaix and Koijen (2021). When the stock market is inelastic, the central bank stock
purchases cannot be undone by private agents. This implies that the demand for stock rises,
22

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

stock price, pinterest rate, R
supplysupplydemanddemandstock quantity in the market, Sbond quantity in the market, B

Figure 11: Frictionless Benchmark
which drives up the stock price. In contrast, with an elastic bonds market, the interest rate
is pinned down by the consumption Euler equation, which is unaffected by the central bankportfolio.
The third result is our focus, and it provides a much more nuanced view than the previous
two cases. When both bonds and stock markets are inelastic, the interest rate unambiguously
rises, and the effect on stock prices is ambiguous. The interest rate rises because as the central
bank supplies more bonds, the households value less of it, which in turn puts downward pressure on bond prices. This rise in the interest rate can counteract the rise in the stock price. Threesufficient statistics, κs , κb , and γr , govern which of these forces dominate in a intuitive manner.
In fact, it boils down to the comparison of stock market inelasticity, κs , and the multiplicative ofbonds market inelasticity and the interest rate sensitivity of stock prices, κb γr . When the stock
market is more inelastic (high κs ), the stock price is more likely to rise. When the bonds market
is relatively inelastic (high κb ) and the sensitivity of stock prices to interest rates is high (highγr ), stock price is more likely to fall.
Figure 11 graphically illustrates the case where both stock and bond markets are inelastic. Unlike the frictionless benchmark, the demand curve for stock is downward sloping withrespect to price, and the demand curve for bond is upward sloping with respect to interest
rate. Moreover, the demand for stock negatively depends on the interest rate. In this economy, central bank stock purchases will have a direct impact of increasing the stock price andinterest rate, as can been seen as the outcome of the shift in supply curve. The increase in ininterest rate, in turn, reduces the demand for stock. Therefore, the net effect on stock price isambiguous.

23

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

stock price, pinterest rate, R

R = R̄(B)R↑

pt = p̄(S; R)
R↑
stock quantity in the market, Sbond quantity in the market, B

Figure 12: Inelastic Stock and Bond market
5.3.2

Fixed Interest Rate

Now we consider the central bank stock purchases financed through lump-sum taxes: dStCB /Pt K ≡dsCB > 0 with dBtCB = 0 and dTt = dStCB for t = 0 and dTt = dStCB − Rst−1 StCB
−1 for t ≥ 1. In thiseconomy, this is the only way that the central bank is able to maintain a constant real interest
rate when it buys stocks. We intend to mimic the central bank stock purchases under the yield
curve control.
In this case, we haved ln pt = κs dsCB ≥ 0(10)d ln Rt = 0.
The fact that the interest rate does not respond stems from the fact that the central bank does notissue bonds to finance the purchases of stocks, which implies the household Euler equation isunaffected. Because the interest rate is fixed, the stock price always (weakly) increases. Figure
13 graphically illsutrates the case with fixed interest rate. Since the interest rate adjustment
is absent, there is downward shift in the demand curve for stock. As a result, the stock pricerobustly rises. Qualitatively, this prediction also corresponds to the case where bond market isperfectly elastic.

24

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

stock price, pinterest rate, Rp↑
R = R̄

pt = p̄(S; R)

stock quantity in the market, Sbond quantity in the market, B

Figure 13: Fixed Interest Rate

5.4

Mapping to the Empirical Analysis and Identification of the StructuralModel

We now connect our structural model to our empirical analysis. Note that equations (9) and (8)are exactly the equations that we estimated in the empirical sections. Taking our benchmark
estimates for the next day (0th row of Table 1), we haveκb = 1.41
γr κb + κs ≈ 0,

(11)
(12)

where we set the latter to zero, since our estimates are noisy and indistinguishable from zero.
In the post-YCC sample, assuming YCC corresponds to the fixed interest rate, equation (10)corresponds to our estimating equation for the stock price. Therefore, we obtainκs = 22.

(13)

Gabaix and Koijen (2021) report empirical estimates of stock market inelasticity of 5. Other
studies (Da et al., 2018; Hartzmark and Solomon, 2021; Li et al., 2021) estimate somewhere
between 1.5 to 6. While our estimates of stock market inelasticity holding the interest rate
fixed, κs , is several times higher than their estimates, the stock market elasticity when interest
rate can freely move, γr κb + κs , is lower than theirs (our point estimates are even negative). The
difference could come from either the difference in identification strategy or the difference inthe underlying macroeconomic environment (e.g., persistently low interest rate environment

25

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

in Japan). However, the robust message of our paper is to underscore the importance of jointly
taking into account stock market and bond market inelasticity.
We close this section by providing an over-identification test. From equations (11), (12), and(13), one can back out the stock price elasticity with respect to the interest rate asγr = −15.6.
This is well in line with existing empirical estimates in the context of Japan. Kubota andShintani (2022a) use high-frequency identification around monetary policy announcements
by the BoJ. They find that unanticipated 1 percentage point monetary policy tightening results in a 10% to 16% drop in stock prices.10 Therefore, our empirical estimates pass the overidentification test, which renders further credibility.

6

Conclusion

By exploiting the unique feature of Bank of Japan’s policy rule, we trace the aggregate impact ofstock market purchases. Taken together, our empirical evidence provides support for a theory
in which both stock and bond markets are substantially inelastic. We believe our results will
prove useful in designing quantitative easting policies around the globe and to understanding
the sources of financial market fluctuations. The natural next step is to assess how the central
bank stock purchases have transmitted to the real economy, which we leave for future research.

10 Kubota and Shintani (2022b) extend the analysis to examine the effect for longer horizons using VAR withexternal instruments. They show that a 1% monetary policy tightening results in around a 25% drop in stock
prices after one year. All these estimates are substantially higher (in absolute terms) than the estimates fromsimilar settings using the US data (e.g., Bernanke and Kuttner, 2005), possibly reflecting the persistently low
interest rate environment in Japan.

26

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

AppendixA

Empirical Appendix

A.1

Local Nonlinear Impulse Response Function

In this section, we allow nonlinearity in the impulse response and show that our estimands canbe interpreted as dynamic “local average treatment effect” in the spirit of Angrist and Imbens(1995).
We first define a potential outcome framework in our context following Rambachan andShephard (2021). For each t ≥ 1, the BoJ decides the amount of ETF purchases ETFt and let
us denote ETF1:t ≡ ( ETF1 , · · · , ETFt ). Let w1:t ≡ (w1 , · · · , wt ) be a potential assignment path
up to t where wt ∈ [0, w̄] for all t. Associated with this potential assignment path, potential
outcome at day t + l time h is Yt+l,h (w1:t+l ).11 Note that for any different assignment paths,
there exist different outcome paths but we only obeserve Yt+l,h ( ETF1:t+l ). For any day t + ltime h, let us denote
Yt+l,h (w) ≡ Yt+l,h ( ETF1:t−1 , |{z}w , ETFt+1:t+l ).
t−thUsing this notation, the observed outcome can be denoted as Yt+l,h ( ETFt ) by definition. Now,
let the outcome be∆yt+l,h ≡ yt+l,h − yt,0 , which is the change in variable y (e.g. stock prices andinterest rates) from the end price of the morning session to at day t to time h at day t + l,
∆yt+l,h = Yt+l,h ( ETFt ).

(14)

We assume that the BoJ’s ETF purchasing policy rule takes the following form, in which theamount of ETF purchase at time t, ETFt , is given by
ETFt = ETF−,t (∆pt )I(∆pt < ct ) + ETF+,t (∆pt )I(∆pt ≥ ct ),(15)

where ∆pt is the log-changes in the TOPIX value in the morning, ct is the cut-off, and ETF−,tand ETF+,t are random functions of ∆pt which represent different policy rules depending onwhether ∆pt is above or below the cutoff at time t. The following assumptions guarantee that
our estimands identify the dynamic local average treatment effect.

11 We assume that the potential outcome depends only on past and contemporaneous assignments. Rambachan

and Shephard (2021) called this assumption Non-antiripating potential outcomes.

27

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

Assumption 1. (i) Yt+l,h (w) is bounded and continuously differentiable in w ∈ [0, w̄] withprobability one and, (ii) ETF−,t (∆p) and ETF+,t (∆p) are bounded and continuous at ct withprobability one.
Assumption 2 (Monotonicity). ETF−,t (ct ) ≥ ETF+,t (ct ) with probability one.
Assumption 3 (Relevance).

R

Pr( ETF−,t (ct ) ≥ w ≥ ETF+,t (ct )|∆pt = ct )dw > 0.

Assumption 4 (Local Independence). For each t + l and h, there exists a neighborhood Nt+l,hof ct such that ∆pt ⊥ ({Yt+l,h (w)}w , ETF−,t (ct ), ETF+,t (ct ))|∆pt ∈ Nt+l,h .
Theorem 1. If Assumptions 1-4 hold, thenlim∆p↑ct E[∆yt+l,h |∆pt = ∆p] − lim∆p↓ct E[∆yt+l,h |∆pt = ∆p]
lim∆p↑ct E[ ETFt |∆pt = ∆p] − lim∆p↓ct E[ ETFt |∆pt = ∆p]∂Yt+l,h (w)|∆pt = ct , ETF−,t (ct ) ≥ w ≥ ETF+,t (ct )]ω̄dw,∂w
R
where ω̄ = Pr( ETF−,t (ct ) ≥ w ≥ ETF+,t (ct )|∆pt = ct )/ Pr( ETF−,t (ct ) ≥ w ≥ ETF+,t (ct )|∆pt =ct )dw.

=

Z

E[

Proof. First, observe thatlim E[∆yt+l,h |∆pt = ∆p] = lim E[Yt+l,h ( ETF−,t (∆p))|∆pt = ∆p]

∆p↑ct

∆p↓ct

= E[Yt+l,h ( ETF−,t (ct ))|∆pt = ct ],where the second equality follows from Assumption 1 and 4. Therefore,lim E[∆yt+l,h |∆pt = ∆p] − lim E[∆yt+l,h |∆pt = ∆p]

∆p↑ct

∆p↓ct

= E[Yt+l,h ( ETF−,t (ct )) − Yt+l,h ( ETF+,t (ct ))|∆pt = ct ]∂Yt+l,h (w)I{ ETF−,t (ct ) ≥ w ≥ ETF+,t (ct )}dw|∆pt = ct ]
∂w
Z
∂Y
(w)
= E[ t+l,h|∆pt = ct , ETF−,t (ct ) ≥ w ≥ ETF+,t (ct )]Pr( ETF−,t (ct ) ≥ w ≥ ETF+,t (ct )|∆pt = ct )dw,
∂w

= E[

Z

where second equality follows from Assumptions 1and 3, and the third equality follows from28

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

1. Similarly,
lim E[ ETFt |∆pt = ∆p] − lim E[ ETFt |∆pt = ∆p] =

∆p↑ct

∆p↓ct

Z

Pr( ETF−,t (ct ) ≥ w ≥ ETF+,t (ct )|∆pt = ct )dw,

and Assumption 2 guarantees that the denominator is positive. Combining these, we have thestated result.
Since Yt+l,h (w) ≡ Yt+l,h ( ETF1:t−1 , w, ETFt+1:t+l ), the local independence assumption requires that falling below the cutoff at day t is not correlated with the future or past ETF purchases. We test this in Appendix A.4.

A.2

Details on Cutoff Estimation

We first split the sample based on six announcements by the BoJ that publicized changes in thetarget amount of ETF purchases on April 4, 2013, October 31, 2014, December 18, 2015, July 29,
2016, July 31, 2018, and March 16, 2020. We then divide each sample based on whether TOPIX
value falls below zero for the past two consecutive days. For the case with consecutive drops
in the past two days, we further split on April 1, 2019, for the reason that we describe below.
In each sample split, we proceed as follows. We take grid points for the cutoff candidates
from -1% to 0% with 0.05% interval, C = {−1.0%, −0.95%, . . . , −0.05%, 0.0%}. For each ofc ∈ C, we estimate the following linear probability model separately on both sides of thecandidate cutoff, c:α + β ∆p for ∆p ∈ [c − k, c]−−tt
Pr−,t ( ETFt > 0|∆pt ) =,(16)
α + β ∆p for ∆p ∈ [c, c + k ]
+

+

t

t

where we take the bandwidth to be 1% around the cutoff, k = 1%. Given the estimates, we cancompute the jump around the cutoff as follows:ct ( ETFt > 0|∆p) − lim Pr
ct ( ETFt > 0|∆p),
Jt (c) ≡ lim Pr
∆p↑c̄

∆p↓c̄

ct denote the fitted value of equation (16). We select the cutoff that maximizes square
where Prof the jump:c∗t ∈ arg max Jt2 (c).
c ∈CWhenever there is a tie, we choose the largest cutoff.
Table B.2 shows the estimated cutoff, and Table B.3 shows the discontinuity in the probability of Bank of Japan’s intervention around the estimated cutoff. As argued in the main text,
the estimated cutoffs align well with what is commonly argued among media. The disconti29

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

nuity around the cutoff is always over 50%, is often over 80%, and they are highly statisticallysignificant. We made a choice to split the sample with consecutive drops in the past two days
at April 1, 2019, because there was a apparent change in the cutoff around this period. If we
do not split the sample at this point in time, the resulting discontinuity is -0.744 . If we split
the sample, the discontinuity is -1.000 in the first half, and it is -0.853 in the second half of thesample. This choice does not materially affect any of our empirical results.
Figure B.1 graphically displays the discontinuity in the probability of intervention for eachperiod. While the magnitude of discontinuity is more apparent in the beginning and the endof the sample period, the sharp discontinuity shows up in all subsamples.

A.3

Manipulation Test

A typical concern in regression discontinuity-based identification strategies is manipulation(McCrary, 2008). We first note that this concern is unlikely in our context since there is little
room for investors to precisely manipulate the stock price index. Having said this, we formallytest the presence of manipulation by examining the continuity of density function of TOPIX
changes in the morning. We estimate the density function using the local polynomial density
estimator by Cattaneo et al. (2020) and test the presence of discontinuity around our estimatedcutoff.
Figure B.2 shows the estimated density and histogram, and Table B.4 reports the estimates
and test statistics for discontinuity. While there is a small mass on the right side of the cutoff,
the p-value of testing the discontinuity is 0.447. Therefore, there is no statistical evidence ofmanipulation.

A.4(Dis)continuity of ETF Purchases across Days

In this section, we argue that the effects we are identifying are the effects of a one-time shockof ETF purchases. As discussed in A.1, yt+l,h is clearly affected by the BoJ’s ETF purchases up
to l days later. Therefore, if falling below the cutoff today is correlated with the future and past
purchases, our empirical estimates cannot be interpreted as the causal effect of one-time BoJ
ETF purchases (Rambachan and Shephard, 2021). In order to address this concern, we estimate
the discontinuity in the amount of ETF purchases around the cutoff across days. Formally, we
estimate the following term,lim E[ ETFt+l |∆pt = ∆p] − lim E[ ETFt+l |∆pt = ∆p].

∆p↑ct

∆p↓ct

(17)

Figure B.3 shows the estimates of discontinuity of the amount of ETF purchases at date t + laround ∆pt = ct . Reassuringly, we find significant discontinuity only at l = 0. Therefore,

30

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

our identified effects are the causal effects of one-time the BoJ’s ETF purchases and are notcontaminated by the future or past ETF purchases.

B

Additional Tables and Figures
Table B.1: Major Announcements by the BoJ
Date
October 28, 2010
October 30, 2012
October 31, 2014
December 18, 2015
July 29, 2016
March 16, 2020

Annoucnement
Intention to purchase 450 billion yen of ETFs
Intention to purchase 2.1 trillion yen of ETFs annually
Annual purchase target increased to 3 trillion yen
Annual purchases target increased to 3.3 trillion yen
Annual purchases target increased to 6 trillion yen
Annual purchases target increased to 12 trillion yen

Notes: Table B.1 shows the six major announcements by the BoJ regarding the target ETF purchase amounts.
Source: Fukuda and Tanaka (2022).

Table B.2: Estimated Cutoff
No Consecutive Drops
PeriodCutoff2010/12/15 - 2013/04/03-1%
2013/04/04 - 2014/10/30 -0.35%
2014/10/31 - 2015/12/17 -0.15%
2015/12/18 - 2016/07/28-0.4%
2016/07/29 - 2018/07/30-0.3%
2018/07/31 - 2020/03/15-0.5%
2020/03/16 - 2020/12/31-0.5%

Consecutive Drops
Period2010/12/15 - 2013/04/03
2013/04/04 - 2014/10/30
2014/10/31 - 2015/12/17
2015/12/18 - 2016/07/28
2016/07/29 - 2018/07/30
2018/07/31 - 2020/03/15
2020/03/16 - 2020/12/31

Notes: Table B.2 shows the estimated cutoff for each of the subsample.

31

Cutoff
-1%
0%
0%
0%
0%
-0.25%-0.25%

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

Table B.3: Discontinuity in Probability of Intervention around the Estimated Cutoff

2010/12/15 - 2013/04/03
2013/04/04 - 2014/10/30
2014/10/31 - 2015/12/17
2015/12/18 - 2016/07/28
2016/07/29 - 2018/07/30
2018/07/31 - 2020/03/15
2020/03/16 - 2020/12/31

No Consecutive Drops
Discontinuity Sample size
estimates
Left Right-1.01143
158(0.012)-0.57660
146(0.100)-0.68372
98(0.099)-0.81120
36(0.119)-0.60478
243(0.069)-0.97849
163(0.022)-0.93029
71(0.070)Consecutive Drops
Discontinuity Sample size
estimates
Left Right-1.00015
47(0.000)-0.93125
30(0.072)-1.12211
17(0.117)-1.0008
11(0.000)-0.94540
33(0.053)-0.74430
34(0.130)-0.98513
14(0.022)

Notes: Table B.3 shows the discontinuity in the probability of the BoJ intervention around the estimated cutoff.
We estimate the discontinuity using the local linear regression with bandwidth 1% around the cutoff and uniform
kernel. The standard errors are reported in parenthesis.

Table B.4: Density Discontinuity Test
Density estimates
Left
Right
0.423
0.506
( 0.065 ) ( 0.067 )Sample size
Bandwidth
Effective sample size

667
0.512
358

Discontinuity test
Difference p-value
0.083
0.373
( 0.093 )

1790
0.512
719

Notes: Table B.4 reports the density estimates on the left and the right of cutoff and test statistics for the discontinuity test. We use the local polynomial density estimator by Cattaneo et al. (2020) with order 2. Robust standard
errors are reported in parenthesis.

32

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

From 2010−12−15 to 2013−04−03

From 2013−04−04 to 2014−10−3050150Probability of BoJ Purchase (%)

100

0

100

50

0
−0.5
0.0
0.5
Changes in TOPIX relative to the cutoff (%)

1.0

From 2015−12−18 to 2016−07−28

−0.5
0.0
0.5
Changes in TOPIX relative to the cutoff (%)

50

1.0

−1.0

From 2016−07−29 to 2018−07−30

50

0

100

50

0
1.0

1.0

150
Probability of BoJ Purchase (%)

Probability of BoJ Purchase (%)

100

−0.5
0.0
0.5
Changes in TOPIX relative to the cutoff (%)

From 2018−07−31 to 2020−03−15

150

−0.5
0.0
0.5
Changes in TOPIX relative to the cutoff (%)

100

0
−1.0

150
Probability of BoJ Purchase (%)

From 2014−10−31 to 2015−12−17150
Probability of BoJ Purchase (%)

Probability of BoJ Purchase (%)

150

100

50

0
−1.0

−0.5
0.0
0.5
Changes in TOPIX relative to the cutoff (%)

1.0

−1.0

−0.5
0.0
0.5
Changes in TOPIX relative to the cutoff (%)

1.0

From 2020−03−16 to 2020−12−31
Probability of BoJ Purchase (%)

150

100

50

0
−0.5
0.0
0.5
Changes in TOPIX relative to the cutoff (%)

1.0

Figure B.1: Discontinuity in the Probability of the BoJ Intervention for each Period
Notes: Figure B.1 shows the discontinuity in the probability of the BoJ intervention around the estimated cutoff
for each period. The blue scatter plot is the binned scatter plot with bin width 0.1%, and the red line indicates the
LOESS fit with shaded gray area being the 95% confidence interval.

33

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

Density

0.6

0.4

0.2

0.0
−2

−1
0
1
Changes in TOPIX relative to the cutoff (%)

2

Figure B.2: Density around the cutoff-.002

BoJ Purchases (% of market cap)
0
.002
.004
.006

.008

Notes: Figure B.2 shows the histogram and the density of changes in TOPIX relative to the cutoff. The shaded area
is 95% confidence interval. We use the local polynomial density estimator by Cattaneo et al. (2020) with order 2.

-4

-3

-2

-1

0
Days

1

2

3

4

Figure B.3: (Dis)continuity of ETF Purchases across Days
Notes: Figure B.3 shows the estimates of (17) across days. The shaded areas represent 90% confidence intervals,
which accounts for heteroskedasticity and autocorrelation.

34

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules".04% of Market Capitalization
.01
.02
.03
0

50

200

(B) Fraction of Market Capitalization

Billion Japanese Yen
100150(A) Absolute Amount

0

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020

Figure B.4: The Amount of the BoJ Purchases
Notes: Figure B.4 plots the amount of stock purchases by the BoJ in each intervention. Figure B.4A shows theabsolute amount of purchases in billion Japanese Yen (approximately 10 million US dollars). B.4B express it as afraction of market capitalization.

35

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

75
50
25
β lh
0
-25
-50
-75
-100

-100

-75

-50

-25

β lh
0

25

50

75

100

(B) Stock Price after YCC

100(A) Stock Price before YCC

.5

.6

.7

.8
.9
1
1.1
1.2
Ratio to the benchmark bandwidth

1.3

1.4

1.5

.5

.6

.8
.9
1
1.1
1.2
Ratio to the benchmark bandwidth

1.3

1.4

1.5

1.4

1.5

β lh
1.5
-.5

-.5

β lh
1.5

3.5

(D) JGB 10-year Yield after YCC

3.5(C) JGB 10-year Yield before YCC

.7

.5

.6

.7

.8
.9
1
1.1
1.2
Ratio to the benchmark bandwidth

1.3

1.4

1.5

.5

.6

.7

.8
.9
1
1.1
1.2
Ratio to the benchmark bandwidth

1.3

Figure B.5: Robustness to Bandwidth Selection
Notes: Figure B.5 shows the robustness of our estimates with respect to the size of the bandwidth. Each dot
represents the point estimates of the response from 11AM of the intervention day to 9AM on the next day. The
vertical line represents the 90% confidence interval, which accounts for heteroskedasticity and autocorrelation.
The dashed green line is the optimal bandwidth proposed by Calonico, Cattaneo, and Titiunik (2014), which
is our benchmark. Figures B.5A and B.5B show the response of stock price before and after YCC, respectively.
Figures B.5C and B.5D show the response of 10-year JGB yield before and after YCC, respectively.

36

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"C

Details on Theoretical Model

The Lagraingian of the consumption part of the household problem ishi∞
L =E0 ∑ βt u(Ct exp(vb ( Bt /Yt )) − λt Ct + Btc + Bti + St − Rt−1 ( Btc−1 + Bti −1 ) − Rst−1 St−1 + Tt ,t =0

where λt is the Lagrangian multiplier for the budget constraint. The first order condition withrespect to Ct and Btc areu0 (Ct exp(vb ( Bt /Yt )) exp(vb ( Bt /Yt )) = λt
1
u0 (Ct exp(vb ( Bt /Yt ))Ct exp(vb ( Bt /Yt ))v0b ( Bt /Yt ) − λt = − βRt λt+1 .
Yt
We can impose market clearing, Ct = Yt , and using the fact that b = Bt /Yt is constant along thebalanced growth, and setting vb (b) = 0,
u0 (Yt ) − u0 (Yt )v0b (b) = βREt u0 (Yt+1 ).

(18)

Suppose the central bank holds a constant fraction of market portfolio, sCB ≡ StCB /Pt K,
and issues the same amount of debt BtCB = −sCB Pt K. Then, the bond market clearing implies
b = sCB Pt K/Yt = sCB p where p = Pt /At . Substituting into 18, we obtainu0 (Yt+1 )0 CB.
1 − vb (s p) = βREt 0u (Yt )
Log-linearizing around sCB = 0, we obtaind ln R =−v00b (0) p CB
ds .
1 − v0b (0)When the central bank does not issue bonds, it is immediate to see d ln R = 0.
The investor part of the household solves
max Etst ,Wt+1[Wt+1 exp(−vs (st (Wt − Ct )/Yt − s̄))]1−γ
1−γ

Wt+1 = ( Rst st + Rt (1 − st )) (Wt − Ct ) − Tt+1 ,where st is the portfolio share on stocks. As central bank rebates back the losses, we have Tt+1 =−( Rst − R)sCB Pt K. From the market clearing, Wt − Ct = Pt K. Substituting these conditions, we

37

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

can rewrite as follows.

[Wt+1 exp(−vs (st p − s̄))]1−γ
max Et1−γ
st ,Wt+1
Wt+1 = Rst (st + sCB ) + Rt (1 − st − sCB ) Pt Kt .

(19)

Following Campbell and Viceira (2002), we assume investors solve an approximate version of
the above portfolio problem in which log portfolio return is normally distributed. Taking log
of the objective function,1p
p
max Et ln Rt + (1 − γ)Vart ln Rt − vs (st p − s̄),s2pwhere Rt ≡ Rst (st + sCB ) + Rt (1 − st − sCB ) is the portfolio return, wherep

Et ln Rt ≈ (1 − st − sCB ) ln R + (st + sCB )( g + ln

1
1+ p 1 2− σ ) + (st + sCB )(1 − st − sCB )σ2
p
2
2

p
Vart ln Rt ≈ (st + sCB )2 σ2 .

Therefore the problem reduces tomax(st + sCB )( g + ln
st

1+ p
γ
− ln R) − (st + sCB )2 σ2 − vs (st p − s̄)p2

The first order condition gives( g + ln

1+ p
− ln R) − γσ2 = pv0s (−sCB p),pwhere we used s = 1 − sCB and s̄ ≡ p. Log-linearizing the above equation around sCB = 0, we
obtain(1 + p )(1 + p) p2 v00s (0) CB
d
ln
R
+
ds
d ln p = −1 + (1 + p)v0s (0)1 + (1 + p)v0s (0)Now we characterize the investor’s portfolio problem when the central bank only buys
stocks so as to keep the interest rate fixed. The central bank purchases the constant share ofmarket capitalization, StCB = sCB Pt Kt , financed with Ft = StCB . The central bank rebates backthe return so that Tt+1 = Rst StCB . The investor’s problem is
max Et

St ,Bt ,Wt+1[Wt+1 exp(−vs (St /Yt − s̄))]1−γ
1−γ

Wt+1 = Rst St + Rt Bt + Rst StCB .

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ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

Because StCB + Bt + St = Pt K, we can rewrite the above problem as[W exp(−vs (st p − s̄))]1−γ
max Et t+1
1−γ
st ,Wt+1
hi
CB
Wt+1 = Rst (st + sCB
)
+
R
(
1
−
s
−
s
)
Pt K.
t
t
t
t
At this point, the problem is equivalent to (19). Therefore we have the exactly the same loglinearized equation:d ln p = −(1 + p )(1 + p) p2 v00s (0) CB
d
ln
R
+
ds ,
1 + (1 + p)v0s (0)1 + (1 + p)v0s (0)with d ln R = 0.

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ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

References
A DACHI , K., K. H IRAKI , AND T. K ITAMURA (2021): “Supplementary Paper Series for the "Assessment" (1): The Effects of the Bank of Japan’s ETF Purchases on Risk Premia in the Stock
Markets,” Tech. Rep. 21-E-3, Bank of Japan.
A NGRIST, J. D. AND G. W. I MBENS (1995): “Two-Stage Least Squares Estimation of Average
Causal Effects in Models with Variable Treatment Intensity,” Journal of the American Statistical
Association, 90, 431–442.
A NGRIST, J. D., Ò. J ORDÀ , AND G. M. K UERSTEINER (2018): “Semiparametric Estimates ofMonetary Policy Effects: String Theory Revisited,” Journal of Business & Economic Statistics,
36, 371–387.
B ARBON , A. AND V. G IANINAZZI (2019): “Quantitative Easing and Equity Prices: Evidence
from the ETF Program of the Bank of Japan,” The Review of Asset Pricing Studies, 9, 210–255.
B ERNANKE , B. S. AND K. N. K UTTNER (2005): “What Explains the Stock Market’s Reaction toFederal Reserve Policy?” The Journal of Finance, 60, 1221–1257.
C ALONICO , S., M. D. C ATTANEO , AND R. T ITIUNIK (2014): “Robust Nonparametric Confidence Intervals for Regression-Discontinuity Designs,” Econometrica : journal of the Econometric Society, 82, 2295–2326.
C AMPBELL , J. Y. AND L. M. V ICEIRA (2002): Strategic Asset Allocation: Portfolio Choice for LongTerm Investors, Clarendon Lectures in Economics, Oxford: Oxford University Press.
C ATTANEO , M. D., M. J ANSSON , AND X. M A (2020): “Simple Local Polynomial Density Estimators,” Journal of the American Statistical Association, 115, 1449–1455.
C HAROENWONG , B., R. M ORCK , AND Y. W IWATTANAKANTANG (2021): “Bank of japan equity
purchases: The (non-) effects of extreme quantitative easing,” Review of Finance, 25, 713–743.
C HODOROW-R EICH , G. (2014): “Effects of Unconventional Monetary Policy on Financial Institutions,” .
C HUNG , J.-C. (2020): “The Bank of Japan’s Exchange-Traded Fund Purchase Program and Its
Impact on Daily Stock Market Returns,” 53.
C OCHRANE , J. H. AND M. P IAZZESI (2002): “The Fed and Interest Rates - A High-Frequency
Identification,” American Economic Review, 92, 90–95.
D A , Z., B. L ARRAIN , C. S IALM , AND J. T ESSADA (2018): “Destabilizing Financial Advice:
Evidence from Pension Fund Reallocations,” The Review of Financial Studies, 31, 3720–3755.
40

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

F UKUDA , S.- I . AND M. TANAKA (2022): “The Effects of Large-scale Equity Purchases during
the Coronavirus Pandemic,” 30.
G ABAIX , X. AND R. S. J. K OIJEN (2021): “In Search of the Origins of Financial Fluctuations: The
Inelastic Markets Hypothesis,” SSRN Scholarly Paper ID 3686935, Social Science Research
Network, Rochester, NY.
G ORODNICHENKO , Y. AND W. R AY (2017): “The effects of quantitative easing: Taking a cue
from treasury auctions,” Tech. rep., National Bureau of Economic Research.
H AHN , J., P. T ODD , AND W. VAN DER K LAAUW (2001): “Identification and Estimation of Treatment Effects with a Regression-Discontinuity Design,” Econometrica : journal of the Econometric Society, 69, 201–209.
H ARADA , K. AND T. O KIMOTO (2021): “The BOJ’s ETF Purchases and Its Effects on Nikkei 225
Stocks,” International Review of Financial Analysis, 77, 101826.
H ARTZMARK , S. M. AND D. H. S OLOMON (2021): “Predictable Price Pressure,” SSRN Scholarly Paper 3853096, Social Science Research Network, Rochester, NY.
I MBENS , G. W. AND T. L EMIEUX (2008): “Regression Discontinuity Designs: A Guide to Practice,” Journal of Econometrics, 142, 615–635.
K ATAGIRI , M., K. TAKAHASHI , AND J. S HINO (2022): “Bank of Japan’s ETF purchase programand equity risk premium: a CAPM interpretation,” .
K RISHNAMURTHY, A. AND A. V ISSING -J ORGENSEN (2011): “The Effects of Quantitative Easingon Interest Rates: Channels and Implications for Policy,” .
——— (2012): “The Aggregate Demand for Treasury Debt,” Journal of Political Economy, 120,
233–267.
K UBOTA , H. AND M. S HINTANI (2022a): “High-Frequency Identification of Monetary Policy
Shocks in Japan,” The Japanese Economic Review.
——— (2022b): “Macroeconomic Effects of Monetary Policy in Japan an Analysis Using Interest
Rate Futures Surprises,” SSRN Scholarly Paper 4015068, Social Science Research Network,
Rochester, NY.
K UERSTEINER , G. M., D. C. P HILLIPS , AND M. V ILLAMIZAR -V ILLEGAS (2018): “Effective Sterilized Foreign Exchange Intervention? Evidence from a Rule-Based Policy,” Journal of International Economics, 113, 118–138.

41

ESRI Discussion Paper Series No.376"The Impact of Central Bank Stock Purchases:
Evidence from Discontinuities in Policy Rules"

L I , J. J., N. D. P EARSON , AND Q. Z HANG (2021): “Impact of Demand Shocks on the Stock Market: Evidence from Chinese IPOs,” SSRN Scholarly Paper 3731219, Social Science Research
Network, Rochester, NY.
M C C RARY, J. (2008): “Manipulation of the Running Variable in the Regression Discontinuity
Design: A Density Test,” Journal of Econometrics, 142, 698–714.
N AKAMURA , E. AND J. S TEINSSON (2018): “High-Frequency Identification of Monetary NonNeutrality: The Information Effect*,” The Quarterly Journal of Economics, 133, 1283–1330.
P ORTER , J. (2003): “Estimation in the Regression Discontinuity Model,” Unpublished
Manuscript, Department of Economics, University of Wisconsin at Madison, 2003, 5–19.
P ORTER , J. AND P. Y U (2015): “Regression Discontinuity Designs with Unknown Discontinuity
Points: Testing and Estimation,” Journal of Econometrics, 189, 132–147.
R AMBACHAN , A. AND N. S HEPHARD (2021): “When Do Common Time Series Estimands Have
Nonparametric Causal Meaning?” .
R OMER , C. D. AND D. H. R OMER (2004): “A New Measure of Monetary Shocks: Derivation
and Implications,” American Economic Review, 94, 1055–1084.
S HIROTA , T. (2018): “Evaluating the Unconventional Monetary Policy in Stock Markets : A
Semi-parametric Approach,” 24.
VAYANOS , D. AND J.-L. V ILA (2021): “A Preferred-Habitat Model of the Term Structure ofInterest Rates,” 100.
WALLACE , N. (1981): “A Modigliani-Miller Theorem for Open-Market Operations,” The American Economic Review, 71, 267–274.

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