Download A Sentiment-based Explanation of the Forward Premium Puzzle*

A Sentiment-based Explanation of the Forward
Premium Puzzle*
Jianfeng Yu†
March 2013
Abstract
A sentiment-based model of the exchange rate is proposed to understand the
forward premium puzzle. Agents over- or underestimate the growth rate of the
economy. All else equal, when perceived domestic growth is higher than perceived
foreign growth, the domestic interest rate is higher than the foreign interest rate. At
the same time, an econometrician would expect an increase in the home currency
value. Together, the model with investor misperception can account for the forward
premium puzzle. In addition, it helps lower the correlation between consumption
growth differentials and exchange rate growth. Finally, this paper provides empirical
evidence supporting the mechanism in the sentiment-based explanation.
JEL Classification: G12, F31, G14, G15
Keywords: investor sentiment, forward premium puzzle, misperception, long-run risk,
Backus-Smith Puzzle
*I would like to thank Raj Aggarwal, Ravi Bansal, Frederico Belo, John Boyd, Zhi Da, Paul Gao,
Bob Goldstein, Jeremy Graveline, David Hirshleifer, Philippe Jorion, Hanno Lustig, Terry Odean, Stavros
Panageas, Tracy Wang, Jeffrey Wurgler, Wei Xiong, Xiaoyun Yu, and seminar participants at the University
of Minnesota, University of Notre Dame, UC-Irvine, the Federal Reserve Bank of Dallas, Peking University,
Tsinghua University, 2010 Econometric Society World Congress, and 2011 WFA for comments. I especially
thank Riccardo Colacito (WFA discussant), Urban Jermann (the editor), an anonymous referee, and Jian
Wang for extremely insightful and detailed comments. I also thank Malcolm Baker, Jeffrey Wurgler, and
Yu Yuan for making their investor sentiment data available. I gratefully acknowledge financial support from
Dean’s small research grant at the University of Minnesota. All errors are my own.
† Corresponding Author: Jianfeng Yu, Department of Finance, Carlson School of Management,
University of Minnesota. Address: 321 19th Avenue South, Suite 3-122, Minneapolis, MN 55455, Tel:
612-625-5498, Email: [email protected].
1.
Introduction
Uncovered interest rate parity (UIP) implies that the expected change in exchange rates
(the foreign price of domestic currency) should equal the interest rate differential between
the foreign and domestic countries (i.e., the forward premium). Therefore, the regression
coefficient of future changes in exchange rates on interest rate differentials should equal one.
However, Fama (1984) and subsequent studies consistently find a negative coefficient in such
regressions. That is, an increase in the foreign interest rate forecasts an appreciation of the
foreign currency. The violation of UIP is often referred as the ”forward premium puzzle” in
the literature. This evidence also implies that there are predictable returns from investing
in currency markets. The forward premium puzzle has inspired a vast theoretical work that
arises to shed light on this puzzle.
This paper offers a sentiment-based explanation of the forward premium puzzle. In
particular, a representative agent endowment economy for each country with complete home
bias is considered. Agents in different countries share a common but subjective belief about
future fundamentals, and agents can be optimistic (i.e., high sentiment) or pessimistic (i.e.,
low sentiment) about future endowment growth. When domestic sentiment is high, it means
that both the home and foreign agents are optimistic about future U.S. endowment growth.1
When agents are optimistic about the domestic economy, and all else is equal, the interest
rate at home is higher than the foreign interest rate due to a strong desire to borrow to
increase current consumption. Thus, the key for our model to reproduce the failure of UIP
is that high sentiment in one country can forecast an appreciation of this country’s currency.
In a complete market, the (log) change in exchange rates, measured in units of the foreign
goods per domestic goods, equals the difference between the (log) stochastic discount factors
of the domestic and foreign countries. Hence, with recursive preferences, the exchange
rate has to adjust to reflect differences in both current and future relative consumption
across countries. When domestic sentiment is high today, high future growth at home is
anticipated. Hence, from the economic agents’ point of view, the home currency is expected
to depreciate to reflect the anticipated difference in consumption across two countries in
the future. The magnitude of this anticipated depreciation depends on the intertemporal
elasticity of substitution (IES). However, an econometrician anticipates that the economic
agent at home will receive a lower-than-expected endowment tomorrow, raising the marginal
1
It is certainly interesting to consider heterogeneous beliefs as in Scheinkman and Xiong (2003), Dumas,
Kurshev, and Uppal (2009), and Xiong and Yan (2010). However, the purpose of the current paper is to
illustrate that investor misperception can lead to the failure of UIP in the simplest possible model. Thus, a
simpler modeling approach is chosen.
1
utility of consumption next period, and hence the home currency appreciates. The magnitude
of this appreciation depends on the home agent’s risk aversion since risk aversion determines
the reduction in the marginal utility. Therefore, the econometrician is likely to observe a
smaller depreciation or even an appreciation of the home currency in the data. In particular,
when risk aversion is greater than the inverse of IES, the second effect dominates and high
domestic sentiment predicts an appreciation of the home currency from the econometrician’s
point of view. All together, when domestic sentiment is relatively high, the home interest
rate is pushed up, and at the same time an appreciation of the domestic currency is expected.
This can qualitatively account for the violation of the UIP condition in the data.
For reasonable parameterizations, the model can indeed produce a negative UIP
coefficient. Meanwhile, since our model can be viewed as a misperception-augmented version
of Colacito and Croce (2011), it can simultaneously match many asset pricing moments as
well as several other stylized facts in the international markets. For example, the model can
account for the high correlation of pricing kernels and the low correlation of fundamentals
(see, e.g., Brandt, Cochrane, and Santa-Clara (2006)). Interestingly, the model can also help
produce a relatively low correlation between consumption growth differentials and foreign
exchange growth (i.e., the Backus and Smith (1993) puzzle). Exchange rates in our model
are driven by both fundamentals and investor sentiment. As a result, investor sentiment
breaks the link between changes in foreign exchange and consumption growth differentials,
and hence reduces their correlation. In sum, our misperception-augmented model improves
upon Colacito and Croce (2011) by producing a negative UIP coefficient and by lowering the
correlation between consumption growth differentials and foreign exchange growth.2
The key refutable predictions of the model are tested using proxies for investor sentiment.
The evidence in the data suggests that investor sentiment has significant power to forecast
changes in exchange rates and returns on foreign exchange, with the same sign as predicted
by the model. The predictive power of the ”true” expected growth rate is also explored.
Compared with sentiment, the ”true” expected growth rate has opposite power in predicting
changes in exchange rates. Whereas a high domestic-foreign sentiment differential predicts
a higher future spot rate, a high true expected growth differential between the domestic and
foreign countries predicts a lower spot rate in the future. This evidence highlights the distinct
role of sentiment and its special ability to account for a negative UIP coefficient. In a standard
2
As shown in Section 3.4. below, the model also helps produce a large equity premium with lower risk
aversion and IES. This is because sentiment can serve as additional ”long-run risk”. Since sentiment is
misperception, the realized consumption growth is still fairly unpredictable. As a result, the model can
potentially feature a large amount of long-run risk without increasing the predictability of actual consumption
growth.
2
rational model, a high true expected growth differential between the domestic and foreign
countries predicts depreciation of the home-country currency. Thus, our evidence supports
this prediction. However, without the sentiment effect, this would lead to a positive UIP
coefficient. On the other hand, a high domestic-foreign sentiment differential is associated
with a low forward premium, but predicts an appreciation of the home currency. Therefore,
misperception is crucial in generating a negative UIP coefficient.
Considerable prior work on the forward premium puzzle has been done. A number
of earlier studies have provided insightful analysis on the role of investor irrationality in
foreign exchange markets.3 Frankel and Froot (1987, 1990a) provide an early application of
irrationality to currency markets. Mark and Wu (1998) develop a model in the noise trading
setting of De Long, Shleifer, Summers, and Waldmann (1990), in which traders overweigh
the forward premium when predicting future changes in the exchange rate. Gourinchas and
Tornell (2004) provide an explanation based upon a distortion in investors’ beliefs about the
interest rate process. Beber, Breedon, and Buraschi (2010) find differences in beliefs impact
implied volatility of currency options and currency returns. Finally, in a closely related
study, Burnside, Han, Hirshleifer, and Wang (2011) offer an insightful explanation based on
investor overconfidence on inflation.
This paper adds to the previous literature by showing that investors’ misperception
on the growth rate can potentially induce a negative UIP coefficient.4 Our modeling
approach also allows us to utilize investor sentiment data to test the mechanisms inside
our model. Indeed, our empirical evidence lends support for a sentiment-based explanation.
In addition, our model highlights the importance of recursive preferences in generating a
negative UIP coefficient. More important, the same mechanism appears to help account
for many salient features in international markets simultaneously. In particular, investor
sentiment can potentially generate substantial long-run risk under recursive preferences,
3
There are numerous other studies that attempt to explain this puzzle under rational expectations.
Notable recent papers include Alvarez, Atkeson, and Kehoe (2009) who examine a segmented market,
Bacchetta and Van Wincoop (2009) who use a rational inattention framework, Burnside, Eichenbaum, and
Rebelo (2007) who include microstructure frictions, Farhi and Gabaix (2008) who employ a Rietz (1988)
rare disaster framework, Verdelhan (2010) who uses habit formtion, and Lustig and Verdelhan (2007) and
Lustig, Roussanov, and Verdelhan (2011, 2012) who perform a cross-sectional analysis of foreign exchange
portfolios. Earlier equilibrium models that attempt to explain the forward premium puzzle include Backus,
Gregory, and Telmer (1993) and Bekaert (1996).
4
Overconfidence a la Burnside et al (2011) is a natural ingredient to produce investor misperception.
However, as shown in the example in the next section, the difference between investor perception and the
objective belief does not have to be derived from investor overconfidence. It could be due to other effects
such as extrapolation, conservatism, or even rational learning. More important, our asset pricing approach
to exchange rates allows us to impose many asset pricing moments to discipline the model parameters. These
asset pricing moments include the first two moments of stock returns, riskfree rate, and their correlations
across countries.
3
which helps account for several asset pricing moments with lower risk aversion and less
predictable actual consumption growth.
Section 2 provides an illustrative example to convey the main idea. Section 3 presents
the model, and Section 4 presents empirical evidence. Section 5 concludes.
2.
An Illustrative Example
This section presents a simple example to convey the main idea of the paper. To make the
presentation transparent, complete home biases and complete markets are imposed. The two
countries are symmetric and representative investors in both countries have power preferences
1−γ
P
t Ct
and maximize their life-time utility: ∞
t=1 δ 1−γ .
Below only domestic variables are considered, and the corresponding foreign variables are
denoted with a superscript ”*”. Assume that the data generating process for consumption
growth is i.i.d. normal: gt+1 ≡ log (Ct+1 ) − log (Ct ) = µg + σg g,t+1 . However, all
investors (homogeneously) believe that they can forecast consumption growth according
to gt+1 = µg + st + σg ˆg,t+1 , where ˆt is i.i.d. standard normal under investors’ perception.
Here, st can be interpreted as investor sentiment. It is worth emphasizing that the term
“sentiment” is used in a very broad sense. It simply represents the difference between
economic agents’ (i.e., investors) belief and econometricians’ belief. This difference in beliefs
could be due to either rational or irrational learning. For example, st could be a result of
agents’ rational learning on news shocks to technology or production, which is not observed
by the econometrician. Moreover, no further restriction is imposed on the dynamics of the
sentiment process in this example. According to investors’ belief, risk-free rates are:
rf,t = − log δ + γµg + γst −
γ2 2
σ .
2 g
(1)
Thus, the risk-free rate differentials are:
∗
rf,t
− rf,t = γ (s∗t − st ) .
(2)
Under a frictionless complete market, the (log) exchange rate et is linked to domestic and
4
foreign discount factors mt and m∗t by the following equation:5
∗
et+1 − et = mt+1 − m∗t+1 = −γ gt+1 − gt+1
,
(3)
where the (log) exchange rate is defined as the foreign price per home consumption unit.
Under investors’ perception
Êt (et+1 − et ) = γ (s∗t − st ) ,
(4)
where Êt (·) is the expectation under economic agents’ subjective belief conditional on
information available up to time t. It follows from at equations (2) and (4) that UIP holds
ex ante under investors’ perception. However, ex-post exchange rates evolve according to
the actual data generating process, which is i.i.d. according to equation (3). Hence, an
econometrician regressing equation (3) on equation (2) would estimate a coefficient zero.
Thus, the UIP condition fails in this simple example.
However, in this simple example, the correlation between consumption growth
differentials and exchange rate growth is one (see equation (3)), whereas this correlation
is very low in the data. The next section shows that under recursive preferences, the UIP
coefficient can be negative and the correlation between consumption growth differentials and
exchange rate growth can also be substantially reduced.
3.
An Exchange Rate Model with Investor Sentiment
This section presents a simple model of exchange rates with investor sentiment. To highlight
the effect of investor sentiment on the forward premium puzzle, the model abstracts from
money, and irrational belief is the only deviation from a standard two-country model.
Hollifield and Yaron (2003) find that inflation risk is virtually unrelated to currency returns,
and they conclude that currency models should focus on real risk. Therefore, inflation is
abstracted from our analysis and the model focuses on the real side of the economy. A few
recent studies on currency markets (e.g., Verdelhan (2010), Colacito and Croce (2011), and
Engel (2011)) also shift the focus to real variables. In particular, Engel (2011) shows that
Fama’s (1984) results also hold in real terms.
5
See Backus, Foresi, and Telmer (2001) for a rigorous argument. In particular, the argument in Backus,
Foresi, and Telmer (2001) still follows even if the agents do not have correct beliefs. Among others, Backus,
Foresi, and Telmer (2001), Brandt, Cochrane and Santa-Clara (2006), and Colacito and Croce (2011) exploit
this equation to relate the dynamics of the exchange rate to the dynamics of the domestic and foreign
discount factors.
5
3.1.
Consumption Dynamics and Preferences
We analyze an economy with two countries that are denoted as domestic and foreign. To
simplify the setup, a separate pure exchange economy for each country is specified by
following Colacito and Croce (2011). Consumption in the two countries is exogenously given
and there is a complete home bias, meaning that the representative agent in each country only
consumes the good with which he is endowed. Asset markets are complete. This setup allows
us to analyze the asset prices in each country separately. For convenience, endowments,
preferences, and prices are characterized only for the domestic country. Identical expressions
indexed by a ”*” apply to the foreign country.
A discrete-time economy with infinite horizons is considered. Following the long-run risk
literature, under the econometrician’s view (i.e, the objective belief or the data-generating
process), the dynamics for consumption growth gt are modeled as
gt+1 = µg + xt + σg g,t+1
(5)
xt+1 = ρx xt + σx x,t+1 ,
(6)
where (g,t , x,t ) is i.i.d. joint standard normal under the data-generating process. The
representative agent in the domestic economy (and the foreign economy) is sentimental and
has a subjective belief that is different from the data-generating process observed by an
econometrician. Under economic agents’ subjective belief, the dynamics of consumption
growth are as follows:
gt+1 = µg + xt + st + σg ˆg,t+1
(7)
st+1 = ρs st + σs s,t+1
(8)
xt+1 = ρx xt + σx x,t+1 ,
(9)
where (ˆg,t , x,t , s,t ) is i.i.d. joint standard joint normal under economic agents’ belief. It is
further assumed that the sentiment dynamics are the same with equation (8) under the datagenerating process.6 To simplify the setting, all the shocks are joint normal and independent
of each other throughout the paper. Since the true expected growth is xt , one can interpret
st as investor sentiment. When st is positive (negative), the agent is optimistic (pessimistic).
6
Strictly speaking, one needs to show that there exist objective and subjective probability beliefs such
that the dynamics of consumption follow those given here under these two probability measures. These
are purely technical requirements, and the proof is just a simple application of the Girsanov theorem. The
learning model in the Internet Appendix provides such an example.
6
The above specification on consumption and sentiment represents the main departure
from rational expectation models and hence deserves further discussion. There are
potentially many ways to obtain a time-varying misperception on the growth rate (i.e.,
sentiment) endogenously. One natural approach is a learning model with overconfidence.
Scheinkman and Xiong (2003) and Dumas, Kurshev, and Uppal (2009) elegantly provide
such models. Nonetheless, an exogenous sentiment process is specified because modeling the
origin of sentiment does not add economic insights for the purpose of this study, while it
does complicate the analysis. Our overall approach is to keep the model as simple as possible
while retaining the key ingredients needed to generate the failure of the UIP condition.7
Moreover, the Internet Appendix shows that all the results remain when the sentiment
process is modeled endogenously.
To complete the model, the investors’ preferences over uncertain consumption stream Ct
is specified as the recursive utility function (e.g., Epstein and Zin (1989))
1−γ
θ
Ut = (1 − δ) Ct
+δ
1−γ
Êt Ut+1
θ
θ1 1−γ
,
(10)
where Êt (·) is the expectation under economic agents’ subjective belief conditional on
information available up to time t, the parameter 0 < δ < 1 is the time discount factor,
γ ≥ 0 is the risk-aversion parameter, ψ ≥ 0 is the intertemporal elasticity of substitution
1−γ
(IES) parameter, and θ = 1−
1 . When risk aversion is larger than the reciprocal of IES,
ψ
agents prefer early resolution of uncertainty of the consumption path (e.g., Kocherlakota
(1990)).
3.2.
The Stochastic Discount Factor
As shown in Epstein and Zin (1989), the logarithm of the intertemporal marginal rate of
substitution (IMRS) is given by
mt+1 ≡ log (Mt+1 ) = θ log δ −
θ
gt+1 + (θ − 1) ra,t+1 ,
ψ
(11)
where ra,t+1 is the logarithm of the gross return on an asset that delivers aggregate
consumption as its dividends each period. For any continuous return rt+1 = log (Rt+1 ) ,
7
The Internet Appendix provides an exchange rate model based on learning with overconfidence. In such
a model, a mean-reverting sentiment process can be generated endogenously. Empirically, the sentiment
data from Baker and Wurgler (2006) also suggest that the sentiment process is mean-reverting. Thus, the
sentiment process is specified as an AR(1) in equation (8).
7
including the one on the consumption claim,
Êt [exp (mt+1 + rt+1 )] = 1.
(12)
With log-linear approximation, the return on wealth can be solved explicitly, and hence the
IMRS can be obtained explicitly as well.
3.3.
The Foreign Exchange Market
A foreign country is introduced into the model. The setup is similar to that used in Colacito
and Croce (2011). For tractability, complete symmetry is imposed, and all the model
parameters are identical across countries. Furthermore, it is assumed that the agents at
home and abroad share the same belief, although their beliefs could be wrong. Thus, when
domestic sentiment is high, it means that all the representative agents are homogeneously
optimistic about the U.S. economy.
3.3.1.
The Exchange Rate and the Forward Premium
Again, under a frictionless complete market, the (log) exchange rate et is linked to domestic
and foreign discount factors mt and m∗t by the following equation:
et+1 − et = mt+1 − m∗t+1 .
(13)
To solve for the exchange rate, the standard log-linearization argument in the long-run risk
literature is used (e.g, Bansal and Yaron (2004) and Colacito and Croce (2011)). First
compute the return on wealth, ra,t+1 , in the pricing kernel equation (11). It follows from the
Campbell-Shiller log-linear approximation that
ra,t+1 ≈ κ0 + κ1 zt+1 − zt + gt+1 .
(14)
The price-consumption ratio can be further approximated as a linear function of the state
variables st and xt . That is,
zt ≈ A0 + A1 st + A2 xt ,
(15)
where A0 , A1 , and A2 are constants. Substituting equations (14) and (15) back into the
Euler equation (12) yields A1 =
1
1− ψ
1−κ1 ρs
and A2 =
1
1− ψ
1−κ1 ρx
.
Furthermore, substituting equations (14) and (15) back into equation (11) gives the
8
equilibrium solution to the IMRS,
mt+1
1
= −γgt+1 + γ −
(xt + st )+κ1 (θ − 1) A1 σs s,t+1 +κ1 (θ − 1) A2 σx x,t+1 +C0 , (16)
ψ
where C0 is a constant. Thus, the risk-free rate, which is determined by the agent’s subjective
belief, is given by
1
rf,t = − log Êt exp (mt+1 ) = (xt + st ) + C1 ,
ψ
(17)
where C1 is a constant. As a result, the forward premium (f pt ), which equals the foreigndomestic interest rate differential, is given by
∗
f pt ≡ rf,t
− rf,t = −
1
1
(st − s∗t ) − (xt − x∗t ) .
ψ
ψ
(18)
Thus, the first testable prediction is obtained as follows.
Prediction 1 The forward premium (i.e., the foreign-domestic interest rate differential) is
negatively associated with the sentiment differential (st − s∗t ) and the true expected growth
diferential (xt − x∗t ). In addition, the interest rate is positively associated with both sentiment
and true expected growth.
Furthermore, it follows from equations (13) and (16) that the solution to changes in
exchange rates is given by
et+1 − et
1
(st − s∗t + xt − x∗t ) + κ1 (θ − 1) A1 σs s,t+1 − ∗s,t+1
=
− γgt+1 + γ −
ψ
+κ1 (θ − 1) A2 σx x,t+1 − ∗x,t+1 .
(19)
∗
γgt+1
Notice that under the data-generating process, all the shocks s,t+1 , x,t+1 , ∗s,t+1 , and ∗x,t+1
∗
have mean zero, Et (gt+1 ) = xt , and Et gt+1
= x∗t . Here, Et (·) is the expectation under
the econometrician’s view (i.e., the objective belief). Therefore, taking the conditional
expectation under the data-generating process yields
1
1
· (st − s∗t ) − (xt − x∗t ) .
Et (et+1 − et ) = γ −
ψ
ψ
(20)
When γ > 1/ψ, the true expected consumption growth and the misperceived growth have
opposite predictive power for the change in the exchange rate. This highlights the distinct
9
role of the misperception in consumption growth and the true expected consumption growth.
To emphasize the unique prediction of the model on the relation between foreign exchange
growth and sentiment differentials, the second testable prediction is summarized as follows.
Prediction 2 If γ > 1/ψ, the change in the exchange rate, et+1 − et , is positively associated
with the sentiment differential (st − s∗t ), but negatively associated with the true expected
growth differential (xt − x∗t ).
3.3.2.
The UIP Coefficient
First, it is worthy of note that under economic agents’ belief, the following equalities obtain
Êt (et+1 − et ) = −
1
∗
· (st − s∗t + xt − x∗t ) = rf,t
− rf,t .
ψ
(21)
Hence, when the sentiment about the domestic growth rate is high and all else equal,
economic agents expect an appreciation of the foreign currency. Further, economic agents
expect a UIP coefficient of one. This result is the same with traditional asset pricing models.
The exchange rate has to reflect the differences in growth rates between the two countries.
If agents in the model expect a high growth rate in the future, they also anticipate the home
currency to depreciate to reflect the differences in growth rates between the two countries.
More important, equation (21) is consistent with the findings in Frankel and Froot (1987,
1990b) that when ex-ante measures of expected exchange rate changes (based on survey data)
instead of the ex post realizations are used as the dependent variable in UIP regression, the
UIP coefficient is in the vicinity of one.
To examine the traditional UIP regression of foreign exchange growth on the forward
premium, one need to combine equations (18) and (20). The regression coefficients of nperiod changes in the exchange rate on the n-period forward premium (i.e., the long-horizon
UIP coefficients) are given by
ψγ
βU IP,n = 1 − 1−ρn
x
1−ρx
2
1−ρn
s
1−ρs
var(xt −x∗t )
var(st −s∗t )
2
+
1−ρn
s
1−ρs
2
(22)
With straightforward algebra, the following proposition is obtained.
Proposition 1 The UIP coefficient βU IP,n is increasing in horizon n, as long as ρx > ρs > 0.
var(x −x∗ )
Further, if the variance ratio var(stt −s∗t ) < γψ − 1, the one-period UIP coefficient βU IP,1 is
t
10
negative.
var(x −x∗ )
If xt = x∗t as in Colacito and Croce (2011), then var(stt −s∗t ) < γψ − 1 always holds, as
t
long as investors prefer early resolution of uncertainty. Thus, the UIP coefficient is negative.
The Internet Appendix shows that through learning with overconfidence, ρx > ρs > 0 holds
for the endogenous sentiment process.8 Hence, Proposition 1 implies that the UIP condition
holds better at longer horizons, which is consistent with the empirical findings in Gourinchas
and Tornell (2004) and Meredith and Chinn (2004). In particular, at longer horizons of five
and ten years, Chinn and Meredith (2004) find much less negative forward premium bias.
The intuition on the failure of the UIP condition is elaborated below. As discussed
earlier, from economic agents’ point of view, when domestic sentiment is higher than foreign
sentiment today, the home currency is expected to depreciate in the future to reflect the
difference in consumption across two countries. Moreover, equation (21) implies that the
magnitude of this anticipated depreciation depends on the IES. The easier it is to substitute
consumption intertemporally, the less the anticipated depreciation.
However, to an econometrician who analyzes the historical data, there is another counter
effect. The econometrician anticipates that the economic agent at home will receive a lowerthan-expected endowment tomorrow, thus raising the marginal utility of consumption next
period. Through this effect, the econometrician expects the home currency to appreciate.
The magnitude of this appreciation depends on the agent’s risk aversion since risk aversion
determines the reduction in the marginal utility due to the lower-than-expected realized
consumption endowment (e.g., equation (16)). Consequently, the econometrician is likely to
observe a smaller depreciation or even an appreciation of the home currency. In particular,
when risk aversion is greater than the inverse of IES, the second effect dominates and high
domestic sentiment predicts an appreciation of home currency. Meanwhile, high domestic
sentiment implies high domestic interest rates. Taken together, the high interest rate
currency is expected to appreciate in the econometrician’s view, and hence a negative UIP
coefficient obtains.
3.3.3.
The Return on Foreign Exchange
Finally, the failure of the UIP condition also implies predictable returns in currency markets.
The return on foreign exchange is defined as the excess return of a domestic investor who
8
This result is intuitive. Misperception is the difference between the true expected growth rate and the
perceived expected growth rate. Hence, the persistence of the difference process should be smaller than that
of the original processes.
11
borrows at the domestic risk-free rate (in logs), rf,t , converts the funds into foreign currency,
∗
, and then converts his earnings back to
lends at the foreign risk-free rate (in logs), rf,t
domestic currency. That is, the return on foreign exchange is just the return on a carry
trade strategy. Hence, in logs, the return on foreign exchange, rtF X , is given by
∗
FX
− rf,t + et − et+1 .
= rf,t
rt+1
(23)
It follows from equations (18), (20), and (23) that under the econometrician’s view, the
expected returns on foreign exchange are given by9
FX
Et rt+1
= −γ · (st − s∗t ) .
(24)
Note that the true expected growth has no effect on returns on foreign exchange. The above
argument leads to the third testable prediction from our model.
Prediction 3 Returns on foreign exchange are negatively associated with the sentiment
differential (st − s∗t ).
Moreover, combining equations (18) and (24), one can show that the regression coefficient
of returns on foreign exchange on the forward premium is given by:
βRF X =
ψγ
var(xt −x∗t )
var(st −s∗t )
.
(25)
+1
Under the special case where the true expected growth is a constant or xt = x∗t , equations
(22) and (25) imply that βU IP,1 = (1 − ψγ) and βRF X = ψγ. The above result is stated as
another proposition below, which is consistent with the empirical evidence in Fama (1984).
var(x −x∗ )
Proposition 2 As long as var(stt −s∗t ) < γψ−1, the regression coefficient of returns on foreign
t
exchanges on the forward premium is greater than one. Under the special condition that the
true expected growth is constant or that xt and x∗t are equal, the regression coefficient of
returns on foreign exchanges on the forward premium is given by γψ and the UIP coefficient
is given by 1 − ψγ.
9
Notice that from the economic agents’ perspective, the expected return on foreign exchange is a constant
(zero) and independent of st . This follows from equation (21) and the definition of the return of foreign
exchange in equation (23).
12
In sum, our simple model can reproduce the forward premium puzzle. However, to
further test whether the proposed mechanism is in the data, one needs to test the refutable
Predictions 1-3. Section 4 provides evidence in support of Predictions 1-3.
3.4.
Additional Implications for Correlations and Volatilities
Although the main purpose of this paper is to provide a simple sentiment-based explanation
for the forward premium puzzle, we also explore the model’s ability to account for other
major puzzles in international markets. More important, these additional moment conditions
impose restrictions on the structural parameters of our model.
Apart from the first two moments of the stock market, three additional stylized facts are
considered here. First, it is well documented that consumption growth is smooth and poorly
correlated across countries, whereas asset market returns are volatile and highly correlated
across countries. Second, it is also well known that consumption growth differentials and
foreign exchange growth are poorly correlated (e.g., the Backus-Smith (1993) puzzle). Third,
more recently, Brandt et al. (2006) argue that high equity premia imply highly volatile
pricing kernels. In addition, consumption growth is poorly correlated across countries.
Within a power utility setting, this suggests a low correlation between pricing kernels across
countries. Together, it implies that the volatility of exchange rates should be much larger
than the 10 ∼ 15% value observed in the data (see equation (13)). In other words, pricing
kernels should be much more correlated than is suggested by consumption data, and hence
quantities and prices have dramatically different implications for the correlations between
the pricing kernels.10
To solve the exchange volatility puzzle a la Brandt et al (2006), Colacito and Croce (2011)
cleverly break the link between the correlation of consumption growth and the correlation
of the pricing kernel by introducing highly correlated long-run consumption. In their
model, pricing kernels are mostly driven by the small but persistent long-run consumption
components, which are perfectly correlated across countries. Thus, a high correlation
between pricing kernels can be reconciled with a low consumption correlation. Their model
also reproduces many other stylized facts in both macroeconomic and financial variables.
By incorporating investor misperception into Colacito and Croce (2011), our model can help
partially account for several additional stylized facts. These include the forward premium
10
For our model with investor misperception, the exact Hansen-Jagannathan bound actually does not
apply. Hence, in theory the pricing kernels in our model do not need to be as volatile as the maximum
observed Sharpe ratio. Thus, the price-implied international correlation between the pricing kernels does
not need to be as high as in Brandt et al. (2006). This already makes the issue less puzzling.
13
puzzle and the low correlation between consumption growth differentials and exchange rate
changes. In particular, equation (19) implies that exchange rate growth and consumption
growth differentials are not perfectly correlated in this model. This imperfect correlation
is mainly due to fluctuation in sentiment and can help partially resolve the Backus-Smith
(1993) puzzle.
3.5.
Calibration
This section provides quantitatively calibration of the model to account for several stylized
facts simultaneously.
3.5.1.
The Parameter Choice
Table 1 lists the main parameter values for our calibration. The model is simulated at a
monthly frequency. Most of the parameter values are borrowed directly from the long-run
risk literature. Following Bansal and Yaron (2004), the mean consumption growth µg is set
to be 0.0015, and the IES ψ is set to be 1.5. The risk aversion γ is set to be 3.5, much
smaller than that in Bansal and Yaron (2004), but closer to the standard value in the macro
literature. The stock market’s dividend is modeled as a levered claim on consumption.11 The
leverage parameter φ is set to be 3, the same as that in Bansal and Yaron (2004). Following
Colacito and Croce (2011), we set the correlation between short-run consumption innovation
ρ g , ∗g = 0.3, the correlation between short-run dividend innovation ρ (d , ∗d ) = −0.1, the
persistence ρx = 0.987, and the average dividend growth µd = 0.0007. Finally, the volatility
of short-run consumption growth innovation σg = 0.006, and the volatility of short-run
dividend growth innovation σd = 0.027. These values are chosen to match the consumption
and dividend dynamics. Long-run consumption volatility σx is set to be 0.00042 , such that
the volatility of the true expected growth (xt ) has roughly the same value as that in Bansal
and Yaron (2004). Finally, the time discount δ is set to be 0.999, so that the model produces
a low riskfree rate.
There are a few less standard parameters. The first one is the volatility of investor
sentiment. Most measures of investor sentiment are unit free, and the absolute level of
sentiment has no particular meaning. Thus, it is hard to gauge the magnitude of sentiment
11
That is, the dividend growth is specified as gd,t+1 = µd + φ (xt + st ) + σd ˆd,t+1 under the agent’s
perception and ˆd,t+1 is i.i.d. normal under agents’ perception. However, under the data-generating process,
gd,t+1 = µd + φxt + σd d,t+1 , where the data-generating process for d,t is i.i.d. normal. As mentioned earlier,
all the shocks are independent of each other.
14
volatility relative to consumption volatility.12 In the main calibration, the relative volatility
of sentiment to consumption growth is set to be less than 5%, so that the model is very close
to its rational benchmark. The second key parameter is the correlation of sentiment indices
across countries. This parameter is especially hard to estimate due to the unobservability
of investor sentiment. Using the proxies in Baker, Wurgler, and Yuan (2012), the average
correlation across countries is 35%. Using additional survey data on international sentiment
indices, this correlation is found to be even lower, around 25%. Given potentially large
amount of noises in the measured sentiment indices, the correlation calculated based on
sentiment proxies can significantly underestimate the true correlation. For example, even
if the true sentiment correlation is perfect and the variance ratio of noise to signal is one,
the measured sentiment correlation would be only 50%. Thus, in the calibration, three
different values, 25%, 50%, and 90%, are chosen to provide a sensitivity analysis. Finally,
the sentiment persistence parameter ρs is set to be 0.965, which is calculated based on Baker
and Wurgler’s sentiment index.
3.5.2.
The Main Calibration Results
Table 2 reports the calibration analysis based on 1, 200, 000 months of artificial data. Column
1 reports the results for a model with i.i.d. consumption growth and without investor
misperception. As expected, the model performance is poor. Column 2 introduces a
perfectly correlated long-run risk component as in Colacito and Croce (2011). The model
can successfully reproduce many stylized facts. For example, the equity premium is large
despite low exchange rate volatility, and the correlation of stock market returns is also large
despite a low fundamental correlation.
However, with perfect correlation between long-run consumption components across
countries and without investor sentiment, all of the exchange rate movements come from
the consumption growth differential (see equation (19)). Thus, the correlation between
consumption growth differentials and exchange rate growth is perfect in this setting.13 In
addition, in this model the forward premium is a constant. Thus, the model cannot address
the UIP puzzle. Column 3 relaxes this perfect correlation by setting ρx ,∗x = 0.98. In this
case, the model still implies a UIP coefficient of exactly one. However, with imperfectly
12
One could potentially use professional forecasts on GDP growth to obtain a rough estimation of
misperceived growth. However, one would need to estimate the true expected GDP growth before obtaining
the misperceived growth. There is no consensus on how to estimate true expected GDP growth.
13
Colacito and Croce (2012) show that by relaxing the assumption of complete home bias, this correlation
can be reduced. Instead, our study shows that introducing investor sentiment is an alternative mechanism
for reducing this correlation.
15
correlated expected growth, the correlation between consumption growth differentials and
exchange rate growth is reduced to 82%, still large but closer to the data.
In column 4, which is our benchmark calibration, investor misperception is introduced.
The international correlation of sentiment is set to be 50%. Compared with column 3, a
few important changes are noticed. First, due to this additional long-run risk component,
the equity premium is now larger and stock returns are also more volatile. Second, the
correlation between consumption growth differentials and exchange rate growth is reduced
even further to 60%, closer to the counterpart in the data. Third, the model produces a
larger volatility in foreign exchange growth, but it is still within the conventional range of
10˜15 percent per year. Fourth, the UIP coefficient is negative (−3.55). This coefficient is
a bit larger in absolute value than its empirical counterpart, which is typically in the range
of −2 to 0. However, the 95% confidence interval for the UIP coefficient is (−11.51, −0.61) ,
obtained from 1, 000 repeated small sample simulations, each with 30 years of simulated
data. Finally, the UIP regression has an R2 of 1%, which is largely consistent with the data.
Columns 5 and 6 provide a sensitivity analysis by choosing different values for the crosscountry sentiment correlation (i.e., 25% and 90%). The UIP coefficient remains negative.
Meanwhile, the model can simultaneously keep other asset pricing moments close to the
data. However, there are a few changes. On the one hand, when the cross-country
sentiment correlation is large, the magnitude of the UIP coefficient is smaller and foreign
exchange growth is smoother, but the correlation between consumption growth differentials
and exchange rate growth is too high. On the other hand, when sentiment correlation is
low, the correlation between consumption growth differentials and exchange rate growth is
reduced, but the exchange rate is a bit too volatile and the UIP coefficient is too negative.
In addition, the misperception-augmented model cannot fully account for the Backus-Smith
puzzle since the model-implied correlation is still larger than the empirical counterpart in
the data, which is typically around 30% or smaller.
Finally, column 7 reports the calibration for a pure misperception-based model with
i.i.d. consumption growth. This calibration helps us understand how far the sentiment
channel can go without the standard long-run risk channel. The variation of sentiment is
set to be about 30% of the variation of the realized consumption growth. Compared with
column 1, the model performance improves in many dimensions. For example, the equity
premium increases from a negative value to 2.52% per annum despite low risk aversion.
The mechanism of this model is similar in spirit to Colacito and Croce (2011) since the
highly correlated sentiment serves as highly correlated long-run risk. Moreover, due to the
misperception effect, the UIP coefficient is negative and the correlation between the observed
16
consumption growth differentials and the exchange rate growth is not perfect anymore. In
addition, differing with Colacito and Croce (2011), even if the sentiment process is volatile
and persistent, the observed consumption growth still behaves as an i.i.d. process, consistent
with the consumption data. Thus, sentiment can potentially produce large long-run risk in
the model while keeping realized consumption growth close to an i.i.d. process.14
Obviously, sentiment alone cannot quantitatively account for all of the patterns observed
in the data. Nonetheless, it is comforting to know that the same mechanism can potentially
help reproduce many stylized facts in international markets simultaneously.
4.
Empirical Tests
In this section, we first introduce notation for the predictive variables, then discuss the
empirical design and data used in the analysis and present summary statistics. The empirical
tests of the key model predictions (Predictions 1-3) follow.
4.1.
Definition of Variables and Empirical Design
denote the log nominal exchange rate (expressed as the foreign price of domestic
Let enom
t
currency) and pt denote the log consumer price index at home. Following Engel (2011),
the real exchange rate is calculated as et = enom
− p∗t + pt . Further define changes in real
t
exchange rates as
∆et = et − et−1 .
(26)
In the empirical test that follows, 3-month forward rates are used as in Engel and West
(2005). It is one of the most popular and easily available forward rates for all of the industrial
countries. The 3-month forward exchange rate Ft is observed at the end of month t for a
delivery at the end of month t + 3. Let ft = log(Ft ). The 3-month forward premium is
defined as
f pt = ft − enom
.
(27)
t
14
One challenge faced by the traditional long-run risks model is that the long-run risk component is hard
to identify using consumption data alone since actual consumption growth behaves like an i.i.d. process.
With high-quality survey data available in the future, one could potentially use survey data to identify
the perceived long-run risk, which is also highly priced under recursive preferences. Alternatively, recent
studies (e.g., Bansal et al. (2009) and Colacito and Croce (2011)) have started to use financial variables
to estimate the long-run consumption component. Asset prices can reflect the misperceived consumption
growth, whereas the realized consumption growth cannot. Thus, our model provides another justification
for looking at the financial data, rather than actual consumption data, to identify the long-run risk.
17
By covered interest parity, f pt equals the difference between the 3-month risk-free rate (from
the end of month t to the end of month t + 3) in the foreign country at the end of month t
and the 3-month domestic risk-free rate the end of month t. The forward premium in real
term is defined as the ex-post real interest rate differential. The return on foreign exchange
is therefore given by
FX
(28)
= ft − enom
rt+3
t+3 .
Finally, the true expected growth rate, xt , is measured as the trailing average of the past
3-year real consumption growth rates as in Bansal, Dittmar, and Lundblad (2005).
The empirical specifications are standard long-horizon predictive regressions (e.g., Fama
and French (1988) and Hodrick (1992)). All variables are observed monthly. The future
changes in spot rates and returns on foreign exchange rates are regressed onto investor
sentiment, st . Therefore, the dependent variable is either the log change in foreign exchange
FX
.
rate, et+h − et , with different horizons or the 3-month log return on foreign exchange, rt+3
The independent variables are investor sentiment, st , or other predictive variables. For
example, to examine the ability of investor sentiment to predict changes in spot rates, the
main regression is
et+h − et = αh + βs Xt + t,t+h ,
(29)
where the predictor Xt is either U.S. sentiment st or the sentiment differential, st − s∗t . The
error term t,t+h is an element of the time t + h, and is autocorrelated due to overlapping
observations when the forecast horizon, h, is greater than one. Newey-West (1987) standard
errors are used to adjust for this autocorrelation and potential heteroscedasticity. The tstatistics resulting from the adjustment test the null hypothesis that a given slope coefficient
equals zero.
4.2.
Data
The data set on foreign exchange includes monthly spot and 3-month forward exchange
rates between the U.S. dollar and currencies in 19 industrial countries: Australia,
Austria, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, the
Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, and the U.K.
The data sample is between 1973m1 and 2007m12 for most countries and may vary in some
countries due to data availability. Legacy exchange rates derived by Haver Analytics are
used for currencies of Euro-area member countries after they switched to the euro.
Monthly CPI data for these 19 countries and the United States over 1973m1 to 2007m12
18
are obtained from the IMF. Annual real consumption per capita from 1970 to 2007 is obtained
from the OECD. Quarterly real U.S. consumption per capita comes from the BEA. Spot
exchange rates come from the Federal Reserve, and 3-month forward exchange rates come
from the International Financial Statistics (IFS). The spot and forward exchange rates are
used to calculate interest rate differentials between the U.S. dollar and other currencies. For
several countries, some of the forward exchange rates are missing in the IFS. In these cases,
the forward rate is obtained from Bloomberg or is calculated from the 3-month interest rate
obtained from the G-10 data set. The choice of data source is dictated by data availability.
The U.S. investor sentiment is obtained from Baker and Wurgler (2006). Their index
serves as a good proxy for misperception since it is orthogonalized against a set of
fundamental variables. Moreover, it is a measure based on market participants rather than
the general population. The monthly sentiment index spans over 42 years, from July 1965
to December 2007. A detailed data description is provided in the Internet Appendix.
As to international sentiment, annual sentiment data for Canada, France, Germany,
Japan, and the U.K. are obtained from Baker, Wurgler, and Yuan (2012). These data span
from 1980 to 2005. Following Baker and Wurgler (2006), the annual sentiment indices are
transferred into a monthly frequency by assigning the same value for the corresponding
subsequent 12 months. In addition, sentiment data for Australia, Belgium, Denmark,
Italy, the Netherlands, New Zealand, and Switzerland are collected. These additional seven
countries are selected purely for data availability reasons. The sentiment data for the rest of
countries are either too short or unavailable. A more detailed data description is provided
in the Internet Appendix. In particular, these sentiment indices are based on surveys and
are related to expected business conditions. Thus, these indices are orthogonalized against
expected consumption growth before they are used in our regression analysis. Ideally,
sentiment indices should be constructed using the same approach as in Baker and Wurgler
(2006). However, data limitations make this approach hard to pursue. On the other hand, it
is plausible that the survey-based sentiment should be correlated with market participants’
perceptions. Thus, these survey-based measures could serve as imperfect proxies for investor
sentiment in our model.
4.3.
Summary Statistics
Because the absolute magnitude of sentiment indices typically has no meaning, all the
sentiment indices are standardized to mean zero and unit standard deviation. All 13
sentiment indices are plotted in Figure 1. As expected, all of indices are quite persistent
19
except Australia. They also tend to move together with a positive average correlation. For
example, most indices are at their highs during late 1990s.
To facilitate comparison with prior literature, we report summary statistics for exchange
rates in nominal terms. Table 3 reports summary statistics for the monthly change in spot
rate (∆enom
). The means and standard deviations of variables are given in percentages. The
t
volatility of the monthly change in spot rate is about 3% for most of the sample countries,
which is consistent with previous studies. The autocorrelation of the change in spot rate is
very low, which suggests that spot rates behave like a random walk.
Table 3 shows that the currencies of Austria, Denmark, Germany, Japan, the Netherlands
and Switzerland appreciated the most against the U.S. dollar from 1973m1 to 2007m12.
Precisely these countries (except Denmark) had lower interest rates than the United States
during this period (or equivalently, a negative forward premium). This implies that the
expectation hypothesis in currency markets seems to hold on average or in the long run. On
the other hand, Table 3 also indicates that the countries with a high average interest rate also
yield a slightly higher return on foreign exchange. This suggests that sorting on the interest
rate differential could yield a significant spread (e.g., Lustig and Verdelhan (2007)). The
volatility of returns on foreign exchange is much more volatile than the forward premium,
consistent with the finding in Fama (1984).
4.4.
Main Empirical Evidence
In the model, all the results pertain to real exchange rates and real interest rates. Thus,
the majority of our empirical analysis is in real terms. However, all of our empirical results
are quantitatively similar if nominal exchange rates are used. Those results are omitted
for brevity, and available upon request. In addition, because a majority of foreign exchange
transactions involve U.S. dollars, the U.S. dollar is used as the domestic currency throughout
the paper.
Sentiment is elusive to measure, and international data usually have poor quality. Thus,
the state variables of our model are measured with inevitable noise. Due to this issue, two sets
of empirical tests are presented. One is based on the domestic-foreign sentiment differential.
Since the model has direct implications on the relation between observables (such as returns
on foreign exchange) and sentiment differentials, this is a natural test. However, sentiment
indices are not available for all the 19 industrial foreign countries, and the international data
are not of the best quality. Thus, another set of tests using U.S. sentiment only are also
20
presented since U.S. sentiment is arguably better measured than its foreign counterparts.
With U.S. sentiment only, the second test can be applied to all 19 countries. However, the
second test is based on indirect predictions of the model regarding the relations between
foreign exchange growth (or returns on foreign exchange) and domestic sentiment. Thus, it
is intended to be used as suggestive and corroborating evidence only.
4.4.1.
Evidence Based on U.S. and Foreign Sentiment
First consider Prediction 1, which predicts that the forward premium in real terms (i.e., the
real interest rate differential) is negatively related to domestic-foreign sentiment differentials
and expected growth differentials. Panel A of Table 4 shows some weak support for the
sentiment channel. In addition, our model and standard rational models share a common
prediction regarding the negative relation between the true expected growth differential and
the forward premium. Panel A of Table 4 shows very weak evidence, probably due to the
intrinsic difficulty of measuring the true expected growth from international consumption
data. Nonetheless, the evidence on the model’s new and unique prediction related to
sentiment is more supportive.
Panel B tests the sharper prediction of the model on the relation of country-specific real
interest rates with sentiment and expected growth. The evidence is not supportive of either
our model or the traditional rational models. This weak result is actually in line with the
existing literature. The property of real interest rates is extremely hard to establish.15 There
are substantial measurement errors on both sides of the regression equation in Panel B. On
the other hand, the stronger result based on interest rate differentials in Panel A appears to
be consistent with the notion that measurement errors might be correlated across countries.
For example, if inflation is overstated due to quality adjustment bias or substitution bias
(see, e.g., Lebow and Rudd (2003)), then these measurement errors could vary over time and
be correlated across countries. Thus, focusing on differentials could potentially alleviate the
measurement error problem.
Next consider Prediction 2, which predicts that changes in real exchange rates are
positively related to sentiment differentials between domestic and foreign countries. The
left part of Panel A of Table 5 reports results from the monthly overlapping predictive
15
Dotsey, Lantz, and School (2003), for example, show the behavior of the real interest rate in US is exactly
the opposite of the stylized facts presented in Stock and Watson (1999). The result is typically sensitive to
the sample period and the choice of inflation indices. More recently, in the context of habit models, Wachter
(2006) argues for countercyclical real interest rates, whereas Verdelhan (2010) argues for the opposite. Thus,
this empirical failure might be due to the intrinsic difficulty in measuring real interest rates, sentiment, and
expected growth, especially in the international context.
21
regression of 3-month changes in real exchange rates on investor sentiment differentials.
The evidence is supportive. In particular, 8 out of 12 countries are statistically significant,
and one more is marginally significant. In addition, our model and rational models share
the same prediction that changes in real exchange rates are negatively related to expected
growth differentials between domestic and foreign countries. The right part of Panel A
tests this common prediction by including both investor sentiment differentials and expected
consumption growth differentials as regressors.
Consistent with Prediction 2, the evidence in the right part suggests that sentiment is
positively related to changes in exchange rates, whereas the true expected growth rate is
slightly negatively related to changes in exchange rates. However, the evidence does not
provide strong support on the negative relation between foreign exchange growth and true
expected growth differentials. Nonetheless, it does appear to support the model’s distinct
prediction on the relation between sentiment and foreign exchange. After all, to solve the
UIP puzzle, one could completely shut down the standard long-run risk channel as in column
7 of Table 2.
Panel B of Table 5 reports the results by regressing exchange rate changes on individual
sentiment indices and true expected growth, rather than differentials. The evidence is not as
strong as that in Panel A. However, it generally is supportive. For example, the coefficients
for domestic sentiment are significant for 9 out of 12 cases, and 2 of them are marginally
significant. The sign for foreign sentiment is generally consistent with the model prediction,
albeit less significant. The sign for domestic expected growth is also largely consistent with
the model prediction. Lastly, the evidence for foreign expected growth is quite weak.
Finally, consider Prediction 3, which predicts that returns on foreign exchange are
negatively related to sentiment differentials between domestic and foreign countries. Panel
A of Table 6 provides empirical support. In particular, the regression coefficients for all
12 countries have the correct sign, and more than half of them are statistically significant.
Although neither our model nor the standard rational model has implications on the relation
between returns on foreign exchange and true expected growth, these results are reported
in the right panel for completeness. There is some weak evidence that true expected growth
differentials positively predict returns on foreign exchange. This might be due to other forces
beyond misperception and the standard long-run risk channel. Lastly, Panel B of Table 6
reports the results of regressing returns on foreign exchange on individual sentiment indices
and true expected growth, rather than differentials. These results are again less supportive
than those based on differentials.
22
4.4.2.
Evidence Based on U.S. Sentiment Only
The evidence in Tables 5 and 6 provides preliminary support for the key channel in the
sentiment-based explanation. As mentioned earlier, because sentiment data are not available
for all 19 industrial countries, the analysis so far is based on 12 countries only. The second set
of tests based on U.S. sentiment alone are now presented as corroborating evidence. These
tests use the better-measured monthly sentiment data built by Baker and Wurgler (2006).
Table 7 reports the results. An indirect implication of Prediction 2 is that all else equal,
changes in real exchange rates are positively related to domestic sentiment and negatively
related to true expected growth. The first two panels of Table 7 provide support for this
indirect implication. All 19 countries show correct signs. About two thirds of the countries
are statistically significant. Similarly, an indirect implication of Prediction 3 is that all else
equal, returns on foreign exchange are negatively related to domestic sentiment. The last
panel of Table 7 provides support for this indirect implication as well. All 19 countries show
correct signs and most of them are statistically significant.
Previous studies have proposed alternative models to reproduce the failure of UIP.
Typically, these explanations work in the following manner. The forward premium is
approximately a linear function of some state variable (e.g., investor sentiment in our model
or surplus ratios in habit models), and the expected change in exchange rates and the
expected return on foreign exchange are also approximately linear functions of the same
state variable. If the coefficients on the forward premium and on the expected change in
exchange rates are opposite, then the model yields a negative coefficient in the UIP regression.
However, these models operate through different mechanisms and constitute fundamentally
different views of the sources and the pricing of risk. Thus, it is particularly important
to test the underlying model-implied relations between the state variable in the model and
the exchange rate growth or returns on foreign exchange. However, it is usually difficult to
implement such tests due to the unobservable state variable.
Our evidence suggests that misperception is an empirically relevant channel in explaining
the forward premium puzzle. Of course, other forces such as habit-formation (e.g., Verdelhan
(2010)) and long-run risk (e.g., Bansal and Shaliastovich (2011)) should also play significant
roles in the foreign exchange market. Our study simply provides preliminary evidence that
investor sentiment is a relevant force in the determination of exchange rates.16
16
It is well known that the forward premium strongly predicts returns on foreign exchange. The Internet
Appendix further documents that investor sentiment has significant additional power in forecasting returns
on foreign exchange. For many currencies, the R2 of a regression in which both the forward premium and
U.S. investor sentiment are included is more than double the R2 of a regression in which the forward premium
23
5.
Conclusion
A sentiment-based model of foreign exchange is proposed to account for the forward premium
puzzle. In addition, the model also helps reproduce many observed patterns in international
markets, such as the low correlation between consumption growth differentials and exchange
rate changes. This paper shows that sentiment induced long-run risk is potentially useful
in reproducing many key asset pricing moments. This observation could be applied to
resolve other puzzles in asset markets in future research. Moreover, this paper also provides
evidence in support of the key mechanism in the model. Our empirical evidence suggests
that investor sentiment has substantial predictive power for the currency market. Thus, this
study provides “out-of-sample” support for the important role of investor sentiment in the
stock market noted by previous studies (see, e.g., Baker and Wurgler (2006) and Stambaugh,
Yu, and Yuan (2012)).
The evidence of expectation-driven fluctuations in the currency market has policy
implications. Exchange rate movements can change the relative prices of imports and exports
when prices are fixed in the exporter’s currency in the short run. A major benefit of exchange
rate flexibility is that it facilitates adjustment of the relative price in response to a countryspecific real shock. However, as argued by Devereux and Engel (2007), if exchange rate
movements are mainly driven by changes in expectations rather than by current demand
and supply conditions in the goods market, exchange rate fluctuations may cause price
distortions in the goods market rather than facilitate price adjustments. This is especially
true if exchange rates are driven by irrational changes in expectations as documented in this
paper.
References
Alvarez, F., Atkeson, A., Kehoe P.J., 2009. Time-varying risk, interest rates, and exchange
rates in general equilibrium. Review of Economic Studies 76, 851–878.
Bacchetta, P., Wincoop, E., 2006. Incomplete information processing: a solution to the
forward discount puzzle. Working Paper. University of Lausanne.
Baker, M., Wurgler, J., 2006. Investor sentiment and the cross-section of stock returns.
Journal of Finance 61, 1645–1680.
Baker, M., Wurgler, J., Yuan, Y., 2012. Global, local, and contagious investor sentiment.
Journal of Financial Economics 104, 272–287.
is the only predictor. Therefore, investor sentiment contains information that is not captured by the forward
premium.
24
Backus, D., Foresi, S., Telmer, C.I., 2001. Affine term structure models and the forward
premium anomaly. Journal of Finance 56, 279–304.
Backus, D., Gregory, A., Telmer, C.I., 1993. Accounting for forward rates in markets for
foreign currency. Journal of Finance 48, 1887–1908.
Backus, D., Smith, G., 1993. Consumption and real exchange rates in dynamic exchange
economies with non-traded goods. Journal of International Economics 35, 297-316.
Bansal, R., Dittmar, R., Lundblad, C., 2005. Consumption, dividends, and the cross section
of equity returns. Journal of Finance 60, 1639–1672.
Bansal, R., Kiku, D., Yaron, A., 2009. Risks for the long run: Estimation and inference.
Working Paper. Duke University and University of Pennsylvania.
Bansal, R., Shaliastovich, I., 2011. A long-run risk-based explanation of predictability
puzzles in bond and currency markets. Working Paper. Duke University.
Bansal, R., Yaron, A., 2004. Risks for the long run: A potential resolution of asset pricing
puzzles. Journal of Finance 59, 1481–1509.
Beber, A., Breedon, F., Buraschi, A., 2010. Differences in beliefs and currency risk premiums.
Journal of Financial Economics 98, 415–438.
Bekaert, G., 1996. The time variation of risk and return in foreign exchange markets: a
general equilibrium perspective. Review of Financial Studies 9, 427–470.
Brandt, M., Cochrane, J., Santa-Clara, P., 2006. International risk sharing is better than you
think, or exchange rates are too smooth. Journal of Monetary Economics 53, 671–698.
Burnside, C., Han, H., Hirshleifer, D., Wang, Y.T., 2011. Investor overconfidence and the
forward premium puzzle. Review of Economic Studies 78, 523–558.
Burnside, C., Eichenbaum, M., Rebelo, S., 2007. Understanding the forward premium
puzzle: A microstructure approach. NBER Working Paper.
Campbell, J.Y., Cochrane, J.H., 1999. By force of habit: a consumption-based explanation
of aggregate stock market behavior. Journal of Political Economy 107, 205–251.
Colacito, R., Croce, M.M., 2011. Risks for the long run and the real exchange rate. Journal
of Political Economy 119, 153–181.
Colacito, R., Croce, M.M., 2012. International asset pricing with recursive preferences.
Working Paper. University of North Carolina.
De Long, B., Shleifer, A., Summers, L.H., Waldmann, R., 1990. Positive feedback investment
strategies and destabilizing rational speculation. Journal of Finance 45, 379–395.
Devereux, M., Engel, C., 2007. Expenditure switching versus real exchange rate stabilization:
competing objectives for exchange rate policy. Journal of Monetary Economics 54, 2346–
2374.
Dumas, B., Kurshev, A., Uppal, R., 2009. Equilibrium portfolio strategies in the presence
25
of sentiment risk and excess volatility. Journal of Finance 64, 579–629.
Engel, C., 2011. The real exchange rate, real interest rates, and the risk premium. Working
Paper. University of Wisconsin.
Engel, C., West, K., 2005. Exchange rates and fundamentals. Journal of Political Economy
113, 485–517.
Epstein, L.G., Zin, S., 1989. Substitution, risk aversion and the temporal behavior of
consumption and asset returns: a theoretical framework. Econometrica 57, 937–969.
Fama, E., 1984. Forward and spot exchange rates. Journal of Monetary Economics 14,
319–338.
Fama, E., French, K., 1988. Dividend yields and expected stock returns. Journal of Financial
Economics 22, 3–25.
Farhi, E., Gabaix, X., 2008. Rare disasters and exchange rates: a theory of the forward
premium puzzle. Working Paper. Harvard University.
Frankel, J., Froot, K., 1987. Using survey data to test standard propositions regarding
exchange rate expectations. American Economic Review 77, 133–153.
Frankel, J., Froot, K., 1990a. Chartists, fundamentalists, and trading in the foreign exchange
market. American Economic Review 80, 181–185.
Frankel, J., Froot, K., 1990b. Exchange rate forecasting techniques, survey data, and
implications for foreign exchange market. NBER Working Paper No.3470.
Froot, K., Frankel, J., 1989. Forward discount bias: is it an exchange risk premium?
Quarterly Journal of Economics 104, 139–161.
Gourinchas, P.O., Tornell, A., 2004. Exchange rate puzzles and distorted beliefs. Journal of
International Economics 64, 303–333.
Hansen, L., Hodrick, R., 1980. Forward exchange rates as optimal predictors of future spot
rates: an econometric analysis. Journal of Political Economy 88, 829–853.
Hollifield, B., Yaron, A., 2003. The foreign exchange risk premium: real and nominal factors.
Working Paper. Carnegie Mellon University and University of Pennsylvania.
Hodrick, R., 1992. Dividend yields and expected stock returns: alternative procedures for
inference and measurement. Review of Financial Studies 5, 357–386.
Kocherlakota, N., 1990. Disentangling the coefficient of relative risk aversion from the
elasticity of intertemporal substitution: an irrelevance result. Journal of Finance 45,
175–90.
Lebow, D., Rudd, J., 2003. Mearsurement error in the consumer rrice index: where do we
stand? Journal of Economic Literature 41, 159-201.
Lustig, H., Verdelhan, A., 2007. The cross section of foreign currency risk premia and
consumption growth risk. American Economic Review 97, 89–117.
26
Lustig, H., Roussanov, N., Verdelhan, A., 2011. Common risk factors in currency markets.
Review of Financial Studies 24, 3731–3777.
Lustig, H., Roussanov, N., Verdelhan, A., 2012. Counter-cyclical currency risk premia.
Journal of Financial Economics, forthcoming.
Mark, N., Wu, Y., 1998. Rethinking deviations from uncovered parity: the role of covariance
risk and noise. Economic Journal 108, 1686–1706.
Meredith, G., Chinn, M., 2004. Monetary policy and long-horizon uncovered interest rate
parity. IMF Staff Papers 51, 409–430.
Newey, W.K., West, K.D., 1987. A simple, positive semi-definite, heteroskedasticity and
autocorrelation consistent covariance matrix. Econometrica 55, 703–708.
Rietz, T., 1988. The equity risk premium: a solution. Journal of Monetary Economics 22,
117–131.
Scheinkman, J., Xiong, W., 2003. Overconfidence and speculative bubbles. Journal of
Political Economy 111, 1183–1219.
Stambaugh, R., Yu, J., Yuan, Y., 2012. The short of it: investor sentiment and anomalies.
Journal of Financial Economics 104, 288–302.
Verdelhan, A., 2010. A habit-based explanation of the exchange rate risk premium. Journal
of Finance 65, 123–145.
Xiong, W., Yan, H., 2010. Heterogeneous expectations and bond markets. Review of
Financial Studies 23, 1433–1466.
27
Figure 1: Investor Sentiment Indices
This figure plots investor sentiment indices for the U.S. and 12 international countries. The monthly US
investor sentiment index is from 1965m07 to 2007m12, obtained from Baker and Wurgler (2006). The
annual sentiment indices from 1980 to 2005 for Canada, France, Germany, Japan, and U.K. are obtained
from Baker, Wurgler, and Yuan (2012). The Australia sentiment is from 1974m10 to 2007m12, obtained
from Westpac Banking Corporation. Belgium sentiment is from 1988m6-2007m12, obtained from National
Bank of Belgium. Denmark sentiment is from 1974m10-2007m12, obtained from Statistics Denmark. Italy
sentiment is from 1982m1-2007m12, obtained from ISAE/ISTAT. New Zealand sentiment is from 1988m62007m12, obtained from WESTPAC Banking Corporation. Netherlands sentiment is from 1987m1-2007m12,
obtained from Dutch Statistics Office. Switzerland sentiment is from 1972m10-2007m12, obtained from Swiss
National Bank.
US
Australia
Belgium
4
4
4
2
2
2
0
0
0
-2
-2
-2
-4
-4
1970 1980 1990 2000
1980
Canada
4
1990
2000
1990 1995 2000 2005
Denmark
France
4
2
2
2
-4
0
0
0
-2
-2
-2
1980
1990
2000
-4
Germany
4
1980
1990
2000
1990
Italy
4
2000
Japan
4
2
2
-4
1980
2
0
0
0
-2
-2
1980
1990
2000
-4
Netherlands
1985 1990 1995 2000 2005
-2
1980
2
2
0
0
0
-2
-2
-2
1990 1995 2000 2005
-4
1990 1995 2000 2005
UK
4
2
0
-2
1980
1990
2000
28
2000
Switzerland
2
-4
1990
New Zealand
-4
1980
1990
2000
Table 1: Parameter Choices
This table reports the parameter choices for the sentiment-augmented long-run risk models. Most of the
parameter values are directly taken from Bansal and Yaron (2004) and Colacito and Croce (2011). As in
Bansal and Yaron (2004), these parameter values are at a monthly frequency.
Statistics
Variable
Risk aversion coefficient
γ
IES coefficient
ψ
Subjective discount factor
δ
Mean consumption growth (%)
µg
Mean dividend growth (%)
µd
Standard deviation of short-run consumption innovations (%)
σg
Standard deviation of long-run consumption innovations (%)
σx
Persistence coefficient in true expected growth
ρx
Standard deviation of dividend innovations
σd
Dividend exposure to long-run consumption (leverage)
φ
Cross-country correlation of consumption innovations
ρg ,∗g
Cross-country correlation of dividend innovations
ρd ,∗d
Persistence coefficient in the sentiment process
ρs
29
Value
3.5
1.5
0.999
0.15
0.07
0.6
0.042
0.987
0.027
3
0.3
-0.1
0.965
Table 2: Calibration
This table reports the key moments (in annual terms) of international markets for our symmetric calibration.
The model is calibrated at a monthly frequency. The dividend growth is specified as gd,t+1 = µd + φ(xt +
st ) + σd ˆd,t+1 under agents’ perception and ˆd,t is i.i.d. normal under agents’ perception. However, under
the data-generating process, gd,t+1 = µd + φxt + σd d,t+1 , where the data-generating process for d,t is i.i.d.
normal. As shown in Table 1, the following parameters are fixed through all calibrations: µg = 0.0015,
µd = 0.0007, σg = 0.006, σd = 0.027, ρx = 0.987, ρs = 0.965, δ = 0.999, ρ(g , ∗g ) = 0.3, ρ(d , ∗d ) = −0.1,
and all the unspecified correlations are set to zero.
variable
ρ(s , ∗s )
ρ(x , ∗x )
σs (basis point)
σx (basis point)
(1)
0.90
1.00
0
0
(2)
0.90
1.00
0
4.2
Corr. b/t log pricing kernel: ρ(m, m∗ )
0.30 0.96
Volatility of FX growth σ(∆et+1 )
0.09 0.09
AC(1) of FX growth ρ(∆et+1 , ∆et )
0.00 0.00
Mean excess stock return (%)
-0.46 3.89
Volatility of excess stock return (%)
9.36 22.76
Average interest rate E(rf ) (%)
2.29 1.80
Volatility of interest rate σ(rf ) (%)
0.00 0.60
Interest rate correlation ρ(rf , rf∗ )
1.00 1.00
Corr. b/t excess market returns
-0.10 0.81
∗
Corr. b/t consumption growth ρ(g, g ) 0.30 0.41
Volatility of consumption growth
2.08 2.27
∗
Corr. b/t dividend growth ρ(gd , gd )
-0.10 -0.01
Volatility of dividend growth
9.36 9.74
Volatility of pricing kernel
0.07 0.31
Ratio of var(s)/var(g)
0.00 0.00
Ratio of var(x)/var(g)
0.00 0.16
∗
ρ(gt+1
− gt+1 , ∆et+1 )
UIP coefficient
R2
1.00
NA
NA
30
1.00
NA
NA
(3)
0.90
0.98
0
4.2
(4)
0.50
0.98
3.6
4.2
(5)
0.25
0.98
3.6
4.2
(6)
0.90
0.98
3.6
4.2
(7)
0.90
1.00
9.0
0
0.94 0.90 0.88 0.94 0.85
0.10 0.14 0.16 0.11 0.14
0.00 -0.01 -0.01 0.00 -0.01
3.89 4.32 4.32 4.32 2.52
22.76 23.69 23.69 23.69 21.62
1.80 1.75 1.75 1.75 1.93
0.60 0.68 0.68 0.68 0.79
0.98 0.88 0.82 0.96 0.90
0.80 0.76 0.74 0.80 0.71
0.41 0.41 0.41 0.41 0.30
2.27 2.27 2.27 2.27 2.08
-0.02 -0.02 -0.02 -0.02 -0.10
9.74 9.74 9.74 9.74 9.36
0.31 0.32 0.32 0.32 0.26
0.00 0.04 0.04 0.04 0.33
0.16 0.16 0.16 0.16 0.00
0.82
1.00
0.00
0.60
-3.55
0.01
0.54
-3.76
0.01
0.76
-1.98
0.00
0.61
-4.20
0.01
31
Country
Australia
Austria
Belgium
Canada
Denmark
Finland
France
Germany
Ireland
Italy
Japan
Netherlands
New Zealand
Norway
Portugal
Spain
Sweden
Switzerland
U.K.
E(∆enom
) σ(∆enom
)
t
t
0.09
2.95
-0.21
3.19
-0.11
3.19
-0.00
1.56
-0.07
3.12
-0.01
2.95
-0.03
3.10
-0.20
3.23
0.06
2.99
0.20
3.00
-0.24
3.31
-0.18
3.20
0.11
3.23
-0.04
2.91
0.39
3.15
0.14
3.03
0.08
3.07
-0.28
3.52
0.04
2.95
ρ(∆enom
) E(f pt ) σ(f pt ) ρ(f pt ) E(rtF X ) σ(rtF X ) ρ(rtF X )
t
-0.00
0.90
1.04
0.69
0.52
5.29
0.69
0.04
-0.26
0.84
0.85
0.32
6.17
0.71
0.05
0.19
0.74
0.80
0.36
6.17
0.71
0.02
0.19
1.50
0.91
0.20
3.13
0.73
0.03
0.44
0.90
0.79
0.61
5.72
0.71
0.05
0.90
1.34
0.89
0.67
5.57
0.71
0.03
0.24
0.69
0.80
0.07
5.86
0.71
0.05
-0.49
0.76
0.93
0.06
6.20
0.70
0.04
0.41
0.93
0.82
0.92
5.49
0.71
0.09
1.47
1.36
0.51
0.74
5.81
0.74
0.04
-0.72
1.06
0.80
-0.07
6.17
0.72
0.03
-0.25
0.96
0.49
0.09
6.11
0.69
0.05
1.03
0.99
0.96
0.58
5.99
0.75
0.02
0.60
1.48
0.34
0.69
5.34
0.67
0.08
1.67
1.02
0.99
1.50
5.76
0.72
0.10
1.29
1.92
0.25
0.42
6.17
0.79
0.09
0.44
1.15
0.64
0.20
5.69
0.72
0.05
-0.82
0.87
0.81
-0.05
6.54
0.71
0.07
0.55
0.78
0.75
0.42
5.50
0.70
), the 3-month
This table reports mean (E), standard deviation (σ), and autocorrelation (ρ) for monthly changes in nominal exchange rates (∆enom
t
forward premium (f pt ) (e.g., the difference between the foreign interest rate and the U.S. rate), and 3-month (overlapping) returns on foreign exchange
(rtF X ) (e.g., borrow from the U.S. dollar and invest in the foreign bond). All variables are observed at a monthly frequency. The sample periods vary
across countries based on data availability. The Internet Appendix provides a detailed description on the data. The means and standard deviations
are on a percentage basis.
Table 3: Summary Statistics
Table 4: The Relation Between Interest Rates and Sentiment
Panel A reports results from the monthly overlapping regression of 3-month (ex-post) real interest rate
∗
differentials (rf,t
− rf,t ) on sentiment differentials (st − s∗T ) and true expected growth differentials (xt − x∗t ).
Panel B reports results from the monthly overlapping regression of 3-month (ex-post) real interest rate on
sentiment (st ) and true expected growth (xt ). A constant is always included in the regressions, but not
reported. The sample periods vary across countries based on data availability. The Internet Appendix
provides a detailed description of the data. β is the coefficient of the corresponding regression. Similarly,
t is the t-statistic, and R2 is the adjusted r-squared for the predictive regression. Newey-West standard
errors are used to adjust for heteroscedasticity and autocorrelation in error terms. The coefficient β for the
sentiment is in percentage.
Australia
Belgium
Canada
Denmark
France
Germany
Italy
Japan
Netherlands
New Zealand
Switzerland
U.K.
Australia
Belgium
Canada
Denmark
France
Germany
Italy
Japan
Netherlands
New Zealand
Switzerland
U.K.
USA
∗
Panel A: Regress rf,t
− rf,t on st − s∗t and xt − x∗t
β(st − s∗t ) t(st − s∗t ) β(xt − x∗t ) t(xt − x∗t )
R2
-0.16
-1.92
-0.06
-0.77 0.04
0.03
0.49
-0.39
-5.82 0.38
0.05
0.32
0.40
1.89 0.06
-0.04
-0.45
-0.01
-0.08 -0.00
-0.14
-0.60
0.15
2.31 0.11
-0.25
-2.04
0.00
0.52 0.03
-0.11
-0.81
0.00
0.06 0.01
-0.15
-1.62
0.01
0.12 0.02
0.03
0.24
-0.16
-1.78 0.05
-0.10
-2.13
-0.24
-6.77 0.20
-0.12
-1.11
-0.04
-0.65 0.03
-0.15
-1.66
-0.08
-0.68 0.03
Panel
β(st )
-0.17
0.22
-0.11
-1.33
-0.19
-0.12
1.21
0.04
-0.32
-0.61
-0.20
-1.14
1.71
B: Regress rf,t on st and xt
t(st )
β(xt )
t(xt )
-1.00
-0.02
-0.03
1.37
1.67
12.89
-0.36
-1.50
-3.91
-4.05
-0.65
-2.63
-0.31
-0.14
-0.71
-0.56
-0.50
-1.79
2.79
-0.63
-7.43
0.17
-0.33
-1.34
-0.88
0.88
3.65
-2.26
1.41
6.66
-0.65
-0.19
-1.21
-3.81
0.91
2.02
5.63
0.19
0.53
32
R2
0.00
0.47
0.16
0.18
0.02
0.02
0.35
0.01
0.11
0.46
0.01
0.08
0.16
33
Australia
Belgium
Canada
Denmark
France
Germany
Italy
Japan
Netherlands
New Zealand
Switzerland
UK
Expected Sign
0.40
0.89
0.36
0.84
2.06
1.01
0.86
0.31
-0.13
0.73
1.17
0.85
+
2.05
2.01
1.67
3.40
3.28
0.81
2.26
0.69
-0.30
2.32
2.76
2.18
+
0.01
0.03
0.01
0.05
0.10
0.01
0.04
0.00
0.00
0.05
0.04
0.03
0.38
1.02
0.43
0.88
3.29
1.27
0.94
0.35
-0.14
0.96
1.24
1.01
+
1.97
2.41
1.78
3.29
4.59
1.07
2.39
0.76
-0.32
3.21
2.84
2.29
+
17.64
-50.71
29.61
-10.38
-73.13
-110.06
-15.42
29.21
-30.25
-88.47
-20.24
-85.20
-
0.62
-1.10
1.43
-0.39
-4.31
-2.09
-0.80
0.69
-0.89
-3.16
-0.76
-1.62
-
Panel A: Regress Changes in Exchange Rates on on (st − s∗t ) and (xt − x∗t )
et+3 − et on st − s∗t and xt − x∗t
et+3 − et on st − s∗t
2
∗
∗
∗
R
β(st − st ) t(st − s∗t ) β(xt − x∗t ) t(xt − x∗t )
β(st − st ) t(st − st )
0.01
0.04
0.02
0.05
0.17
0.04
0.04
0.00
0.00
0.10
0.04
0.05
R2
The left part of Panel A reports results from the monthly overlapping predictive regression of 3-month changes in real exchange rates on sentiment
differentials between domestic and foreign countries. The right part of Panel A reports results from the monthly overlapping predictive regression of
3-month changes in real exchange rates on sentiment differentials and true expected growth differentials. Panel B reports results from the monthly
overlapping predictive regression of 3-month changes in real exchange rates on individual sentiment and true expected growth. A constant is always
included in the regressions, but not reported. The sample periods vary across countries based on data availability. The Internet Appendix provides a
detailed description of the data. β is the coefficient of the corresponding regression. Similarly, t is the t-statistic, and R2 is the adjusted r-squared for
the predictive regression. Newey-West standard errors are used to adjust for heteroscedasticity and autocorrelation in error terms. The coefficient β
for the sentiment differential is in percentage.
Table 5: Forecasting Exchange Rate Changes with both US and Foreign Sentiment
34
Australia
Belgium
Canada
Denmark
France
Germany
Italy
Japan
Netherlands
New Zealand
Switzerland
U.K.
Expected Sign
Panel B: Regress Exchange Rate Changes
β(xt ) t(xt )
β(st ) t(st ) β(s∗t ) t(s∗t )
0.74 2.30 -0.07 -0.27 -29.85 -0.65
1.22 2.29 -0.95 -1.99 -109.79 -1.74
0.77 2.72 -0.16 -0.71
2.36 0.08
0.73 1.72 -0.90 -2.33 -114.28 -2.48
2.33 4.23 -2.11 -2.77 -78.40 -1.35
1.14 1.54 0.37 0.55 -85.48 -1.52
0.99 2.20 -0.95 -1.74 -113.28 -1.85
0.96 2.04 0.23 0.41
24.66 0.42
0.36 0.66 0.55 1.10 -109.15 -1.85
1.27 2.12 -0.68 -1.56 -126.28 -1.76
1.49 3.08 -0.79 -1.49 -144.14 -2.61
1.39 2.76 -0.55 -1.07 -88.84 -1.48
+
+
-
on st , s∗t ,
β(x∗t )
-40.91
10.69
-61.01
-30.45
49.45
58.01
4.85
-42.63
3.67
92.10
-3.23
45.62
+
xt , and x∗t
t(x∗t )
R2
-1.17 0.02
0.18 0.04
-2.43 0.06
-0.90 0.07
2.33 0.12
0.81 0.06
0.26 0.05
-0.88 0.02
0.09 0.02
1.89 0.10
-0.11 0.07
0.54 0.05
+
35
Australia
Belgium
Canada
Denmark
France
Germany
Italy
Japan
Netherlands
New Zealand
Switzerland
UK
Expected Sign
-0.57
-0.89
-0.40
-0.88
-1.95
-1.25
-1.13
-0.46
-0.38
-0.87
-1.29
-1.02
-
-2.42 0.02
-1.55 0.05
-1.57 0.01
-3.29 0.05
-2.80 0.08
-0.98 0.01
-2.51 0.07
-0.97 0.00
-0.52 -0.00
-2.78 0.07
-2.82 0.05
-2.45 0.04
-
-0.55
-0.92
-0.38
-0.92
-3.43
-1.52
-1.65
-0.50
-0.39
-1.05
-1.33
-1.16
-
-2.35
-1.66
-1.38
-3.22
-4.35
-1.25
-2.93
-1.03
-0.54
-3.41
-2.85
-2.50
-
-21.84
8.12
10.51
10.15
88.06
114.18
54.64
-28.45
3.00
67.95
13.12
77.15
NA
R2
-0.70 0.02
0.16 0.04
0.49 0.01
0.37 0.05
4.72 0.18
2.12 0.05
1.70 0.10
-0.64 0.00
0.06 -0.01
2.46 0.10
0.46 0.04
1.44 0.05
NA
Panel A: Regress Returns on Foreign Exchange on (st − s∗t ) and (xt − x∗t )
FX
FX
on st − s∗t and xt − x∗t
rt+3
on st − s∗t
rt+3
2
∗
∗
∗
R
β(st − st ) t(st − s∗t ) β(xt − x∗t ) t(xt − x∗t )
β(st − st ) t(st − st )
The left part of Panel A reports results from the monthly overlapping predictive regression of 3-month returns on foreign exchange on sentiment
differentials between domestic and foreign countries. The right part of Panel A reports results from the monthly overlapping predictive regression
of 3-month returns on foreign exchange on sentiment differentials and true expected growth differentials. Panel B reports results from the monthly
overlapping predictive regression of 3-month returns on foreign exchange on individual sentiment and true expected growth. A constant is always
included in the regressions, but not reported. The sample periods vary across countries based on data availability. The Internet Appendix provides a
detailed description of the data. β is the coefficient of the corresponding regression. Similarly, t is the t-statistic, and R2 is the adjusted r-squared for
the predictive regression. Newey-West standard errors are used to adjust for heteroscedasticity and autocorrelation in error terms. The coefficient β
for the sentiment differential is in percentage.
Table 6: Forecasting Returns on Foreign Exchange with both US and Foreign Sentiment
36
Australia
Belgium
Canada
Denmark
France
Germany
Italy
Japan
Netherlands
New Zealand
Switzerland
U.K.
Expected Sign
Panel
β(st )
-1.02
0.13
-0.88
-1.30
-4.28
-1.97
-1.50
-1.20
-0.44
-1.43
-1.84
-1.71
-
B: Regress Returns on Foreign
t(st ) β(s∗t ) t(s∗t ) β(xt )
-2.44 0.14 0.52 40.62
0.09 1.37 1.89 104.15
-3.15 -0.13 -0.48 -22.17
-3.03 0.52 1.31 118.14
-4.08 3.09 3.66 75.61
-1.64 -0.45 -0.33 86.87
-1.54 1.69 2.49 135.76
-2.52 -0.22 -0.38 -46.37
-0.37 -0.20 -0.32 125.55
-2.40 0.72 1.74 109.80
-3.57 0.73 1.20 143.76
-3.24 0.52 1.01 77.36
+
+
NA
Exchange on st ,
t(xt )
β(x∗t )
0.79
52.73
0.96
99.93
-0.65 -29.03
2.48
11.81
0.95 -76.21
1.11 -135.59
1.54 -39.26
-0.76
33.39
1.28
28.86
1.51 -74.36
2.50 -14.90
1.29 -28.38
NA
NA
s∗t , xt , and x∗t
t(x∗t )
R2
1.40
0.04
1.17
0.06
-0.94
0.09
0.34
0.08
-3.16
0.19
-1.66
0.08
-1.03
0.10
0.65
0.03
0.61
0.01
-1.55
0.10
-0.41
0.08
-0.34
0.07
NA
37
Australia
Austria
Belgium
Canada
Denmark
Finland
France
Germany
Ireland
Italy
Japan
Netherlands
New Zealand
Norway
Portugal
Spain
Sweden
Switzerland
U.K.
Expected Sign
(et+3 − et ) on st
β(st ) t(st ) R2
0.75 2.07 0.02
0.97 2.31 0.02
1.16 2.62 0.03
0.08 0.37 0.00
0.98 2.31 0.02
0.64 1.78 0.01
0.92 2.07 0.02
1.01 2.33 0.02
1.13 2.00 0.02
0.60 1.30 0.01
1.18 2.76 0.03
1.02 2.34 0.02
0.82 1.49 0.01
0.51 1.31 0.01
0.49 1.03 0.00
1.07 2.36 0.03
1.09 2.62 0.03
0.93 1.79 0.02
0.88 1.68 0.02
+
+
(et+3 − et ) on st
β(st ) t(st ) β(xt )
0.77 2.16 -0.89
1.09 2.53 -3.92
1.29 2.89 -4.46
0.05 0.24 0.96
1.10 2.52 -4.23
0.75 1.96 -3.88
1.07 2.62 -4.91
1.14 2.59 -4.30
1.35 2.47 -4.99
0.75 1.75 -4.82
1.22 2.80 -1.21
1.16 2.62 -4.67
0.98 1.83 -5.32
0.60 1.51 -2.92
0.56 1.20 -2.46
1.19 2.53 -3.81
1.17 2.65 -2.78
1.11 2.14 -5.84
1.00 2.12 -4.02
+
+
and xt
t(xt )
-0.47
-2.04
-2.26
0.95
-2.15
-2.05
-2.49
-2.17
-2.33
-2.36
-0.58
-2.38
-2.48
-1.63
-1.06
-1.70
-1.40
-2.64
-1.89
R
0.01
0.04
0.05
0.00
0.04
0.03
0.05
0.04
0.05
0.03
0.03
0.05
0.04
0.02
0.01
0.04
0.04
0.05
0.04
2
FX
) on
(rt+3
β(st ) t(st )
-1.93 -3.98
-1.47 -2.75
-1.42 -2.64
-0.42 -1.72
-1.39 -3.14
-1.28 -2.96
-1.15 -1.95
-1.31 -2.51
-1.09 -1.27
-1.55 -1.80
-1.39 -3.14
-1.48 -2.75
-1.38 -2.27
-0.98 -2.32
-2.71 -0.91
-1.16 -1.57
-1.40 -2.99
-1.38 -2.45
-0.91 -1.51
-
R2
0.07
0.05
0.05
0.01
0.05
0.04
0.03
0.04
0.01
0.03
0.04
0.05
0.04
0.03
0.02
0.03
0.05
0.04
0.02
st
This table reports results from the monthly overlapping predictive regression of 3-month changes in real exchange rates and 3-month returns on
foreign exchange on investor sentiment, st , respectively. The sample is monthly from 1973m1 to 2007m12 during which investor sentiment data
are available. β is the coefficient of the corresponding regression. Similarly, t is the t-statistic, and R2 is the adjusted r-squared for the predictive
regression. Newey-West standard errors are used to adjust for heteroscedasticity and autocorrelation in error terms. The coefficient β for sentiment
is in percentage.
Table 7: Forecasting Exchange Rate Changes and Returns on Foreign Exchange by Sentiment