KIER DISCUSSION PAPER SERIESKYOTO INSTITUTEOFECONOMIC RESEARCH
Discussion Paper No.1094“Time-varying ambiguity shocks and business cycles”
Takao Asano, Xiaojing Cai, and Ryuta Sakemoto

August 2023

KYOTO UNIVERSITY
KYOTO, JAPANTime-varying ambiguity shocks and business cycles∗
Takao Asano†, Xiaojing Cai‡, and Ryuta Sakemoto§August 16, 2023

Abstract
This study investigates how ambiguity has driven output and inflation in the
U.S. over the past 70 years. We adopt the recently developed techniques that
disentangle ambiguity from risk and assess the responses of output and inflation to ambiguity shocks. We observe that an increase in ambiguity led to
an increase in output during high inflation periods, indicating the ambiguity
lover behavior. We also uncover that ambiguity and risk estimated by realized
volatility have the opposite impacts on business cycles, which is consistent withthe prevailing asset pricing literature.
Keywords: Ambiguity, Risk premiums, Uncertainty, TVP-VAR
JEL Codes: E32, E44∗

This research was financially supported by International Joint Research Center of Advanced Economic Research of KIER and the Grants-in-Aid for Scientific Research, JSPS (22K13430, 20K01745).
†
Faculty of Economics, Okayama University, Tsushimanaka 3-1-1, Kita-ku, Okayama 700-8530,
Japan. E-mail: takaoasano@okayama-u.ac.jp‡
Faculty of Economics, Okayama University, Tsushimanaka 3-1-1, Kita-ku, Okayama 700-8530,
Japan. E-mail: caicai@okayama-u.ac.jp§
Corresponding author: Faculty of Economics, Okayama University, Tsushimanaka 3-1-1, Kitaku, Okayama 700-8530, Japan. Keio Economic Observatory, Keio University, Minato-ku, Tokyo,
Japan. E-mail: rsakemoto@okayama-u.ac.jp

1

Introduction

It is diﬃcult to predict what will happen in the future. For example, the randomnessof asset returns is one of the most important elements in portfolio selection problems
and asset pricing, as it makes it troublesome for investors to determine how much andwhich stocks should be invested. It is also hard to pin down the probability distributions of asset returns. When a decision maker does not have a unique probability
distribution of random variables, we say that the decision maker perceives ambiguity
in random variables. The importance of the distinction between risk and ambiguity
has long been recognized in economics, particularly in decision theory, since Knight(1921).1 This paper sheds light on the role of ambiguity and empirically estimatesthe eﬀects of such ambiguity on output and inflation in the U.S. post-World War II
era. Although “uncertainty” is often referred to as risk or ambiguity, we employ thenotion of uncertainty in a broader sense; that is, uncertainty includes the notions ofrisk and ambiguity and entails all situations except for deterministic cases.
Axiomatizations of rational behaviors under ambiguity have been proposed toovercome Ellsberg’s paradox (1961), which experimentally shows that individuals’
behaviors cannot be explained by the standard expected utility theory. Gilboa andSchmeidler (1989) propose the Maxmin expected utility theory (MEU) and Schmeidler (1989) introduces the Choquet expected utility theory (CEU) to provide rigorous
behavioral foundations of decision makers under ambiguity. Klibanoﬀ et al. (2005)
present the smooth ambiguity model that can diﬀerentiate decision makers’ attitudes
toward ambiguity from their perception of ambiguity.2 The applications of MEU,
CEU, and the smooth ambiguity model to portfolio choice and asset pricing highlight the importance of ambiguity (Dow and Werlang, 1992; Epstein and Wang, 1994;
Cao et al., 2005; Gollier, 2011). However, empirical studies have only recently begunin this area owing to the diﬃculty of estimating ambiguity.
Recently, a series of works based on the theoretical model by Izhakian (2020)
1

In decision theory, risk is the situation in which individuals have a unique probability. However,in this paper, risk indicates the volatility of random variables.
2
For a textbook presentation, see, for example, Gilboa (2009), and also Nishimura and Ozaki(2017).

1disentangle ambiguity from risk and evaluate how these two elements are linked toasset prices (Izhakian and Yermack, 2017; Brenner and Izhakian, 2018; Augustin and
Izhakian, 2020). These studies present empirical evidence that risk and ambiguity
have opposite impacts on early exercises of employee options, stock market expected
returns, and prices of credit default swaps. However, in contrast to these asset pricing
contexts, how ambiguity drives macroeconomic output and inflation over a longer
period remains an open question. The previous literature uncovers that risk plays animportant role in business cycles (Bloom et al. 2009; Basu and Bundick, 2017). Our
first contribution is that we investigate whether ambiguity has diﬀerent impacts on
output and inflation in comparison with risk.
This paper adopts the approach proposed by Brenner and Izhakian (2018) andextract ambiguity from the market risk premium.3 Estimating ambiguity using otherrisk premiums including size, value, and momentum, is our second contribution.4
These risk premiums reflect investors’ expectations and sentiment since many investors invest in money based on these investment styles (Barberis and Shleifer, 2003;
Stambaugh and Yuan, 2016). The previous literature demonstrates that these risk
premiums contain future information about economic growth (Liew and Vassalou,
2000) and function as proxies of investment opportunities (Petkova, 2006). These results support the view of Fama and French (1995) and Petkova and Zhang (2005) who
highlight that three- and four-factor models are interpreted by a risk-based story. We
extend this literature by estimating ambiguity from these risk premiums and explore
whether it contains diﬀerent information from the market risk premium ambiguity.
Our third contribution is that we deploy a time-varying parameter vector autoregressive model with stochastic volatility (TVP-VAR-SV) to evaluate the timevarying relationships between ambiguity shocks and macroeconomic variables (e.g.,
Primiceri, 2005; Galı́ and Gambetti, 2009; Baumeister and Peersman, 2013). The
estimation period in this study exceeds 70 years, and it is reasonable to consider
that the underlying economic structure has changed. For instance, Del Negro et al.
3

We refer to this ambiguity as market risk premium ambiguity when we need to distinguish itfrom ambiguity obtained from other risk premiums. See Table A2.
4
Fama and French (1993) and Carhart (1997) reveal that including these factors in the Capital
Asset Pricing Model (CAPM) reduces the pricing errors of cross-sectional asset pricing models.

2

(2019) suggest that demographic trends lower real interest rates and Clarida et al.
(2000) and Bernanke (2020) highlight that monetary policies play an important rolein stabilizing the economy. New information technologies lead to improvements ininventory management (Kahn et al., 2002) and financial innovations that allow firms
and households to mitigate their financial constraints (Dynan et al., 2006). To reflect
these structural changes, the TVP-VAR-SV model is employed.
Our main findings are fourfold. First, the responses of output and inflation to
ambiguity shocks vary over time. An increase in ambiguity led to an increase in output during the high inflation period in the 1970s and the 1980s, and this is consistent
with the ambiguity lover behavior in Brenner and Izhakian (2018). This suggests that
economic agents prefer ambiguity with an increase in the probability of unfavorable
outcomes, such as high inflation.
Second, the responses of inflation to ambiguity shocks were negative in the 1950s.
The inflation rate in the 1950s was relatively high and an increase in ambiguity wasa positive signal for economic agents, and hence the high value of ambiguity led to
low inflation. However, the response of inflation was unclear in the 1970s and the1980s, indicating that positive ambiguity shocks were not suﬃcient in reducing thehigh inflation during this period.
Third, ambiguity estimated from the size, value, and momentum risk premiums
is weakly linked to output and inflation, suggesting that these risk premiums contain
information about future economic growth, but do not capture information about
the entire market.
Finally, our estimated ambiguity diﬀers from stock market volatility. Our realized
and mixed data sampling (MIDAS) volatilities show that positive volatility shocks
reduced output during the high inflation period in the 1970s and the 1980s.5 Moreover,
we deploy the uncertainty measure proposed by Jurado et al. (2015) and find that
positive uncertainty shocks reduce output and raise inflation during the high inflation
period. These indicate that ambiguity and risk have the opposite impacts on businesscycles.
The rest of this paper is organized as follows: Section 2 reviews the related liter5

The MIDAS model is proposed by Ghysels et al. (2005).

3

ature, Section 3 describes our theoretical background and methodologies, Section 4
explains the dataset, Section 5 presents our main empirical results, Section 6 conducts
further analysis, and Section 7 concludes.

2

Literature Review

Since the pioneering works by Gilboa and Schmeidler (1989) and Schmeidler (1989)
that provide an answer to the Ellsberg paradox, we have recognized the role of ambiguity through the applications of CEU and MEU. Dow and Werlang (1992) explain
the existence of portfolio inertia (or no trade). Portfolio inertia means price ranges
over which investors neither buy nor sell stocks. Arrow (1965) demonstrates that
portfolio inertia cannot be explained under the standard expected utility theory. Under a representative agent model, Epstein and Wang (1994) show that equilibrium
prices are characterized by Euler inequalities, in stark contrast to Lucas (1978) who
presents that the equilibrium price is uniquely determined by Euler equality. Thisimplies that ambiguity leads to the indeterminacy of equilibria.
From the empirical viewpoint, previous studies have clarified the importance of
ambiguity. Epstein and Schneider (2008) investigate the eﬀects of bad and good news
on investors’ behaviors, and provide the empirical result that investors overvalue negative information and undervalue positive information under ambiguity. Analyzing
the U.S. stock market and accounting data, Kelsey et al. (2010) find that negative momentum is greater than positive momentum in terms of the magnitude andpersistence of portfolio returns, which indicates that ambiguity plays a part in suchasymmetric patterns. Driouchi et al. (2018) analyze the behavior of U.S. index put
option holders during the pre-crisis and credit crunch period in 2006–2008 and findevidence of ambiguity in the U.S. index options market during the credit crunch
period. Ilut and Schneider (2014) consider shocks to confidence about future totalfactor productivity (TFP) as changes in ambiguity, and empirically show that TFP
and confidence shocks play a role in explaining business cycle fluctuations.6
Risk has been intensively investigated by the literature and plays a key role in6

See Bianchi et al. (2018) for further analysis

4economic growth. It triggers a sudden drop in aggregate outputs and then induces
a strong recovery in economic activities, which impacts firms’ decisions about newinvestments. The recent economic literature proposes measures of risk and assesses
how they influence business cycles. For instance, realized and implied volatilities on
the stock market are widely employed by previous studies (e.g., Bloom et al. 2009;
Byrne et al., 2013; Basu and Bundick, 2017). Several studies decompose volatility
into good and bad volatilities. Patton and Sheppard (2015) adopt high frequency
data of stock prices and uncover that negative volatility is more strongly associatedwith future volatility. Segal et al. (2015) estimate good and bad volatilities frommacroeconomic data and find that they have opposite impacts on economic growth.
Berger et al. (2020) distinguish realized risk from forward looking risk, and find thatthe former drives economic activities. Several other studies use a large number ofmacroeconomic variables and deploy factor models in estimating risks (Jurado et al.,2015; Carriero et al., 2018).
TVP-VAR-SV models are employed to investigate whether response of shocks
is time-varying, which is crucial in exploring economic relationships over a longer
horizon. These include the response of interest rates to inflation shocks (Primiceri,2005), the response of output to nontechnology shocks (Galı́ and Gambetti, 2009),
the response of GDP to real price of crude oil shocks (Baumeister and Peersman,2013), the response of inflation in the U.K. to monetary policy shocks (Ellis et al.,2014), the response of GDP to financial condition shocks (Abbate, et al., 2016), andthe response of food production to food supply shocks (Peersman et al., 2021).

3

Methodologies

3.1

Measuring ambiguity

In this paper, we use the measure of ambiguity, ℧2 , proposed by Izhakian (2020), as
explained below. This measure allows us to separate risk, attitudes toward risk, andattitudes toward ambiguity, and to obtain the degree of ambiguity. This provides theempirical value of ambiguity, which has been widely adopted in the literature; see
Izhakian and Yermack (2017), Brenner and Izhakian (2018), Augustin and Izhakian
5

(2020), and Izhakian et al. (2022).7
Let S be a state space, let E be a σ-algebra on S, and let P : E → [0, 1] be aprobability measure. Let (S, E, P ) be a probability space, let r : S → X be a randomvariable (measurable function) on S, where X ⊆ R denotes an interval of outcomes(consequences) in R that contains the interval [0, 1]. Let P be a set of probability
measures on (S, E) and let Pr (x) = P ({s ∈ S|r(s) ≤ x}) denote the cumulative
probability (cumulative distribution function). Izhakina (2020) defines the expectedcumulative probability of x ∈ X over P by∫
E[Pr (x)] =Pr (x)dξ,P

and the variance of the cumulative probability of x ∈ X over P by8∫
Var[Pr (x)] = (Pr (x) − E[Pr (x)])2 dξ.
P

The measure of ambiguity is defined as∫2℧ [r] ≡
E[Pr (x)]Var[Pr (x)]dx,(1)

X

which represents the weighted average of the variances of the probabilities. In thefollowing analysis, random variable r is considered to be the rates of returns of stock
prices and X = R.
We divide the range of daily returns, from −6% to +6%, into ten intervals witheach width 1%, and add two intervals, (−∞, −6%] and (+6%, ∞), which yields
12 intervals. We discretize the return distributions into 12 bins Bi = (ri−1 , ri ]of equal sizes for i = 0, 1, . . . , 11, that is, B0 = (r−1 , r0 ] = (−∞, −6%], B1 =(r0 , r1 ] = (−6%, −5%], . . . , B11 = (r10 , r11 ] = (6%, ∞). We compute the mean
and the variance of the probabilities for each interval, P (Bi ) for i = 0, 1, . . . , 11.
Because some bins may not have return realizations, it is not easy to compute
Fu et al. (2023) point out that there are some caveats to adopting ℧2 as a measure of ambiguity, but Kostopoulos et al. (2022) also present empirical evidence that an alternative measure ofambiguity obtained from the expected volatility in the implied volatility and ℧2 generate similar
patterns. See Online Appendix for details.
8∫ Similarly, Izhakian (2020) defines the expected probability density over P by E[φ∫ r (x)] =
φ
(x)dξ
and
the
varianceoftheprobability
density
over
P
by
Var[φ
(x)]
=
(φr (x) −r
r
PP
E[φr (x)])2 dξ, where φr (x) = P ({s ∈ S|r(s) = x}) is the probability density.
7

6

the probabilities.

Therefore, similar to Augustin and Izakian (2020) and Bren-ner and Izhakian (2018), we alternatively assume a normal distribution, that is,
P [Bi ] = [Φ(ri ; µ, σ) − Φ(ri−1 ; µ, σ)], where Φ(·) denotes the cumulative normal distribution function. Note that P (B0 ) = Φ(r0 ; µ, σ) − Φ(r−1 ; µ, σ) = Φ(r0 ; µ, σ) andP (B11 ) = Φ(r11 ; µ, σ)−Φ(r10 ; µ, σ) = 1−Φ(r10 ; µ, σ). Note that µ and σ are estimatedat a monthly frequency using daily data.
We estimate the degree of ambiguity of each month based on the following discrete
form of Equation (1):℧2 [r]= E[Φ(r0 ; µ, σ)]Var[Φ(r0 ; µ, σ)]10∑+E[Φ(ri ; µ, σ) − Φ(ri−1 ; µ, σ)]Var[Φ(ri ; µ, σ) − Φ(ri−1 ; µ, σ)]i=1+ E[1 − Φ(r10 ; µ, σ)]Var[1 − Φ(r10 ; µ, σ)].

(2)

Note that Φ(r−1 ; µ, σ) = 0 and Φ(r11 ; µ, σ) = 1. The right-hand side of Equation (2)does not include a scaling factor in Brenner and Izhakian (2018), since our interval
size is larger than that of Brenner and Izhakian (2018). We confirm that this change
does not impact our main results since the degree of ambiguity obtained by Brenner
and Izhakian (2018) and our method in Equation (2) exhibit similar fluctuations, asreported in Figure A11.

3.2

Time-varying parameter vector autoregressive (TVP-VAR)
model

The TVP-VAR model is an extension of the standard VAR model and allows us toobtain time-varying parameters. In this study, we employ the TVP-VAR to capture
the time-varying relationships between ambiguity and business cycles. This model
includes time variation of the coeﬃcients and the multi-variate SV (Primiceri, 2005).
Ellis et al. (2014) and Peersman et al. (2021) highlight that the SV captures nonlinearities in the simultaneous relations between the variables in the system. The
previous literature presents empirical evidence that the eﬀects of inflation, the realoil price, and the food price vary over time (Primiceri, 2005; Baumeister and Peers7

man, 2013; Peersman et al., 2021). We focus on a period of more than 70 years andassume that the economic structure has changed due to new information technologies,
financial innovations, and low interest rates (Kahn et al. 2002; Dynan et al., 2006;
Del Negro et al., 2019). Taken together, we consider that the eﬀects of ambiguity
vary over time.
We follow Primiceri (2005) and describe a basic TVP-VAR model as follows:
A t Yt =k∑Ψi,t Yt−i + ut , i = 1, ..., k, t = k + 1, ..., T

(3)

i=1

ut ∼ N (0, Σt ),where Yt is a n × 1 vector of variables. In this study, we employ output (∆y),
inflation (∆p), and ambiguity (℧2 ); that is, n = 3. The TVP-VAR model has many
parameters, and we restrict the number of variables (e.g., Peersman et al., 2021). At
is the n×n simultaneous relations; Ψi,t is the n×n matrix of coeﬃcients; ut represents
a n × 1 vector of error terms and follows the N (0, Σt ) distribution; Σt is the n × nvariance-covariance matrix of error terms. The number of lags in the VAR is denoted
as k and we adopt k = 2 based on the Bayesian information criterion (BIC).
We then multiply both sides of Equation (3) by A−1
to obtain a reduced formt
representation of the TVP-VAR model:
Yt =

k
∑

Bi,t Yt−i + A−1
t Vt ϵt(4)

i=1

ϵt ∼ N (0, In ),where the coeﬃcient matrix Bi,t = A−1t Ψit ; ϵt is the n × 1 residual vector and follows
the N (0, In ) distribution. The variance-covariance matrix Σt in Equation (3) is de′′composed as Σt = At−1 Vt Vt (A−1t ) where Vt is the n × n diagonal matrix of standard
deviations and written as:


σ1,t


 0
Vt =  .
 ..
0

0
σ2,t
..
.
···
8


0
.. 
···
. 

..
. 0 
0 σn,t···The simultaneous relations At is assumed to be a lower triangular matrix as follows:


1
0
···
0
 a21,t 1
···
0


At =  ..

.
.
.
.
 .
.
.
0
an1,t · · · ann−1,t 1
The coeﬃcient matrix Bt , the simultaneous relations At and the standard deviation
matrix Vt are all time-varying matrices. According to Primiceri (2005), these are all
assumed to follow a random walk process as follows:
Bt = Bt−1 + ξtB
at = at−1 + ξta
ht = ht−1 + ξth

 

ϵt
In 0 0 0
ξtB   a  ∼ N 0,  0 Q 0 0  ,
 ξt 
  0 0 S 0 hξt
0 0 0 W

where at is the element of the lower-triangular matrix At , and ht = log σt2 where σtis the diagonal element of Vt .
Following Primiceri (2005), Koop and Korobilis (2010), and Ellis et al. (2014),
we employ Bayesian methods for estimation. We implement the Gibbs sampler that
evaluates the posterior distributions of the parameters. This entire algorithm is
executed 30,000 times, with the first 10,000 draws discarded as a burn-in.
Our identification scheme follows Bernanke et al. (2005) and assumes that a slowmoving variable does not have a contemporaneous impact on a fast-moving variable.
Our variable order of the base model is: output (∆y), inflation (∆p), and ambiguity(℧2 ). A change in output is relatively slow while that in ambiguity is fast, since it
is based on stock market information. Bekaert et al. (2013) employ the uncertainty
variable obtained from the stock market and adopt the same identification scheme. In
the robustness section, we present that our results do not depend on the identification
scheme.

9

4

Data

We adopt a market risk premium that is calculated as the excess return on the valueweighted return of Center for Research in Security Prices (CRSP) firms (e.g., Fama
and French, 1993). We employ the market risk premium at a daily frequency andestimate ambiguity at a monthly frequency. Although Brenner and Izhakian (2018)
and Augustin and Izhakian (2020) use intraday data, this paper deploys daily data,
which enables us to investigate the eﬀects of ambiguity on output and inflation in
the U.S. over a long period of time.
We also use size, value, and momentum premiums which are calculated as spread
returns at a daily frequency (Fama and French, 1993; Carhart, 1997). We construct
six value-weighted portfolios based on firm size and book-to-market ratio. The size
premium is calculated as the average spread return between three small and threelarge groups. The value premium is obtained as the average spread return between
two value and two growth groups. We also deploy the momentum premium that is
calculated as the average spread return between two high prior and two low prior return groups. The prior return is measured from the last 12 to two months but we skip
the most recent month to avoid the reversal eﬀect (Carhart, 1997). These data aredownloaded from Kenneth French’s website. Liew and Vassalou (2000) demonstratethat the size and value premiums contain information to predict future economic
growth. Petkova (2006) adopts financial variables and shows that these premiums
are linked to investment opportunity sets. The momentum premium is generated byheterogeneous exposures to past winner and past loser firms (Johnson, 2002; Liu andZhang, 2008). These results suggest that the size, value, and momentum premiums
diﬀer from the market risk premium, and hence the ambiguity obtained from theserisk premiums also has diﬀerent impacts on business cycles.
We employ the industrial production index for the U.S. to capture output at amonthly frequency (Bekaert et al., 2013; Jurado et al., 2015). We measure inflation
by adopting the consumer price index (CPI) for all items (Bekaert et al., 2013; Jurado
et al., 2015). Following Bekaert et al., (2013), we employ the log diﬀerence of thesevariables and evaluate growth rates. We denote the growth rate of output ∆y, and10

that of inflation ∆p. The data are obtained from the Federal Reserve Bank of St.
Louis. Our data period extends from January 1947 to December 2021, and thestarting point depends on the availability of the monthly CPI.9

5

Empirical results

5.1

Estimated ambiguity

We begin our discussion with the fluctuations of ambiguity and Figure 1 depicts
time series changes in ambiguity. The upper panel indicates ambiguity (left y-axis)
and National Bureau of Economic Research (NBER) recessions, and the lower panel
demonstrates ambiguity (left y-axis) and the market risk premium (right y-axis) thatis calculated using the excess return of all CRSP firms. We employ the market risk
premium at daily frequency and obtain monthly ambiguity. We observe a relatively
high average value of ambiguity in the 1950s and the 1960s. The upper panel illustrates that the average value of ambiguity tends to be low during recessions. Brenner
and Izhakian (2018) find that the average value of ambiguity becomes high whenmarket volatility is relatively low. The lower panel focuses on longer-term results
than those in Brenner and Izhakian (2018), but we can observe the same pattern.

5.2

Time-varying impulse responses

Next, we move onto the results of the TVP-VAR. Figure 2 demonstrates the timevarying impulse responses of output to ambiguity shocks. The results include periods
from one- to four-months. We present the median responses with 68% bootstrap confidence bands, which is standard in the literature (e.g., Primiceri, 2005; Baumeister
and Peersman, 2013). First, we focus on the two-month result in the upper rightpanel of Figure 2, since the one-month result does not show a clear pattern, indicating that the ambiguity shocks have lagged eﬀects. We note that one standard
deviation of the ambiguity shocks raises output, whereas there is clear time variation.
The confidence intervals of the responses are above zero between the middle of the
1970s and the beginning of the 1990s, suggesting that high ambiguity is positive for9

See Tables A1 and A2 for details.

11output. This period includes high inflation triggered by oil price shocks (e.g., Hamilton, 1983; Edelstein and Kilian, 2009). An increase in ambiguity suggests that highinflation does not last, which is positive for aggregate output. The median responseof output achieved the highest value (0.43%) around 1979 when the U.S. experienced
the highest inflation in our sample period.10
Our results are consistent with the ambiguity lover behavior. Brenner and Izhakian(2018) report that when the probability of unfavorable outcomes increases, agents prefer ambiguity in the stock market. In our situation, unfavorable outcomes correspondto the high inflation period. In contrast, Ilut and Schneider (2014) adopt confidence
about TFP and reveal that confidence shocks lower output, which is opposite to our
results. They deploy a diﬀerent approach and consider that an increase in ambiguity
leads to more cautious behavior of economic agents, and this accounts for diﬀerence
in results. Note that the positive responses of output to ambiguity shocks diﬀer fromthe responses of that to uncertainty shocks. Basu and Bundick (2017) and Berger etal. (2020) present that stock market return volatility has negative impacts on output. Combining these results, our findings suggest that ambiguity contains diﬀerent
information from risk estimated by stock market return volatility.
We also observe the time-varying response of inflation to ambiguity shocks in
Figure 3. The upper left panel shows that the responses at the one-month were
negative from the 1940s to the middle of the 1960s. The median response in 1950
was the lowest (−0.08%), but it reached around zero in 1966. The inflation ratein the 1950s was relatively high and an increase in ambiguity was a positive signal
for economic agents, and hence the high value of ambiguity led to low inflation.
The response of inflation was unclear in the 1970s and the 1980s, which suggeststhat positive ambiguity shocks were not suﬃcient in reducing high inflation and thattight monetary policy was needed, as reported by the literature (Clarida et al., 2000;
Mavroeidis, 2010).

10

In addition, the lower panels in Figure 2 illustrate that the positive ambiguity eﬀects persist at
the three- and four-months. The results at the eight- and 12-months are reported in Figure A4 andthe impacts became marginal.

12

5.3

Ambiguity of other risk premiums

Having found the significant ambiguity impacts, this section estimates ambiguity fromother risk premiums. We focus on size, value, and momentum premiums proposed by
Fama and French (1993) and Carhart (1997), since the risk premiums are available atdaily frequency. Liew and Vassalou (2000) uncover that the size and value premiums
contain information about future economic growth, and Vassalou (2003) reports thatthey are linked to future news about economic growth. Petkova (2006) and Bali andEngle (2010) address that they are associated with investment opportunity sets andare considered risk premiums. Moreover, the literature reports the strong relationshipbetween the momentum risk premium and business cycles since past winner firms havehigh exposure to business cycles compared with past loser firms (e.g., Johnson, 2002;
Liu and Zhang, 2008).
Figure 4 demonstrates the responses of output and inflation to the size ambiguity
shocks. The upper two panels in Figure 4 reveal that the positive size ambiguity
shocks lead to an increase in output, except for the 1950s, and that these impacts
increase gradually. The lower two panels display that the positive size ambiguity
shocks raise inflation from the middle of the 1980s. The literature reports that thesize risk premiums decayed since many investors employed the size factor.11 Figure A1
also illustrates that the average size ambiguity became lower after the 1980s. These
are related to the results that output and inflation responses are more sensitive tochanges in size ambiguity. However, the median values of responses are much smaller
than those to the market risk premium ambiguity shocks in Figures 2 and 3.
Figures A7 and A8 present the responses of output to the value and momentum
ambiguity shocks. We can see that the responses diﬀer from the results in Figures 2
and 3 and that the median values are small, which suggests that ambiguity of the size,value and momentum risk premiums contains diﬀerent information from ambiguityof the market risk premium. The size and value premiums contain information about
future economic growth, but do not capture information about the entire market.
For instance, not all investors focus on the size and value factors and some investors
11

See Asness et al. (2018) and references therein.

13

employ only a specific investment style (Cronqvist et al., 2015). Therefore, ambiguityof these premiums has relatively smaller impacts on output and inflation comparedwith the ambiguity of the market risk premium.

6

Further analysis

6.1

Diﬀerent identification scheme

The empirical results of impulse responses depend on the chosen identification scheme.
In our main results, we adopted a recursive identification scheme and assumed thatslow-moving variables did not have any contemporaneous impacts on fast-moving
variables (e.g., Bernanke et al., 2005). This section investigates whether our main
results are sensitive to variable order. Identification II uses the following order:
ambiguity, inflation, and output, and Identification III employs the following order:
output, ambiguity, and inflation. Figures A5 and A6 present empirical evidence thatthe time-varying impulse responses of output and inflation are robust against thechange in variable order.

6.2

Responses to volatility shocks

This section investigates how the eﬀects of ambiguity shocks diﬀer from those ofvolatility shocks. Izhakian and Yermack (2017) and Brenner and Izhakian (2018)present empirical evidence that volatility and ambiguity have the opposite impacts
on equity risk premiums and option values. We employ the following three measuresof volatility and replace ambiguity with one of them. First, we estimate monthly
realized volatility (V OLRV ) using daily excess returns as in Bloom (2009) and Byrne
et al. (2013). Second, we deploy a MIDAS model and estimate monthly volatility(V OLM IDAS ) using daily realized volatility. Ghysels et al. (2005) address that thesize of rolling window size is crucial for volatility forecast and the MIDAS model
provides eﬀects of the optimal window size. Finally, we adopt total macroeconomic
uncertainty (UNC) proposed by Jurado et al. (2015). This measure is obtained as theforecast error variance of macro economic indicators and the forecast model includes
latent common factors and stochastic volatility.
14

Figure 5 shows the time-varying responses of output and inflation to the volatility
shocks. The upper two panels show that the volatility shock lowers output. Thisis consistent with the results reported by Basu and Bundick (2017) who employ
implied volatility and estimate a constant VAR model. We uncover that the volatility
eﬀects were greater during the high inflation period in the 1970s and 1980s. Thesefindings are opposite to the eﬀects of the ambiguity shocks reported by the previous
section, and linked to the findings reported by Brenner and Izhakian (2018) and
Izhakian and Yermack (2018) who focus on asset prices. Figure A10 displays that
we observe a similar pattern using volatility estimated by the MIDAS model. Figure
A9 demonstrates that the total macroeconomic uncertainty shock reduced output for
the entire period and raised inflation at the end of the 1970s.
In summary, we observe that the eﬀects of the ambiguity shocks are not capturedby the volatility shocks.

6.3

Excluding eﬀects of stochastic volatility

In this section, we explore the responses of output and inflation to ambiguity shocks
using a constant VAR model. The TVP-VAR model includes many parameters andstochastic volatility. This complexity of the model may aﬀect our estimation results.
In particular, stochastic volatility is strongly linked to risk in modern finance theory(e.g., Harvey et al., 1994) and generates dynamic interactions across macroeconomic
variables (Mumtaz and Zanetti, 2013). To deal with these concerns, we follow Giannone et al. (2015) and employ the constant Bayesian VAR (BVAR) model. We
split the data into two periods: Period I covers February 1947 to December 1989,
and Period II extends from January 1990 to December 2021.
Figure 6 presents the impulse response results obtained by the BVAR. The upperand lower left panels demonstrate that the response of output was positive and thatof inflation was negative in Period I. The output result is consistent with the resultsin Figure 2, indicating that the positive ambiguity shocks led to an increase in output
in the 1970s and 1980s. The negative inflation result is linked to the results in Figure
3, which illustrates that the positive ambiguity shocks lowered inflation in the 1950s.
The right two panels in Figure 6 also show the consistent results in Figures 215and 3. The ambiguity shocks have less impact on output and inflation after 1990.
The possible reasons for these weak ambiguity eﬀects are that forward guidance of
the Federal Reserve Board (FRB) plays an important role (e.g., Bernanke, 2020).
For instance, Gürkaynak et al. (2005) uncover that Federal Open Market Committee statements have persistent impacts on long-term interest rates. Campbell et al.
(2012) report that forward guidance that is tied with action by the FRB is informativefor economic agents in reducing economic uncertainty.

7

Conclusion

This paper conducts an empirical assessment of how ambiguity is associated with
output and inflation for the U.S. during the post-World War II era. Estimating
ambiguity has long been considered a diﬃcult task, while we adopt the recently
developed method proposed by Izhakian (2020) and Brenner and Izhakian (2018) who
disentangle ambiguity from risk. This paper estimates ambiguity from the market,
size, value, and momentum risk premiums at a monthly frequency over the past 70
years.
It is observed that the ambiguity obtained from the market risk premium hastime-varying impacts on output and inflation. The positive ambiguity shocks raised
output between the middle of the 1970s and the beginning of the 1990s. An increasein ambiguity was positive during the high inflation period because high ambiguity
indicated that the high inflation period was not persistent. In contrast, it is foundthat market volatility shocks reduced output in the same period, which is consistent
with the asset pricing findings reported by Izhakian and Yermack (2017), Brenner
and Izhakian (2018), and Augustin and Izhakian (2020). Our results provide useful
information for policymakers. Focusing solely on market volatility may lead to an
overestimation of the negative impacts on output, as ambiguity occasionally has
positive eﬀects on output. Moreover, the impacts of ambiguity shocks on output andinflation vary over time. Therefore, policymakers should update their understandingof the impact driven by ambiguity when assessing eﬀective policies to address changes
in output and inflation.

16

It is also uncovered that ambiguity estimated from the size, value, and momentum
risk premiums has weaker impacts on output and inflation than that obtained fromthe market risk premium. This is because the size, value, and momentum risk premiums do not capture information about the entire market, since some investors do notchange their investment style (Cronqvist et al., 2015). Investigating the information
content of ambiguity obtained by these premiums is a task for future work.

17References[1] Abbate, A., S. Eickmeier, L. Wolfgang, and M. Marcellino (2016): “The Changing International Transmission of Financial Shocks: Evidence from a ClassicalTime-Varying FAVAR,” Journal of Money, Credit and Banking 48, 573-601.
[2] Arrow, K. J. (1965): Aspects of the Theory of Risk-Bearing, Helsinki: Yrjo
Jahnsonin Saatio.
[3] Asness, C., A. Frazzini, R. Israel, T.J. Moskowitz, and L.H. Pedersen (2018):“Size Matters, If You Control Your Junk,” Journal of Financial Economics 129,479-509.
[4] Augustin, P. and Y. Izhakian (2020): “Ambiguity, Volatility, and Credit Risk,”
Review of Financial Studies 33, 1618-1672.
[5] Bali, T.G. and R.F. Engle (2010): “The Intertemporal Capital Asset Pricing
Model with Dynamic Conditional Correlations,” Journal of Monetary Economics
57, 377-390.
[6] Basu, S. and B. Bundick (2017): “Uncertainty Shocks in a Model of Eﬀective
Demand,” Econometrica 85, 937-958.
[7] Baumeister, C. and G. Peersman (2013): “Time-Varying Eﬀects of Oil Supply
Shocks on the US Economy,” American Economic Journal: Macroeconomics 5(4): 1-28.
[8] Bekaert, G., M. Hoerova, and M. Lo Duca (2013): “Risk, Uncertainty and Monetary Policy,” Journal of Monetary Economics 60(7): 771-778.
[9] Berger, D., I. Dew-Becker, and S. Giglio (2020): ”Uncertainty Shocks as SecondMoment News Shocks,” Review of Economic Studies 87, 40-76.
[10] Bernanke, B.S. (2020): “The New Tools of Monetary Policy,” American Economic Review 110, 943-83.

18

[11] Bernanke, B.S., J. Boivin, and P. Eliasz (2005): “ Measuring the Eﬀects of Monetary Policy: A Factor-Augmented Vector Autoregressive (FAVAR) Approach,”
Quarterly Journal of Economics 120: 387-422.
[12] Bianchi, F., C.L. Ilut, and M. Schneider (2018): “Uncertainty Shocks, Asset
Supply and Pricing over the Business Cycle,” Review of Economic Studies 85,
810-854.
[13] Bloom, N. (2009): “The Impact of Uncertainty Shocks,” Econometrica 77, 62385.
[14] Brenner, M. and Y. Izhakian (2018): “Asset Pricing and Ambiguity: Empirical
Evidence,” Journal of Financial Economics 130, 503-531.
[15] Byrne, J.P., G. Fazio, and N. Fiess (2013): “Primary Commodity Prices: Comovements, Common Factors and Fundamentals,” Journal of Development Economics 101, 16-26.
[16] Campbell, J., C.L. Evans, J. Fisher, and A. Justiniano (2012): “Macroeconomic
Eﬀects of Federal Reserve Forward Guidance,” Brookings Papers on Economic
Activity 43(1), 1-80.
[17] Cao, H.H, T. Wang, and T. Zhang (2005): “Model Uncertainty, Limited Market
Participation, and Asset Prices,” Review of Financial Studies 18, 1219-1251.
[18] Carhart, M. (1997): “On Persistence in Mutual Fund Performance,” Journal of
Finance 52, 57-82.
[19] Carriero, A., T.E. Clark, and M. Marcellino (2018): “Measuring Uncertainty and
Its Impact on the Economy,” Review of Economics and Statistics 100, 799-815.
[20] Clarida, R., Galı́, J., and Gertler, M. (2000): “Monetary Policy Rules andMacroeconomic Stability: Evidence and Some Theory,” Quarterly Journal of
Economics 115(1), 147-180.

19

[21] Cronqvist, H., A. Siegel, and F. Yu (2015): “Value versus Growth Investing:
Why Do Diﬀerent Investors Have Diﬀerent Styles?,” Journal of Financial Economics 117(2), 333-349.
[22] Del Negro, M., D. Giannone, M. Giannoni, and A. Tambalotti (2019): “Global
Trends in Interest Rates,” Journal of International Economics 118, 248-262.
[23] Dow, J. and S.R.C. Werlang (1992): “Uncertainty Aversion, Risk Aversion, andthe Optimal Choice of Portfolio,” Econometrica 60, 197-204.
[24] Driouchi, T., L. Trigeorgis, and R.H.Y. So (2018): “Option Implied Ambiguity
and Its Information Content: Evidence from the Subprime Crisis,” Annals of
Operations Research 262, 463-491.
[25] Dynan, K.E., D.W. Elmendorf, and D.E. Sichel (2006): “Can Financial Innovation Help to Explain the Reduced Volatility of Economic Activity?,” Journal of
Monetary Economics 53(1), 123-150.
[26] Edelstein, P. and L. Kilian (2009): “How Sensitive are Consumer Expenditures
to Retail Energy Prices?,” Journal of Monetary Economics 56(6), 766-779.
[27] Ellis, C., H. Mumtaz, and P. Zabczyk (2014): “What Lies Beneath? A TimeVarying FAVAR Model for the UK Transmission Mechanism,” Economic Journal
576, 668-699.
[28] Ellsberg, D. (1961): “Risk, Ambiguity, and the Savage Axioms,” Quarterly Journal of Economics 75, 643-669.
[29] Epstein, L.G. and M. Schneider (2008): “Ambiguity, Information Quality, andAsset Pricing,” Journal of Finance 63, 197-228.
[30] Epstein, L.G. and T. Wang (1994): “Intertemporal Asset Pricing under Knightian Uncertainty,” Econometrica 62, 283-322.
[31] Fama, E.F. and K.R. French (1993): “Common Risk Factors in the Returns on
Stock and Bonds,” Journal of Financial Economics 33, 3-56.
20

[32] Fama, E.F. and K.R. French (1995): “Size and Book-to-Market Factors in Earnings and Returns,” Journal of Finance 50, 131-155.
[33] Fu, R., B. Melenberg, and N. Schweizer (2023): “Comment on “A Theoretical
Foundation of Ambiguity Measurement”,” Journal of Economic Theory 207,
105573.
[34] Galı́, J. and L. Gambetti (2009): “On the Sources of the Great Moderation,”
American Economic Journal: Macroeconomics, 1 (1): 26-57.
[35] Ghysels, E., P. Santa-Clara, and R. Valkanov (2005): “There is a Risk-Return
Trade-oﬀ after All,” Journal of Financial Economics 76, 509-548.
[36] Giannone, D., M. Lenza, and G.E. Primiceri (2015): “Prior Selection for Vector
Autoregressions,” Review of Economics and Statistics 97, 436-451.
[37] Gilboa, I. (2009): Theory of Decision under Uncertainty, Cambridge University
Press.
[38] Gilboa, I. and D. Schmeidler (1989): “Maxmin Expected Utility with NonUnique Priors,” Journal of Mathematical Economics 18, 141-153.
[39] Gollier, C. (2011): “Portfolio Choice and Asset Prices: The Comparative Statics
of Ambiguity Aversion,” Review of Economic Studies 78, 1329-1344.
[40] Gürkaynak, R.S., B. Sack, and E.T. Swanson (2005): “Do Actions Speak Louder
than Words? The Response of Asset Prices to Monetary Policy Actions andStatements,” International Journal of Central Banking 1(1), 55-93.
[41] Hamilton, J.D. (1983): “Oil and the Macroeconomy since World War II,” Journal of Political Economy 91 (2), 228-248.
[42] Harvey, A., E. Ruiz, and N. Shephard (1994): “Multivariate Stochastic Variance
Models,” Review of Economic Studies 61(2), 247-264.
[43] Ilut, C. L. and M. Schneider (2014): “Ambiguous Business Cycles,” American
Economics Review 104, 2368-2399.
21[44] Izhakian, Y. (2020): “A Theoretical Foundation of Ambiguity Measurement,”
Journal of Economic Theory 187, 105001.
[45] Izhakian, Y. and D. Yermack (2017): “Risk, Ambiguity, and the Exercise of
Employee Stock Options,” Journal of Financial Economics 124, 65-85.
[46] Izhakian, Y., D. Yermack, and J.F. Zender (2022): “Ambiguity and the Tradeoﬀ
Theory of Capital Structure,” Management Science 68, 4090-4111.
[47] Johnson, T.C. (2002): “Rational Momentum Eﬀects,” Journal of Finance 57,
585-608.
[48] Jurado, K., S.C. Ludvigson, and S. Ng (2015): “Measuring Uncertainty,” American Economic Review 105, 1177-1216.
[49] Kahn, J.A., M.M. McConnell, and G. Perez-Quiros (2002): “On the Causes of
the Increased Stability of the U.S. Economy,” Economic Policy Review, Federal
Reserve Bank of New York, 8, 183-202.
[50] Kelsey, D., R. Kozhan, and W. Pang (2010): “Asymmetric Momentum Eﬀects
under Uncertainty,” Review of Finance 15, 603-631.
[51] Klibanoﬀ, P., M. Marinacci, and S. Mukerji (2005): “A Smooth Model of Decision Making under Ambiguity,” Econometrica 73, 1849-1892.
[52] Knight, F. H. (1921): Risk, Uncertainty and Profit, Hart, Schaﬀner and Marx.
[53] Koop, G. and D. Korobilis (2010): “ Bayesian Multivariate Time Series Methods
for Empirical Macroeconomics,” Foundations and Trends in Econometrics 3:
267-358.
[54] Kostopoulos, D., S. Meyer, and C. Uhr (2022): “Ambiguity about Volatility andInvestor Behavior,” Journal of Financial Economics 145, 277-296.
[55] Liew, J. and M. Vassalou (2000): “Can Book-to-Market, Size and Momentum be
Risk Factors That Predict Economic Growth?,” Journal of Financial Economics
57, 221-245.
22

[56] Liu, L.X. and L. Zhang (2008): “Momentum Profits, Factor Pricing, and Macroeconomic Risk,” Review of Financial Studies 21(6), 2417-2448.
[57] Lucas Jr., R.E. (1978): “Asset Prices in an Exchange Economy,” Econometrica
46, 1429-1445.
[58] Mavroeidis, S. (2010): “Monetary Policy Rules and Macroeconomic Stability:
Some New Evidence,” American Economic Review 100(1), 491-503.
[59] Mumtaz, H. and F., Zanetti (2013): “The Impact of the Volatility of Monetary
Policy Shocks,” Journal of Money, Credit and Banking 45, 535-558.
[60] Nishimura, K.G. and H. Ozaki (2017): Economics of Pessimism and Optimism,
Springer.
[61] Patton, A.J. and K. Sheppard (2015): “Good Volatility, Bad Volatility: Signed
Jumps and the Persistence of Volatility,” Review of Economics and Statistics 97,
683-697.
[62] Peersman, G., S. Rüth, and W. Van der Veken (2021): “The Interplay between
Oil and Food Commodity Prices: Has It Changed over Time?,” Journal of International Economics 133, S0022199621001203.
[63] Petkova, R. (2006): “Do the Fama-French Factors Proxy for Innovations in
Predictive Variables?,” Journal of Finance 61, 581-612.
[64] Petkova, R. and L. Zhang (2005): “Is Value Riskier than Growth?,” Journal of
Financial Economics 78(1), 187-202.
[65] Primiceri, G.E. (2005): “Time Varying Structural Vector Autoregressions andMonetary Policy,” Review of Economic Studies 72, 821-852.
[66] Segal, G., I. Shaliastovich, and A. Yaron (2015): “Good and Bad Uncertainty:
Macroeconomic and Financial Market Implications,” Journal of Financial Economics 117, 369-397.

23

[67] Schmeidler, D. (1989): “Subjective Probability and Expected Utility without
Additivity,” Econometrica 57, 571-587.
[68] Vassalou, M. (2003): “News Related to Future GDP Growth as a Risk Factor
in Equity Returns,” Journal of Financial Economics 68(1), 47-73.

24

Figure 1. Ambiguity of the market risk premium
Notes: This figure presents the ambiguity of the market risk premium, which is obtained by Equation(2). The upper panel indicates ambiguity (left y-axis) and NBER recessions and the lower panel
demonstrates ambiguity (left y-axis) and the market risk premium (right y-axis, %) that is calculatedby the excess return of all CRSP firms.

25

Figure 2. Impulse responses of output
Notes: This figure plots the median impulse responses of output to one standard deviation of theambiguity shocks with 68% error bands. The TVP-VAR model employs the following variables:
output (∆y), inflation (∆p), and ambiguity (℧2 ). The caption of each panel demonstrates the time
horizon for the response.

26

Figure 3. Impulse responses of inflation
Notes: This figure plots the median impulse responses of inflation to one standard deviation of theambiguity shocks with 68% error bands. The TVP-VAR model employs the following variables:
output (∆y), inflation (∆p), and ambiguity (℧2 ). The caption of each panel demonstrates the time
horizon for the response.

27

Figure 4. Impulse responses to size ambiguity shocks
Notes: This figure plots the median impulse responses of output and inflation to one standard
deviation of the size ambiguity shocks with 68% error bands. The TVP-VAR model employs thefollowing variables: output (∆y), inflation (∆p), and size ambiguity (℧2size ). The caption of eachpanel demonstrates the response and the time horizon.

28

Figure 5. Impulse responses to realized volatility shocks
Notes: This figure plots the median impulse responses of output and inflation to one standard deviation of the realized volatility of shocks with 68% error bands. The realized volatility is calculatedusing the daily excess returns of the CRSP value-weighted index. The TVP-VAR model employs
the following variables: output (∆y), inflation (∆p), and realized volatility (V OLRV ). The captionof each panel demonstrates the response and the time horizon.

29

Figure 6. Impulse responses obtained by the constant BVAR model
Notes: This figure plots the median impulse responses of output and inflation to one standard
deviation of the ambiguity shocks with 68% error bands. We employ the constant BVAR model.
The BVAR model employs the following variables: output (∆y), inflation (∆p), and ambiguity (℧2 ).
Period I is from February 1947 to December 1989 and Period II is from January 1990 to December
2021. The BVAR model is estimated based on the method of Giannone et al. (2015).

30

Time-varying ambiguity shocks andbusiness cycles
August 16, 2023

Online Supplement, Not for Publication
List of contents
A Estimation of time-varying parameter vector autoregressive (TVP-VAR) model
B Measure of ambiguity
Figures A1-A3 Ambiguity of other risk premiums
Figure A4 Impulse responses over a longer horizon
Figures A5-A6 Diﬀerent identification
Figures A7-A7 Impulse responses to diﬀerent shocks
Figure A10 Comparison of ambiguity
Table A1 Descriptive statistics
Table A2 Data sources

1

A

Estimation of time-varying parameter vector autoregressive (TVP-VAR) model

A.1

PriorsWe follow Primiceri (2005), Koop and Korobilis (2010), and Baumeister and Peersman (2013) and the prior for the coeﬃcients Bt , the simultaneous relations At , andthe log of variances ht are set toB0 ∼ N (BOLS , 4 · V ar(BOLS ))A0 ∼ N (AOLS , 4 · V ar(AOLS ))h0 ∼ N (hOLS , 4 · Ik ),where the subscript OLS indicates the point estimates obtained by ordinary leastsquares (OLS) and V ar denotes the covariance matrix.
The prior for the hyper-parameters are set to the inverse-Wishart distributionsas:
Q0 ∼ IW(kq2 · τ · V ar(BOLS ), τ )W0 ∼ IW(kw2 · 3 · I3 , 2)
S1,0 ∼ IW(ks2 · 2 · V ar(A1,OLS ), 2)
S2,0 ∼ IW(ks2 · 3 · V ar(A2,OLS ), 3),
where S1 and S2 indicate the two blocks of S, τ = 12 denotes the sample size using
the constant OLS, kq = 0.01, kw = 1, and ks = 0.1.

A.2

Estimation

We implement the Gibbs sampler that evaluates the posterior distributions of thefour blocks of parameters: the coeﬃcients B T ; the simultaneous relations AT ; thestandard deviations V T , where the superscript T refers to the whole sample; and thehyper-parameters M that contain Q, W , S1 , and S2 .1
The estimation procedure is summarized as follows:
Step 1. Initialize B T , AT , V T , and M by setting up training sample prior.
Step 2. Draw B T from the conditional posterior distribution p(B|Y T , AT , V T , M ).
Step 3. Draw AT from the conditional posterior distribution p(A|Y T , B T , V T , M ).
Step 4. Draw V T from the conditional posterior distribution p(V |Y T , B T , AT , M ).
Step 5. Draw M by sampling Q, W, S1 , and S2 from p(V |Y T , B T , AT , V T ) =p(Q|Y T , B T , AT , V T )·p(W |Y T , B T , AT , V T )·p(S1 |Y T , B T , AT , V T )·p(S2 |Y T , B T , AT , V T ),since Q, W , S1 , and S2 are independent of the other blocks.
Step 6. Go to Step 2.
1

See Primiceri (2005) and Baumeister and Peersman (2013) for details.

2

B

Measure of Ambiguity

This paper employs a measure of ambiguity, ℧2 , proposed by Izhakian (2020). As Fu
et al. (2023) point out, some caveats are in order. Although Izhakian (2020) showsthat the ordering based on ℧2 can represent the decision maker’s preference order,
Fu et al. (2023) cast doubt on the validity of Izhakian’s (2020) main theorems, inparticular, Theorems 5, 6, and 7 that relate the decision maker’s preference order tothe measure of ambiguity ℧2 . Despite these problems, we adopt ℧2 as a measureof ambiguity for two reasons. First, as discussed by Fu et al. (2023, p. 3), “While
the strong connection between preferences under ambiguity and ℧2 seems lost, ℧2
may well measure ambiguity–just like there are many other measures of risk besides
the variance”, this measure is worth adopting. Second, Kostopoulos et al. (2022),
employing the V-VSTOXX as a measure of ambiguity, adopt ℧2 as an alternative
measure and obtain similar results.

3

Figure A1. Ambiguity of the size premium
Notes: This figure presents ambiguity of the size premium, which is obtained by Equation (2). The
upper panel indicates ambiguity (left y-axis) and NBER recessions and the lower panel demonstrates
ambiguity (left y-axis) and the size premium (right y-axis, %).

4

Figure A2. Ambiguity of the value premium
Notes: This figure presents ambiguity of the value premium, which is obtained by Equation (2). The
upper panel indicates ambiguity (left y-axis) and NBER recessions and the lower panel demonstrates
ambiguity (left y-axis) and the value premium (right y-axis, %).

5

Figure A3. Ambiguity of the momentum premium
Notes: This figure presents ambiguity of the momentum premium, which is obtained by Equation(2). The upper panel indicates ambiguity (left y-axis) and NBER recessions and the lower panel
demonstrates ambiguity (left y-axis) and the momentum premium (right y-axis, %).

6

Figure A4. Impulse responses to ambiguity: 8- and 12-months ahead
Notes: This figure plots the median impulse responses of output and inflation to one standard
deviation of the ambiguity shocks with 68% error bands. The TVP-VAR model employs the following
variables: output (∆y), inflation (∆p), and ambiguity (℧2 ). The caption of each panel demonstrates
the response and the time horizon.

7

Figure A5. Impulse responses to ambiguity: Identification II
Notes: This figure plots the median impulse responses of output and inflation to one standard
deviation of the ambiguity shocks with 68% error bands. The TVP-VAR model employs the following
order: ambiguity (℧2 ), inflation (∆p), and output (∆y). The caption of each panel demonstrates
the response and the time horizon.

8

Figure A6. Impulse responses to ambiguity: Identification III
Notes: This figure plots the median impulse responses of output and inflation to one standard
deviation of the ambiguity shocks with 68% error bands. The TVP-VAR model employs the following
order: output (∆y), ambiguity (℧2 ), and inflation (∆p). The caption of each panel demonstrates
the response and the time horizon.

9Figure A7. Impulse responses to value ambiguity shocks
Notes: This figure plots the median impulse responses of output and inflation to one standard
deviation of the value ambiguity shocks with 68% error bands. The TVP-VAR model employs thefollowing variables: output (∆y), inflation (∆p), and value ambiguity (℧2value ). The caption of eachpanel demonstrates the response and the time horizon.

10

Figure A8. Impulse responses to momentum ambiguity shocks
Notes: This figure plots the median impulse responses of output and inflation to one standard
deviation of the momentum ambiguity shocks with 68% error bands. The TVP-VAR model employs
the following variables: output (∆y), inflation (∆p), and momentum ambiguity (℧2mom ). The captionof each panel demonstrates the response and the time horizon.

11

Figure A9. Impulse responses to macro uncertainty shocks
Notes: This figure plots the median impulse responses of output and inflation to one standard
deviation of the total macro uncertainty index (Jurado et al., 2015) of shocks with 68% error bands.
The TVP-VAR model employs the following variables: output (∆y), inflation (∆p), and macro
uncertainty (UNC). The caption of each panel demonstrates the response and the time horizon.

12Figure A10. Impulse responses to MIDAS volatility shocks
Notes: This figure plots the median impulse responses of output and inflation to one standard
deviation of the MIDAS volatility shocks with 68% error bands. The TVP-VAR model employs thefollowing variables: output (∆y), inflation (∆p), and MIDAS volatility (V OLM IDAS ). The captionof each panel demonstrates the response and the time horizon.

13

Figure A11. Comparison of ambiguity
Notes: This figure presents the comparison of ambiguity. Ambiguity (left) is obtained by Equation(2) and BI (right) is estimated by the original procedures in Brenner and Izhakian (2018) using
daily data.

14

Table A1: Descriptive statistics

Panel A
Market risk premiums
Size premiums
Value premiums
Momentum premiums
Panel B
Market risk premiums
Size premiums
Value premiums
Momentum premiums℧2℧2size℧2value℧2mom∆y∆p
V OLRVV OLM IDAS
UNC

Daily
Mean
Std.Dev. Max
0.032
0.945
11.350
0.003
0.490
6.240
0.014
0.520
6.740
0.029
0.695
7.120
Monthly
Mean
Std.Dev. Max
0.683
4.277
16.1000.122
2.806
21.4200.289
2.749
13.5600.690
3.880
18.200
0.200
0.373
5.315
1.304
2.058
21.3301.415
1.920
20.6401.097
2.189
41.614
0.223
1.092
6.233
0.286
0.342
1.945
0.800
0.505
5.845
0.800
0.332
3.946
0.648
0.107
1.268

Min
Obs.
−17.440 19053−11.630 19053−5.000 19053−14.370 19053
Min−23.240−17.230−13.970−34.300
0.000
0.000
0.002
0.000
−14.152−1.786
0.174
0.295
0.523

Obs.
899
899
899
899
899
899
899
899
899
899
899
899
738

Notes: This table reports the means, standard deviations, maximums,
minimums, and the number of observations. Following Brenner andIzhakian (2018), ℧2 , ℧2size , ℧2value , and ℧2mom are obtained by Equation(2). V OLRV is monthly realized volatility using daily excess returns.
V OLM IDAS is estimated by the MIDAS model (Ghysels et al., 2005).
UNC is total macroeconomic uncertainty proposed by Jurado et al.
(2015).

15

16
Ambiguity obtained from the size premiums
Ambiguity obtained from the value premiums
Ambiguity obtained from the momentum premiums

Monthly realized volatility
Monthly volatility estimated by
the MIDAS model (Ghysels et al., 2005)
Total macroeconomic uncertainty (Jurado et al., 2015)℧2℧2size℧2value℧2mom

V OLRVV OLM IDAS

Notes: This table shows the data definition and source.

UNC

Log of consumer price index for
all items in U.S. city average
Ambiguity obtained from the market risk premiumspyMomentum premiums

Value premiums

Size premiums

Market risk premiums

Definition
The excess return on the value weighted
return of CRSP firms
The average spread return between threesmall and three large groups
The average spread return between twovalue and two growth groups
The average spread return between two highprior and two low prior return groups
Log of industrial production index

Table A2: Data sources

No

No
No

No

No

No

No

Yes

Yes

No

No

No

First DiﬀerenceNo

Sydney Ludvigson’s web site

Federal Reserve Bank of
St. Louis
Federal Reserve Bank of
St. Louis
Our estimation by
Equation (2)Authors’ estimation by
Equation (2)Authors’ estimation by
Equation (2)Authors’ estimation by
Equation (2)Authors’ estimation
Authors’ estimation

Kenneth French’s website

Kenneth French’s website

Kenneth French’s website

SourceKenneth French’s website