An investigation of future spot rates
Kaifeng Wang
Introduction
As a global leader in home appliances and appliances for professional use, Electrolux is selling more than 40 million products to customers in 150 coun-tries every year. Meanwhile, its purchasing activities also take place world-wide. Due to these tremendous international transactions Electrolux Group is highly exposed to fluctuations in the foreign currency exchange market.
In particular, along with the launch of its globalization strategy, Elec-trolux is attempting to manage the purchasing activities of major raw ma-terials (e.g. plastic foam, carbon steel) as well as commodities (e.g. motors, compressors) centrally. By doing so, as a powerful buyer, Electrolux is able to utilize its advantages (e.g. discount, negotiation power) sufficiently, hence reducing the production costs and consequently increasing the products com-petitiveness in the appliance market. In addition, purchasing globalization is a premise for Electrolux to refine and unify the standards of its purchasing components and suppliers.
Normally, an innovation needs to be followed by the addition and/or ad-justment of regulations so that the expected positive outcome can be reached and reinforced. Let us imagine that a purchaser has to choose one supplier between two candidates one period earlier than the transaction takes place (which is a common case in the real manufacturing world). Further, to sim-plify the problem, assume that he only takes into account the price aspect. He has two quotes on his table but in two kinds of currencies, the negotia-tion has to be closed today and the company will use a third currency to pay
the price stated in the contract1. More than that, the company will have to
hedge this third currency by buying it in the forward market soon after the deal. Therefore, the purchaser has to think of what the right solution is and which exchange rate should be applied so that two offers can be compared. It is not an easy question since no one knows what the future spot rates will be and therefore they have to be estimated. As the market expectation of the future spot rates, the forward exchange rates naturally have the potential to be a good predictor. This assumption has been tested by many researchers and is well known as the forward rate unbiasedness hypothesis (FRUH) in econometrics.
This study reviews the previous FRUH researches and decides on a most
1The spot rates corresponding to the contract maturity day should be used for the
appropriate model/approach, which is further implemented to examine the actual spot and forward rates data to see if forward rates are the unbiased estimates of the future spot rates in the statistical sense and therefore solves the problem. In addition, the FRUH analysis yields a by-product—a vector error correction model (VECM) which includes current spot and forward rates as regressors. This estimated model can generate spot rates forecasts and can therefore become another solution of the spot rates prediction.
Another common exchange rate hypothesis—the random walk hypothesis is also taken into consideration by this analysis, which states that exchange rates follow a random walk process. The theory implies that the current spot rates summarize all the relevant information that determines future exchange rates, therefore it could simply be the best reflection of the future spot rates. In order to assess the best alternative to predict future spot rates, we utilize VECM and the lagged spot rates respectively to generate spot rates estimates and compare these estimates with the actual spot rates data. The comparison is made via various forecast error measures and the option with the better performance should be regarded as the better method to make the spot rates prediction.
This research aims at finding an optimal estimate of future spot rates, therefore helping Electrolux refine its existing purchasing guidelines. More-over, this study extends the previous FRUH studies by utilizing its testing outcome—VECM to preform the spot rates estimation. Last but not least, this study also considers the random walk hypothesis and puts it into prac-tice.
The paper consists of five sections; section 1 reviews the exchange rate literature, of which the econometric part focuses on the development of the FRUH study; section 2 illustrates the theoretical models of testing the FRUH further in detail; section 3 describes the empirical FRUH and random walk analysis processes and the respective outcomes, as well as, different estimates and the actual data are compared via a number of statistical tests; section 4 concludes the findings; section 5 summarizes the policy implications from this research.
1. Review of the exchange rates literature
Theorem (IRP) is introduced. The theorem indicates that an equilibrium is present amongst spot rates, forward rates, risk-free interest rates at the corresponding domestic and foreign markets. Based on this observation, the further econometric exploration is motivated. In the econometric part, the forward rate unbiasedness and random walk hypotheses are illustrated, which provide the main technical frame of the empirical study later in this paper.
1.1 Economic theorem
The Interest Rate Parity Theorem (IRP) states that the interest rate differ-ential between two countries is equal to the differdiffer-ential between the forward exchange rate and the spot exchange rate. This relation can be illustrated by following the two methods via which an investor can convert a foreign currency into US Dollars
The first option is to invest a certain amount of foreign currency X locally
for a time period T with the risk-free domestic interest rate rt,T; and
simul-taneously arrange a forward contract to convert the proceeds of this local
investment into US Dollar at the forward exchange rate Ft,T. This process
can be shown as
X −→ X (1 + rt,T) −→
X (1 + rt,T)
Ft,T
The second option is to convert X into US Dollars at the spot exchange
rate St and invest these Dollars at the US market with the risk-free interest
rate rt,T∗ for a time period T . This process can be shown as
X −→ X St −→ X St 1 + r∗t,T
IRP assumes that agents of the foreign exchange market are risk-neutral and that they use all information about the market rationally so that no unexploited profit (arbitrage) remains in the market. In other words, the cash flows from both options should be equal—investors cannot make risk-free profits. This equilibrium can be shown as
X (1 + rt,T)
Ft,T
= X
St
(1 + rt∗, T ) (1)
Ft,T = St
1 + rt,T
1 + rt,T∗ (2)
which, however, does not imply that the forward rate is a “dependent” variable, determined by the spot rate and the two interest rates. Rather, what the IRP says is that the four variables are determined jointly, and that the equilibrium should be satisfied. Therefore, a correlation is expected between spot and forward rates in accordance with their economic definition, which is the premise for the econometric analysis later on.
1.2 Econometric theorem
21.2.1 The forward rate unbiasedness hypothesis
The forward rate unbiasedness hypothesis (FRUH) has attracted many re-searchers during the past two decades. The results are mixed and the asso-ciated econometric methodologies are under constant development.
Fama (1984) stated that any forward rate can be interpreted as the sum
of a risk premium and an expected future spot rate ft= Pt+E(st+1). E(st+1)
is the rational or efficient forecast, conditional on all information available at
t and a model is required to determine Pt. This equation is no more than a
particular definition of the premium component of the forward rate, therefore it is not directly estimated but used to interpret the findings. He analyzed
the regression of the change in the spot rate ∆st+1 on the forward premium
ft− st and found that 1) the variation of both components varies over time
and most of the variation in forward rates is the variation in the premium; 2) the premium and expected future spot rate components of forward rates are negatively correlated.
Later Chiang (1988) argued that previous tests based on full-sample esti-mations might not be accurate because these tests imply an implicit assump-tion that the structure of the behavior relaassump-tion between spot and forward exchange rates is constant over time. Therefore he instead implemented a stochastic coefficient model to investigate FRUH by including the possibility that the estimated coefficients may be sensitive to the sample period used in the estimation. He concluded that FRUH is not rejected by full-sample
2s
tand ftin the following sections indicate the natural logarithm of spot and forward
estimation but the results from the sub-sample study using joint-rolling re-gression reject FRUH in most cases. Furthermore, he also indicated that it is possible to improve the accuracy of the exchange rate prediction with ef-fective use of information underlying the stochastic pattern of the estimated parameters in forecasting.
Scott W. Barnhart and Andrew C. Szakmary (1991) further discussed the causes of conflicting results upon FRUH. In addition to Chiang’s argument, they claimed that the time series properties of spot and forward exchange rate data rule out certain econometric specifications used to test FRUH. They found that 1) both spot and forward exchange rates have unit roots and are co-integrated; 2) the co-integrating parameter in the regression of the realized
future spot rates st+1 on the current forward rates ft is approximately one;
3) FRUH is clearly rejected for the full modern floating exchange rate era and the coefficients of the test are unstable.
Thereafter, different data sets were also examined by other researchers. Their approaches can be mainly attributed to two categories: Error correc-tion model or rolling regression. Despite of the conflicting arguments over whether or not the FRUH should be rejected, the non-stability of the es-timated coefficients and the non-stationarity of spot and forward exchange rates were both confirmed.
The co-integrating relation between st+1 and ft has been commonly
rec-ognized and utilized by most researchers in establishing their error correction models (ECM). Nevertheless, at the same time, there are also a few others who based their studies on the ECM using the co-integrating relation
be-tween contemporaneous spot and forward rates (i.e. st and ft). In the paper
“Cointegration and forward and spot exchange rate regression”, Eric Zivot
(2000) argued that simple models of co-integration between st and ft can
more easily capture the stylized facts of typical exchange rate data than
simple models of co-integration between st+1 and ft.
1.2.2 The random walk hypothesis
and Grilli and Kaminsky (1991) however found evidence against the random walk hypothesis.
The hypothesis can be tested by examining the existence of a unit root in the data or the serial correlation of the exchange rates increment. The random walk hypothesis cannot be rejected if there is a unit root or the exchange rates increment is serially independent. However, the conventional statistical tests for the existence of unit roots on the exchange rates are not very satisfactory (Sims, 1988). Thus, various serial correlation tests of the exchange rate increment are preferred by most researchers (e.g. Christina 1991, In Choi 1999).
Amongst these serial correlation tests, the variance ratio test appears more frequently in the exchange rate studies. The foundation of this test is that the variance of the increments in a random walk is linear in the sampling interval (Lo and MacKinlay, 1988). In other words, if a series follows a random walk process, the variance of its q-th differences will be q times the variance of its first differences. The testing methodology is presented as follows.
Assume that there are a total of nq + 1 observations that are specified as
s0, s1, . . . , snq. The variance ratio is calculated as
VR(q) = σ
2
q(q)
σ21(q)
where σq2 is an unbiased estimator of 1/q of the variance of the q-th
differences of st and σ12 is an unbiased estimator of the first differences of st.
They can be derived as
σ2q(q) = 1 m nq X t=q (st− st−q − q ˆµ) 2 σ21(q) = 1 nq − 1 nq X t=1 (st− st−1− ˆµ)2 where m = q(nq − q + 1)(1 − q/nq), ˆµ = nq1 (snq− s0).
Furthermore, the asymptotic variance of the variance ratio Φ(q) is ob-tained as
Φ(q) = 2(2q − 1)(q − 1)
Finally, the statistic z(q) asymptotically follows a standard normal dis-tribution under the null hypothesis (the series is random walk), i.e.
z(q) = VR(q) − 1
[Φ(q)]1/2 ∼ N (0, 1)
Hence, if the z(q) derived from the data is not larger than the critical value (e.g. 1.96 at the 5% significance level), the hypothesis that the series under test is a random walk cannot be rejected.
2. Theoretical tests on FRUH
In this section, the FRUH testing methods are discussed. They consist of two regression equations which are commonly applied in the earlier studies and two vector error correction models which are frequently utilized in recent research. Further, it shows that these methods are not isolated but connected to each other.
2.1 Two regression equations
In the early FRUH studies, two econometric specifications were used exten-sively by researchers. One is called “Level regression” and the other one is called “Differences regression” or “Percent change” specification.
• Level regression
st+1= µ + βfft+ υt+1 (3)
This form is to regress the future spot rates directly on the corresponding current forward rates. The null hypothesis that FRUH is true requires µ = 0,
βf = 1 and that υt+1 is stationary with E[υt+1] = 0. Assuming st+1 and ft
have unit roots, i.e. st+1, ft ∼ I(1), the FRUH requires that st+1 and ft are
co-integrated with co-integrating vector (1, -1).
To understand this regression from a theoretical perspective we express
the relationship between st+1 and ft in another form as
st+1= ft− Pt+ ζt+1 (4)
where Pt = ft − E[st+1] represents the rational expectations risk
is stationary, but ft will be a biased predictor of st+1 if the risk premium is
correlated with variables in the current information set. • Differences regression
∆st+1 = µ∗+ αs(ft− st) + υ∗t+1 (5)
This form is to regress the change of the realized spot rates on the forward premium or the difference between forward and spot rates. The null
hypoth-esis that FRUH is true requires µ∗ = 0, αs = 1 and E[υ∗t+1] = 0. In fact,
the typical estimates of αs derived in many empirical studies are significantly
negative. To explain this result we rewrite the differences regression Eq.(5) from Eq.(4) as
st+1 = ft− Pt+ ζt+1
st+1− st = ft− st− Pt+ ζt+1
∆st+1 = (ft− st) − Pt+ ζt+1 (6)
Therefore Eq.(5) could be misspecified if risk neutrality fails. I.e., given
that all variables in Eq.(6) are I(0), if Pt is correlated with ft − st, then
the OLS estimate of αs will be biased away from one even when the FRUH
holds. Thus the negative estimate of αs is a result from the omitted variable
bias, moreover it is proved that Cov(E[st+1] − st, Pt) < 0 and Var(Pt) >
Var(E[st+1] − st).
2.2 Vector error correction model (VECM)
Since neither “Level regression” nor “Differences regression” is sufficient for the FRUH test, another model is required. This model should be able to catch all the econometric properties of spot and forward rates. Vector error correc-tion model is rather ideal in this situacorrec-tion since it considers co-integracorrec-tions as well as short-run deviations. The VECM with alternative co-integrations are examined below, i.e. the model includes either the co-integration
be-tween stand ft or the co-integration between st+1 and ft. Finally, the better
specification is concluded.
2.2.1 Model with co-integration between st and ft
The general VECM used to describe spot and forward rates is
∆yt= µ + αβ 0 yt−1+ t (7) where yt= ft st ! , µ = µf µs ! , α = αf αs ! , β = 1 −βs ! and t= f t st ! . We can write Eq.(7) down more explicitly as
∆ft = µf + αf(ft−1− βsst−1) + f t (8)
∆st = µs+ αs(ft−1− βsst−1) + st (9)
If the intercepts µf and µs are restricted to the error correction term
as µf=−αfµc and µs=−αsµc, µc represents the mean of the co-integrating
equation ft−1− βsst−1. Eq.(8) and (9) become
∆ft = αf(ft−1− βsst−1− µc) + f t (10)
∆st = αs(ft−1− βsst−1− µc) + st (11)
Given that β0yt− µc=ft− βsst− µc, Eq.(10) and (11) can be regarded as
an AR(1) process by rewriting them as
Hence, the co-integrating residual β0yt− µc=ft− βsst− µc is stationary
when
β
0
α + 1
= |αf − βsαs+ 1| < 1, which also implies that st and ft are
co-integrated. In particular, if we let βs=1, then the forward premium ft− st
is I(0) and follows an AR(1) process. Eq.(8) and (9) become
∆ft = µf + αf (ft−1− st−1) + f t (12)
∆st = µs+ αs(ft−1− st−1) + st (13)
Now we can see that, actually, Eq.(13) is the “Differences regression” used to test FRUH. Therefore, it is an incomplete form of this special VECM
(βs=1).
Based on Eq.(10) and (11), the FRUH requires that 1) st and ft are
co-integrated; 2) βs=1; 3) µc=0 (meanwhile it must hold that αs=1 so that Eq.
(11) can hold, implying that |αf| < 1).
Under the FRUH, Eq.(10) and (11) are simply
∆ft = αf(ft−1− st−1) + f t (14)
∆st = (ft−1− st−1) + st (15)
where we can see that, in fact, the FRUH requires that the expected change of spot rates equals to the forward premium, whereas the change of
forward rates is explained by ft−1− st−1 with αf as the coefficient.
2.2.2 Model with co-integration between st+1 and ft
For the convenience of our inference, instead of the VECM specification used
above, we start with the triangular form here. The general form of the
triangular representation for yt is
ft = µc+ βsst+ uf t (16) st = st−1+ ust (17) Let β 0 α + 1
< 1 and βs = 1, then uf t = ft− st− µc and it follows an
AR(1) process; led Eq.(11) one period, therefore we have
∆st+1 = αsuf t+ s,t+1
st+1− st = αsuf t+ s,t+1
st+1 = αsuf t+ ft− µc− uf t+ s,t+1
Further, Eq.(10) can be rewritten as
∆ft= αfuf,t−1+ f t (19)
Eq.(18) and (19) are the triangular representation of st+1 and ft.
Gener-ally, the forecast error st+1− ft+ µc is an AR(1) process plus a noise process.
The autocorrelation disappears when the FRUH is true or the forward
pre-mium ft − st is not autocorrelated. Since uf,t and s,t+1 are uncorrelated,
st+1 − ft+ µc can be seen as an ARMA(1,1) process, which implies that
(st+1− ft+ µc, ∆ft)’ cannot be given by a simple VAR model because the
ARMA(1,1) can only be expressed as a pure AR(MA) process of infinite order.
Therefore, the VAR/VECM regarding the contemporaneous spot and for-ward rates is the appropriate model to utilize in this investigation.
3. Empirical study
The Euro (EUR), Swedish Kronor (SEK) and Australian Dollar (AUD) spot rates and three-month forward rates are tested in this paper. The data are the average of bid and ask rates on the last trading day of each month from 1996 January to 2006 July and expressed in U.S dollars. Their respective natural logarithms are calculated and analyzed, which is suggested in all ex-change rate research. The main content of this empirical experiment includes the forward rate unbiasedness hypothesis test, random walk test and the fore-cast evaluation. The outcome and results derived from these processes are demonstrated in details through the section.
3.1 Statistical facts
Figure 1 illustrates the line graphs of spot rates (forward rates are similar),
the forward premiums ft− st and the forecast errors st+1− ft.
Some statistics are included in Table 1. The standard deviation of ∆st+1,
∆ft+1 and st+1− ft are similar; and all of them are roughly ten times higher
than the standard deviation of ft− st for EUR. For SEK and AUD, all the
them are highly autocorrelated for AUD. ∆st+1 and st+1− ft are negatively
correlated with ft− stfor EUR and SEK, however only st+1− ftis negatively
correlated with ft− st for AUD.
Quite a few of the facts found here are not exactly consistent with Zivot’s (2000) observations on the spot and 1-month forward rates. He found that
∆st+1, ∆ft+1and st+1−fthave approximately ten times larger variance than
ft− st; the forward premium is highly autocorrelated but not the forecast
error; ∆st+1, ∆ft+1 and st+1− ft are all negatively correlated with ft− st.
These discrepancies may be due to the different properties of the 1-month and 3-month forward rates.
3.2 Co-integration
3.2.1 Unit root test
In order to test whether or not spot and forward exchange rates are station-ary, the correlogram and augmented Dickey-Fulller (ADF) statistical test are utilized in this study.
The correlograms of spot and forward rates for three currencies show that the autocorrelation coefficients (ac) damp out slowly and the corresponding partial autocorrelation coefficients (pac) cut off after the first lag, which resembles a non-stationary AR(1) behavior.
The correlograms of the first differences of their spot and forward rates show that neither ac or pac at any lag is significantly different from zero, which indicates white noise processes. Therefore, the correlogram inspection provides the visual evidence that spot and forward rates are non-stationary at their levels and become stationary after first differencing, hence they are integrated of order one, i.e. I(1) processes.
The result from the formal ADF statistical test verifies the same conclu-sion. Table 2 shows the ADF statistics and the corresponding null hypothesis probability of spot and forward rates as well as of their first differences. The existence of a unit root in spot and forward rates cannot be rejected at any significance level, which however is easily rejected for the first differences.
Table 2: Augmented Dickey-Fulller unit root test
Euro Swedish Kronor Australian Dollar
t-statistics Prob t-statistics Prob t-statistics Prob
st -1.070 0.726 -1.693 0.432 -1.126 0,704
ft -1.083 0.721 -1.536 0.512 -1.592 0.484
∆st+1 -9.687* 0.000 -14.32* 0.000 -10.97* 0.000
∆ft+1 -9.918* 0.000 -16.05* 0.000 -10.89* 0.000
Prob denotes the probability of the null hypothesis that the series under the test has a unit root. * indicates that the Augmented Dickey-Fuller statistic calculated for the data is larger than the critical value at 1%, 5% and 10% significance levels, which suggests the existence of a unit root in the series under the test.
3.2.2 The autoregressive distributed lag relation
Since spot and forward rates are both I(1) processes, the direct regression will yield biased and inefficient estimates. However, it is possible to establish an error correction model (ECM) instead, since a co-integration may exist between them. A co-integration is an I(0) linear combination of two I(1) variables, its zero mean and constant variance prevent the two I(1) variables from drifting too far apart. A co-integration (i.e. a long-term relation) is likely to exist between spot and forward rates due to the fact that they have the link shown by E.q. (1).
The general ADL(p,q) scheme is
A(L)st = c + B(L)ft+ t
with
A(L) = 1 − α1L − α2L2− · · · − αpLp
B(L) = β0+ β1L + β2L2+ · · · + βqLq
where L is the lag operator, i.e.
L(st) = st−1
L2(st) = st−2
.. .
Ln(st) = st−n
Replacing L by 1 and using ADL(5,5) scheme gives
A(1)st = c + B(1)ft+ t
Namely, the ADL relation between spot rates st and forward rates ft is
established by regressing st on its lagged values st−1 to st−5 and ft and its
lagged values ft−1 to ft−5. The estimation is meaningless if A(1) and B(1)
are not significantly different from zero, hence we can test the co-integrating relation by testing whether A(1) and B(1) are statistically zero.
A(1) is derived as one minus the sum of all coefficients on the lagged s terms and B(1) is the sum of all coefficients on the f terms. Table 3 shows the estimation results of all corresponding ADL tests of three currencies.
For EUR, the sum of coefficients A(1) is 0.3623 with p-value 0.006 for the null hypothesis and B(1) is 0.3537 with p-value 0.007; for SEK, the sum of coefficients A(1) is 0.5865 with p-value 0.0004 and B(1) is 0.5730 with p-value 0.0005; for AUD, the sum of coefficients A(1) is 0.0461 with p-value 0.0276 and B(1) is 0.0597 with p-value 0.0038. The residuals of the estimated ADL relations are stationary, which ensures the efficacy of this test. Therefore,
there is evidence that st and ft are co-integrated.
3.2.3 Johansen Co-integration Test
The Johansen method is also used here to confirm the assumption of a
• Estimate Unrestricted VARs of the first differences of st and ft by
choosing different lag lengths;
• Compare Schwarz criterion values of these VARs;
• The lag length of the VAR with the smallest Schwarz criterion value is used in Johansen Co-integration Test.
Eventually, a lag of length one is appropriate for all three currencies in this study. No deterministic trend is assumed in the data and the hypotheses of whether or not an intercept should be included in the co-integration equation are both tested (Table 4). The results from either option indicate that there exists one co-integrating equation between spot and 3-month forward rates of EUR and SEK at the 5% significance level, while AUD has one co-integration only when the intercept is restricted to the error correction term.
3.3 The VECM of (s
t, f
t)
0The existence of a co-integration between spot and 3-month forward rates has been confirmed for all three currencies, which makes VECM estimation possible. Furthermore, the FRUH then can be tested by checking whether three conditions given in the theoretical part are fulfilled at the same time.
This means that if FRUH is true, then Eq.(15) ∆st = (ft−1 − st−1) + st
should hold.
Table 5 shows the results of the restricted VECM estimation. Two coef-ficients of the co-integrating equation (CE) are restricted as β=(-1,1). In ad-dition, no lagged difference is included in the estimated specification. There-fore, the FRUH is approved if the intercept included in the CE is statistically
zero and the coefficient of the CE in ∆st is not significantly different from
one.
When applying the T-test, neither of these two conditions is fulfilled for EUR. The null hypothesis that the intercept in the CE is zero cannot be
rejected for SEK and AUD, however, αs is significantly different from one
3.4 Random walk
As indicated by the correlogram and ADF unit root test in the beginning of this section, spot rates are an I(1) process, which implies that exchange rates in this study could be random walk processes and therefore the current spot rates are good estimates of the future rates.
We utilize the hypothesis concept from In Choi (1999), who defined the
null hypothesis as ∆st is serially uncorrelated (i.e. spot rates are random
walk) and the alternative hypothesis as ∆stis serially correlated. ∆stdenotes
the first differences of spot rates as before. The hypothesis is tested by the Serial correlation LM test , the Durbin-Watson statistic (DW statistic) and the variance ratio test in this study (Table 6). In the Serial correlation LM test, the Obs*R-squared (the number of observations times the R-square) has
an asymptotic χ2 distribution under the null hypothesis. The distribution of
the F-statistic is not known, but is often used to conduct an informal test of the null. The DW statistic will be around 2 if there is no serial correlation, will fall below 2 if there is positive serial correlation (in the worst case, it will be near zero). If there is negative correlation, the statistic will lie somewhere between 2 and 4 (Table 6). The principle of the variance ratio test has been illustrated explicitly in the literature review part.
In the Serial correlation LM test, the null hypothesis cannot be rejected at any significance level for the Euro and Australian Dollar whereas it can be only bearly rejected at 1% level for the Swedish Kronor. The observation is also confirmed by DW statistic, which is very close to 2 for the Euro and Australian Dollar, but relatively far from 2 for the Swedish Kronor (still very close to 2 though).
Table 6: Serial correlation LM, DW and variance ratio tests of spot rates
Euro Swedish Kronor Australian Dollar
Serial correlation LM test
F-statistic 0.781 3.597* 0.445
Obs*R-squared 3.170 13.39* 1.826
Durbin-Watson stat 1.998 1.967 1.992
Variance ratio test
q=2 1.072 0.697 0.948
q=7 0.311 0.205 0.273
q=9 0.241 0.160 0.212
* denotes that the null hypothesis is rejected at the 5% and 10% significance levels. The null hypothesis of the Serial correlation LM test is that there is no serial correlation in the residuals up to order 4.
3.5 Model evaluation
3.5.1 The evaluation of VECM
Firstly, the residuals of the VECM based on full-sample data estimation are tested to see if they are serially independent and normally distributed. The model is reasonable if two criteria are both fulfilled. Figure 2 shows the correlograms and Q-Q graphs of these residuals. It gives a strong indication that the residuals of the full-sample VECM estimation for the Euro and Australian Dollar are serial uncorrelated and following a normal distribution, whereas for the Swedish Kronor the residuals are serially independent but not likely normally distributed.
the in-sample VECM; subsequently, we add one observation at a time and re-estimate the VECM by using the expending data (t = 1, 2, . . . , n, n + 1); derive the one-step forecast (t = n + 2) from the new VECM; this step-wise process is repeated until we obtain the forecast of the last observation (t = n + m). In the end, this approach yields n in-sample forecasts and m out-of-sample forecasts. Then, the Chow forecast test (Chow, 1960) is ap-plied by using these forecasts to test the constancy of the estimated VECM parameters.
The Chow statistic is calculated as
(SSE1 − SSE2) /n2
SSE2/ (n1− k)
∼ F (n2, n1 − k)
where SSE1 and SSE2 are the sum of the squared forecast errors from
the full-sample and in-sample estimations respectively. n1 and n2 are their
corresponding sample sizes and k is the number of regressors in the estimated model. The hypothesis of parameter constancy is rejected if the Chow statis-tic exceeds the chosen F cristatis-tical value.
The Chow statistic is 0.961 for EUR, 0.796 for SEK and 0.733 for AUD, none of them is larger than the corresponding F -statistic at the 5% signifi-cance level. Therefore, the test says nothing to suggest that the parameters of the estimated VECM is not constant.
The model is also evaluated by CUSUM and CUSUMSQ tests (Brown, Durbin, and Evans, 1975) and the outcome is shown in Figure 3. Both CUSUM and CUSUMSQ statistics are lying within the 5% significance bands for all three currencies, therefore the test does not indicate the existence of model specification errors. This message is also confirmed in the Ramsey RESET test (Ramsey ,1969). By including the second power of the predicted values of the dependent variable, the Ramsey RESET statistic is 1.227 for EUR, 0.711 for SEK and 1.131 for AUD. None of them is larger than the corresponding F critical value at the 5% significance level.
3.5.2 The comparison of VECM and st−13
In order to decide which one is a better estimate of the future spot rates
between VECM and st−1, some statistics are calculated. Theil’s U and the
3The methodologies applied here refer to the study outcome in the paper composed
mean absolute relative error are scale-independent measures for the forecast error and should be as close to zero as possible. The mean squared error (MSE) is also calculated and furthermore is decomposed to bias, variance and covariance ratios to compensate its data-size dependence. The bias ratio tells the distance between the mean of forecasts and the mean of the actual values. The variance ratio indicates the variation of the forecasts relative to the variation of actual spot rates. The covariance measures the remaining unsystematic forecasting errors. These three percentages in theory should sum up to 100%. A good model should have small bias and variance ratios.
The formulars of these forecast error measures are stated below. ˆsi denotes
forecasts, si denotes the actual spot rates, ¯ˆs and ¯s are their respective means,
σsˆ and σs are their respective standard deviations. ρ is the correlation
coef-ficient between forecasts and the actual data. Theil’s U U = 1 n n X i=1 (ˆsi− si)2 s2 i
The mean absolute relative error (MARE)
MARE = 1 n n X i=1 ˆ si− si si
The mean squared error (MSE)
MSE = 1 n n X i=1 (ˆsi− si) 2
The bias, variance and covariance ratios of the MSE
BR = ¯ˆs − ¯s2 Pn i=1(ˆsi− si) 2 /n VR = (σsˆ− σs)2 Pn i=1(ˆsi− si) 2 /n CR = 2 (1 − ρ) σˆsσs Pn i=1(ˆsi− si) 2 /n
Table 7 summarizes the prediction error statistics of the VECM in-sample,
out-of-sample and st−1 forecasts. For the Euro, VECM forecasts are more
qualified in terms of Theil’U and MSE decomposition, equal or worse in
terms of other statistics. For Swedish Kronor, st−1 forecasts always have
forecasts are more accurate according to the mean absolute relative error and MSE decomposition. It appears to be quite difficult to conclude an absolutely better choice for EUR and AUD but SEK.
4. Conclusion
As most other recent researches revealed, this study again rejected the for-ward rate unbiasedness hypothesis. However, some statistical truths showed in this paper are not completely consistent with the previous studies. This is mainly due to the property difference between monthly and quarterly for-ward rates. Apparently, the longer period forfor-ward rates are more distorted than those shorter period ones because the new information is added to the market over time.
The study reviewed the previous exchange rates analyses and the VECM method is chosen to be implemented in this investigation. By deriving the
triangular representation of st+1 and ft, it is found that (st+1, ft)0 can be
interpreted as an ARMA(1,1) model which consequently implies that a simple VAR/VECM cannot be used when the co-integration is established between
st+1 and ft. Thus, eventually, the VECM with the co-integration of (st, ft)0
is used.
In the empirical part, the Euro, Swedish Kronor and Australian Dollar spot and forward rates against US Dollar are analyzed. Correlograms and Augmented Dickey-Fuller unit root test confirm that both rates of the three currencies follow an I(1) process. The ADL test and Johansen co-integration
test indicate that there exists a long-run equilibrium between stand ft, which
makes the VECM estimation possible. By imposing the co-integration vector
β = (−1, 1)0 and excluding the lagged differences of endogenous variables,
the restricted VECM is estimated. The statistical results from the VECM estimations show that none of the three forward rates can satisfy the two
FRUH conditions αs = 1 and µc = 0 at the same time. The quarterly
forward rates are not unbiased estimates of the future spot rates.
Next, the hypothesis that a spot rates time series is a random walk process is tested by the Breusch-Godfrey Lagrange multiplier, the Durbin-Watson and variance ratio test. The statistical results consistently support this null assumption and therefore suggest that the current spot rates are alternative estimators of the future spot rates.
well as some model evaluation statistical tests such as the Chow forecast test, CUSUM and CUSUMSQ tests. None of them gives a strong evidence that the VECM is inadequate or misspecified. Therefore, the VECM forecasts worth being further evaluated.
In order to compare the forecasts derived from these two approaches, some prediction error statistics are calculated for the VECM in-sample,
out-of-sample and st−1 forecasts respectively. As a result, st−1 forecasts
outper-form the VECM forecasts for Swedish Kronor rates, but it is difficult to say for the Euro and Australian Dollar since the forecast measures give mixed conclusions.
5. Policy implications
This study is dedicated to help the companies such as Electrolux purchasing department establish an internal guideline for its buyers, purchasing analysts and managers regarding exchange rates application. The guideline should be able to assist them in obtaining a more accurate decision while making nego-tiations, analyzing suppliers and participating in other purchasing activities where the usage of the future exchange rates is involved.
Some general remarks can be drawn from this experiment.
• We have no evidence to suggest that forward rates per se are an unbi-ased predictor of the future spot rates. However, in the long term, the contemporaneous spot and forward rates are interlinked and likely to be identical.
• The vector error correction model (VECM) using the contemporaneous spot and forward rates, a by-product of the FRUH test, can generate reasonable spot rate forecasts.
• Spot rates are following random walk process, i.e. the current spot rates are also fair estimates of the future spot rates.
From a practical perspective, it seems that the current spot rates are more appealing because 1) they do not require the forward rates input; 2) they prevent forecasting from the possible change of the estimated model (e.g. Chiang, T.C., 1988); 3) they are easy to use.
Therefore, the current spot rates ought to be stated in the company guideline as the commonly used reference of the future rates at this stage.
In order to improve the accuracy of this guideline, the analogous experi-ment needs to be performed for additional currencies and forward rates with different time horizons. The procedure of this test can be summarized as:
1. Estimate a VECM by using the contemporaneous spot and forward rates as regressors.
2. Derive spot rates forecasts from the estimated VECM.
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