On the correct model specification for estimating the structure of a currency basket

Transcription

1 On the correct model specfcaton for estmatng the structure of a currency basket Jyh-Dean Hwang Department of Internatonal Busness Natonal Tawan Unversty 85 Roosevelt Road Sect. 4, Tape 106, Tawan A strand of lterature has focused on nferrng the weghts of the anchor currences n a country s currency basket. In partcular, wth the growng mportance of Chna s economy, there has been a surgng reseah nterest n the role of the renmnb as an nternatonal anchor currency and n Chna s de facto exchange rate regme. In ths artcle, we show by a smple model and demonstrate usng the observed data on the Specal Drawng Rghts (SDR) basket that the correct specfcaton to estmate the weghts of anchor currences n a currency basket s to use the rates of change n exchange rates and to wrte exchange rates n quantty term. When one s to estmate the structure of a currency basket, close attenton should be pad to usng the rates of change or the levels of exchange rates as well as wrtng exchange rates n quantty or prce term. Keywords: Exchange rate regme, Currency basket, Currency weghts, Currency amounts, renmnb. JEL classfcaton: F31, F36 1

2 I. Introducton There has been a contnued nterest n the dstncton between the de facto and de jure exchange rate regmes. A strand of lterature has focused on nferrng the weghts of the anchor currences n a country s currency basket. These works use a lnear regresson model and attempt to estmate the mplct weghts of the anchor currences. The weght-nference technque s popularzed by Frankel and We (1994) and has been extensvely used n the lterature, for example Kwan (1996), McKnnon and Schnabl (2004), and Bowman (2005). Ths technque has receved tremendous attenton recently. Wth the growng mportance of Chna s economy and the polcy ntatves of the Chnese government to nternatonalze ts currency, there has been a surgng reseah nterest n the role of the renmnb as an nternatonal anchor currency, for example Chen et al. (2009), Ito (2010), Subramanan and Kessler (2013), and Fratzscher and Mehl (2014), and n Chna s de facto exchange rate regme, for example, Ogawa and Sakane (2006), Yamazak (2006), Frankel (2009), Fdrmuc (2010), and Fang et al. (2012). The majorty of these studes employ the weght-nference technque. To nfer the weghts n the currency baskets, Frankel and We (1994) propose a specfcaton n the log dfferenced form: n e e (1), t, t t 1 where stands for the dfference, e s the log of the exchange rate of the regonal currency, e s the log of the exchange rate of anchor currency, all exchange rates are n prce term (currency per unt of numerare), t stands for tme and t for resduals. The estmated coeffcent s nferred as the weght of anchor currency n the currency basket. On the other hand, Yamazak (2006) and Fdrmuc (2010) propose the followng specfcaton n levels: 2

3 E n E (2) re, t, t t 1 where the same notatons apply and the estmated coeffcent weght of anchor currency. s nferred as the Are both of these two specfcatons correct n the sense of allowng one to nfer the weghts of the anchor currences usng the estmated coeffcents? Moreover, are the exchange rates to be expressed n quantty term (numerare per unt of currency) or prce term (currency per unt of numerare)? Ths artcle ams at answerng these two questons. We show that to estmate the weghts of the anchor currences n the basket, the correct specfcaton s to use the rates of change n exchange rates and to wrte exchange rates n quantty term. If the estmaton equaton s based on the levels of exchange rates and exchange rates are n quantty term, the estmated coeffcents are the amounts, not the weghts, of the anchor currences. When the estmaton equaton s based on the levels of exchange rates and exchange rates are n prce term, the estmated coeffcents are anythng but the weghts or amounts of the anchor currences. The rest of ths artcle s as follows. Secton II derves a smple model to estmate the structure of a currency basket and dscusses the mplcatons of dfferent model specfcatons. Secton III uses the observed data on the SDR basket to verfy the mplcatons of dfferent model specfcatons. Secton IV concludes ths artcle. II. The Model For smplcty, suppose the regonal currency (RC) follows a basket peg and the value of one regonal currency s pegged to the value of a basket of currences 1 whch has q 1 unt of US dollar and q 2 unt of euro (q 1 > 0, q 2 > 0), then we have: 1 The exchange rate of the regonal currency (RC) and the basket of currences can be wrtten as 1 RC = X Basket. As the sze of the basket can be rescaled proportonally, t s more convenent to express the exchange rate as 1 RC = 1 Basket. 3

4 Value of 1 RC = Value of 1 basket of currences (3) Values of the currences n Equaton 3 are measured by a common numerare. Usng the Swss franc as the numerare, Equaton 3 can be rewrtten as: E qe q E (4) 1 usd 2 eur where E, E usd, and E eur are the exchange rates of the regonal currency, USD, and EUR aganst CHF, expressed n quantty term (CHF per unt of currency). The left-hand sde of Equaton 4 s the CHF equvalent of one regonal currency and the rght-hand sde s the CHF equvalent of one currency basket. It should be noted that the coeffcent q n Equaton 4 s the amount of currency n the basket, not the weght of currency. The weght of currency (w ) s: w qe qe qe qe E 1 usd 2 eur (5) Wrtten n dfferenced form and dvded by E, Equaton 4 becomes: E E Eeur q q (6) E E E usd 1 2 Rewrte E q E qe (recall that E qe E on the rght-hand sde of Equaton 6 as E E s the weght of currency by Equaton 4), we have: qe and let w E E E Eeur w w (7) E E E usd 1 2 usd eur where w 1 and w 2 are the weghts of the USD and EUR. Equaton 7 can be nterpreted 4

5 ntutvely. If the USD, wth a weght of w 1, apprecates by 1% aganst the CHF and other thngs beng equal, the currency basket wll apprecate by w 1 % aganst the CHF. As the value of the regonal currency s pegged to the value of the basket, the Monetary Authorty must apprecate the regonal currency by w 1 % aganst the CHF to keep ts exchange rate aganst the basket constant. Three mportant ponts can be made on the model specfcatons for estmatng the structure of a currency basket. Frst, to estmate the weghts of the anchor currences, the rates of changes n exchange rates should be used and exchange rates should be wrtten n quantty term. For example, to nfer the weghts of the USD and EUR, one should estmate the followng equaton: E E Eeur, t w w t (8) E E E, t usd, t 1 2, t usd, t eur, t where exchange rates are n quantty term, t stands for tme and t for resduals. The estmated w 1 and w 2 are the weghts of the USD and EUR, respectvely. When the rates of changes n exchange rates are not large, Equaton 8 can be approxmated by the log dfferenced form: e w e w e (9), t 1 usd. t 2 eur, t t where stands for the dfference and e s the log of the exchange rate E. 2 Second, the estmated coeffcents are the amounts of the anchor currences f we regress the level of exchange rate of the regonal currency on the levels of exchange rates of the anchor currences and exchange rates are n quantty term: E E E (10), t 1 usd, t 2 eur, t t 2 When Equaton 9 s estmated, the estmated w wll be the same regardless of the fact that exchange rates are n quantty term or prce term. The specfcaton proposed by Frankel and We (1994) s smlar to Equaton 9. 5

6 By Equaton 4, we know that the estmated 1 and 2 n Equaton 10 are the amounts of the USD and EUR n the basket. Therefore, t s ncorrect to nfer the estmated coeffcents as the currency weghts. Thrd, f we regress the level of exchange rate of the regonal currency on the levels of exchange rates of the anchor currences and exchange rates are n prce term, the estmated coeffcents are anythng but the amounts or weghts of the anchor currences. As exchange rate n prce term s smply the nverse of exchange rate n quantty term, the regresson equaton n ths case s: (1/ E ) (1/ E ) (1/ E ) (11), t 1 usd, t 2 eur, t t By Equatons 4 and 7, the estmated 1 and 2 of Equaton 11 are nether the amounts nor the weghts of the anchor currences. We cannot nfer the estmated coeffcents as the currency weghts. It wll be cumbersome f not mpossble to transform these coeffcents to the currency weghts. Examples that used ths specfcaton are Yamazak (2006), Moosa et al. (2009) and Fdrmuc (2010). The estmaton of currency weghts n the currency basket s analogous to the estmaton of the weghts of assets n a portfolo. Suppose we have a portfolo consstng of 1 share of IBM stock and 2 shares of Ctbank stock and the prces of the IBM and Ctbank stocks are P (USD per share of IBM stock) and P c (USD per share of Ctbank stock), then the value of ths portfolo V (USD per portfolo) s: V 2 P 1 P (12) c If we regress the rate of changes n V on the rates of changes n P and P c, the estmated coeffcents are the average weghts of IMB and Ctbank stocks n ths portfolo for the estmaton perod. If we regress V on P and P c, the estmated coeffcents are the numbers of share of IMB and Ctbank stocks n the portfolo. If we regress 1/V on 1/P and 1/P c, the estmated coeffcents are anythng but the weghts or the numbers of shares of IMB and Ctbank stocks. 6

7 III. Emprcal Illustraton usng Data on the SDR Basket Ths secton uses the observed data of the Specal Drawng Rghts (SDR) basket to test for the mplcatons of dfferent model specfcatons derved n Secton II. The IMF has announced that on January 1, 2011 the ntal weghts assgned to USD, EUR, GPB and JPY n the SDR basket are 41.9%, 37.4%, 11.3% and 9.3%. Effectve from January 1, 2011, the currency amounts of the USD, EUR, GPB and JPY n the SDR basket are 0.660, 0.423, and 12.1, whch wll be n effect to the end of 2015 (confer IMF, Currency amounts n new Specal Drawng Rghts basket). Suppose we only know that the USD, EUR, GBP and JPY are n the SDR basket, how can we use the observed data to estmate the weghts of these four currences? We use three dfferent model specfcatons dscussed n Secton II and daly data from January 1, 2011 to Aprl 30, 2014, wth Swss franc as the numerare, to do the task. Estmaton results are reported n Table 1. All estmated coeffcents n Table 1 are statstcally sgnfcant at the 1% level. In the frst specfcaton, exchange rates are n quantty term and the rate of changes n SDR exchange rate s regressed on the rates of changes n the exchange rates of the four anchor currences. Ths s the correct specfcaton shown n Secton II. The estmated coeffcents of the USD, EUR, GBP and JPY are 0.437, 0.362, and 0.073, respectvely, whch are very close to the ntal weghts of these four currences. And we can use these estmated coeffcents to nfer the weghts of the USD, EUR, GBP and JPY n the SDR basket. In the second specfcaton, exchange rates are n quantty term and the level of exchange rate of SDR s regressed on the levels of exchange rates of the four currences. We have shown n Secton II that the estmated coeffcents from ths specfcaton are the currency amounts. Usng ths specfcaton, the estmated coeffcents of the USD, EUR, GBP and JPY are 0.660, 0.423, and , respectvely, whch are vrtually dentcal to the actual amounts of these four currences. If we use these coeffcents to nfer the currency weghts, we wll erroneously conclude that the JPY s the most mportant currency n the SDR basket 7

8 and t s far more mportant than the other three currences. The thrd specfcaton wrtes exchange rate n prce term and regresses the level of exchange rate of SDR on the levels of exchange rates of the four currences. As shown n Secton II, the estmated coeffcents from ths specfcaton are anythng but the currency weghts or amounts. Usng ths specfcaton, the estmated coeffcents of the USD, EUR, GBP and JPY are 0.273, 0.310, and 0.001, respectvely, whch are ndeed very dfferent from the ntal weghts and amounts of these four currences. If we use these coeffcents to nfer the currency weghts, we wll erroneously conclude that that the EUR s the most mportant currency and the JPY s neglgble n the SDR basket. In sum, estmaton results on the structure of the SDR basket are consstent wth the predctons of our model n Secton II and support the argument that the correct specfcaton to estmate the weghts of the anchor currences n a currency basket s to use the rates of changes n exchange rates and to wrte exchange rates n quantty term. IV. Concluson In ths artcle, we have shown by a smple model and demonstrated usng the observed data on the SDR basket that the correct specfcaton to estmate the weghts of the anchor currences n a currency basket s to use the rates of changes n exchange rates and to wrte exchange rates n quantty term. When the estmaton equaton s based on the levels of exchange rates and exchange rates are n quantty term, the estmated coeffcents are the amounts, not the weghts, of the anchor currences. If the estmaton equaton s based on the levels of exchange rates and exchange rates are n prce term, the estmated coeffcents are anythng but the weghts or amounts of the anchor currences. Based on these fndngs, we suggest that close attenton should be pad to usng the rates of changes or the levels of exchange rates as well as wrtng exchange rates n quantty or prce term when one s to estmate the structure of a currency basket. 8

9 Table 1. Estmaton of the SDR basket usng three dfferent model specfcatons USD EUR GBP JPY Adj. R 2 Panel 1. rates of change n exchange rates are used and exchange rates are n quantty term (104.25) (90.99) (31.06) (25.76) Panel 2. levels of exchange rates are used and exchange rates are n quantty term (898.49) (972.25) (198.87) (800.49) Panel 3. levels of exchange rates are used and exchange rates are n prce term (376.52) (407.38) (96.30) (348.84) Notes: All exchange rates use Swss Franc (CHF) as the numerare. Sample perod s from January 1, 2011 to Aprl 30, 2014 (868 observatons). Fgures n parentheses are t-statstcs. The IMF has announced that on January 1, 2011 changes n the relatve weghts of the four currences n the SDR basket come nto effect. The ntal weghts assgned to USD, EUR, GPB and JPY are 41.9%, 37.4%, 11.3% and 9.3%. Effectve from January 1, 2011, the currency amounts of USD, EUR, GPB and JPY are 0.660, 0.423, and 12.1, whch wll be n effect to the end of

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