STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN

Transcription

1 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) STRUCTURAL CHANGE IN TIME SERIES OF THE EXCHANGE RATES BETWEEN YEN-DOLLAR AND YEN-EURO IN OBARA, Takashi * Absrac The economic srucural change in he daa of he yen exchange raes o US dollar and o euro is invesigaed. We analyse he disribuion of he residuals which are he differences beween he smoohed and original daa. The probabiliy densiy funcion is well approximaed by he Gram-Charlier expansion. The exisence of he fac ha he disribuion of he residuals pars from he normal disribuion indicaes he occurrence of he srucural change. Inroducing a new index, composed by he skewness and he kurosis, we can easily find a sign of he srucural change. I is shown ha boh daa of he yen-dollar rae and he yen-euro rae from January, 001 o May 18, 004 undergo he srucural changes hree imes ogeher on May 3, 001, March 11, 00, and December 7, 00. JEL classificaion: C51 Key words: srucural change, Gram-Charlier expansion, exchange raes, skewness and kurosis * Obara Takashi College of Inernaional Relaions, Nihon Universiy, Japan, kobara@ir.nihon-u.ac.jp 61

2 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) 1.- Inroducion When we are given a series of daa, some of us wan o forecas is value a fuure ime, and apply he well-esablished mehods such as ARIMA or regression models. If here are economic srucural changes in he daa generaing mechanism, a naive applicaion of hose mehods does no give us righ answers. Then he main concern of economericians was o esablish a mehod by which a sign of he srucural change in he daa ough o be deeced. Akiba (1994), Broemeling and Tsurumi (1987), and Hackl (1989) give surveys abou various kinds of mehods o find a sign of he srucural change. Typical mehods ha hey explained in hese books are o examine he sabiliy of he coefficiens of he equaions ha a model such as ARIMA or he regression analysis conains. A subse of he series in which he coefficiens of he equaions are sable agains he recursive esimaion consiues one daa regime. The daa may have several regimes. A node beween he wo regimes is he poin where he srucure changes. In his paper, we will find a sign of economic srucural changes in he ime series of he yen exchange raes agains US dollar and euro by a differen way from he above menioned mehods. In Secion, we inroduce he way of exponenial smoohing which does no depend on a ype of model such as ARIMA or he regression analysis. We will pu sress on residuals associaed wih 6

3 Obara, T. Srucural Change beween Yen-Dollar and Yen-Euro he daa smoohing. I is shown ha he disribuion of he residuals depars from he normal disribuion. In Secion 3, we show ha he Gram-Charlier expansion 1 is useful o approximae he disribuion of he residuals. The hird and he fourh erms of he Gram-Charlier expansion depend on he skewness and he kurosis of he daa, respecively. Then hese higher order momens are imporan o deermine he disribuion of he residuals. In Secion 4, we inroduce a way o generae some ime series from one ime series by consiuing subses recursively from he original daa se. When he skewness and he kurosis evaluaed in each subse, hey are consans. Bu hey are regarded as ime series along all he subses. We define a new index γ made from he skewness and he kurosis. The index γ goes o zero, if he disribuion of he residuals is close o he normal disribuion. A change of he disribuion means ha of he daa srucure. Then when he index γ goes o zero or depars from zero, we know ha he srucural change occurs. Taking cumulaive sum of his index, we find a sign of he srucural change easily as a kink of he cumulaed curve. I is shown ha he yen exchange raes o US dollar and o euro from January, 001 o May 18, 004 undergo he srucural changes hree imes ogeher. 1 See, for example, Kendall and Suar (1979). 63

4 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004).- Residuals for he Daa Smoohing Here are ime series of he exchange raes of Japanese yen o US dollar and euro, which are daily series from January, 001 o May 18, 004. They are illusraed wih Graph 1 and. Can we find any kinds of srucural changes from hese daa? In order o answer affirmaively, someone will apply ARIMA model o he daa. Or oher may supplemen he given daa wih oher ones and formulae regression equaions. Their main concern is o find he break poins in he daa fiing curves calculaed by ARIMA or he regression. Before drawing a conclusion from hese mehods, we mus analyze he propery of he residuals associaed wih hose daa fiings. The validiy of ARIMA or he regression is based on he saionariy of he residuals. For simpliciy we usually posulae ha he residuals are normally disribued. The ARIMA or he regression model is a kind of daa smoohing. The simples daa smoohing is he moving average. Among ohers, he exponenial smoohing is he mos convenien for our presen purpose. I is expressed by a formula (1) x = x 1+ α ( x x 1), Source: Pacific Exchange Rae Service, p://pacific.commerce.ubc.ca/xr. 64

5 Obara, T. Srucural Change beween Yen-Dollar and Yen-Euro Graph 1: Plo of yen-dollar rae and is smoohing Graph : Plo of yen-euro rae and is smoohing where { x } is he original series and { x } is he smoohed series. The characerisic of (1) is ha he lengh of he series x is he same as x. There are wo arbirary consans in (1): one is α wih 0<α <1 and he oher is he iniial value x 1. We will pu α = 0.3 and x1 = x1 for convenience. The smoohed curves for he 65

6 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) yen-dollar rae and he yen-euro rae are given in Graph 1 and, respecively. A residual by he smoohing is defined as () x = x x. If he residuals are no saionary, he smoohing (ARIMA or he regression) will no give he righ resuls. I is convenien o suppose ha he saionary series should be disribued as (3) x N ( µσ, ) where µ and σ are he mean value and he variance of respecively. To examine his hypohesis, we ake a hisogram of { x }. For convenience, we sandardize he daa as x µ (4) z = σ (5) N µ = x / N, σ = ( x µ ) / N, N = 850. = 1 The sandardized densiy funcion for he normal disribuion is (6) N = 1 1 z f () z = exp π. A hisogram for he yen-dollar rae is given in Graph 3 ogeher wih he plo of (6). The disribuion of { z } seems o be apar from N (0,1). The pormaneau es 3 is as follows: Q-value=896.7, χ (44) = Since Q > χ 0.95, we rejec he hypohesis (3). Then a naive applicaion of he well-known mehod such as ARIMA o he yen exchange raes may lead o a wrong conclusion. x, 3 See, for insance, Box, Jenkins, and Reinsel (1994). 66

7 Obara, T. Srucural Change beween Yen-Dollar and Yen-Euro Graph 3: Hisogram of yen-dollar rae and normal disribuion 3.- The Gram-Charlier Expansion In his secion, we will give a skech of approximaion of he probabiliy densiy funcion. The densiy funcion px ( ) may have he following orhogonal expansion in erms of polynomials φ ( x ) (7) px ( ) = anφn( xwx ) ( ) n= 0 wih he orhogonal propery (8) φ n( x) φ m( xwxdx ) ( ) = δ nm,, nm=, 0,1,,... I should be noed ha he orhogonal relaion (8) has he weigh funcion wx. ( ) The coefficiens (9) an = φn( xpxdx ) ( ). a n can be obained by (8) as n 67

8 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) When φ ( x ) is he Hermie polynomial4 H ( x ) and wx ( ) is he n sandardized densiy funcion of he normal disribuion f () z, he expression (7) is said he Gram-Charlier expansion. Here are relaions (10) 1 x µ φn( x) = Hn( ) n! σ n (11) x µ wx ( ) = f( x) = exp π σ σ (1) µ = σ = ( µ ) xp( xdx ), x pxdx ( ). The Hermie polynomials for some orders are given by (13) H ( y) = 1, H ( y) = y, H ( y) = y H ( y) = y 3 y, H ( y) = y 6y From equaions (9)-(13), we have a = 1, a = 0, a = m3 1 m4 a3 = 3, a4 = 4 3 3! σ 4! σ. 3 4 (14) m = ( x µ ) pxdx ( ), m = ( x µ ) pxdx ( ) 3 4 Using all he resuls obained above, we have he Gram-Charlier expansion up o he fourh order: (15) 1 1 px y y y y f x 3! 4! 3 4 ( ) = 1 α(3 ) + β(3 6 + ) ( ) 4 Kendall and Suar (1979). 68

9 Obara, T. Srucural Change beween Yen-Dollar and Yen-Euro where y = ( x µ )/ σ and m3 (16) α = sk = 3 σ m4 (17) β = ku 3, ku =. 4 σ The erms sk and ku are called he skewness and he kurosis, respecively. If he disribuion is normal, from (1), (14), (16) and (17), we have sk =0 and ku =3, herefore α = β = 0. The significance of he expression (15) is ha he magniude of he deviaion of px ( ) from f ( x ) is calculaed wih he knowledge of α and β. Now we go back o he series z in (4) and replace x in his secion wih x in (4) and (5). If we also replace yxpxdx ( ) ( ) for a funcio n yx ( ) by y / 1 N = N, we can calculae α and β numerically. And we can evaluae he Gram-Charlier expansion (15). The hisogram of z and he evaluaed Gram-Charlier expansion are illusraed in Graph 4. Comparing Graph 3 wih Graph 4, we know ha fiing is improved by (15). Granger and Joyeux (1980) poined ou ha some economic ime series had he disribuion wih long memory. Bu his is no he case. The ail of he disribuion is as shor as ha of he normal disribuion. 69

10 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) Graph 4: Hisogram of yen-dollar rae and Gram-Charlier expansion 4.- A New Index γ In his secion, we will closely examine he disribuion of he residuals follows x. For his purpose, we shall make subses of { x } as X = { x, x,..., x } 1 1 X = { x, x,..., x } L 3 n+ 1 L X = { x, x,..., x }, n 1 n where we se n = 00 for convenience. We will calculae he momens in each subse X. Now we pu n 1 n 1 = + k = + k k= 0 k= 0 ( ) µ x / n, σ x µ / n n 1 n = = k= 0 k= ( + k µ ) /, ( + k µ ) /. m x n m x n 70

11 Obara, T. Srucural Change beween Yen-Dollar and Yen-Euro From hese, we shall define he dependence of α and β as α m = = σ m 3 4, β 3 4 σ In case he original series { x } is saionary, all he momens are independen from. If X disribues normally, hen α 0 and β 0 for each. Therefore i is plausible o inroduce a new index γ which indicaes he deviaion of he disribuion from he normal disribuion: γ = α + β. The ime dependence of γ is ploed in Graph 5. Graph 5: Time dependence of γ From his graph, we know ha he index γ has sharp jumps o or from he regions of γ 0 around = 100, 300, and 500. To see hose facs more clearly, i will be convenien o ake he cumulaive sum of γ : 71

12 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) G The ime series = γ = 1. G for he yen-dollar rae and he yen-euro rae are ploed in Graph 6. In his graph, here are kinks around = 100, 300, and 500. The derivaive from he lef is differen from ha from he righ a hose poins. Using he echnique of he cumulaive sum, we are able o compare paerns of a couple of G s in a graph ogeher, in our case, he paerns of he yen-dollar rae and he yen-euro rae. Acually, we know ha boh he yen raes have he kinks a he same daes. Those daes correspond o he real daes; May 3, 001, March 11, 00, and December 7, 00. The plo of G for he yen-euro rae does no have a sharp kink excep for = 300. This is because he value of he parameer α in (1) or he lengh n of he subse X i may no be adequae. Graph 6: Cumulaive sum G When we ake a difference and he resulan residuals are no saionary, we used o ake he second difference. By he same 7

13 Obara, T. Srucural Change beween Yen-Dollar and Yen-Euro reason, we may need he double smoohing for he yen-euro rae. Anyway we shall name hose poins in Graph 6 a = 100, 300, and 500 criical poins. The daa of he yen exchange raes can be divided by he criical poins ino four subses which correspond o four phases of he daa srucure. 5.- Conclusions In his paper, we discussed how o find srucural changes in he ime series of he yen exchange raes o US dollar and euro from January, 001 o May 18, 004. Here are he resuls: 1) The daa are smoohed by he way of he exponenial smoohing. The disribuion of he residuals associaed wih he smoohing deviaes from he normal disribuion. A naïve applicaion of ARIMA or he regression models is no desirable. ) The densiy funcion of he residuals is well approximaed by he Gram-Charlier expansion. 3) The daa are recursively divided ino some subses. The higher order momens, he skewness and he kurosis, are calculaed in each subse. They form ime series along he whole daa. 4) The deviaion from he normal disribuion indicaes ha he daa srucure changes from he saionary sae. The srucural changes are deeced from he behavior of hese momens as he ime series. Inroducing a new index composed by hose momens, we can find a sign of he 73

14 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.1-3(004) srucural change in he yen exchange raes around May 3, 001, March 11, 00, and December 7, 00. Bibliography Akiba, H. (1994). Srucural Changes in Foreign Exchange Markes, Ashish Publishing House, New Delhi. Box, G..E.P., Jenkins, G,.M., and Reinsel, G..C. (1994). Time Series Analysis: Forecasing and Conrol, hird ediion, Prenice-Hall Inernaional, London. Broemeling, L. D. and Tsurumi, H. (1987). Economerics and Srucural Change, Marcel Dekker, New York. Granger, C.W. J. and Joyeux, R. (1980). ``An Inroducion o Long-Memory Time Series Models and Fracional Differencing," Journal of Time Series Analysis, Vol.1, pp Hackl, P. (ed), (1989). Saisical Analysis and Forecasing of Economic Srucural Change, Springer, Berlin. Kendall, M. and Suar, A, (1979). The Advanced Theory of Saisics, Vol. 1, fourh ediion, Charles Griffin, London. Journal IJAEQS published by AEEADE: hp:// 74