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Int. Fin. Markets, Inst. and Money 21 (2011) 617–622
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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
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Possible solutions to the forward bias paradox夽 Richard T. Baillie a,b,c,∗ a b c
Departments of Economics and Finance, Michigan State University, USA School of Economics and Finance, Queen Mary University of London, UK Rimini Center for Economic Analysis, Italy
a r t i c l e
i n f o
Article history: Received 2 May 2011 Accepted 18 May 2011 Available online 27 May 2011 JEL classification: C31 Keyword: Forward premium anomaly
a b s t r a c t This note outlines the economic theory behind the theory of uncovered interest parity and some of the econometric issues involved in testing and interpretation. I illustrate some of the issues involved by estimating a rolling regression of the forward premium regression from 22 years of eight major currencies. I also conclude that Pippenger’s model is not consistent with the theory of UIP and that furthermore there are severe econometric problems in estimating his model. The forward premium anomaly remains a paradox in international finance that is important and worthwhile to understand more fully. © 2011 Elsevier B.V. All rights reserved.
1. Introduction It is always a great pleasure to consider some fresh insights into the long standing issue of the so called forward premium, or forward discount puzzle. Pippenger (2011) has claimed that the paradox disappears when two additional variables are admitted to the familiar forward premium regression. I argue that the “model” that Pippenger presents has nothing to do with the issues surrounding Uncovered Interest Parity (UIP). Furthermore it suffers from extreme mis-specification and multicollinearity issues which render its OLS estimates to be specious from any testing perspective. I also briefly discuss some of the empirical evidence in the literature on UIP and illustrate some of the periods of departure from 22 years of data on eight freely floating currencies. This simple analysis does indeed indicate many issues that are at odds with the simple UIP formulation and provides

I am grateful to Geoff Booth for suggesting that I write this note. ∗ Correspondence address: Department of Finance, Broad School of Business, Michigan State University, East Lansing, MI 48824, USA. Tel.: +1 517 355 1864. E-mail address: [email protected] 1042-4431/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.intfin.2011.05.004
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evidence that there are periods when it does not hold in the short run. While I am thoroughly in favor of resolving mysteries of all kinds and international finance puzzles in particular, I nevertheless think it is premature to claim victory on the forward premium paradox. The forward premium anomaly provides some important empirical evidence that makes us realize that we do not fully understand the workings of some international financial markets. These extreme episodes may be the most important ones in terms of generating new theories and new indights. First, it is important to draw a distinction between the issue of “forward bias” and the the failure of the theory of Uncovered Interest Rate Parity (UIP). The two issues are not quite the same. The theory of UIP is derived from a well formulated economic theory, while the absence of bias in the forward rate is a rather vague approximation to relationships that hold under UIP. Formally, the UIP condition states that Et (0002st+1 ) = (it − it∗ ),
(1)
where Et (·) denotes the conditional expectation based on a sigma field of all relevant information at time t. The variable st is the logarithm of the spot exchange rate and is measured in terms of the number of dollars in terms of a unit of foreign currency at time t; and it and it∗ are the one period to maturity nominal interest rates available on similar domestic and foreign assets respectively. From an economic theory perspective, the UIP hypothesis requires the joint assumptions of rational expectations, risk neutrality, free capital mobility and the absence of taxes on capital transfers. Hence expected real returns in the forward market must be zero, so that, Et
0002S
t+1
− Ft
Pt+1
0003
= 0,
(2)
where Pt+1 is the domestic price level. By a Taylor series expansion of Eq. (3) to second order terms, Et (0002st+1 ) = (ft − st ) −
000210003 2
Vart (0002st+1 ) + Covt (0002st+1 pt+1 )
(3)
Note that, even under rational expectations and risk neutrality the right hand side of Eq. (3) contains the two conditional second moment terms sometimes known as the “Jensen Inequality” terms since they would not appear in a formulation of the theory in levels. Hence the relationship of Et 0002st+1 = (ft − st ) arises naturally as an approximation to the economic theory embodied in the Euler equation in (2) which states that real returns from going long or short in the forward market should average out to zero. This relationship implies that the rate of depreciation (or appreciation) of the future spot rate is approximately the same as the interest rate differential.1 A more general formulation allows the risk premium to be time dependent. The standard discrete time, consumption based asset pricing model of Lucas (1978), provides a risk adjusted equivalent to Eq. (2), which emphasizes real returns over the current and future consumption streams of the representative investor, Et
0002S
t+1
− Ft
Pt+1
0003 0002 U 0003 (C
t+1 ) U 0003 (Ct )
0003
= 0,
(4)
where U0003 (Ct ) is the marginal rate of consumption in period t. Then, Et (0002st+1 ) = (ft − st ) −
000210003 2
Vart (0002st+1 ) + Covt (0002st+1 pt+1 ) + qt
(5)
where qt is the natural logarithm of the intertemporal marginal rate of substitution and is generally called the “risk premium”. The above theory dates back at least to Hansen and Hodrick (1983), and excellent explanations are available in Hodrick (1987) and Engel (1996). So my first comment on Pippenger (2011) is over a lack of clarity concerning the object being tested. His concern over “forward rate unbiasedness” is a rather old fashioned terminology dating back to Frankel (1977) and others in the 1970s. This does not distinguish whether it is the level of
1 Since logged differenced spot prices are approximately equal to the continuously compounded rate of return on the spot rate.
R.T. Baillie / Int. Fin. Markets, Inst. and Money 21 (2011) 617–622
619
the forward rate, or its logarithm that is supposed to be unbiased. Furthermore pure unbiasedness in this sense is not consistent with the various theoeretical forms of UIP in Eqs. (2) and (4) that may include Jensen inequality terms and also time dependent risk premium respectively. A related second criticism concerns Pippenger’s claim that the specification Et 0002st+1 = (ft − st ) arises over the econometric desirability to deal with the quantity of stationary returns rather than the levels one of Et st+1 = ft . It is true that differencing the spot rate does “reduce it to stationarity”, but this is separate from the economic theory. Hence expressing the theory of UIP under rational expectations and a constant risk premium as Et (0002st+1 ) = (ft − st ) = (it − it∗ )
(6)
is always an approximation which neglects the Jensen inequality terms, and possible time dependent risk premium. While possible peso problems, segmented markets, and heterogenous trading behavior have all been suggested as resolutions of the anomaly, the presence of time-dependent risk premia has generally seemed the most persuasive; see Hodrick (1987, 1989) and Mark and Wu (1997). 2. Testing the theory Early articles by Hansen and Hodrick (1980) and Baillie et al. (1983) showed how the simple unbiasedness proposition could be rejected with weekly data and 30 days and 90 days forward rates. However, following Fama (1984) it has become standard to test the theory from the regression equation given by 0002st+1 = ˛ + ˇ(ft − st ) + ut+1
(7)
where the theory of UIP implies ˛ = 0, ˇ = 1 and ut+1 being serially uncorrelated. The forward premium anomaly of a negative slope coefficient has generally ocurred for most freely floating currencies and appears robust to the choice of numeraire currency. This appeared to motivate the analysis of Fama (1984). In a widely cited “meta study” by Froot and Thaler (1990), it is found that the average value of the estimated ˇ across 75 studies was −0.88. However, it is now fairly well known that the size of the slope coefficient estimate tends to be time varying and regime specific; and the Froot and Thaler (1990) study is clearly based on data before 1990. More recent analysis are even more informative. For example, see Baillie and Bollerslev (2000), who analyze the DM-$ exchange rate between March 1978 through November 1995, and who find slope coefficient estimates as low as −17 between 1985 and 1991 and slightly positive coefficient estimates after 1993. In a recent paper examining 200 years of the UIP condition, Lothian and Wu (2011) conclude that “the estimates become negative only when the sample is dominated by the period of the 1980s and . . . large interest rate differentials have significantly stronger forecasting powers for currency movements than smaller interest rate differentials”. Furthermore, Wu and Zhang (1996), Zhou (2002) and Bansal (1997) all noted the asymmetry in the forward premium anomaly, which tended to be more extreme with larger negative slope coefficient estimates, when US interest rates were below foreign equivalents. So, that the sign of the forward premium, (ft − st ), is an important indicator on the magnitude of the anomaly. One way of expressing this is to consider the model with time varying beta, given by 0002st+1 = ˛ + ˇt (ft − st ) + ut+1
(8)
where the very nature of ˇt being time dependent signifies the forward premium anomaly can move from being severe to minimal across a realization. It should be noted that the anomaly is not only present in time series studies, but also is also to be found in panels and cross sections, as in Flood and Rose (1996). In Fig. 1 below, I have graphed estimates of the ˇt derived from a 5-year rolling regression for eight different currencies against the US dollar for data from December, 1988 through October 2010; which gives a sample size of 262 observations. I have used eight freely floating currencies that have not been part of the Eurozone and have therefore existed for the complete 22-year period. They are the Australian dollar (AUD), the Canadian dollar (CAD), Swiss franc (CHF), Danish krone (DKK), Japanese yen (JPY), UK pound (GBP), Norwegian krone (NOK) and the New Zealand dollar (NZD). It can be seen that there are many periods where the ˇt is significantly negative and other periods particularly towards the end of the sample, where its value becomes relatively high and sometimes
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Fig. 1. Slope estimates from Rolling UIP Regressions using 5-year subsamples. The dashed line represents 95% confidence bands.
exceeds unity. For example, the New Zealand dollar has a ˇt that is consistently negative from 2001 through 2008 and is as small as −10 in some periods. However, after 2008 it is large and positive and +10 with two sided 95% confidence intervals easily exceeding +3. In general the financial crisis of the Fall of 2008 with lower nominal interest rates has coincided with the AUD, CHF, NOK and NZD all having large and positive ˇt coefficients from the rolling regression. Another interesting aspect of this admittedly ad hoc procedure, is that the shape of the estimated ˇt graphed over time is remarkably
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621
similar between currencies and also regardless of which is the numeraire currency. Therefore there is strong evidence of predictable and consistent evidence of the forward premium anomaly at specific periods over a range of currencies. The method for estimating the ˇt can alternatively be derived from models involving regime switching where UIP holds for part of the sample, and where the anomaly is only present for some periods. In particular, both Baillie and Kilic¸ (2006) and Baillie and Chang (2011) specify a function of the form ˇt = ˇ1 (1 − G(zt ; 0003, c)) + ˇ2 G(zt ; 0003, c)
(9)
where G(·) is a transition function in the range of (0,1) and chosen to be the logistic function, G(zt ; 0003, c) = (1 + exp(−0003(zt − c)/0004zt ))
−1
,0003 > 0
where zt is the transition variable, 0004zt is the standard deviation of zt , 0003 is a slope parameter, and c is a location parameter. Hence the behavior of the ˇt and hence the extent or duration of the forward premium anomaly, essentially depends on the nature of the transition functions which assigns weights over the sample to move the ˇt from regimes where UIP holds to others where substantial risk premiums occur. Driving the transition function is the variable zt which can be any fundamental that has a role to play in this analysis. Baillie and Kilic¸ (2006) consider transition variables of the interest rate differential, the money growth differential, income differentials, volatility of US money growth rates and deviations from the equilibrium spot rate estimated from a monetary model. While Baillie and Chang (2011) consider a similar model only with the transition variables reflecting carry trade variables. They find that UIP has a tendency to hold precisely when the carry trade appears most profitable might seem to suggest that carry trade profits should be much smaller than previous studies have found. The other explanation of the anomaly focuses on econometric issues. Estimation of Eq. (7) involves a classic problem of regressing the virtually uncorrelated spot returns on the very persistent, highly autocorrelated forward premium. Furthermore the variance of the spot returns is often 20 times that of the forward premium. Contrary to the claim of Pippenger (2011), the recognition of this stylized fact dates back to Cornell (1977) and is mentioned by many subsequent authors. The other non standard aspect of Eq. (7) is that the relatively uncorrelated spot returns are being regressed on the interest rate differential which has very persistent autocorrelation and nonlinearities; see Hai et al. (1997), Baillie and Bollerslev (2000), Maynard and Phillips (2001) and Sakoulis et al. (2010). 3. Pippenger’s tests of UIP Pippenger (2011), so called “solution” of the forward premium anomaly is to take the CIP at time ∗ ) and to write as s ∗ t + 1, i.e. (ft+1 − st+1 ) = (it+1 − it+1 t+1 = ft+1 − (it+1 − it+1 ) and to then subtract st from ∗ ). The regression that Pippenger (2011) estimates is both sides to give, 0002st+1 = ft+1 − st − (it+1 − it+1 obtained by adding and subtracting ft on to the right hand side to give ∗ 0002st+1 = 0002ft+1 + (ft − st ) − (it+1 − it+1 )
The main question is whether this equation is anything to do with UIP. It has been derived from adding and subtracting spot and forward rates into the CIP condition; and as such as nothing to do with UIP. The only point of reference is that if we were to take conditional expectations of only the left hand side and to drop the first and third terms on the right hand side; we would get the UIP condition. But this is basically irrelevant to testing UIP. Also the equation suffers from severe multicollinearity. To see this consider estimation of the regression ∗ 0002st+1 = 00051 (ft − st ) + 00052 0002ft+1 + 00053 (it+1 − it+1 ) + εt+1
where εt+1 is a white noise disturbance. The fact that CIP is an identity leads to the restriction that 00051 + 00053 = 0. In fact Pippenger (2011) continually underestimates the identity nature of the CIP condition. The properties of CIP were originally discussed by Aliber (1973), while studies by Andersen and Bollerslev (1997) and Lyons (2001) using extremely high frequency international finance data have
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confirmed the identity nature of CIP. Furthermore, spot returns are very highly correlated with forward rate returns and consequently 00052 = 1 can be imposed. I checked this by estimating the regression 0002st+1 = 00050002ft+1 + εt+1 for the same monthly data used to generate the results in Fig. 1. For all eight currencies the estimate of 0005 was indistinguishable from unity and the R2 of the regression was 0.99. Given the CIP identity and hence restriction 00051 + 00053 = 0, it is clear that the regression estimated by Pippenger (2011) suffers from perfect multicollinearity. In fact the four regressions estimated by Pippenger in his Table 2, illustrate this very nicely. In all cases the restriction that 00052 = 1 is almost exactly satisfied, and the restriction that 00051 + 00053 = 0 would appear to hold indicating the virtual textbook example of multicollinearity. Hence I conclude that not only is Pippenger’s regression equation unrelated to UIP, it is also statistically fallacious. The forward premium anomaly remains an important issue in international finance that is important and worthwhile to understand more fully. References Aliber, R.Z., 1973. The interest rate parity theorem: a reinterpretation. Journal of Political Economy 81, 1451–1459. Andersen, T.G., Bollerslev, T., 1997. Intraday periodicity and volatility persistence in financial markets. Journal of Empirical Finance 4, 115–158. Baillie, R.T., Lippens, R.E., McMahon, P.C., 1983. Testing rational expectations and market efficiency in the foreign exchange market. Econometrica 51, 553–563. Baillie, R.T., Bollerslev, T., 2000. The forward premium anomaly is not as bad as you think. Journal of International Money and Finance 19, 471–488. Baillie, R.T., Chang, S.S., 2011. Carry trades, momentum trading and the forward premium anomaly. Journal of Financial Markets 14, 441–464. Baillie, R.T., Kilic¸, R., 2006. Do asymmetric and nonlinear adjustments explain the forward premium anomaly? Journal of International Money and Finance 25, 22–47. Bansal, R., 1997. An exploration of the forward premium puzzle in currency markets. Review of Financial Studies 10, 369–403. Cornell, B., 1977. Spot rates, forward rates and exchange market efficiency. Journal of Financial Economics 5, 56–65. Engel, C., 1996. The forward discount anomaly and the risk premium: a survey of recent evidence. Journal of Empirical Finance 3, 123–192. Fama, E.F., 1984. Forward and spot exchange rates. Journal of Monetary Economics 14, 319–338. Flood, R.P., Rose, A.K., 1996. Fixes: of the forward discount puzzle. Review of Economics and Statistics 78, 748–750. Frankel, J.A., 1977. The forward exchange rate, expectations and the demand for money: the German hyperinflation. American Economic Review 64, 653–670. Froot, R.A, Thaler, R.H., 1990. Anomalies: foreign exchange. Journal of Economic Perspectives 4, 179–192. Hai, W., Mark, N.C., Wu, Y., 1997. Understanding spot and forward exchange rate regressions. Journal of Applied Econometrics 12, 715–734. Hansen, L.P., Hodrick, R.J., 1980. Forward exchange rates as optimal predictors of Future spot rates: an econometric analysis. Journal of Political Economy 88, 829–853. Hansen, L.P., Hodrick, R.J., 1983. Risk averse speculation in forward exchange markets: an econometric analysis. In: Frenkel, J.A. (Ed.), Exchange Rates and International Macroeconomics. University of Chicago Press. Hodrick, R.J., 1987. The Empirical Evidence on the Efficiency of Forward and Futures Foreign Exchange Markets. Harwood, London. Hodrick, R.J., 1989. Risk, uncertainty and exchange rates. Journal of Monetary Economics 23, 433–459. Lothian, J.R, Wu, L., 2011. Uncovered interest-rate parity over the past two centuries. Journal of International Money and Finance 30, 448–473. Lucas, R., 1978. Risk, uncertainty and exchange rates. Journal of Monetary Economics 23, 433–459. Lyons, R.K, 2001. The Microstructure Approach to Exchange Rates. MIT Press, Cambridge. Mark, N.C., Wu, Y., 1997. Rethinking deviations from uncovered interest rate parity: the role of covariance risk and noise. Economic Journal 108, 1686–1706. Maynard, A., Phillips, P.C.B., 2001. Rethinking an old empirical puzzle: econometric evidence on the forward discount anomaly. Journal of Applied Econometrics 16, 671–708. Pippenger, J., 2011. The solution to the forward-bias puzzle: Reply. Journal of International Financial Markets, Institutions and Money 21, 629–636. Sakoulis, G., Zivot, E., Choi, K., 2010. Time variation and structural change in the forward discount: implications for the forward rate unbiasedness hypothesis. Journal of Empirical Finance 17, 957–966. Wu, Y., Zhang, H., 1996. Asymmetry in forward exchange rate bias: a puzzling result. Economic Letters 50, 407–411. Zhou, S., 2002. The forward premium anomaly and the trend behavior of the exchange rates. Economic Letters 76, 273–379.
Author: John C. Maxwell
Publisher: HarperCollins Leadership
Release Date: 01 April 2007
ISBN: 9781418508326
Pages: 224 pages
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