regime switches in exchange rate volatility and uncovered interest parity

Journal of International Money and Finance 30 (2011) 1436–1450

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Journal of International Moneyand Finance

journal homepage: www.elsevier .com/locate/ j imf

Regime switches in exchange rate volatility and uncoveredinterest parity

Hibiki Ichiue a,*, Kentaro Koyama b

aMonetary Affairs Department, Bank of Japan, 2-1-1 Nihonbashi-Hongokucho, Chuo-ku, Tokyo 103-8660, JapanbBooz & Company, Roppongi Hills Mori Tower 27F, 6-10-1-27 Roppongi, Minato-ku, Tokyo 106-6127, Japan

JEL classification:G15

Keywords:Uncovered interest rate parityForward discount puzzleCarry tradeMarkov-switching modelBayesian Gibbs sampling

* Corresponding author. Tel.: þ81 3 3277 2688;E-mail addresses: [email protected] (H. Ich

0261-5606/$ – see front matter � 2011 Elsevier Ltdoi:10.1016/j.jimonfin.2011.07.003

a b s t r a c t

We use a regime-switching model to examine how exchange ratevolatility is related to the failure of uncovered interest parity. Mainfindings are as follows. First, exchange rate returns are stronglyinfluenced by regime switches in the relationship between thereturns and interest rate differentials. Second, low-yieldingcurrencies appreciate less frequently, but once it occurs, theirmovements are faster than when they depreciate. Third, depreci-ation of low-yielding currencies and low volatility are mutuallydependent on each other. Finally, these three findings are moreevident for shorter horizons. The second and third results areconsistent with a market participants’ view: short-term carrytrades in a low-volatility environment and their rapid unwindingsubstantially influence exchange rates. We consider the effects offunding liquidity to explain these results.

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1. Introduction

Low-interest-rate currencies tend to depreciate relative to high-interest-rate currencies. Thisobservation is inconsistent with one of the most popular theories, uncovered interest parity (UIP), buthas been confirmed for many currencies and periods in the extensive literature on the subject. Eventhough more than 25 years have passed since Fama (1984) called this inconsistency the “forward

fax: þ81 3 5255 6758.iue), [email protected] (K. Koyama).

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H. Ichiue, K. Koyama / Journal of International Money and Finance 30 (2011) 1436–1450 1437

discount puzzle”, the failure of UIP is still one of themost prominent puzzles in economics. In fact, thereis no consensus on how to explain the puzzle yet, and researchers still continue to tackle the problem.1

In contrast, many market participants including monetary authorities have reached a consensusthat depreciation of low-interest-rate currencies has been influenced by carry trade activities in a low-volatility environment.2 This view is well described in the speech of de Rato (2007), then ManagingDirector of the IMF. He said that the carry trades at that time reflected the environment of low volatilityand wide interest rate differentials, and exerted downward pressure on one of the lowest-interest-ratecurrencies, the Japanese yen. This pressure, in fact, resulted in the gradual depreciation of the yen untilthe middle of 2007. He also warned that unwinding of carry trades could lead to rapid reversalmovements of exchange rates, mentioning the episode that a disruptive reduction in carry tradepositions forced the yen into sharp appreciation in October 1998.3 His warning, in fact, materialized.After the middle of 2007, the yen sharply appreciated several times while volatility increased and carrytrades were unwound. In particular, after the bankruptcy of Lehman Brothers in September 2008, theyen appreciated by more than 15 percent against the U.S. dollar in three months. As illustrated in theepisode of the yen, low-interest-rate currencies have experienced gradual depreciation and sharpappreciation. This behavior of exchange rates is described among currency traders that “exchange ratesgo up by the stairs and down by the elevator”.

Brunnermeier and Pedersen (2009) construct a theoretical model in which volatility and financialpositions influence each other, although this model does not focus on the foreign exchange markets. Inthis model, speculators borrow from financiers, who accept speculators’ positions as collateral butrequire margins to control the risk of losses on the collateral. The margins must be financed with thespeculators’ own capital. So, when financiers set higher margins, speculators are more likely to hitfunding constrains and be forced to reduce their positions. Such forced reductions in speculatorpositions increase volatility and thus the risk of losses on the collateral, which forces financiers to seteven higher margins. Brunnermeier and Pedersen (2009) call the mutually dependent relationshipbetween volatility and speculators’ positions through margins “the margin spiral”. In their model, themarket switches between two equilibriums: one is the low liquidity equilibrium with high volatilityand reduced positions, while the other is the high liquidity equilibrium with low volatility andincreased positions.

Only recently, a few papers including Brunnermeier et al. (2009) empirically study the relationshipbetween exchange rate returns and market volatility. They find that low-interest-rate currencies aremore likely to sharply appreciate when VIX, a stock market volatility measure, is higher. Brunnermeieret al. (2009) also argue that sharp appreciation of low-interest-rate currencies may be the underlyingcause of the failure of UIP. That is, speculators may require risk premiums for taking short positions inlow-interest-rate currencies, with which speculators would face losses when the low-interest-ratecurrencies sharply appreciated. If so, the premiums move depending on the interest rate differen-tials. The time-varying premiums may be a source of the failure of UIP, in which the premiums areassumed to be time-invariant.

We study what role exchange rate volatility, rather than stock market volatility, plays in the failureof UIP. To this end, we apply a regime-switchingmodel to exchange rate data. The idea of using regime-switching models for examining exchange rates is not new. After Hamilton (1989) proposed theregime-switching model to examine the persistency of recessions and booms, many papers, includingEngel and Hamilton (1990), Bekaert and Hodrick (1993), Engel (1994), Bollen et al. (2000), and

1 See Chinn and Meredith (2004), Lusting and Verdelhan (2007), Alvarez et al. (2009), Burnside et al. (2011) and Plantin andShin (2011) for instance.

2 A simple definition of carry trade is “borrowing in a low-interest-rate currency to invest in a higher one to earn the interestdifferential”.

3 de Rato said, in his speech: “As a result, despite a large current account surplus, there has been downward pressure on theyen in the short run. Indeed, in real effective terms, the yen is now at a 20-year low”. “The carry trade is not a consequence ofglobal imbalances. Rather, it reflects the globalization of financial markets and the current environment of low volatility andwide interest rate differentials”. “Moreover, both financial markets and countries are exposed to risks if there is a suddenreversal of financial flows. For example, a disruptive unwinding of carry trade positions occurred in October 1998, when the U.S.dollar fell by 15 percent against the Japanese yen in 4 days”.

H. Ichiue, K. Koyama / Journal of International Money and Finance 30 (2011) 1436–14501438

Dewachter (2001), applied this model to exchange rate data.We extend their regime-switchingmodelsto investigate the relationship among exchange rate returns, volatility, and interest rate differentials.

To estimate the model, we employ a Bayesian Gibbs sampling. Our method is an extension of thoseused in previous studies, such as Albert and Chib (1993), Kim et al. (1998), and Kim and Nelson (1999).There is a new aspect in our method, however. With this estimation method, we can examine high-frequency state transitions and, at the same time, avoid possible estimation biases arising from theuse of high-frequency data.

The empirical results basically support the market participants’ view: low volatility influencesdepreciation of low-interest-rate currencies. In fact, the reverse is also true. That is, depreciation oflow-interest-rate currencies contributes to maintaining a low-volatility environment, which impliesa tendency that volatility does not increase until carry trades start to be unwound. More specifically,our result suggests that when low-interest-rate currencies start to appreciate, exchange rate returnvolatility increases by more than expected by the fast appreciation. This finding may be attributable tofunding liquidity of speculators. That is, when speculators are forced to unwind their positions, theylose their ability to take advantage of investment opportunities and to provide market liquidity. Theabsence of liquidity-providing speculators may lead to high volatility. All these results suggest thata low-volatility environment and depreciation of low-interest-rate currencies are mutually dependenton each other. In addition, the results are more evident for shorter horizons. This implies that short-term carry trade activities play important roles in the mutual dependence.

The rest of this paper is organized as follows. Section 2 uses OLS regressions for finding basicempirical facts on the relationship among exchange rate returns, volatility, and interest rate differ-entials. Section 3 describes our regime-switching model. Section 4 discusses the estimated results.Section 5 concludes this paper.

2. OLS analysis

This section uses OLS regressions for finding basic empirical facts on the relationship amongexchange rate returns, volatility, and interest rate differentials. We first replicate a traditional regres-sion to test UIP with our updated data. UIP is defined as an equation:

Etðytþn � ytÞ ¼ rt;n � r�t;n; (1)

where yt denotes the log exchange rate of the home currency against the foreign currency at time t, andrt,n and r�t;n denote the interests earned by investing in the home and foreign countries, respectively,from time t to time t þ n. The right-hand side of (1) is interpreted as the return from the interest ratedifferential when investors borrow in the foreign currency to invest in the home currency for nmonths.This equation means an arbitrage relationship in which the expected exchange rate return cancels outthe return from the interest rate differential. Therefore, according to this theory, if the foreign interestrate is lower than the home interest rate, the foreign currency should appreciate on average.

A typical test of UIP is conducted using the so-called Fama regression:

ytþn � yt ¼ aþ b�rt;n � r�t;n

�þ εtþn; (2)

with a null hypothesis of a ¼ 0 and b ¼ 1. We use the U.S. dollar (USD) as the home currency, and theJapanese yen (JPY), British pound (GBP), Swiss franc (CHF), and Deutsche mark (DEM) as the foreigncurrency. The data of exchange rates and interest rates (three- and six-month LIBORs) are obtainedfrom the IMF International Financial Statistics. We use end-of-month data over the period from January1980 to April 2000 for the DEM, since it was replaced by the Euro, and to December 2009 for the othercurrencies. We use non-overlapping quarterly and semiannual data for our OLS analyses with three-and six-month interest rates, respectively, to avoid possible estimation biases in standard errors arisingfrom overlapping data.4

4 Using monthly overlapping data with heteroskedasticity and autocorrelation consistent standard errors, however, does notchange the implications.


Table 1 reports the estimation result of the slope coefficient b of the regression. There are twonoteworthy points. First, the slope coefficient is negative for all eight combinations of four currenciesand two horizons. This result confirms that low-interest-rate currencies tend to depreciate, and is inopposition to what is predicted by UIP. In particular, for both three- and six-month horizons, the slopecoefficient of the JPY is the largest in absolute values, and only this is significantly different from zero atthe five-percent-significance level. The JPY is thus the most puzzling currency. Second, the absolutevalue of slope coefficient is larger for shorter horizons, although this is just marginally not the case forthe DEM. This result is broadly consistent with the findings of Alexius (2001) and Chinn and Meredith(2004), who show that the failure of UIP is less evident for longer horizons.

As reviewed in the preceding section, many market participants consider that a low-volatilityenvironment is an important factor for carry trade activities, which cause the puzzling relationship,depreciation of low-interest-rate currencies. To confirm this view, we add another term to the tradi-tional regression (2):

ytþn � yt ¼ aþ ðb0 þ bvvtÞ�rt;n � r�t;n

�þ εtþn; (3)

where vt is the annualized historical volatility calculated using daily exchange rate returns for 20business days up to end of month. b0 þ bvvt in (3), as well as b in (2), reflects how an exchange ratereturn is related to the corresponding interest rate differential. For instance, a positive bv means thata lower volatility leads to a lower b0 þ bvvt, which can be interpreted as a larger deviation fromwhat ispredicted by UIP.

Table 2 shows that bv is positive in seven cases out of eight, and is significantly different from zero atthe one-percent-significance level in five cases. Thus, the puzzling relationship is stronger when thehistorical volatility is lower. In particular, bv of the JPY is the largest for the three-month horizon, and isjust slightly smaller than the largest, that of the GBP, for the six-month horizon. This result does notseem to be just a coincidence with the result seen in Table 1: the JPY really is the most puzzlingcurrency. On the contrary, these results suggest that exchange rate volatility plays some role in thepuzzling relationship.

We confirmed that exchange rate volatility helps predict exchange rate returns. That is, whenvolatility is lower, low-interest-rate currencies are less likely to appreciate, possibly because financiersset lower margins. In fact, the reverse causality may also be true. For instance, fast appreciation of low-interest-rate currencies results in large losses on speculators’ carry trade positions and forces specu-lators to reduce their positions. The speculators then may lose their ability to take advantage ofinvestment opportunities and to provide market liquidity. The absence of liquidity-providing specu-lators may increase exchange rate volatility. The OLS analysis conducted here cannot help us under-stand the interrelationship between exchange rate returns and volatility. We will consider this issueusing a regime-switching model, described in the next section.

3. The regime-switching model

The OLS analysis in the previous section showed that the forward discount puzzle is more evidentfor shorter horizons and under a lower-volatility regime. To investigate these findings, we employ

Table 1The slope coefficient of the Fama regression.

Horizon (months) Currency

JPY GBP CHF DEM

3 �2.66*** (0.86) �1.59* (0.92) �1.28* (0.67) �0.60 (0.80)6 �2.27** (0.90) �1.56* (0.89) �1.00 (0.73) �0.61 (0.88)

Notes: This table reports the regression estimate of the slope coefficient in Equation (2) for four currencies (JPY, GBP, CHF, DEM)and two horizons (3- and 6-months). The standard errors are reported in parentheses. The values with ***, **, and * are differentfrom zero at the one, five, and ten-percent-significance levels, respectively. The sample is January 1980 to April 2000 for theDEM, and to December 2009 for the other currencies.

Table 2The effects of exchange rate volatility on the failure of UIP.


JPY GBP CHF DEM

3 0.54*** (0.13) 0.53*** (0.16) 0.42*** (0.14) 0.22 (0.21)6 0.52*** (0.18) 0.53*** (0.17) 0.23 (0.18) �0.04 (0.28)

Notes: This table reports the regression estimate of bv in Equation (3), which can be interpreted as the effects of historicalvolatility on the relationship between exchange rate returns and interest rate differentials, for four currencies (JPY, GBP, CHF,DEM) and two horizons (3- and 6-months). The standard errors are reported in parentheses. The values with ***, **, and * aredifferent from zero at the one, five, and ten-percent-significance levels, respectively. The sample is January 1980 to April 2000 forthe DEM, and to December 2009 for the others.


a regime-switching model. This section describes the model, and the empirical results are shown in thenext section. Subsection 3.1 illustrates the idea of our model in comparison with models in the liter-ature. Subsection 3.2 describes our model. Subsection 3.3 discusses our estimation strategy.

3.1. Comparison with models in the literature

After Hamilton (1989) proposed the regime-switching model to examine the persistency ofrecessions and booms, many papers have applied this model to exchange rate data. In Engel andHamilton’s (1990) two-regime model, an exchange rate return is specified as

ytþn � yt ¼ ai þ sihtþn; (4)

where i˛f1;2g denotes the regime, ai and si denote the trend of exchange rate and the volatility ofexchange rate return under regime i, respectively, and htþn w N(0,1). Engel and Hamilton (1990)estimate this model, and find that exchange rates have long swings: exchange rates move in onedirection for long periods of time. Engel (1994) applies the same model to a large variety of currencies.

Bekaert and Hodrick (1993) also employ a two-regime model. They, however, add an interest ratedifferential to the Engel and Hamilton’s (1990) model5:

ytþn � yt ¼ ai þ bi

�rt;n � r�t;n

�þ sihtþn: (5)

Note that this two-regime model assumes simultaneous switches of the intercept ai, the slopecoefficient bi, and the volatility parameter si. The assumption of simultaneous switches in this model,as well as in the models of Engel and Hamilton (1990) and Engel (1994), may be too restrictive; wewilldiscuss this issue later.

Bollen et al. (2000) and Dewachter (2001) employ a four-regime model:

ytþn � yt ¼ ai þ sjhtþn; (6)

where ði; jÞ˛fð1;1Þ; ð2;1Þ; ð1;2Þ; ð2;2Þg. Equation (6) seems to be similar to (4), which is employed byEngel and Hamilton (1990) and Engel (1994), but these equations are different in one crucial aspect.Bollen et al. (2000) and Dewachter (2001) assume that i and j are independent from each other, andthus the trend ai and the volatility sj switch independently. On the other hand, Engel and Hamilton(1990) and Engel (1994) assume that the trend and the volatility depend perfectly on each other, i.e.switch simultaneously.

To show the difference between these regime-switching models and ours, let us use the followingnesting model:

5 Bekaert and Hodrick (1993), in fact, employ a little larger specification with a lag of exchange rate return. In addition, theyuse forward exchange rates rather than interest rate differentials. A forward exchange rate and the corresponding interest ratedifferential, however, have a one-on-one relationship according to covered interest parity, which has been confirmed toapproximately hold in the empirical literature. Thus we use interest rate differentials just for convenience in comparison.


ytþn � yt ¼ ai þ bi rt;n � r�t;n þ sjhtþn: (7)
� �
The model employed by Engel and Hamilton (1990) and Engel (1994) is a special case, where bi ¼ 0and i ¼ j. Bekaert and Hodrick’s (1993) model can be interpreted as the case where i ¼ j. Bollen et al.(2000) and Dewachter (2001) assume bi ¼ 0, and i and j are independent.

We employ a four-regime model, in which the two state variables i and j are neither necessarilyperfectly dependent nor independent. In that sense, our model is less restricted than the models in theliterature. We only assume that the intercept ai does not switch, i.e. a h a1 ¼ a2, because of thefollowing reason. According to themarket participants’ view presented in Section 1, regime switches inexchange rate returns should be interpreted as switches in the relationship between the returns andinterest rate differentials, or switches in speculators’ activities between carry trades and theirunwinding, rather than just switches in trends. Thus we focus on the switches in the slope coefficientbi. In fact, our assumption is supported by a statistical test conducted in the next section.

3.2. The model

This subsection describes our model rigorously. First, we define Sbt as the slope regime. The slopecoefficient at time t is bi when Sbt ¼ i, i ¼ 1,2. Similarly, we define Sst as the volatility regime, with thevolatility parameter at time t being sjwhen Sst¼ j, j¼ 1,2. Without losing generality, we assume b1< b2and 0 < s1 < s2. Next, we define a regime indicator variable that spans the regime space for both theslope and volatility regimes as

St ¼

8>><>>:

1 if Sbt ¼ 1 and Sst ¼ 12 if Sbt ¼ 2 and Sst ¼ 13 if Sbt ¼ 1 and Sst ¼ 24 if Sbt ¼ 2 and Sst ¼ 2

(8)

where St evolves according to a first-order Markov process with a transition probability matrix P. The(k,l) element of the transitional matrix Pkl is the probability of transition from the regime l to the regimek in a month, since we use monthly data.

In what follows, we call the regime with Sbt ¼ 1, i.e. the lower slope coefficient b1, the “Negative”regime, while we call the regime with Sbt ¼ 2 the “Positive” regime. These terms are motivated by theempirical results shown later: the estimates of b1 and b2 are negative and positive, respectively, for allcurrencies and horizons examined. Similarly, the regimes with Sst ¼ 1 and Sst ¼ 2, i.e. the low and highvolatility parameters, are called the “Low” and “High” regimes, respectively. In addition, we call, forinstance, the regime with St ¼ 1, or Sbt ¼ 1 and Sst ¼ 1, the “Negative/Low” regime.

Our model can be summarized as follows:

ytþn � yt ¼ aþ bt

�rt;n � r�t;n

�þ sthtþn; (9)

bt ¼ b1ðS1t þ S3tÞ þ b2ðS2t þ S4tÞ; (10)

st ¼ s1ðS1t þ S2tÞ þ s2ðS3t þ S4tÞ; (11)

Skt ¼ 1 if St ¼ k; and Skt ¼ 0 otherwise; k ¼ 1;2; 3;4; (12)

Pr½Stþ1 ¼ kjSt ¼ l� ¼ pkl; k; l ¼ 1; 2;3;4; (13)

X4

k¼1

pkl ¼ 1; (14)


b1 < b2; (15)

0 < s1 < s2: (16)

Here, htþn w N(0,1) is independent from ht, ht�1,., and is identically distributed. The model has 21parameters including a, b1, b2, s1, s2, and a 4 � 4 transition matrix P. We should estimate theseparameters and the unobservable state variable St. We discuss the estimation strategy in the nextsubsection.

3.3. Estimation strategy

To estimate the above model, we employ a Bayesian Gibbs sampling approach, which is used by,among others, Albert and Chib (1993), Kim et al. (1998), and Kim and Nelson (1999). Starting fromarbitrary initial values of the parameters, the Gibbs sampling proceeds by taking:

Step 1: A drawing from the conditional distribution of the state St, t ¼ 1,2,3,., given a, b1, b2, s1, s2, thetransition matrix P, and the monthly data yt, rt,n, and r�t;n, t ¼ 1,2,3,..

Step 2: A drawing from the conditional distribution of P, given the monthly state St, t ¼ 1,2,3,..Step 3: A drawing from the conditional distribution of a, given b1, b2, s1, s2, and the every-n-month

state and data St, yt, rt,n, and r�t;n, t ¼ 1, 1 þ n, 1þ2n,..Step 4: A drawing from the conditional distribution of b1, given a, b2, s1, s2, and the every-n-month

state and data.Step 5: A drawing from the conditional distribution of b2, given a, b1, s1, s2, and the every-n-month

state and data.Step 6: A drawing from the conditional distribution of s1, given a, b1, b2, s2, and the every-n-month

state and data.Step 7: A drawing from the conditional distribution of s2, given a, b1, b2, s1, and the every-n-month

state and data.

By iterating these steps successively, we simulate a drawing from the joint distribution of the statevariable and the parameters of the model, given the data. It is then straightforward to summarize themarginal distributions of any of these, given the data.

Although this Gibbs sampling is just a simple extension of the methods employed in the literature,there is an important new aspect. In steps 1 and 2, we generate the state variable and the transitionmatrix from their conditional distributions, given the monthly data or state. Nevertheless, in steps 3through 7, we generate a, b1, b2, s1, and s2, given the every-n-month, i.e. quarterly or semiannual dataand state. In the later steps, we use the lower-frequency state and data since htþn w N(0,1) is inde-pendent only from ht, ht�1,.. On the other hand, since we use themonthly state and data in steps 1 and2, we can examine the properties of higher-frequency state transitions.

We employ diffuse or non-informative priors for all the parameters of the model as a w N(0,10),b1 w N(�1,10), b2 w N(1,10), s21wIGð4;300Þ, s22=s21wIGð4;8Þ, and (pk1,.,pk4) w Dirichlet (p0,k1,.,p0,k4)where p0,kk ¼ 4 and p0,kl ¼ 1 if k s l. Data that we use in estimating the regime-switching model aremonthly and exactly the same as those used for the OLS analysis in Section 2.

4. Results

This section discusses the estimation results of our regime-switching model. Subsection 4.1statistically confirms that our model is appropriate in comparison with alternative models.Subsection 4.2 interprets the parameter estimates from the point of view of carry trade activities, anddiscusses possible causes of the failure of UIP and the better performance of UIP for longer horizons.Subsection 4.3 investigates the estimated regime probabilities and transition matrix to find that lowvolatility and depreciation of low-interest-rate currencies mutually depend on each other. Thissubsection also considers the effects of funding liquidity to explain these findings.


4.1. Model comparison

This subsection statistically confirms that our model is appropriate in comparison with threealternatives. The first alternative model is called the “unrestricted four-regime model”, which is lessrestricted than our model in the sense that the intercept parameter a is allowed to switch simulta-neously with the slope coefficient b. The second alternative is called the “independent four-regimemodel”, in which the slope and volatility regimes are restricted to be independent. The last one iscalled the “two-regime model”, in which the slope and volatility regimes are restricted to switchsimultaneously.

For the comparison between our model and these alternatives, we use the Bayes factor, the ratio ofthe marginal likelihood of our model to the marginal likelihood of each alternative. With this defini-tion, if the log of the Bayes factor is positive, we can interpret our model as being supported. Tocalculate the Bayes factor, we employ Chib’s (1995) method. Table 3 reports the log of the Bayes factorfor 24 combinations of three alternative models, four currencies, and two horizons. This table showsthat the log of the Bayes factor is positive in 23 cases out of 24, and that our model overwhelms thealternatives. Moreover, in 18 cases out of 24, the log of the Bayes factor is larger than 4.61, which isinterpreted as decisive evidence against the alternative model, according to Jeffreys (1961).

The evidence against the unrestricted four-regime model suggests that allowing the interceptparameter to switch does not improve themodel’s ability to explain exchange rate returns. This impliesthat regime switches in exchange rate returns should be interpreted as switches in the relationshipbetween exchange rate returns and interest rate differentials without switches in the trends ofexchange rates. The evidence against the other two alternative models implies that the slope andvolatility regimes are neither perfectly dependent nor independent.

4.2. Parameter estimates

The preceding subsection confirmed that our model performs better than the alternatives. Here wediscuss the estimates of the parameters except for the transition matrices. Table 4 reports the posteriormeans of these parameters. This table shows that the slope coefficients b1 and b2 are negative andpositive, respectively, and the absolute value of b1 is smaller than that of b2 in all eight cases. Thissuggests that exchange rates move faster when the low-interest-rate currencies appreciate than whenthey depreciate. On the other hand, Table 5 shows that the unconditional probability of the Negativeregime is higher than 50 percent in all eight cases.

All these results imply that low-interest-rate currencies appreciate less frequently, but once itoccurs, the exchange rates move faster thanwhen they depreciate. This is consistent with the currencytraders’ view described as “going up by the stairs and down in the elevator”.

These results also help understand the conventional OLS results on the failure of UIP. Note that thepositive slope coefficient b2 is much larger than one, which is predicted by UIP. For instance, b2 is 5.11 inthe case of JPY for the three-month horizon. The large value of b2 suggests that exchange rates movevery fast when the low-interest-rate currencies appreciate. With the risk of fast appreciation of low-

Table 3The Bayes factor.

Alternative models Horizon (months) Currency

JPY GBP CHF DEM

Unrestricted four-regime model 3 5.07* 2.98 5.10* 4.066 6.49* �2.23 4.62* 1.98

Independent four-regime model 3 3.54 14.03* 15.00* 4.356 9.60* 15.64* 22.34* 20.86*

Two-regime model 3 5.03* 27.06* 11.80* 7.95*6 5.23* 47.38* 24.45* 20.58*

Notes: This table reports the log of the Bayes factor, the ratio of marginal likelihood of our model to that of each alternative. Thevalue with * is interpreted as decisive evidence against the alternative model, according to Jeffreys (1961).

Table 4Parameter estimates of the regime-switching model.

Currency Horizon (months) Parameter

a b1 b2 s1 s2

JPY 3 �5.06* (2.18) �2.81** (0.69) 5.11** (2.04) 14.6** (1.9) 26.9** (4.1)6 �3.96 (2.31) �2.62** (0.79) 3.05* (1.43) 12.5** (1.7) 21.9** (6.1)

GBP 3 2.89 (1.83) �3.75** (0.84) 6.26** (1.73) 13.2** (1.9) 27.5** (4.9)6 3.13 (1.97) �3.42** (0.91) 4.49* (1.75) 8.6** (1.5) 19.3** (4.1)

CHF 3 �3.92 (1.96) �2.87** (0.59) 5.23** (1.33) 7.8** (2.5) 21.2** (2.3)6 �2.98 (2.04) �2.18** (0.69) 3.97** (1.67) 8.0** (1.7) 16.5** (2.6)

DEM 3 �1.92 (2.21) �3.64** (0.93) 4.60** (1.47) 15.2** (2.5) 23.5** (4.1)6 �2.07 (2.21) �2.78** (0.96) 3.81** (1.62) 8.4** (1.9) 16.3** (3.0)

Notes: This table reports the posterior means of parameters, except for the transition matrix, of the estimated regime-switchingmodel described in Section 3. The standard errors are reported in parentheses. The values with ** and * are different from zero atthe one and five-percent-significance levels, respectively.


interest-rate currencies, which is called “currency crash risk”, speculators may require risk premiumsfor taking short positions in low-interest-rate currencies, as discussed by Brunnermeier et al. (2009). Ifso, the premiums move depending on the interest rates. The time-varying premiums may be a sourceof the failure of UIP, in which the premiums are assumed to be time-invariant.

For all four currencies, the positive slope coefficient b2 is smaller for the six-month horizon than forthe three-month horizon. This result implies that the appreciation of low-interest-rate currencies isslower, or the currency crash risk is lower for longer horizons. The slower appreciation may allowspeculators to require less risk premiums for taking short positions in low-interest-rate currencies. Thisis a possible reason for the stylized fact that the failure of UIP is less evident for longer horizons.

Table 4 also shows that the higher volatility s2 is around twice the lower volatility s1. In fact, s2 ands1 are different at the five-percent-significance level in all eight cases. This table also shows that thesevolatilities are lower for longer horizons for all four currencies. This reflects the mean-reverting natureof exchange rates, which may contribute to reducing the risk and thus risk premiums of long-terminvestment. This is another possible reason for the better performance of UIP for longer horizons.

4.3. Regime probabilities and transition matrix

Fig. 1 depicts the estimated probabilities of the Negative and Low regimes for the JPY, the mostpuzzling currency according to the OLS analysis conducted in Section 2. Panel (a) shows that theNegative and Low regimes are highly correlated for the three-month horizon. For instance, theprobabilities of the Negative and Low regimes simultaneously dropped around October 1998, when theJPYappreciated against the USD by 15 percent in four days. These probabilities simultaneously droppedalso after the bankruptcy of Lehman Brothers in September 2008. This suggests that the appreciation ofthe JPY against the USD at that time was caused not only by the narrowed interest rate differential,which reflects the condition that the Federal Reserve quickly lowered its target interest rate while Bankof Japan had only little room to lower the target rate, but also by the regime change in the relationship

Table 5Unconditional probabilities of the negative and low regimes.


JPY GBP CHF DEM

Negative 3 0.68 0.77 0.62 0.606 0.59 0.75 0.58 0.61

Low 3 0.63 0.69 0.21 0.576 0.80 0.65 0.39 0.38

Notes: This table reports the unconditional probabilities of regimes, which are calculated using eigenvectors of estimatedtransition matrix.

The three-month horizon

00.10.20.30.40.50.60.70.80.9

1

80 82 84 86 88 90 92 94 96 98 00 02 04 06 08

Low Negative

The six-month horizon

00.10.20.30.40.50.60.70.80.9

1

80 82 84 86 88 90 92 94 96 98 00 02 04 06 08

Low Negative

(a)

(b)

Fig. 1. Regime Probabilities for the JPY Notes: This figure shows the estimated probabilities of the Negative (thick line) and Low(shadow) regimes for the Japanese yen.


between the interest rate differential and the exchange rate return. Panel (b) of Fig. 1 shows that thecorrelation between the Negative and Low regimes is relatively low for the six-month horizon. That is,the slope coefficient and the volatility parameter have a weaker relationship for longer horizons.

To investigate this relationship, let us look at the transition matrix for the JPY. Panel (a) of Table 6reports the monthly transition matrix for the three-month horizon. The (k,l) element denotes theprobability of transition from the regime l to the regime k in a month. The left and right halves of this

Table 6Transition matrices for the JPY.

Negative/Low Positive/Low Negative/High Positive/High

(a) The three-month horizonNegative/Low 0.86 0.18 0.20 0.10Positive/Low 0.03 0.58 0.07 0.08Negative/High 0.05 0.09 0.63 0.09Positive/High 0.05 0.15 0.09 0.73(b) The six-month horizonNegative/Low 0.91 0.09 0.20 0.07Positive/Low 0.05 0.82 0.12 0.13Negative/High 0.02 0.03 0.57 0.07Positive/High 0.02 0.05 0.11 0.73

Notes: This table reports the estimated monthly transition matrices for the JPY of the regime-switching model described inSection 3. Panels (a) and (b) report the matrices for the three- and six-month horizons, respectively.

JPY GBP CHF DEM

0.30.40.50.60.70.80.91.0

3M 6M

Pr(N/L->N/L)

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3M 6M

Pr(P/L->P/L)

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3M 6M

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3M 6M

JPY GBP CHF DEM

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Pr(N/L->P/L)

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Pr(P/L->N/L)

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JPY GBP CHF DEM

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Pr(N/L->H)

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Pr(P/L->H)

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0.0

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3M 6M

(a)

(b)

(c)

Fig. 2. Transition Probabilities When the Initial Volatility Regime Is Low Notes: This figure reports the estimated transition prob-abilities for the combinations of four currencies (JPY, GBP, CHF, and DEM) and two horizons (3M and 6M) when the initial regime isthe Negative/Low or Positive/Low regime. Panel (a) reports the probabilities of staying in the Negative/Low (Pr(N/L / N/L), whitediamond) and Positive/Low (Pr(P/L / P/L), black square) regimes. Panel (b) reports the probabilities of shifting from the Negative/Low regime to the Positive/Low regime (Pr(N/L / P/L), white diamond), and the probabilities of shifting in the opposite direction(Pr(P/L / N/L), black square). Panel (c) reports the probability of shifting from the Negative/Low regime to the Negative/High orPositive/High regime (Pr(N/L / H), white diamond), and the probability of shifting from the Positive/Low regime to the Negative/High or Positive/High regime (Pr(P/L / H), black square).


matrix correspond to the transition probabilities when the initial volatility regimes are Low and High,respectively. We first focus on the left half.

The probability of staying in the Negative/Low regime is 86 percent as shown in the (1,1) element.This means that the expected duration of this regime is 7.3 (¼ 1/(1�0.86)) months. On the other hand,the probability of staying in the Positive/Low regime is 58 percent, and thus the expected duration isonly 2.3 months. Therefore the Positive/Low regime is much less stable than the Negative/Low regime.That is, the shift from the Positive/Low regime to the other regimes is more likely than that from theNegative/Low regime.

The probability of shifting from the Positive/Low regime to the Negative/Low regime is 18 percent,which is much higher than 3 percent, the probability of shifting in the opposite direction. That is, the


slope regime tends to converge into the Negative regime as long as the volatility regime is Low. Thusthe slope regime tends to be Negative in a low-volatility environment, not only because the Negativeregime is likely to continue but also because the shift from the Positive regime to the Negative regime ismore likely than the shift in the opposite direction. This implies that the speculators continue or startto take carry trade positions when volatility is low.

The other reasonwhy the probability of staying in the Positive/Low regime is lower than that in theNegative/Low regime is that the volatility parameter shifts to High more easily under the Positive/Lowregime. The probability of shifting from the Positive/Low regime to the Positive/High or Negative/Highregime, which is calculated as the sum of (3,2) and (4,2) elements, is 25 percent, and is much higherthan 10 percent, the probability of shifting from the Negative/Low regime to the Positive/High orNegative/High regime. This result implies that low volatility is easier to be maintained when low-interest-rate currencies depreciate than when they appreciate.

JPY GBP CHF DEM

0.30.40.50.60.70.80.91.0

3M 6M

Pr(N/H->N/H)

0.30.40.50.60.70.80.91.0

3M 6M

Pr(P/H->P/H)

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3M 6M

0.30.40.50.60.70.80.9

3M 6M

CHF DEMJPY GBP

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Pr(N/H->P/H)

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(a)

(b)

(c)

Fig. 3. Transition Probabilities When the Initial Volatility Regime Is High Notes: This figure reports the estimated transitionprobabilities for the combinations of four currencies (JPY, GBP, CHF, and DEM) and two horizons (3M and 6M) when the initialregime is the Negative/High or Positive/High regime. Panel (a) reports the probabilities of staying in the Negative/High (Pr(N/H/ N/H), white diamond) and Positive/High (Pr(P/H / P/H), black square) regimes. Panel (b) reports the probabilities of shifting from theNegative/High regime to the Positive/High regime (Pr(N/H / P/H), white diamond), and the probabilities of shifting in the oppositedirection (Pr(P/H / N/H), black square). Panel (c) reports the probability of shifting from the Negative/High regime to the Negative/Low or Positive/Low regime (Pr(N/H / L), white diamond), and the probability of shifting from the Positive/High regime to theNegative/Low or Positive/Low regime (Pr(P/H / L), black square).


The sharp contrast in stability between the Negative and Positive regimes, however, is not seenclearly in the right half of Panel (a) and thewhole Panel (b), i.e. when the volatility is higher or horizonsare longer. Panels (a) and (b) of Table 6, thus, are summarized by saying that depreciation of low-interest-rate currencies and a low-volatility environment are mutually dependent on each other,particularly for shorter horizons.

Figs. 2 and 3 confirm that this implication of Table 6 holds not only for the JPY but also for the othercurrencies. Fig. 2 depicts the estimated transition probabilities when the initial volatility regime is Low.Panel (a) shows that the probability of staying in the Negative/Low regime in the next month is higherthan that in the Positive/Low regime for all four currencies and two horizons. For all currencies, thisdifference in the probabilities is larger for the three-month horizon than for the six-month horizon.Panels (b) and (c) show that the probabilities of shifting from the Positive/Low regime to the otherregimes are higher than those from the Negative/Low regime, especially for the three-month horizon.In contrast, Fig. 3 shows that this clear asymmetry between the Negative and Positive regimes for the 3-month horizon is not observed when the initial volatility regime is High. All these results of Figs. 2 and3 confirm the implication of Table 6: depreciation of low-interest-rate currencies and a low-volatilityenvironment are mutually dependent on each other, particularly for shorter horizons.

The causality from a low-volatility environment to depreciation of low-interest-rate currencies canbe explained with the theoretical model of Brunnermeier and Pedersen (2009), in which low-volatilityleads to low margins and enables speculators to finance their positions.

Let us consider the reverse causality form depreciation of low-interest rate currencies to a low-volatility environment. This implies a tendency that volatility does not increase until carry tradepositions start to be unwound. Brunnermeier et al., (2009) argue that a reduction in speculators’positions increases exchange rate return volatility because unwinding is rapid. This is consistent withour result: exchange rate return volatility increases when the low-interest-rate currency starts toappreciate, because the positive slope coefficient b2 is greater than the absolute value of the negativecoefficient b1. More specifically, our result suggests that exchange rate return volatility tends toincrease not only because unwinding is rapid but also because exchange rate return volatility tends torise additionally. This empirical fact cannot be explained with the existing theories, and it is beyond thescope of this paper to construct a new theoretical model. Instead we here provide a possible expla-nation based on the argument of Brunnermeier and Pedersen (2009). That is, once low-interest-ratecurrencies start to appreciate, speculators who have carry trade positions are suffered from losseson the positions and are forced to reduce the positions rapidly to meet the margin requirements. Thenthe speculators lose their ability to take advantage of investment opportunities and to provide marketliquidity. Brunnermeier and Pedersen (2009) argue that the absence of liquidity-providing speculatorsleads to low asset prices. We argue that the absence of such speculators may also lead to high volatility.

5. Conclusion

Low-interest-rate currencies tend to depreciate relative to high-interest-rate currencies. Thisobservation is in opposition to what is predicted by UIP, and, according to a many market participants’view, this is caused by carry trade activities in a low-volatility environment. We use OLS regressionsand a regime-switching model to examine how exchange rate volatility and depreciation of low-interest-rate currencies are related to each other. The regime-switching model allows the sloperegime, which governs the relationship between exchange rate returns and interest rate differentials,and the volatility regime to be neither necessarily perfectly dependent nor independent. This propertyof the model enables us to investigate the relationship among exchange rate returns, volatility, andinterest rate differentials.

We estimate this model using a Bayesian Gibbs sampling approach, in which the state and thetransition matrix are generated conditional on the monthly data and/or state, while the parameters ofintercept, slope coefficient, and volatility are generated conditional on the quarterly or semiannual dataand state. With this method, we can examine the state transitions in a high frequency, while avoidingpossible estimation biases arising from overlapping data.

The main findings are as follows. First, a statistical test using the Bayes factor supports our model incomparisonwith the alternatives. The evidence suggests that regime switches in exchange rate returns


should be interpreted as switches in the relationship between the returns and interest rate differentialswithout switches in the trends of exchange rates. The evidence also implies that the slope and volatilityregimes are partially dependent; we should not assume that the regimes are perfectly dependent orindependent.

Second, low-interest-rate currencies appreciate less frequently, but once it occurs, their movementsare faster than when they depreciate, as described among currency traders that “exchange rates go upby the stairs and down by the elevator”. This may be because low-interest-rate currencies appreciatewhen speculators are forced to unwind their carry trade positions rapidly to meet their marginrequirements.

Third, low-volatility tends to cause low-interest-rate currencies to depreciate. This may be becauselower volatility results in lower margins and enables speculators to take more carry trade positions. Infact, the reverse causality is also true. That is, depreciation of low-interest-rate currencies contributestomaintaining a low-volatility environment, which implies a tendency that volatility does not increaseuntil carry trades start to be unwound. Our result suggests that exchange rate return volatility increasesnot only because unwinding of carry trades is rapid. Since speculators are forced to unwind or liquidatetheir positions, they lose their ability to take advantage of investment opportunities and to providemarket liquidity. This may increase exchange rate return volatility additionally. These results imply thata low-volatility environment and depreciation of low-interest-rate currencies are mutually dependenton each other, and the stability would be lost once one of these two underling conditions is failed tomeet.

Finally, the second and third findings are more evident for shorter horizons. This implies that theexchange rate behavior of slow depreciation and fast appreciation of low-interest-rate currencies, andthe mutual dependence between depreciation of low-interest-rate currencies and a low-volatilityenvironment are influenced by short-term carry trade activities.

Acknowledgments

We thank Naohiko Baba, Markus K. Brunnermeier, James D. Hamilton, Keiji Kono, Teppei Nagano,Masao Ogaki, Mototsugu Shintani, Toshiaki Watanabe, Tomoyoshi Yabu, and the staff at Bank of Japanfor their helpful comments. The views expressed here are solely ours, and do not necessarily reflectthose of Bank of Japan.

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regime switches in exchange rate volatility and uncovered interest parity

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