The Forward-Discount Anomaly and Risk: Allowing for Piece-Wise
Linearity
Olesia Kozlova∗
University of New Hampshire
E-mail: [email protected]
June 2013
Abstract
This paper reconsiders the inability of empirical researchers to account for the widespread finding that
the forward exchange rate appears to be a biased predictor of spot rates with a time-varying risk premium.
The study follows Froot and Frankel (1989) and others in using survey data on exchange rate expectations
to decompose the bias into two components, the correlations of the forward discount with the risk premium
and with forecast errors. However, in doing so, it allows for these correlations to undergo shifts at discreet
points of time, that is, for piece-wise linear relationships. The paper finds that the forward rate bias is indeed
temporally unstable. It also finds within the separate subperiods of stability that this bias is in part due
to the presence of a time-varying risk premium for all three currency markets examined. For two of these
markets, the risk premium component accounts for nearly half of the forward rate bias.
1
Introduction
Dozens of studies in international macroeconomics report that the slope coefficient in a regression of the future
change in the spot exchange rate on the forward premium, which we call the Bilson (1981)-Fama (1998) (BF)
regression, is not only significantly less than one, but less than zero. Interpreting their results as implying a
stable relationship in the data, researchers have concluded from this evidence that the forward rate is a biased
predictor of future changes in the spot rate and that “one can make predictable profits by betting against the
forward rate” (Obstfeld and Rogoff, 1996, p. 589).
To account for these predictable profits, researchers have developed a variety of risk premium models based
on the rational expectations hypothesis (REH). Fama (1984) shows that in order for these models to account for
the negative slope coefficient, the risk premium needs not only be time-varying but highly variable. However,
empirical studies generally find that REH models do not produce nearly enough variation in the premium without
implausibly large estimates of the degree of risk aversion.1
The forward-discount anomaly, as it is called, has led many researchers to appeal to behavioral-finance
models in which a negative bias arises because market participants fall prey to systematic forecasting biases and
technical trading. In these models, speculators could earn greater profits simply by betting against the forward
rate, but they pass up this obvious opportunity. Such gross irrationality arises because speculators are assumed
to underreact or overreact to news, underestimate or overestimate economic growth, or make use of chartists rules
in a way that remains fixed over time.2 Empirical evidence also seems supportive of this explanation.3 Indeed,
∗
1 For
reviews of this literature, see Lewis (1995) and Engel (1996).
instance, Yu (2011), Mark and Wu (1998), Gourinchas and Tornell (2004) and Burnside et al. (2011).
3 For example, see Mark and Wu (1998) and Gourinchas and Tornell (2004).
2 For
1
when no allowance is made for temporal instability, studies using survey data on exchange rate expectations find
that “the [forward-discount anomaly] is entirely due to expectational errors” (Froot and Thaler, 1990, p. 190).4
In this paper, we advance an alternative explanation of the discount anomaly: it is a byproduct of presuming
that a single conditional probability distribution can account for the process underlying exchange-rate movements
over many decades. In currency and other asset markets, participants revise their forecasting strategies, at least
from time to time, as their understanding of the market process develops, and as economic policy and other
features of the social context within which they make their trading decisions also change. Such change would
lead to shifts in the exchange-rate process, and thus to instability in the BF regression. Indeed, when researchers
look for such instability, they typically find it. For example, Frydman and Goldberg (2007) examine the three
largest currency markets and find that there are stretches of time during which the slope coefficient is largely
negative, while in others it is positive and less than, equal to, or greater than one. This instability – involving
subperiods of both positive and negative biases – implies that contrary to the widespread belief in the field, a
strategy of merely betting against the forward rate is unlikely to deliver “predictable profits.”
The performance of the “naı̈ve carry trade” strategy, a speculation on the existence of the forward discount
anomaly, exactly confirms this point. It involves buying a high-interest-rate currency against a low-interest-rate
one. If the forward rate biasedness is a stable phenomenon, the carry trade strategy should generate on average
large systematic profits. However, empirical evidence suggests that returns to a single-currency carry trade
are highly unstable and subject to sadden crashes (Brunnermeier et al. (2008); Baillie and Chang (2011); and
Richard T. Baillie (2013)). There are periods of time when it is profitable, but dramatic change in the exchange
rate movements can lead to periods where the carry trade delivers huge losses that could wipe out earlier profits.
One of many examples of such events would be the financial crisis of 2008 that has led to a reversal of movements
of many currencies and the apparent collapse of a profit from the carry trade. 5
There is also evidence in the literature that the portfolio carry trade performs better than a single-currency
carry trade. For example, recent papers by Burnside et al. (2006), Eichenbaum et al. (2007), and Hochradl and
Wagner (2010) document that realized cumulative returns to investing in the portfolio of currencies are as high
as cumulative returns to investing in the S&P500. However, other studies find that portfolio carry trade profits
are highly volatile, have negative skew and low Sharpe ratios, which makes it unattractive to investors (Òscar
Jordà and Taylor (2009); and Clarida et al. (2009)). These results are in conflict with those of Burnside et al.
(2006) and Eichenbaum et al. (2007), suggesting that the carry trade profitability is only a matter of time. It
largely depends on the sample period examined and the composition of currencies included in the portfolio. All
this provides additional evidence that the returns from carry trade are highly unstable.6
Although the evidence of instability goes a long way in resolving the puzzle, it leaves open the question of
whether a time-varying risk premium plays a role in accounting for the forward-rate bias that is found once the
empirical analysis recognizes the instability in the data. In this paper, we examine this question.
We first test the BF regression for points of structural change using monthly data for the British pound,
Japanese yen, and German mark exchange rate relative to the U.S. dollar. We rely on two procedures: the
CUSUM squares (CUSQ) test due to Brown et al. (1975) and the one-step Chow test due to Hendry (1979).7
Both tests employ recursive estimation, which enable us to search the data for break points rather than imposing
them a priori. The results of this analysis indicate that the BF regression undergoes shifts at discrete points
of time, implying that the relationship between the future change in the spot exchange rate and the forward
premium is piece-wise linear: we find that each currency market is characterized by at least three extended
4 For
more evidence see Frankel and Froot (1987) and Bacchetta et al. (2009).
instance, in the late 2008 the yen rose by 60% in just two months against the high-yielding Australian dollar, resulting in
huge losses for investors who had to repay back expensive yen-denominated debt (The Economist, December 2009).
6 Òscar Jordà and Taylor (2009) in their study also provide some evidence that returns from carry trade are improved if the risk
premium is taken into account. This is consistent with another argument in our paper that profits in separate regimes are in part
due to risk.
7 Frydman and Goldberg (2007) use the CUSUM test of Brown et al. (1975) instead of the CUSQ test. However, Deng and
Perron (2008) have recently shown that the CUSQ test is superior to the CUSUM test for dynamic regressions, which is the case
that Frydman and Goldberg (2007) and we consider.
5 For
2
subperiods or regimes in which the null of parameter constancy cannot be rejected. Like Frydman and Goldberg
(2007), we find linear pieces of the data for which the forward rate bias is negative and others for which it is
positive for all three currencies.
To examine whether a time-varying risk premium helps in accounting for this bias, we follow Froot and
Frankel (1989) and others and use survey data on exchange rate expectations to decompose the forward rate
bias in each regime into two components, one that depends on a time-varying risk premium and the other on
systematic forecast errors. Our analysis relies on survey data from Money Market Services International (MMSI),
which provides the median of market participants’ point forecasts at the one-month forecast horizon. We find
that the bias in many of the separate regimes of stability is due in part to a time-varying risk premium for all
three currencies examined. For the Japanese yen and German mark currencies, the premium accounts for nearly
half of the forward bias. Although we find that a correlation between forecast errors and the forward premium
also plays a role in most regimes for mark and pound currency markets, this finding is not an indication of any
irrationality: to be useful, one would need to anticipate when structural change occurs and what the correlation
will be when it does.
The remainder of the paper is structured as follows. Section 2 replicates the usual findings of the forward
discount anomaly in the literature for our data. Section 3 shows that the relationship between the future change
in the exchange rate and the forward premium is best approximated as a piece-wise linear. Section 4 explores the
relative contributions of a time-varying risk premium and systematic forecast errors in explaining the forward
rate bias in separate regimes of parameter stability. Section 5 offers concluding remarks.
2
The Standard Test of the Forward Discount Bias
The forward discount anomaly is based on a regression of the actual future change in the spot exchange rate on
the forward discount:
st+1 − st = +β(ft/t+1 − st ) + εt+1
(1)
where εt+1 = ∆st+1 − ∆ŝt+1 is the forecast error, ∆ŝt+1 is the expected change in the future exchange rate,
f pt/(t+1) = ft/(t+1) − st is the forward premium, and ∆st+1 = st+1 − st is the future change in the spot exchange
rate. If investors are assumed to be risk neutral and their expectations are portrayed with REH, then the forward
rate should be an unbiased predictor of the future spot rate, that is α = 0, β = 1, and ηt+1 is a mean-zero white
noise process.
The forward rate unbiasedness hypothesis (FRUH) has been rejected by Bilson (1981), Fama (1984) and
many other researchers who have examined a wide range of developed-country currency markets, and most seem
to agree on the direction of the bias. Indeed, in reviewing the literature, Froot and Thaler (1990) state that
the average point estimate of β in 75 published articles is -0.88 and appears robust to the choice of numeraire
currency. Researchers have concluded from this evidence that the forward rate is a biased predictor of the future
change in the spot rate and systematic profits can be made in the foreign exchange market.
In this study, we estimate equation (1) by ordinary least squares (OLS) for three most commonly traded
currencies: dollar rates of the British pound, German mark and Japanese yen. The data on the forward rates,
spot exchange rates and Eurocurrency interest rates are averages of bid and ask rates at the London close of
trading taken on days corresponding to the survey forecast days obtained from Data Resources Incorporated
(DRIFACS). The sample period spans from December 1982 to February 1997 at monthly frequency. All variables
are transformed into a logarithm form, except for the interest rates.
The results of estimating the BF regression over the full sample are presented in Table 1. Like other studies,
we find that the estimated slope coefficients are not only less than one, but have negative signs in all three cases,
confirming well-known relationship between the change in the exchange rate and the forward premium. While
most slope estimates are not statistically significant, we find that the null of the forward rate unbiasedness is
rejected by the data for all currencies expect the German mark. Test of the null hypothesis that the slope
coefficient equals to one can be rejected for the case of the British pound and Japanese yen currencies at the 1%
3
level.
Table 1 also reports the results from fitting equation (1) with pooled regression model over the whole sample
and also for the sample period considered by Froot and Frankel (1989). The point estimate of the slope coefficient
β from pooling across all currencies is negative, but not statistically different from zero in both cases. We also
estimate pooled regression model with GMM standard errors. The results do not change significantly - the
forward rate unbiasedness hypothesis, α = 0, β = 1, is strongly rejected at the 1% or 5% significance levels in
all cases.
2.1
Expectational Errors vs. the Risk Premium
One possible explanation for the negative forward bias is the existence of a time-varying risk premium in exchange
rates. If foreign exchange market participants are risk averse, they would demand higher rate of return to be
compensated for holding open positions in a risky foreign exchange market. In order for this omitted variable to
cause the slope coefficient in the BF regression to be negative, the risk premium needs not only be time-varying
but negatively correlated with the interest rate differential as shown by Fama (1984).
While such a risk premium could explain the forward bias from a statistical perspective, empirical analysis of
REH-based risk premium models has shown that it is hard to explain excess returns in forward foreign exchange
on the basis of a time-varying risk premium alone: either the coefficient of risk aversion, must be incredibly
large, or else the conditional covariance of consumption and the spot rate must be incredibly high.8 This led
researchers to consider another possibility that market participants in currency markets are irrational - they do
not use efficiently all the available information at time time t, which leads to systematic correlations between
forecast errors and the forward premium.
With the help of the survey data on exchange rate expectations, we can directly determine whether deviations
of β from unity are due to either one of the alternative explanations: a time-varying risk premium or systematic
expectational errors. 9 Following Froot and Frankel (1989), the probability limit of the slope coefficient in the
regression of the future change in the exchange rate on the forward premium can be written as:
β=
cov(st+1 − st , ft/t+1 − st )
var(ft/t+1 − st )
(2)
Defining the risk premium as rpt = f pt −∆ŝt/t+1 and forecast errors as εt = 4st+1 −4ŝt/t+1 , and substituting
these expressions into the expression for the BF slope coefficient, given by equation (1), we get
cov(4ŝt/t+1 + εt , f pt )
var(f pt )
(3)
cov(f pt − rpt/t+1 , f pt ) + cov(εt , f pt )
var(f pt )
(4)
β=
or
β=
Rearranging some terms in equation (3), we obtain the well-known Fama (1984) decomposition:
β = 1 − βrp − βre
cov(rp
(5)
,f p )
t
t/t+1
t+1 ,f pt )
where βrp =
and βre = − cov(η
var(f pt )
var(f pt ) . That is, β is equal to one minus the term arising due
to the existence of a time-varying risk premium, βrp , and minus the term that represents the portion of the
bias caused by systematic expectational errors, βre . Under REH, the forecast errors should be orthogonal to
the information available at time t. Since the forward premium is included as a part of that information set,
βre should be zero. On the other hand, under the hypothesis that the correlation of the risk premium with the
8 For
9 See
reviews of this literature, see Lewis (1995) and Engel (1996).
also MacDonald and Torrance (1990), Frankel and Chinn (1993), Chinn and Frankel (2000)and Verschoor and Wolff (2001).
4
forward discount is zero (no time-varying risk premium), βrp will also be zero.
The existence of a time-varying risk premium is formally tested by fitting a regression of the expected change
in the spot exchange rate on the forward discount:
∆ŝt/t+1 = α2 + β2 f pt/t+1 + vt
(6)
where vt is the random measurement error in the surveys. Under the hypothesis that the risk premium and
the forward discount are uncorrelated, β2 will equal one, while the null of perfect substitutability implies that
α2 = 0 and β2 = 1. It should be noted that β2 coefficient is precisely equal to 1 − βrp in equation (5), reflecting
a portion of the forward discount bias due to the existence of a time-varying risk premium. Further, as Froot
and Frankel (1989) point out, the hypothesis of a zero mean risk premium can be tested by examining whether
the α2 coefficient is significantly different from zero.
Table 2 reports empirical results from estimating pooled regression model across all three currencies over the
whole sample. The slope coefficient is positive and statistically significant at the 1% level. The null hypothesis
of β2 = 1 cannot be rejected at the 5% significance level, which is in accordance with Froot and Frankel’s (1989)
finding that most of the bias in the forward premium can be attributed to expectational errors. Estimating
equation (6) via OLS with GMM standard errors over Froot and Frankel’s sample period that runs from December
1982 through December 1986 produces even stronger rejection of the existence of the risk premium. In fact,
we are unable to reject the hypothesis that all of the bias is due to systematic forecast errors and none due
to a time-varying risk premium. Froot and Frankel (1989) attribute this finding to irrationality that stems
from the presence of heterogeneous traders in the market. More recent study by Bacchetta et al. (2009) has
strengthen their argument by providing additional evidence of systematic expectational errors across different
financial markets, such as the foreign exchange (FX), bond, and stock markets. Using survey data, they find
that forecast errors are also predicable with the same sign and magnitude as excess returns in all three markets.
However, Cavaglia et al. (1994) point out that in the pooled regression estimation the slope coefficient of one
could result from mixing one group of countries where the risk premium is negatively correlated with the forward
premium, with another group of countries where the correlation is positive. Thus, pooling data across countries
might lead to conclude that risk premium is invariant over time or does not play any role for the forward premium
bias. Examining a set of 10 developed-countries currency markets, Cavaglia et al. (1994) are able to show the
existence of a time varying risk premium for some developed countries currency markets when the data are not
pooled across countries.10 Frankel and Chinn (1993) and Chinn and Frankel (2000) also find some evidence that
the bias in the forward rate is attributable to both the existence of a time-varying risk premium and systematic
expectational errors in a larger set of survey data, but mostly over long forecast horizons. However, all these
studies report that the magnitude of forecast errors is much larger than that of a time-varying risk premium.
Table 2 also reports regression results from estimating equation (6) via OLS for each currency individually.
It shows that the slope coefficient estimate for the DM/$ rate over the full sample is still insignificantly different
from unity. Moreover, the intercept is insignificantly different from zero indicating a mean-zero risk premium.
However, the hypothesis of perfect substitutability is rejected for the other two currencies. We find that slope
coefficients for the British pound and Japanese yen relative to the U.S. dollar rate are less than one implying
that there is some evidence of a time-varying risk premium, which corroborates the results of Cavaglia et al.
(1994) and Verschoor and Wolff (2001).
In order to examine the role of systematic forecast errors in explaining the forward bias, we estimate the
regression of forecast errors, Et ŝt+1 − st+1 , on the forward premium following Froot and Frankel (1989) methodology, that is:
Et ŝt+1 − st+1 = α3 + β3 f pt/t+1 + νt+1
(7)
where νt+1 is the random measurement error in the survey data or regression error. If forecast errors are
10 Verschoor
and Wolff (2001) find similar results for Scandinavian currency markets.
5
uncorrelated with the forward premium β3 will equal to 0. Under the null hypothesis of rational expectations
both coefficients should equal zero: α = 0, and β = 0. This test is also known as forecast errors “orthogonality
test” – expectational errors should be orthogonal to the information set available at the time the expectations
are formed when agents use all available information efficiently.
Table 3 summarizes the test results for each currency separately as well as pooled regression estimates across
all currencies. The rejections against the null hypothesis that the slope coefficient is equal to zero occurs at the
10% level for the BP/$ and at the 5% level for the JY/$ rates. On the other hand, the estimate of the slope
coefficient for the GM/$ exchange rate is not statistically different from zero. This can be put as: in two out of
three cases we cannot reject the hypothesis that all of the forward bias is due to systematic expectational errors
and none is due to a time-varying risk premium. The point estimate of βre from the pooled regression model is
statistically significant at the 1% level and the null hypothesis of α3 = 0, β3 = 0 is easily rejected. This provides
a strong evidence of correlations between the forward premium and expectational errors.
The main conclusion arising form Tables 2 and 3 is that deviations from the unbiasedness hypothesis for the
examined currencies seems to be mainly due to systematic forecast errors when estimated for the full sample and,
especially, when the currencies are stacked together in a pooled regression model. By relaxing the assumption
that the risk premium coefficient is the same across currencies and performing the analysis for individual currency
markets, we are able to uncover some evidence of the existence of a time-varying risk premium and its correlation
with the forward premium. Although the magnitude of the bias attributable to expectational errors is estimated
to be much larger than the one due to a time-varying risk premium.
However, there is still a question mark over our conclusions based on figures in Tables 2 and 3 as well as
Froot and Frankel’s (1989) and other researcher’s empirical findings due to the common practice among the
researchers of ignoring the issue of temporal instability of the data. The vast majority of empirical studies that
show the supposed predictability of currency returns are based on estimating time-invariant linear regression
models which assume that the process generating each series is stable in a sample that involves two decades of
data or more. However, when the return history used in estimating the beta is that long, regime changes can
influence the resultant slope coefficient estimate and ignoring this problem will lead to misleading statistical
inference.
3
The Temporal Instability of the BF Regression
In fact, currency returns in the foreign exchange market do not unfold in accordance with a pre-specified
mechanical rule. The knowledge that underpins the market’s forecast grows. Popper (1957) pointed out that if
knowledge grows, then there is no way for any human being to successfully predict the future, since that would
require him to know “today” that which he will only learn, discover, and know “tomorrow”. Hence, as market
participants’ knowledge grows, they would revise their forecasting strategies at least intermittently.
Frydman and Goldberg (2013) show how arguments in Popper (1982, 1990) provide a theoretical foundation
for the proposition in their Keynes-IKE model that market participants nor economists have perfect knowledge
and full understanding of the true model of the economy. Agents test their models and update expectations
functions as new information becomes available. Such revisions of individuals’ forecasting strategies may occur
due to changes in a country’s institutions, political, economic and policy environments, technological shocks to
productivity, and other shifts in social or psychological factors.
Since the social context also changes in ways that cannot be fully foreseen, revisions will involve not just
different betas but also different variables.11 Mangee (2013) and Sullivan (2013) show that in real-world markets,
the way fundamentals, psychology, and social context matter changes over time. There are sub-periods during
which the relationship is relatively stable, but eventually knowledge or some other facet of the process changes at
points in time and in ways that do not conform to a mechanical rule, leading to a new relationship. Therefore, in
the foreign exchange market not only are the fundamental processes unstable, but different sets of fundamentals
11 See
Goldberg and Frydman (1996) for more details.
6
matter in different subperiods.
Thus, the correlations between the future change in the exchange rate and the forward premium will depend
on the fundamental variables that agents use to forecast and the way they interpret this information when
forming their expectations about future returns, that is it will depend on their forecasting strategies. Market
participants sometimes revise their forecasting strategies as new information on fundamentals becomes available,
and, when they do, they cause the correlations in the data to shift.
To see this, we can write the projection of the risk premium, rp
ˆ t/t+1 = 4ŝt/t+1 − f pt , on the forward
premium, f pt , as follows:
E(rp
ˆ t/t+1 , f pt ) = cov(4ŝt/t+1 , f pt ) − var(f pt )
(8)
From equation (8) it can be seen that revisions in forecasting strategies will lead to shifts in the relationship
between market participants’ expectations of 4ŝt/t+1 and the causal variables that enter into their forecasting
strategies. This in turn, will cause the correlations in the data to shift. So we would not expect the slope
coefficient in the BF regression to be constant over time.
Consequently, empirical studies aimed at explaining the forward discount puzzle should allow for correlations
in the data to be temporarily unstable. Ignoring structural change may obscure empirical results. Indeed, Baillie
and Bollerslev (2000), Maynard and Phillips (2001), and some other studies have recognized that when structural
breaks are ignored or high persistence in the data is not taken into account, spuriously unfavorable empirical
results may be obtained. Frydman and Goldberg (2007) and many other studies that have used developed
countries’ data provide some evidence of such structural instability of the Bilson Fama regressions (see Chinn
(2006); Flood and Rose (2002); Sakoulis and Zivot (2000); Lewis (1995), and Engel (2011)). They find that
forward premium bias is significantly greater than one during some periods, while less than one, or zero during
others. Moreover, several other researchers document that the slope coefficient in the regression of excess returns
on the forward premium can vary considerably over subsamples.12 More particularly, the empirical evidence
provided by these studies suggests that there is a significant negative correlation between the forward premium
and the future change in the spot rate in the 1980s, and simply no significant relation between the two during
1970s and 1990s.
3.1
A Piece-Wise Linear Relationship
This study also looks at the problem of the structural instability of the correlations between actual returns and
the forward premium in the FX market, but utilizes more powerful tests for structural change. Under Imperfect
Knowledge Economics (IKE), market participants typically revise their forecasting strategies in a guardedly
moderate way. This means that unless individuals have specific reasons to change their forecasting strategies,
they will adhere to their existing strategy or only alter it in a gradual fashion. In other words, the impact of
these revisions on forecasts does not outweigh the influence stemming from trends in fundamentals themselves.
This implies that there are stretches of time when traders leave their forecasting rules unaltered or slightly
modified, so that the relationship between the forward premium and the risk premium stays relatively constant
within the regimes.
To allow for such type of structural instability in the analysis, the relationship between excess returns and
the forward premium is approximated as piece-wise linear, that is, there are long stretches of time (the linear
pieces) during which correlations are stable, but across linear pieces the correlations are different. In each of the
identified regimes, the relationship between the forward premium and realized returns is estimated with constant
parameters.
If a market participant does have reasons to suspect or anticipate a genuine change, he cannot be sure about
his beliefs, let alone about the precise date or nature of the change. Therefore, the Keynes-IKE model fully
pre-specifies neither when the trends in fundamentals will change nor when market participants will revise their
forecasting strategies. This leads us to consider structural change tests that are based on recursive estimation
12 See
Velasco and Moon (2010), Chinn (2006), Baillie and Bollerslev (2000), Engel (1996), and Naka and Whitney (1995).
7
in order to determine de-facto structural breaks endogenously, rather than imposing de-jure structural breaks a
priori.
3.2
Testing for Structural Breaks with Unknown Timing
Before doing formal structural change analysis, we estimate rolling regressions at 4 year horizon for all three
currencies. Figures 1 a-c represent the results of the rolling regression with the US dollar being the numeraire
currency vis a vis the British pound, Japanese yen, and German mark. For all the results solid line depicts
the OLS estimator of the rolling slope coefficient and the broken lines are the conventional two standard error
bands around the point estimates of the time-varying parameter beta coefficient. These graphs are informal
tests of structural change, but they clearly indicate that the parameter estimates in the regression of actual
change in the spot exchange rate on the forward premium apparently change over time for all three currencies.
They suggest the presence of at least three different regimes of relative stability in the slope coefficient for each
currency market. The classic large negative values of β coefficient estimates mostly occur in samples ending in
the mid-1980s, with the estimated beta coefficients being in the range of -5 to -12 for the GM/$, BP/$, and
JY/$. The rejection of the forward unbiasedness hypothesis during 1990s is less severe for all three currencies
and the slope coefficients become somewhat positively biased in the late 1990s.
Although the estimates from the rolling regressions provide some evidence of structural instability, they do
not tell us exactly where the break has occurred. Therefore, in order to formally examine the stability of the
slope coefficient in the Bilson Fama regression, we need to implement structural break tests that allow us to
search for unknown break points in time. A simple and direct method for deriving approximate estimates of
structural break dates is to use tests based on recursive estimation. In this paper, we rely on ADL specification
and CUSUM of squares test (CUSQ) along with one-step Chow test. The ADL model of order 2 is applied to
account for the possibility of non-stationary or near unit root (I(1)) processes of the series while investigating the
pattern of the structural stability of the relationship between the forward premium and foreign excess returns.
Frydman and Goldberg (2007) used CUSUM test rather than CUSQ test to determine structural breaks
in the Bilson Fama regression. However, there exists evidence in the literature that CUSUM of squares test
outperforms CUSUM test in a regression model with mixing assumptions about the regressors and residuals. For
instance, Deng and Perron (2008) show that the null distribution of the CUSQ test remains asymptotically the
same, regardless of whether or not the errors are serially correlated, while Ploberger (1989) also demonstrates
that the CUSQ has non-trivial local power against a wide range of patterns of heteroskedasticity. Nielsen and
Sohkanen (2011) extend the work of Deng and Perron (2008) to cover non-stationary dynamic modeling and
show that the test remains valid in the linear autoregressive distributed lag model.
One of the limitations of the CUSQ test is that it can only tell us whether the structural change has occurred
or not. This test cannot be used for the purpose of identifying the exact timing of break points (Ploberger and
Kramer, 1992, 1996). Therefore, we use it in combination with one step recursive Chow test proposed by Hendry
(1979). This test is based on the sequence of studentized recursive residuals, which is also appropriate for unit
root process modeled in the ADL framework. The one step Chow test is essentially a prediction test, which is
computed recursively over the whole sample and tests one-step-ahead forecast failure at each time step. This
test is available in PcGive package (Hendry, 1986) as part of the model misspecification diagnostics.
Another disadvantage of the CUSQ test is that it is developed for a single structural break and might not be
effective when there are multiple breaks in the data (Hansen, 1992; Perron, 2005; and Deng and Perron, 2008).
To deal with this shortcoming, we use a sequential CUSQ procedure to examine the stability of the parameters.
We start with a full sample of the data. The CUSUM of squares test begins with some initialization period
and then rolls the regression forward adding one observation at a time and computing recursive residuals for
each regression. Once the plot of the cumulative sum of squared residuals crosses the boundary (associated with
the 5 percent significant level of the test), the null hypothesis of stability overtime of the intercept and slope
parameters is rejected. To determine the exact timing of the first breakpoint, we split the sample at the point of
8
intersection of CUSQ test with its critical bound and run one-step Chow tests to identify when the first break
has occurred. To test for the presence of additional breaks we split the data into two subsamples at the point of
structural break, and re-run the CUSQ test for each subperiod. If the test identifies that the break has occurred,
one-step Chow test is used to produce the location of the second break. This procedure is repeated in a similar
manner until no additional structural break points can be identified.
However, if CUSQ remains within its critical bounds failing to detect any structural break when estimated
over the whole sample, we split our sample in half which leaves around 7 years of observations for each subsample
(it is true for the case of mark and yen currency markets). Then CUSUM of squares test is applied to both
of the subsamples starting with the first observation and until the test indicates that the first break point has
occurred. After one-step Chow tests have determined the exact location of the break, the sample is split at this
point and the rest of the analysis proceeds in the similar fashion.
3.3
Interpreting Estimation Results
Table 4 and figures 2 a-c summarize the results of iterative CUSQ test applied to ADL(2) model. Our structural
change analysis reveals that the relationship between the forward premium and excess returns is temporarily
unstable. The table reports numerous points of structural breaks for all three currencies. Most of the break
dates, like in Frydman and Goldberg (2007), are related to major policy shifts or changes in the exchange rate
arrangements. Table 5 summarizes major events surrounding break points. For instance, structural breaks in
late 1985 and early 1990s come about because of the 1985 Plaza agreement and reunification of Germany in
October 1990 respectively.13
For practical reasons we examine the relationship between excess returns and the forward premium within
the regimes of relative parameter stability that include two years of data or more. Table 6 summarizes the
results of piece-wise linear estimation of the Bilson-Fama regression.
The slope coefficients in the subsample BF regressions exhibit large fluctuations form period to period. For
each currency market there are at lest three separate regimes of relative parameter stability. Inspection of Table
6 reveals that the forward premium biases are generally persistent and switch sign occasionally. Most of the
estimated slope coefficients seem to move in a similar fashion - they change from being significant and negative
in the 1980s, insignificant in the early 1990s to significant and positive in the second half of the 1990s. The
forward rate unbiasedness hypothesis is consistently rejected across all of the regimes of the BP/$ and GM/$
rates. However, we fail to reject the FRUH hypothesis in most of the regimes of the JY/$ rate. The rejection
of the FRUH in the first and third regimes occurs mostly due to standard errors being so large (we are able to
reject the H0 : β̂ = 1), while the rejection of the FRUH in the fourth subsample is due to the slope coefficient
being close to one (both H0 : β̂ = 1 and H0 : α̂ = 0, β̂ = 1 are rejected). Another interesting finding is that in
the case of the BP/$ rate, beta averages out to be significantly positive and greater than 1. This contrasts well
with the time-invariant estimates of the slope coefficient over the whole sample.
Significant estimates of the beta slope coefficient in different regimes and their wide range indicate that
the restriction of parameters to be constant in conventional regression models is inappropriate. The standard
deviations of beta are large, in particular in the case of the BP/$ and GM/$ rates, suggesting large variations
in the time-varying forward bias. These findings confirm those of Frydman and Goldberg (2007) implying that
there exists no stable systematic correlation between future excess returns and the forward premium.
Given the evidence that beta is not uniformly negative, what needs to be explained is not the statistically
significant negative bias of the forward premium, but rather why the correlations between the forward premium
and excess returns are sometimes negative, sometimes positive, less than one, greater than one, or insignificantly
different form zero. Such high instability of the forward discount predictions implies that an easy forecasting rule
like betting against the forward premium cannot generate constant profits in a world of imperfect knowledge. It
13 In order to check the robustness of our results we also use other procedures of testing for structural change. In particular, we
consider Andrews-Quandt test and Fluctuation test. The results do not change significantly and our main conclusions stays the
same. The robustness check results are not reported in this study but are available upon request.
9
can be profitable during some periods, but deliver losses during others. No one can foresee when these temporary
correlations will occur or disappear, whether they will be positive or negative and how long they will last.
4
Subsample Analysis: More Evidence for a Time-Varying Risk Premium
In this section, we examine the relative contributions of nonzero correlations between the forward premium and
the risk premium and/or systematic forecast errors to the variation of the slope coefficient β allowing for a piecewise linearity. Frydman and Goldberg (2007) present some evidence that there is a time-varying risk premium.
They show that compensation for risk or uncertainty plays an important role in the structural instability of the
Bilson Fama regression. The authors also relate the movements in the risk premium to the movements in the
“aggregate gap” – the difference between the historical benchmark value and the expected value of the exchange
rate that have an effect on the revisions of forecasting strategies. However, Frydman and Goldberg (2007) leave
open the question of how much of the variation in β is due to the correlation in the risk premium and the
correlation in forecast errors. This paper addresses this issue.
We start our analysis by calculating components of the failure of the unbiasedness hypothesis for each regime
of relatively stable forecasting strategies. We only consider those regimes where either hypothesis α = 0, β = 1
or β = 1 has failed. The fifth and sixth columns of Table 7 capture the shares of the bias attributable to a
time-varying risk premium, βrp , and systematic errors, βre , while the third column reports the implied Fama
beta coefficient, 1 − βrp − βre . The constructed values for the percentage of the bias explained by βre and βrp
(columns seven and eight) suggest the following: on the one hand, the forecast errors contribute significantly to
a forward discount bias for all three currency rates. On the other hand, the risk premium contributes largely
to a forward bias for the JY/$ and GM/$ exchange rates but not for the BP/$ rate. Interestingly, the risk
premium and forecast errors are of opposite sign in one of the regimes for the case of the BP/$ exchange rate.
This implies that the risk premium is contributing to the forward bias by pushing the estimate of β above 1.
This result is in line with Froot and Frankel (1989) and Frankel and Chinn (1993), who find that “in these cases,
risk premiums do not explain a positive share of the forward discount’s bias”. Therefore, we record it in the
table as all 100% of the bias is due to the systematic forecast errors. By taking average values of the columns
seven and eight across the regimes of parameter stability for each currency, we can see that for the dollar rate of
German mark and Japanese yen a time-varying risk premium explains nearly half (43% and 34% respectively)
of the deviation of β from 1. By contrast, in case of the British pound the proportion of the bias explained by
the risk premium is only 12%, while the other 82% is explained by forecast errors.
These findings are in stark contrast with the results obtained by Froot and Frankel (1989). They report that
the portion of the bias attributable to systematic forecast errors is much larger in magnitude compared with
the risk premium for different currencies, from which they infer that most of the bias in the forward premium is
due to the expectational errors. However, the figures in Table 7 suggest the opposite to be true - at least for the
monthly GM/$ and JY/$ rates a time-varying risk premium plays an important role in explaining the forward
bias. We also find that the correlations between prediction errors and the forward premium and between the
risk premium and the forward premium are positive during some periods, while negative during others. This
implies that both terms, βre and βrp , are highly unstable.
4.1
Testing for the Presence of a Time-Varying Risk Premium
In order to put some significance level on the point estimates of the relative contribution of the risk premium
and forecast errors to the variation in the Fama’s beta calculated in Table 7, formal regression analysis needs
to be conducted. The portion of the bias attributable to a time-varying risk premium can be directly tested by
running the following regression within the separate regimes of relative parameter stability:
rpt = α4 + β4 f pt + ut
10
(9)
A test whether the β4 = 0 is equivalent to testing whether the values of the βrp in Table 7 are statistically
different from zero. Alternatively, the relationship between the risk premium and the forward premium can be
examined by fitting equation (6) for each of the identified subperiods. These two regressions will provide the
same answer, because the slope estimate in the regression of the expected change in the future spot rate on the
forward premium is precisely equal to 1 − βrp .
Table 8 reports the results of relative importance of the risk premium component by fitting equation (9) for
each linear piece of the data. The results indicate quite strongly the presence of a time-varying risk premium we find that estimated coefficients are statistically different from zero across different regimes, as opposed to the
results of Froot and Frankel (1989). The null of βrp = 0 is rejected at the 1% significance level in all subsamples
for the GM/$ exchange rate and at the 10% level in one of the regimes for both BP/$ and JY/$ exchange rates.
That βrp is significantly different from zero may be stated differently: we cannot reject the hypothesis that all
of the forward bias is due to the risk premium, or equivalently we cannot reject the hypothesis that none of the
forward bias is due to systematic expectational errors. The null of perfect substitutability is consistently rejected
in the data in all subperiods across three currencies. It should be also noted that the estimated slope coefficient
βrp is sometimes negative, sometimes positive, greater than one, or less than one, implying a time-varying nature
of the premium.
For the case of the BP/$ rate the risk premium is statistically significant but has the wrong sign in one out of
three regimes. This suggests that expectational errors are responsible for most of the bias in all subsamples for
pound market. On the contrary, most of the risk premium estimates are statistically significant and all of them
have the correct sign in the case of the GM/$ and JY/$ rates providing strong evidence of correlations between
the risk premium and the forward premium, which explains an important portion of the forward bias. Our
findings are consistent with Cavaglia, Verschoor and Wolf’s (1994) findings. However, we are able to uncover
more evidence of a time-varying risk premium once the structural instability of the parameter estimates is taken
into account. The results in Table 8 indicate the existence of a constant or time-varying risk premium in every
subperiod across all three currency markets (the null of perfect substitutability is rejected in all cases).
Similarly, the portion of the bias explained by systematic errors can be examined by regressing expectational
errors on the forward premium for every regime of statistical parameter constancy given by equation (7), and
testing whether β3 = 0, which is equivalent to testing βre = 0. The results are reported in Table 9. For pound
and mark currencies there is some evidence that the forecast errors are systematically related to the forward
premium, although some of the individual coefficients are insignificantly different from zero. The expectational
errors are statistically significant in most of the regimes for the British pound against the dollar rate. However,
in all of the subperiods of the Japanese yen and German mark markets we fail to reject the hypothesis that
βre = 0, or equivalently, we cannot reject the hypothesis that all of the bias is due to a time-varying risk
premium. This suggests that there is little evidence that expectational errors play any role or are correlated
with the risk premium for yen and mark currency markets.
Table 10 combines the estimation results of Tables 6, 8 and 9. Clearly, the estimates of systematic forecast
errors, βre , are indeed of much larger magnitude than those on the risk premium, βrp , for the case of the dollar
rate of the British pound. However, this should not necessarily be interpreted as evidence that investors’ forecasts
are irrational. Under IKE, temporal correlations between systematic errors and available information, which
includes the forward premium, may appear from time to time when the market, coping with imperfect knowledge,
occasionally mispredicts the movements in the future change in the exchange rate. Eventually, it will correct
itself and investors will revise their forecasting strategies while exploiting information in the forecast errors. The
IKE implies that, if prediction errors are correlated with the forward premium during some subperiods, they
should be highly unstable, occur and disappear at unanticipated points in time.
A different conclusion is reached for the case of the Japanese yen and German mark currencies. The estimated
slope coefficients of βrp and βre are of similar magnitudes. However, βre is insignificantly different from zero
in most of the regimes. We are unable to reject the hypothesis that forecast errors account for no portion
11
of the forward rate biasedness in all regimes of the GM/$ and JY/$ exchange rates. On the contrary, βrp is
statistically significant across all subsamples for GM/$ exchange rate and in one of three regimes for the JY/$
rate, suggesting that a time-varying risk premium appears to play at least as equally important role as forecast
errors in explaining the forward bias for the JY/$ rate and be mostly a dominant cause of the failure of the
forward rate unbiasedness hypothesis in the GM/$ currency market.
To summarize, we find that accounting for structural instability of market participants’ forecasting strategies
gives rise to more evidence on the existence of the risk premium and its important role in driving currency
returns. We are able to show that it is significant in many of the regimes across three currencies, and accounts
for a larger portion of the forward premium bias in mark and yen currency markets. This finding is particularly
interesting since it indicates that it is hazardous to draw conclusions, as most researchers have done, about the
sign, magnitude and importance of the risk premium from estimating time-invariant correlations in the data and
pooling it across currencies.
5
Concluding Remarks
In this paper, we show that the forward discount puzzle stems not from the presence of irrational market
participants, but from the practice among researchers of ignoring the problem of temporal instability. We point
out that correlations between the future change in the spot exchange rate and the forward premium are likely
to be temporally unstable. They depend on how market participant form their market expectations about
future returns, that is, on their forecasting strategies. Under IKE, rational market participants sometimes revise
their strategies as new information on fundamentals becomes available due to changes in policies, institutions,
exchange rate arrangements, and social context. But, when they do, they cause the correlations in the data to
shift.
Our structural break analysis of the Bilson-Fama regression for three mostly traded currencies provides
evidence of multiple points of instability in all three developed-country currency markets examined. This study’s
result indicate that correlation between the future change in the exchange rate and today’s forward premium is
not negative as is widely believed. Rather, the bias is sometimes positive, sometimes negative, and sometimes
zero. This implies that successive speculation in the foreign exchange market is not as simple as suggested by
the voluminous literature on international finance. Indeed, the “predictable” profits cannot be made by simply
betting against the forward rate. Although, this rule delivers profits in some subperiods for some currencies, it
stops being profitable at moments of time that cannot be foreseen. No one can precisely specify ahead of time
when the correlation might be negative and for how long, so no one can know in advance when it might be
profitable to bet against or with the forward rate. In order for investors to systematically make money by taking
either short or long positions in the forward exchange market based on the value of the forward premium, they
would need to anticipate when the structural change will occur, otherwise it will not lead to significant money
making opportunities.
This instability of the BF regression contradicts behavioral-finance models’ assumption that speculators
invariably overreact or underreact in a fixed way, as well as their prediction of a negative forward-rate bias. As
Fama has observed, “apparent overreaction to information is about as common as underreaction, and post-event
continuation of pre-event abnormal returns is about as frequent as post-event reversals” (Fama, 1998, p. 283).
In addition to this finding, our paper makes another contribution to the empirical assessment of the source
of the forward premium bias. This paper uses a survey data on exchange rate expectations to decompose the
bias into two components: a time-varying risk premium and systematic forecast errors within separate regimes
of relative parameter constancy.
Our results indicate that allowing for the structural instability reveals more evidence for the existence of a
time-varying risk premium. Using several econometric specifications, the study finds that the correlations in
the data in all three currency markets can be traced to the importance of risk. Moreover, the risk premium is
responsible for a larger portion of the bias in the case of the Japanese yen and German mark currency markets.
12
These findings are in contrast with those from Froot and Frankel (1989) who find estimates of the risk premium
component that are insignificantly different from zero, concluding that it does not account for a positive share
of the bias. Our findings imply that estimating time-invariant correlations in the data can be misleading, and
that when analyzed properly they provide evidence not of irrationality but of the importance of risk.
These results have important implications for economic theory and for re-regulating financial system. In
particular, the significance of the analysis is supported by the intense debate over the appropriate degree of
government interventions into the foreign exchange market. Monetary authorities in many developed countries
are concerned about pronounced exchange rate swings that is why they tend to intervene or regulate their
exchange rate markets. If returns in financial markets were generally driven by irrationally on the part of
market participants, there would be a significant role for government regulators to limit this irrationality, for
example, by actively intervening in these markets to limit price fluctuations. However, if returns are driven by
risk and other fundamentals, as we show in this study, then the need for official intervention would be much
more limited.
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15
Table 1: The Standard Bilson Fama Regression
Country
Obs
UK
171
Japan
171
Germany
171
Pooled OLS
513
Pooled with
GMM
Pooled OLS
513
Pooled with
GMM
FF
FF
α̂
β̂
0.005
(0.004)
-0.011***
(0.004)
-0.002
(0.003)
-2.128
(1.305)
-3.999***
(1.334)
-0.361
(0.920)
-0.002
(0.001)
-0.002
(0.001)
-0.006*
-0.639
(0.506)
-0.672
(0.509)
-1.763
(0.001)
-0.003
(0.001)
(1.114)
-1.853
(1.152)
t-stat
β̂ = 1
t-stat
β̂ = 0
F-prob
α̂ = 0, β̂ = 1
R²
DW
-2.39**
-1.63
0.0398
0.015
1.81
-3.75***
-2.99***
0.0009
0.050
1.77
-1.48
-0.39
0.3057
0.001
1.83
-3.24***
-1.26
0.0028
0.003
1.78
-3.29***
-1.32
0.0024
0.003
1.78
-2.48**
-1.58
0.0319
0.017
1.81
-2.48**
-1.61
0.0258
0.018
1.95
***, ** and * denote statistical significance at the 1%, 5% and 10% levels respectively. The number in parenthesis indicates robust
standard errors. The critical values of dL and dU for null hypothesis of no autocorrelation with over 100 observation and one explanatory
variable are 1.654 and 1.694, respectively at the 5% significance level. FF stands for Froot and Frankel’s (1989) sample 1982/12 -1986/12.
Table 2: Test of Perfect Substitutability
Country
Obs
α̂2
UK
171
Germany
171
Japan
171
(0.001)
(0.487)
Pooled OLS
513
0.0013***
0.732***
(0.0004)
(0.163)
0.0013***
0.766***
(0.0004)
(0.161)
0.0021**
1.191***
(0.0009)
(0.352)
0.0012
0.947**
(0.0010)
(0.379)
Pooled with
513
GMM SE
Pooled OLS
Pooled with
GMM SE
FF
FF
0.004***
β̂2
0.208
(0.001)
(0.381)
0.001
1.108***
(0.001)
(0.284)
-0.002
-0.769
t-stat
t-stat
F-prob
α̂2 = 0
β̂2 = 1
α̂2 = 0, β̂2 = 1
3.05***
-2.08**
1.30
R²
DW
0.0101
0.002
0.90
0.37
0.4310
0.082
1.07
-1.66*
-3.63***
0.0006
0.015
1.17
2.78***
-1.63
0.0048
0.038
1.03
2.92***
-1.46
0.0057
0.042
1.03
2.20**
0.54
0.0859
0.073
1.13
1.19
-0.14
0.2820
0.041
1.20
***, ** and * denote statistical significance at the 1%, 5% and 10% levels respectively. The number in parenthesis indicates robust
standard errors. The critical values of dL and dU for null hypothesis of no autocorrelation with over 100 observation and one explanatory
variable are 1.654 and 1.694, respectively at the 5% significance level. FF stands for Froot and Frankel’s (1989) sample 1982/12 -1986/12.
16
Table 3: Test of Rational Expectations
Country
Obs
α̂3
β̂ 3
UK
171
Germany
171
(0.003)
(0.918)
Japan
171
-0.009**
-3.230**
(0.004)
(1.375)
Pooled OLS
513
-0.003**
-1.373***
(0.001)
(0.518)
Pooled with
513
-0.003**
-1.397***
(0.001)
(0.521)
-0.008**
-2.953**
(0.003)
(1.169)
-0.003**
-1.008**
(0.001)
(0.486)
GMM SE
Pooled OLS
Pooled with
GMM SE
FF
FF
0.002
-2.335*
(0.004)
(1.383)
-0.003
-1.468
t-stat
t-stat
F-prob
α̂3 = 0
β̂ 3 = 0
α̂3 = 0,β̂ 3 = 0
0.34
-1.69*
-1.23
R²
DW
0.5205
0.017
1.72
-1.60
0.1997
0.015
1.93
-2.55**
-2.35**
0.0325
0.032
1.76
-2.14**
-2.65***
0.0039
0.014
1.80
-2.28**
-2.68***
0.0038
0.014
1.80
-2.45**
-2.52**
0.0139
0.042
1.99
-2.07**
-2.07**
0.0064
0.007
1.78
***, ** and * denote statistical significance at the 1%, 5% and 10% levels respectively. The number in parenthesis indicates robust
standard errors. The critical values of dL and dU for null hypothesis of no autocorrelation with over 100 observation and one explanatory
variable are 1.654 and 1.694, respectively at the 5% significance level. FF stands for Froot and Frankel’s (1989) sample 1982/12 -1986/12.
Table 4: Structural Break Test Results
Currency
Break Points
(sample period)
Sub-periods of
Number
relative stability
of obs
BP/$
1985/1
1982/12-1985-1
26
(1982/12- 1997/2)
1991/1
1985/2-1991/1
72
1992/8
GM/$
(1982/12- 1997/2)
1984/1
1991/2-1992/8
19
1992/9 - 1997/2
54
1982/12-1984/1
14
1991/1
1984/2-1991/1
84
1992/9
1991/2-1992/9
20
1994/9/1995/6
1992/10 - 1994/9
24
1995/7 - 1997/2
20
JY/$
1985/7
1982/12-1985/7
32
(1982/12- 1997/2)
1988/3
1985/8-1988/3
32
1990/12
1988/4-1990/12
33
1995/1/6
17
1991/1-1995/1
49
1995/7-1997/2
20
Figure 1: Rolling Regression Estimates of Beta - 48 month sample window
(a) BP/$
(b) GM/4
(c) JY/$
18
Figure 2: Structural Change Test Results
BP/$ Spot Exchange Rate
1.0
0.9
0.8
0.7
0.6
0.5
82 83
84
85
86
87
88
89
90
91
92
93
94
95
96
94
95
96
94
95
96
(a) BP/$ Structural Break Points
GM/$ Spot Exchange Rate
3.6
3.2
2.8
2.4
2.0
1.6
1.2
82 83
84
85
86
87
88
89
90
91
92
93
(b) GM/$ Structural Break Points
JY/$ Spot Exchange Rate
280
240
200
160
120
80
82 83
84
85
86
87
88
89
90
91
92
93
(c) JY/$ Structural Breakpoints
19
Table 5: Historical Events Surrounding Structural Breaks
Country
Structural
(sample period)
Breaks
Germany
1984/1
UK
1985/1
Economic and historic events corresponding to breakpoints
ˆ The dollar set new records against the British pound,
the German mark and some other European countries
ˆ German bank and other European central banks,
as well as Federal Reserve, intervened heavily
to halt appreciation of the dollar (1985/01-05).
Japan
1985/7
ˆ Plaza Agreement. G-5 Banks intervened actively
in the foreign exchange market. July 1985 corresponds
approximately to the end of the rising dollar
(the USD peaked in February 1985)
ˆ Louvre accord agreement. G-7 central banks’
Japan
1988/3
Japan
1990/12
Germany, UK
1991/1
Germany, UK
1992/8
Germany
1994/9
ˆ Fed FX market intervention in collaboration
Japan
1/6/1995
with the Bank of Japan to lower the yen.
intervention to halt dollar depreciation (1987/06).
ˆ 14 times lowering rate during 1990/09-1991/12
(FFR, 8.20% -4.43%)
ˆ The reunification of Germany: monetary on 1 July 1990
and political on 3 October 1990.
ˆ The crisis in the exchange rate mechanism that brought
down the Pound and Mark among other European currencies.
Table 6: The Structural Instability of the Bilson Fama Regression
Currency
Subperiods
of Stability
BP/$
1982/12-1985-1
1985/2-1991/1
JY/$
β̂bf
0.015**
-0.661
(0.006)
(4.466)
0.013
-6.028**
(0.011)
(2.942)
-0.009*
8.644***
(0.004)
(2.983)
1984/2-1991/1
-0.017**
-4.77*
0.007
2.485
1992/10 - 1994/9
-0.020
6.787**
0.005
2.516
1982/12-1985/7
-0.014
-5.468
0.011
3.711
1985/8-1988/3
-0.025
-2.563
0.016
7.949
1988/4-1990/12
-0.013724
-6.611*
0.010709
3.825
-0.006599
1.141
0.003794
2.815
1992/9 - 1997/2
GM/$
α̂bf
1991/1-1995/1
t-tests
F-test
β̂bf = 1
α̂bf = 0, β̂bf = 1
-0.37
R²
DW
0.0552
0.001
2.19
-2.39**
0.0024
0.057
1.83
2.56**
0.0457
0.139
1.98
-2.32**
0.0411
0.042
2.12
2.30**
0.0034
0.177
2.03
-1.74*
0.1493
0.068
1.97
-0.45
0.0238
0.003
1.92
-1.99*
0.1293
0.088
1.98
0.05
0.2257
0.003
1.91
***, ** and * denote statistical significance at the 1%, 5% and 10% levels respectively. The number in parenthesis indicates robust
standard errors. The critical values of dL and dU for null hypothesis of no autocorrelation with less than 50 observation and one
explanatory variable are 1.503 and 1.584. (for less than 30 obs the numbers are 1.352 and 1.489), respectively at the 5% significance level.
20
Table 7: Causes of the Forward Premium Bias
Currency
Sub-periods of
β bf =
(sample period)
relative stability
1 − β rp − β re
BP/$
1982/12-1985-1
-0.661
1985/2-1991/1
-6.028
7.028
-1.96
8.988
0
100
1992/9 - 1997/2
8.644
-7.644
-0.110
-7.534
1.44
98.56
12.89
87.11
GM/$
bias
β rp
β re
β rp
β re
%
%
1.661
0.618
1.043
37.23
62.77
1984/2-1991/1
-4.766
5.766
2.229
3.537
38.66
61.34
1992/10 - 1994/9
6.787
-5.787
-2.671
-3.116
46.15
53.85
42.41
57.59
1982/12-1985/7
-5.468
6.468
1.212
5.256
18.75
81.25
1985/8-1988/3
-2.563
3.563
2.025
1.538
56.82
43.18
1988/4-1990/12
-6.611
7.611
1.918
5.693
JY/$
25.20
74.80
33.59
66.41
Table 8: Sub-sample Analysis of the Risk Premium Component
Currency
Subperiods
of Stability
α̂4
β̂ 4 (β̂rp )
t-stat
α̂4 = 0
t-stat
β̂ 4 = 0
F-prob
α̂4 = 0,β̂ 4 = 0
R²
DW
BP/$
1983/4-1985-1
-0.004**
(0.002)
0.009
(0.004)
-0.006***
(0.001)
0.618
(1.168)
-1.959*
(1.001)
-0.110
(0.803)
-2.75**
-0.33
0.0350
0.012
1.29
2.30**
-1.96*
0.0705
0.052
0.73
-4.34***
-0.14
0.000
0.0004
1.50
0.008***
(0.002)
0.002
2.229***
(0.723)
-2.671***
4.07***
3.082***
0.0005
0.104
1.46
0.60
-3.15***
0.0000
0.340
1.99
(0.003)
(0.847)
1982/12-1985/7
0.001
(0.004)
1.212
(1.304)
0.15
0.93
0.0887
0.028
1.15
1985/8-1988/3
0.009
(0.005)
0.012***
(0.002)
2.025
(2.705)
1.918*
(1.016)
1.59
0.75
0.0719
0.018
2.16
4.17***
1.89*
0.0001
0.103
1.84
1985/2-1991/1
1992/9 - 1997/2
GM/$
1984/2-1991/1
1992/10 - 1994/9
JY/$
1988/4-1990/12
***, ** and * denote statistical significance at the 1%, 5% and 10% levels respectively. The number in parenthesis indicates robust
standard errors. The critical values of dL and dU for null hypothesis of no autocorrelation with less than 50 observation and one
explanatory variable are 1.503 and 1.584. (for less than 30 obs the numbers are 1.352 and 1.489), respectively at the 5% significance level.
21
Table 9: Sub-sample Analysis of the Importance of Forecast Errors Component
Currency
Sub-periods of
relative stability
BP/$
1982/12-1985-1
GM/$
JY/$
-0.011*
1.043
(0.006)
(4.408)
1985/2-1991/1
-0.0217*
8.987***
(0.012)
(3.244)
1992/9 - 1997/2
0.015***
-7.534**
(0.005)
(3.034)
1984/2-1991/1
0.009
3.537
(0.007)
(2.525)
1992/10 - 1994/9
0.018**
-3.116
(0.006)
(2.776)
1982/12-1985/7
1985/8-1988/3
1988/4-1990/12
R2
DW
0.1996
0.002
2.31
2.77***
0.004
0.099
1.54
-2.81***
0.0149
0.106
1.97
1.40
0.36
0.023
2.19
-1.12
0.0098
0.041
1.82
-1.26
0.4465
0.051
1.74
-0.20
0.0998
0.001
2.15
-1.37
0.1033
0.057
1.92
β̂re
α̂re
-0.013
-5.256
(0.012)
(4.157)
-0.016
-1.538
(0.015)
(7.523)
-0.002
-5.693
(0.012)
(4.141)
t-stat
F-prob
βˆre = 0
α̂re = 0, β̂re = 0
0.24
***, ** and * denote statistical significance at the 1%, 5% and 10% levels respectively. The number in parenthesis indicates robust
standard errors. The critical values of dL and dU for null hypothesis of no autocorrelation with less than 50 observation and one
explanatory variable are 1.503 and 1.584. (for less than 30 obs the numbers are 1.352 and 1.489), respectively at the 5% significance level.
Table 10: Summary of Test Results of Two Competing Alternatives
Currency
BP/$
Regime
1982/12-1985-1
1985/2-1991/1
GM/$
JY/$
β̂ =
α̂2
β̂ 2
β̂ rp
β̂er
1 − β̂rp − βˆre
(α̂2 = 0)
(β̂ 2 =1)
(β̂rp = 0)
(β̂er = 0)
-0.661
0.004***
0.382
0.618
-1.043
(2.76)
(-0.53)
(0.523)
(0.24)
-0.009**
-2.959***
-1.959*
-8.987***
(-2.30 )
(1.96 )
(-1.96)
(2.77)
1.110
-0.110
7.534**
-6.028**
1992/9 - 1997/2
8.644***
0.006***
(4.34)
(0.14 )
(-0.14)
(-2.48)
1984/2-1991/1
-4.77*
-0.008***
-1.229***
2.229***
-3.537
1992/10 - 1994/9
6.787**
1982/12-1985/7
1985/8-1988/3
1988/4-1990/12
-5.468
-2.563
-6.611*
(-4.07)
(-3.08)
(3.08)
(1.40)
-0.002
-3.671***
-2.671***
3.116
(-0.77)
(3.36)
(-3.36)
(-0.97)
-0.0005
-0.212
1.212
-5.256
(-0.16 )
(-0.93)
(0.93)
(1.26)
-0.009**
-1.025
2.025
-1.538
(-1.59)
(-0.75)
(0.75)
(0.20)
-0.012***
-0.918*
1.918*
-5.693
(-4.17)
(-1.89)
(1.89)
(1.37)
***, ** and * denote statistical significance at the 1%, 5% and 10% levels. The number in parenthesis indicate t-statistics.
22