gdp growth and currency valuation: the case of the dollar

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Journal of International Money and Finance 26 (2007) 1424e1449www.elsevier.com/locate/jimf

GDP growth and currency valuation: Thecase of the dollar

Peter J.G. Vlaar*

Research Department, De Nederlandsche Bank, P.O. Box 98, 1000 AB Amsterdam, The Netherlands

Abstract

One popular view on the strength of the US dollar around the turn of the century is that the highergrowth in the US compared to Europe had stimulated foreigners to buy American assets, thereby drivingup the exchange rate. In this paper a modified portfolio balance model is presented, in which it is shownthat the impact of output growth on the exchange rate depends crucially on the origin of this growth. Animprovement of the output gap is shown to actually depress the exchange rate whereas an increase in po-tential output growth leads to an appreciation, especially if this improvement is likely to be persistent. Inan empirical example, it is shown that the equilibrium real dollar rate is indeed positively affected by hightrend growth in the US, whereas it is negatively affected by a positive output gap. The model outperformsthe random walk in forecasting future real dollar rates one to eight quarters ahead.� 2007 Elsevier Ltd. All rights reserved.

JEL classification: C32; C61; E32; F31

Keywords: Rational expectations; Portfolio balance model; Taylor rule; Kalman filter; Foreign direct investment

1. Introduction

Many observers have been surprised by the continuous rise of the US dollar between 1995 and 2001despite the large deficit on the US current account. One popular explanation for this strength is the so-called prosperous economy view. According to this view the dollar has been appreciating because in-ternational investors preferred American assets in order to profit from the presumed higher growth

* Tel.: þ31 20 5243189; fax: þ31 20 5242529.

E-mail address: [email protected]

0261-5606/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jimonfin.2007.09.008

mailto:[email protected]

http://www.elsevier.com/locate/jimf

1425P.J.G. Vlaar / Journal of International Money and Finance 26 (2007) 1424e1449

potential in the US compared to Europe or Japan.1 The theoretical foundation for this view is not clear,however, as the higher growth potential of the US should already be reflected in the price of US assets(Sinn and Westermann, 2001). Moreover, the theory does not explain the ever increasing willingnessof foreigners to hold American assets, necessary to finance the deficit on the US current account.

In this paper an attempt is made to theoretically justify the presumed link between outputgrowth and real appreciation. For this purpose a modified portfolio-balance rational-expectationsexchange rate model is developed. As usual, any surplus or deficit on the current account has to bebalanced by the capital account. The capital account contains two elements in our model: net for-eign direct investment and net portfolio investment. All of these elements are affected by devel-opments in output growth, the extent to which depends crucially, however, on the composition ofthis growth. In the theoretical model a distinction is made between changes in the output gap on theone hand and changes in potential output on the other. With respect to the latter two different kindsof shocks are modeled: one-time shocks and persistent trend growth shocks. The current account isassumed to depend on the real exchange rate and the output gap, net direct investment dependscrucially on trend growth, whereas desired net portfolio investment depends on expected devia-tions from uncovered interest parity. After calibrating the model for the US economy, the modelis solved using model-consistent forward-looking expectations for the exchange rate. It turns outthat positive demand shocks e that affect the output gap, but not potential output e result in a de-preciation of the dollar, whereas supply shocks, especially persistent ones, lead to an appreciation.

The theoretical model can thus offer two explanations for the persistent strength of the dollardespite the large deficit on the current account. First, the continuous outflow of dollars due to thedeficit on the current account is partly compensated by the inflow due to net foreign direct in-vestment. Second, as the output gap and trend growth are not observable, the prolonged periodof high growth without much inflationary pressures might have resulted in a gradual update onthe likely composition of output growth.2 Indeed, as the prosperous period of high growth pro-longed, more and more people started to believe in a New Economy in which the output gapwould be less pronounced and potential output growth would be permanently higher than before.Both the lower presumed output gap and the higher trend growth positively affected the dollar.

In an empirical application on the real dollar rate, it is shown that the equilibrium real valueof the dollar is indeed positively affected by high trend growth in the US, whereas a positiveoutput gap has a negative impact. The model outperforms the random walk in forecastingreal exchange rate changes one to eight quarters ahead. The explanatory power of the relation-ship is relatively weak, however. The real exchange rate may occasionally deviate from its pre-dicted value by more than 20%. At the end of 2004 the value of the US dollar seemed to beclose to equilibrium.

The rest of the paper is structured as follows. Section 2 provides some background on recentdevelopments, primarily in the US. In Section 3 the theoretical model will be presented. Themodel will be calibrated for the US and the results are shown by means of impulse responsefunctions. In Section 4 the predictions of the theoretical model will be confronted with

1 For the euro dollar rate, Corsetti and Pesenti (1999) were among the first to show the influence of growth differen-

tials. They conclude that cyclical divergence seems to be the root cause for the slide of the euro, but do not present

a formal theory explaining this phenomenon. Previously, Tatom (1987, 1995) has shown the importance of productivity

differentials in explaining exchange rate changes in the 1980s.2 This argument has previously been used by Brunner et al. (1980). They show that if agents cannot observe the per-

sistence of shocks as they occur, a permanent reduction in productivity leads to stagflation and persistent unemployment

even if all expectations are rational and all markets clear.

1426 P.J.G. Vlaar / Journal of International Money and Finance 26 (2007) 1424e1449

realizations of the real dollar rate. Section 5 concludes. Appendix A describes the state spacerepresentation of the theoretical model, and Appendix B provides the data sources.

2. Stylized facts

Fig. 1 shows the annual real growth in the US (up until 1991 West-), Germany and Japan. In his-torical perspective, the real growth rates in the US during the 1990s were not exceptional, even forUS standards. Higher US growth rates were occasionally achieved both during the 1960s, 1970s andthe 1980s. What was exceptional, though, was the length of the prosperous period. Moreover, at thesame time, Germany was experiencing an exceptionally long period of slow growth in the aftermathof the German unification. For Japan, economic conditions during the 1990s were even worse.Consequently, for a prolonged period the US outperformed both Germany and Japan, in terms ofgrowth. Also during the mid 1980s, growth in the US was substantially higher than abroad, espe-cially in 1984. As shown in Fig. 2, during this period the dollar also seemed highly overvalued.Therefore, there seems to be some empirical support for the prosperous economy view.

Fig. 3 sheds some more light on this issue. It shows that the bulk of the large deficits on the cur-rent account resulting from an overvalued dollar is financed by portfolio investment. However,within this category, net portfolio investment in equity is relatively unimportant. For instance, dur-ing much of the 1990s the net capital flow related to equity was actually negative for the US, mean-ing that US investors bought more foreign stocks than foreigners bought US ones. Foreign directinvestment seems more directly related to growth differentials, although with a lag.

3. The theoretical model

As a starting point for our theoretical model the dynamics of output, inflation and interest ratesare modeled similar to the closed economy models of Gerlach and Smets (1997), Rudebuschand Svensson (1999) and Smets (1999), supplemented by some open economy features:

yrthyp

t þ ht; ð1Þ

ypt ¼ yp

t�1þ gt þ 3supt ; ð2Þ

-3

0

3

6

9

12

15

1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004

US Germany Japan

Fig. 1. Annual real growth US, Germany and Japan (%).


gt ¼ a0 þ a1gt�1þ 3gt ; ð3Þ

ht ¼ f0þf1ht�1þf2ht�2 þfiðit�1�pt�1Þ þfsup

�3sup

t þ 3gt

�þ 3d

t ; ð4Þ

pt ¼ b0þ b1pt�1þ b2pt�2 þ b3pt�3þ b4pt�4þ bhht þ bDsDst þ 3pt ; ð5Þ

Dit ¼ gDhDht þ gDpDpt þ giðit�1 �g0� gh1ht�1� gp1pt�1Þ þ 3it; ð6Þ

where ytr is log real GDP; yt

p is log potential output; ht is the output gap; gt is trend growth ofpotential output; it is the quarterly average nominal short term interest rate; pt is quarterly CPIinflation (not annualized), pth

P3i¼0 pt�i is annual inflation; st is the quarterly average log

nominal exchange rate expressed as the domestic currency value of foreign currency; D denotesthe first difference operator Dxt h xt� xt�1.

Output is decomposed in potential output and the output gap (Eq. (1)). With respect to potentialoutput the main difference between our model and the one in Smets (1999) is that the trend growth of

70

80

90

100

110

120

130

1970 1974 1978 1982 1986 1990 1994 1998 2002

Fig. 2. CPI-based real dollar rate.

-5

-4

-3

-2

-1

0

1

2

3

4

5

1970 1974 1978 1982 1986 1990 1994 1998 2002

Current account Direct investment Portfolio investment Equity investment

Fig. 3. US balance of payment flows (% of GDP).


potential output changes over time. The trend growth process (Eq. (3)) is supposed to be stationary(a1< 1) as there is ample evidence of output growth being stationary.3 Eq. (4) describes the dynamicsof the output gap as being related to its lagged values and the lagged real interest rate, demand shocks(3t

d), and one-time or persistent supply shocks (3tsup, respectively, 3t

g).4 The influence of supply shockson the output gap seems natural as the output gap is defined as the difference between demand andsupply. If fsup is restricted to zero, any shock to supply has to result in an equal rise in demand. More-over, supply shocks would not have any influence on inflation in that case. Eq. (5) can be interpreted asa Phillips-curve which relates inflation to the output gap and to lags in inflation. Moreover, as thissystem is part of an open economy model, current depreciation is also supposed to increase inflation.Finally, Eq. (6) represents the familiar Taylor rule (Taylor, 1993), expressed in error correction form.

Our exchange rate model follows the tradition of the portfolio balance approach (Bransonet al., 1977) in the sense that domestic and foreign bonds are viewed as imperfect substitutes,and in that the equilibrium on the balance of payments is given a central role. We distinguishthree main elements on the balance of payments: the trade balance, net foreign direct invest-ment and portfolio investment. The exchange rate related equations are the following:

FA�t ¼ duip

�ift � it

4þEtDstþ1

�þ nfa

t with nfat ¼ d1nfa

t�1þ 3fat ; ð7Þ

FDIt ¼ c0þ cg

1

4

X5

i¼2

gt�iþ 3fdit ; ð8Þ

DTBt ¼ ttb

�TBt�1 � t0� tqqt�1

�þ tDhDht þ 3tb

t ; ð9Þ

FArevtjt�1 ¼

�FAass

t�1

St

St�1

�1þ if

t�1

4

�� FAlia

t�1

�1þ it�1

4� m3i3

it

��GDPt

GDPt�1

þ 3revt ; ð10Þ

FAt ¼ FArevtjt�1þTBt þ FDIt; ð11Þ

FAt ¼ FA�t ; ð12ÞDsthpt �pf

t �Dqt; ð13Þwhere FAt

* and FAt denote, respectively, the desired and actual stock of net foreign portfolioassets, that is to say foreign assets (FAt

ass) minus foreign liabilities (FAtlia), both measured in

dollars and scaled by nominal US GDP (GDPt); FAtjt�1rev denotes the revalued scaled dollar

value, including interest or dividend payments, of the time t� 1 stock of net foreign assets;FDIt and TBt represent, respectively, the US net flow of foreign direct investment and theUS trade balance, both measured in dollars and scaled by nominal GDP; qt is the log CPI-basedreal exchange rate; it

f and ptf denote foreign short term interest rate and quarterly inflation rate,

respectively; Et represents the expectations operator, conditional on time t information.The desired stock of foreign assets is usually related to the stock of wealth in a portfolio

balance model. For the US demand of foreign assets, US wealth would probably indeed bethe theoretically preferred scaling variable. However, for the foreign demand of US assets,the net flow of foreign direct investment or the US trade balance, US GDP seems more appro-priate. Moreover, using wealth would be problematic when calibrating the model as wealth isnot easily observable. The choice of the scaling variable is not essential in our model.

3 Gerlach and Smets (1997) model a random walk process for trend growth.4 The impact of the real exchange rate was also investigated, but was found to be insignificant.


The desired amount of net foreign assets (Eq. (7)) e which comprises the US demand for for-eign assets minus foreign demand for US assets e is assumed only to depend systematically onexpected deviations from uncovered interest parity (UIP). According to the prosperous economyview the strength of the dollar is primarily due to the better growth potential for the US than forEurope or Japan. Consequently, international investors want to buy more US assets, thereby driv-ing up the exchange rate. This would mean a direct influence of growth potential on foreign assetdemand. However, the presumed higher growth potential of the US would not only trigger for-eigners to buy US stocks, but US investors as well (Sinn and Westermann, 2001). Therefore,the higher profit potential should already be priced in the US stocks, unless other US investmentopportunities, including bond market investments, show a similar improvement. A higher demandfor capital (to finance new profitable investments) should indeed also lead to higher interest rates inthe high productivity country (Tatom, 1987). This way, symmetric preferences at home and abroaddo not prevent productivity differentials to have an impact on the exchange rate. This argument,however, leads to the conclusion that for portfolio investment, increased growth potential is pri-marily important because of its influence on interest rates or future exchange rates. The factthat the UIP relationship is central to the portfolio investment decision does not mean, however,that all portfolio investments are done in one period bonds. It merely states that the relative attrac-tiveness for foreign assets depends on the interest rate differential and expected exchange rate de-velopments. In order to model persistent deviations of desired asset values from UIP-implied onesthe disturbance term (nt

fa) in Eq. (7) is given an AR(1) structure.5

In our model, net foreign demand for US assets is not primarily triggered by higher interest ratesdue to higher investment, but by the expected appreciation due to foreign direct investment flows.The argument that higher growth potential as such does not make equity investment more inter-esting for foreigners than for domestic investors is less convincing when foreign direct investmentis concerned. For foreign direct investment not only the profit opportunities of the acquired firmare important, but also the extent to which the investment improves the performance of the acquir-ing firm. Synergy effects and, for instance, improved marketing and distribution channels due to anacquisition are likely to be higher if the acquired firm is located in a fast growing economy. There-fore, it is assumed that net foreign direct investment is a function of the trend growth of the econ-omy (Eq. (8)). As direct investment is a long term decision, the e relatively short lived e outputgap is not considered a relevant variable. The two to five quarters’ lag in the influence of trendgrowth seems reasonable given information and decision lags. Apart from trend growth, other var-iables might have been included in the specification as well. One obvious candidate would be thereal exchange rate as a highly overvalued dollar would make foreign direct investment to the USmore expensive. On the other hand in as far as foreign direct investment is complementary to for-eign trade, an overvalued dollar stimulates FDI if the overvaluation is likely to last. Campa andGoldberg (1999) reveal that the real exchange rate does have little or no effect on direct investmentin high mark-up sectors which absorb much of the actual exchange rate changes in their mark-ups,whereas the reverse holds for low mark-up industries. We found the influence of the real exchangerate on FDI to be highly instable, with a significant positive effects during some periods, and a sig-nificant negative impact on others. Also the effects of interest rates, see Stokman and Vlaar (1996),primarily representing intra-firm capital transfers, turned out to be unstable.

5 Apart from financial considerations, net foreign asset positions might also depend on real factors. Lane and Milesi-

Ferretti (2002) find a significant impact of relative output levels, the stock of public debt and demographic factors. The

fit of their model for the US is rather poor, however.


Eq. (9) describes the trade balance as being determined in the long run by the real exchange rateand in the short run by changes in the output gap. The real exchange rate only starts to affect importsand exports with a lag as trade volumes are only adjusted gradually, the so-called J-curve effect. Theoutput gap is included to account for the increased import demand in economic upswings.6 The cap-ital income part of the current account is directly related to the lagged stocks of foreign assets. Thispart is modeled together with the revaluation of net foreign assets (Eq. (10)). The time t scaled dollarvalue of t� 1 net foreign assets equals the time t� 1 stock of foreign assets of US investors, subjectto possible dollar depreciations and quarterly interest rate payments, minus the time t� 1 US lia-bilities, subject to interest payments and possible revaluation due to unexpected interest ratechanges.7 Both US foreign assets and liabilities are moreover adjusted for nominal GDP changesas this is the denominator of the fraction. The revaluation effect of unexpected interest rate changesis included as investors do not only buy short term foreign bonds, but also long term bonds or stocks.Foreign variables are not supposed to be subject to unexpected changes.

The last three equations are identities. Eq. (11) represents the equilibrium on the balance ofpayments, Eq. (12) states that desired foreign asset demands are always fulfilled, and Eq. (13)relates nominal to real depreciations.

The required equality of desired and actual foreign asset stocks is the main driving force for theexchange rate in the model. Given the expected future exchange rate, the solution for the currentexchange rate is obvious. Following Dornbusch (1976) we will assume that expectations areformed fully rationally e that is to say model-consistent e and that the system will evolve towardsa long-run steady state in which the real exchange rate reaches an equilibrium level that is consistentwith current account balance. In the portfolio balance literature the concept of model-consistentexpectations is hardly ever pursued. An exception is Dooley and Isard (1982), who use an iterativeprocedure to derive model-consistent one month ahead predictions. Also in their model, expecta-tions for two or more months ahead are not rational, however. Our model is solved by means of themethod of undetermined coefficients (Blanchard and Kahn, 1980). See Appendix A for details.

3.1. Calibration

In order to find the parameters for the model, we follow Kuttner (1994), Gerlach and Smets (1997)and Smets (1999) in using the unobserved components (UC) technique, see Harvey (1989). The mainreason to estimate potential output and output gap equations as opposed to relying on published seriesis that it is easier to match our model characteristics. The potential output series published by the Con-gressional Budget Office, for instance, is very smooth and contains a double unit root. Therefore, itwould violate the steady state assumption for output growth, and one-time supply shocks would beabsent. Moreover, by applying the one-sided Kalman filter to estimate the unobserved state variables,one can be sure that indeed no future information is used in calculating these variables.8

6 According to the intertemporal theory of the current account, see Sachs (1981) and Obstfeld (1986), temporarily

high growth rates associated with the positive output gap should conversely lead to higher savings, and therefore a cur-

rent account surplus. The empirical evidence on this issue is ambiguous. Backus and Kehoe (1992) find net export po-

sitions to be countercyclical. Vredin and Warne (1991) on the other hand find the Swedish current account to be

positively correlated with the transitory component of GDP.7 In this formula it is implicitly assumed that the average quarterly return on all foreign assets equals the interest rate.

Under the assumption that domestic and foreign equity risk premia are equal, this simplifying assumption has no con-

sequences for the solution of the model.8 In the UC methodology, the state variables refer to the unobserved components. This should not be confused

with the endogenous state variables referred to in Appendix A, as there e in the tradition of the real business cycle

literature e state variables stand for the endogenous variables that drive the dynamics of the system.


Some small modifications had to be made relative to the theoretical model in order to betteraccount for the empirical regularities. Both output, inflation, the trade balance and net foreigndirect investment turned out not to be free of seasonal patterns. For direct investment this wasremedied by including seasonal dummies, whereas the seasonal pattern of the other series wasmodeled by means of three extra state variables (Harvey, 1989) as their seasonal patterns werenot stable over time. The inflation equation was extended to include the supply factors oil andimport prices. These factors pick up systematic changes in inflation that are not due to the out-put gap. Real oil price changes were also included in the supply shock equation to account forthe loss of economic capacity related to higher real energy prices, see Rasche and Tatom (1977,1981).9 The estimated system has the following state space representation:

26666664

Dyrt

pt

Dit

DTBt

FDItþ2

37777775¼

26666664

1 �1 1 0 0 0 1 1�1 0 0 0 0 0 0 0

bh 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

gDh�gDh�gigh1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

tDh �tDh 0 0 0 0 0 0 0 0 0 0 0 1�1 0

0 0cg

4

cg

4

cg

4

cg

40 0 0 0 0 0 0 0 0 0

37777775

26666666666666666666666666666666664

ht

ht�1

gt

gt�1

gt�2

gt�3

ft

s1yt

s2yt

s3yt

s1pt

s2pt

s3pt

s1tbt

s2tbt

s3tbt

37777777777777777777777777777777775

þ

26666664

0

b0þb1pt�1þb2pt�2þb3pt�3þb4pt�4þboilDpoilt þbimp1Dpimp

t�1þbimp2Dpimpt�2

gDpDptþgiðit�1�g0�gp1pt�1Þttb

�TBt�1�t0�tqqt�1

�c0þcs1s1þcs2s2þcs3s3

37777775

þ

26666664

0

3pt

3it

3tbt

3fdit

37777775; ð14Þ

9 The impact of oil prices on the output gap was also investigated, but was found not to be significant.


26666666666666666666666666666666664

ht

ht�1

gt

gt�1

gt�2

gt�3

ft

s1yt

s2yt

s3yt

s1pt

s2pt

s3pt

s1tbt

s2tbt

s3tbt

37777777777777777777777777777777775

¼

26666666666666666666666666666666664

f1 f2 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 a1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 �1 �1 �1 0 0 0 0 0 0

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 �1 �1 �1 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 �1 �1 �1

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

37777777777777777777777777777777775

26666666666666666666666666666666664

ht�1

ht�2

gt�1

gt�2

gt�3

gt�4

ft�1

s1yt�1

s2yt�1

s3yt�1

s1pt�1

s2pt�1

s3pt�1

s1tbt�1

s2tbt�1

s3tbt�1

37777777777777777777777777777777775

þ

26666666666666666666666666666666664

fiðit�1�pt�1Þ0

a0

0

0

0

droil

�poil

t �poilt�4�pt

�0

0

0

0

0

0

0

0

0

37777777777777777777777777777777775

þ

26666666666666666666666666666666664

1 fsup fsup 0 0 0

0 0 0 0 0 0

0 1 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 1 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

0 0 0 0 0 0

37777777777777777777777777777777775

2666666664

3dt

3gt

3supt

3syt

3spt

3stbt

3777777775; ð15Þ

where ft represents the one-time supply shocks, including the real impact of oil price shocks;s1t

x, s2tx and s3t

x are state variables representing the seasonal pattern of variable x; ptoil and pt

imp

represent log oil and import prices, respectively; s1, s2 and s3 are demeaned seasonal dummies;the real interest rate is demeaned in order to derive a zero-mean output gap.

Eq. (14) is the measurement or observation equation, which states how the vector of ob-served variables is related to the vector of unobserved components. Eq. (15) is the transition


equation, which specifies the time-series process that governs the evolution of the unobservablevariables. The disturbances in both the measurement and the transition equations are assumedto be uncorrelated.

In order to be optimally able to identify the output gap and changes in trend output, the max-imal available data set is used to estimate the parameters as preferably several cycles should beincluded. For output, prices and interest rates this means data for 1957:I until 2004:III, whereastrade balance and direct investment data start in 1973. Apart from the availability of data, it isimportant to account for changes in behavior over time. With respect to interest rates, the as-sumption that the same reaction function was valid for four decades is probably unwarranted.Given the extremely low real interest rates and high inflation in the 1970s we restrict the inter-est rate equation to the period starting from 1980. Table 1 gives the estimated maximum likeli-hood parameters.10

The estimate of fsup turned out to be very imprecise and quite sensitive for slight changes inthe model. For none of the specifications a value of �0.5 could be rejected. A higher real in-terest rate indeed leads to a lower output gap although the parameter is only borderline signif-icant. Fig. 4 shows the resulting output gap (black line), using only current and lagged data,together with a �2 standard deviation confidence band. Also, the output gap according tothe Congressional Budget Office (CBO) is shown (grey line) as a reference value. Althoughthe overall pattern of the two series is similar in terms of turning points, substantial differenceswith respect to the level of the gap are also apparent. Especially during the high inflation periodfrom the mid 1970s to the beginning of the 1980s, the UC method predicts a significantlyhigher output gap. During the second half of the 1960s and from 1997 until 2002 on the otherhand the CBO estimate is higher. These results are comparable to Smets (1999).

Regarding potential output, our estimates indicate that the persistence of growth shocks isvery significant, though fairly limited in size as the half-time of growth shocks is about sixquarters. Moreover, one-time shocks to potential output seem to dominate persistent shocksas ssup is about 11 times larger than sg. Moreover, the one-time shocks are also significantlyaffected by real oil price changes. These results contrast with the Congressional Budget Officeestimate of potential output, which is smooth and seems integrated of order two. The UC andCBO trend growth series are both plotted in Fig. 5. The UC estimate of potential growth (blackline) is more volatile. Moreover, the two trend growth measures sometimes deviate consider-ably for years, although hardly ever significantly so.

The coefficients for the inflation equations all have the expected sign and most are very sig-nificant. With respect to the exchange rate effect on inflation (bDs in the theoretical model), theestimated equation does not give direct guidance. Exchange rate effects pass through via the oiland imports prices. Due to pricing to market, however, the relationship between the two is prob-ably not one to one. Based on these results we calibrate bDs¼ 0.06 in the baseline model, andinvestigate the results for bDs¼ 0 as well.

The direct inflation effect on the interest rate turned out to be negative, though not significantlyso, and was restricted to zero. In the long run the inflation effect is significantly positive, althoughthe estimated coefficient (1.16) is somewhat smaller than the standard value (1.5). Equivalence ofthe short- and long-run inflation effects had to be strongly rejected. Therefore this restriction,which is most common in the literature, was not imposed. For the output gap on the otherhand, the direct interest rate effect was very similar to the long-run impact and similarity was

10 The model is estimated in Gauss, with the use of the Kalman filter procedure written by Paul Soderlind. This pro-

cedure follows the syntax of Harvey (1989).

Table 1

State space

droil sd2 sg

2 ssup2 ssy

2

ytr �0.007 0.40 0.004 0.49 0.001

(3.0) (4.1) (0.8) (3.0) (1.0)

boil bimp1 bimp2 sp2 ssp

2

pt 0.013 0.043 0.047 0.065 0.0002

(7.6) (3.3) (3.5) (7.4) (1.4)

si2

it 0.27

(1.4)

TBt

FDIt

Notes: Effe ation 1980:Ie2004:III, for the trade balance 1973:IIe2004:II and for

foreign dir

14

34

P.J.G

.V

laar

/Journal

ofInternational

Money

andF

inance26

(2007)1424e

1449

results US

f1 f2 fi fsup a1 a0

1.29 �0.36 �0.057 �0.5 0.88 0.09

(10.9) (2.8) (1.7) (e) (17.0) (2.1)

bh b0 b1 b2 b3 b4

0.11 0.59 0.21 �0.24 0.26 0.13

(3.2) (3.4) (2.6) (3.4) (3.4) (1.9)

gDh gDp gi g0 gh1 gp1

1.22 0 �0.12 2.94 1.22 1.16

(4.1) (e) (2.1) (1.1) (4.1) (1.9)

tDh ttb t0 tq stb2 sstb

2

�0.14 �0.053 �3.36 �0.21 0.074 0.0013

(2.8) (2.4) (5.1) (2.7) (6.2) (2.1)

c0 cg cs1 cs2 cs3 sfdi2

�3.54 4.32 �0.21 0.33 �0.33 0.50

(1.6) (1.7) (1.2) (1.6) (1.8) (3.4)

ctive sample for the output and inflation equations 1958:IIe2004:III, for the interest rate equ

ect investment 1973:Ie2004:II. Absolute t-values in parenthesis.


therefore imposed. Here, the coefficient (1.22) is considerably bigger than what is usuallyimposed (0.5). The sensitivity with respect to these parameters will be investigated later on.

With respect to the impact of growth on the trade balance and the foreign direct investment,we do indeed find a significant impact of the output gap on the former, whereas the latter isaffected by trend growth. Besides the output gap, the trade balance is significantly affectedby the real exchange rate.

The remaining parameters to be calibrated are duip, d1, m3i and mfa. These parameters have to dowith the (composition of the) stock of foreign assets. Unfortunately, data on these matters arescarce and probably not very reliable. Fig. 6 shows the stock of US assets and liabilities as reportedin the international investment position of the IFS. We include both portfolio and other investmentstocks as the latter form a non-negligible part of the balance of payments. The other investmentsare dominated by banks, for which portfolio considerations might also play a significant role. Thestock of assets and liabilities as a percentage of GDP hardly converges to a steady state as it is con-stantly increasing over time. Moreover, it seems that the stock of liabilities is growing faster thanthe stock of assets. Based on these results, the ‘steady state’ value for the stock of assets and lia-bilities (mfa) was calibrated at 0.5 as current values are most representative. With respect to thecomposition of foreign assets nothing much can be said. Equity comprises a non-negligiblepart of foreign assets (about one-third), but nothing is known about the maturity of bonds. Theinterest rate shock elasticity of foreign assets (m3i) is calibrated at 1.

As can be seen from Fig. 6, the net foreign assets of the US display a clear negative trend. Un-fortunately, the impact of interest rate differentials or future depreciations could hardly be detected.When a time trend was included in the regression, the estimate of the interest rate effect became assmall as 0.36 and not significant at all. When the trend was not included or when the ex-post devi-ation from uncovered interest parity was modeled, no positive sign was detected.11 Therefore, duip

was calibrated at 0.36 in the standard scenario, whereas an alternative was investigated withduip¼ 100. In order to model the very persistent unexplained residual, d1 was given the value 0.99.12

-10

-5

0

5

10

1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004

Fig. 4. US output gap according to UC method and Congressional Budget Office.

11 The results were slightly worse if the net other investment positions were excluded.12 Formally, d1 needs to be smaller than one in order for the steady state to exist.


3.2. Impulse response analysis

Fig. 7 gives the responses of a 1% demand shock on several key variables. The output gapinitially improves before declining after the second quarter. It takes almost four years for theoutput gap to reach zero again. The interest rate immediately reacts aggressively to the demandshock, on a more than proportional basis. Consequently, the impact on inflation is relativelysmall, although it should be realized that the inflation rate in the model is not annualized. Giventhe firm rise in interest rates it is surprising that the real exchange rate is depreciating in firstinstance. The expected future depreciations more than compensate for the higher interest rate,leading to a net increase in foreign assets held by US investors. The expected depreciationmight partly be explained by the trade deficit. The real exchange rate reaches its lowest valueof �1.8% after almost two years. In nominal terms (not shown here) the maximum depreciation

0.4

0.6

0.8

1

1.2

1.4

1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000 2004

Fig. 5. US growth potential according to UC method and Congressional Budget Office.

0

10

20

30

40

50

60

70

80

1980 1983 1986 1989 1992 1995 1998 2001

Portfolio assets Portfolio liabilities Equity assets Equity liabilities

Fig. 6. US foreign investment position (% of GDP).


is about 3.1%, also reached after two years. The negative impact of the output gap on the ex-change rate runs counter to the arguments of Corsetti and Pesenti (1999) that cyclical differ-ences were to blame for the slide of the euro at the end of the 1990s.

The responses of a 1% trend growth shock are given in Fig. 8. This shock results in a substantialreal appreciation, about 31% after two and a half years (35% in nominal terms). This is primarily dueto a substantial increase of net foreign direct investment. Also the demand of foreigners for US port-folio assets exceeds the US demand for foreign assets as the net foreign asset position of the US de-clines. Again, the expected appreciation effect more than compensates for the somewhat lowerinterest rate. Inflation declines, both due to the appreciation and the negative output gap. In the longerrun, the real appreciation deteriorates the trade balance (after five years the deficit is about 2.8% ofGDP), which in turn leads to an outflow of foreign assets and, in as far as this is not desired, to a grad-ual depreciation. These results confirm the findings by Tatom (1987, 1995) that productivity in-creases boost the exchange rate. The main reason for this effect in our model (FDI) differs fromthe one in Tatom (higher interest rates due to higher capital demand), however.

10 20 30 40

1.51

0.5

0.51

1.5

real exchange rate

10 20 30 40

0.10.05

0.050.1

foreign assets

10 20 30 40

0.150.1

0.05

0.050.1

0.15

trade

10 20 30 40

0.20.40.60.8

11.2

output gap

10 20 30 40

0.050.1

0.150.2

inflation

10 20 30 40

0.250.5

0.751

1.251.5

interest rate

Fig. 7. Impact of a demand shock.

10 20 30 40

2010

102030

real exchange rate

10 20 30 40

21.5

10.5

0.51

1.5

foreign assets

10 20 30 40

0.51

1.52

2.53

3.5

direct investment

10 20 30 40

0.60.40.2

0.20.4

output gap

10 20 30 40

0.40.2

0.20.4

inflation

10 20 30 40

1.51

0.5

0.51

1.5

interest rate

Fig. 8. Impact of a trend growth shock.


The impact of a one-time supply shock is not shown as the only impact of this shock is viathe negative impact on the output gap. Consequently, the impulse responses are equivalent tothe ones for a demand shock, except for the sign and the size.

Fig. 9 provides the impulse responses of an inflation shock. As the interest rate only re-sponds to inflation with a lag, real interest rates decline and the output gap improves. Neverthe-less, the inflation rate quickly returns to low level. The real exchange rate initially rises onlyslightly less than inflation leaving the nominal exchange rate almost unchanged. In the longerrun the nominal exchange rate depreciates up to 2.8% after four years.

The responses to an interest rate shock are shown in Fig. 10. An interest rate shock leads toan immediate appreciation of the dollar. This appreciation has two causes. Firstly, absence ofany exchange rate effects the higher interest rate would induce both US citizens and foreignersto hold more dollar assets, thereby reducing US net foreign assets. On top of that, the interestrate rise depreciates the current stock of US assets held by foreigners. As the trade balance andforeign direct investment are hardly affected by the interest rate shock, the exchange rate has toappreciate initially in order to attain desired assets holdings. On the one hand this will revalue

10 20 30 400.5

0.5

1

real exchange rate

10 20 30 40

0.040.02

0.020.04

foreign assets

10 20 30 40

0.060.040.02

0.020.040.06

trade

10 20 30 400.1

0.05

0.050.1

0.150.2

output gap

10 20 30 40

0.20.40.60.8

1inflation

10 20 30 40

0.20.40.60.8

interest rate

Fig. 9. Impact of an inflation shock.

10 20 30 400.2

0.20.40.60.8

10 20 30 40

0.00250.002

0.00150.001

0.0005

0.0005

10 20 30 400.001

0.0010.0020.0030.004

trade

10 20 30 40

0.20.15

0.10.05

output gap

10 20 30 40

0.060.050.040.030.020.01

0.01

inflation

10 20 30 40

0.20.40.60.8

1

real exchange rate foreign assets

interest rate

Fig. 10. Impact of an interest rate shock.


the current stock of US assets held by foreigners, whereas on the other hand the higher initiallevel induces future depreciation expectations that reduce the desired US asset holdings.

The final scenario shown is that of a desired foreign assets shock. As shown in Fig. 6, foreignasset positions can deviate substantially and persistently from UIP-induced values. The gap be-tween net foreign assets and liabilities can be as much as 25% of GDP, whereas our model as-sumes equality in steady state. A 1% of GDP increase in net foreign asset demand leads to animmediate real depreciation of about 1.8%, see Fig. 11. This depreciation leads to an improve-ment of the trade balance, thereby accumulating foreign assets over time. Consequently, despitethe fact that the increased extra demand for foreign assets is very persistent (d1¼ 0.99), the realexchange rate returns to zero in three years.

To summarize, the model can explain fluctuations in real exchange rates, though the size ofthe predicted movements is relatively small compared to observed fluctuations (Fig. 2). A 1%shock in trend growth ultimately leads to a 31% real appreciation. However, a 1% change inquarterly trend growth is highly unrealistic as the standard deviation of these shocks for theUS is only 0.062%. Although larger effects might be caused by subsequent shocks with thesame sign in the US (for instance between 2001 and 2003, see Fig. 5) or shocks in the rest ofthe world of opposite sign, one can hardly justify real exchange rate changes of more than,say, 10% by these shocks.13 The impact of demand shocks is probably limited to at most 5%,given the standard deviation for the US of 0.64%. Inflation or interest rate shocks can explaineven less. The desired foreign asset shocks can explain some deviation, but in order to explainsustained periods of overvaluation, a growing foreign demand for US assets seems necessary.

3.3. Sensitivity analysis

Not all the parameters in the model are without doubt. In this section some alternativescenarios will be presented. Thereby, we will concentrate on the real exchange rate. The firstscenario considered is the one in which the assets of different countries are almost perfect

10 20 30 40

1.51

0.5

0.51

real exchange rate

10 20 30 40

0.750.8

0.850.9

0.95

foreign assets

10 20 30 400.05

0.025

0.0250.05

0.0750.1

trade

10 20 30 40

0.020.01

0.010.02

output gap

10 20 30 400.02

0.020.040.060.08

0.10.12

inflation

10 20 30 40

0.0750.05

0.025

0.0250.05

0.075

interest rate

Fig. 11. Impact of a shock to the desired foreign assets.

13 Larger effects can probably be generated by distinguishing tradables from non-tradables. If productivity changes are

concentrated in the tradables sector, the real exchange rate is permanently affected by supply shocks. This is the well

known BalassaeSamuelson effect (see Balassa, 1964 and Samuelson, 1964).


substitutes. In the model, this is reflected in a calibrated value for duip of 100. Fig. 12 shows theimpact of a 1% demand, trend growth, and interest rate shock on the real exchange rate. Sur-prisingly enough, the real exchange rate response to an interest rate shock is not very muchaffected by the abandonment of this crucial assumption of the portfolio balance model.

The impact of a demand shock on the other hand changes radically under perfect substitu-tion. The real exchange rate now appreciates initially 1.4% after which it gradually depreciatesto zero in three years. The initial appreciation is due to the fact that the relative demand for USassets increases in response to the implied interest rate rise in the US. The outflow of funds dueto the trade balance deficit is less important under perfect substitution. As perfect substitutabil-ity is the maintained assumption in most of the theoretical literature, this scenario shows whatbrings about our atypical exchange rate result for a demand shock.

The impact of a trend growth shock on the real exchange rate also changes substantially as itis hardly hump-shaped any more, but more persistent. The initial appreciation of almost 6%seems negligible given the low standard deviation of trend growth shocks. In contrast to thestandard model, the real exchange rate does not appreciate further as the flow of capital dueto net foreign direct investment is easily absorbed. However, also the pace of subsequent de-preciation is much slower under perfect substitution. Concluding one might say that allowingfor perfect substitution, the variability of real exchange rates is even less understood.

The second scenario investigates the impact of a different Taylor rule parameterization. Theabsence of a direct interest rate effect of inflation changes might have consequences for the im-pulse responses. Therefore, we also computed them under the more common parameterization:gDp¼ 0.36; gp1¼ 1.5; gDh¼ 0.12; gh1¼ 0.5; gi¼�0.24. This calibration implies a combina-tion of interest rate smoothing and similar short- and long-run targets. The results of this exer-cise are shown in Fig. 13. Hardly any changes are visible for any of the shocks. Consequently,the results are robust in this respect.

The final two scenarios check the sensitivity with respect to the direct price effect of depre-ciations (bDs¼ 0), see Fig. 14, and the influence of supply shocks on the output gap (fsup¼ 0),see Fig. 15. The real exchange rate dynamics turns out not to be sensitive for these adjustments.

10 20 30 400.5

0.25

0.250.5

0.751

1.25

demand shock

10 20 30 40

3

4

5

6trend shock

10 20 30 400.2

0.20.40.60.8

interest rate shock

Fig. 12. Response of real exchange rate under almost perfect substitution.

demand shock trend shock interest rate shock

10 20 30 40

1.51

0.5

0.51

1.5

10 20 30 40

2010

102030

10 20 30 40

0.20.40.60.8

Fig. 13. Response of real exchange rate under conventional Taylor rule parameters.


4. Empirical results for the US dollar

The empirical support for the portfolio balance model has not been very positive (Taylor, 1995).One of the reasons for the poor results is probably the lack of reliable data on wealth and, espe-cially, foreign investment positions. Usually, accumulated current account balances are used toapproximate foreign asset holdings. This procedure, however, does not take into account directinvestment and revaluations of assets other than due to exchange rate developments.

We will not try to derive an exact testable equation from our theoretical model. Data limi-tations, especially with respect to the stock of foreign assets, or even bilateral capital flows,would make this almost impossible. Moreover, the convenient assumption that the rest of theworld is always in equilibrium is clearly not realistic. Consequently, a multi-country modelneeds to be derived first. This is clearly beyond the scope of this paper. Also, as is quite clearfrom Fig. 6, the relationships modeled in this paper cannot tell the full story, and therefore can-not be expected to forecast well. The purpose of this section is more modest. It merely seeksempirical evidence for our model predictions with respect to the influence of different kinds ofoutput shocks on the exchange rate. Therefore, an ad hoc empirical model on the real (CPI-based) US dollar rate is estimated. The maintained simplifying assumption is that the rest ofthe world does, on average, not experience trend growth shocks or a non-zero output gap.As real exchange rates and real oil prices contain (near) unit roots, the equation is estimatedin error correction form to prevent a spurious regression:

Dqt ¼ 0:22ð2:6Þ

Dqt�1� 0:09ð3:4Þ

�qt�1 � 99:6

ð2:7Þgt�1 þ 2:27

ð1:8Þht�1 � 0:20

ð2:7Þ

�poil

t�1� pcpit�1

�þ 3:97ð13:6Þ

�Sample : 1973 : Ie2004 : III; R2 ¼ 0:14; LM

�1�¼ 0:09½0:76�

; LM�4�¼ 1:72½0:15�

; ð16Þ

where LM(1) and LM(4) are BreuscheGodfrey serial correlation LM tests in F-form; absolutet-values in parenthesis, p-values in brackets.

demand shock trend shock interest rate shock

10 20 30 40

21.5

10.5

0.51

1.5

10 20 30 40

2010

102030

10 20 30 400.2

0.20.40.60.8

1

Fig. 14. Response of real exchange rate under absence of direct price effect depreciation.

10 20 30 400.2

0.2

0.4

output gap

10 20 30 40

10.5

0.51

1.5

interest rate

10 20 30 40

2010

102030

real exchange rate

Fig. 15. Impact of trend shock absent direct gap effect.


In order to get a reasonable fit, including the real oil price in the equation seemed necessary.Higher real oil prices increase the real value of the dollar. This might be due to the fact that theUS is less dependent on oil imports than most other OECD countries. One of the main predic-tions of the theoretical model is confirmed by this empirical exercise: higher trend growth in theUS significantly increases the real value of the dollar, whereas a higher output gap has the op-posite (and much smaller) effect.14

In order to investigate the empirical relevance of this relationship, recursive out of sampleone to eight quarters ahead predictions for the real US dollar were made, see Table 2. Forthis purpose, real oil prices were assumed to be constant over the forecasting horizon. The cur-rent output gap and trend growth are based on the one-sided Kalman filter results as shown inFigs. 4 and 5, whereas future output gaps and trend growth are forecasted, based on the coef-ficients in Table 1, and assuming constant real interest rates over the forecast horizon.

With the exception of the two quarters ahead prediction, the model outperforms the randomwalk, despite the fact that oil prices and real interest rates were assumed constant over the fore-cast horizon. The outperformance does not hold for every subperiod, however. Especially thehuge appreciation of the dollar in the mid-1980s and the subsequent depreciation are not fullyforeseen by the model. Consequently, over this period the random walk model performs better.

Fig. 16 shows the realized real dollar rate together with the dynamically simulated rate,based on an exogenous output gap, trend growth and real oil prices. Most of the swings inthe real dollar rate are captured by the model, but substantial differences still remain. At times,the real exchange rate may deviate as much as 20% from the predicted value. Based on thesehistorical results, the perceived real overvaluation of just over 10% in the first quarter of 2002was not exceptional. The subsequent depreciation seems to be of the right order, as at the end of2004, the real exchange rate was almost in equilibrium according to our model.

5. Conclusions

One of the main purposes of this exercise was to find out whether the prolonged strength of thedollar until 2002 and the subsequent substantial depreciation could be explained by real outputgrowth in the US. Based on the theoretical model, the answer to this question must probablybe: partly. Trend growth indeed can explain real appreciation, primarily due to the expected netforeign direct investment it attracts. A positive output gap on the other hand seems to depressthe real exchange rate as the resulting deficit on the trade balance depletes foreign asset holdings.

Table 2

Root mean squared forecast error real US dollar, 1980e2004

Forecast horizon (quarters) Model Random walk Ratio

1 2.92 2.93 0.998

2 4.69 4.62 1.015

3 5.87 5.91 0.993

4 7.04 7.27 0.969

5 8.26 8.62 0.959

6 9.12 9.79 0.932

7 9.91 10.84 0.915

8 10.71 11.80 0.908

14 If the CBO trend growth and output gap measures are used instead, similar effects are found, although both coef-

ficients are not significantly different from zero (t-values of 1.2) in that case.


The size of these effects are, however, quite small compared to real life fluctuations in realexchange rates. This should not come as a complete surprise as the bulk of the changes in thecapital account are due to portfolio investments, whereas our empirical success in explainingdesired foreign asset holdings was disappointing. In order to better understand exchange ratedevelopments, a better understanding of foreign asset holdings seems essential. This wouldprobably imply not only modeling investment decisions (where to invest), but also savings de-cisions (when to invest). The current theoretical framework might be a useful starting point forsuch an analysis.15

So far, our empirical successes are encouraging. We do find significant effects of trendgrowth and output gaps, as predicted by the theoretical model, and the model outperformsthe random walk, especially over longer horizons. Even though major deviations between pre-dicted and actual real exchange rates still remain, the model seems to provide a good theoreticaland empirical justification for major shifts in real exchange rates. Regarding recent fluctuation,our empirical model implies that the dollar was overvalued by about 10% at the beginning2002, whereas a situation of near equilibrium was attained at the end of 2004.

Acknowledgements

Useful comments by two anonymous referees and by seminar participants at the Univer-sity of Amsterdam, the fourth annual research conference ‘Understanding Exchange Rates’

70

80

90

100

110

120

130

1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003

Realisation Dynamic simulation

Fig. 16. Actual and dynamically simulated real dollar rate.

15 A related relevant literature is the one on the natural real exchange rate (NATREX), see Stein and Allan (1995). As

in our model, the real exchange rate brings about equilibrium on the balance of payments, and productivity shocks are

an important source for real exchange rate fluctuations. Moreover, the approach explicitly models savings and invest-

ment decisions. However, the NATREX approach is only concerned with the medium to long run. Moreover, as the

NATREX model does not assume a constant steady state, the methods used in our model are not directly applicable.


at De Nederlandsche Bank, the European Finance Association 2002 conference, and the 8thInternational Conference on Macroeconomic Analysis and International Finance are grate-fully acknowledged. In particular the suggestions by Anders Vredin, Christoph Fischer andJuha Junttila contributed positively to this paper. The views in this paper are those ofthe individual author and do not necessarily reflect official positions of De NederlandscheBank.

Appendix A. Solving the model

We will use the toolkit of Uhlig (1999), which is based on Blanchard and Kahn (1980), inorder to derive a fully consistent path for the variables in our model in response to shocks.Within this method, all variables are expressed in deviation from steady state. The variablesare classified in three groups: exogenous variables (denoted zt), endogenous state variables(xt), and jump variables ( yt). The exogenous variables are not influenced by any endogenousvariable in the system. The distinction between the state and jump variables is that laggedjump variables do not enter the system. The system has to be rearranged such that it can beexpressed as follows:

0¼ Axt þBxt�1þCyt þDzt; ð17Þ

0¼ EtðFxtþ1þGxt þHxt�1þ Jytþ1þKyt þ Lztþ1 þMztÞ; ð18Þ

ztþ1 ¼ Nzt þ 3tþ1; Et3tþ1 ¼ 0: ð19Þ

One of the necessities of this system is its linearity. This is clearly not fulfilled for the re-valuation (Eq. (10)). This equation can be log-linearized under the assumption that there existsa steady state stock of scaled foreign assets and that this steady state value equals the one forforeign liabilities (denoted here mfa):

FArevtjt�1¼

�FAass

t�1

St

St�1

�1þ if

t�1

4

��FAlia

t�1

�1þ it�1

4�m3i3

it

��GDPt

GDPt�1

þ3revt

zmfa

��1þfaass

t�1

�ð1þDstÞ

�1þ if

t�1

4

��1þfalia

t�1

��1þ it�1

4�m3i3

it

��

��1þDyr

tþpt

�þ3rev

t

zmfa

�faass

t�1�faliat�1þDstþ

ift�1� it�1

4þm3i3

it

�þ3rev

t

zFAt�1þmfa

�Dstþ

ift�1� it�1

4þm3i3

it

�þ3rev

t ð20Þ

where fat�1ass and fat�1

lia denote the log deviations of foreign assets and liabilities from steady statevalues. The main approximation in the first step is replacing actual stock values by log


deviations from steady state: FA h mfaefa z mfa(1þ fa). In the second step all second order ef-

fects are supposed to be negligible, and in the last step the first step is reversed.The next thing to do is to get rid of variables that do not converge to a steady state. In our

model these are actual and potential output, and the nominal exchange rate. The latter can bereplaced by the real exchange rate (Eq. (13)), whereas the former two are not essential for thedynamics of the system. The steady state deviations for the remaining variables are simplygiven by subtracting their unconditional mean. In order to reduce the number of variables fur-ther, the revalued net foreign assets (20) can be substituted in Eq. (11), and the desired foreignassets (7) can be replaced by the actual foreign assets (12). As only one period lagged variablesare allowed in systems (17)e(19), additional variables need to be introduced for every variablethat enters an equation with more than one lag.

The final system contains one jump variable, 10 state variables, and 15 exogenousvariables16:

yt ¼ ½FDIt�;

xt ¼ ½ht pt it qt FAt TBt ht�1 pt�1 pt�2 pt�3 �0;

zt ¼�

gt gt�1 gt�2 gt�3 gt�4 gt�5 3dt 3

supt 3p

t 3it 3tb

t 3fdit 3rev

t nfat nfa

t�1

�0:

The matrices for the non-forward-looking equations in system (17) are

A¼

2666666666666664

�1 0 0 0 0 0 0 0 0 0bh �1þ bDs 0 �bDs 0 0 0 0 0 0gDh gDp �1 0 0 0 0 0 0 00 mfa 0 �mfa �1 1 0 0 0 0

tDh 0 0 0 0 �1 �tDh 0 0 00 0 0 0 0 0 �1 0 0 00 0 0 0 0 0 0 �1 0 00 0 0 0 0 0 0 0 �1 00 0 0 0 0 0 0 0 0 �10 0 0 0 0 0 0 0 0 0

3777777777777775

;

B¼

2666666666666664

f1 �fi fi fq 0 0 f2 �fi �fi �fi

0 b1 0 bDs 0 0 0 b2 b3 b4

�gh1gi�gDh �gigp1 1þgi 0 0 0 0 �gigp1 �gigp1 �gDp�gigp1

0 0 �mfa

4mfa 1 0 0 0 0 0

0 0 0 �t1tq 0 1þt1 0 0 0 01 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0

3777777777777775

;

16 As the subtraction of steady state values is not material for the results, the same notation will be pursued for the

deviations from steady state. Alternatively one could add vectors of intercepts related to the steady states to systems

(17)e(19).


C¼ ½0 0 0 1 0 0 0 0 0 �1 �0;

D¼

2666666666666664

fsup �a1fsup 0 0 0 0 1 fsup 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1 0 0 0 0 0 00 0 0 0 0 0 0 0 0 1 0 0 0 0 00 0 0 0 0 0 0 0 0 mfam3i 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0

cg

4

cg

4

cg

4

cg

40 0 0 0 0 1 0 0 0

3777777777777775

:

The AR(1) matrix for the exogenous process (Eq. (19)) is

N ¼

26666666666666666666666664

a1 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 d1 00 0 0 0 0 0 0 0 0 0 0 0 0 1 0

37777777777777777777777775

:

The matrices of the forward-looking equations (Eq. (18)) have the following structure:

F¼ ½0 duip 0 �duip 0 0 0 0 0 0 �;

G¼h

0 0 �duip

4duip �1 0 0 0 0 0

i;

H ¼ ½0 0 0 0 0 0 0 0 0 0 �;

J ¼ ½0�;

K ¼ ½0�;

L¼ ½0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 �;

M ¼ ½0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 �:

What one is looking for in order to solve the system is the recursive law of motion:

xt ¼ Pxt�1þQzt; ð21Þ


yt ¼ Rxt�1þ Szt: ð22Þ

The solution is given in Theorem 3.2 of Uhlig (1999). The tedious part of this solution concernsthe P-matrix which is shown to satisfy the following (matrix) quadratic equations:

0¼ C0APþC0B; ð23Þ

0¼�F� JCþA

�P2�

�JCþB�GþKCþA

�P�KCþBþH; ð24Þ

where Cþ is the pseudo-inverse of the matrix C, and C0 denotes a matrix whose rows form a ba-sis for the null space of C0.

Uhlig (1999) solves this system by means of a generalized eigenvalue problem. In ourcase, the solution can also be found directly. The linear equations (23) are related to thenon-forward-looking equations in model (17) and can be solved analytically. The quadraticequations (24) refer to the forward-looking part of the system (Eq. (18)), which in ourcase is only the UIP relationship (7). Given the 10 state variables in our model, this resultsin 10 quadratic equations in 10 elements of P, which can be solved numerically with the helpof Mathematica (Wolfram, 1999). The system results in 11 solutions, only one of which isstationary. This is the only relevant solution as the other (explosive) solutions do not con-verge to a steady state.

Appendix B. Data sources

Most data come from the December 2004 International Financial Statistics cd-rom of theIMF. The following lines were used:

00176AAZZF. Spot oil price

111..RECZF. Real effective exchange rate, based on relative CPI

11160B..ZF. Federal funds rate US

11164.ZF. Consumer price index US

11175..DZF. Import prices US

11178AFDZF. Balance of payments: balance on goods and services US

11178AIDZF. Balance of payments: balance on goods, services and income US

11178BDDZF. Balance of payments: direct investment abroad US

11178BEDZF. Balance of payments: direct investment in rep. econ., N.I.E. US

11178BFDZF. Balance of payments: portfolio investment assets US

11178BGDZF. Balance of payments: portfolio investment liabilities US

11178BKDZF. Balance of payments: portfolio investment equity assets US

11178BMDZF. Balance of payments: portfolio investment equity liabilities US

11179ACDZF. International investment position: portfolio investment assets US

11179ADDZF. International investment position: portfolio investment assets US, equity

11179AFDZF. International investment position: other investment assets US

11179LCDZF. International investment position: portfolio investment liabilities US

11179LDDZF. International investment position: portfolio investment liabilities US, equity

11179LFDZF. International investment position: other investment liabilities US

11199B.CZF. Gross domestic product US

11199BVRZF. GDP volume (2000¼ 100) US

The potential output and corresponding real output series are taken from the Federal ReserveBank of St. Louis database. The real potential GDP series is by the US Congress, Congressional


Budget Office. The corresponding seasonally adjusted real GDP series is by the US Departmentof Commerce, Bureau of Economic Analysis.

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