Do asymmetric and nonlinear adjustments explain the forward premium anomaly?

Transcription

1 Journal of International Money and Finance 25 (2006) 22e47 Do asymmetric and nonlinear adjustments explain the forward premium anomaly? Richard T. Baillie a,b,c, *, Rehim Kiliç d a Department of Economics, Michigan State University, East Lansing, MI , USA b Department of Finance, Michigan State University, East Lansing, MI , USA c Department of Economics, Queen Mary University of London, London E1 4NS, UK d School of Economics, Georgia Institute of Technology, 781 Marietta Street, Atlanta, GA , USA Abstract This paper explores some of the asymmetries and nonlinearities in an attempt to throw light on the forward premium anomaly, where spot exchange rate returns are typically found to be negatively correlated with the lagged interest rate differential and lead to an apparent rejection of Uncovered Interest Parity (UIP). The approach in this paper is motivated by some recent theoretical literature on the limits to speculation and hysteresis. The paper estimates Logistic Smooth Transition Dynamic Regression (LSTR) models with a variety of transition variables, including the lagged forward premium, monetary and income fundamentals and also variables associated with time varying risk premium, including the conditional variances of some fundamentals. Results are reported for nine different currencies. Many of the estimated LSTR models provide evidence for the existence of an outer regime that is consistent with UIP. Estimation of the standard forward premium regression on observations falling in these regimes across the sample is moderately supportive of UIP holding in the outer regime. A simulation experiment also suggests that an LSTR dgp can produce data consistent with the anomaly. However, parameter estimation issues leads to considerable uncertainty with the estimated transition functions and hence imprecise definitions of regimes. The results are an interesting step in the direction of understanding the nonlinear dimension of the problem without fully resolving the anomaly. Ó 2005 Elsevier Ltd. All rights reserved. JEL classification: C22; F31; F41 Keywords: Forward premium anomaly; Uncovered Interest Parity; Nonlinearity; LSTR models * Corresponding author. Department of Economics and Department of Finance, Michigan State University, Marshall Hall, East Lansing, MI , USA. Tel.: þ ; fax: þ addresses: baillie@msu.edu (R.T. Baillie), rk114@mail.gatech.edu (R. Kiliç) /$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:1016/j.jimonfin

2 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e Introduction The theory of uncovered interest rate parity (UIP) states that the expected return, or rate of appreciation on a currency equals the interest rate differential, or equivalently the forward premium. One popular method for testing the theory has been to regress the rate of appreciation of the spot rate on the lagged forward premium. A test for UIP is then to test if the slope coefficient is unity, the intercept zero and the residuals serially uncorrelated. The forward premium anomaly is the widespread empirical finding of a negative slope coefficient in the above regression, so that the rate of appreciation of the spot exchange rate is negatively correlated with the lagged forward premium. This phenomenon has been consistently found for most freely floating currencies in the current float and appears robust to the choice of numeraire currency. Hence the forward premium anomaly implies that the country with the highest interest rate will have an appreciating currency, and not a depreciating currency, as implied by the theory of UIP. Many possible theories have been proposed to attempt to explain the anomaly. For example, considerable previous work has modeled time dependent risk premia, e.g., Hodrick (1987, 1989) and Mark and Wu (1997). Other work has considered possible peso problems, segmented markets and heterogenous trading behavior; and excellent surveys of the forward premium anomaly and suggested resolutions have been provided by Hodrick (1987) and Engel (1996). More recent work has emphasized the econometric issues involving unbalanced regressions where the approximate martingale spot returns are being regressed on the lagged forward premium, which is highly autocorrelated and possibly a long memory process. These issues are analyzed by Baillie and Bollerslev (2000) and Maynard and Phillips (2001). Baillie and Bollerslev (2000) also show that the magnitude and sign of the estimated slope coefficient in the forward premium regression appears to be slowly time varying, particularly in small sample sizes. This current paper also deals with some of the econometric aspects of the anomaly and focuses on some nonlinear and asymmetric aspects of the relationship. The need for such an approach can be justified from several theoretical papers that consider transaction costs from closing arbitrage conditions in financial markets. Also, some recent applied research, e.g. Bansal and Dahlquist (2000), has found the size and extent of the anomaly to be related to the interest rate differential. A convenient starting point for the focus of this paper, is the logistic smooth transition dynamic regression (LSTR) model, which is related to the LSTAR and other nonlinear models introduced by Granger and Terasvirta (1993) and Terasvirta (1994). Such models have begun to be extensively applied to various issues in financial economics. This study estimates LSTR models for a variety of transition variables, including the forward premium, money supply differentials, income differentials, combinations of fundamentals, squared forward premium and conditional variances of fundamentals that are theoretically postulated to be important elements of the time varying risk premium. For some currencies and some transition variables, the estimated LSTR models appear to show the existence of two major regimes, with an inner regime that is consistent with the forward premium anomaly, and an outer regime where the conditions associated with UIP cannot be rejected. The results for nine currencies are generally supportive of the proposition that nonlinearity is an important aspect of the forward premium anomaly, with a range of transition variables appearing to be related to the bands or regimes. Furthermore, many of the estimated LSTR models have an outer regime where the conditions for UIP cannot be rejected. The paper also reports results of estimating the classic forward premium anomaly regression across all the observations falling into a particular regime. In several cases such regressions in the outer band are often supportive of UIP holding; and these results are apparent for a variety of transition variables including the

3 24 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 forward premium, money supply differentials, and the variability of US money growth. The transition variables involving income differentials, income velocity and some further quantities associated with risk premium, are found to be less informative. Further supportive evidence is provided from a simulation experiment which finds that an LSTR data generating process is consistent with data that give rise to the forward premium anomaly. In contrast to the previous positive statements, and the clear intuitive appeal of the LSTR models, more detailed investigations reduce the optimism that the LSTR specification is completely appropriate. In particular, the properties of the outer or inner regime can often depend on how they are defined in terms of closeness to the theoretical boundaries of one and zero. The relatively small number of observations falling in the bands can cause difficulties for inference. Also, inherent sampling variability from parameter estimation gives rise to further uncertainty in the estimated transition functions. This implies that the identification of precise regimes when UIP holds, or conversely when the anomaly is dominant, is quite difficult. Hence, while the LSTR models appear quite useful for identifying sources of nonlinearity in the forward premium anomaly, there appear to be additional types of nonlinearity that are not captured very well by these models. Further refinements of the specifications and links with theory are important to make further progress on understanding the anomaly. The rest of the paper is organized as follows: Section 2 briefly discusses some of the more important background literature. Section 3 discusses the econometric model and Section 4 presents the empirical results, including detailed analysis of the estimated models and the properties of the regimes. Section 5 reports the results of a simulation study and Section 6 provides a brief conclusion. 2. Uncovered interest rate parity and nonlinearity The theory of Uncovered Interest Rate Parity (UIP) is of central importance in international finance and states that E t ðds tþk Þ¼ði t;k i t;k Þ; ð1þ where E t ($) denotes the conditional expectation based on a sigma field of all relevant information at time t. The variable s t 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; i t,k and i t;k are the k periods to maturity nominal interest rates available on similar domestic and foreign assets, respectively, while Ds tþk hs tþk s t. Since covered interest rate parity is known to hold virtuously continuously, Eq. (1) can also be written as E t ðds tþk Þ¼ f t;k s t ¼ðit;k i t;k Þ: ð2þ Hence the expected future k period rate of appreciation (depreciation) is equal to the current forward premium (discount), denoted by ( f t,k s t ). Following Fama (1984), and assuming k ¼ 1 for simplicity, it has been common to test the UIP hypothesis by embedding Eq. (2) into the regression framework of Ds tþ1 ¼ a þ b f t;1 s t þ utþ1 ; ð3þ where UIP implies that u tþ1 is a disturbance. The null hypothesis of UIP being valid, implies a ¼ 0 and b ¼ 1 and u tþ1 being serially uncorrelated. The forward premium anomaly refers to the fact that estimation of Eq. (3) has generally led to a negatively valued slope coefficient. This

4 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 25 finding is widespread regardless of choice of numeraire currency and sample period, although the finding is most extreme with data from 1973 through to the mid 1980s. In one survey, Froot and Thaler (1990) report that the mean value of ^b across 75 published studies is 8. Hence the forward premium anomaly implies that the country with the higher rate of interest has an appreciating currency, rather than a depreciating currency, as implied by the theory of UIP. Some theoretical motivation for focusing on asymmetries and nonlinearities in the forward premium anomaly is available from the work of Baldwin (1990), Dumas (1992), Sercu and Wu (2000), Lyons (2001) and others. Baldwin s (1990) partial equilibrium model has homogenous, risk neutral currency market traders who have small transaction costs, with the optimal level of cross currency bond market speculation being marked by a first order hysteresis band. Spot returns are only influenced by interest rate differentials outside a band. Dumas (1992) developed a general equilibrium model of exchange rate determination in spatially separated markets with significant costs of international trade. The model implies that the nominal exchange rate will depend nonlinearly on the fundamentals, with the speed of adjustment to parity increasing in proportion to the deviation from parity. Hollifield and Uppal (1997) also derive nonlinear relationships resulting from agents closing arbitrage conditions in financial markets. More recently, Sercu and Wu (2000) proposed a model in which transactions cause a bias in the forward premium anomaly regression, regardless of the possible existence of a risk premium. While Kilian and Taylor (2003) consider a nonlinear relationship between the level of a nominal exchange rate and its fundamental value. The form of nonlinearity is driven by the presence of heterogenous traders and the strength of reversion to the fundamental level. A further rationalization for the introduction of nonlinear dynamics in the UIP condition comes from the limits to speculation, or arbitrage hypothesis of Lyons (2001). This model emphasizes the importance of the Sharpe ratio in determining whether certain investment strategies are followed. 1 Lyons (2001) argues that if UIP holds exactly, so that a ¼ 0, and b ¼ 1 in Eq. (3), then the associated Sharpe ratio is zero. As the slope coefficient departs from unity, the Sharpe ratio becomes positive. It is only when the Sharpe ratio is higher than a threshold level that the deviation from UIP will be high enough to be viewed as an arbitrage opportunity. Hence UIP is not expected to hold when the Sharpe ratio on a currency strategy is higher than a threshold level. In contrast to the above theoretical contributions, some previous empirical studies have found apparent nonlinearities in the adjustment process for UIP. Bilson (1981), Flood and Rose (1996) and Huisman et al. (1998) all consider examples where extreme observations of the forward premium have proportionately more influence on forcing UIP to hold. A related study by Wu and Zhang (1996) finds evidence that the forward premium anomaly is asymmetric; while Zhou (2002) shows that UIP does not hold between 1980 and Bansal (1997) and Bansal and Dahlquist (2000) provide evidence that indicates that the spot rate appreciation when dollar denominated responds differently to positive and negative interest rate differentials. The work suggests that the forward premium anomaly is far more likely to occur during periods in which US interest rates are less than foreign interest rates. 3. Dynamic logistic UIP regression This section considers the application of transition regression models to attempt to gain insights into the forward premium anomaly. Given the nature of adjustment in financial markets with learning and possibly segmented markets, and also due to the limits to arbitrage 1 The Sharpe ratio is defined as E R s R rf ss, where E[R s ] is the expected return on the strategy, R rf is the risk-free interest rate, and s s, is the standard deviation of the returns to the strategy.

5 26 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 arguments, it seems intuitively plausible to examine smooth asymmetric adjustment, rather than discrete adjustment. For this reason a dynamic logistic smooth transition regression (LSTR) modeling approach is implemented in this study. The LSTR model is related to the Logistic Smooth Transition Auto-Regressive (LSTAR) models introduced by Granger and Terasvirta (1993) and by Terasvirta (1994). The adjustment process in the LSTR model occurs in every period and the speed of adjustment is governed by the values of a transition variable. While there are many possible choices of transition functions, the logistic function specification is attractive in the present context since it allows for relatively sharp asymmetries in the adjustment processes. The LSTR model for the forward premium anomaly regression is then Ds tþ1 ¼ a 1 þ b 1 ft;1 s t þ a2 þ b 2 ft;1 s t Fðzt ; g; cþþu tþ1 ; ð4þ where u tþ1 is a zero mean, stationary I(0) disturbance term, and F($) is the transition function which determines the speed of reversion and is chosen to be the logistic function, Fðz t ; g; cþ¼ð1 þ expðgðz t cþ=s zt ÞÞ 1 with g > 0; ð5þ where z t is the transition variable, s zt is the standard deviation of z t, while g is a slope parameter and c is a location parameter. The parameter restriction g > 0 is an identifying restriction. The logistic function (5), is bounded between 0 and 1, and depends on the transition variable z t so that Fðz t ; g; cþ/0asz t / N, Fðz t ; g; cþ ¼0:5 for z t ¼ c, and Fðz t ; g; cþ/1asz t / þn. When g/n, Fðz t ; g; cþ becomes a step function, such that the LSTR model becomes effectively a threshold model. Therefore, the LSTR model nests a two-regime threshold model. For g ¼ 0, Fðz t ; g; cþ ¼0:5 for all z t, in which case the model reduces to a linear regression model with parameters a ¼ a 1 þ 0:5a 2, and b ¼ b 1 þ 0:5b 2. The exponent in Eq. (5) is normalized by dividing by s zt, which allows the parameter g to be approximately scalefree. This is particularly useful for the initial estimates for the nonlinear optimization used to estimate the parameters in Eq. (4). The values taken by the transition variable and the transition parameter g will determine the speed of reversion to UIP. For any given value of z t, the transition parameter g determines the slope of the transition function and hence the speed of transition between extreme regimes, with low values of g implying slower transitions. 2 This study considers several possible transition variables that may plausibly be expected to influence the validity of the UIP condition. In particular, lagged forward premia, money supply differentials, income differentials, combinations of fundamentals, squared forward premia, and conditional variances of fundamentals that are theoretically linked to time varying risk premium, are all contained in the menu of transition variables being considered. These variables are considered likely to influence the speed of adjustment to equilibrium as a function of the size of the deviation from equilibrium. The parameter c can be interpreted as the threshold between the two regimes corresponding to Fðz t ; g; cþ ¼0 and Fðz t ; g; cþ ¼1, in the sense that the logistic function changes monotonically from 0 to 1 as z t increases, while Fðc; g; cþ ¼0:5. Note that the inner regime 2 Alternatively, the LSTReUIP model can be thought as a model in the spirit of a Markow switching regression where the probability of switch between extreme regimes is a function of an observed-predetermined switching variable, rather than an unobserved state variable.

6 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 27 corresponds to z t ¼ c, where Fðz t ¼ 0; g; cþ ¼1=2 and Eq. (4) becomes a UIP regression of the form Ds tþ1 ¼ ða 1 þ 0:5a 2 Þþðb 1 þ 0:5b 2 Þ f t;1 s t þ utþ1 : ð6þ The lower regime corresponds for given g and c to lim zt /NFðz t ; g; cþ where Eq. (4) becomes a standard linear UIP regression Ds tþ1 ¼ a 1 þ b 1 ft;1 s t þ utþ1 ; ð7þ while the upper regime corresponds to lim zt /þnfðz t ; g; cþ where Eq. (4) becomes a different UIP regression Ds tþ1 ¼ ða 1 þ a 2 Þþðb 1 þ b 2 Þ f t;1 s t þ utþ1 : ð8þ Hence model (4) nests three regimes with different dynamics. Under the restrictions of a 1 þ a 2 ¼ 0 and b 1 þ b 2 ¼ 1, the upper regime corresponds to a domain in which UIP has a high probability of holding. As the transition variable increases, so does the probability of the UIP condition being valid. Conversely, smaller values of the transition variable move towards a band or region where the anomaly has a high probability of occurring. One test for the success of the model is to test UIP from an estimated forward premium regression on all the observations lying within the outer or inner bands. This is implemented in the following section of this paper. If nonlinear model (4) approximates the true dgp of the UIP relationship, then the slope coefficient in the forward premium anomaly regression (3) can be expected to be different to unity. This conjecture turns out to be supported by the simulation evidence presented in Section 5 of this paper. A further point about Eq. (4) is that it has the interpretation of being an error correction model with the lagged forward premium being the deviations from a cointegrating relationship, with the transition function F($) playing the role of a nonlinear filter applied to the transition variable. There is voluminous evidence that spot and forward rates are cointegrated with a coefficient of unity; and Baillie and Bollerslev (1994) and Maynard and Phillips (2001) argue that the forward premium is well approximated as a long memory process, which suggests a form of fractional cointegration as developed by Granger (1986). It should be noted that model (4) implies a yet more complex form of cointegration. An important consideration in subsequent analysis turns out to be the choice of the parametric transition function. The logistic transition function of the LSTR and LSTAR models appears considerably more general and flexible in this situation, than the ESTAR model, Gðz t ; gþ ¼1 exp g z 2 t, with zt again representing the transition variable. The ESTR model inevitably imposes strong restrictions of symmetry that are not consistent with the data employed in this study. 3 A far simpler and less appealing set up would be to allow discrete switching from one period to another. This can be expressed as a regression of spot returns on 3 Since the first draft of this paper was written, the authors became aware of a working paper by Sarno et al. (2004), which uses ESTR models and deviations from UIP for a transition variable. Instead of a priori excluding the ESTR representation, we have also estimated models with exponential transition function and found out that the data in this study reject the symmetry assumption imposed by the ESTR model in most cases. Details of the results can be obtained from the authors.

7 28 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 an intercept and two separate variables involving the positive lagged forward premia and the negative lagged forward premia. The regression is then Ds tþ1 ¼ a þ b þ f t;1 s t þ þb f t;1 s t þu tþ1 ; ð9þ where ft;1 s t þ ¼ ft;1 s t ; if ft;1 s t > 0 0; if f t;1 s t < 0 and ft;1 s t ¼ 0; if ft;1 s t > 0 ft;1 s t ; if ft;1 s t 0 : where the variables ð f t s t Þ þ and ð f t s t Þ represent positive and negative forward premium, respectively. This approach is essentially equivalent to separating the periods into those with positive and negative interest rate differentials, and was implemented by Bansal (1997) and Bansal and Dahlquist (2000). This type of immediate switching between regimes does not seem as appealing as smooth transition functions when dealing with the dynamics of financial markets. 4. Empirical results This study uses monthly data on spot exchange rates, and one month forward exchange rates for the Belgian Franc (BF), Canadian Dollar (CD), Dutch Guilder (DG), French Franc (FF), German Mark (GM), Italian Lira (IL), Japanese Yen (JY), Swiss Franc (SF) and UK Pound (UKP) vis-à-vis the US Dollar. The data are provided by the Bank of International Settlements and are end of month mid rates, from December 1978 to December 1998 for the Eurozone currencies of the BF, DG, FF, GM and IL; and are from December 1978 to January 2002 for the other currencies of CD, JY, SF and UKP. Additionally monthly observations on money supply, and income measured by the index of industrial production were provided by the Federal Reserve Bank of Cleveland. 4 Panel A of Table 1 reports the results from the conventional forward premium regression (3), which realizes the typical anomalous results with estimates of the intercept a being close and generally not significantly different from zero, and with estimated slope coefficients b, which are negative in all cases except for the FF and the IL. Robust t-statistics of the null hypothesis that b ¼ 1 indicate rejection at conventional significance levels in all cases except for the FF. Panel B of Table 1 relates to the estimation of the simple discrete switching model (9) with the forward premium being used as the transition variable. When the premium on the US dollar is positive, then the estimated slope coefficients are all positive, except for the CD and the UKP. When the forward premium is negative, the slope coefficients are all negative, and significantly different from unity in all cases except in the cases for the CD and the IL. The t-test for the hypothesis that the slope coefficient is unity is rejected only in three cases (CD, JY, and UKP) at the 5 significance level in the state where the forward premium is positive. While the same null hypothesis is rejected in seven out of nine cases in the state in which 4 A narrow M1 definition of money supply was used in order to maintain comparability across countries.

8 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 29 Table 1 Forward premium anomaly regressions BF CD DG FF GM IL JY SF UKP Panel A a (01) b 24 (03) (78) (30) 32 (50) 94 (69) 48 (51) (84) (05) (19) t b¼ Panel B a (01) b þ (51) (51) (1.513) (81) (1.467) 46 (69) (2.117) (2.768) (37) þ t b ¼ b (1.798) 68 (85) (1.144) (1.628) (97) (6.973) (61) (65) (1.993) t b ¼ W Key: Asymptotic standard errors are in parentheses along with the corresponding parameter estimates. W is the robust Wald test statistic for testing H 0 : b þ ¼ b in Eq. (9) and has an asymptotic c 2 distribution with two degrees of freedom. Panel A reports the standard UIP regression results. Panel B displays Bansal (1997) type regression results. the forward premium is negative. The results suggest that the forward premium anomaly is related to situations in which the US interest rate is less than foreign interest rates (i.e. when US dollar is quoted at a discount). The robust Wald test for the equality of the slope coefficient f t;1 s t þ and ft;1 s t is rejected for five out of the nine currencies. These results are broadly consistent with those of Wu and Zhang (1996), Bansal (1997) and Bansal and Dahlquist (2000). Nonlinearities in the forward premium regression can be more formally analyzed by implementing the LSTReUIP regression (4). Following Terasvirta (1994) the model is estimated by nonlinear least squares, with the starting values obtained from a grid search over g and c. Tables 2e5 provide details of the estimation of LSTR models when the transition variable is the lagged forward premium. Hence z t ¼ f t;1 s t szt, which can be interpreted as the Risk Adjusted Forward Premium (RAFP). 5 The results of estimating the LSTReUIP model (4) unrestrictedly are given in Table 2; while the model estimated subject to the restrictions of a 1 þ a 2 ¼ 0 and b 1 þ b 2 ¼ 1 are reported in Table 3. The robust Wald test of these restrictions in Table 2 fails to reject in all cases at the 5 level except for the CD when one uses the asymptotic critical value of On the other hand, bootstrapped critical values indicate that restrictions are rejected for both the CD and SF at the.05% level while they are not rejected at.025% level for the SF (although not reported in the table,.025% critical value for the SF is while 5 A further specification that was attempted was to use the deviations from UIP; i.e. z t ¼ s tþ1 f t;1. Although the results were qualitatively quite similar to the forward premium being used, diagnostic tests suggested the deviations from UIP were statistically inferior to the forward premium. This is partly due to the variability of z tþ1 ¼ s tþ1 f t;1 ¼ðstþ1 s t Þ f t;1 s t being dominated by the spot rate appreciation (noise) compared to the forward premium (signal). Full details of the results from using deviations from UIP can be obtained from the authors on request.

9 30 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 Table 2 Unrestricted estimation of LSTReUIP model when the transition variable is the forward premium Ds tþ1 ¼ a 1 þ b 1 ft;1 s t þ a2 þ b 2 ft;1 s t Fðzt ; g; cþþu tþ1 ; where Fð$Þ¼½1 þ expðgðz t cþ=s zt ÞŠ 1 BF CD DG FF GM IL JY SF UKP a 1 04 (08) 04 (06) (07) 12 (05) 06 (05) 08 (19) b (2.742) 58 (2.793) (1.412) (2.430) (1.448) (1.699) (2.403) (57) (78) a 2 03 (10) 02 (12) 11 (10) 09 (10) 09 (14) 34 (29) 07 (38) 64 (45) 80 (90) b (2.689) (1.641) (2.613) (2.371) (2.660) (3.252) 181 (5.278) (11.203) (13.836) g (2.709) (3.697) (3.045) (2.061) (3.605) (32) (71) (35) (1.70) c $ 02 $ 01 $ 07 $ $ 05 (01) Wald W 95% LM (4) LM (8) prnl AIC SIC T Key: Asymptotic standard errors are in parentheses below the corresponding parameter estimates. The transition variable is the lagged risk adjusted forward premium. Wald denotes the robust Wald statistic for the null hypothesis that a 2 þ a 1 ¼ 0 and b 2 þ b 1 ¼ 1 which is c 2 distributed with two degrees of freedom. W 95% denotes the 95 percentile of the bootstrapped distribution of the Wald statistic; while prnl is the p value for the test of no remaining nonlinearity in the residuals; and LM (4) and LM (8) are LM tests for testing for the presence of serial correlation in the residuals up to lags 4 and 8, respectively. These tests are constructed as in Eitrheim and Terasvirta (1996). AIC and SIC are the Akaike and Schwartz Information Criteria, respectively. T refers to the sample size. for the CD it is 5.660). 6 The unrestricted results in Table 2 indicate that apart from the CD, all the estimates of the transition parameter g, are significantly different from zero. Also, with the exception of the CD, all estimates of b 1 are negative and generally significantly different from zero, while estimates of b 2 are positive. The results in Table 2 generally indicate that the anomaly tends to occur for small and/or negative forward premium on the US dollar, while large forward premia are generally consistent with the UIP condition being less likely to be rejected. 7 The restricted LSTReUIP results in Table 3 have estimated threshold parameters ĉ, which are statistically different from zero for three currencies. For all cases the estimated value of b 1 is negative and that of b 2 is large and positive indicating that the UIP condition is more likely to hold as higher values of the RAFP are associated with F($) moving towards unity. The 6 Since the asymptotic distribution of Wald statistic would possibly depend on other parameters of the LSTR model (g and c essentially), the critical values denoted by W 95%, are simulated with the appropriate sample sizes by calibrating on the estimated LSTR models under the null hypothesis a 1 þ a 2 ¼ 0 and b 1 þ b 2 ¼ 1 in Eq. (4) with innovations bootstrapped from the residuals of the estimated models. 7 Estimation of some unrestricted LSTR models for the numeraire currencies of the Japanese Yen and the UK pound did not change the basic results in Table 2.

10 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 31 Table 3 Restricted estimation of LSTReUIP model when the transition variable is the forward premium Ds tþ1 ¼ a 1 þ b 1 ft;1 s t þ a2 þ b 2 ft;1 s t Fðzt ; g; cþþu tþ1 ; where a 2 ¼a 1 ; b 1 ¼ 1 b 2 and Fð$Þ¼ ½1 þ expðgðz t cþ=s zt ÞŠ 1 BF CD DG FF GM IL JY SF UKP a 1 ¼a 2 07 (10) 02 (01) (01) 13 (05) (05) 06 b 1 ¼ 1b (4.261) 85 (90) (1.469) (2.250) (1.449) (77) (81) (82) (69) g (1.374) (3.483) (1.893) (1.378) (2.533) (3.046) (2.171) (2.584) (1.899) c $ 03 (01) $ 02 $ 08 $ $ 05 (00) LM (4) LM (8) prnl AIC SIC T Key: As for Table 2. estimated level of the forward premium for attainment of the upper regime varies from approximately % for the BF and the CD; % for the DG; 5% for the GM; % for the JY; % for the FF; 5% for the UKP; and % for the IL. 8 Conversely, the forward premium anomaly is likely to exist when the forward premium on the US dollar is approximately less than % for the DG, GM, JY and SF; % for the BF; and % for the FF. Analogously the anomaly is likely to occur when the US dollar is quoted at a premium of approximately less than % for the CD; % for the IL; and 5% for the UKP. For most currencies the transition function attains values closer to unity between 1989 and 1994 and closer to zero between 1979 and 1989 and also after This finding may explain why conventional UIP regressions tend to reject the UIP hypothesis less severely, when data from the 1990s are used to test it; see the discussions in Baillie and Bollerslev (2000) and Flood and Rose (2002). Fig. 1 displays the estimated transition function graphed against the RAFP transition variable. The high levels of uncertainty involved in estimating the transition parameter have been documented by a number of authors, e.g. van Dijk et al. (2002). These issues carry over to estimation of the transition functions themselves. The estimated transition functions are plotted against time for each currency in Fig. 2. Each estimated transition variable has its 95% confidence intervals plotted as broken lines around the solid line of the transition function. For some currencies such as the FF, the sampling variability of the estimated transition function is particularly acute. In turn this affects the precise definition of when regimes occur. Further analysis of the validity of the LSTR models is conducted by computing a forward premium anomaly regression for all the observations lying in each regime. This seems a reasonable procedure since the residuals in regression (3) appear to be approximately serially uncorrelated, so that each observation in a particular regime is randomly drawn throughout the sample. 8 These results can be seen from the estimated transition function graphs by looking at the region for the forward premium for which the transition takes value unity.

11 32 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 Table 4 Forward premium anomaly regressions estimated for different regimes identified from unrestricted LSTR models Ds tþ1 ¼ a þ bð f t s t Þþu tþ1 Regime a b W a¼0, b¼1 t b¼1 b 95% a 95% n 1. BF Outer 05 (05) (1.270) , , Middle (7.240) , , Lower 06 (05) (2.523) , , CD Outer (13) , 88 02, Middle 03 (01) 01 (1.987) , , Lower 03 (01) 31 (4.398) , , DG Outer 03 (07) 66 (2.094) , , Middle 06 (06) (13.146) , , Lower (1.280) , , FF Outer (16) , , Middle (7.712) , , Lower 01 (07) (2.496) , , GM Outer 01 (13) 87 (3.451) , , Middle (2.713) , , Lower 11 (05) (1.330) , , IL Outer 25 (29) 70 (2.532) , , Middle 36 (10) (1.375) , , Lower (1.499) , , JY Outer 64 (22) 291 (9.203) , , 21 3 Middle (2.566) , , Lower 13 (06) (1.190) , , SF Outer 81 (27) (7.995) , , Middle 11 (26) (16.315) , , Lower (63) , , UKP Outer 78 (71) (13.016) , , Middle 48 (04) (48.367) , , Lower (78) , , Key: W a¼0, b¼1 stands for the robust Wald test for testing the null hypothesis that a ¼ 0, b ¼ 1. The.05% critical value for two degrees of freedom is t b¼1 is the standard t-ratio statistics for testing b ¼ 1 in the forward premium anomaly regression. b 95% and a 95% stands for the 95% confidence intervals for the slope and intercept parameters in the same regression. n is the sample size for the corresponding regime. In the results reported in Tables 3 and 4, the outer regime of the transition function is defined for p > 0, the inner regime for p < 0, and the central (middle) regime defined to be between them. This definition of regimes is clearly arbitrary, and is necessary due to the uncertainty in the estimation of the transition function. The choice of p values adopted here was

12 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 33 Table 5 Forward premium anomaly regressions estimated for different regimes identified from restricted LSTR models Ds tþ1 ¼ a þ bð f t s t Þþu tþ1 Regime a b W a¼0, b¼1 t b¼1 b 95% a 95% n 1. BF Outer 02 (08) 66 (1.715) , , Middle (2.152) , 84 07, Lower 07 (08) (2.934) , 58 23, CD Outer 09 (05) (1.517) , 88 01, Middle 04 (05) (2.336) , , Lower 02 (01) (49) , 42 00, DG Outer 01 (08) 33 (2.393) , , Middle 16 (05) (17.755) , , Lower (1.303) , , FF Outer 06 (09) (34) , , Middle (1.767) , , Lower (1.436) , , GM Outer 01 (13) 87 (3.451) , , Middle (2.757) , , Lower (1.318) , , IL Outer 10 (25) (2.228) , , Middle 92 (19) 120 (2.019) , , Lower (95) , 39 02, JY Outer 01 (07) 56 (4.193) , , Middle 14 (06) 46 (14.875) , , Lower (90) , , SF Outer 04 (11) (4.525) , , Middle (107) , , Lower (74) , 67 17, UKP Outer 68 (93) (18.800) , , Middle 09 (49) (9.423) , , Lower (66) , , Key: As for Table 4. found to be consistent across countries in order to achieve a reasonable number of observations in each regime. To restrict the outer regime to values of the transition function to say p > 5 greatly reduces the number of observations for some currencies, which renders the forward premium regression very sensitive to estimation error. Also, given the very uncertainty of estimating the transition function, it seems reasonable to err on the side of inclusivity rather than

13 34 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47-06 Belgian Franc Canadian Dollar Dutch Guilder Fig. 1. Estimated transition functions over transition variable. Key: estimated transition functions over lagged forward premium are displayed. The vertical lines are approximate bands for which F($) z 0 and F($) z 1. exclusivity by using reasonably wide bands. Analogous results for Tables 3 and 4 for a full grid of p values are available from the authors on request, but are omitted for reasons of conserving space. The results of estimating the forward premium regression from the unrestricted LSTR model are given in Table 4 and from the restricted LSTR model are given in Table 5. It can be seen that the estimated regressions for the outer regime are apparently quite supportive of the theory of UIP, with positive slope coefficients for six of the nine currencies, and robust Wald tests that fail to reject UIP for four of the nine currencies. Not surprisingly, these figures are even more favorable in Table 5 for the restricted LSTReUIP model with the robust Wald test failing to reject at the 5 level for eight of the nine currencies.

14 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e French Franc German Mark Italian Lira Fig. 1 (continued). While the RAFP is in some sense an obvious choice for transition variable, it is plausible that fundamentals in an asset pricing, monetary model framework could be equally informative. Consequently, Tables 6e8 report unrestricted LSTReUIP models for money supply differentials, income differentials and the income differential velocity, respectively. The results in Table 6 from using the money supply differential as a transition variable are qualitatively very similar to that of the forward premium in Table 2. For eight of the nine currencies, a robust Wald test applied to the unrestricted LSTReUIP model with the transition variable being the money supply differential is unable to reject the UIP restrictions at the 5 level. Only the SF leads to a rejection of the UIP conditions. Similarly to Table 2, the growth of foreign

15 36 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47-08 Japanese Yen Swiss Franc UK Pound Fig. 1 (continued). money relative to US money growth leads to the transition function approaching the outer regime. 9 It generally seems that money supply differentials are equally important for defining transition variables as the forward premium. Analysis of individual regimes as in Tables 4 and 5 were also computed for the money supply differential models similarly to the forward premium. The results were qualitatively the same as in Table 4 and are not reported for reasons of saving space. The use of other fundamentals for the transition variable is noticeably less 9 A potential explanation is that in some time periods, the growth of foreign money relative to US money leads to the expectation that the dollar may appreciate and of possible increases in US interest rates. Hence the forward premium on the US dollar increases, which in turn leads to more investors buying US denominated assets. Thus periods in which US money growth is less than foreign money growth may create a tendency for UIP to hold.

16 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e /80 12/84 BF 12/88 12/92 12/96 CD 12/80 12/84 12/88 12/92 12/96 12/00 DG 12/80 12/84 12/88 12/92 12/96 Fig. 2. Estimated transition functions over time. Key: panels display estimated transition functions with 95% confidence intervals from the unrestricted LSTReUIP regression over the sample period.

17 38 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 12/80 12/84 FF 12/88 12/92 12/96 12/80 12/84 GM 12/88 12/92 12/96 IL 12/80 12/84 12/88 12/92 12/96 Fig. 2 (continued). successful. The use of the income differential and also relative income velocity as transition variables in Tables 7 and 8, respectively, leads to the robust Wald test rejecting the UIP constraints on five occasions based on the bootstrapped critical values. Hence the only fundamental variables which are generally useful in defining transition variables are interest rate differentials and money supply differentials.

18 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 39 JY 12/80 12/84 12/88 12/92 12/96 12/00 SF 12/80 12/84 12/88 12/92 12/96 12/00 UKP 12/80 12/84 12/88 12/92 12/96 12/00 Fig. 2 (continued).

19 40 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 Table 6 Unrestricted estimation of LSTReUIP model when the transition variable is the money growth rate differential Ds tþ1 ¼ a 1 þ b 1 ft;1 s t þ a2 þ b 2 ft;1 s t Fðzt ; g; cþþu tþ1 ; where Fð$Þ¼ ½1 þ expðgðz t cþ=s zt ÞŠ 1 ; with z t ¼ m t m t BF CD DG FF GM IL JY SF UKP a (05) (06) (09) 07 b (66) (09) (1.459) (1.548) (1.156) 48 (1.511) (81) (1.600) (1.135) a (08) 01 (05) 08 (05) 02 (05) 12 (12) 17 (20) 05 (10) 03 b (1.420) 84 (2.561) (2.092) (1.639) (1.590) (2.011) (3.890) (1.746) (2.064) g (18.912) (2.600) (3.882) (1.398) 254 (4.535) (2.353) (2.264) 105 (3.462) (1.419) c $ 07 $ 04 $ $ $ (15) (01) (09) Wald W 95% LM (4) LM (8) prnl AIC SIC T Key: As for Table 2. A further interesting possibility is that the forward premium anomaly is fundamentally associated with levels of time varying risk premium rising above a certain threshold. As noted by Engel (1996) and Baillie and Bollerslev (2000) among others, the Euler equation for risk averse investor equalizing real returns in the spot and forward markets can be approximated as Ds tþ1 ¼ f t;1 s t 0:5 Vart ðs tþ1 ÞþCov t ptþ1 s tþ1 þ Covt qtþ1 s tþ1 þ utþ1 ; ð10þ where u tþ1 is a disturbance and q tþ1 represents the logarithm of the inter-temporal marginal rate of substitution between consumption in different time periods. Since monthly spot returns have virtually no ARCH effects, the Jensen inequality term is usually omitted from consideration; see Bekaert and Hodrick (1992, 1993) and Engel (1996). Hence the term Cov t (q tþ1 s tþ1 )in Eq. (11) is generally considered to be the focus for modeling time dependent risk premium. There is a voluminous literature on the estimation of risk in such models. Several authors such as Giovannini and Jorion (1987) and Hodrick (1989) have used the lagged squared forward premium to proxy the risk premium. Table 9 accordingly reports estimates of the unrestricted LSTReUIP model with the lagged squared forward premium as the transition variable. The results are generally unsupportive of the model with the robust Wald test strongly rejecting for six of the nine currencies. Other favored possibilities for modeling the Cov t qtþ1 s tþ1 term arises from using linear combinations of the conditional variances of the fundamentals as a proxy for this term. A

20 R.T. Baillie, R. Kiliç / Journal of International Money and Finance 25 (2006) 22e47 41 Table 7 Unrestricted estimation of LSTReUIP model when the transition variable is the income growth rate differential Ds tþ1 ¼ a 1 þ b 1 ft;1 s t þ a2 þ b 2 ft;1 s t Fðzt ; g; cþþu tþ1 ; where Fð$Þ¼ ½1 þ expðgðz t cþ=s zt ÞŠ 1 ; with z t ¼ y t y t BF CD DG FF GM IL JY SF UKP a (09) $ 16 (12) b (50) (95) (1.690) (64) (1.202) (12) (1.432) $ (3.847) a (10) 05 (06) (06) 06 (17) $ 12 (13) b (1.353) (1.262) (7.496) (2.072) 84 (1.455) (1.114) 61 (2.650) $ (4.172) g (3.405) (70) 459 (3.220) 187 (1.563) (4.564) 3152 (13.652) (4.632) $ (2.366) c $ 14 (07) (01) $ $ $ $ 16 (11) Wald $ W 95% $ LM (4) $ LM (8) $ prnl $ 01 AIC $ SIC $ T $ 278 Key: As for Table 2. theoretical derivation from a cash in advance, overlapping generations model is available in Hodrick (1989); while Baillie and Osterberg (1997) extend the model to include the effects of central bank intervention and Kaminsky and Peruga (1990) use a multivariate GARCH model to capture the risk premium with the conditional covariances of the monetary model s fundamentals playing a role in modeling risk. Accordingly, in this study the conditional variances of various fundamentals were generated from appropriate univariate ARIMAeGARCH models. The precise formulation of these models can be obtained from the authors on request and are omitted here. After considerable experimentation it was found that one of the most useful transition variables was the conditional variance of US money growth rates. The results are given in Table 10 and for seven out of the nine currencies the robust Wald test indicates that the UIP conditions cannot be rejected. The transition variable in this case is defined as the unconditional variance minus the conditional variance; so that periods of high volatility in US money growth are associated with an inner regime where the forward premium anomaly is more likely to occur. While periods of relative stability and low volatility of US money growth move the transition function to the outer regime where there is a high probability of the UIP conditions not being rejected. The results from using the conditional variances of other fundamentals as transition variables were generally more mixed. The use of the conditional variances of other countries money supplies was less consistently favorable to the LSTReUIP model and is not reported. Also, for many countries the conditional variance of income did not generally contain sufficient time variation to make it useful as a transition variable. We also attempted to estimate LSTR models with a vector of transition variables; see van Dijk et al. (2002) for a discussion of some of the possible models. However, the resulting