Jul 6, 2007 - with low interest rates and the purchase of currencies with high interest rates. ... Similar roles are played by momentum trading strategies, which are .... When either (3a), (3b), or (3c) hold,there is an increased probability that UIP ... The link between trend chasing and ...... Marcel Dekker, New York, 507&552.
Carry Trades, Momentum Trading and Nonlinear Adjustments to Uncovered Interest Parity Richard T Baillie Michigan State University, USA and Queen Mary University of London, UK Sanders S Chung Michigan State University, USA July 6, 2007
Abstract This paper examines the role of recent currency trading strategies, known as carry trading and momentum trading and their implications for risk premium in the FX market. The risk premium arises from the forward premium anomaly, where the slope coef…cient in a regression of spot returns on the lagged interest rate di¤erential is negative and signi…cantly di¤erent to unity. The paper estimates Logistic Smooth Transition Dynamic Regression (LSTR) models with a variety of transition variables, including di¤erent carry trading interest rates and also volatility of spot rates associated with a momentum trading strategy. There is some evidence for the existence of an outer regime, that is consistent with uncovered interest parity holding when US interest rates exceed corresponding rates for the carry trade. There is conversely evidence for an inner regime that is consistent with the forward premium anomaly, where risk premia are statistically signi…cant. JEL Classi…cation: C22, F31, F41. Keywords: Forward premium anomaly, Uncovered Interest Parity, Non-linearity, LSTR models.
Corresponding author.
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1
Introduction
Many commentators and practitioners in …nancial markets have emphasized the recent phenomenon of the growth of carry trading, which involves the borrowing or selling of currencies with low interest rates and the purchase of currencies with high interest rates. From an international …nance perspective this provides fascinating corroboration that FX market participants are well aware and able to exploit the well known forward premium anomaly. Hence, a substantial proportion of FX market agents appear willing to exploit the empirical fact that currencies associated with relatively high interest rates have tended to appreciate and low interest rate currencies to depreciate. Clearly, the presence of signi…cant risk premium compensate investors for betting against uncovered interest rate parity. Similar roles are played by momentum trading strategies, which are essentially bandwagon e¤ects of joining existing trends, and hence further reinforce the appreciating currencies associated with high interest rates. Clearly, such trading strategies are short term and risky since they ignore fundamental’s evaluation of currencies and are vulnerable to sudden unanticipated changes in exchange rates. It has been suggested that the relative calmness in the world’s FX markets since the Asian crisis of 1998, have misled many FX traders into believing that momentum strategies are successful. The Economist (2006) recently stated that “the reasons for the success of the carry trade remain a bit of a mystery”. The aim of this paper is to incorporate the carry trading and momentum trading approaches into models for risk premium as an explanation for the failure of uncovered interest rate parity and to provide further understanding of the forward premium anomaly. While the new carry trading and momentum strategies have drawn attention from the …nancial press and media, there have hitherto been remarkably few academic studies of the phenomenon. A notable exception is Galati and Melvin (2004, p. 67), who refer to a survey by the Bank for International Settlements (BIS), which concludes that the substantially increased turnover in the FX market between 2001 and 2004 "seems to have been driven by momentum trading and carry trades in a global search for yield. . . ". The current study attempts to bridge the gap between this relatively new trading behaviour and the academic literature on risk and the failure of UIP. The theory of Uncovered Interest rate Parity (UIP) implies that the expected return, or rate of appreciation on a currency equals the interest rate di¤erential, 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 coe¢ cient is unity, the intercept zero and the residuals serially uncorrelated.
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The forward premium anomaly is the widespread empirical …nding of a negative slope coe¢ cient 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 ‡oating currencies in the current ‡oat and appears robust to the choice of numeraire currency. Froot and Thaler …nd that the average estimated coe¢ cient across 75 published to be -0.88. 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 uncovered interest rate parity. There are several proposed explanations of the anomaly. For example, considerable previous work has modeled time dependent risk premia, e.g., Hodrick (1987, 1988) 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 Maynard and Phillips (2001) and Baillie and Bollerslev (2000), who also show that the magnitude and sign of the estimated slope coe¢ cient in the forward premium regression appears to be slowly time varying, particularly in small sample sizes. This current paper applies similar methodology to that of Baillie and Kilic (2006), namely the logistic smooth transition dynamic regression (LSTR) model, which is related to the LSTAR and other non linear models introduced by Granger and Terasvirta (1993) and Terasvirta (1994). For most of the currencies considered, the transition variables derived from the estimated LSTR models appear to show the existence of three 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 detected. While Baillie and Kilic (2006) considered transition variables based on interest rate di¤erentials, money supply di¤erentials, income di¤erentials, combinations of fundamentals, and conditional variances of fundamentals; the present study focuses on carry trade and momentum strategies. The inner regime, where UIP breaks down is associated with transition variables that are associated with loose or uncertain US monetary policy; i.e. relatively low US interest rates, relatively US monetary growth and also high volatility of US money growth. This study also examines the e¤ect on the transitions of carry trading and momentum trading strategies. The rest of this paper is organized as follows. The next section brie‡y reviews the UIP condition and the forward premium anomaly, while Section 3 describes the currency-trading strategies and discuss their implications for nonlinear reversion to UIP. Section 4 then
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describes the econometric methodology, and the next section deals with the interpretation of the empirical …ndings. Section 6 then provides a brief conclusion.
2
The forward premium anomaly
The theory of UIP is a condition in international …nance and open economy macroeconomics. In an informationally e¢ cient market, prices should fully re‡ect all available information and it should not be possible, on average, to earn excess returns from speculation. For currency markets, this is stated as follows: Et st+1 = it
it ;
(1)
where Et is the conditional expectations operator on a sigma …eld of all relevant information up to and including time t, st is the logarithm of the spot exchange rate quoted as the foreign price of domestic currency, and it and it are the one-period risk-free domestic and foreign interest rates, respectively. Under UIP, expected returns from currency speculation are equal to the interest di¤erential, and the country with the higher interest rate is expected to have a depreciating currency. Using covered interest parity, which is an arbitrage condition that holds virtuously continuously, equation (1) can be expressed as Et st+1 = it ft
it =
st where ft is the logarithm of the forward rate for a one period ahead transaction.
Then, equivalently, (1) states that the expected rate of depreciation is equal to the forward premium (discount) on the domestic (foreign) currency. A standard test of UIP is to estimate the regression st+1 = Under UIP, the null hypothesis is that
+ (ft = 0,
st ) + ut+1:
(2)
= 1, and that the error term, ut+1 is serially
uncorrelated. The forward premium anomaly refers to the widespread …nding of a negative slope coe¢ cient that is signi…cantly di¤erent from unity. For example, Table 1 shows the results from estimating (2) by ordinary least squares (OLS) for a variety of exchange rates when the numeraire currency is (a) the dollar, (b) the yen, and (c) the Swiss franc. With a few exceptions, the estimated slope coe¢ cients are generally negative, regardless of numeraire currency. This result implies that the forward premium, or equivalently the interest di¤erential, is a biased predictor of future exchange-rate returns, and that this bias consistently points in the wrong direction.
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3
Carry trades and momentum trading.
There are some similarities between the carry trade and momentum trading strategies and the limits to speculation hypothesis of Lyons (2001). These ideas …t in well with the methodological approach of Baillie and Kilic (2006), where nonlinear threshold relationships are found to quite well describe the adjustments to UIP in certain regimes. Clearly, pro…t maximizing investors prefer to fund carry trades with the lowest cost currency (i.e., the currency with the lowest interest rate). Under covered interest arbitrage, the carry trade is equivalently implemented by selling forward currencies that are at a forward premium and buying currencies that are at a forward discount. In such a trade, investors are essentially exploiting the forward bias (thus, it is also known as bias trading), betting that the target currency will not depreciate so as to o¤set the interest di¤erential. In fact, the most basic carry trade involves borrowing in low interest rate currencies (so-called funding currencies) to fund investments (i.e., lend) in higher-yielding currencies (or target currencies). The lower the interest rate on this preferred funding currency relative to alternative funding currencies, then the more attractive it is to fund carry trades with this particular currency. Arbitrage considerations imply that excess returns from such a strategy will be eliminated and reversion to UIP will occur. This study focuses on three alternative funding currencies that have had the lowest interest rates among all developed country currencies over the past 30 years; namely the US dollar (USD), the Japanese yen (JPY), and the Swiss franc (SF). The USD is said to be the preferred funding currency if Y SF minfiJP ; it g t
SD iU >0 t
(3a)
so that ceterus paribus when evaluating the attractiveness of the dollar as a funding currency, the most important comparison presumably is between the US and the next-lowestcost currency. Similarly, the yen is the preferred funding currency if SD SF minfiU ; it g t
Y iJP >0 t
(3b)
and the Swiss franc is the preferred funding currency if SD JP Y minfiU ; it g t
iSF >0 t
(3c)
When either (3a), (3b), or (3c) hold,there is an increased probability that UIP will be valid for exchange rates expressed with the USD, JPY, or SF as the numeraire currency
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respectively. Conversely, the breakdown of (3a), (3b), or (3c) will lead to an increased probability of the forward premium anomaly1 . The carry trade hypothesis can also be motivated by the approach of Lyons (2001) where the limits to speculation hypothesis is associated with the existence of higher than usual pro…t opportunities from conducting carry trades; and induces agents to trade these pro…ts away. When speculative capital is not allocated to exploiting the forward bias with a particular funding currency, then the forward premium anomaly would arise.
3.1
Momentum trading and volatility
As noted by Galati and Melvin (2004), the BIS survey indicated "the presence of clear trends and higher volatility in foreign exchange markets led to investments in currencies that experienced a persistent trend of appreciation”. The link between trend chasing and higher volatility centers on the notion that momentum trading is a type of positive feedback investment rule. If the trend chasers respond to past price movements rather than expectations about fundamentals, an observed increase (decrease) in prices, (whether due to rational speculation, noise, or a combination thereof), is expected to trigger a round of buying (selling), which in turn causes prices to rise (decline) further, and so forth, so that momentum trading strategies can exaggerate price upswings and downswings. DeLong et al (1990) show that if momentum traders are active in the market, then rational speculation can actually have a destabilizing e¤ect on asset prices, contrary to Friedman’s (1953) notion that rational speculation is necessarily price stabilizing2 . The key observation is that rational speculators may anticipate the actions of positive feedback traders. Hence, upon receiving and trading on some fundamental news, rational speculators will make further trades in the same direction with the knowledge that the resulting price movements will spur momentum trading tomorrow. Hence it may be rational and pro…table for speculators to be on the trend chasing bandwagon, and to then leave when irrational momentum traders have also joined the bandwagon. Thus rational speculation in the presence of momentum traders may destabilize prices, amplifying deviations from equilibrium3 1
Indeed, the carry trade is quite episodic. For example, from 1995 to 1998, the yen carry trade was en vogue, as the Bank of Japan e¤ectively pursued a zero-interest rate policy (BIS Quarterly Review, 1999). Although outside the range of our data (see Section 4), market commentators refer to the period 2001-2004 as ‘the dollar carry trade’, as dollar short rates dipped to around 1%. Most recently, in the post-2005 period, the yen carry trade has become fashionable. 2 Friedman’s view is that rational speculation corrects deviations from fundamentals, thus dampening the ‡uctuations caused by ‘destabilizing’ noise traders. These noise traders, who on average buy high and sell low, are quickly eliminated from the market. 3 See Pojarliev (2005) for a study on the pro…tability of trend following rules.
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Hong and Stein (1999) show that slow di¤usion of private information across the population of ‘news watching’traders causes initial under-reaction to news, allowing momentum traders to pro…t from trend chasing as the news gets incorporated gradually into prices. However, due to positive feedback, trend chasing ultimately leads to overreaction in the long run. When subsequent groups of momentum traders enter the market in later stages of the ‘momentum cycle’, prices will have already overshot their equilibrium values so that further momentum trading becomes unpro…table, since by this time, agents are fully informed and prices necessarily revert to equilibrium. As a result, momentum trading ampli…es the cycle of overshooting and reversion, and hence leads to increased price volatility. In practice, the carry trade and momentum trading strategies can be closely related, since speculators implementing carry trades, will short sell funding currencies to buy target currencies. This causes high interest-rate currencies to appreciate against low interest-rate currencies. But such exchange-rate movements also stimulate momentum traders, who in turn might use the carry trades to trend chase and thereby amplify and prolong the appreciation of high-interest-rate currencies. According to Cavallo (2006), market participants attribute the dramatic 18% upswing in the dollar against the yen over most of 2005 to precisely this sort of behavior. Clearly, there are limits on such appreciation, and between March and May of 2006, the dollar depreciated by 10% on expectations of higher Japanese interest rates. This pattern of continuation and reversal is also consistent with Cutler, Poterba, and Summers (1991) notion of ‘speculative dynamics’.
In particular, they document that
monthly excess returns for exchange rates exhibit positive autocorrelation (continuation) up to two years, but negative autocorrelations (reversal) at longer lags. The interpretation in Cutler et al (1990, 1991), DeLong et al (1990), and Hong and Stein (1999) is that in the short run, prices may deviate from equilibrium values due to positive feedback trading, but in the long run there is a reversion to fundamentals.
4
Dynamic Logistic UIP Regression
This section considers the application of various threshold models to attempt to characterize some aspects of the forward premium anomaly. Given the nature of adjustment in …nancial markets, and the limits to arbitrage 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 Teräsvirta (1993) and by Teräsvirta (1994).
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An excellent survey of STAR models is available in van Dijk et al. (2002). Similar to STAR models, 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. The use of a logistic function speci…cation allows for sharp asymmetries in the adjustment processes. The LSTR model for the forward premium anomaly regression is then st+1 = [
1
+
1 (ft
st )] + [
2
+
2 (ft
st )]F (zt ; ; c) + ut+1 ;
(3)
where ut+1 is a zero mean, stationary I(0) disturbance term, and F (:) is the transition function which determines the speed of reversion. In this study F (:) is chosen to be the logistic function, F (zt ; ; c) = (1 + exp( where zt is the transition variable,
zt
(zt
c)=
1
zt ))
with
> 0;
(4)
is the standard deviation of zt , while
parameter and c is a location parameter. The parameter restriction
is a slope
> 0 is an identifying
restriction. The logistic function (4), is bounded between 0 and 1, and depends on the transition variable zt so that F (zt ; ; c) ! 0 as zt !
1 as zt ! +1. When
1 , F (zt ; ; c) = 0:5 for zt = c, and F (zt ; ; c) !
! 1, F (zt ; ; c) becomes a step function, such that the LSTR
model becomes e¤ectively a threshold model. Therefore, the LSTR model nests a tworegime threshold model. For
= 0, F (zt ; ; c) = 0:5 for all zt , in which case the model
reduces to a linear regression model with parameters The exponent in (4) is normalized by dividing by
= zt ,
1
+ 0:5
2,
and
=
1
+ 0:5
which allows the parameter
2.
to
be approximately scale-free. This is particularly useful for the initial estimates for the nonlinear optimization used to estimate the parameters in (3). The values taken by the transition variable and the transition parameter
will deter-
mine the speed of reversion to UIP. For any given value of zt , the transition parameter determines the slope of the transition function and hence the speed of transition between extreme regimes, with low values of
implying slower transitions.
This study considers carry trades and momentum strategies as transition variables, which may increase the probability for the exchange rate moving to a threshold where UIP is valid. These variables are also hypothesized to theoretically determine the speed of adjustment to equilibrium is 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 F (zt ; ; c) = 0 and F (zt ; ; c) = 1, in the sense that the logistic function changes monotonically from 0 to 1 as zt increases, while F (c; ; c) = 0:5. Note that the inner regime 8
1 2
corresponds to zt = c, where F (zt = 0; ; c) =
and equation (3) becomes a UIP regression
of the form st+1 = [(
1
+ 0:5
2)
+(
The lower regime corresponds to for given a standard linear UIP regression st+1 = [
1
1
+ 0:5
2 )(ft
and c to lim zt !
+
1 (ft
st )] + ut+1 : 1 F (zt ;
(5)
; c) where (3) becomes
st )] + ut+1 ;
(6)
while upper regime corresponds to limzt !+1 F (zt ; ; c) where (3) becomes a di¤erent UIP regression
st+1 = [(
1
+
2)
+(
1
+
2 )(ft
st )] + ut+1 :
(7)
Hence the model in (3) nests three regimes with di¤erent dynamics. Under the restrictions of
1
+
2
= 0 and
1
+
2
= 1, the upper regime corresponds to a domain where 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 the anomaly occurring. The conditions for the validity of the UIP regime are tested later from the estimated models. There are several formulations of the spot and forward rate cointegrating relationship. There is voluminous evidence that spot and forward rates are cointegrated with a coe¢ cient 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 (3) implies a yet more complex form of cointegration An important consideration in subsequent analysis turns out to be the choice of parametric transition function. The logistic transition function of the LST R and LST AR models appears considerably more general and ‡exible in this situation, than the EST AR model, G(zt ; ) = 1
exp(
(zt2 )), with zt again being the transition variable. The EST R
model inevitably imposes strong restrictions of symmetry, which is avoided by the LSTR formulation. The data in this study are from the BIS and consist of the spot and one month forward exchange rates for the Belgian franc (BF), Canadian dollar (CD), Dutch guilder (DG), French franc (FF), German mark (GM), Japanese yen (JPY), Swiss franc (SF), and UK pound (UKP) against the US dollar (USD). The spot and forward rates are end-of-themonth mid rates. For the BF, DG, FF, and GM, the data are from December 1978 to December 1998, for a total of 241 monthly observations. For the CD, JPY, SF, and UKP,
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the data are from December 1978 to January 2002, for a total of 277 monthly observations. In estimating (4) and (5), the transition variable zt is standardized so that it is the original variable divided by its standard deviation and hence makes the transition variable scale free.
5
Empirical results
Table 1 provides results on the standard forward premium anomaly regression with spot returns regressed on the lagged interest rate di¤erential for the three di¤erent numeraire currencies of the USD, JY and the SF. Consistent with the …ndings of previous studies, the slope coe¢ cient is generally negative and statistically signi…cantly di¤erent from both zero and unity. As noted by Baillie and Bollerslev (2000), the so called anomaly holds regardless of numeraire currency. This study focuses on the proposition that relatively pro…table carry trades, with a large di¤erential between the interest rate on the preferred funding currency and the nextlowest-cost funding currency; then the stronger is the reversion to UIP for transactions denominated in the preferred funding currency. The transition variables are then the variables de…ned in (3a), (3b), and (3c). The e¤ects of momentum trading is examined through increases in volatility of the spot exchange rate, as measured by the conditional variance of a GARCH(1,1) model; so that the transition variable is denoted as ^ 2s;t+1 . For these models the threshold parameter c is generally the estimated unconditional variance of spot returns, ^ 2s . Table 2 presents the results from estimating the LSTR in (4) and (5) using the interest Y ; iSF g di¤erential (3a), minfiJP t t
SD , as the transition variable and the dollar as the iU t
numeraire currency. For all exchange rates, the estimated slope coe¢ cients in the lower regime are negative and the forward premium anomaly is evident. The theory of UIP will hold in the outer regime when the restrictions of 1
+
2
1+
2
= 0 and
= 1 are valid. So that,the upper regime corresponds to a domain where UIP has
a high probability of holding. As the transition variable increases, so does the probability of the UIP condition being valid. Robust Wald tests reported in table 2 are unable to reject the hypothesis for the DG, GM, IL, JPY, and UKP. However, the restrictions can be rejected for the BF, FF, and SF. However, it appears that the rejection for these three exchange rates is mostly associated with the intercept estimates, which are signi…cantly di¤erent from zero. For all exchange rates except the SF, the estimates of the smoothness parameter
are
fairly small, indicating a gradual transition between regimes, and are statistically signif-
10
icantly di¤erent from zero, except for the CD. .The relatively large estimate for the SF suggests more abrupt type of switching. From a carry trade perspective, movement into the upper regime occurs when US interest rates are slightly higher than those of alternative funding currencies, so that the USD is not strictly the preferred funding currency. However, this could re‡ect the degree of substitutability of the SF and JPY for the USD in terms of, e.g., lower transactions costs when implementing carry trades with the dollar. Figure 1 shows the estimated transition functions for the CD, FF, GM, JPY, SF, and UKP, plotted against time. The estimated transition functions all appear to be in the upper regime during the …rst half of the 1990’s, which corresponds to a period when the dollar was the preferred funding currency and hence UIP is more likely to hold. All transition functions attain values close to or approaching the upper bound of F ( ) = 1 and implies that the strength of reversion to UIP depends on the size of the interest di¤erential between the dollar and the next lowest cost funding currency, These results are consistent with those of Baillie and Kilic (2006), Baillie and Bollerslev (2000) and Flood and Rose (2002), who …nd that the rejection of UIP during the 1990’s is less clear. SD ; iSF g The results from estimating (4) and (5) using minfiU t t
Y , as the transition iJP t
variable are reported in Table 3. The strongest evidence for a carry trade interpretation for non-linear adjustments to UIP for JY denominated currencies is obtained for the BF, DG, GM, and UKP, and the estimated smoothness parameters are all quite small, suggesting gradual rather than discrete transition between regimes. Figure 2, plots the estimated transition function over time for the GM and UKP (which are fairly representative of all currencies), shows that the upper regime is attained several times during the sample period. Notably, the transition function is quite close to the upper bound between 1995 and 1998, which corresponds precisely to a period market commentators have deemed the ‘yen carry trade’(BIS Quarterly Review, 1999). The SF is an interesting case and has an estimated slope coe¢ cient which is positive in the lower regime, where Japanese interest rates are higher than Swiss rates.4 Lastly, for the USD, it appears that the forward bias is present in both regimes, suggesting that the yen carry trade against the US dollar might be somewhat of a ‘money tree’, at least in our sample. While this …nding does not conform to our priors concerning the carry trade, it is nevertheless consistent with Baz et al (2001) and Villanueva (2007) who …nd that, on average, portfolios of carry trades that include long-short positions in the USD -JPY pair can earn positive excess returns. 4 Given their historically low interest rates the JPY and SF are primarily funding currencies and are rarely, if ever, targets of carry trades. For example, Baz et al (2001) show that in a currency portfolio meant to exploit the forward bias, it would have been optimal to be short both JPY and SF throughout the 1990’s.
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Table 4 shows the results from using the Swiss franc as the numeraire currency and SD ; iJP Y g the interest di¤erential (3c), minfiU t t
iSF t , as the transition variable. The robust
Wald test rejects the null hypothesis of UIP for all currencies except the FF and JPY. However, there are some interesting features of the SF due to some special considerations for the SF. In particular, Switzerland had the lowest interest rates of all industrialized nations during most of the 1980s decade, and is frequently regarded as a safe haven currency, which is in part due to the reputation of the Swiss banking industry and the currency’s explicit gold backing. This may explain how the SF has exhibited a tendency to appreciate during times of crisis. Indeed, Kugler and Weder (2005) attribute the anomalous low returns on SF assets relative to assets denominated in other major currencies to an insurance premium against catastrophic events. Speci…cally, investors are willing to hold low yielding Swiss assets with the expectation that the Swiss franc will appreciate during severe crisis situations. Given that large scale catastrophes rarely occurred in reality, Kugler and Weder (2005) point to a peso problem as the likely reason for the historical under performance of Swiss assets. Halliday (2001) has noted that during the 1980’s, which saw a resurgence in Cold War tensions, the peso problem would have been particularly acute and the safe haven premium would have been highly valuable. Speci…cally, investors’ex ante probability weight on the
magnitude and frequency of major crisis situations, and hence of Swiss-franc appreciation, would have been unusually high (though unwarranted, ex post).5 As such, peso problems and safe-haven e¤ects would have made the Swiss franc a particularly undesirable funding currency for conducting carry trades, despite the fact that Swiss interest rates were the lowest among alternative funding currencies over most of the 1980’s.
After the collapse
of the Soviet Union in 1991, an important issue faced by Swiss monetary authorities was the role of the SF alongside the ERM. Prior to the introduction of the single currency, the German mark was the most important cross rate for Swiss economic activity, and in the past the Swiss National Bank (SNB) and Bundesbank had pursued highly compatible exchangerate policies (Fischer, 2002). This resulted in an extremely stable GM-SF exchange rate during the 1990’s6 . At the same time, formal convergence of the ERM currencies began in 1990 with the …rst stage of the Economic and Monetary Union (EMU). Galati (1999) found that during 5 To illustrate the severity of the peso problem related to the Cold War during the 1980’s, Kugler and Weder (2005) …nd that the SF was particularly sensitive to events such as the death of Soviet leader Chernenko (1985), the Chernobyl disaster (1986), and the fall of the Berlin Wall (1989). In contrast, the Persian Gulf War (1990) had a considerably less decisive e¤ect. 6 Genberg and Kadareja (2001) characterize the period 1980-1999 as an explicit target zone where the SNB tried to maintain the exchange rate in a range of 80 to 90 GM/SF.
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the 1987-92 period, the co-movement of ERM currencies with the GM against the dollar was particularly strong. The link weakened between 1992 and 1995 following ERM crises when the UKP and IL exited the ERM; but the co-movements became stronger again between 1995 and 1998. Since Swiss interest rates were the lowest among EMU member states during most of the 1990s, the SF appeared to have been an ideal funding currency for this period. Baz et al (2001) use an optimal mean-variance approach to study a portfolio of carry trades and …nd that over the 1989-99 period the optimal weight on the short position in Swiss francs is consistently larger than the short weight on the JY7 Hence the relative attractiveness of SF carry trades, implies an increased probability for UIP to hold in currencies denominated in terms of the SF. The evidence presented in Table 5 is consistent with this hypothesis and the lower regime, which mainly corresponds to the 1990’s is when the SF was not the preferred funding currency but SF carry trades were nevertheless very attractive. The following variant of the LSTR model in equations (4) and (5) is intended to capture this phenomenon:: st+1 = [
1
+[
+ 1 (ft + 2 + 2 (ft
+
where (ft
1
st )+ +
st ) =
(
ft
st ) =
(
ft
+
and (ft
st )+ +
(ft 2
st ) ](1
(ft
st ; ft 0;
0;
(10)
st ) ]G(zt ; ; c) + ut+1 ;
st > 0
otherwise st ; ft
G(zt ; ; c))
st < 0
otherwise
;
(11a)
;
(11b)
again using (3c) as the transition variable. The distinction between positive and negative states of the forward premium in (11a) and (11b) is made since the analysis is for when the SF is no longer the preferred funding currency, and hence does not necessarily have a lower interest rate than the target currency. The rationale is that SF carry trades would be implemented under (11a), where SF interest rates are lower than those of the target currency, but not in (11b), where the cost of borrowing SF exceeds the target currency’s risk free rate. The nested model in (10), (11a), and (11b) exhibits discrete switching between two 7 The portfolio in Baz et al (2001) consists of the GM, JPY, SF, UKP, and USD. Most tellingly, the long (short) position in GM (SF) o¤set each other almost exactly. According to the authors, “this re‡ects the high correlation between these two currencies and the lower [SF] interest rates (p. 8).”
13
smooth transition models based on the sign of the interest di¤erential, st+1 = [
1
+
+ 1 1 (ft
st )+ ](1
G(zt ; ; c)) + [
2
+
+ 1 2 (ft
st )+ ]G(zt ; ; c) + ut+1 : (12)
The interest rate di¤erentials between the BF, FF, and IL against the SF are positive throughout the entire sample period, so for these particular currencies (12) is equivalent to (4) and
+ 1
=
1.
Furthermore, the results in Table 4 indicate that the estimates of
1
for these particular currencies are consistent with UIP. Thus, for the nested model, these considerations suggest that we pinpoint our analysis on the coe¢ cient + 1
+ 1.
Speci…cally,
corresponds to a sub regime where the SF is not the preferred funding currency but
SF-funded carry trades are still pro…table (on the basis of interest di¤erentials). The results from the estimation of (10), (11a), and (11b) are reported in Table 5. For the CD, UKP, and the …ve ERM currencies the robust Wald test does not reject the null hypothesis of UIP in the outer regime, and Figure 3 shows the estimated transition functions over time for the representative currencies of the FF, GM, and UKP. For the FF and GM, the transition functions are consistently in the lower regime for the entire decade of the 1990’s, while for the UKP, the transition function attains the lower bound of zero numerous times during the …rst half of the 1990’s. At the same time, interest di¤erentials on the CD, UKP, and ERM currencies were positive during most of this decade. Hence, these results suggest that UIP is more likely to hold when the SF is not the preferred funding currency but SF funded carry trades are nevertheless pro…table, and that such conditions prevailed during the 1990’s. Therefore, a carry-trade explanation of nonlinear adjustments to UIP in Swiss-franc cross-rates might still be viable.
5.1
Momentum trading and volatility
Table 6 shows the results from estimating the LSTR model in (4) and (5) with the transition variable being the conditional variance of the spot exchange rate. Importantly, for all exchange rates the LM tests of the parameter …xes do not reject the restrictions we have placed on the threshold parameters. Furthermore, for all currencies except the JPY, the estimated slope coe¢ cients in the upper regime are positive and for the most part very close to one. The robust Wald test does not reject the null hypotheses of UIP and the estimated smoothness parameters are all fairly small, suggesting a slow speed of transition between low-volatility and high-volatility regimes. Figure 4 plots the estimated transition functions over time for the CD, FF, GM, JPY, SF, and UKP, which are fairly representative for these estimated models. For the CD, SF, and UKP the transition functions spike intermittently into the upper regime, while for the FF and GM they attain and stay at the upper bound
14
for longer periods of time. In either case, a possible intepretation is that the upper regime corresponds to the later stages of Hong and Stein’s (1999) momentum cycle, which is characterized by overreaction and reversion about equilibrium of UIP. The spikes in the CD and SF transition functions occur on average every two to three years, while the average length of time during which the FF and GM are moving near the upper bound is roughly three to four years. This pattern is consistent with the speculative dynamics of Cutler et al (1990, 1991), who …nd that monthly excess returns are positively correlated for up to two years, but negatively correlated (i.e., reverting to UIP) at longer horizons, notably three to four years. Also, between 1991 and 1994, the transition functions for momentum trading correspond with those for the dollar carry trade (Figure 1), with both attaining the upper regime during this period (albeit in a ‘spiked’manner for the CD, SF, and UKP). This …nding is consistent with the view that the two currency-trading strategies can be related, as discussed in Section 3. Finally, for the UKP the most prominent features are the two large spikes occurring in 1985-86 and in 1992-93.8 Notably, the timing of the latter spike corresponds to the ERM currency crisis, which led to the pound’s exit from the EMU, and is consistent with the …ndings in Flood and Rose (2002) regarding ‘UIP in crisis’. Overall, the results in Table 5 and Figure 4 suggest that during periods of high volatility, UIP has a higher probability of holding with the strength of reversion depending on the level of volatility.
6
Conclusion
This paper has studied the forward premium anomaly from the perspective of carry trade and momentum trading strategies. On applying the LSTR models and a similar methodology as Baillie and Kilic (2006), evidence is found for the carry trade to be an important variable in determining whether the exchange rate is moving towards the outer regime where UIP is valid. The carry trade hence determines the speed of adjustment and also where signi…cant risk premia may occur leading to a violation of UIP. The empirical evidence presented in the paper …nds support for this hypothesis, particularly for the US dollar and Japanese yen carry trades against the UK pound and the EMU currencies. However, yen carry trades against the US dollar do not appear to exhibit reversion to UIP. The results also indicate that UIP is more likely to hold when the Swiss franc is not the preferred funding currency; although there is evidence that Swiss franc carry trades are still 8 This is due to the fact that the threshold c had to be set rather high relative to the unconditional variance of spot returns in order to achieve convergence.
15
pro…table. This may be due to the safe haven status of this currency and its relationship with the ERM. Further results accounting for these features indicate that the pattern of nonlinear adjustments to UIP observed in the SF cross-rates are also consistent with an explanation based on the carry trade. The paper also considered momentum trading strategies and found that UIP is more likely to hold in a regime where volatility is unusually high, which is consistent with momentum trading; although clearly other explanations are possible. However, the timing of these regimes is consistent with the pattern of continuation and reversal of excess returns documented in Cutler et al (1990, 1991), thus providing support for their theory of speculative dynamics. Finally, the results presented above suggest that these currency trading strategies may well have a substantial role in the marked deviations from UIP that have been observed in recent times in the world’s FX market. Future research may usefully be directed at incorporating trading strategies with more conventional models of time varying risk premium.
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[8] Burnside, C., M. Eichenbaum, I. Kleshchelski, S. Rebelo (2006), “The returns to currency speculation,” National Bureau of Economic Research (NBER), Working Paper 12489, August. [9] Cavallo, M. (2006), “Interest rates, carry trades, and exchange rate movements,” FRBSF Economic Letter, 2006-31, Federal Reserve Bank of San Francisco, November. [10] Chinn, M.D. (2006), “The (partial) rehabilitation of interest rate parity in the ‡oating rate era: Longer horizons, alternative expectations, and emerging markets,” Journal of International Money and Finance, 25, 7-21. [11] Cutler, D.M., J.M. Poterba, and L.H. Summers (1990), “Speculative dynamics and the role of feedback traders,” AEA Papers and Proceedings, 80, 63-68. [12] Cutler, D.M., J.M. Poterba, and L.H. Summers (1991), “Speculative dynamics,” Review of Economic Studies, 58, 529-546. [13] DeLong, B.J., A. Shleifer, L.H. Summers, and R. Waldmann (1990), “Positive feedback investment strategies and destabilizing rational speculation,” Journal of Finance, 45, 379-395. [14] Dolan, B. (2005), “Currency personalities,” Currency Trader, September, 36-42. [15] Engle, R.F. (1982), “Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom in‡ation,” Econometrica, 50, 987-1007. [16] Engel, C. (1996), “The forward discount anomaly and the risk premium: A survey of recent evidence,” Journal of Empirical Finance, 3, 123-192. [17] Fama, E.F. (1984) “Forward and spot exchange rates,” Journal of Monetary Economics, 14, 319-338. [18] Fischer, A.M. (2002), “Fluctuations in the Swiss Franc: What has changed since the euro’s introduction?” Journal of Public Policy, 22, 143-159. [19] Flood, R.B. and A.K. Rose (2002), “Uncovered interest parity in crisis,” IMF Sta¤ Papers, 49, 252-266. [20] Froot, R.A., and R.H. Thaler (1990), “Anomalies: Foreign exchange,”Journal of Economic Perspectives, 4, 179-192
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[21] Franses, P.H. and D. van Dijk (2000), Non-linear Time Series Models in Empirical Finance, Cambridge University Press, Cambridge. [22] Friedman, M. (1953), “The case for ‡exible exchange rates,” Essays in Positive Economics, University of Chicago Press, Chicago. [23] Galati, G. (1999), “The dollar-mark axis,” BIS Working Papers, No. 74, Bank for International Settlements, August. [24] Galati, G. and M. Melvin (2004), “Why Has FX Trading Surged? Explaining the 2004 Triennial Survey,” BIS Quarterly Review, December, 67-74. [25] Genberg, H. and A. Kadareja (2001), “The Swiss franc and the euro,” in Baldwin, R. and A. Brunetti (eds.), Economic Impact of EU Membership on Entrants: New Methods and Issues, Kluwer Academic Publishers, Boston, 119-148. [26] Granger, C.W.J. and T. Terasvirta (1993), Modeling Nonlinear Economic Relationships. Oxford University Press, Oxford, New York. [27] Halliday, F. (2001), "Cold war," in Krieger, J. (ed.), The Oxford Companion to Politics of the World, Oxford University Press, Inc., New York. [28] Hodrick, R.J. (1987), The Empirical Evidence on the E¢ ciency of Forward and Futures Foreign Exchange Markets, Harwood, London. [29] Hong, H. and J.C. Stein (1999), “A uni…ed theory of underreaction, momentum trading, and overreaction in asset markets,” Journal of Finance, 54, 2143-2184. [30] Kugler, P. and B. Weder (2005), “Why are returns on Swiss franc assets so low?” Applied Economics Quarterly, 51, 231-246. [31] Lyons, R.K. (2001), The Microstructure Approach to Exchange Rates, MIT Press, Cambridge. [32] Maynard, A. and P.C.B. Phillips (2001), “Rethinking an old empirical puzzle: econometric evidence on the forward discount anomaly,” Journal of Applied Econometrics, 16, 671-708. [33] Newey, W.K. and K.D. West (1987), “A simple, positive semi-de…nite, heteroskedasticity and autocorrelation consistent covariance matrix,” Econometrica, 55, 703-708. [34] Phillips, K.L. and K. Snow (1998), “The forward bias: Is it a money tree?” Economics Letters, 61, 373-379. 18
[35] Pojarliev, M. (2005), “Performance of currency trading strategies in developed and emerging markets: Some striking di¤erences,” Financial Markets and Portfolio Management, 19, 297-311. [36] Sarno, L. (2005) “Viewpoint: Towards a solution to the puzzles in exchange rate economics: Where do we stand?” Canadian Journal of Economics, 38, 673-708. [37] Sarno, L., G. Valente, and H. Leon (2006), “Nonlinearity in deviations from uncovered interest parity: An explanation of the forward bias puzzle,” Review of Finance, 10, 443-482. [38] Terasvirta, T. (1998), “Modeling economic relationships with smooth transition regressions,”in Giles, D.E.A. and A. Ullah (eds.), Handbook of Applied Economic Statistics, Marcel Dekker, New York, 507-552. [39] Tong, H. (1990), Non-linear time series: a dynamical system approach, Oxford University Press, Oxford. [40] Villanueva, O.M. (Forthcoming) "Forecasting currency excess returns: Can the forward bias be exploited?" Journal of Financial and Quantitative Analysis; Available at SSRN: http://ssrn.com/abstract=712681 [41] Wu Y. and H. Zhang (1996), “Asymmetry in forward exchange rate bias: a puzzling result,” Economics Letters, 50, 407-411.
19
Table 1 Standard Uncovered Interest Parity (UIP) Regressions st+1 =
+ (ft1
st ) + ut+1
(a) USD Numeraire
t
BF
CD
0.001
0.002
(0.002)
FF
GM
IL
JPY
SF
-0.002
0.001
-0.002
0.001
-0.010
-0.004
0.005
(0.001)
(0.003)
(0.002)
(0.003)
(0.004)
(0.003)
(0.003)
(0.002)
-0.834
-1.132
-1.601
0.023
-0.895
0.425
-2.728
-1.395
-2.526
(0.834)
(0.360)
(0.904)
(0.732)
(0.821)
(0.904)
(0.750)
(0.728)
(1.055)
-2.199
-5.920
-2.877
-1.335
-2.307
-0.636
-4.974
-3.291
-3.341
241
277
241
241
241
241
277
277
277
=1
T
DG
UKP
(b) JPY Numeraire
t
BF
CD
DG
FF
GM
IL
SF
0.004
0.013
0.007
0.002
-0.004
0.003
0.002
0.026
0.010
(0.003)
(0.004)
(0.002)
(0.003)
(0.004)
(0.005)
(0.002)
(0.006)
(0.003)
-0.352
-3.257
-2.832
0.427
-1.623
0.331
-2.128
-4.946
-2.728
(0.680)
(0.868)
(1.086)
(0.547)
(1.093)
(0.688)
(0.971)
(1.158)
(0.750)
-1.988
-4.903
-3.530
-1.048
-2.400
-0.972
-3.221
-5.134
-4.974
241
277
241
241
241
241
277
277
277
=1
T
UKP
USD
(c) SF Numeraire
t
BF
CD
DG
FF
GM
IL
0.000
0.006
(0.001)
0.002
-0.001
-0.002
0.002
-0.002
0.026
0.010
(0.004)
(0.001)
(0.001)
(0.001)
(0.003)
(0.002)
(0.006)
(0.003)
0.506
-1.394
-1.068
0.679
-1.550
0.213
-2.128
-4.946
-2.728
(0.368)
(0.792)
(0.433)
(0.316)
(0.670)
(0.358)
(0.971)
(1.158)
(0.750)
-1.345
-3.021
-4.775
-1.014
-3.808
-2.198
-3.221
-5.134
-4.974
=1
SF
UKP
USD
T 241 277 241 241 241 241 277 277 277 Notes: Robust (Newey-West) standard errors in parentheses below corresponding parameter estimates. t
=1
is the robust t-statistic to for testing H0 :
20
= 1. T is the sample size.
Table 2 LSTR-UIP Regressions for the Dollar Carry Trade Numeraire Currency: USD Transition Variable: [min(JPY,SF)-USD] Interest Di¤erential
st+1 = [
1
+
1 1 (ft
G(zt ; ; c) = [1 + exp( xt =
Y ; iSF g minfiJP t t
BF
st )](1 (zt U it SD
CD
G(zt ; ; c)) + [ c))]
DG
1; z
t
2
= xt =
+
1 2 (ft
st )]G(zt ; ; c) + ut+1
x;
FF
GM
IL
JPY
SF
UKP
Lower regime: G( ) = 0 1
1
0.012
0.001
-0.009
0.013
-0.001
0.017
-0.014
0.008
0.006
(0.005)
(0.001)
(0.007)
(0.005)
(0.009)
(0.009)
(0.006)
(0.007)
(0.002)
-1.304
-1.888
-4.880
-0.408
-2.362
-1.843
-3.633
-0.128
-4.760
(1.355)
(0.464)
(1.478)
(1.032)
(1.754)
(1.101)
(1.188)
(1.208)
(1.098)
Upper regime: G( ) = 1 2
2
-0.017
0.012
-0.013
-0.016
-0.029
-0.035
-0.005
-0.009
-0.011
(0.005)
(0.004)
(0.026)
(0.005)
(0.024)
(0.023)
(0.014)
(0.003)
(0.010)
4.115
-3.332
8.713
3.478
9.892
6.602
4.296
2.224
3.744
(1.791)
(1.133)
(5.939)
(1.550)
(4.420)
(4.422)
(10.438)
(1.730)
(3.158)
Transition parameters: 6.267
5.633
1.814
6.209
1.407
2.438
2.695
77.812
3.429
(3.840)
(9.382)
(0.594)
(2.802)
(0.392)
(1.297)
(2.044)
(30.542)
(1.081)
-1.357
-0.453
-1.424
-1.100
-1.334
(0.119)
(0.210)
(0.118)
(0.589)
(0.016)
2 =1
1.740
-3.822
1.299
1.599
2.012
1.145
0.316
0.708
0.869
W ald
13.113
14.611
2.885
11.782
4.052
4.594
0.111
8.801
1.094
c
t
T 241 277 241 241 241 241 277 277 277 Notes: Robust (Newey-West) standard errors in parentheses below corresponding parameter estimates. t
2 =1
is the robust t-statistic for testing H0 :
for testing H0 :
2
= 0;
2
= 1; it is asymptotically
2 2
= 1. W ald is the robust Wald statistic
distributed with two degrees of fredom. T
is the sample size. For UKP, transition variable is (SF-USD) interest di¤erential
21
Table 3 LSTR-UIP Regressions for the Yen Carry Trade Numeraire Currency: JPY Transition Variable: [min(USD,SF)-JPY] Interest Di¤erential
st+1 = [
1
+
1 1 (ft
st )](1
G(zt ; ; c) = [1 + exp( xt =
SD ; iSF g minfiU t t
BF
(zt JP it Y
CD
G(zt ; ; c)) + [ c))]
1; z
t
2
= xt =
+
1 2 (ft
st )]G(zt ; ; c) + ut+1
x;
DG
FF
GM
IL
SF*
UKP
USD
Lower regime: G( ) = 0 1
1
0.018
0.024
0.011
0.008
0.008
0.009
0.010
0.042
0.014
(0.009)
(0.006)
(0.004)
(0.007)
(0.004)
(0.013)
(0.009)
(0.018)
(0.006)
-2.186
-4.918
-3.660
0.453
-2.035
0.003
1.046
-7.914
-2.829
(1.647)
(1.304)
(1.776)
(0.604)
(1.694)
(1.207)
(2.666)
(3.717)
(1.411)
Upper regime: G( ) = 1 2
2
-0.011
0.003
-0.011
-0.005
-0.014
-0.001
0.002
-0.015
0.003
(0.009)
(0.007)
(0.010)
(0.008)
(0.008)
(0.011)
(0.018)
(0.033)
(0.009)
1.169
-1.073
3.081
0.097
4.342
0.333
-6.403
2.313
-1.950
(1.224)
(1.352)
(4.120)
(0.995)
(3.165)
(1.436)
(6.217)
(5.337)
(1.804)
Transition parameters: 1.839
2.520
2.548
2.616
3.050
2.962
2.170
1.561
1.772
(1.418)
(1.559)
(1.110)
(1.965)
(1.413)
(4.057)
(2.350)
(0.929)
(1.087)
c
0.175 (1.382)
t
2 =1
0.138
-1.534
0.505
-0.908
1.056
-0.464
-1.191
0.246
-1.635
W ald
3.290
6.030
2.771
6.023
3.715
1.414
9.988
1.794
7.679
T 241 277 241 241 241 241 277 277 Notes: As for Table 2. For SF, transition variable is (SF-JPY) interest di¤erential
277
22
Table 4 LSTR-UIP Regressions for the Swiss Franc Carry Trade Numeraire Currency: SF Transition Variable: [min(USD,JPY)-SF] Interest Di¤erential
st+1 = [
1
+
1 1 (ft
st )](1
G(zt ; ; c) = [1 + exp(
(zt
SD ; iJP Y g minfiU t t
iSF t
xt =
BF
CD
G(zt ; ; c)) + [ c))]
1; z
DG*
t
= xt =
FF
+
2
1 2 (ft
st )]G(zt ; ; c) + ut+1
x;
GM
IL
JPY
UKP
USD
Lower regime: G( ) = 0 1
1
-0.002
0.004
0.002
0.002
0.023
-0.001
-0.058
0.007
0.001
(0.002)
(0.005)
(0.002)
(0.004)
(0.013)
(0.007)
(0.058)
(0.004)
(0.004)
1.838
-0.892
-1.700
1.627
-17.801
1.782
-17.231
-1.229
-0.991
(0.683)
(1.662)
(1.128)
(1.362)
(10.981)
(1.278)
(8.306)
(1.065)
(1.309)
Upper regime: G( ) = 1 2
2
0.001
0.024
0.007
-0.008
0.002
-0.004
0.048
0.017
0.020
(0.004)
(0.009)
(0.005)
(0.005)
(0.001)
(0.010)
(0.059)
(0.008)
(0.008)
-0.105
-3.984
-1.969
0.844
-1.182
0.018
-4.040
-2.733
-3.911
(0.654)
(1.313)
(0.804)
(0.527)
(0.714)
(0.675)
(3.041)
(1.162)
(1.274)
Transition parameters 16.096
14.800
17.918
1.163
10.929
1.092
0.668
9.017
8.654
(8.145)
(11.159)
(33.830)
(0.805)
(7.858)
(0.660)
(0.479)
(4.494)
(7.061)
c
t
1.341
-1.338
(0.482)
(0.126)
2 =1
-1.689
-3.796
-3.692
-0.297
-3.056
-1.454
-1.657
-3.212
-3.856
W ald
19.672
22.104
30.857
3.912
11.053
6.662
2.747
18.128
20.867
T 241 277 241 241 241 241 277 277 Notes: As for Table 2. For DG, transition variable is (USD-SF) interest di¤erential
277
23
Table 5 Nested LSTR-UIP Regressions for the Swiss Franc Carry Trade Numeraire Currency: SF Transition Variable: [min(USD,JPY)-SF] Interest Di¤erential
st+1 = [
1
+
+ 1 1 (ft
+[
st )+ + 2
+ 1 2 (ft
+
G(zt ; ; c) = [1 + exp(
(zt
SD ; iJP Y g minfiU t t
iSF t
xt =
BF
1
CD
(ft1 st
c))]
st ) ](1
)+
+
1; z
t
G(zt ; ; c))
1 2 (ft
= xt =
st ) ]G(zt ; ; c) + ut+1
x;
DG
FF
GM
IL
JPY
UKP
USD
Lower regime: G( ) = 0 1 + 1
-0.002
-0.002
-0.002
0.002
-0.003
-0.001
.
0.001
.
(0.002)
(0.005)
(0.002)
(0.004)
(0.003)
(0.007)
.
(0.006)
.
1.838
0.726
1.888
1.627
2.712
1.782
.
1.598
.
(0.683)
(1.690)
(1.542)
(1.362)
(1.874)
(1.278)
.
(2.021)
.
-14.116
-21.417
-15.265
.
4.079
.
(3.509)
(5.781)
(5.110)
.
(3.205)
.
1
Upper regime: G( ) = 1 2 + 2
0.001
0.024
0.002
-0.008
0.002
0.048
.
0.014
.
(0.004)
(0.010)
(0.002)
(0.005)
(0.002)
(0.059)
.
(0.005)
.
-0.105
-3.993
-1.050
0.844
-1.670
-4.040
.
-2.392
.
(0.654)
(1.338)
(0.516)
(0.527)
(0.817)
(3.041)
.
(0.857)
.
-14.398
-3.123
-99.850
.
-7.842
.
(14.406)
(49.817)
(16.228)
.
(2.030)
.
2
Transition parameters: 16.096
14.079
13.540
1.163
13.292
1.092
.
314.260
.
(8.145)
(14.413)
(9.728)
(0.805)
(5.149)
(0.660)
.
(248.877)
.
.
-0.840
.
.
(0.007)
.
c
t
+ 1 =1
1.227
-0.162
0.576
0.460
0.914
0.612
.
0.296
.
W ald
1.559
0.683
0.618
0.597
1.145
0.502
.
0.308
.
241
277
241
241
241
241
.
277
.
T Notes: t
+ 1 =1
is the robust t-statistic for testing H0 :
for testing H0 :
1
= 0;
+ 1
= 1; it is asymptotically
Rest same as for Table 2.
24
+ 1 = 1. W ald is the robust Wald statistic 2 distributed with two degrees of fredom.
Table 6 LSTR-UIP Regressions for Momentum Trading (Volatility) Numeraire Currency: USD Transition Variable: Conditional variance of spot returns
st+1 = [
1
+
1 1 (ft
st )](1
G(zt ; ; c) = [1 + exp( xt =
(zt
G(zt ; ; c)) + [ 1; z
c))]
t
2
= xt =
+
1 2 (ft
st )]G(zt ; ; c) + ut+1
x;
^ 2s;t BF
CD
DG
FF
GM
IL
JPY
SF
UKP
Lower regime: G( ) = 0 1
1
0.001
0.002
-0.005
-0.001
-0.004
-0.003
-0.007
-0.004
0.006
(0.005)
(0.001)
(0.005)
(0.003)
(0.005)
(0.003)
(0.003)
(0.003)
(0.002)
-2.956
-1.280
-4.397
-4.241
-3.145
-0.517
-2.539
-1.795
-2.743
(1.894)
(0.383)
(1.437)
(1.371)
(1.383)
(0.701)
(0.755)
(0.785)
(1.010)
Upper regime: G( ) = 1 2
2
-0.001
-0.003
-0.004
-0.003
-0.004
-0.006
-0.046
-0.013
-0.107
(0.005)
(0.003)
(0.004)
(0.004)
(0.004)
(0.020)
(0.006)
(0.014)
(0.054)
0.516
1.485
1.128
1.279
1.508
4.196
-5.847
5.361
31.609
(1.227)
(2.103)
(1.996)
(0.693)
(1.339)
(2.943)
(3.272)
(3.290)
(15.169)
Transition parameters
c
2.924
4.813
3.480
15.660
3.510
7.827
9.409
1.345
4.360
(2.099)
(11.433)
(4.836)
(9.949)
(3.422)
(1.873)
(8.238)
(0.332)
(2.656)
11.481
6.543
14.341
8.545
14.840
4.968
6.474
3.670
4.150
-
LM
-
-
-
-
-
-
-
-
0.142
0.110
0.187
0.260
1.676
0.314
1.636
0.502
0.955
2 =1
-0.394
0.231
0.064
0.403
0.379
1.086
-2.093
1.326
2.018
W ald
0.743
1.417
1.176
0.415
1.232
3.854
57.575
4.388
4.073
t
T
241 277 241 241 241 241 277 277 277 Notes: LM is the Lagrange Multiplier test of the null hypothesis that the parameter restriction on c is true; it is asymptotically variance of spot returns,
^ 2s;t ,
2
with one degree of freedom.
Estimates of the conditional
are obtained from …tting a GARCH(1,1) model on each respective
spot exchange rate return series, except for JPY (ARCH(1) model) and SF (lagged squared returns). Rest same as for Table 2.
25
Figure 1 Estimated Transition Functions for the Dollar Carry Trade
Canadian Dollar 1
French Franc 1
Weight onRegime1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/96
01/98
01/00
01/02
0
Weight onRegime1
01/80
01/82
01/84
01/86
German Mark 0.7
01/88
01/90
01/92
01/94
01/96
01/98
Japanese Yen 0.9
Weight onRegime1
Weight onRegime1
0.8
0.6
0.7 0.5
0.6
0.4
0.5
0.3
0.4 0.3
0.2
0.2 0.1 0
0.1 01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/96
0
01/98
01/80
01/82
01/84
01/86
Swiss Franc 1
01/88
01/90
01/92
01/94
01/96
01/98
01/00
01/02
01/96
01/98
01/00
01/02
UK Pound 1
Weight onRegime1
Weight onRegime1
0.9 0.8
0.8
0.7 0.6
0.6
0.5 0.4
0.4
0.3 0.2
0.2
0.1 0
01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/96
01/98
01/00
01/02
0
01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
Notes: Regime 1 refers to upper regime as in (6). Plot shows value of G(zt ; ; c) against time.
26
Figure 2 Estimated Transition Functions for the Yen Carry Trade
German Mark 1
UK Pound 1
Weight onRegime1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/96
0
01/98
Notes: As for Figure 1.
27
Weight onRegime1
01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/96
01/98
01/00
01/02
Figure 3
Estimated Transition Functions for the Swiss Franc Carry Trade
(Nested Model)
French Franc 1
German Mark 1
Weight onRegime1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/96
0
01/98
Weight onRegime1
01/80
01/82
01/84
01/96
01/98
01/86
01/88
UK Pound 1
Weight onRegime1
0.8
0.6
0.4
0.2
0
01/80
01/82
01/84
01/86
01/88
01/90
Notes: As for Figure 1.
28
01/92
01/94
01/00
01/02
01/90
01/92
01/94
01/96
01/98
Figure 4 Estimated Transition Functions for Momentum Trading (Volatility)
Canadian Dollar 1
French Franc 1
Weight onRegime1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/96
01/98
01/00
01/02
0
Weight onRegime1
01/80
01/82
01/84
01/86
German Mark 1
1
Weight onRegime1
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/96
0
01/98
01/80
1
Weight onRegime1
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1 01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/94
01/96
01/98
01/82
01/84
01/86
01/88
01/90
01/92
01/94
01/96
01/98
01/00
01/02
01/96
01/98
01/00
01/02
UK Pound
0.9
0
01/92
Weight onRegime1
Swiss Franc 1
01/90
Japanese Yen
0.9
0
01/88
01/96
01/98
01/00
01/02
Notes: As for Figure 1.
29
0
Weight onRegime1
01/80
01/82
01/84
01/86
01/88
01/90
01/92
01/94
