Consistent Expectations and the Behavior of Exchange Rates
Kevin J. Lansingy
FRB San Francisco and Norges Bank
Jun Maz
University of Alabama
September 25, 2013
Abstract
This paper introduces a form of boundedly-rational expectations into a standard assetpricing model of the exchange rate, where cross-country interest rate di¤erentials are governed by Taylor-type rules. We postulate that agents augment a simple random walk
forecast with news about fundamentals. We solve for a “consistent expectations equilibrium,” in which the coe¢ cient on fundamental news in the agent’s forecast rule is pinned
down using the observed covariance between exchange rate changes and fundamental news.
This learnable equilibrium delivers the result that the forecast errors observed by the agent
are close to white noise, making it di¢ cult for the agent to detect any misspeci…cation of
the subjective forecast rule. We show that the model generates volatility and persistence
that is broadly similar to that observed in monthly bilateral exchange rate data (relative
to the U.S.) for Canada, Japan, and the United Kingdom over the period 1974 to 2012.
Moreover, we show that regressions performed on model-generated data can deliver the
well-documented forward premium anomaly whereby a high interest rate currency tends
to appreciate, thus violating the uncovered interest rate parity condition.
Keywords: Exchange rates, Uncovered interest rate parity, Forward premium anomaly,
Expectations, Excess volatility.
JEL Classi…cation: D83, D84, E44, F31, G17.
For helpful discussions, comments and suggestions, we would like to thank Richard Baillie, George Evans,
Cars Hommes, Reuven Glick, and session participants at the 2012 Annual Symposium of the Society for
Nonlinear Dynamics and Econometrics, the 2013 Meeting of the Society for Computational Economics, the
2013 Norges Bank Spring Research Retreat at Venastul, and the 2013 conference on “Expectations in Dynamic
Macroeconomic Models,” hosted by the Federal Reserve Bank of San Francisco.
y
Research Department, Federal Reserve Bank of San Francisco, P.O. Box 7702, San Francisco, CA 941207702, (415) 974-2393, FAX: (415) 977-4031, email: [email protected]
z
Department of Economics, Finance and Legal Studies, University of Alabama, Box 870224, 200 Alston
Hall, Tuscaloosa, AL 35487, (205) 348-8985, email: [email protected]
1
Introduction
1.1
Overview
E¤orts to explain movements in exchange rates as a rational response to economic fundamentals have, for the most part, met with little success. Thirty years ago, Meese and Rogo¤
(1983) demonstrated that none of the usual economic variables (money supplies, real incomes,
trade balances, in‡ation rates, interest rates, etc.) could help forecast future exchange rates
better than a simple random walk forecast. With three decades of additional data in hand,
researchers continue to con…rm the Meese-Rogo¤ results.1 While some tenuous links between
fundamentals and exchange rates have been detected, the empirical relationships are generally
unstable (Bacchetta and Van Wincoop 2004, 2013), hold only at 5 to 10 year horizons (Chinn
2006), or operate in the wrong direction, i.e., exchange rates may help predict fundamentals
but not vice versa (Engel and West 2005).2 The failure of fundamental variables to improve
forecasts of future exchange rates has been called the “exchange rate disconnect puzzle.”
Another puzzle relates to the “excess volatility” of exchange rates. Like stock prices, exchange rates appear to move too much when compared to changes in observable fundamentals.3
Engel and West (2006) show that a standard rational expectations model can match the observed persistence of real-world exchange rates, but it substantially underpredicts the observed
volatility. West (1987) makes the point that exchange rate volatility can be reconciled with
fundamental exchange rate models if one allows for “regression disturbances,” i.e., exogenous
shocks that can be interpreted as capturing shifts in unobserved fundamentals. Similarly,
Balke, Ma and Wohar (2013) …nd that unobserved factors (labeled “money demand shifters”)
account for most of the volatility in the U.K./U.S. exchange rate using data that extends
back more than a century. An innovative study by Bartolini and Gioginianni (2001) seeks to
account for the in‡uence of unobserved fundamentals using survey data on exchange rate expectations. The study …nds “broad evidence of excess volatility with respect to the predictions
of the canonical asset-pricing model of the exchange rate with rational expectations.”
A third exchange rate puzzle is the so-called “forward-premium anomaly.” In theory, a
currency traded at a premium in the forward market predicts a subsequent appreciation of
that currency in the spot market. In practice, there is a close empirical link between the
observed forward premium and cross-country interest rate di¤erentials, consistent with the
covered interest parity condition. Hence, theory predicts that a low interest rate currency
should, on average, appreciate relative to a high interest rate currency because the subsequent
appreciation compensates investors for the opportunity cost of holding a low interest rate
bond. The theoretical slope coe¢ cient from a regression of the observed exchange rate change
on the prior interest rate di¤erential is exactly equal to one. This prediction of the theory,
known as the uncovered interest parity (UIP) condition, is grossly violated in the data. As
shown by Fama (1984), regressions of the observed exchange rate change on the prior interest
1
For surveys of this vast literature, see Rossi (2013), Cheung, Chinn and Pascual (2005), and Sarno (2005).
Speci…cally, Engel and West (2005) …nd Granger causality running from exchange rates to fundamentals.
Recently, however, Ko and Ogaki (2013) demonstrate that this result is not robust after correcting for the
small-sample size.
3
Early studies along these lines include Huang (1981) and Wadhwani (1987).
2
1
rate di¤erential yield estimated slope coe¢ cients that are typically negative and signi…cantly
di¤erent from one. Like the other two puzzles, the forward-premium anomaly has stood the
test of time.4
The wrong sign of the slope coe¢ cient in UIP regressions can be reconciled with noarbitrage and rational expectations if investors demand a particular type of risk premium to
compensate for holding an uncovered currency position. By failing to account for the risk
premium, the standard UIP regression (which assumes risk-neutral investors) may deliver
a biased estimate of the slope coe¢ cient.5 Recently, Lustig and Verdelhan (2007) provide
some evidence that carry-trade pro…ts (excess returns from betting against UIP) may re‡ect
a compensation for risk that stems from a negative correlation between the carry-trade pro…ts
and investors’consumption-based marginal utility. However, Burnside (2011) points out that
their empirical model is subject to weak identi…cation such that there is no concrete evidence
for the postulated connection between carry-trade pro…ts and fundamental risk. Moreover,
Burnside, et al. (2012) show that there is no statistically signi…cant covariance between carrytrade pro…ts and conventional risk factors. Verdelhan (2010) develops a rational model with
time-varying risk premiums along the lines of Campbell and Cochrane (1999). The model
implies that rational domestic investors will expect low future returns on risky foreign bonds
in good times (due to an expected appreciation of the domestic currency) when risk premia
are low and domestic interest rates are high relative to foreign interest rates. Hence, the model
delivers the prediction that the domestic currency will appreciate, on average, when domestic
interest rates are high— thus violating the UIP condition, as in the data. Unfortunately,
the idea that investors expect low future returns on risky assets in good times is strongly
contradicted by a wide variety of survey evidence from both stock and real estate markets.
The survey evidence shows that real-world investors typically expect high future returns on
risky assets in good times, not low future returns.6 Overall, the evidence suggests that rational
risk premiums are not a convincing explanation for the empirical failure of UIP.
This paper develops a model that can account for numerous quantitative features of realworld exchange rates, including the anomalies described above. The key aspect of our approach
is the way in which agents’expectations are modeled. Starting from a standard asset-pricing
model of the exchange rate, we postulate that agents augment a simple random walk forecast
with news about fundamentals. Fundamentals in our model are determined by cross-country
interest rate di¤erentials which, in turn, are described by Taylor-type rules, along the lines of
Engel and West (2005, 2006). We solve for a “consistent expectations equilibrium,” in which
the coe¢ cient on fundamental news in the agent’s subjective forecast rule is pinned down using
the observed covariance between exchange rate changes and fundamental news. This learnable
equilibrium delivers the result that the forecast errors observed by the agent are close to white
noise, making it di¢ cult for the agent to detect any misspeci…cation of the forecast rule.7
4
For recent evidence, see Baillie and Kilic (2006) and Baillie and Chang (2011).
See Engel (1996) for a survey of this large literature and Engel (2013) for a more recent review of new
developments in this …eld.
6
For additional details, see Amromin and Sharpe (2013), Greenwood and Shleifer (2013), Jurgilas and
Lansing (2013), and Williams (2013).
7
The consistent expectations equilibrium concept was developed by Hommes and Sorger (1998). A closely5
2
We demonstrate that our model can generate volatility and persistence that is broadly similar to that observed in monthly bilateral exchange rate data (relative to the U.S.) for Canada,
Japan, and the United Kingdom over the period 1974 to 2012. We show that regressions performed on model-generated data can deliver the forward-premium anomaly, whereby a high
interest rate currency tends to appreciate, thus violating the UIP condition. Moreover, the
estimated slope coe¢ cient in the model regressions varies over a wide range when estimated
using a 15-year (180 month) rolling sample period. This result is consistent with the wide
range coe¢ cient estimates observed across countries and time periods in the real-world data.8
In our model, agents’perceived law of motion (PLM) for the exchange rate is a driftless
random walk that is modi…ed to include an additional term involving fundamental news,
i.e., the innovation to the AR(1) driving process that is implied by the Taylor-rule based
interest rate di¤erential. To maintain generality regarding agents’information sets, we allow a
fraction of market participants to construct the random walk component of the forecast using
the contemporaneous exchange rate while the remainder employ the lagged exchange rate.
The agents’PLM has only one parameter that can be estimated within the model by running
a regression of the exchange rate change on fundamental news, both of which are observable
to the agent.9 The agent’s primary misspeci…cation is to postulate a unit root in the law of
motion for the exchange rate, whereas the rational expectations (RE) solution is characterized
by a AR(1) law of motion the exchange rate which is inherited from the fundamental driving
process.
We show that regardless of the starting value for the coe¢ cient on fundamental news in the
subjective forecast rule, a standard learning algorithm will converge to the …xed point which
de…nes the unique consistent expectations (CE) equilibrium. The equilibrium is characterized
by an actual law of motion (ALM) which is stationary but highly persistent. Hence, a large
amount of data would be needed for the agent to reject the hypothesis of a unit root in the
observed exchange rate. Intuitively, the boundedly-rational equilibrium exploits the fact that
the expected future exchange rate enters the relevant no-arbitrage condition with a near-unity
coe¢ cient. This feature causes the agent’s perception of a unit root in the exchange rate
to become close to self-ful…lling. The backward-looking nature of the agent’s forecast rule is
crucial for generating the forward-premium anomaly. By introducing a large weight on the
lagged exchange rate in the agent’s forecast rule, the CE model shifts the temporal relationship
between the expected exchange rate change and the interest rate di¤erential, thus ‡ipping the
sign of the slope coe¢ cient in the UIP regression.
Our setup is motivated by two important features of the data: (1) real-world exchange rates
exhibit near-random walk behavior, and (2) exchange rates and fundamentals do exhibit some
related concept is the “restricted perceptions equilibrium” described by Evans and Honkopohja (2001, Chapter
13). For other applications of the consistent expectations concept to asset pricing or in‡ation, see Sögner and
Mitlöhner (2002), Branch and McGough (2005), Evans and Ramey (2006), Lansing (2009, 2010), and Hommes
and Zhu (2013).
8
For evidence of variability in estimated UIP slope coe¢ cients, see Bansal (1997), Flood and Rose (2002),
Baillie and Kilic (2006), Baillie and Chang (2011), and Ding and Ma (2013).
9
Lansing (2010) employs a similar random walk plus fundamentals setup to account for numerous quantitative features of long-run U.S. stock market data.
3
tenuous empirical links, as noted above. A random walk forecast can be viewed as boundedlyrational because it economizes on the costs of collecting and processing information. As noted
by Nerlove (1983, p. 1255): “Purposeful economic agents have incentives to eliminate errors
up to a point justi…ed by the costs of obtaining the information necessary to do so...The most
readily available and least costly information about the future value of a variable is its past
value.”A forecast rule that employs both the lagged exchange rate and fundamental news can
be interpreted as summarizing a consensus market forecast that arises from the interaction
of chartist and fundamentalist-type traders.10 When we apply the model’s forecast rule to
the real-world data, we …nd that the inclusion of fundamental news together with the lagged
exchange rate can often improve forecast accuracy relative to an otherwise similar random
forecast that omits the fundamental news term.
Using survey data on 3-month ahead forecasts of the Canada/U.S. exchange rate, we show
that changes in the interest rate di¤erential (a proxy for fundamental news) are helpful in
explaining forecasted changes in the Canada/U.S. exchange rate— a result that is consistent
with our postulated forecast rule. In particular, the data shows that survey respondents tend
to forecast a currency appreciation during periods when the interest rate di¤erential increases.
here Our postulated forecast rule is also consistent with survey data which shows that the
vast majority of professional forecasters use both technical analysis (chart patterns of past
exchange rate movements) and fundamental economic data to construct their forecasts.
1.2
Related Literature
This paper relates to some previous literature that has employed models with distorted beliefs
to account for observed features of exchange rates. Gourinchas and Tornell (2004) postulate
that agents have distorted beliefs about the law of motion for fundamentals. They assume that
the perceived interest rate di¤erential is an exponentially-weighted moving average of the true
di¤erential. The crucial moving average parameter in their model is calibrated, rather than
endogenized within the model itself as is done here. Moreover, di¤erent values for the movingaverage parameter in their model are needed to account for di¤erent exchange rate anomalies.
A related mechanism is proposed by Burnside et al. (2011). In their model, distorted beliefs
arise because agents overweight a private signal about future monetary fundamentals. In
contrast, we postulate that agents have distorted beliefs about the law of motion for exchange
rates, not fundamentals. These distorted beliefs not only a¤ect the dynamics of the exchange
rate, but they also a¤ect the dynamics of equilibrium interest rate di¤erential via the Taylortype rule.
Bacchetta and Van Wincoop (2009) introduce “random walk expectations” into an exchange rate model with risk aversion and infrequent portfolio adjustments. Unlike our setup,
the agent’s subjective forecast in their model completely ignores fundamentals. While their
model can account for the forward-premium anomaly, it relies on exogenous shocks from ad
hoc noise traders to account for the observed volatility of exchange rate changes.
10
There is a large literature on forecast switching models with heterogenous interacting agents. For examples
applied to exchange rates, see Frankel and Froot (1990), De Grauwe and Grimaldi (2006), and Markiewicz
(2012), among others.
4
Figure 1: The slope of the …tted relationship between the observed exchange rate change and the prior
month’s interest rate di¤erential is negative, illustrating the forward-premium anomaly.
Chakraborty and Evans (2008) introduce constant-gain learning about the reduced-form
law of motion for the exchange rate. The agent in their model employs the correct (i.e., rational) form for the law of motion, but the estimated parameters are perpetually updated
using recent data. They show that statistical variation in the estimated parameters may
cause the UIP condition to be violated, particularly in small samples. However, their model
does not account for excess volatility of the exchange rate. Mark (2009) develops a model
with perpetual learning about the Taylor-rule coe¢ cients that govern the cross-country interest rate di¤erential. He shows that the model can account for major swings in the real
deutschemark/euro-dollar exchange rate over the period 1976 to 2007. A drawback of perpetual learning models is that the gain parameter (which governs the weight on recent data in the
agent’s forecast rule) is not pinned down within the model itself. In contrast, our approach
employs no free parameters in the agent’s subjective forecast rule.
2
Data on Exchange Rates and Fundamentals
According to the UIP theory, the cross-country interest rate di¤erential should be a key explanatory variable for subsequent exchange rate changes. Figure 1 plots scatter diagrams
of monthly bilateral exchange rate changes (in annualized percent) versus the prior month’s
average short-term nominal interest rate di¤erential (in percent) for three pairs of countries,
namely, Canada/U.S., Japan/U.S. and U.K./U.S. The data covers the period from January
5
Figure 2: The volatility of exchange rate changes is around 10 to 30 times higher than the volatility
of the interest rate di¤erential.
1974 through October 2012.11 The bottom right panel of Figure 1 shows a scatter diagram
of the pooled data. Figure 2 shows time series plots of the same data, with the bottom right
panel depicting the relative volatility of exchange rate changes to interest rate di¤erentials,
where volatilities are computed using a 15-year rolling sample period.
Three aspects of Figures 1 and 2 standout. First, there is no tight systematic relationship between monthly exchange rate changes and the prior month’s interest rate di¤erential
(exchange rate disconnect puzzle). Second, the volatility of exchange rate changes is around
10 to 30 times higher than the volatility of the interest rate di¤erential (excess volatility
puzzle). Third, the dashed lines in Figure 1 show a negative slope in the …tted relationship
between the observed exchange rate change and the prior month’s interest rate di¤erential
(forward-premium anomaly).
The top left panel of Figure 3 plots the interest rate di¤erentials that are used to predict
exchange rate changes via a standard UIP regression that takes the form:
st+1 =
0
+
1 (it
it ) + "t+1 ;
(1)
where st log (St ) is the logarithm of the nominal exchange rate and st+1 st+1 st is the
monthly percent change (annualized) from period t to t + 1: The short-term nominal interest
11
Exchange rate changes are computed as the log di¤erence of sequential end-of-month values and then
annualized. Interest rates are annualized. All data are from the IMF’s International Financial Statistics
database.
6
Figure 3: Estimated UIP slope coe¢ cients lie mostly in mostly in negative territory and exhibit
substantial time variation.
rate di¤erential is it it , where it is the rate for either Canada, Japan, or the U.K. and it
is the U.S. interest rate. Under UIP (which assumes rational expectations and risk-neutral
investors), we have 0 = 0 and 1 = 1; with the error term "t+1 re‡ecting white-noise rational
forecast errors.
The remaining panels of Figure 3 plot the estimated slope coe¢ cients 1 for each country
together with the associated standard error bands using a 15-year rolling sample period. The
horizontal dashed lines mark the values 1 = 1 and 1 = 0: For all three countries, the rolling
estimates of 1 lie mostly in negative territory, thus violating the UIP condition. Table 1 below
provides the full-sample estimates for 1 : The point estimates are all negative, consistent with
…tted lines shown in Figure 1. Moreover, the 95 percent con…dence intervals based on the
reported standard errors exclude the theoretical prediction of 1 = 1 in their coverages.12
Table 1. Full-Sample UIP Slope Coe¢ cients
Pooled
Canada
Japan
U.K.
Data
^
0:29
1:86
1:21
0:16
1
Std: Error
(0.65)
(0.75)
(0.76)
(0.29)
Note: Sample period is 1974.m1 to 2012.m10.
12
Using a dataset of 23 countries for the sample period of the 1990s, Flood and Rose (2002) obtain positive
estimated values of 1 using pooled data. However, they acknowledge (p. 257) that “pooling is a dubious
procedure” given the heterogeneity in the individual country estimates of 1 :
7
The consistently negative estimates for 1 reported here and elsewhere in the literature have
interesting economic implications. In practice, the results imply that a carry-trade strategy
(taking a long position in high-interest currency while shorting a low-interest currency) can
deliver substantial excess returns, where excess returns are measured by (it it )
st+1 : When
<
1;
the
future
excess
return
can
be
predicted
using
the
current
interest
rate
di¤erential
1
13
it it ; which raises serious doubts about market e¢ ciency.
E¤orts to account for the
predictability of excess returns in the data can be classi…ed into two main categories: (1)
linking excess returns to some form of compensation for bearing risk, or (2) departures from
fully-rational expectations. Empirical evidence using conventional risk factors argues against
the …rst approach (Burnside, et al. 2012). In this paper, we adopt the second approach to
explaining the behavior of exchange rates.
Another notable feature of the UIP regressions, evident in Figure 3, is the substantial time
variation in the estimated slope coe¢ cient for a given country. Baillie and Kilic (2006) and
Baillie and Chang (2011) employ nonlinear time series models with regime shifting regression
coe¢ cients to capture this feature of the data. Bansal (1997) shows that the sign of 1 appears
to be correlated with the sign of the interest rate di¤erential, but his results do not generalize
to other sample periods or countries. Ding and Ma (2012) develop a model of cross-border
portfolio reallocation that can help explain a time-varying 1 estimate. Our model can also
account for a time-varying 1 estimate.
3
Model
The framework for our analysis is a standard asset-pricing model of the exchange rate. Fundamentals are given by cross-country interest rate di¤erentials which, in turn, are described
by Taylor-type rules, along the lines of Engel and West (2005, 2006). Given our dataset, the
home country in the model is intended to represent either Canada, Japan, or the U.K. while
the foreign country represents the United States (denoted by variables).
We postulate that the home country central bank sets the short-term nominal interest rate
according to the following reaction function
it =
it
1
+ (1
)[ g
t
+ gy yt + gs (st
st ) ] +
t;
(2)
where it is the interest rate, t is the in‡ation rate, yt is the output gap (log deviation
of actual output from potential output), st is the log of the nominal exchange rate (home
currency relative to foreign currency), and st is the log of the central bank’s exchange rate
target. Hence, st st represents the percentage deviation of the nominal exchange rate from
its target level.14 The term t represents an exogenous monetary policy shock. In contrast to
Engel and West (2005, 2006), we allow for interest-rate smoothing on the part of the central
bank, as governed by the parameter > 0: For the remaining reaction function parameters,
13
Cochrane (2001, p. 394) points out that return predictability is directly related to the phenomenon of
excess volatility.
14
We omit constant terms from equation (2) because our empirical application of the central bank reaction
function makes use of demeaned data.
8
we follow standard practice in assuming g > 1; and gy ; gs > 0: In other words, the central
bank responds more than one-for-one to a change in in‡ation and raises the nominal interest
rate in response to a larger output gap or a weaker home currency (larger st ) relative to the
target.15
The target value for the exchange rate is pinned down by the purchasing-power-parity
(PPP) condition
s t = pt pt ;
(3)
where pt and pt are the aggregate price levels, in logarithms, for the home and foreign country,
respectively.
The foreign (i.e., U.S.) central bank sets the short-term nominal interest rate according to
it =
it
1
+ (1
)[ g
t
+ g y yt ] +
t;
(4)
where we assume that the reaction function parameters ; g ; and gy are the same across
countries. Subtracting equation (4) from equation (2) and imposing (3) yields the following
expression for the cross-country interest rate di¤erential
it
it
=
(it
+
it
1
1)
+ (1
t:
t
)fg (
t)
t
+ gy (yt
yt ) + gs [st
(pt
pt )] g
(5)
Assuming risk-neutral, rational investors, the uncovered interest rate parity condition implies
Et st+1 st = it it ;
(6)
where Et st+1 is the rational forecast of next period’s log exchange rate.16 The UIP condition
says that a negative interest rate di¤erential it it < 0 will exist when rational investors expect
a home currency appreciation, i.e., when Et st+1 < st : The expected appreciation compensates
investors for the opportunity cost of holding a low interest rate domestic bond rather than
a high interest rate foreign bond. Rational expectations implies Et st+1 = st+1 "t+1 , where
"t+1 is a white-noise forecast error. Hence, theory predicts that, on average, a low interest rate
currency should appreciate relative to a high interest rate currency such that E (st+1 st ) < 0:
Substituting the cross-country interest rate di¤erential (5) into the UIP equation (6) and
solving for st yields the following no-arbitrage condition that determines the equilibrium exchange rate
st = bEt st+1 + xt;
(7)
where the e¤ective discount factor is given by b
variable xt is de…ned as
xt
b (it
b(
t
1
it
t)
1)
b(1
)[ g (
:
1= [1 + (1
t
t)
)gs ] : The fundamental driving
+ gy (yt
yt )
gs (pt
pt ) ]
(8)
15
Empirical studies …nd evidence that central banks in many industrial countries respond to movements in
exchange rates or exchange rate changes. See, for example, Clarida, et al. (1998), Lubik and Schorfheide (2007),
and Justiniano and Preston (2010).
16
More precisely, the UIP condition is Et St+1 =St = (1 + it ) = (1 + it ) : Following standard practice, we take
logs of both sides and ignore the Jensen’s inequality term such that log (Et St+1 ) ' Et log (St+1 ) :
9
The no-arbitrage condition (7) shows that the equilibrium exchange rate st depends on
both the fundamental driving variable xt and the agent’s conditional forecast Et st+1 : The
general form of equation (7), whereby the current value of an endogenous variable depends in
part on its own expected value, appears in a wide variety of economic models. Given values
for xt and st ; we can recover the current-period interest rate di¤erential as follows
it
it =
1
xt +
b
1
b
st :
b
(9)
Empirical estimates of central bank reaction functions typically imply values near one and
small values for gs such that b ' 1: In this case, the equilibrium dynamics for it it will be
very similar to the equilibrium dynamics for xt :
3.1
Rational Expectations
Under rational expectations, the equilibrium exchange rate st is pinned down by the agent’s
forecast of the discounted future values of the driving variable xt . To derive the unique rational
expectations solution, we iterate equation (7) forward to substitute out st+i for i = 1; 2; 3:::
Applying the law of iterated expectations and imposing a transversality (i.e., no bubbles)
condition yields the following present-value exchange rate equation
st =
1
X
bi Et xt+i ;
(10)
i=0
where Et is the mathematical expectation operator which presumes knowledge of the law of
motion for the driving variable xt : Engel and West (2005) emphasize that the form of equation
(10) is same for a variety of di¤erent speci…cations about monetary policy. Balke, et al. (2013)
derive a similar formula for the exchange rate using the classical monetary exchange rate model
in which the central bank is presumed to control the money supply rather than the short-term
nominal interest rate.
The macroeconomic variables that enter the de…nition of xt exhibit a high degree of persistence in the data. We therefore model the behavior of the fundamental driving variable using
the following stationary AR(1) process
xt =
xt
1
+ ut ;
ut
N 0;
2
u
;
j j < 1;
(11)
where the parameter governs the degree of persistence. While some studies allow for a unit
root in the law of motion for fundamentals, we maintain the assumption of stationarity for
consistency with most of the literature. In practice, it is nearly impossible to distinguish
between a unit root process and one that is stationary but highly persistent (Cochrane 1991).
The following proposition provides an analytical solution for the equilibrium exchange rate
under rational expectations.
Proposition 1. When fundamentals are governed by equation (11), the rational expectations
(RE) solution for the equilibrium exchange rate is given by
xt
st =
;
1 b
10
Proof : Repeatedly substituting the law of motion (11) for xt+i for i = 1; 2; 3;... into the
present-value equation (10) yields st = xt 1 + b + b2 2 + b3 3 + ::: = xt = (1 b ). :
From Proposition 1, the agent’s conditional forecast is given by
xt
;
1 b
Et st+1 =
(12)
which shows that the rational forecast depends only on fundamentals and not on past exchange
rates.
Given the RE solution, straightforward computations yield the following unconditional
moments:
V ar(st ) =
1
(1
V ar( st ) =
Corr(st ; st
b )
V ar(xt );
(13)
2(1
)
V ar(xt );
(1 b )2
1)
=
;
(15)
Cov ( st+1 ; it
it ) =
V ar (it
(1
(1
it ) =
(14)
(1
(1
)2
V ar(xt );
b )2
)2
V ar(xt );
b )2
(16)
(17)
2 and we have used the expression for i
it given by equation
where V ar(xt ) = 2u = 1
t
(9). Equation (15) shows that the rational exchange rate inherits the persistence properties
of the fundamental driving variable. When ' 1; the exchange rate rate will be close to a
random walk such that Corr(st ; st 1 ) ' 1 and Corr( st ; st 1 ) = (
1) =2 ' 0:
From equations (16) and (17), the analytical slope coe¢ cient from a UIP regression is
given by
Cov ( st+1; it it )
= 1;
(18)
1 =
V ar (it it )
which shows that UIP holds exactly under rational expectations.
3.2
Consistent Expectations
Real-world exchange rates exhibit near-random walk behavior. A forecast rule that uses only
the most recently-observed exchange rate almost always outperforms a fundamentals-based
forecast (Rossi 2013). In addition to its predictive accuracy, a random walk forecast has
the advantage of economizing on computational and informational resources. Still, there is
evidence that market participants pay attention to fundamentals. A recent study by Dick and
Menkho¤ (2013) uses survey data to analyze the methods of nearly 400 professional exchange
rate forecasters. The data shows that the vast majority of forecasters use both technical
analysis (chart patterns of past exchange rate movements) and fundamental economic data to
11
construct their forecasts. To capture these ideas, we postulate that agents’perceived law of
motion (PLM) for the exchange rate is given by
st+1 = st +
ut+1 ;
(19)
where ut+1 represents “fundamental news,” as measured by the innovation to the AR(1) fundamental driving process (11). The PLM can be interpreted as representing an aggregation of
the perceptions of chartist and fundamentalist-type traders. The presence of the past exchange
rate st captures the chartist in‡uence while the news term ut+1 captures the fundamentalist
in‡uence. There is only one parameter that can be readily estimated by agents in model by
running a regression of st on ut :
bt st+1 which takes the
The PLM is used by agents to construct a subjective forecast E
place of the rational forecast Et st+1 in the no-arbitrage condition (7). Since the no-arbitrage
condition implies that st depends on the agent’s own subjective forecast, it is questionable
whether an agent could make use of st when constructing a forecast in real-time, particularly
at higher frequencies. To deal with this issue, models that employ adaptive learning or other
forms of boundedly-rational expectations typically assume that agents can only make use of the
lagged realization of the forecast variable (in this case st 1 ) when constructing the subjective
forecast at time t.17 A lagged-information setup avoids simultaneity in the determination of
the actual and expected values of the forecast variable. Here we wish to maintain generality
regarding agents’information. We postulate that a fraction 2 [0; 1] of market participants
(labeled Type-1 agents) employ contemporaneous information about the exchange rate while
the remaining fraction 1
(labeled Type-2 agents) employ lagged information.18 As in
the rational solution, we assume that both types of agents have access to contemporaneous
information about the fundamental variable xt : Notice that according to the de…nition (8), xt
does not not depend on the contemporaneous exchange rate st and hence it can be included
in the information sets of both types of agents. A long history of observations of xt would
allow agents to discover the stochastic process (11). With this knowledge, agents could infer
the fundamental news ut from sequential observations of xt and xt 1 :
From the PLM (19), the subjective forecast of Type-1 agents is given by
b 1;t st+1 = st ;
E
(20)
which coincides with a random walk forecast because agents’knowledge of fundamentals imb 1;t ut+1 = 0: To compute the subjective forecast of the Type-2 agents, the PLM is
plies E
iterated ahead two periods to obtain
b 2;t st+1
E
=
=
=
b 2;t [st + ut+1 ] ;
E
b 2;t st+1 [st 1 + ut + ut+1 ] ;
E
st
1
+ ut ;
17
(21)
For an overview of these methods, see Evans and Honkapohja (2001), Hommes (2013), and Hommes and
Zhu (2013).
18
Adam, Evans and Honkapohja (2006) employ a similar generalization of agents’information sets in a model
of adaptive learning applied to hyperin‡ation.
12
which makes use of the lagged exchange rate st 1 : Finally, we assume that aggregate market
forecast is given by the following population-weighted average of the two agent types:
bt st+1
E
=
=
=
b 1;t st+1 + (1
E
st + (1
st
1
) (st
+
1
st + (1
b 2;t ;
)E
+ ut ) ;
) ut ;
(22)
where
st is a momentum term that derives from aggregating across agents with di¤erent
information sets.
Substituting the aggregate market forecast (22) into the no-arbitrage condition (7) and
solving for st yields the following actual law of motion (ALM) for the exchange rate
st =
b(1
)
st
1 b
1
b(1
1
+
)
ut +
b
xt
;
1 b
(23)
where xt is governed by (11). When b; 2 (0; 1) ; the ALM implies that st remains stationary.
But since b ' 1; the coe¢ cient on st 1 will be close to unity for any 2 (0; 1) such the agents’
perception of a unit root in the exchange rate will be close to self-ful…lling.
3.2.1
De…ning the Consistent Expectations Equilibrium
We now de…ne a “consistent expectations equilibrium” in the spirit of Hommes and Sorger
(1998). Speci…cally, the parameter
in the agents’ PLM (19) is pinned down using the
19
moments of observable data.
Since the PLM presumes that st exhibits a unit root, it is
natural to assume that agents’estimate as follows
=
Cov ( st ; ut )
2
u
;
(24)
where Cov ( st ; ut ) and 2u = V ar (ut ) can both be computed from observable data. An
analytical expression for the observable covariance can be derived from the ALM (23) which
implies:
st =
b
1
1
st
b
Cov ( st ; ut ) =
1
+
b(1
1
b(1
1
)
b
) +1
b
ut +
2
u:
1
xt
;
b
(25)
(26)
Equations (24) and (26) can now be combined to form the following de…nition of equilibrium.
19
In Hommes and Sorger (1998), the PLM is linear, whereas the ALM is nonlinear. Here, the PLM is
nonstationary, whereas the ALM is stationary but highly persistent. In both cases, the PLM and the ALM
exhibit an identical covariance statistic.
13
De…nition 1. A consistent expectations (CE) equilibrium is de…ned as a perceived law of
motion (19), an actual law of motion (23), and a subjective forecast parameter ; such that
the equilibrium value
is given by the unique …xed point of the linear map
=
1
=
where b
1= [1 + (1
b(1
T( )
1
b
) +1
;
b
1
;
)gs ] < 1 is the e¤ ective discount factor.
Interestingly, the …xed point value
depends only on the e¤ective discount factor b and
not on which measures the fraction of Type-1 agents. But as we shall see, does in‡uence
other model results, such as the theoretical slope coe¢ cient from a UIP regression. The slope
of the map T ( ) determines whether the equilibrium is stable under learning. Since the slope
T 0 ( ) = b(1
)= (1 b ) 2 (0; 1) ; the CE equilibrium is globally stable. In Section 4,
we demonstrate that a real-time learning algorithm always converges to the …xed point
regardless of the starting value for :
Using the ALM (23), straightforward (but tedious) computations yield the following unconditional moments.
8
9
2 )]
2 ) + 2 b(1 )[1+b (1 )(1
=
< 1 + [b2 2 (1
) + 2b ](1
)(1
1 b
b(1 )
V ar(xt );
V ar(st ) =
;
:
1 b2 2b(1 b)
(27)
2(1 b)
V ar( st ) =
(1 b )
(
1
+ [b2
2 (1
](1
)+2b
)(1
(1 b)[1+b (1
)(1
1 b
b(1
)
2)
2 )]
1 b2 2b(1 b)
)
V ar(xt )
(28)
Corr(st ; st
1) =
Cov ( st+1 ; it
V ar(st )
b(1
)
+
1 b
(1
it )
=
b )
n
1+b (1
1 b
n
(1 b)2
1
(1
+
b(1 b )
b(1 b )
V ar (it it )
(1 b)2
1 n
=
+
1
V ar(st )
b2
b2
2)
)(1
b(1
)
o V ar(x )
t
;
V ar(st )
b)(1+ )[1+b (1
1 b
b(1
2(1 b)[1+b (1 )(1
1 b
b(1 )
2 )]
)(1
)
o V ar(x )
t
;
V ar(st )
(29)
2 )]
o V ar(x )
t
V ar(st )
(30)
(31)
where = 1= (1 b) in equilibrium. A comparison of the above expressions with the corresponding RE moments from equations (13) through (17) shows that CE moments are considerably more complicated. This is due to the presence of the lagged exchange st 1 in the ALM
(23) which gives rise to additional covariance terms.
14
Some insights can be obtained by considering the case when the e¤ective discount factor
approaches unity. When b ! 1 the equilibrium exchange rate exhibits a unit root such that
V ar (st ) ! 1: As b ! 1; the analytical slope coe¢ cient from a UIP regression is given by
1
= lim
b!1
Cov ( st+1; it it )
=
V ar (it it )
1
1
;
(32)
which has the opposite sign from the RE model as given by (18). When all agents employ
lagged information, we have = 0 and the slope coe¢ cient equals 1: However, when some
agents employ contemporaneous information, we have 0 < < 1 and the slope coe¢ cient
can become more negative than 1, which is consistent with some of the empirical estimates
reported in Table 1. When 0 < b < 1; the slope coe¢ cient remains negative, but is smaller in
magnitude than in the limiting case of b ! 1: These analytical results demonstrate that the
CE model can deliver the well-documented forward-premium anomaly. We will con…rm these
results numerically in the quantitative analysis presented in Section 5.
The intuition for the forward-premium anomaly in the CE model is straightforward. For
simplicity, let us consider the case where all agents are Type-2 such that = 0: From equation
bt st+1 = st 1 + ut : Subtracting st from both
(22), the aggregate market forecast becomes E
bt st+1 st =
sides of this expression yields E
st + ut . In contrast, the RE model implies
Et st+1 = st+1 "t+1 , where "t+1 is the white noise forecast error. Again subtracting st from
both sides yields Et st+1 st = st+1 "t+1 . The UIP condition (6) relates the forecasted
change in the exchange rate to the prior interest rate di¤erential it it : By introducing some
positive weight on st 1 in the aggregate market forecast, the CE model shifts the temporal relationship between the forecasted change of the exchange rate and the interest rate di¤erential,
thus ‡ipping the sign of the slope coe¢ cient in the UIP regression.
Additional insight can obtained be substituting the ALM for st (23) into the Taylor-rule
based interest rate di¤erential (9) to obtain the following equilibrium expression for the interest
rate di¤erential:
it
it
=
1
xt +
b
=
(1
)
=
(1
)(
1
b
b
1 b
st
1 b
st +
|
1
b(1
)
st
1 b
1
+
b(1
)
1 b
{z
ut +
xt
1 b
st
+
(1
1
ut )
b)
b
ut
xt
1 b
;
(CE model);
;
}
(33)
where we have made use of the ALM for st (25). From equation (1), the sign of the UIP
slope coe¢ cient 1 is governed by the sign of Cov ( st+1 ; it it ) : Using the above result, we
have st+1 =
it+1 it+1 = (1
) + ut+1 which in turn implies
Cov ( st+1 ; it
it ) =
1
1
Cov it+1
it+1 ; it
it
(CE model):
(34)
The right-side of the above expression will be negative so long as the interest rate di¤erential
exhibits positive serial correlation, as it does in both the model and the data. Intuitively,
15
since the interest rate di¤erential depends on the exchange rate via the Taylor rule, and the
exchange rate depends on agents’ expectations via the …rst order condition (7), a departure
from rational expectations can shift the dynamics of the variables that appear on both sides
of UIP regression (1).
The corresponding derivation in the RE model is
it
it
=
1
xt +
b
1
xt
;
1 b
| {z }
b
b
st
=
(1
1
=
st+1
)
b
xt ;
ut+1
1 b
(RE model);
(35)
where we have made use of st+1 = xt+1 = (1 b ) from Proposition 1. Using the above
result, we have st+1 = (it it ) + ut+1 = (1 b ) which in turn implies
Cov ( st+1 ; it
it ) = V ar (it
it )
(RE model);
(36)
such that the right-side is always positive.
4
Applying the Model’s Methodology to the Data
Using the de…nition of the fundamental variable xt in equation (8), we construct times series for
xt in Canada, Japan, the United Kingdom using monthly data on the consumer price index,
industrial production, and the short-term nominal interest rate di¤erential relative to the
United States. Our data is from the International Monetary Fund’s International Financial
Statistics (IFS) database and covers the period January 1974 through October 2012. To
construct measures of the output gap for each country, we estimate and remove a quadratic
trend from the logarithm of the industrial production index.20 In constructing the time series
for xt ; we use the following calibrated values for the Taylor-rule parameters: = 0:8; g = 1:5;
gy = 0:5; and gs = 0:1: These values are consistent with those typically employed or estimated
in the monetary literature.21 Given the Taylor-rule parameters, the e¤ective discount factor
in our model is b 1= [1 + (1
)gs ] = 0:98:
Table 1 reports summary statistics for the nominal interest rate di¤erential (relative to the
U.S.) and the constructed time series for xt . As noted earlier in the discussion of equation (9),
the equilibrium dynamics for it it will be very similar to the equilibrium dynamics for xt
whenever b ' 1; as is the case here.
20
21
Similar results are obtained if industrial production is detrended using the Hodrick-Prescott …lter.
See, for example, Taylor (1993) and Clarida, Gali, and Gertler (1998).
16
Table 2. Summary Statistics of Data Fundamentals
Canada Japan
U.K.
Std Dev(it it )
1:62% 2:35%
2:18%
Corr(it it ; it 1 it 1 ) 0:956
0:972
0:954
Std Dev(xt )
1:65% 2:53%
2:21%
Corr(xt ; xt 1 )
0:959
0:976
0:956
Corr(it it ; xt )
0:987
0:953
0:992
Note: Sample period is from 1974.m1 to 2012.m10. The fundamental
variable
xt is de…ned by equation (8).
Before proceeding with simulations from the theoretical model, we wish to examine the
performance of the postulated forecast rules (20) and (21) using real-world exchange rate
data. Table 3 reports the average root mean squared forecast errors (RMSFE) for four di¤erent
forecast rules. The RMSFE statistics are computed sequentially using a 15-year rolling sample
period and then averaged. The initial sample period is from 1974.m1 to 1988.m1. The …rst
two rows of the table show the results for our postulated forecast rules (20) and (21) which are
associated with the Type-1 and Type-2 agents, respectively.22 The third row shows the results
of a random walk forecast using only lagged-information about the exchange rate, i.e., no
fundamental news. The fourth row shows the results for a fundamentals-only forecast where
the coe¢ cients f0;t and f1;t are estimated sequentially for each rolling sample period using data
for each country’s fundamental variable xt :
Consistent with previous studies, the top row of Table 3 shows that a random walk forecast
using contemporaneous information (Type-1 agent forecast) has the lowest RMSFE and thus
outperforms the other three forecast rules. The fundamentals-only forecast in the bottom row
of the table exhibits the worst performance with the highest RMSFE. For this exercise, our
main interest is asking whether the use of fundamental news can improve the performance of
a random walk forecast that uses lagged information about the exchange rate. The answer
turns out to be mixed. Comparing the second and third rows of Table 3 shows that the use
of fundamental news slightly degrades forecast performance (higher RMSFE) when averaged
over the entire sequence of rolling sample periods. However, Figure 4 shows that the use of
fundamental news can lower the RMSFE statistics (relative to a random walk forecast with
lagged information) for Japan and the U.K. This result can be observed for the more recent
15-year sample periods. For Canada, the …gure shows that the use of fundamental news
has very little impact on forecast performance relative to a lagged information random walk
forecast. This may be because Canada’s exchange rate behaves closest to a pure random walk
of the three countries. Evidence of this can be seen in the bottom right panel of Figure 4
where we plot the 15-year rolling autocorrelation of exchange rate changes Corr ( st ; st 1 ) :
The rolling autocorrelation statistic for Canada remains closest to zero over the entire sample
period, indicating a consistent, near-random walk behavior of Canada’s exchange rate.
22
To construct the Type-2 agent’s forecast, we sequentially estimate the law of motion for fundamentals (11)
for the rolling sample period and use the resulting parameter estimates to identify a sequence of fundamental
innovations ut for that sample period. We then estimate the response coe¢ cient to fundamental news using
only lagged information about the exchange rate, i.e., t 1 = Cov ( st 1 ; ut 1 ) =V ar (ut ) :
17
Figure 4: For Japan and the U.K., the use of fundamental news can improve forecast performance
relative to a random walk forecast with lagged information. For Canada, the use of fundamental news
has little impact on forecast performance.
Table 3. Average 15-year Rolling RMSFEs
Forecast Rule
Canada Japan
UK
b 1;t st+1 = st
E
19:0% 39:8%
36:2%
b 2;t st+1 = st 1 + t 1 ut
E
26:8% 57:8%
53:3%
b
Et st+1 = st 1
26:6% 58:6%
52:7%
bt st+1 = f0;t + f1;t xt
E
122%
343%
157%
Notes: Root mean squared forecast errors (RMSFE) are computed
sequentially using a 15-year rolling sample period and then averaged.
The initial sample period is from 1974.m1 to 1988.m1.
While the preceding exercise showed that the use of fundamental news can sometimes
improve exchange rate forecasts, it is interesting to consider direct evidence on whether professional forecasters behave in a way that is consistent with our model. We noted earlier that
the vast majority of professional forecasters report that they use both chart patterns and fundamentals when constructing their exchange rate forecasts (Dick and Menkho¤ 2013). Figure
5 provides some additional evidence that professionals construct forecasts in a way that is
consistent with our model.23
The top panel of Figure 5 plots the 3-month ahead survey forecast for the Canada/U.S.
exchange rate together with the lagged value of the actual exchange rate, i.e., the exchange
23
The survey forecasts for the Canada/U.S. exchange rate are from Consensus Economics via Datastream.
18
Figure 5: Survey forecasts track well with the lagged exchange rate, consistent with the subjective
forecast rule (21). Moreover, survey forecasts appear to respond to contemporaneous news about
changes in the interest rate di¤erential— a proxy for fundamental news.
rate at time t 1: The two lines are almost on top of each other, suggesting that the forecasts
are strongly in‡uenced by lagged information about the exchange rate, consistent with the
forecast rule for Type-2 agents in the CE model.24
The bottom panel of Figure 5 shows a scatter plot of the forecasted 3-month ahead exchange
rate change from the survey versus the change in the Canada/U.S. interest rate di¤erential— a
proxy for fundamental news. The scatter plot shows that survey respondents tend to forecast
a negative change in the exchange rate (a forecasted appreciation) in response to a higher
interest rate di¤erential. This pattern shows that professional forecasters appear to respond
to contemporaneous news about fundamentals, consistent with our postulated PLM (19).
5
Quantitative Analysis
5.1
Numerical Solution for the Equilibrium
We now examine a numerical solution for the consistent expectations equilibrium by plotting
the map T ( ) from De…nition 1. As noted earlier, the calibrated Tayor-rule parameters imply
an e¤ective discount factor of b = 0:98: The parameters of the fundamental driving process
24
A similar pattern can be observed when plotting survey forecasts for U.S. in‡ation versus lagged actual
in‡ation. See Lansing (2009).
19
Figure 6: At the …xed point equilibrium, the volatility of exchange rate changes in the CE model
is nearly three times higher than in the RE model. The forecast errors observed by agents in the
CE model are close to white noise, making it di¢ cult for them to detect any misspeci…cation of their
subjective forecast rules.
(11) are chosen to achieve Std Dev(xt ) = 0:02 (i.e., 2%) and Corr(xt ; xt 1 ) = 0:96; which are
close to the values observed in the data (see Table 2). This procedure yields u = 0:0056 and
= 0:96: We calibrate the fraction of Type-1 agents so that the volatility of the interest rate
di¤erential in the CE model is identical to that in the RE model. In this way, we ensure that the
exchange rate volatility generated by the CE model is driven solely by expectations and not by
excess volatility in the underlying macroeconomic fundamentals. Our calibration procedure
yields = 0:531 which implies that about one-half of the agents in the CE model employ
contemporaneous observation st ; while the remainder employ the lagged observation st 1 . For
comparison, Carroll (2003) uses survey data on U.S. in‡ation forecasts to estimate a value of
0:27 for the fraction of agents who update their forecasts to the most recent information. One
might expect the fraction of up-to-date agents to be higher in foreign exchange markets which
are characterized by large transaction volumes and high frequency trading. When = 0:531;
both models imply Std Dev(xt ) = 1:4%: Larger values of deliver a more negative UIP slope
coe¢ cient in the CE model, as shown by equation (32).
The top-left panel of Figure 6 plots the map T ( ) from De…nition 1 using the calibrated
parameter values. The map crosses the 45-degree line at
= 1= (1 b) = 51: The slope of
0
the map is T ( ) = b(1
)= (1 b ) = 0:96: Since the slope is less than unity, the …xed point
is stable under learning, as we demonstrate in the model simulations below. Still, convergence
20
to the equilibrium can be slow since the T-map lies very close to the 45-degree line. The
top-right panel of Figure 6 plots the theoretical standard deviation of st in the CE model as
varies from 0 to 100. The horizontal dashed line shows the corresponding standard deviation
in the RE model. At the equilibrium value of
= 51; the volatility of st in the CE model is
nearly three times higher than in the RE model (Table 4). Recall that the agents’subjective
forecasts in the CE model embed a unit root assumption. Due to the self-referential nature of
the no-arbitrage condition (7), the agents’subjective forecasts in‡uence the dynamics of the
object that is being forecasted. In equilibrium, the agents’perception of a unit root becomes
close to self-ful…lling.
The bottom-left panel of Figure 6 shows the persistence of the forecast errors for the Type-1
and Type-2 agents. At the …xed point equilibrium, both subjective forecast rules perform quite
well in the sense that they produce near-zero autocorrelations, making it extremely di¢ cult
for the agents to detect any misspeci…cation errors.25 The bottom-right panel of Figure 6 plots
the theoretical standard deviation of the interest rate di¤erential it it . As noted above, our
calibration procedure for ensures that the volatility of it it in the CE model is identical
to that in RE model at the …xed point equilibrium.
Table 4 shows the theoretical moments of st in the CE model as predicted by the perceived
law of motion (19) and the actual law of motion (23). The moments observed by the agents
are very close to those predicted by their perceived law of motion, giving no indication of any
misspeci…cation. In addition, the autocorrelation structure of st remains close to zero at all
lags, reinforcing the agents’perception of unit root in the law of motion for st :
Table 4. Theoretical Moments of st
Rational Expectations
Consistent Expectations
Statistic
PLM = ALM
PLM (predicted) ALM (observed)
Std Dev ( st )
9:62%
28:56%
28:66%
Corr ( st ; st 1 )
0:0200
0
0:0058
Corr ( st ; st 2 )
0:0192
0
0:0051
Corr ( st ; st 3 )
0:0184
0
0:0191
Note: Parameter values are b = 0:98; = 0:96; u = 0:0056; = 0:531:
5.2
Forecast Error Comparison
Proposed departures from rational expectations are often criticized on the grounds that intelligent agents would eventually detect the misspeci…cation of their subjective forecast rule.
We counter this criticism here in two ways. First, we show that the forecast errors observed
by the Type-1 and Type-2 agents in the CE model are close to white noise, making it di¢ cult
for these agents to detect any misspeci…cation. Second, we show that the agents will perceive
no accuracy gain from switching to a fundamentals-based forecast and therefore would tend
to become “locked-in” to the use of their subjective forecast rule.26
25
We present a more comprehensive analysis of the agents’forecast errors in Section 5.2 below.
Lansing (2006, 2009) examines the concept of forecast lock-in the context of boundedly-rational models of
asset prices or in‡ation.
26
21
From equations (20) and (21), the forecast errors observed by the Type-1 and Type-2
agents in the CE model are given by
err1;t+1
err2;t+1
b 1;t st+1 ;
E
=
st+1
=
st+1
st ;
=
st+1
=
st+1
b 2;t st+1 ;
E
st
1
(37)
ut ;
(38)
where the evolution of st is governed by the ALM (23).
Now consider a Type-1 or Type-2 agent who is contemplating a switch to a fundamentalsbased forecast. In deciding whether to switch forecasts, the agent keeps track of the forecast
errors associated with each forecast under consideration. Before any switch occurs, the evolution of st is still governed by (23). As noted earlier, we assume that enough time has gone
by to allow all agents to have discovered the stochastic process (11) which governs the fundamental driving variable xt . Under these assumptions, the fundamentals-based forecast for the
exchange rate is given by equation (12). The resulting forecast error is
errf,t+1 = st+1
1
xt
;
b
(39)
where the subscript “ f ” denotes variables associated with the fundamentals-based forecast.
Given the forecast error expressions, it is straightforward to derive the associated
autocorq
relations and the root mean squared forecast error, as given by RM SF Ei
E[(erri;t+1 )2 ]
for i = 1; 2; f. In all cases, the forecasts are unbiased such that M ean (erri;t+1 ) = 0:
In the RE model, the forecast errors observed by the rational agent are given by
errre,t+1
=
st+1
=
xt+1
1 b
=
ut+1
;
1 b
Et st+1
xt
1 b
(40)
where have made use of the rational solution from Proposition 1. The above expression implies
RM SF Ere = u = (1 b ) :
Table 5 compares the moments of the various forecast errors de…ned above, as computed
from model simulations. In the CE model, the autocorrelation of the forecast errors are close
to zero at all lags for both the Type-1 and Type-2 agents, giving no signi…cant indication
of a misspeci…cation in their subjective forecast rules. A comparison of the RM SF E values
shows that the RE model exhibits the most accurate forecasts (lowest RM SF E), as expected.
However, it is important to realize that an individual agent inhabiting the CE model could
never achieve this kind of forecast performance even if that agent were to switch to the same
fundamentals-based forecast that is being used by the rational agent in the RE model. This is
22
because the actual law of motion in the CE model is permanently shifted by the presence of the
Type-1 and Type-2 agents. As a result, the fundamentals-based forecast performs very poorly
in the CE model, as shown by the last column of Table 5. Hence, from the individual agent’s
perspective, switching to the fundamentals-based forecast would appear to reduce forecast
accuracy, so there is no incentive to switch.
Table 5. Comparison of Forecast Errors
CE Model
Corr(errt+1 ; errt )
Corr(errt+1 ; errt 1 )
Corr(errt+1 ; errt 2 )
q
RE Model
0:001
0:009
0:000
2
M ean errt+1
9:51%
Type-1 Agent
0:006
0:014
0:004
Type-2 Agent
0:036
0:043
0:031
28:6%
29:1%
Fundamental
Forecast
0:975
0:964
0:953
134:7%
Notes: Forecast errors are de…ned by equations (37) through (40). Model statistics are computed from
a 10,000 period simulation. RE = rational expectations, CE = consistent expectations.
5.3
Model Simulations
Figure 7 shows the results of model simulations using the same parameter values shown in
Table 4. The top-left panel plots twenty-four separate real-time learning paths (grouped by
starting value) for the fundamental news response parameter . We employ four separate
starting values 0 2 f31; 41, 61, 71g that initially enter the ALM (19), with each learning
path subject to a di¤erent set of draws for the fundamental innovations ut : Each period, a new
value for is computed from past observable data using equation (24) and then substituted
into the ALM and so on. To speed up the learning process, we assume that agents in the
CE model compute the relevant covariance in equation (24) using a 15-year rolling sample
period. The simulations con…rm that the consistent expectations equilibrium is learnable; the
estimated value of eventually converges to the …xed point value
= 51; regardless of the
starting value.
The top-right panel of Figure 7 plots simulated values for st in the CE model at the
equilibrium …xed point. The numerical simulations con…rm the theoretical results presented
earlier in Table 4: the CE model generates considerably more volatility than the RE model.
This is true despite the fact that both models exhibit identical volatilities for the interest
rate di¤erential it it : The interest rate di¤erentials for the two models are plotted in the
bottom-left panel of Figure 7.
Table 6 provides a more complete comparison of the moments observed in the data versus
those generated by the models. Both models produce near-random walk behavior for the
exchange rate such that Corr( st ; st 1 ) ' 0 and Corr( 2 st ; 2 st 1 ) ' 0:5: However,
the RE model signi…cantly underpredicts the volatility observed in the data. In particular,
the RE model produces Std Dev( st ) ' 10% and Std Dev( st ) ' 14% whereas the data is
characterized by Std Dev( st ) ' 23 - 38% and Std Dev( 2 st ) ' 33 - 53%. In contrast, the
23
Figure 7: Simulations con…rm that the consistent expectations equilibrium is learnable. The CE
model generates considerably more exchange rate volatility than the RE model, despite having the
same volatility for the interest rate di¤erential. The CE model delivers a UIP slope coe¢ cient that is
negative and time-varying, as in the data.
CE model does a much better job of matching both of the volatility statistics in the data, with
Std Dev( st ) ' 29% and Std Dev( st ) ' 40%:
The middle rows of Table 6 con…rm that the moments of the interest rate di¤erential
it it in both models are close those observed in the data. Hence, the assumption of boundedrationality in the CE model generates “overreaction” of the exchange rate relative to the RE
model. Finally, the bottom row of Table 5 shows that the CE model produces a negative correlation between st+1 and the prior interest rate di¤erential it it : The negative correlation
is consistent with the data in all three countries and with the typical negative sign of the slope
coe¢ cient in empirical UIP regressions, as discussed further below.
24
Table 6. Unconditional Moments: Data versus Model
RE
Canada
Japan
U.K.
Model
Std Dev( st )
22:7%
38:3%
35:6%
9:60%
Corr( st ; st 1 )
0:054
0:055
0:093
0:022
2
Std Dev
st
33:0%
52:5%
48:0%
13:7%
Corr( 2 st ; 2 st 1 )
0:535
0:493
0:467
0:505
Std Dev(it it )
1:62%
2:35%
2:18%
1:40%
Corr(it it ; it 1 it 1 )
0:956
0:972
0:954
0:962
Corr( st+1 ; it it )
0:0209
0:1145
0:0736
0:137
CE
Model
28:6%
0:006
40:4%
0:504
1:40%
0:999
0:093
Notes: Data sample period is from 1974.m1 to 2010.m10. RE = rational expectations,
CE = consistent expectations. Model statistics are computed from a 10,000 period simulation.
5.4
UIP Regressions
Table 7 compares the results of UIP regressions using both the real-world data and the modelgenerated data. In addition to the full-sample estimates reported earlier in Table 1, we now
also report the mean results from 15-year rolling regressions. The bottom-right panel of Figure
7 plots the rolling estimates of the UIP slope coe¢ cient in both the RE model and the CE
model.
Table 7 shows that the estimated UIP slope coe¢ cients for Canada, Japan, and the U.K.
are consistently negative, illustrating the forward-premium anomaly. The R2 statistics from
these regressions are all close to zero, illustrating the exchange rate disconnect puzzle. The
RE model produces a positive UIP slope coe¢ cient of 0:94 in the full-sample regression. The
95% con…dence interval based on the associated standard error of 0:07 would include the
theoretical prediction of unity. The mean point estimate in the 15-year rolling regressions
is higher at 1.57. Interestingly, the RE model can account for the exchange rate disconnect
puzzle as it produces a very small R2 statistic in the regressions. This result is consistent with
the arguments put forth by Engel and West (2005, 2006) who show that a fully-rational model
with persistent fundamentals and a discount factor close to unity can deliver near-random walk
behavior of the exchange rate such that changes in the exchange rate are nearly unpredictable
using fundamentals. Still, the RE model cannot account for the forward premium anomaly
and woefully underpredicts the volatility of real-world exchange rates.
The CE model produces negative UIP slope coe¢ cients in both the full-sample and rolling
regressions. The 95% con…dence interval around the full-sample point estimate of 1:85 would
lie entirely in negative territory. The R2 statistics from the regressions are even smaller than
those in the RE model, so the exchange rate is even more disconnected from fundamentals
in the CE model, ceteris paribus. Another realistic feature of the CE model, evident in the
bottom-right panel of Figure 7, is that it produces considerable time variation in the estimated
slope coe¢ cient when the UIP regressions are run using a 15-year rolling sample period. Similar
large time variation in the estimated slope coe¢ cients can be seen in the country-data UIP
regressions plotted earlier in Figure 3. In contrast, the rolling-regression estimates in the RE
model are almost always positive and exhibit much smaller time variation, clustering around
25
Figure 8: The dashed lines show the …tted relationships from the full-sample UIP regressions in the
RE model (top panel) and the CE model (bottom panel). While the relationship between st+1 and
it it is weak in both models, only the CE model produces a negative slope coe¢ cient, consistent with
the country data plotted in Figure 1.
1.0. For both models, the larger standard errors obtained from the rolling regressions are
partly due to the smaller sample size and partly due to estimation bias that can arise in the
presence of a persistent regressor.27
Full Sample
Estimates
b
1
Std Error
R2
Mean 15-year
Rolling Estimates
b
1
Std Error
R2
Table 7. UIP Regressions: Data versus Model
RE
Canada
Japan
U.K.
Model
0:29
1:86
1:21
0:94
(0:65)
(0:75)
(0:76)
(0:07)
0:007
0:013
0:005
0:019
CE
Model
1:85
(0:20)
0:009
RE
Model
1:57
(0:64)
0:032
CE
Model
4:93
(2:50)
0:022
Canada
1:68
(1:08)
0:024
Japan
2:79
(1:48)
0:020
U.K.
0:78
(1:44)
0:020
Notes: The table shows the results of an OLS regression in the form of equation (1). The data
covers the period from 1974.m1 to 2010. RE = rational expectations, CE = consistent expectations.
Model statistics are computed from a 10,000 period simulation.
Figure 8 shows scatter plots of the simulated exchange rate change st+1 versus the prior
interest rate di¤erential it it in both models. The dashed lines show the …tted relationships
from the full-sample UIP regressions. While the relationship between st+1 and it it is weak
27
See Stambaugh (1986) and Nelson and Kim (1993) for detailed discussions of estimation bias in the presence
of a persistent regressor.
26
in both models, only the CE model produces a negative slope coe¢ cient, consistent with the
country scatter plots shown earlier in Figure 1.
6
Concluding Remarks
A reading of the last three decades of academic research on exchange rates highlights numerous
puzzles and anomalies that have stubbornly resisted explanation, particularly in the context
of models where all agents are fully-rational. This paper showed that a plausible departure
from full rationality— in the form of boundedly-rational agents who augment a simple random
walk forecast with news about Taylor-rule fundamentals— can account for several empirical
exchange rate puzzles, including the apparent disconnect from fundamentals, the extremely
high volatility relative to fundamentals, and the failure to satisfy the uncovered interest rate
parity condition. Given that real-world exchange rates exhibit near-random walk behavior, it
makes sense that agents would adopt a forecast rule that allows for a unit root. Our model
setup is also consistent with survey data which shows that professional exchange rate forecasts
track well with the lagged value of the exchange rate and respond to movements in fundamental
news, as measured by changes in the cross-country interest rate di¤erential.
27
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