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The forward-bias puzzle: Still unsolved
Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
Contents lists available at ScienceDirect
Journal of International Financial Markets, Institutions & Money j o ur na l ho me pa ge : w w w . e l s e v i e r . c o m / l o c a t e / i n t f i n
Short communication
The forward-bias puzzle: Still unsolved Christian Müller ∗ Zurich University of Applied Sciences, School of Management and Law, CH-8401 Winterthur, Switzerland
a r t i c l e
i n f o
Article history: Received 13 May 2011 Accepted 1 June 2011 Available online 12 June 2011 JEL classification: B5 C2 C3 F3
a b s t r a c t This article argues that Pippenger’s (2011) “Solution to the forwardbias puzzle” must be regarded as an econometric explanation of the famous puzzle, although it does not offer an exhaustive economic answer to it. Some of Pippenger’s (2011) findings are reproduced and established in a cointegrated multiple time series model. It is suggested that economists should stop trying to working out the forward-bias puzzle and start looking for fundamentally better models of foreign exchange rate determination. A tentative alternative is provided. © 2011 Elsevier B.V. All rights reserved.
Keywords: Puzzles Rational expectations Uncertainty
1. Introduction To economists, economic puzzles are painful reminders that some of their treasured theories are questionable, if not outright wrong, which is why economists dislike puzzles. But economists are also fascinated by puzzles. A puzzle affords its discoverer(s) long lasting fame, countless citations, and to all those who try to tackle the problem abundant publication opportunities. This is why there is just one thing that is worse than a puzzle: its solution. 2. CIP vs. SFB Pippenger (2011) tried to solve a puzzle, and his solution to the standard forward bias puzzle (henceforth SFB) is this: ∗ ) ± et+1 , st+1 = 0 + 1 (ft − st ) + 2 (ft+1 − ft ) − 3 (it+1 − it+1
∗ Corresponding author. Tel.: +41 58 934 68 87; fax: +41 58 935 68 87 E-mail addresses: [email protected], [email protected] 1042-4431/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.intfin.2011.06.001
(1)
606
C. Müller / Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
where s denotes spot rates, f the forward rate, i and i∗ domestic and foreign interest rates respectively. The subscript t denotes time. Put simply (1) presents a general model of CIP that nests the standard forward bias regression model. If we establish that 2 = 0 and 3 = 0 Eq. (1) boils down to the well-known spot rate–forward rate–regression that has so many times been shown to lack empirical support. I re-ran the analysis for the US–UK data (2nd subsample: 1983–1993) to test whether or not (1) can be reduced to the SFB model. The F-statistic of the hypothesis H0 : 2 = 3 = 0 clearly indicates rejection (F (2, 2557) = 4.7 with a p-value of 0.0 when corrected for heteroscedasticity), and hence evidence against the SFB model. Guessing the outcome of such a test for the remaining examples on the basis of the individual t-statistics one can expect a clear rejection of the null and hence support for the suggested solution to the puzzle. 3. A cautionary note Some caution is in order, however, I also conducted some experiments with multiple time series and cointegration analysis. Stationarity of the terms st+1 and (ft − st ) must be considered a necessary condition for the SFB model and can be said to support it. Pippenger’s (2011) econometric exercise took stationarity of (ft − st ) for granted while it should have been established econometrically. Defining yt = [st , ft ] , a suitable multiple time series reads yt = A0 Dt +
p
Ai yt−i + εt
i=1
where εt is a 2 × 1 vector of white noise innovations, Dt is a vector of constants, and Ai , i = 0, . . ., p are 2 × 2 coefficient matrices. Deviating from Pippenger (2011), and for convenience’s sake, I shifted the time subscript backwards by one period. This shift does not affect the validity of the model. Applying Johansen’s (1991) techniques allows us to test for covariance stationarity. In short, if there were a linear combination of the endogenous variables that is stationary, the model could be re-written as follows ∗ yt = yt−1 +
p−1
i yt−i + εt
i=1
with yt∗ = (yt , 1) and = ˛ˇ is the product of two matrices ˛ and ˇ with column rank r. The matrices ˛ and ˇ are of dimension 2 × r and 2 × r + 1 matrices, respectively. Here, we restrict the model’s constant to belong to the stationary space in order to prevent it from picking up a time trend. If we allowed for a time trend in st or ft , these trends could lead to odd results like ever-increasing exchange rates. By choosing p = 23, we can estimate the model and test the number of cointegrating relationships. The lag order is chosen to span the time period of the forward contract’s maturity. It turns out that there is a linear combination of the variables that is stationary. Obviously, the most plausible candidate for such a relation is (ft − st ). The Johansen approach allows us to check whether or not this is indeed the case. We simply need to test whether or not the matrix ˇ can be appropriately restricted. We choose the restrictions to correspond to the SFB model (Eq. (1) in Pippenger’s paper) and set ˇ = (− 1, 1, c) under H01 . In other words, the hypothesized stationary relationship reads: st = ft + c, where we let the constant assume any value while it remains restricted to the stationary space Table 1. The restriction of H01 is binding and can be used to construct a statistical test, which – upon rejection – would lend support to Pippenger’s argument. Table 2 shows the results. It turns out that the difference between st and ft cointegrates to a stationary variable. Therefore, a necessary condition for the SFB regression to hold true is met. For the sake of convenience, Appendix A presents a more accessible representation of the various hypotheses. The rejection of H00 in conjunction with accepting H01 proves a systematic, structural link between st and ft . Moreover, this link is robust with respect to additional information such as ft and (it − it∗ ). Therefore, the only way left to establish the SFB puzzle is by scrutinizing the adjustment process.
C. Müller / Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
607
Table 1 Johansen rank test in the SFB regression: the case of UK–US. ∗ Model: yt = yt−1 +
22
i=1
i yt−i + εt
Sample: 11-2-1983 to 9-30-1993 H00 r0 = 0
H10 r≥1
Johansen trace test 29.35
p-Value 0.00
Conclusion r=1
Note: since we need to restrict the constant to belong to the stationary space spanned by ˛, there is no point in also testing: H0 : r0 = 1 against H1 : r = 2. Therefore, the test sequence stops after the first test. For those interested in that hypothesis test: the trace statistic amounts to 6.45 with a p-value of .16. The test results are also robust with respect to changing the number of lags to 18. At this lag length all lags are included that are significant at the ten-percent level. Table 2 Coefficient tests: the case of UK–US. ∗ + Model: yt = yt−1
22
i=1
i yt−i + εt
Sample: 11-2-1983 to 9-30-1993 Null H01 : ˇ = (−1, 1, c) H02 : H01 ∩ ˛ = (1/22, ˛2 )
Alternative ˇ free ˛, ˇ free
LR-test 0.36 12.70
Distribution 2 (1) 2 (2)
p-Value 0.55 0.00
Conclusion Accept H01 Reject H02
According to Pippenger’s (2011) argument and econometric model, adjustment to deviations from the long-run relationship would be made by the spot rate (only). Therefore, we should find, firstly, that at each point in time 1/22 of any deviation would be corrected on average and, secondly, that ft does not adjust. Division by 22 is appropriate because the forward contracts in Pippenger’s (2011) data last for one month, the data is daily data, and a month has 22 working days, on average. Again, the Johansen framework allows us to set up corresponding tests. Next, we test if the adjustment coefficient which measures the correction of st can be restricted to 1/22 (see hypothesis H02 in Table 2). Roughly speaking, this restriction corresponds to setting the ˇ1 in Pippenger’s (2011) Eq. (1) to one. One should be aware of the fact that t stands for a single day in my regression model, while it refers to one month in Pippenger’s (2011) article. The unrestricted estimate of the adjustment coefficient is negative, while it should actually be +1. It is therefore no surprise that the test leads to a clear rejection.1 This finally replicates the puzzle. To sum up, the puzzle also survives in the multiple time series context that permits a much richer dynamic structure of the interactions between forward and spot rate. However, it should be pointed out that further, sophisticated arguments could be used to save the simple model: instead of looking at restrictions on the adjustment coefficients, one might want to look at the whole short-run adjustment pattern. This could be done by impulse-response analysis (IR-A), for example. IR-A in turn requires identification of shocks and the calculation of confidence bands, all of which provide ample opportunities for further discussion and ways to rescue the SFB regression. Moreover, time series analyses are plagued by all sorts of short-run noise such as generated by temporal aggregation (Marcellino, 1999; Weiss, 1984; Wei, 2006), which can lead to serious distortions of the inference on feedback channels. Therefore, we should be cautious and not jump to the conclusion that the literature will make a U-turn anytime soon and accept Pippenger’s (2011) line of argument straight away. 4. Yet another puzzle Pippenger (2011) argues that the puzzling forward-bias regression suffers from an omitted variable bias and demonstrates this argument in a series of regressions. The omission of key explanatory variables is held responsible for the failure of the SFB regression.
1 In multiple time series model adjustment may also take place through ft , which is at odds with the standard SFB regression. Shutting down this adjustment channel would imply ˛2 = 0 on top of all previous hypotheses. A corresponding hypothesis is also rejected.
608
C. Müller / Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
Pippenger’s (2011) explanation is also in line with Wang and Jones’s (2003) emphasis on the mismatch between the variance of the dependent and the independent variables in the SFB model. Pippenger (2011) goes one important step further, however. By pointing out that CIP nests the SFB regression he establishes that both are mutually exclusive. The puzzle of this reasoning lies – in my view – in its interpretation. Pippenger (2011) claims that the omitted variable bias is the solution to the forward-bias puzzle. However, in their original article Obstfeld and Rogoff (2000) do not define the puzzle in econometric but in economic terms: “no model seems to be very good at explaining exchange rates even ex post.” (p.40) Although Pippenger provides the econometric cause why the SFB equations cannot work if CIP is considered to be holding at the same time, we do not yet understand the economics of this observation. In other words, a proper solution to the puzzle is still missing.
5. Economic solutions to the puzzle? Those looking for solutions will certainly appreciate the tailwind of the recent financial crisis. While the economy has suffered, economics may benefit as the crisis underlines the urgent need for innovative suggestions for tackling puzzles that rest on restrictive rationality assumptions, like the SFB. Among the many potential explanations, like black swans, ambiguity, or psychology, my personal favorite is the possible lack of objective price processes and hence the impossibility to converge to a “true” exchange rate as the number of investors grows (see e.g., Mueller-Kademann, 2009). Note that irrespective of any particular model entertained, economists tend to assume that there is an objective function describing the price process of stocks and foreign exchange. Those functions are independent of the researcher, or any other person, which ultimately permits its investigation by anybody interested. Any investigator who is clever enough should be able to identify the true price process. Therefore, by the law of large numbers, independent and eager researchers should at least on average converge to the true process. Moreover, the more people are working on the problem, the more should be revealed about the true price. Now, let us assume for a moment that the price process does not follow objective rules. For example, one might assume that many different people negotiate prices, and the more people are involved, the more diverse their opinions become. Since any asset price is the upshot of negotiations, we might conclude that prices are subjective but not objective. As a result, the more individuals are involved in pricing, the more volatile the price should become and vice versa. Interestingly, the “lunch-break puzzle” and the “gone-fishing-effect”, like some other similar observations, are easily explained by this subjective approach. Surely, besides this personal favourite there are probably many more exciting innovative ideas around which deserve our attention; the forward bias puzzle has lost its appeal.
6. Conclusion To sum up, the economic profession seems to me to be indebted to Pippenger (2011) for another, exceptionally strong argument against the SFB regression industry. It is now about time to finally reject the underlying theory and hence drop the notion of the “forward-bias puzzle” in favor of going back to the drawing board and coming up with a better, probably fundamentally different, economic model for spot exchange rate determination.
Acknowledgment I do thank the editor and an anonymous referee for their comments and help. John Pippenger provided the data and advise on how to use it. Danielle Adams and Katrin Meyer Leu helped with the use of English.
C. Müller / Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
609
Appendix A. In this appendix the multiple time series model is presented in more detail to illustrate the test approach. The documentation focuses on the structural relation between spot and forward rate. Therefore, the cointegration relationship is documented. Since the cointegration coefficients determine the long-run properties of the model and hence of the variables’ interactions, all other coefficients are of secondary relevance. The model is first presented in symbolic form. The notation follows the most widely used conventions in the literature. Owed to the multi-equation approach, the notation cannot be directly linked to Pippenger (2011) and it is therefore not directly comparable to it. A tentative link to Pippenger’s (2011) Eq. (1) is made below.
st ft
=
˛1 ˛2
ˇ1
ˇ2
c
st ft 1
+
22
1,1,i
2,1,i
i=1
1,2,i 2,2,i
st−i ft−i
+
es,t ef,t
(2)
1,1,i 1,2,i and one should note that ˛1 plays the role of ˇ1 in 2,1,i 2,2,i Pippenger’s (2011) Eq. (1) if there was no feedback from st to ft . Otherwise, this is only approximately true. The constant term in Pippenger’s (2011) Eq. (1) is denoted c here and the long-run equilibrium is defined by ˇ1 and ˇ2 . We now estimate the free parameters under H01 . This hypothesis was accepted. It is convenient to simplify i =
st ft
=
=
−0.075 −0.089
−1
1 −0.003
st ft 1
+
22 st−i es,t i
i=1
ft−i
+
ef,t
−0.075(ft−1 − st−1 ) − 0.0002 + . . . −0.089(ft−1 − st−1 ) − 0.0002 + . . .
(3)
The estimates ˛1 and ˛2 are both significant (standard errrors 0.03 in each case) yet ˛1 exhibits the wrong sign. The second hypothesis, H02 , concerns this coefficient. Under this hypothesis the adjustment 1 coefficient ˛1 was restricted to 22 in order to match ˇ1 = 1 in the case of single equation modelling.
st ft
=
=
1/22 0.0589
−1
1 −0.002
st ft 1
+
1 (ft−1 − st−1 − 1 × 10−5 ) + . . . 22 0.0323(ft−1 − st−1 − 1 × 10−4 ) + . . .
22 st−i es,t i
i=1
ft−i
+
ef,t
(4)
Hypothesis H02 was firmly rejected, which can be considered evidence for the existence of the
22
st−i is able to approximately capture omitted variable ft−i effects and hence produce a statistically acceptable model. Therefore, and in contrast to Pippenger’s (2011) approach, rejection of H02 is what really matters. puzzle. It should be noted that
i=1 i
References Johansen, S., 1991. Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59 (6), 1551–1581. Marcellino, M.G., 1999. Some consequences of temporal aggregation in empirical analysis. Journal of Business & Economic Statistics 17 (1), 129–136. Mueller-Kademann, C., 2009. Puzzle solver. MPRA Paper 19852. University Library of Munich, Germany. http://ideas.repec.org/p/pra/mprapa/19852.html. Obstfeld, M. Rogoff, K., 2000. The six major puzzles in international macroeconomics: is there a common cause? Working Paper 7777. National Bureau of Economic Research.
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C. Müller / Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
Pippenger, J., 2011. The solution to the forward-bias puzzle. Journal of International Financial Markets, Institutions and Money 21 (2), 296–304. Wang, P., Jones, T., 2003. The impossibility of meaningful efficient market parameters in testing for the spot-forward relationship in foreign exchange markets. Economics Letters 81, 81–87. Wei, W.W.S., 2006. Time Series Analysis, 2nd edn. Addison-Wesley Publishing Inc., New York. Weiss, A.A., 1984. Systematic sampling and temporal aggregation on parameter estimation in distributed lag models. Journal of Econometrics 26, 271–281.
Contents lists available at ScienceDirect
Journal of International Financial Markets, Institutions & Money j o ur na l ho me pa ge : w w w . e l s e v i e r . c o m / l o c a t e / i n t f i n
Short communication
The forward-bias puzzle: Still unsolved Christian Müller ∗ Zurich University of Applied Sciences, School of Management and Law, CH-8401 Winterthur, Switzerland
a r t i c l e
i n f o
Article history: Received 13 May 2011 Accepted 1 June 2011 Available online 12 June 2011 JEL classification: B5 C2 C3 F3
a b s t r a c t This article argues that Pippenger’s (2011) “Solution to the forwardbias puzzle” must be regarded as an econometric explanation of the famous puzzle, although it does not offer an exhaustive economic answer to it. Some of Pippenger’s (2011) findings are reproduced and established in a cointegrated multiple time series model. It is suggested that economists should stop trying to working out the forward-bias puzzle and start looking for fundamentally better models of foreign exchange rate determination. A tentative alternative is provided. © 2011 Elsevier B.V. All rights reserved.
Keywords: Puzzles Rational expectations Uncertainty
1. Introduction To economists, economic puzzles are painful reminders that some of their treasured theories are questionable, if not outright wrong, which is why economists dislike puzzles. But economists are also fascinated by puzzles. A puzzle affords its discoverer(s) long lasting fame, countless citations, and to all those who try to tackle the problem abundant publication opportunities. This is why there is just one thing that is worse than a puzzle: its solution. 2. CIP vs. SFB Pippenger (2011) tried to solve a puzzle, and his solution to the standard forward bias puzzle (henceforth SFB) is this: ∗ ) ± et+1 , st+1 = 0 + 1 (ft − st ) + 2 (ft+1 − ft ) − 3 (it+1 − it+1
∗ Corresponding author. Tel.: +41 58 934 68 87; fax: +41 58 935 68 87 E-mail addresses: [email protected], [email protected] 1042-4431/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.intfin.2011.06.001
(1)
606
C. Müller / Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
where s denotes spot rates, f the forward rate, i and i∗ domestic and foreign interest rates respectively. The subscript t denotes time. Put simply (1) presents a general model of CIP that nests the standard forward bias regression model. If we establish that 2 = 0 and 3 = 0 Eq. (1) boils down to the well-known spot rate–forward rate–regression that has so many times been shown to lack empirical support. I re-ran the analysis for the US–UK data (2nd subsample: 1983–1993) to test whether or not (1) can be reduced to the SFB model. The F-statistic of the hypothesis H0 : 2 = 3 = 0 clearly indicates rejection (F (2, 2557) = 4.7 with a p-value of 0.0 when corrected for heteroscedasticity), and hence evidence against the SFB model. Guessing the outcome of such a test for the remaining examples on the basis of the individual t-statistics one can expect a clear rejection of the null and hence support for the suggested solution to the puzzle. 3. A cautionary note Some caution is in order, however, I also conducted some experiments with multiple time series and cointegration analysis. Stationarity of the terms st+1 and (ft − st ) must be considered a necessary condition for the SFB model and can be said to support it. Pippenger’s (2011) econometric exercise took stationarity of (ft − st ) for granted while it should have been established econometrically. Defining yt = [st , ft ] , a suitable multiple time series reads yt = A0 Dt +
p
Ai yt−i + εt
i=1
where εt is a 2 × 1 vector of white noise innovations, Dt is a vector of constants, and Ai , i = 0, . . ., p are 2 × 2 coefficient matrices. Deviating from Pippenger (2011), and for convenience’s sake, I shifted the time subscript backwards by one period. This shift does not affect the validity of the model. Applying Johansen’s (1991) techniques allows us to test for covariance stationarity. In short, if there were a linear combination of the endogenous variables that is stationary, the model could be re-written as follows ∗ yt = yt−1 +
p−1
i yt−i + εt
i=1
with yt∗ = (yt , 1) and = ˛ˇ is the product of two matrices ˛ and ˇ with column rank r. The matrices ˛ and ˇ are of dimension 2 × r and 2 × r + 1 matrices, respectively. Here, we restrict the model’s constant to belong to the stationary space in order to prevent it from picking up a time trend. If we allowed for a time trend in st or ft , these trends could lead to odd results like ever-increasing exchange rates. By choosing p = 23, we can estimate the model and test the number of cointegrating relationships. The lag order is chosen to span the time period of the forward contract’s maturity. It turns out that there is a linear combination of the variables that is stationary. Obviously, the most plausible candidate for such a relation is (ft − st ). The Johansen approach allows us to check whether or not this is indeed the case. We simply need to test whether or not the matrix ˇ can be appropriately restricted. We choose the restrictions to correspond to the SFB model (Eq. (1) in Pippenger’s paper) and set ˇ = (− 1, 1, c) under H01 . In other words, the hypothesized stationary relationship reads: st = ft + c, where we let the constant assume any value while it remains restricted to the stationary space Table 1. The restriction of H01 is binding and can be used to construct a statistical test, which – upon rejection – would lend support to Pippenger’s argument. Table 2 shows the results. It turns out that the difference between st and ft cointegrates to a stationary variable. Therefore, a necessary condition for the SFB regression to hold true is met. For the sake of convenience, Appendix A presents a more accessible representation of the various hypotheses. The rejection of H00 in conjunction with accepting H01 proves a systematic, structural link between st and ft . Moreover, this link is robust with respect to additional information such as ft and (it − it∗ ). Therefore, the only way left to establish the SFB puzzle is by scrutinizing the adjustment process.
C. Müller / Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
607
Table 1 Johansen rank test in the SFB regression: the case of UK–US. ∗ Model: yt = yt−1 +
22
i=1
i yt−i + εt
Sample: 11-2-1983 to 9-30-1993 H00 r0 = 0
H10 r≥1
Johansen trace test 29.35
p-Value 0.00
Conclusion r=1
Note: since we need to restrict the constant to belong to the stationary space spanned by ˛, there is no point in also testing: H0 : r0 = 1 against H1 : r = 2. Therefore, the test sequence stops after the first test. For those interested in that hypothesis test: the trace statistic amounts to 6.45 with a p-value of .16. The test results are also robust with respect to changing the number of lags to 18. At this lag length all lags are included that are significant at the ten-percent level. Table 2 Coefficient tests: the case of UK–US. ∗ + Model: yt = yt−1
22
i=1
i yt−i + εt
Sample: 11-2-1983 to 9-30-1993 Null H01 : ˇ = (−1, 1, c) H02 : H01 ∩ ˛ = (1/22, ˛2 )
Alternative ˇ free ˛, ˇ free
LR-test 0.36 12.70
Distribution 2 (1) 2 (2)
p-Value 0.55 0.00
Conclusion Accept H01 Reject H02
According to Pippenger’s (2011) argument and econometric model, adjustment to deviations from the long-run relationship would be made by the spot rate (only). Therefore, we should find, firstly, that at each point in time 1/22 of any deviation would be corrected on average and, secondly, that ft does not adjust. Division by 22 is appropriate because the forward contracts in Pippenger’s (2011) data last for one month, the data is daily data, and a month has 22 working days, on average. Again, the Johansen framework allows us to set up corresponding tests. Next, we test if the adjustment coefficient which measures the correction of st can be restricted to 1/22 (see hypothesis H02 in Table 2). Roughly speaking, this restriction corresponds to setting the ˇ1 in Pippenger’s (2011) Eq. (1) to one. One should be aware of the fact that t stands for a single day in my regression model, while it refers to one month in Pippenger’s (2011) article. The unrestricted estimate of the adjustment coefficient is negative, while it should actually be +1. It is therefore no surprise that the test leads to a clear rejection.1 This finally replicates the puzzle. To sum up, the puzzle also survives in the multiple time series context that permits a much richer dynamic structure of the interactions between forward and spot rate. However, it should be pointed out that further, sophisticated arguments could be used to save the simple model: instead of looking at restrictions on the adjustment coefficients, one might want to look at the whole short-run adjustment pattern. This could be done by impulse-response analysis (IR-A), for example. IR-A in turn requires identification of shocks and the calculation of confidence bands, all of which provide ample opportunities for further discussion and ways to rescue the SFB regression. Moreover, time series analyses are plagued by all sorts of short-run noise such as generated by temporal aggregation (Marcellino, 1999; Weiss, 1984; Wei, 2006), which can lead to serious distortions of the inference on feedback channels. Therefore, we should be cautious and not jump to the conclusion that the literature will make a U-turn anytime soon and accept Pippenger’s (2011) line of argument straight away. 4. Yet another puzzle Pippenger (2011) argues that the puzzling forward-bias regression suffers from an omitted variable bias and demonstrates this argument in a series of regressions. The omission of key explanatory variables is held responsible for the failure of the SFB regression.
1 In multiple time series model adjustment may also take place through ft , which is at odds with the standard SFB regression. Shutting down this adjustment channel would imply ˛2 = 0 on top of all previous hypotheses. A corresponding hypothesis is also rejected.
608
C. Müller / Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
Pippenger’s (2011) explanation is also in line with Wang and Jones’s (2003) emphasis on the mismatch between the variance of the dependent and the independent variables in the SFB model. Pippenger (2011) goes one important step further, however. By pointing out that CIP nests the SFB regression he establishes that both are mutually exclusive. The puzzle of this reasoning lies – in my view – in its interpretation. Pippenger (2011) claims that the omitted variable bias is the solution to the forward-bias puzzle. However, in their original article Obstfeld and Rogoff (2000) do not define the puzzle in econometric but in economic terms: “no model seems to be very good at explaining exchange rates even ex post.” (p.40) Although Pippenger provides the econometric cause why the SFB equations cannot work if CIP is considered to be holding at the same time, we do not yet understand the economics of this observation. In other words, a proper solution to the puzzle is still missing.
5. Economic solutions to the puzzle? Those looking for solutions will certainly appreciate the tailwind of the recent financial crisis. While the economy has suffered, economics may benefit as the crisis underlines the urgent need for innovative suggestions for tackling puzzles that rest on restrictive rationality assumptions, like the SFB. Among the many potential explanations, like black swans, ambiguity, or psychology, my personal favorite is the possible lack of objective price processes and hence the impossibility to converge to a “true” exchange rate as the number of investors grows (see e.g., Mueller-Kademann, 2009). Note that irrespective of any particular model entertained, economists tend to assume that there is an objective function describing the price process of stocks and foreign exchange. Those functions are independent of the researcher, or any other person, which ultimately permits its investigation by anybody interested. Any investigator who is clever enough should be able to identify the true price process. Therefore, by the law of large numbers, independent and eager researchers should at least on average converge to the true process. Moreover, the more people are working on the problem, the more should be revealed about the true price. Now, let us assume for a moment that the price process does not follow objective rules. For example, one might assume that many different people negotiate prices, and the more people are involved, the more diverse their opinions become. Since any asset price is the upshot of negotiations, we might conclude that prices are subjective but not objective. As a result, the more individuals are involved in pricing, the more volatile the price should become and vice versa. Interestingly, the “lunch-break puzzle” and the “gone-fishing-effect”, like some other similar observations, are easily explained by this subjective approach. Surely, besides this personal favourite there are probably many more exciting innovative ideas around which deserve our attention; the forward bias puzzle has lost its appeal.
6. Conclusion To sum up, the economic profession seems to me to be indebted to Pippenger (2011) for another, exceptionally strong argument against the SFB regression industry. It is now about time to finally reject the underlying theory and hence drop the notion of the “forward-bias puzzle” in favor of going back to the drawing board and coming up with a better, probably fundamentally different, economic model for spot exchange rate determination.
Acknowledgment I do thank the editor and an anonymous referee for their comments and help. John Pippenger provided the data and advise on how to use it. Danielle Adams and Katrin Meyer Leu helped with the use of English.
C. Müller / Int. Fin. Markets, Inst. and Money 21 (2011) 605–610
609
Appendix A. In this appendix the multiple time series model is presented in more detail to illustrate the test approach. The documentation focuses on the structural relation between spot and forward rate. Therefore, the cointegration relationship is documented. Since the cointegration coefficients determine the long-run properties of the model and hence of the variables’ interactions, all other coefficients are of secondary relevance. The model is first presented in symbolic form. The notation follows the most widely used conventions in the literature. Owed to the multi-equation approach, the notation cannot be directly linked to Pippenger (2011) and it is therefore not directly comparable to it. A tentative link to Pippenger’s (2011) Eq. (1) is made below.
st ft
=
˛1 ˛2
ˇ1
ˇ2
c
st ft 1
+
22
1,1,i
2,1,i
i=1
1,2,i 2,2,i
st−i ft−i
+
es,t ef,t
(2)
1,1,i 1,2,i and one should note that ˛1 plays the role of ˇ1 in 2,1,i 2,2,i Pippenger’s (2011) Eq. (1) if there was no feedback from st to ft . Otherwise, this is only approximately true. The constant term in Pippenger’s (2011) Eq. (1) is denoted c here and the long-run equilibrium is defined by ˇ1 and ˇ2 . We now estimate the free parameters under H01 . This hypothesis was accepted. It is convenient to simplify i =
st ft
=
=
−0.075 −0.089
−1
1 −0.003
st ft 1
+
22 st−i es,t i
i=1
ft−i
+
ef,t
−0.075(ft−1 − st−1 ) − 0.0002 + . . . −0.089(ft−1 − st−1 ) − 0.0002 + . . .
(3)
The estimates ˛1 and ˛2 are both significant (standard errrors 0.03 in each case) yet ˛1 exhibits the wrong sign. The second hypothesis, H02 , concerns this coefficient. Under this hypothesis the adjustment 1 coefficient ˛1 was restricted to 22 in order to match ˇ1 = 1 in the case of single equation modelling.
st ft
=
=
1/22 0.0589
−1
1 −0.002
st ft 1
+
1 (ft−1 − st−1 − 1 × 10−5 ) + . . . 22 0.0323(ft−1 − st−1 − 1 × 10−4 ) + . . .
22 st−i es,t i
i=1
ft−i
+
ef,t
(4)
Hypothesis H02 was firmly rejected, which can be considered evidence for the existence of the
22
st−i is able to approximately capture omitted variable ft−i effects and hence produce a statistically acceptable model. Therefore, and in contrast to Pippenger’s (2011) approach, rejection of H02 is what really matters. puzzle. It should be noted that
i=1 i
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