exchange rates and interest arbitrage - university of...

Exchange rates and interest arbitrage∗

Anella MunroReserve Bank of New Zealand

[email protected]

February 8, 2013

Rather than asking whether uncovered interest parity (UIP) holds, this paperestimates the contribution of interest arbitrage to exchange rate fluctuations.Forecasts of future relative interest returns are constructed based on AR1models and on forecasts embedded in interest rate swap contracts. Thoseforecasts are used to construct UIP-consistent exchange rates (a forwardsum of relative returns), and to decompose exchange rate fluctuations intoforecast revisions, reversion towards UIP and residual shocks (unrelatedto risk-neutral interest arbitrage). The decomposition associates interestarbitrage with the former two which account for 27 to 40% of exchange ratefluctuations for a set of advanced country USD exchange rates. The bulk ofthat contribution comes from the within-period ‘‘Dornbusch jump” responserather than the 1-period return that has been a focus of the parity literature.Negative covariance of the forecast revision and the shock implies severedownward bias in estimation of that contribution using standard techniques.The paper discusses the potential role of capital flows in generating thatnegative covariance and the continued appreciation of high interest ratecurrencies.

JEL classification: F31, G12

∗This paper has benefitted from comments from Charles Engel, Leo Krippner, ChrisMcDonald, and Konstantinos Theodoris.

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1 Introduction

A vast literature considers the empirical failure of the uncovered interest parity(UIP) hypothesis for developed country currencies (see, for example, Fama(1984); and literature surveys in Engel (1996), Thornton (2009) and Engel(2012)). UIP implies that risk-neutral investors should be indifferent betweenholding domestic and foreign currency assets with similar risk characteristics.Excess profits should have been arbitraged away. Empirically, high interestrate currencies tend to yield excess short term returns: not only is the higherinterest return not fully offset by currency depreciation, but the high interestcurrency often appreciates in the short term.

The rejection of UIP has been explored in many dimensions. The role of riskpremia has been a prominent explanation (Fama (1984), Froot and Frankel(1989)), but is difficult to reconcile with two stylised facts. First, a highinterest currency tends to be a strong currency consistent with UIP (Engel2012) while a high risk premium implies a high interest rate, but a weakcurrency. Second, in a dynamic sense, it is unclear why a high risk premiumcurrency should continue to appreciate. Burnside (2011) argues that it fitswell with tail events.

Another prominent explanation is expectations bias. A key assumption inthe one-period test of UIP is rational expectations: expectations about futurespot rates are not always right, but on average they should be. According toUIP, the expected future spot rate is the forward rate (the spot rate net of theone-period relative interest return). In contrast, Meese and Rogoff (1983)’sfinding that a random walk forecast is hard to beat suggests that a no-changeexchange rate expectation may be rational and the forward premium may bea biased forecast. If the exchange rate is a random walk, then the forwardpremium must be inversely related to wither the expectations bias or theshock (Engel and West (2005), van Wincoop and Bacchetta (2007),Engel(2012)). Chinn and Frankel (1994) find greater support for UIP when usingex-ante market expectations of future spot rates.

Finally, interaction of interest arbitrage with other factors can explain thefailure of the UIP condition. Froot and Thaler (1990) argue that partial ad-justment to shocks can generate the UIP puzzle. van Wincoop and Bacchetta(2007) generate the UIP puzzle in a model in which interest arbitrage is activebut expectations are subject to infrequent adjustment.

Central to the approach in this paper is the (Dornbusch 1976) asset priceview of exchange rates. In the presence of sticky prices, if the home country

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interest rate rises, the home currency should immediately appreciate relativeto a future equilibrium rate, so that it can subsequently depreciate to equalisereturns from the two currencies each period. As such the exchange rate is asum of expected future relative returns.

Engel and West (2005) construct forecasts of future fundamentals based onAR1 models and VARs for a range of exchange rate models for advancedcountry USD exchange rates. They show that the forecasts are near-randomwalks, suggesting the rational expectations assumption in the Fama equationmay be problematic. They find correlation between the forecasts and theexchange rate and some evidence that the exchange rate leads fundamen-tals, somewhat counterbalancing the usual bleak view of the role of interestarbitrage.

This paper extends Engel and West (2005) by (i) constructing forecasts thatemploy long-term interest rate swaps,1 that provide a market-based forecastof short term benchmark interest rate paths, and (ii) using the forecasts todecompose exchange rate fluctuations into factors associated with interestarbitrage and shocks that are not. Interest arbitrage is reflected in theDornbusch jump and the 1-period return (that together make up the forecastrevision), and in the reversion to a UIP-consistent level. The decompositionprovides a useful basis to consider sources of estimation bias.

While the Dornbusch jump (a response to news) should be unpredictableex-ante, it is of interest when examining interest arbitrage empirically for atleast three reasons. First, for a floating currency, the jump reflects interestarbitrage, so it should be included a measure of interest arbitrage.

Second, the change in fundamentals, and so change in the forecast interestpath, can be observed at the time the exchange rate is measured at theend of the period. Bjørnland (2009) and Bjørnland and Halvorsen (2010)show empirical support for a Dornbusch jump for a range of advanced smallopen economy currencies using a VAR identification that allows (as does ourdecomposition) for a contemporaneous interaction between interest rates andthe exchange rate.

Third, there is good reason to believe that the within-period jump is endoge-

1Chinn and Meredith (2004) and Chinn and Quayyum (2012) relate the multi-periodexchange rate changes to multi-period bond differentials and find some support for UIPover those longer horizons. The approach here is conceptually different to those papers intwo ways: (i) the swap rate is used here to reflect a long sequence of short-term returnsrather than the long-period return on a bond; and (ii) that sequence of short-term returnsis related to 1-period exchange rate movements.

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nously related to the shock (Fama 1984), and we know from the standard UIPtest result that the jump and/or the shock is correlated with the one-periodreturn.

The results show some evidence of cointegration between the forecasts andexchange rates for a range of advanced country USD exchange rates. That co-movement appears to be increasing over time. Forecast revisions are positivelycorrelated with changes in exchange rates consistent with interest arbitragebeing a significant part of the data generating process. The contribution ofinterest arbitrage comes mainly from the Dornbusch jump component whichhas a variance two orders of magnitude larger than the one-period relativereturn in the standard parity test. In the decomposition, interest arbitrage(forecast revisions plus the error correction term) accounts for an estimated27 to 40% of the variance of a range of advanced country USD exchangerates. Covariances among the shock and the interest arbitrage componentssuggest severe bias in using standard estimation techniques to examine therole of interest arbitrage in exchange rate fluctuations.

I argue that capital flows help to explain two stylised facts: the negativecovariance between forecast revisions and shocks; and the continued appre-ciation of high interest rate currencies. Capital flows unrelated to interestarbitrage create interest arbitrage opportunities, so influence exchange ratesin a way that covaries negatively with the effect of interest arbitrage if thelatter is incomplete ((Froot and Thaler 1990)). Second, capital should flowto a high return currency to equalise returns to capital over the mediumterm. That flow is consistent with continued appreciation of a high interestcurrency because the equalisation of returns to capital may take decadesunless countries run very large current account deficits or have relatively highsavings rates over an extended period. Capital flows are driven by a rangeof factors unrelated to risk-neutral interest arbitrage (eg. risk premia, carrytrade, portfolio shifts, trade flows, and central bank intervention). Empiricallyfinancial flows are large and volatile and have explanatory power for exchangerates (Evans and Lyons 2002).

In the next section, forecasts of fundamentals are derived analytically. Section3, examines UIP-consistent exchange rates based on those forecasts and coin-tegration with observed developed country US dollar exchange rates. Section4 uses the decomposition of exchange rate changes to examine covarianceamong the factors and sources of estimation bias. Section 5 concludes.

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2 Interest arbitrage and real interest returns

UIP is derived as an optimality condition that equalises the expected returnson domestic and foreign currency assets. Risk-neutral investors (who careonly about the mean of the risk premium) should be indifferent betweenholding domestic and foreign currency assets with similar risk characteristics.A higher real interest return in the home country rt relative to the foreigncountry r∗t should be offset by an expected depreciation (rise) of the homecurrency so returns are equal and there is no excess return to holding thehome or foreign currency: 2

Et(qt+1)− qt = (rt − r∗t ) + λt (1)

where qt is the log of the nominal exchange rate, defined as the home priceof a unit of foreign currency (a fall in qt represents an appreciation of thehome currency), Et(qt+k) is the expected spot exchange rate at t+ k basedon information at time t, and λt is the excess home currency return (Engel2012).3

2.1 The standard one-period test

Assuming rational expectations (on average, the expected future spot rateshould equal the observed future spot rate), the standard test for UIP writtenin real terms is: 4

∆qt+1 = α + β(rt − r∗t ) + εt+1 (2)

H0 : β = 1

2UIP can be expressed in real or nominal terms. Assuming stationarity of the real exchangerate, it is convenient to express UIP in real terms so that the equilibrium exchange rate isconstant. If the nominal interest differential in part reflects an inflation differential, thenthe nominal equilibrium must adjust to satisfy purchasing power parity (PPP) in the longterm. See Appendix A

3λt may reflect a variety of factors including capital flows that are not driven by risk-neutralinterest arbitrage, relative default risk, liquidity risk, measurement error, forecast error etc.

4The same hypothesis can be (and was originally) expressed as a test of the ”unbiasdenss”of the forward exchange rate ft,t+1 as a predictor of the future nominal spot rate, st+1:∆st+1 = α + β(ft,t+1 − st) + εt+1; H0 : α = 0, β = 1. The forward rate ft,t+1 isthe certainty-equivalent expected future spot rate according to covered interest parity((ft,t+1 − st) ∼= it − i∗t) whereby the entire transaction is contracted ex-ante, so there isno uncertainty and arbitrage tends to hold in normal times. See Baba and Packer (2009)on deviations from covered interest parity during the global financial crisis.

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The hypothesis H0 : β = 15 has been widely rejected in both nominal andreal terms. The overwhelming evidence is that, not only is β not equal toone, but it is often of the opposite sign with unity outside the 90% confidenceband (despite large standard errors). The high interest rate currency offersnot only a higher yield, but also tends to appreciate over the higher yieldperiod.

2.2 The exchange rate as an asset price

Equation 1 can be substituted forward to express the exchange rate as arelative asset price that is an infinite forward sum of future relative returns:

qt = −Et∞∑k=0

(rt+k − r∗t+k)− Et∞∑k=0

λt+k + limk→∞

Et(qt+k) (3)

Assuming that the final term in 3, is a constant purchasing power parity(PPP) equilibrium, q̄, and using the notation of (Engel 2012),

qt − q̄ = −Et∞∑k=0

(rt+k − r∗t+k)︸︷︷︸EtRt

−Et∞∑k=0

λt+k︸︷︷︸Λt

(4)

The deviation of the real exchange rate from PPP equilibrium is the expectedsum of real relative returns EtRt consistent with risk-neutral UIP, plus theexpected sum of excess returns Λt that drives a wedge between the exchangerate and its risk-neutral UIP-consistent level. At t+ 1,

qt+1 − q̄ = −Et+1Rt+1 −Λt+1 (5)

Subtracting 4 from 5,

∆qt+1 = −[Et+1Rt+1 − EtRt]− [Λt+1 −Λt]

= (rt − r∗t )− [Et+1Rt+1 − EtRt+1] + (1− ρΛ)Λt − λt+1 (6)

The term [Et+1Rt+1 − EtRt] is the forecast revision. It comprises the one-period relative return (rt − r∗t ) plus the change in the expected path ofsubsequent relative returns (the Dornbusch jump).6

5The hypothesis is sometimes stated as H0 : α = 0, β = 1 but in practice only β = 0matters (McCallum (1994)).

6See Appendix C for a graphical exposition.

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While the revision of the UIP-implied exchange rate at t + 1 should notbe forecastable ex-ante, it is important for understanding the contributionof interest arbitrage in three respects. First, the within-period response tonews-driven forecast revisions is an integral part of interest arbitrage. Second,empirically, the within-period response from the decomposition is much largerthan the subsequent adjustment embodied in the 1-period return. Finally,there is reason to believe that it covaries endogenously with the shock. Forexample, a capital inflow that appreciates the home currency and depressesthe path of home interest rates creates an interest arbitrage opportunity.Arbitrage should respond, leading to an offsetting capital outflow and/or adownward adjustment in the value of the home currency (repricing may occurwithout flows).

The revision to the UIP-consistent path can be estimated at time t+ 1 whenwe measure ∆qt+1 using data available at t + 1 − ε, ε → ∞ (interest ratesare persistent). By the time the markets have set qt+1, arbitrageurs haverevised expectations about the forward path of relative returns based on dataavailable up to t+ 1− ε, and their response to that revised forecast shouldinfluence ∆qt+1.

The final term in equation 6 is the change in expected excess returns. If Λt+1,the deviation of the real exchange rate from the UIP-consistent equilibrium,is stationary, then the final term includes an error correction term and ashock since Λt+1 = ρΛΛt︸︷︷︸

EtΛt+1

+ λt+1︸︷︷︸shock

.

The forecast revision and the ex-ante deviation from parity create incentivesfor interest arbitrage opportunities. If shocks were small relative to interestarbitrage, then changes in the forecast plus the error correction term wouldkeep the exchange rate near parity; the forecasts would be cointegratedwith the observed exchange rate; and forecast revisions would be positivelycorrelated with exchange rates. As shocks become large relative to theresources devoted to interest arbitrage, the exchange rate deviates furtherfrom parity, is less cointegrated with the forecast path, and forecast revisionsbecome less correlated with exchange rate fluctuations.

In the rest of this section, specific expressions for EtRt are derived, based onforecasts of future interest rate paths. These are based on AR1 models (as in(Engel and West 2005)) and on the nominal interest rate forecasts embeddedin the 10 year interest rate swap.

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2.2.1 Expectations based on an AR1 model

Suppose agents have a simple AR1 forecasting model for interest rates,inflation rates and λt:

xt+1 = ρx xt + εt, where 1 > ρx > 0 and εt+k ∼ N(0, σ2) (7)

Et(xt+1) = ρx xt

forxt ∈ [rt, r∗t , πt, π

∗t , λt] (8)

Using the geometric sum approximation 1 + a+ a2 + a3 + ... ' 1/(1− a), theexpected sum of the future path of xt is:

Et

∞∑k=0

xt+k 'xt

1− ρx

Substituting AR1 forecasts, based on variables observed at time t, intoequation 6, the level of the real exchange rate consistent with UIP is:

− EtRt ' −it

1− ρi+

i∗t1− ρi∗

+(ρπ)2πt−1

1− ρπ−

(ρπ∗)2π∗t−1

1− ρπ∗(9)

where the forecast of inflation is based on observed inflation at t− 1.

Small but persistent changes in relative returns may have large effects whensummed over a long horizon. Short-term interest differentials are persistent(AR1 coefficients for monthly rates are typically about 0.98) so the coefficientρ

1−ρ is expected to be large ( ρ1−ρ = 49 for ρ = 0.98).

Similarly, at t+ 1, the level of the exchange rate consistent with UIP is:

− Et+1Rt+1 ' ' − it+1

1− ρi+

i∗t+1

1− ρi∗+ +

(ρπ)2πt−1

1− ρπ−

(ρπ∗)2π∗t−1

1− ρπ∗(10)

Substituting (9) and (10) into 6, the change in the exchange rate consistentwith UIP between t and t+ 1 can be written:

∆qt+1 = −∆it+1

1− ρi+

∆i∗t+1

1− ρi∗+ +

(ρπ)2πt−1

1− ρπ−

(ρπ∗)2π∗t−1

1− ρπ∗+ (1− ρλ)Λt − λt+1

Since interest rates are persistent, and ρ1−ρ is large so the forecast revision is

expected to be large. Moreover, the Dornbusch jump (∆Rt+1 − (rt − r∗t )) ispotentially correlated with (rt − r∗t ) if Rt is near a random walk (Engel andWest 2005).

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2.2.2 Expectations embedded in long-term swap rates

Long-term swap rates provide an alternative, and probably considerably moreaccurate, proxy for the expected sum of future nominal interest rates than anAR1 model. The swap rate is a fixed interest rate equal to the ex-ante expectedaverage 90-day (or 180 day) rate over the 10-year duration. Therefore, the10-year swap rate differential provides a measure of the expected sum ofrelative short-term interest returns over ten years.

An advantage of using swap rates rather than long-term bond rates here isthat they carry mainly interest rate risk and relatively little credit risk. Theinterest rate risk is desirable for our purpose: exposure to interest rate riskmeans that the forecasts implied by the swap curve need to be as accurate aspossible. Also, the forecasts embedded in interest rate swaps are the basis fora vast volume of transactions. The Bank for International Settlement reportsthat the notional amount outstanding in June 2011 was over $400 trillion.7

The fact that a swap has little credit risk is also desirable. It means thatthe swap differential is mainly a forecast of the difference in short termrisk-free interest rates. The credit risk embedded in a swap comes from twomain sources. One is counter-party risk (associated with the swap beingout-of-the-money due to movements in interest rates) which is commonlymitigated by posting collateral.

The second is the credit risk embodied in the underlying 90- or 180-daybenchmark (Libor or equivalent) rate8 that tends to be small because it isrelatively short-term and because central bank liquidity operations contributeto keeping short-term spreads modest.9 In contrast government or privatebonds can carry a large ‘‘specialness” discount or credit premium.

Limiting the forecasting horizon to 10 years (after 10 years, the interest ratesand exchange rate are assumed to be at the long-run equilibrium), the level

7BIS Quarterly Bulletin, June 2011.8Libor is not without problems: it is a quoted (not necessarily transacted) rate based onsubmissions from a panel of banks. The panel is dominated by large AA to A-rated banks.The panel varies from currency to currency so there are credit risk differences between thedomestic and foreign interest rate series (unless quotes from a matched panel are used).There may also be differences in liquidity risk and country-currency risk in the domesticand foreign rates, and quotes have at times been subject to manipulation.

9During the global financial crisis the Libor spread to the OIS curve (expected averagepolicy rate) did rise sharply, but the rise was short lived compared to longer term bondspreads to swap, and Libor quickly convereged toward the policy rate.

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of the exchange rate consistent with UIP:

− EtRt = −119∑k=0

(it+k − i∗t+k) + Et

∞∑k=1

(πt+k − π∗t+k)

≈ − 120 (i10St − i∗10S

t ) + +(ρπ)2πt−1

1− ρπ−

(ρπ∗)2π∗t−1

1− ρπ∗︸︷︷︸EtRt

(11)

where i10S and i∗10S are home and foreign 10-year swap rates (% per month,the expected average monthly return on the short-term rate for the subsequent120 months) and the expected inflation differential is based on an AR1 forecastas before.10

Substituting equation (11), at t and t+ 1 into 6, the change in the exchangerate consistent with interest parity is:

∆qt+1 = −120∆(i10St+1 − i∗10S

t+1 ) +ρ2π∆πt

1− ρπ− ρ2

π∗∆π∗t

1− ρπ∗− (1− ρλ)Λt − λt+1(12)

= (rt − r∗t ) + [∆Rt+1 − (rt − r∗t )] + (1− ρλ)Λt − λt+1 (13)

Again, ∆Rt+1 is potentially large and volatile, being based on large or infinitesums; and ∆Rt+1 − (rt − r∗t ) is potentially correlated with (rt − r∗t ), if Rt+1

is a near-random walk, so may bias the estimate of β in equation 2.

3 UIP exchange rates and cointegration

3.1 Data and forecasts

The data set here includes eight USD currencies: the Australian dollar(AUD), Canadian dollar (CAD), Swiss Franc (CHF), euro (EUR), Britishpound (GBP), Japanese yen (JPY), New Zealand dollar (NZD) and SwedishKrona (SEK). The sample period is January 1990 to July 2012, or the sub-sample for which the relevant data are available (see Appendix B for details).The data are monthly, so the sample size is about 260 for most countries,and shorter for the euro and for currencies where long term interest swapmarkets developed later. This covers the bulk of the period for which reliable

10The sum of relative inflation could be based on break-even inflation rates from inflation-indexed government bonds, but inflation-indexed bonds are only systematically issued in afew jurisdictions, and tend to be traded in much less liquid markets than nominal bonds.

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swap rates are available. Estimation of equation 2 (see Appendix 3) confirmthe standard result of a β coefficient of less than the theoretical value of 1and usually less than zero.

Forecasts of fundamentals (equations 9 and 11) are constructed using cal-ibrated coefficients based on estimated univariate autoregressions, and acoefficient of 120 on the 10-year swap differential. The discount factor isunity (the forecasts are approximations to undiscounted infinite sums).11

The forecasts are shown in Figure 1 with the observed real exchange ratedeviation from the sample mean.

3.2 Integration and cointegration

If interest arbitrage is active then the real exchange rate qt and the forecasts−EtRt should be be co-integrated. Given the simple theoretical co-integratingstructure between the two series, an Engel and Granger (1987) approachto testing for co-integration is used: residuals Λt = qt + Rt are tested forstationarity.

Unit root tests of real exchange rates, the forecasts, and the deviation of theexchange rate from the forecasts are shown in Table 1. The real exchangerate tests as stationary for one (GBP) of the eight USD exchange rates.

The forecasts, −EtRt, test as stationary about half the time: for the CAD,GBP, NZD and SEK for simple AR1 forecasts, and for the EUR, GBP andNZD when the ten-year swap rate forecast is used as a nominal interest rateforecast. That result is encouraging in view of (i) the relatively short sampleand (ii) the discount factor of one (infinite un-discounted sum). (Engel andWest 2005) show that an asset price manifests near random walk behavior iffundamentals are I(1) and the factor for discounting future fundamentals isnear one.

The deviations between the real exchange rate and the forecasts qt − EtRt,test as stationary for the GBP and borderline stationary for the CHF andSEK. While the gap between the exchange rate and UIP-implied level doesnot test as stationary in most cases, it is almost always less integrated (morenegative ADF statistic) than the real exchange rate (when the role for interestarbitrage is not accounted for). That result suggests weak evidence of co-

11(Engel and West 2005) consider discount factors of 0.5,0.9 and 0.98 and find strongestresults for the latter, consistent with a theoretical value of one.

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integration. Factors other than an integrated rational forecast appear to bematerial in the more severe random walk behaviour of real exchange rates.

If estimated rather than calibrated coefficients are used (see Tables 2 and3), then the residual is less integrated again, as might be expected from afitting exercise. If a trend is included as a proxy for terms of trade or BalassaSamuelson effects), then the AUD, CAD, CHF, GBP and SEK residuals testas stationary.12

Correlation coefficients between the exchange rate and the forecasts are shownat the bottom of Table 1. In all cases the correlations are positive and themagnitude ranges from 0.05 to 0.60 for the simple AR1 forecast and from0.17 to 0.76 for the forecast that uses the 10-year swap differential.

In view of the stronger results for the forecast that employs the 10-year swapdifferential, in the interest of parsimony, the remaining results are restrictedto using those forecasts.

4 Decomposition of exchange rate changes

A natural next step would be to estimate the contribution of forecast revisions∆Rt to exchange rate changes ∆qt. However, there is good reason to think thatestimation using standard techniques may be biased. Fama (1984) shows thatvariation in the the rational forecast path (in our notation Rt) is negativelycorrelated with the excess return Λt (Fama’s premium). Without separatingthose effects, standard estimation techniques are likely to be problematic.

The problem is that the data generating process looks a lot like a classicalsupply-demand interaction that is very difficult to identify empirically. Oneapproach to resolving such a problem is the use of instruments that areuncorrelated with one component. The approach here is to use a decompositionbased on the rational expectations forecasts to separate the two. The variancesof the components are used to estimate the role of interest arbitrage inexchange rate fluctuations and their covariances used to inform on potentialestimation bias for different specifications.

If interest arbitrage is active, then there should be a positive relationshipbetween changes in the exchange rate, the forecast revision and an error

12See Thoenissen and Benigno 20XX for a decomposition of the real exchange rate into alaw of one price gap, a relative terms of trade effect, and a Balassa Samuelson effect.

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correction term. In the limit that interest parity holds, the correlation coeffi-cient should converge to unity and the decomposition accurately measuresthe interest arbitrage contribution.

4.1 Theoretical decomposition

Using the forecasts of relative real interest returns, exchange rate fluctuationsare decomposed into changes related to interest arbitrage (forecast revisionsand revision toward the UIP-consistent path), and changes unrelated tointerest arbitrage (shocks). These components are directly related to risk-neutral arbitrage opportunities, so are used as proxied for the interest arbitragecontribution.13

The forecast revision ∆Rt is:

∆Rt = Et+1Rt+1 − EtRt (14)

= Et+1Rt+1 − EtRt+1 − (rt − r∗t ) (15)

where the first two terms reflect news (the change in the forecast for qt+1)and the last term (the current relative return) is observed at time, t.

The error correction term is estimated from:

∆qt+1 + (Rt+1Rt) = φΛt + λt (16)

where φ is the error correction coefficient and λt is the shock. Estimatedequations are shown in Table 4. The estimated error correction coefficientis the expected sign in all cases consistent with adjustment of the exchangerate toward the UIP-implied path. The estimated coefficient is significant tothe 10% level for 5 of 8 currency pairs and to the 14% level for the others.The magnitude of the term ranges from 2.3% per month for the CAD (halflife of 30 months) to 7.1% per month for the GBP (half life of 9 months).

13This is not the only potential decomposition. Effectively it assumes that shocks affect theinterest rate path and the exchange rate, but interest arbitrage only reprices the exchangerate (eg. a floating exchange rate). An alternative decomposition, that attributes theprojection of (∆qt,−∆Rt) onto the ∆qt = ∆Rt plane, attributes a slightly higher shareof exchange rate fluctuation to interest arbitrage. If only half of the forecast revision isattributed to UIP, its share goes up by about 5 percentage points (the variance of theshock falls by more).

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4.2 Variances

Table 5 reports the variances of the components relative to the varianceof the one-period return. Variances of the forecast revisions, ∆Rt and ofthe shock λt, and changes in the exchange rate ∆qt are all two orders ofmagnitude larger than the variance of the one-period return. The variance ofthe estimated error correction term is one order of magnitude larger.

Figure 2 shows the components of exchange rate fluctuations from the decom-position for the eight USD currency pairs: the 1-period return, the forecastrevision net of the one-period return (Dornbusch jump), the error correctionterm and the residual. The magnitudes of the Dornbusch jump and the resid-ual help to explain the large standard errors in a standard Fama equation,but not the systematic bias in the coefficient.

If the decomposition is a reasonable approximation of the truth, then thecontribution of interest arbitrage can be estimated from its contribution tothe exchange rate variance. Table 5 suggests that the contribution is 26 to42% for the eight USD exchange rates. On average interest arbitrage accountsfor about a third of exchange rate variance, with the shock accounting forthe remaining two thirds.

4.3 Covariance with the exchange rate

A potential problem is that, if interest arbitrage is not active, that variancedecomposition infers a role for interest arbitrage where there is none. Ifinterest arbitrage is not active, then there should be no material positiverelationship between forecast revisions (−∆Rt) and exchange rate changes.

Table 6 shows correlations between changes in the real exchange rate andits components. The correlation coefficients are divided by the variance ofthat component so that the reported coefficients are comparable to regressioncoefficients. The correlation between the forecast revision, −Rt, and thechange in the exchange rate is positive for seven of the eight currency pairs.It is negative for the CAD with a coefficient of -0.13. For the other currenciesit is lowest for the AUD at 0.11 and highest for the SEK at 0.39 and theNZD at 0.37. Figure 3 shows changes in the real exchange rate together withforecast revisions for the eight USD currency pairs.

Mean reversion (error correction term) is positively correlated with theexchange rate, consistent with the estimated reversion toward the UIP-consistent path (Table 4).

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The shock component is positively correlated with exchange rate changes inall cases with an average implied regression coefficient of 0.66. Again, thedecomposition implies that that the shock is to be about twice as importantas the forecast revision in exchange rate fluctuations.

4.4 Covariance with the shock

Covariance of the interest arbitrage components with the shock can informon potential estimation bias. If equation 6 is estimated directly and theexplanatory variables are correlated with the residual, then the estimatedparameters will be biased:

β̂ = β +cov(xt, εt)

var(xt)(17)

where β̂ is the estimated coefficient and β is the true value.

Table 7 shows the correlation between the shock and the other componentsbased on the decomposition. The shock is negatively correlated with theforecast revision and (by construction) uncorrelated with the error correctionterm. The forecast revision can be decomposed into the 1-period return andthe news-driven Dornbush jump. On average, both are negatively correlatedwith the shock, but the result is more consistent for the jump component.This result is consistent with Fama (1984)’s interpretation of an estimatedcoefficient of less than 0.5 in the standard UIP test equation 2: that both therational forecast path and the shock are time varying, that they are negativelycorrelated, and that the shock is the larger of the two.

Negative correlation with the forecast revision implies that the shock bothappreciates (depreciates) the home currency and depresses (increases) expectedinterest returns. The effect on returns drives the forecast revision and createsan interest arbitrage opportunity. In turn, the interest arbitrage responsemust be less than complete (Froot and Thaler 1990). If adjustment werecomplete, UIP would hold and the correlation would be unity.14 Capitalflows and risk premia are examples of shocks that would generate negativeco-movement. However, risk premia do not open (risk adjusted) arbitrageopportunities, so may not elicit an interest arbitrage response.

Negative correlation of the shock with the 1-period return, which is knownex-ante, implies that the high return currency continues to appreciate. Again

14Dornbusch ”overshooting” implies complete adjustment to the rational forecast path.

15

capital flows provide a potential explanation. Capital should flow to a highreturn to capital until returns are equalised. However, equalisation of returnsbetween a low capital (high returns to capital) economy and a high capital(low return to capital) economy may imply either large current accountdeficits or strong relative savings performance over an extended period. Asa result capital may rationally continue to flow towards the high return tocapital in a persistent manner (eg carry trade). In contrast, it is unclear whya high risk currency should continue to appreciate.

Capital flows should generate opposite movement in the exchange rate andthe forecast path. For example, an inflow should appreciate the exchange rateand put downward pressure on home interest rates. The lower home interestpath and stronger home currency create an interest arbitrage opportunity.

So one interpretation of the negative correlation between the exchange rate andthe forecast revision is the role for shocks that generate negative comovement,combined with less than complete adjustment, via interest arbitrage, to theUIP-consistent path.

Estimated equations of the form:

∆qt+1 = α + β∆Rt+1 + φΛt − λt+1 (18)

are shown in Tables 8 and 9. The R2 statistics of 0.02 to 0.13 in the estimatedequations imply a considerably lower interest arbitrage contribution thanimplied by the decomposition. We know, however, from (Fama 1984) that thecoefficient estimates for the terms in ∆Rt are biased downward for estimatedFama coefficients of 0.5 or less (all of the currency pairs for this data set).An interesting feature is the rise in the estimated contribution over time foralmost all of the currency pairs as shown in Figure 4 (except the GBP). Sothe contribution of interest arbitrage may be increasing, perhaps as moreaccurate forecasts of future relative returns become available. The estimates,while improving, are still consistent with the idea that interest arbitrageresponds to arbitrage opportunities created by factors such as capital flows,but less than fully offsets their effect on exchange rates.

4.5 Covariance with the interest differential

In the standard Fama equation, the residual incorporates the revision to theforecast of subsequent returns (Dornbusch jump), the error correction termand the shock. The forecast revision and shock should be unforecastableex-ante, unless there is a systematic response to the higher return (eg capital

16

flows). If the 1-period return covaries with any of those, then the coefficientestimate will be biased (equation 17 above).

Table 10 shows covariances with the 1-period return divided by the varianceof the 1-period return to give an idea of the magnitude of the estimationbias in the standard equation. On average, the 1-period return is negativelycorrelated with all three components of the shock of the Fama equation.

The shock component can be understood, as before, in terms of capital flows:if capital continues to flow to a high return to capital to equalise returns oncapital (eg carry trade) and is not fully offset by interest arbitrage, then thehigh return currency should continue to appreciate, rather than depreciatingin line with a theoretical coefficient of one.

The negative correlation with the Dornbusch jump is more difficult to ra-tionalise. In theory, since the 1-period return is known ex-ante, the jumpshould correspond to the news component of the forecast and be uncorrelatedwith the 1-period return. An endogenous response of interest arbitrage to theshock should contribute a positive bias (partly offsetting the negative bias ofthe shock). That leaves the empirical explanation of (Engel and West 2005):if the UIP-consistent path −Rt is a near random walk (Rsum in Table 1),the jump (the forecast revision net of the 1-period return: −∆Rt − (rt − r∗t ))is likely to be negatively correlated with the one-period return. Estimatedequations that account for the jump (that can be measured by the time theexchange rate is set) illustrate that relationship (Appendix 3).

The bias from the error correction term is the smallest in magnitude and leastconsistent in sign.

5 Conclusions and discussion

The exchange rate, as a relative asset price, should reflect the sum of futurerelative returns. If the home currency is expected to yield a higher return,then the currency should immediately appreciate (Dornbusch jump), andthen depreciate as those returns are realised. That initial adjustment -- theDornbusch overshooting effect -- is an integral part of uncovered interestarbitrage.

This paper constructed forecasts of relative interest returns to calculate a UIP-consistent exchange rate path. As in Fama (1984) and Engel and West (2005),the exchange rate is split into the rational forecast component consistent

17

with UIP, and an excess return unrelated to interest arbitrage. There is weakevidence of co-integration between the exchange rate and the rational forecastpath: the deviation between the two is less integrated than the exchange rate,but only stationary for the GBP/USD; and error correction terms are of thecorrect sign and are mostly significant.

Changes in exchange rates were decomposed into forecast revisions, an er-ror correction term and shocks. The former two are related to incentivesfor interest arbitrage and are taken as proxies for the interest arbitragecontribution.

Consistent with active interest arbitrage, countries forecast to have high rela-tive real interest returns show strong real exchange rates; forecast innovationsare positively correlated with exchange rate changes; and there is evidence ofreversion toward the UIP-consistent path. The decompositions suggest thatinterest arbitrage accounts for about a third of exchange rate innovations forthe eight advanced country USD exchange rates examined.

Shocks (unrelated to interest arbitrage) account for the remaining two thirdsof exchange rate changes. A role for capital flows in the shock process isconsistent with two important stylised facts: continued appreciation of highinterest currencies and negative covariance between variation in the rationalforecast path and the shocks ((Fama 1984)). The continued appreciationof high interest currencies (wrong sign Fama coefficient) makes sense ifcapital flows seek to equalise returns on capital (a process that may takedecades), and those flows are less than fully offset by interest arbitrage (Frootand Thaler 1990). Since capital inflows appreciate the home currency andput downward pressure on home interest rates, their exchange rate impactcovaries negatively with that implied by interest arbitrage (a weaker homecurrency associated with the lower home interest rate path). Capital flows arelarge and volatile, and reflect a variety of driving forces other than interestarbitrage (eg. various risk premia, portfolio shifts, carry trade, central bankintervention). The driving factors may vary over time as in (Bacchetta andWincoop 2004)’s ”scapegoat” model. A key role for capital flows is supportedby the empirical order flow literature (Evans and Lyons 2002) that finds astrong role for FX market net buy-sell orders on exchange rate fluctuations.

References

Bacchetta, P and E V Wincoop (2004), ‘‘A scapegoat model of exchange-rate fluctuations,’’ American Economic Review, 94(2), 114--118.

18

Bjørnland, H (2009), ‘‘Monetary policy and exchange rate overshoot-ing: Dornbusch was right after all,’’ Norges Bank, Working Paper,2009/09.

Bjørnland, H and J Halvorsen (2010), ‘‘How does monetary policy respondto exchange rate movements? new international evidence,’’ Centre forApplied Macroeconomic Research, CAMAR Working Paper Series,1/2010.

Burnside, C (2011), ‘‘Carry trades and risk,’’ National Bureau of Eco-nomic Research, Inc, NBER Working Papers, 17278.

Chinn, M and J Frankel (1994), ‘‘Patterns in exchange rate forecasts for25 currencies,’’ National Bureau of Economic Research, Inc, NBERWorking Papers, 3807.

Chinn, M and G Meredith (2004), ‘‘Monetary policy and long-horizonuncovered interest parity,’’ Staff Papers, 513.

Chinn, M and S Quayyum (2012), ‘‘Long horizon uncovered interestparity re-assessed,’’ Working Paper, 18482.

Dornbusch, R (1976), ‘‘Expectations and exchange rate dynamics,’’Journal of Political Economy, 84(6), 1161--76.

Engel, C (1996), ‘‘The forward discount anomaly and the risk premium:A survey of recent evidence,’’ Journal of Empirical Finance, (3).

Engel, C (2012), ‘‘The real exchange rate, real interest rates, and therisk premium,’’ Paper presented at Victoria University of Wellingtonmacroworkshop October 29 2012.

Engel, C and K D West (2005), ‘‘Exchange rates and fundamentals,’’Journal of Political Economy, 113(3), 485--517.

Engel, R and C Granger (1987), ‘‘Co-integration and error correction:Representation, estimation and testing,’’ Econometrica, 55.

Evans, M D D and R K Lyons (2002), ‘‘Order flow and exchange ratedynamics,’’ Journal of Political Economy, 110(1), 170--180.

Fama, E F (1984), ‘‘Forward and spot exchange rates,’’ Journal ofMonetary Economics, 14(3), 319--338.

Froot, K and J Frankel (1989), ‘‘Forward discount bias: Is it an exchangerisk premium?’’ Quarterly Journal of Economics, 104.

Froot, K and R Thaler (1990), ‘‘Anomalies: Foreign exchange,’’ Journalof Economic Perspectives, (4), 179--192.

McCallum, B T (1994), ‘‘A reconsideration of the uncovered interestparity relationship,’’ Journal of Monetary Economics, 33(1), 105--132.

Meese, R A and K Rogoff (1983), ‘‘Empirical exchange rate models ofthe seventies : Do they fit out of sample?’’ Journal of InternationalEconomics, 14(1-2), 3--24.

Thornton, D L (2009), ‘‘Resolving the unbiasedness puzzle in the foreign

19

exchange market,’’ Federal Reserve Bank of St. Louis, WorkingPapers, 2009-002.

van Wincoop, E and P Bacchetta (2007), ‘‘Random walk expectationsand the forward discount puzzle,’’ American Economic Review, 97(2),346--350.

20

Table 1: Cointegration: unit root tests

q_t 0.89 1.17 2.00 1.30 2.66 * 1.84 1.19 1.92 2.34 ***

AR1 forecasts

R(t) 2.55 3.09 ** 1.82 2.42 2.91 ** 1.84 3.78 *** 5.04 *** 2.54

q(t) R(t) 1.05 1.39 2.10 1.30 2.70 * 1.89 1.16 2.48 3.30 **

AR1 residual 1.76 2.03 2.25 1.77 3.14 ** 2.08 1.49 2.48 3.85 ***

SW10 forecasts

R(t) 2.19 2.31 1.83 2.63 * 3.38 ** 1.86 2.95 ** 2.19 2.42

q(t) R(t) 1.95 1.63 2.64 * 1.48 3.07 ** 1.76 1.47 2.54 3.14 **

SW10 residual 2.73 * 1.21 2.86 * 2.28 3.04 ** 1.75 2.06 2.74 * 4.32 ***

ADF unit root test. *** indicates rejection of a unit root at the 1% significance level; ** the 5% level and * the 10% level.

Correlation of real exchange rate and forecast

AUD/USD CAD/USD CHF/USD EUR/USD GBP/USD JPY/USD NZD/USD SEK/USD NZD/AUD

AR1 forecasts 0.60 0.27 0.22 0.11 0.39 0.05 0.12 0.36 0.66

SW10 forecasts 0.74 0.17 0.54 0.35 0.35 0.21 0.55 0.67 0.49

JPY/USD NZD/USD SEK/USD NZD/AUDAUD/USD CAD/USD CHF/USD EUR/USD GBP/USD

21

Table 2: Level equations and residual unit root tests (using AR1 forecasts ofnominal interest rates)

qt = α + β1i30t + β2i

30∗t + β3π

12t−1 + β4π

12∗t−1 + εt


Coef. Coef. Coef. Coef. Coef. Coef. Coef. Coef. Coef.

(A) AR1 MODEL

c 0.18 ** 0.03 0.03 0.12 * 0.11 *** 0.04 0.11 * 0.17 *** 0.02

i30* 114.6 *** 13.9 10.1 33.4 25.1 *** 43.4 *** 32.7 * 36.7 *** 39.4 ***

i30 89.1 *** 46.8 *** 26.9 ** 26.7 7.7 14.7 49.6 *** 8.4 32.6 **

pi12* 49.1 *** 7.3 9.5 56.7 6.6 22.0 31.7 2.1 0.8

pi12 8.4 59.2 ** 17.4 57.1 4.6 17.4 14.3 16.0 12.5 **

Rsquared 0.40 0.24 0.09 0.19 0.35 0.14 0.17 0.42 0.58

Adjusted Rsq. 0.39 0.23 0.07 0.17 0.34 0.13 0.16 0.41 0.57

Log likelihood 123.0 184.4 166.2 80.2 324.8 148.4 93.3 169.9 344.0

DurbinWatson 0.06 0.06 0.08 0.08 0.15 0.06 0.05 0.19 0.27

q t ADF unit root test 0.89 1.17 2.00 1.30 2.66 * 1.84 1.19 1.92 2.34 ***

ϵ t ADF unit root test 1.76 2.03 2.25 1.77 3.14 ** 2.08 1.49 2.48 3.85 ***

No obs 255 258 257.0 150 258 257 254 257.0 201

Sample

HAC standard errors (Bartlett kernel, NeweyWest fixed bandwidth = 5.0000)

1991M02 1991M021991M04 1991M02 1991M02 2000M02

2012M06 2012M07 2012M06 2012M07 2012M07 2012M062012M06 2012M06 2012M06

1991M05 1991M02 1995M10

22

Table 3: Level equations and residual unit root tests (using 10Y swap forecastsof nominal interest rates)

qt = α + β1(i10St − i∗10S

t ) + β2π12t−1 + β3π

12∗t−1 + εt


Coefficient Coefficient Coefficient Coefficient Coefficient Coefficient Coefficient Coefficient Coefficient

(B) 10YEAR SWAP MODEL, dependent variable: qt

c 0.19 *** 0.08 0.15 *** 0.18 *** 0.08 *** 0.08 * 0.36 *** 0.13 *** 0.01

idiff10 179.5 *** 29.9 125.9 *** 189.1 *** 37.2 ** 29.9 ** 216.8 *** 85.9 *** 91.5 ***

pi12* 30.9 ** 10.5 31.1 *** 176.6 *** 3.1 14.5 2.1 7.6 13.7

pi12 17.2 22.6 31.8 ** 99.3 *** 22.4 ** 2.7 26.6 36.6 ** 23.7 **

Rsquared 0.56 0.08 0.30 0.24 0.20 0.06 0.34 0.49 0.44

Adjusted Rsquared 0.56 0.07 0.29 0.22 0.19 0.04 0.33 0.48 0.43

Log likelihood 164.2 158.4 199.5 84.1 298.1 135.7 73.5 185.7 306.8

DurbinWatson stat 0.13 0.04 0.11 0.13 0.14 0.05 0.10 0.10 0.18

q t unit root test (ADF) 0.89 1.17 2.00 1.30 2.66 * 1.84 1.19 1.92 2.34 ***

ϵ t unit root test (ADF) 2.73 * 1.21 2.86 * 2.28 3.04 ** 1.75 2.06 2.74 * 4.32 ***

HAC standard errors (Bartlett kernel, NeweyWest fixed bandwidth = 5.0000)

No. Obs. 255 258 257 150 258 257 196 257 196

Sample

2012M06

1991M02

2012M07

1991M02 2000M02

2012M06 2012M07

1991M02

2012M06

1996M03

2012M062012M07 2012M06 2012M06

1991M02 1991M02 1996M03 1991M02

23

Table 4: Error correction estimates

∆qt+1 + ∆Rt+1 = c+ φΛt + εt

EUR_USD

Variable

constant 0.00 0.00 0.01 ** 0.00 0.00 0.01 0.00 0.00

1.05 0.25 2.08 0.58 1.60 1.51 0.93 1.54

ϕ 0.036 * 0.023 0.054 *** 0.028 0.071 *** 0.025 * 0.024 0.050 **

1.95 1.63 2.64 1.48 3.07 1.76 1.47 2.54

Rsquared 0.01 0.01 0.03 0.01 0.04 0.01 0.01 0.02

Adj. Rsquared 0.01 0.01 0.02 0.01 0.03 0.01 0.01 0.02

S.E. of regression 0.04 0.03 0.04 0.03 0.03 0.04 0.04 0.04

SSR 0.40 0.26 0.33 0.17 0.28 0.34 0.32 0.39

Log likelihood 457.2 517.6 484.3 289.4 510.8 481.7 347.6 464.3

Fstatistic 3.82 2.67 6.98 2.19 9.44 3.11 2.15 6.48

Prob(Fstatistic) 0.05 0.10 0.01 0.14 0.00 0.08 0.14 0.01

DurbinWatson 2.20 2.24 2.07 2.17 2.03 2.01 2.15 1.91

No. Obs. 253 256 255 148 256 255 195 255

Sample 1991M06 1991M04 1991M04 2000M04 1991M04 1991M04 1996M04 1991M04

2012M06 2012M07 2012M06 2012M07 2012M07 2012M06 2012M06 2012M06

Half life /b 19 30 13 24 9 27 28 14

tstat tstattstat tstat tstat tstat tstat tstat

coef. coef.coef. coef. coef. coef. coef. coef.

SEK_USDNZD_USDAUD_USD CAD-USD CHF-USD GBP_USD JPY_USD

24

Table 5: Variances

∆qt+1 = −∆Rt+1 − φΛt + εt

UIP

Δqt+1 (rt - r*t) ΔRt+1 -ϕ(qt +Rt) residual contribution

"Fama" ( a ) ( b ) ( c ) (a+b)(a+b+c)

AUD/USD 408.0 1 199.4 8.4 570.8 0.27CAD/USD 267.3 1 223.6 5.1 554.8 0.29CHF/USD 195.3 1 130.5 6.6 227.9 0.38EUR/USD 273.2 1 115.0 2.9 319.6 0.27GBP/USD 277.5 1 220.4 12.7 355.5 0.40JPY/USD 162.5 1 116.1 2.3 207.7 0.36NZD/USD 574.4 1 317.8 12.9 811.9 0.29SEK/USD 156.4 1 119.6 3.7 184.8 0.40NZD/AUD 314.9 1 230.2 21.6 349.6 0.42avg (USD) 289.3 1 180.3 6.8 404.1 0.32

var(x) / var(rt-r*t)

25

Table 6: Correlations with exchange rate movements (∆qt)

∆qt+1 = −∆Rt+1 − φΛt + εt

Cov( x , Δqt+1 ) / var( x )

Δqt+1 -ΔRt+1 (rt - r*t) -ΔRt+1 + (rt - r*t) -ϕ(qt +Rt) Shock

AUD 1 0.11 -1.46 0.12 0.71 0.69CAD 1 -0.13 -0.32 -0.13 0.39 0.55CHF 1 0.30 -0.51 0.29 0.93 0.68EUR 1 0.33 0.73 0.33 0.91 0.76GBP 1 0.30 0.19 0.29 0.35 0.59JPY 1 0.29 -0.77 0.30 1.23 0.60NZD 1 0.37 -1.21 0.37 0.85 0.76SEK 1 0.39 0.73 0.38 0.56 0.61NZDAUD 1 0.43 -0.18 0.42 0.36 0.66avg (USD) 1 0.24 -0.33 0.24 0.74 0.66

26

Table 7: Correlations with the residual (εt)

∆qt+1 = −∆Rt+1 − φΛt + εt

-ΔRt+1 (rt - r*t) -ΔRt+1 - (rt - r*t) -ϕ(qt +Rt)

AUD/USD -0.87 -1.97 -0.85 0.000CAD/USD -1.13 -1.22 -1.14 0.000CHF/USD -0.58 0.01 -0.57 0.000EUR/USD -0.66 -0.94 -0.67 0.000GBP/USD -0.70 0.91 -0.70 0.000JPY/USD -0.70 0.16 -0.69 0.000NZD/USD -0.64 -1.64 -0.63 0.000SEK/USD -0.62 -0.02 -0.63 0.000NZD/AUD -0.49 -0.21 -0.49 0.000avg (USD) -0.74 -0.59 -0.74 0.000

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27

Tab

le8:

Dif

fere

nce

equat

ions

bas

edon

AR

1fo

reca

sts

∆q t

=α

++φ

(qt−

1−qeqt−

1)

+β

1(∆i t−

∆i∗ t

)+β

2(∆π

12t−

∆π

12∗

t)

+ε t

AR

1 E

quat

ion

AU

D_U

SDC

AD

_USD

CHF_

USD

EUR

_USD

GBP

_USD

JPY_

USD

NZD

_USD

SEK_

USD

Coe

ffici

ent

Coe

ffici

ent

Coe

ffici

ent

Coe

ffici

ent

Coe

ffici

ent

Coe

ffici

ent

Coe

ffici

ent

Coe

ffici

ent

(t-s

tat)

(t-s

tat)

(t-s

tat)

(t-s

tat)

(t-s

tat)

(t-s

tat)

(t-s

tat)

(t-s

tat)

C0.

00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.23

0.65

-0.3

3-0

.32

0.67

-0.2

20.

711.

311.

04

Coe

ffici

ent

(t-s

tat)

NZD

_AU

D

0.23

0.65

-0.3

3-0

.32

0.67

-0.2

20.

711.

311.

04

EG_A

R1_

RES

(-1)

-0.0

24

-0.0

21*

-0.0

32*

-0.0

10

-0.0

66**

-0.0

34**

-0.0

26*

-0.0

32*

-0.1

29**

*-1

.37

-1.6

8-1

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0-2

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6-3

.47

DI3

0-D

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61**

*10

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46

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25

-45.

46**

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0.98

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0

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25

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0.98

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0

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1.77

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2.37

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4-7

.00

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.10

DPI

DIF

F12(

-1)

8.63

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36**

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72

6.53

-13.

35**

*7.

56

3.41

-1.3

5

a/

1.46

0.89

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7-1

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1.40

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21.

530.

64-0

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040.

040.

020.

180.

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07A

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020.

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030.

030.

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030.

040.

02S

um s

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ed re

sid

0.28

0.12

0.27

0.15

0.17

0.25

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0.33

0.08

Sum

squ

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d0.

280.

120.

270.

150.

170.

250.

310.

330.

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g lik

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061

4.2

509.

729

9.5

575.

552

2.4

493.

248

5.8

494.

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stat

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5.28

3.06

3.57

1.19

18.5

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902.

113.

234.

87P

rob(

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atis

tic)

0.00

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0.00

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0.02

0.00

Dur

bin-

Wat

son

2.16

2.07

1.98

2.19

1.94

1.94

2.10

1.89

1.90

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

No.

Obs

.25

425

625

514

825

625

525

425

520

0.00

Sam

ple:

Impl

ied

valu

e of

:

1995

M11

2012

M06

1991

M05

1991

M04

2012

M07

2012

M06

2012

M07

2012

M08

2012

M06

2012

M06

2012

M06

1991

M05

2012

M06

1991

M04

1991

M04

2000

M04

1991

M04

1991

M04

Impl

ied

valu

e of

:rh

o (id

iff)

0.97

1.10

1.68

0.94

0.98

0.91

0.80

0.81

0.92

rho(

pidi

ff)0.

88

0.75

1.06

1.09

0.85

1.07

0.87

0.71

1.74

Hal

f life

b/

2932

2171

1020

2622

5a/

For

the

AU

D a

nd N

ZD in

flatio

n, th

e la

g is

3 ra

ther

than

one

to a

llow

for q

uarte

rly in

flatio

n re

leas

es.

a/ F

or th

e A

UD

and

NZD

infla

tion,

the

lag

is 3

rath

er th

an o

ne to

allo

w fo

r qua

rterly

infla

tion

rele

ases

.b/

Hal

f life

is c

alcu

late

d fro

m c

oeffi

cien

t on

erro

r cor

rect

ion

term

and

is m

easu

red

in m

onth

s.

Not

es:

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orco

rrec

tion

term

use

sre

sid

ual

from

AR

1le

vel

seq

uati

on

inta

ble

2

28

Tab

le9:

Dif

fere

nce

equat

ions

bas

edon

nom

inal

10-y

ear

swap

inte

rest

rate

fore

cast

san

dA

R1

infl

atio

nfo

reca

sts

∆q t

+1

=α

+φ

(qt−

1−qeqt−

1))

+β

1∆

(i10S

t+1−i∗

10S

t+1

)+β

2∆

(π12t+

1−π

12∗

t+1)

+ε t

+1

EUR

_USD

Var

iab

le

C0

.00

0

.00

0

.01

**0

.00

0

.00

0

.01

*0

.00

0

.00

0

.00

**

0.7

90

.25

2.1

60

.80

1.1

71

.94

0.8

11

.65

2.0

6

EG_C

_RES

(1

)0

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29

Table 10: Correlations with the 1-period return (r−r∗t )

∆qt+1 = (rt − r∗t )− (Et+1Rt+1 − EtRt+1)︸︷︷︸∆Rt+1+(rt−r∗t )

−φΛt + εt

ΔRt+1 (rt - r*t) ΔRt+1 - (rt - r*t) -ϕ(qt +Rt) shock

err. corr.AUD/USD -0.38 1 -1.38 0.82 -1.88 -1.17CAD/USD 0.86 1 -0.14 -0.33 -0.83 -0.32CHF/USD -0.58 1 -1.58 -0.11 -0.09 -0.51EUR/USD 1.49 1 0.48 0.02 -0.78 0.72GBP/USD 0.31 1 -0.69 -0.15 0.03 -1.49JPY/USD -0.41 1 -1.41 -0.51 0.14 -1.37NZD/USD -0.80 1 -1.99 0.36 -1.23 -2.09SEK/USD 0.95 1 -0.06 -0.29 0.07 0.00NZD/AUD -0.38 1 -1.37 0.95 -0.62 -0.43avg (USD) 0.18 1 -0.85 -0.03 -0.57 -0.78Note: The β coefficient is the estimated coefficient from equation (2). The theoretical value of 1 plus the implied bias may not add to the estimated coefficient because of correlation among the components of the equation residual.

estimated β coefficient

��, �� ∗� �/�� ∗�

30

Figure 1: Real exchange rates and forecast sums of future relative returns

-50%

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%Jan-90 Jan-95 Jan-00 Jan-05 Jan-10

% deviation RER AUD-USD

real exchange rate (dev. from sample mean)

Forecast (AR)

Forecast (swap)

-30%

-20%

-10%

0%

10%

20%


% deviation RER CAD-USD

Real exchange rate (dev from sample mean)

Forecast (AR)

Forecast (swap)

-40%

-30%

-20%

-10%

0%

10%

20%

30%


% deviation RER CHF-USD

real exchange rate (dev. from sample mean)

Forecast (AR)

Forecast (swap)

-40%

-30%

-20%

-10%

0%

10%

20%

30%Jan-99 Jan-04 Jan-09

% deviation RER EUR-USD

real exchange rate (dev from sample mean)

Forecast (AR)

Forecast (swap)

-30%

-20%

-10%

0%

10%

20%


% deviation RER GBP-USD


Forecast (AR)

Forecast (swap)

-120%

-100%

-80%

-60%

-40%

-20%

0%

20%

40%


% deviation RER JPY-USD

real exchange rate (dev from sample mean)Forecast (AR)

Forecast (swap)

-60%

-50%

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%


% deviation RER NZD-USD

Real exchange rate (dev from sample mean)

Forecast (AR)

Forecast (swap)

-40%

-30%

-20%

-10%

0%

10%

20%

30%

40%

50%


% deviation RER SEK-USD


Forecast (AR)

Forecast (swap)

Notes: Real exchange rates (red lines) are % deviation from sample mean so are subject to alevel shift if the sample mean differs from the long-run real equilibrium. Dotted black linesshows UIP-consistent exchange rate (−Rt) constructed as 120 times the 10-year nominalswap differential (monthly rate), net of an AR1-based forecast of the relative inflationpaths. The dashed grey line shows the forecast constructed solely from AR1 forecasts ofthe relative nominal 30-day interest rates and relative annual inflation.

31

Figure 2: Decomposition of ∆qt

-30%

-25%

-20%

-15%

-10%

-5%

0%

5%

10%


% change AUD-USD

1-period return

D(Exp returns)

Error correction

Residual

USD appreciation

-20%

-15%

-10%

-5%

0%

5%

10%


% change CAD/USD

1-period return

D(Exp returns)

Error correction

Residual

USD appreciation

-15%

-10%

-5%

0%

5%

10%


% change CHF/USD

1-period return D(Exp returns)

Error correction Residual

USD appreciation

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%Jan-99 Jan-04 Jan-09

% change EUR/USD

1-period return

D(Exp returns)

Error correction

Residual

USD appreciation

-25%

-20%

-15%

-10%

-5%

0%

5%

10%


% change GBP/USD



USD appreciation

-15%

-10%

-5%

0%

5%

10%

15%


% change JPY/USD



USD appreciation

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%Jan-96 Jan-01 Jan-06 Jan-11

% change NZD/USD



USD

appreciation

-15%

-10%

-5%

0%

5%

10%

15%


% change SEK/USD



USD appreciation

32

Figure 3: Forecast revisions (−∆Rt) and exchange rate changes (∆qt)

-30%

-25%

-20%

-15%

-10%

-5%

0%

5%

10%


% change AUD-USD

Exchange rate change

Forecast revision

USD appreciation

-20%

-15%

-10%

-5%

0%

5%


% change CAD/USD


Forecast revision

USD appreciation

-15%

-10%

-5%

0%

5%

10%


% change CHF/USD


Forecast revision

USD appreciation

-15%

-10%

-5%

0%

5%

10%

15%Jan-99 Jan-04 Jan-09

% change EUR/USD


Forecast revision

USD appreciation

-20%

-15%

-10%

-5%

0%

5%


% change GBP-USD


Forecast revision

-15%

-10%

-5%

0%

5%

10%

15%


% change JPY-USD


Forecast revision

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%Jan-96 Jan-01 Jan-06 Jan-11

% change NZD/USD


Forecast revisionUSD

appreciation

-20%

-15%

-10%

-5%

0%

5%

10%


% change SEK/USD


Forecast revision

USD appreciation

33

Figure 4: R2 statistic for equation in Table 9 estimated difference equationsover different time horizons

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

AUD CAD CHF EUR GBP JPY NZD SEK NZD/AUD

full sample

2000 onwards

2005 onwards

34

A Real and nominal UIP

The exchange rate is an asset price that can be expressed as an infinite forward sum offuture relative returns:

st = −Et

∞∑k=0

(it+k − i∗t+k)− Et

∞∑k=0

λt+k + limk→∞

Et(st+k) (A.1)

where st is th nominal exchange rate (a rise is a depreciation), it and i∗t are the home andforeign nominal interest rate and λt is the excess return to holding the home currency. Ina one-period recursive form:

st = −(it − i∗t )− λt + Et(st+1) (A.2)

UIP can be expressed in real or nominal terms, but if we assume stationarity of the realexchange rate in the longer term, then we need to be careful about stationarity whenworking in nominal terms, and in particular about the final term in equation A.1. Tothe extent that the nominal interest differential reflects higher home inflation, the homecurrency must depreciate to satisfy PPP in the long term.

Assuming stationarity of the real exchange rate (see Engel (2012)), limk→∞Et(st+k) inequation 3 needs to be expressed in terms of the long-term real exchange rate, q, impliedby purchasing power parity (PPP).

Defining the real exchange rate as:

qt = st − pt + p∗t

where qt, pt, p∗t are logs of the real exchange rate, home and foreign prices respectively,

then

limk→∞

(Etst+k) = q̄ + pt − p∗t + Et

∞∑k=1

(πt+k − π∗t+k) (A.3)

Substituting A.3 into equation A.1,

st − pt + p∗t︸︷︷︸qt

−q̄ = −Et

∞∑k=0

(it+k − i∗t+k) + Et

∞∑k=1

(πt+k − π∗t+k)− Et

∞∑k=0

λt+k

= −Et

∞∑k=0

(rt+k − r∗t+k)︸︷︷︸Rt

−Et

∞∑k=0

λt+k︸︷︷︸Λt

(A.4)

where the left hand side is the deviation of the real exchange rate at time t from equilibrium,rt = it−E(πt+1) is the real exchange rate from the Fisher equation and Rt is the expectedsum of future one-period real relative returns (nominal relative interest returns net ofinflation). Higher expected nominal home returns appreciate the home currency, but tothe extent that they reflect relative inflation, the nominal equilibrium rate must depreciateto be consistent with the long-run real equilibrium.

In steady-state, 0 = r − r∗ + λ.

35

B Data

Exchange rates and nominal interest rates are end-month rates. Real exchange rates aremeasured ex-post. The inflation component of real interest rates is forecast on the basis ofdistributed lag equations. CPI data is assumed to be released with in a month.

Nominal 30-day interest rates and spot exchange rates are end-month rates from Bloomberg:

currency 90-day rate 10-year swap exchange rateAUD ADBB1M Curncy ADSW10 Curncy AUD CurncyCAD CD001M Curncy CDSW10 Curncy CAD CurncyEUR EU001M Curncy EUSa10 Curncy EUR CurncyGBP BP001M Curncy BPSW10 Curncy GBP CurncyJPY JY001M Curncy JYSW10 Curncy JPY CurncyNZD NDBB1M Curncy NDSW10 Curncy NZD CurncyUSD US0001M Index USSW10 Curncy 1

Nominal 30-day interest rates are Libor rates or a local equivalent rate where the localbenchmark rate is more heavily traded (eg Australia and New Zealand bank bill rates).10-year swap rates are available for the whole sample (1990-2012) for the Australian dollar,Canadian dollar, Japanese yen and US dollar, from November 1990 for the Swiss franc andUK pound, from Feb 1991 for the Swedish krona, from March 1996 for the New Zealanddollar, and from January 1999 for the Euro.

Consumer price indices are from national sources and the IMF International FinancialStatistics. CPI data is assumed to be released within a month after the relevant period.For Australia and New Zealand, quarterly price indices are interpolated so that inflation isthe same for the three months of the quarter and additional lags are used in the differenceequations so that they are based only on data observed at the time.

The one-period real interest rate is constructed as it − i∗t +Et(πt+1 − π∗t+1). For countrieswith monthly inflation data, Et(πt+1) and Et(π

∗t+1) are constructed as a 2-step ahead

forecast using 3 lags of monthly inflation and 2 lags of annual inflation:

πt = c0 + c1πt−2 + c2πt−3 + c3πt−4 + c4π12t−2 − c5π12

t−3)

For countries with quarterly inflation data (Australia and New Zealand), the 2-steps aheadinflation forecast is constructed using:

πt = c0 + c1πt−3 + c2πt−6 + c3π12t−3

Similarly, ∆Rt is calculated based on inflation data observed at time t.

36

C Dornbusch overshooting

Dornbusch’s overshooting model is described in nominal terms based on a money demandequation to describe interest rates, uncovered interest parity and sticky prices. It is easierto think about Dornbusch overshooting behaviour with interest rates set directly (mostmodern monetary policy regimes) and in real terms.

Figures 5 and 6 illustrate the role of changing expectations on the exchange rate. In Figure5, the solid blue line represents the expected path, at t = 0, of the interest differentialover the future. When agents expect domestic interest rates to rise above foreign interestrates by 1% p.a. for 2 years, the exchange rate is immediately bid up to 2% above itsequilibrium level. This initial rise (Dornbusch overshooting) equalises returns on domesticand foreign assets: the higher domestic interest return over the subsequent two periods isoffset by the expected ”Fama” depreciation of the domestic currency.

Figure 5: Exchange rate adjustment to change in expectations

-1 0 1 2 3 4

-2

-1

0

1

2%

time

Dornbusch � jump � Fama depreciation

ft,t+1

ft,t+2

E

t (r

t-r*

t) (% per period)

qt

Now suppose that, in response to news between time 0 and 1, the interest differentialis expected to persist for an additional period (Figure 6). Initially, the higher domesticinterest rate was expected to last for two periods (solid blue line), but by t = 1 is itexpected to last for an additional period from t = 2 to t = 3 (dashed blue line). At t = 1the exchange rate consistent with UIP is 2% above its long-run equilibrium. The changein the exchange rate from time 0 to 1, is the sum of the ex-ante expected depreciation(forward premium) and the change in the expected path of relative returns rates fromt = 0 to t = 1 (Et+1 − Et(st+1)). In the example, the two effects call them the Famadepreciation and Dornbusch overshooting, sum to zero.

In a dynamic sense, both the initial jump and subsequent gradual depreciation illustratedhere occur continuously, in response to any news that leads to a revision of the future stateof the economy. Both the initial jump and subsequent depreciation are directly related tointerest arbitrage.

37

Figure 6: Exchange rate adjustment to change in expectations

-1 0 1 2 3 4

-2

-1

0

1

time

Dornbusch jump �

Dornbusch jump�

Fama depreciation �

� Fama depreciation

E

t (r

t-r*

t)

qt

Et+1

(rt-r*

t)

qt+1

D Estimated equations

38

Table 11: Fama equation estimates: nominal variables∆st+1 = α + β(it − i∗t ) + εt+1

theoretical Rsqvalue coef t-stat low high adjRsq

AUD_USD C 0 0.00 -0.23

( 0.00 , 0.00 ) * 0.002

iDIFF30(-1) 1 -0.79 -0.78

( -2.47 , 0.88 )

-0.001

CAD_USD C 0 0.00 0.42

( 0.00 , 0.00 ) * 0.000

iDIFF30(-1) 1 -0.11 -0.10

( -1.85 , 1.63 ) * -0.004

CHF_USD C 0 0.00 1.24

( 0.00 , 0.01 ) * 0.004iDIFF30(-1) 1 -0.91 -1.00

( -2.40 , 0.59 )

0.000

EUR_USD C 0 0.00 0.22

( 0.00 , 0.01 ) * 0.001iDIFF30(-1) 1 -1.20 -0.47

( -5.45 , 3.05 ) * -0.005

GBP_USD C 0 0.00 0.56

( 0.00 , 0.01 ) * 0.004iDIFF30(-1) 1 1.77 0.97

( -1.23 , 4.77 ) * 0.000

JPY_USD C 0 0.00 2.22 ** ( 0.00 , 0.00 )

0.014iDIFF30(-1) 1 -0.02 -1.92 * ( -0.03 , 0.00 )

0.010

NZD_USD C 0 0.00 0.34

( 0.00 , 0.01 ) * 0.014iDIFF30(-1) 1 0.08 0.08

( -1.54 , 1.70 ) * 0.011

SEK_USD C 0 0.00 0.51

( 0.00 , 0.00 ) * 0.000iDIFF30(-1) 1 0.21 1.97 * ( 0.03 , 0.39 )

-0.004

NZD_AUD C 0 0.00 -0.08

( 0.00 , 0.00 ) * 0.001iDIFF30(-1) 1 0.54 0.50

( -1.26 , 2.33 ) * -0.004

90\% C.I.

39

Table 12: Fama equation estimates: real variables∆qt+1 = α + β(rt − r∗t ) + εt+1

theoretical Rsqvalue coef t-stat low high adjRsq

AUD_USD C 0 0.00 -1.64

( -0.01 , 0.00 ) * 0.031

RDIFF30(-1) 1 -3.51 -2.84 *** ( -5.55 , -1.47 )

0.027

CAD_USD C 0 0.00 0.09

( 0.00 , 0.00 ) * 0.000

90\% C.I.

CAD_USD C 0 0.00 0.09

( 0.00 , 0.00 ) * 0.000

RDIFF30(-1) 1 0.18 0.17

( -1.56 , 1.92 ) * -0.004

CHF_USD C 0 0.00 0.08

( 0.00 , 0.00 ) * 0.013RDIFF30(-1) 1 -1.64 -1.84 * ( -3.11 , -0.17 )

0.009

EUR_USD C 0 0.00 0.44

( 0.00 , 0.01 ) * 0.001RDIFF30(-1) 1 -0.57 -0.42

( -2.78 , 1.65 ) * -0.006RDIFF30(-1) 1 -0.57 -0.42

( -2.78 , 1.65 ) * -0.006

GBP_USD C 0 0.00 -1.27

( -0.01 , 0.00 ) * 0.008RDIFF30(-1) 1 -1.49 -1.41

( -3.23 , 0.25 )

0.004

JPY_USD C 0 0.00 0.12

( 0.00 , 0.00 ) * 0.028RDIFF30(-1) 1 -1.37 -1.71 * ( -2.69 , -0.05 )

0.024RDIFF30(-1) 1 -1.37 -1.71 * ( -2.69 , -0.05 )

0.024

NZD_USD C 0 0.00 -1.51

( -0.01 , 0.00 ) * 0.028RDIFF30(-1) 1 -2.09 -2.71 *** ( -2.91 , -0.32 )

0.024

SEK_USD C 0 0.00 -0.59

( -0.01 , 0.00 ) * 0.000RDIFF30(-1) 1 0.00 0.00

( -1.30 , 1.30 ) * -0.004

NZD_AUD C 0 0.00 -0.61

( 0.00 , 0.00 ) * 0.001RDIFF30(-1) 1 -0.43 -0.34

( -2.53 , 1.67 ) * -0.004

40

Table 13: Adjusted Fama equation estimates: AR1 forecasts∆qt+1 = α + β1(rt − r∗t ) + β2(∆RAR1

t + (rt − r∗t )) + φ(qt−1 − qUIP,t−1) + εt+1

AR1_R theoretical Rsqvalue coef t-stat low high adjRsq

AUD_USD C 0 0.00 0.76

( 0.00 , 0.01 ) *

RDIFF30(-1) 1 -1.33 -1.07

( -3.37 , 0.72 )

ecm(-1) <0 -0.02 -1.15

( -0.05 , 0.01 )

0.057

dexpAR1 1 0.70 3.40 *** ( 0.36 , 1.04 ) * 0.046

CAD_USD C 0 0.00 0.04

( 0.00 , 0.00 ) *RDIFF30(-1) 1 -0.37 -0.35

( -2.11 , 1.37 ) *ecm(-1) <0 -0.02 -1.49

( -0.04 , 0.00 )

0.020dexpAR1 1 -0.30 -1.50

( -0.63 , 0.03 )

0.009

CHF_USD C 0 0.00 -0.21

( 0.00 , 0.00 ) *RDIFF30(-1) 1 -0.76 -0.86

( -2.23 , 0.70 )

ecm(-1) <0 -0.04 -2.29 ** ( -0.06 , -0.01 ) * 0.024dexpAR1 1 -0.09 -0.86

( -0.26 , 0.08 )

0.013

EUR_USD C 0 0.00 -0.45

( 0.00 , 0.00 ) *RDIFF30(-1) 1 -0.30 -0.22

( -3.00 , 1.17 ) *ecm(-1) <0 -0.02 -0.79

( -0.08 , 0.02 )

0.009dexpAR1 1 0.17 0.76

( -0.14 , 0.44 )

-0.011

GBP_USD C 0 0.00 -0.06

( 0.00 , 0.00 ) *RDIFF30(-1) 1 0.83 0.84

( -0.80 , 2.46 ) *ecm(-1) <0 -0.08 -2.87 *** ( -0.13 , -0.03 ) * 0.141dexpAR1 1 0.74 6.02 *** ( 0.54 , 0.95 )

0.130

JPY_USD C 0 0.00 -0.20

( 0.00 , 0.00 ) *RDIFF30(-1) 1 -0.68 -0.85

( -1.99 , 0.63 )

ecm(-1) <0 -0.04 -2.05 ** ( -0.06 , -0.01 ) * 0.028dexpAR1 1 0.05 1.13

( -0.02 , 0.13 )

0.016

NZD_USD C 0 0.00 -0.72

( -0.01 , 0.00 ) *RDIFF30(-1) 1 0.85 1.05

( -0.49 , 2.20 ) *ecm(-1) <0 -0.02 -1.38

( -0.04 , 0.00 )

0.019dexpAR1 1 0.39 1.72 * ( 0.01 , 0.76 )

0.008

SEK_USD C 0 0.00 0.06

( 0.00 , 0.00 ) *RDIFF30(-1) 1 1.13 1.46

( -0.15 , 2.41 ) *ecm(-1) <0 -0.03 -1.66 * ( -0.06 , 0.00 ) * 0.034dexpAR1 1 0.40 2.46 ** ( 0.13 , 0.67 )

0.023

NZD_AUD C 0 0.00 0.38

( 0.00 , 0.00 ) *RDIFF30(-1) 1 0.16 0.13

( -1.90 , 2.22 ) *ecm(-1) <0 -0.13 -3.36 *** ( -0.19 , -0.07 ) * 0.063dexpAR1 1 0.23 1.89 * ( 0.03 , 0.42 )

0.049

90% C.I.

Notes: Error correction term is residual from equation (A) in table 2

41

Table 14: Adjusted Fama equation estimates: AR1 forecasts∆qt+1 − (∆RAR1

t + (rt − r∗t )) = α + β1(rt − r∗t ) + φ(qt−1 − qUIP,t−1) + εt+1

AR1_L theoretical Rsqvalue coef t-stat low high adjRsq

AUD_USD C 0 0.00 0.82

( 0.00 , 0.01 ) *

RDIFF30(-1) 1 -1.15 -0.93

( -3.19 , 0.89 )

0.008

ecm(-1) <0 -0.02 -1.10

( -0.05 , 0.01 )

0.000

CAD_USD C 0 0.00 0.30

( 0.00 , 0.00 ) *

RDIFF30(-1) 1 0.23 0.20

( -1.64 , 2.10 ) * 0.019ecm(-1) <0 -0.03 -2.20 ** ( -0.05 , -0.01 ) * 0.011

CHF_USD C 0 0.00 -0.33

( 0.00 , 0.00 ) *RDIFF30(-1) 1 0.55 0.52

( -1.20 , 2.30 ) * 0.007ecm(-1) <0 -0.02 -1.21

( -0.06 , 0.01 )

-0.001

EUR_USD C 0 0.00 -0.19

( -0.01 , 0.00 ) *RDIFF30(-1) 1 0.12 0.09

( -2.19 , 2.44 ) * 0.002ecm(-1) <0 -0.02 -0.55

( -0.07 , 0.04 )

-0.012

GBP_USD C 0 0.00 -0.20

( 0.00 , 0.00 ) *RDIFF30(-1) 1 1.17 1.20

( -0.44 , 2.79 ) * 0.042ecm(-1) <0 -0.09 -3.09 *** ( -0.13 , -0.04 ) * 0.035

JPY_USD C 0 0.00 -0.61

( -0.01 , 0.00 ) *RDIFF30(-1) 1 -0.52 -0.41

( -2.61 , 1.57 ) * 0.003ecm(-1) <0 0.02 0.77

( -0.02 , 0.07 )

-0.005

NZD_USD C 0 0.00 -0.77

( -0.01 , 0.00 ) *RDIFF30(-1) 1 1.51 1.91 * ( 0.20 , 2.82 ) * 0.024ecm(-1) <0 -0.02 -1.54

( -0.04 , 0.00 )

0.016

SEK_USD C 0 0.00 0.40

( 0.00 , 0.01 ) *RDIFF30(-1) 1 1.28 1.62

( -0.02 , 2.59 ) * 0.037ecm(-1) <0 -0.05 -2.65 *** ( -0.08 , -0.02 ) * 0.029

NZD_AUD C 0 0.00 0.29

( 0.00 , 0.00 ) *RDIFF30(-1) 1 0.68 0.50

( -1.58 , 2.94 ) * 0.077ecm(-1) <0 -0.17 -4.02 *** ( -0.23 , -0.10 ) * 0.068

90% C.I.

Notes: Error correction term is residual from equation (A) in table 2

42

Table 15: Adjusted Fama equation estimates: 10Y swap forecasts∆qt+1 = α + β1(rt − r∗t ) + β2(∆R10s

t − (rt − r∗t )) + φ(qt−1 − qUIP,t−1) + εt+1

SW10 R theoretical Rsqvalue coef t-stat low high (adj Rsq)

AUD_USD C 0 0.00 0.76

( 0.00 , 0.01 ) *

RDIFF30(-1) 1 -1.81 -1.45

( -3.87 , 0.25 )

ecm(-1) <0 -0.05 -2.13 ** ( -0.08 , -0.01 ) * 0.040dexp SW10 1 0.18 1.93 * ( 0.02 , 0.33 )

0.028

CAD_USD C 0 0.00 0.10

( 0.00 , 0.00 ) *RDIFF30(-1) 1 -0.28 -0.27

( -2.01 , 1.44 ) *ecm(-1) <0 -0.03 71.54 * ( -0.05 , 0.00 ) * 0.028dexp SW10 1 -0.12 -1.67 * ( -0.23 , 0.00 )

0.017

CHF_USD C 0 0.00 -0.11

( 0.00 , 0.00 ) *RDIFF30(-1) 1 -0.11 -0.13

( -1.53 , 1.30 ) *ecm(-1) <0 -0.05 -2.71 *** ( -0.08 , -0.02 ) * 0.092dexp SW10 1 0.34 4.27 *** ( 0.21 , 0.47 )

0.081

EUR_USD C 0 0.00 -0.32

( -0.01 , 0.00 ) *RDIFF30(-1) 1 -0.48 -0.36

( -2.67 , 1.71 ) *ecm(-1) <0 -0.01 -0.36

( -0.06 , 0.04 )

0.047dexp SW10 1 0.34 2.63 *** ( 0.13 , 0.55 )

0.027

GBP_USD C 0 0.00 0.22

( 0.00 , 0.00 ) *RDIFF30(-1) 1 0.19 0.19

( -1.47 , 1.86 ) *ecm(-1) <0 -0.04 -1.88 * ( -0.07 , 0.00 ) * 0.084dexp SW10 1 0.32 4.68 *** ( 0.20 , 0.43 )

0.073

JPY_USD C 0 0.00 -0.04

( 0.00 , 0.00 ) *RDIFF30(-1) 1 0.58 0.72

( -0.76 , 1.92 ) *ecm(-1) <0 -0.06 -3.21 *** ( -0.09 , -0.03 ) * 0.100dexp SW10 1 0.30 4.27 *** ( 0.18 , 0.42 )

0.089

NZD_USD C 0 0.00 -0.33

( -0.01 , 0.00 ) *RDIFF30(-1) 1 0.64 0.70

( -0.88 , 2.17 ) *ecm(-1) <0 -0.03 -1.77 * ( -0.06 , 0.00 ) * 0.078dexp SW10 1 0.40 3.56 *** ( 0.21 , 0.58 )

0.063

SEK_USD C 0 0.00 -0.16

( 0.00 , 0.00 ) *RDIFF30(-1) 1 1.20 1.63

( -0.02 , 2.41 ) *ecm(-1) <0 -0.04 -2.04 ** ( -0.07 , -0.01 ) * 0.128dexp SW10 1 0.39 5.68 *** ( 0.28 , 0.50 )

0.118

NZD_AUD C 0 0.00 0.29

( 0.00 , 0.00 ) *RDIFF30(-1) 1 0.84 0.70

( -1.15 , 2.82 ) *ecm(-1) <0 -0.09 -3.03 *** ( -0.15 , -0.04 ) * 0.164dexp SW10 1 0.48 5.88 *** ( 0.35 , 0.62 )

0.151

90% C.I.

Notes: Error correction term is residual from equation (C) in table 3

43

Table 16: Adjusted Fama equation estimates: 10Y swap forecasts∆qt+1 − (∆R10s

t + (rt − r∗t )) = α + β1(rt − r∗t ) + φ(qt−1 − qUIP,t−1) + εt+1

SW10 L theoretical Rsqvalue coef t-stat low high adjRsq

AUD_USD C 0 0.00 0.76

( 0.00 , 0.01 ) *

RDIFF30(-1) 1 -1.31 -0.92

( -3.68 , 1.05 ) * 0.015

ecm(-1) <0 -0.05 -1.80 * ( -0.09 , 0.00 ) * 0.008

CAD_USD C 0 0.00 0.13

( 0.00 , 0.00 ) *

RDIFF30(-1) 1 0.01 0.13

( -2.26 , 2.64 ) * 0.022ecm(-1) <0 -0.04 -2.40 ** ( -0.08 , -0.01 ) * 0.015

CHF_USD C 0 0.00 0.11

( 0.00 , 0.00 ) *RDIFF30(-1) 1 0.01 0.89

( -0.73 , 2.43 ) * 0.030ecm(-1) <0 -0.05 -2.67 *** ( -0.09 , -0.02 ) * 0.022

EUR_USD C 0 0.00 0.06

( 0.00 , 0.00 ) *RDIFF30(-1) 1 -0.71 -0.49

( -3.08 , 1.66 ) * 0.004ecm(-1) <0 -0.02 -0.51

( -0.07 , 0.03 )

-0.010

GBP_USD C 0 0.00 -0.07

( 0.00 , 0.00 ) *RDIFF30(-1) 1 1.03 0.86

( -0.94 , 2.99 ) * 0.039ecm(-1) <0 -0.07 -3.02 *** ( -0.11 , -0.03 ) * 0.031

JPY_USD C 0 0.00 0.11

( 0.00 , 0.00 ) *RDIFF30(-1) 1 0.03 1.90 * ( 0.24 , 3.36 ) * 0.041ecm(-1) <0 -0.07 -3.15 *** ( -0.10 , -0.03 ) * 0.034

NZD_USD C 0 0.00 -0.47

( -0.01 , 0.00 ) *RDIFF30(-1) 1 1.57 1.61

( -0.04 , 3.17 ) * 0.028ecm(-1) <0 -0.03 -1.58

( -0.06 , 0.00 )

0.018

SEK_USD C 0 0.00 0.02

( 0.00 , 0.00 ) *RDIFF30(-1) 1 1.28 1.51

( -0.12 , 2.67 ) * 0.027ecm(-1) <0 -0.05 -2.30 ** ( -0.08 , -0.01 ) * 0.019

NZD_AUD C 0 0.00 0.20

( 0.00 , 0.00 ) *RDIFF30(-1) 1 1.56 1.19

( -0.60 , 3.73 ) * 0.090ecm(-1) <0 -0.14 -4.20 *** ( -0.19 , -0.08 ) * 0.081

90% C.I.

Notes: Error correction term is residual from equation (C) in table 3

44

Figure 7: Fama coefficients: nominal real and adjusted

3 coefficient

Estimated coefficient on 1-period interest differential (beta)

2

3

-

1

-1

-FAMA

nominalFAMA real

Eq11 (AR1)

Eq 12 (AR1)

Eq 11 (SW)

Eq 12 (SW)

-2

-1 nominal real (AR1) (AR1) (SW) (SW)

AUD CAD CHF-3

AUD CAD CHF

EUR GBP JPY

NZD SEK NZD_AUD-4

NZD SEK NZD_AUD

Notes: Estimates on interest differential from nominal and real FAMA equations and from adjusted Fama equations. All equations estimated in real terms except bars marked "FAMA nominal'.

45

exchange rates and interest arbitrage - university of...

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