Int. Fin. Markets, Inst. and Money 21 (2011) 623– 628

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Journal of International FinancialMarkets, Institutions & Money

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Short communication

A comment on: “The solution to the forward-bias puzzle”

Alan King ∗,1

Department of Economics, University of Otago, PO Box 56, Dunedin 9054, New Zealand

a r t i c l e i n f o

Article history:Received 13 March 2011Accepted 12 May 2011Available online 19 May 2011

JEL classification:E44F31G14G15

Keywords:Forward-bias puzzleCovered interest parityEfficient markets hypothesis

a b s t r a c t

Pippenger (2011) recently proposed a solution to the longstand-ing forward-bias puzzle. He argues that the puzzling estimatesobtained using the standard equation for the efficient marketshypothesis are due to omitted variable bias. He identifies the miss-ing variables as the future change in the forward exchange rateand the future interest differential. When these are added to thestandard equation, he finds a one-to-one relationship between thefuture change in the spot rate and the forward premium. However,we argue that his equation can only test covered interest parity andoffers no insight into the forward-bias puzzle.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

In a recent article in this journal, Pippenger (2011) presents a solution to the longstanding forward-bias puzzle. This puzzle relates to the prediction that, in efficient financial markets with rational agents,the current forward premium (the difference between the natural logs of the forward, ft, and spot, st,exchange rates) should be an unbiased predictor of the actual future change in the spot rate:

�st+1 = st+1 − st = ˛1 + ˇ1(ft − st) (1)

where ˛1 = 0 and ˇ1 = 1.The numerous empirical studies published over the last three decades rarely find estimates of ˇ1

that conform with the efficient markets hypothesis, instead it typically takes a value that is closer to

∗ Tel.: +64 3 4798686; fax: +64 3 4798174.E-mail address: alan.king@otago.ac.nz

1 I wish to thank Nathan Balke and Mark Wohar for providing their dataset.

1042-4431/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.intfin.2011.05.003

624 A. King / Int. Fin. Markets, Inst. and Money 21 (2011) 623– 628

Table 1Pippenger’s (2011) regression estimates based on Balke and Wohar’s (1998) dataset.

ˆ̨ 1 ˆ̌ 1 R̄2/DW

(A) �st+1 = ˛1 + ˇ1(ft − st)US–UK2 Jan 1974 to 1 Nov 1983 0.657 (0.069) −1.425 (0.166) 0.033

0.087US–UK2 Nov 1983 to 30 Sep 1993 0.930 (0.129) −3.034 (0.406) 0.025

0.075

�̂0 �̂1 �̂2 �̂3 R̄2/DW

(B) �st+1 = �0 + �1(ft − st ) + �2(ft+1 − ft ) − �3(it+1 − i∗t+1)

US–UK2 Jan 1974 to 1 Nov 1983 0.007 (0.001) 0.990 (0.006) 1.000 (0.000) −1.009 (0.008) 0.9997

1.468US–UK2 Nov 1983 to 30 Sep 1993 0.003 (0.001) 0.992 (0.011) 0.999 (0.000) −1.002 (0.011) 0.999

1.776

Notes: Standard errors are in parentheses. Source: Pippenger’s (2011) Tables 1 and 2.

zero than unity and not infrequently negative (see the surveys of this literature cited by Pippenger,2011). As Frankel and Poonawala (2010) note, most previous explanations for the empirical rejectionof the hypothesis relate to either the presence of a risk premium or to systematic errors in agents’forecasts of the spot rate.

Pippenger (2011) argues that the puzzle is a case of omitted variable bias and identifies twovariables (neither of which can be interpreted as representing either a risk premium or errors inexpectations) that do not appear in Eq. (1). When these variables are added to a regression of theforward premium against the future change in the spot rate his estimates of the coefficient on (ft − st)becomes significantly positive, numerically much closer to one than zero and, when using Balke andWohar’s (1998) dataset for the US–UK case, statistically indistinguishable from unity. (Table 1 repro-duces Pippenger’s estimation results for Eq. (1) [panel A] and his extended equation [panel B] in theUS–UK case.) Pippenger (2011, 303) concludes that “[for] my data, those two omitted variables appearto explain the bias in the forward bias puzzle.” Unfortunately, appearances can be deceiving and thisis a case in point.

2. The two “omitted” variables

Pippenger’s two new variables are identified through the derivation of an equation that beginswith the assumption that covered interest parity holds at all times, subject only to those errors (et)that arise because of the thresholds created by the presence of transaction costs:

(ft − st) − (it − i∗t ) = ±et, (2)

where it − i∗t is the interest rate differential between domestic and foreign risk-free securities thathave a period to maturity matching that associated with the forward exchange rate.

Advancing Eq. (2) by one period and solving for the spot rate gives the following expression:

st+1 = ft+1 − (it+1 − i∗t+1) ± et+1 (3)

Subtracting st from both sides of Eq. (3) and simultaneously adding and subtracting ft from the right-hand side of Eq. (3), producing Pippenger’s new equation:

�st+1 = st+1 − st = (ft − st) + (ft+1 − ft) − (it+1 − i∗t+1) ± et+1 (4)

Eq. (4) includes two terms, namely (ft+1 − ft) and (it+1 − i∗t+1), that do not appear in Eq. (1). Pippenger(2011, 299) claims that these two variables “are the econometric source of the forward-bias puzzle”and then states that, with reference in particular to the results presented in Table 1, “at least for my

A. King / Int. Fin. Markets, Inst. and Money 21 (2011) 623– 628 625

data, the bias due to omitting (ft+1 − ft) and (it+1 − i∗t+1) explains the bias in the forward-bias puzzle.”(My emphasis.)

3. The problem with the solution

As Pippenger (2011, 299) quite correctly states, Eq. (4) “is simply a restatement of covered interestparity”. However, it is misleading to claim that it offers an explanation for the forward-bias puzzle.Because Eq. (4) is an expression of covered interest parity and because it happens to contain theefficient market hypothesis, the two additional terms, by construction, merely measure (or accountfor) the extent to which the latter differs from the former in any given period.

The key point here is that, to the extent that covered interest parity is empirically valid, any restate-ment of its expression must, logically, be equally valid. Hence, there is nothing unique or special aboutEq. (4); it is merely one of a potentially infinite number of ways of re-expressing the covered interestparity relationship by taking any two data series and simultaneously adding them to and subtractingthem from Eq. (3).

The only question that can be answered by estimating Eq. (4) is whether covered interest parityholds empirically. Pippenger’s results indicate that it performs very well in all the cases he considers.In particular, when applied to Balke and Wohar’s (1998) data for the US and the UK, covered interestparity accounts for (or “explains” in a purely econometric sense) at least 99.9% of the variation in�st+1 and the estimated coefficients for all variables are almost identical to their expected values (seeTable 1, panel B).

The very high statistical significance and near unit value of the estimated coefficient on (ft − st),however, is not a sign that the forward-bias found when estimating Eq. (1) has been corrected; it ispurely a consequence of the near-perfect fit obtained when using Balke and Wohar’s (1998) dataset.If any other data series were to take the place of ft and st in Eq. (4), the outcome would be the same.This can be demonstrated by means of a simple empirical exercise.

For the purposes of this exercise, six different artificial data series are constructed and arrangedinto pairs:

Pair 1 (x1t, y1t): Comprises two random series drawn (separately) from a Normal distribution with amean of zero and a standard deviation of 0.1.Pair 2 (x2t, y2t): Both series are constructed from (different) random series such that their first-difference is a first-order autoregressive process. The autocorrelation coefficient is set to +0.5 for�x2t and −0.5 for �y2t.Pair 3 (x3t, y3t): These are dummy variables that take a value of one on a single date each year (26June for x3t and 27 November for y3t) and equal zero otherwise. If markets were closed on the datein question, the following working day is used.

Each pair of artificial series is first used to replace ft and st in Eq. (1) (Balke and Wohar’s, 1998,real measure of st+1 is retained) and the regression results obtained are reported in Table 2.2 Notethat these show some parallels with Pippenger’s results derived from the real measures of ft and st.In particular, the explanatory power and Durbin-Watson statistics are quite low, and the estimates ofˇ1 are in all cases significantly less than unity and on one occasion negatively signed.

However, it is important to note that the only significance to be placed on these similarities is thatthey indicate that the artificial variables have not been contrived to produce a one-to-one relationshipbetween (st+1 − yit) and (xit − yit). In other words, the regressions with the artificial series exhibit thesame “bias” (i.e., the estimates of ˇ1 all deviate from unity) as that observed in the actual test of theefficient market hypothesis. Of course, the fact that the estimates of ˇ1 reported in Table 2 deviate

2 Balke and Wohar’s (1998) dataset contains daily observations of the bid and ask values of the spot and forward exchangerates and one-month eurodollar and eurosterling interest rates. Consequently, the geometric mean of each series’ bid and askvalues are employed for estimation purposes and the future value of a series (e.g., st+1) is defined as t plus 22 observations. Forfurther details, see Balke and Wohar (1998).

626 A. King / Int. Fin. Markets, Inst. and Money 21 (2011) 623– 628

Table 2Regression estimates for Eq. (1), where (ft − st) has been replaced with artificial series.

ˆ̨ 1 ˆ̌ 1 R̄2/DW

st+1 − y1t = ˛1 + ˇ1(x1t − y1t)US–UK2 Jan 1974 to 1 Nov 1983 −0.6719 (0.0134) 0.5029 (0.0229) 0.1631/0.3974US–UK2 Nov 1983 to 30 Sep 1993 −0.4627 (0.0126) 0.5167 (0.0228) 0.1853/0.4357

st+1 − y2t = ˛1 + ˇ1(x2t − y2t)US–UK2 Jan 1974 to 1 Nov 1983 −2.2115 (0.4868) −0.3795 (0.0590) 0.1950/0.0068US–UK2 Nov 1983 to 30 Sep 1993 7.8714 (0.2197) 0.3595 (0.0214) 0.6042/0.0116

st+1 − y3t = ˛1 + ˇ1(x3t − y3t)US–UK2 Jan 1974 to 1 Nov 1983 −0.6746 (0.0134) 0.5165 (0.1210) 0.0847/0.1755US–UK2 Nov 1983 to 30 Sep 1993 −0.4651 (0.1167) 0.5017 (0.1167) 0.0947/0.2137

Notes: Standard errors are in parentheses.

from unity is not at all puzzling; there are absolutely no grounds for expecting to see a one-to-onerelationship between (st+1 − yit) and (xit − yit). Had such an estimate of ˇ1 been observed, it wouldhave been purely by chance.

Nonetheless, a one-to-one relationship is exactly what we do see in Table 3 when Eq. (4) is re-estimated using Balke and Wohar’s (1998) measures of all variables except ft and st, which are againreplaced with the artificial series. Regardless of the pair of artificial series employed, the estimate of�1 obtained is highly significant and barely distinguishable from unity.

How can this be? The explanation is as follows. Each artificial (and hence irrelevant) series appearstwice in the regression equation; once partnered with one of the relevant variables from the coveredinterest parity relationship (xit with ft+1; yit with st+1) and once as part of a separate variable (for which�1 is the assigned coefficient). Therefore, the regression process is forced to assign each artificial seriesthe same coefficient3 as that corresponding to its relevant partner variable, specifically, −1 for yit and−�2 for xit. The regression process will then, in effect, attempt to compensate for this by assigningthe same coefficient (but with the opposite sign) to the artificial series at its other appearance in theregression equation.

The extent to which an OLS regression can do this is potentially complicated by the fact that, attheir second appearance in the regression equation, xit and yit are constrained to have a coefficientof the same magnitude (i.e., �1). If in practice �2 /= 1 (i.e., covered interest parity does not actuallyhold), �1 will have to be a compromise between �2 and unity – i.e., the constraint is binding. However,should �2 = 1 (i.e., covered interest parity does hold, which according to Akram et al. (2008) is typicallythe case when using daily or lower frequency data), then the constraint becomes non-binding and �1can be set to unity without compromising the entire regression.

Hence, the value and significance of each estimate of �1 in Table 3 is entirely derived from thevalue and significance of the coefficients on the two variables from the original covered interest parityexpression, st+1 and ft+1. Whenever Eq. (4) is applied to a sample period over which covered interestparity holds, �1 will receive a value that is close, if not equal, to unity regardless of what actual dataseries are used to represent ft and st.

3 As each artificial series is being subtracted from the (relevant) variable it is partnered with, the sign of this coefficient willchange.

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Table 3Regression estimates for Eq. (4), where ft and st have been replaced with artificial series.

�̂0 �̂1 �̂2 �̂3 R̄2/DW

st+1 − y1t = �0 + �1(x1t − y1t ) + �2(ft+1 − x1t ) − �3(it+1 − i∗t+1)

US–UK2 Jan 1974 to 1 Nov 1983 −0.00007 (0.00004) 0.99993 (0.00008) 1.00000 (0.00007) −1.02795 (0.00493) 0.99999

1.5347US–UK2 Nov 1983 to 30 Sep 1993 0.00002 (0.00003) 0.99990 (0.00006) 0.99997 (0.00008) −1.01794 (0.00558) 0.99999

1.6045

st+1 − y2t = �0 + �1(x2t − y2t ) + �2(ft+1 − x2t ) − �3(it+1 − i∗t+1)

US–UK2 Jan 1974 to 1 Nov 1983 0.00019 (0.00006) 1.00005 (0.00001) 1.00003 (0.00001) −1.02039 (0.00592) 1.00000

1.5546US–UK2 Nov 1983 to 30 Sep 1993 0.00019 (0.00011) 0.99998 (0.00001) 0.99997 (0.00001) −1.01032 (0.00579) 1.00000

1.6122

st+1 − y3t = �0 + �1(x3t − y3t ) + �2(ft+1 − x3t ) − �3(it+1 − i∗t+1)

US–UK2 Jan 1974 to 1 Nov 1983 0.00009 (0.00005) 0.99993 (0.00012) 1.00002 (0.00008) −1.02786 (0.00491) 0.99999

1.5351US–UK2 Nov 1983 to 30 Sep 1993 0.00002 (0.00003) 0.99992 (0.00005) 0.99995 (0.00008) −1.01821 (0.00567) 0.99999

1.6061

Notes: Standard errors are in parentheses.

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4. Conclusion

Whenever a regression of Eq. (1) produces a coefficient estimate for the forward premium thatdeparts from unity and so rejects the efficient market hypothesis, that same forward premium isguaranteed to receive a coefficient close to unity in equation (4) providing only that covered interestparity holds. Such an outcome, however, tell us nothing about why the efficient market hypothesisfailed to hold in the first place, as literally any variables used in place of ft and st in equation (4)would have produced the same outcome. Regressions based on Eq. (4) are nothing more than a testof covered interest parity. The proximity of �1 to unity is purely an indication of how well coveredinterest parity performs and it does not represent an unbiased estimate of the ˇ1 coefficient from theefficient markets hypothesis. Forward-bias remains a puzzle without a satisfactory solution.

References

Akram, Q., Rime, D., Sarno, L., 2008. Arbitrage in the foreign exchange market: turning on the microscope. Journal of InternationalEconomics 76, 237–253.

Balke, N., Wohar, M., 1998. Nonlinear dynamics and covered interest rate parity. Empirical Economics 23, 535–559.Frankel, J., Poonawala, J., 2010. The forward market in emerging currencies: less biased than in major currencies. Journal of

International Money and Finance 29, 585–598.Pippenger, J., 2011. The solution to the forward-bias puzzle. Journal of International Financial Markets, Institutions & Money

21, 296–304.