Monthly Forecasting of the Dollar to the Ruble Exchange Rate with Adaptive Kalman Filter

Transcription

1 Inernaional Journal of Sysems Science and Applied Mahemaics 8; (): 4-9 hp:// doi:.648/j.ijssam.8. ISSN: (Prin); ISSN: (Online) Monhly Forecasing of he Dollar o he Ruble Exchange Rae wih Adapive Kalman Filer Sergei Borodachev Graduae School of Economics and Managemen, Ural Federal Universiy, Ekaerinburg, Russia address: To cie his aricle: Sergei Borodachev. Monhly Forecasing of he Dollar o he Ruble Exchange Rae wih Adapive Kalman Filer. Inernaional Journal of Sysems Science and Applied Mahemaics. Vol., No., 8, pp doi:.648/j.ijssam.8. Received: June, 8; Acceped: June 9, 8; Published: July, 8 Absrac: The goal: o develop a model ha allows you o forecas he dollar o he ruble exchange rae for a monh ahead based on macroeconomic daa, published a monhly inervals. Proposed srucural model of he dynamics of he ruble and dollar masses ha deermine he exchange rae, depending on changes in foreign exchange reserves, he balance of foreign rade, he moneary base, he MICEX index, he price of oil. Wih he help of he Kalman filer (KF), he model parameers, he dynamics of he money masses were esimaed, and forecasing of he dollar exchange rae was done. Monhly daa were used from he beginning of 5 o mid-7. The esimaion of he capaciy of dollar marke was found in abou half he capaciy of he MICEX index funds. Average error of forecass, based on informaion available one sep before he forecased momens (RMSEA) was.99. Adapive form of KF was developed when, similarly o he EM algorihm, he phases of KF esimaion in he window and minimizaion of average predicion error o deermine he opimal esimaes for he sysem model parameers in his momen are sequenially alernaed. Wih his RMSEA became.9. Keywords: Currency Marke, Adapive Kalman Filer, Exchange Raes, Predicion Error of Adequacy. Inroducion Many sudies have been devoed o he forecasing of exchange raes [, ]; in paricular, a model of daily forecasing has been proposed in []. In his paper, an aemp is made o build monhly forecass. Paricularly, such forecass are necessary when invesing in Dual Currency Deposis. Of course, he changes in he dollar-ruble exchange rae, especially shor-erm ones, are grealy influenced by informaion shocks and crises, so a relaively quie period was chosen for he raining of he model and is esing from he end of 4 o mid-7. Furher in he aricle - he momens of he beginning of he monhs, ake he values from o T- ha is from January, 5 o May, 7 (oal T = 9 monhs).. Model and Daa Rd denoe he rubles mass, and Le D he corresponding mass of dollars paricipaing in he exchange and deermining he proper dollar / ruble rae a ime : S Rd =. () D We will assume ha changes in hese masses depend on changes in he following macroeconomic variables. Rd = Rd + β M + β M + β R, () + D = D + γ B + γ Ca + γ M + γ Sa, () + Here all he differences are backward O = O O. M - MICEX index, R - a moneary base in a narrow definiion (includes cash issued by he Bank of Russia (including balances in cash deparmens of credi insiuions) and balances on accouns of mandaory reserves for funds borrowed by credi insiuions in he naional currency deposied wih he Bank of Russia) bn., B - he price of a barrel of oil $, Ca - cash currency and deposis included in he iem "Inernaional reserve asses and oher liquidiy in foreign currency of he Russian Federaion" $ rillion, Sa - is he monhly balance of he rade balance of he "Foreign

2 5 Sergei Borodachev: Monhly Forecasing of he Dollar o he Ruble Exchange Rae wih Adapive Kalman Filer Trade of he Russian Federaion wih goods" $ bn. All hese daa, as well as he exchange rae Y - "Nominal raes of foreign currencies o he ruble (rubles per uni of foreign Table. Dollar-ruble exchange rae and macroeconomic variables affecing i. currency)" are available on he Bank of Russia websie hp: / / See Table Y M R B Ca Sa The inclusion of precisely hese conrol acions is dicaed by common sense, heir availabiliy and he resuls of some selecions for he bes descripion. The esimaion of he muliple correlaion coefficien of he change in he dollar exchange rae Y wih all he conrols B, Ca, M, Sa, R (i is equal o he coefficien of deerminaion R in he regression analysis) was The Wilks-Barle crierion rejecs he hypohesis of pairwise independence of all hese six values wih a significance level of.7.. Kalman Filer The Kalman filer is widely used in sudies boh in macroeconomics [4, 5, 6] and microeconomics [7]. Logarihm () o obain a linear connecion s = rd d of he proper value of he oupu quaniy wih he componens of he sae vecor. The lower case leers denoe he logarihms of he corresponding quaniies. Taking logarihms () and (), imply ha ln( + x) x a small x. As shown by numerical experimens wih hese daa, i is useful o inroduce a "cobweb" cycle in he supply of rubles and dollars o he exchange marke. As is known, he volume of he nex delivery would necessarily be beween he wo previous ones. For his, in he logarihmic equaions () and () in righ hand pars replace rd rd + rd and 4 4 d d + d. Then, similarly o [8], we have a model 4 4 of he sysem in he sae space x + = F x + G u + w, y = H x + v ; where w, v noises of sysem and observaion, Mw =, Mv =, T Q = Mw w, T R = Mv v. Sae vecor x ( rd rd d d M ) =, conrol T vecor ( ) T u = B Ca M Sa R, operaors of evoluion and conrol F β / 4 / 4 Rd = / 4 / 4, β β Rd Rd G = γ γ γ γ, observaion operaor D D D D H = ( ). The esimaion of he vecor of model parameers θ = ( σ ) T v σw β β β γ γ γ γ was carried ou by minimizing he mean square error of he Kalman filer (KF) predicion of he dollar rae RMSE( θ) = [ Y exp( y )] Tu Tl ˆ θ = arg min RMSE( ˆ θ ). RMSE ˆ θ Tu + + ˆ, ( ) = Tl Here y ˆ + is he forecas a ime of he logarihm of he dollar rae a he nex momen. Tl is he iniial momen, Tu is he final momen of he raining (fi) window. Then, using he esimaed parameers ˆRMSE θ, saring wih he las sae esimae, is covariance marix and u, all a he momen Tu, perform a momens from Tu + o T- (geing y and u ) he Kalman filer forecasing for a monh ahead. The roo-mean-square error of adequacy characerizes he average error of hese forecass RMSEA = Y y T T + + [ exp( ˆ )] (he forecas Tu = Tu+ a ime T- has no comparaive; i is he goal of calculaion). We choose he esimae of he iniial sae ˆx = ( ) T, and he covariance marix of his esimae equal o uni marix I. R = σvi, Q = σwi. Calculaions are carried ou in PTC Mahcad Prime. If o rain (esimae) from Tl = o Tu =, hen he esimaed parameers ˆv σ = 4.9, ˆw σ =, he posiiviy of he esimaes ˆβ =.76, ˆ γ =.6, ˆ γ =.74 does no require commens, ˆ γ =.47 i.e. wih he growh of foreign exchange reserves, he Cenral Bank, in a monh, increases he sale of dollars, while wih he decay i reduces i. This

3 Inernaional Journal of Sysems Science and Applied Mahemaics 8; (): leads o severe flucuaions in he rae. ˆβ =.46 can mean fixaion of profi in he sock marke and is flow ino he currency in a monh, ˆ γ =.49 - he inflow of foreign invesmens a monh afer he growh of he MICEX index. The raio of hese coefficiens is approximaely equal o he dollar-ruble exchange rae. ˆβ = wha has grown in he sock marke monhs ago aracs rubles from he foreign exchange marke. You can roughly esimae he raio of he ruble capaciy of he dollar marke and he ruble capaciy of he MICEX index ˆβ + in he MICEX index by uni (.5%) causes a change funds. If we ake he effecive ˆβ, hen he change Rd of unis of his calculaion, which for absolue values roughly equal e means.%. Thus if we assume ha exacly his money is ransferred from he sock marke o he foreign exchange marke, hen he capaciy of he dollar marke is half he capaciy of he MICEX index funds. The achieved RMSE =.9 S. To check he adequacy of he model remains 7 pairs of forecass + he acual dollar rae (from o 8). RMSEA, obviously, is no expeced o be less han RMSE and was equal o.99 S. See Figure. Figure. Monhly daa from January, 5 o May, 7. The observed rae of USD / RUR (solid curve), adjused o i he forecas for informaion a monh ago (poin curve - raining), he forecas for he esimaed model (dashed curve). I can be seen ha he dashed curve unduly reflecs large flucuaions in he exchange rae, which were during he raining period and are associaed wih specific causes for ha ime. In pracical forecasing, i is naural o become aached o he daa of he mos recen period, including he momen of forecasing. Therefore, perform raining from Tl = 6 o Tu = T- = 7, i.e. ake ino accoun he las rae observaion T 8 available Y =. The resuls in Figure. Figure. Monhly daa from June, 6 o May, 7: he observed rae of USD / RUR (solid curve) and adjused o i he forecas for informaion a monh ago (poin curve - raining). The forecas for he esimaed model for he fuure momen on June, 7 (aserisk). ˆv σ =, ˆw σ =.8, he esimae ˆ γ = -.45 changed sign reflecs he effors of he Cenral Bank o reain he course during his period. The achieved RMSE =.8. The forecas for he esimaed model for he fuure momen on June, 7 is For he same period of raining, he behavior of he ruble and dollar masses paricipaing in he exchange was sudied. Esimaes of hese quaniies are available hrough esimaes

4 7 Sergei Borodachev: Monhly Forecasing of he Dollar o he Ruble Exchange Rae wih Adapive Kalman Filer of he saes of he sysem. Trading volumes resuls on he MICEX (currency marke) can confirm he correcness of hese esimaes. See Figure. The daa is aken from he sie hp:// Figure. Monhly esimaes from June, 6 o May, 7: ruble mass (solid curve), dollar mass (dashed curve). Half sum of he rading volumes on he MICEX (currency marke) for he preceding and subsequen monhs (poin curve). All values are normalized o heir values on June, 6. I is ineresing o compare he adequacy of forecasing for his model and he model proposed in []. If in ha model (wih he replacemen of he ime inerval from day o monh) we use he daa of his work, hen a Tl = and Tu = RMSE = 4.6, RMSEA = Adapive Kalman Filer Ofen he parameers of he sysem vary wih ime, so he ask arises of heir adapive esimaion and use. Noe ha someimes his dynamics of he sysem parameers is a cenral (main) ineres. In his paper, we propose he idea of a possible soluion of his problem. The basic idea is similar o he EM mehod [9], when he phases of esimaion and minimizaion are sequenially alernaed. The algorihm is as follows.. The sae esimae and is covariance marix are assumed a ime =, and also w - he number of window poins (widh) of he primary esimaion of he (vecor) parameer θ.. For =,..., (w ) KF esimaion in he window is performed and compose ˆ + + ˆ ( ) = w RMSE θ [ y yˆ ( θ )], where y - w = observed sysem oupu, y ˆ + - forecas on informaion a he ime, hen, saring from a given iniial parameer esimae ˆ θ, he parameer esimae ˆi θ = arg min RMSE( ˆ θ ) is calculaed and will be ˆ θ applied a any momen i w of clause.. The ime i is specified - he iniial posiion of he beginning of he window of he regular parameer esimaion and is widh v. Wih value ˆi θ we calculae ˆii x and i P. 4. For every = i,..., (T - v ) a. he KF esimaion in he window,..., + v- is carried ou and compose ˆ ˆ ( ) = v τ τ τ RMSE θ [ y yˆ ( θ )], hen w τ = saring wih he ˆ θ parameer esimaion, he parameer esimae 垐 + θ = arg min RMSE ( θ ) is calculaed. Wih his value we calculae x ˆ + + and. These values will be applied for he nex P + when moving he window sep forward. b. a he poin + v, using he found wih ˆ θ + sae esimaion x ˆ+ v + v and conrol v u +, an adapive forecas y ˆ + v+ + v is made. To his poin he resulan (by he immediaely preceding window) parameer esimae ˆ+ v ˆ+ rθ = θ is assigned. 5. The dynamics of esimaions of he parameer rθ ˆ+ v, ˆ θ

5 Inernaional Journal of Sysems Science and Applied Mahemaics 8; (): sae x ˆ+ v + v and forecas y ˆ + v+ + v of he oupu quaniy is colleced from he momen i + v up o he las momen of he available daa T-. Since he las one w y is used o esimae he parameers in he primary window, he final predicion y ˆ + v+ + v mus be considered as independen no earlier han a he momen + v w, and since min = i, hen i + v w. Le us apply his adapive KF o improve he RMSEA of dollar-ruble exchange rae in he range =,..., 8. If we now choose w = 7, i = 4, v =, hen RMSEA is.9 S. This v urned ou o be he bes, ha is, he bes predicions were obained wih he adjusmen of he parameers from he daa of he las momens preceding he momen of forecasing. See Figure 4, which shows he observed daa and forecass on informaion a he ime of he monh before. Figure 4. Monhly daa from November, 6 o May, 7. Observed USD / RUR rae (solid curve), KF forecass from Fig. (dashed curve), and forecass for adapive KF (poin curve). The dynamics of he esimaed parameers of he model is shown in Figure 5. All esimaes are normalized o heir absolue values a = 7: ˆβ = -5.6, ˆβ = 6.9, ˆβ =.67, ˆ γ = -., ˆ γ =.77, ˆ γ =., ˆ γ =.. Figure 5. Monhly esimaes of parameers from June, 6 o May, 7. ˆβ - he change in he moneary base in recen monhs has sligh effec on he invesmen of rubles in dollars. ˆ γ - in he

6 9 Sergei Borodachev: Monhly Forecasing of he Dollar o he Ruble Exchange Rae wih Adapive Kalman Filer summer of 6, he increase in he price of oil reduced he inflow of dollars ino he currency marke (!), recenly significanly increases. ˆ γ - shows he effors of he Cenral Bank o reain he exchange rae, especially a he end of 6 and early 7. ˆ γ - he influx of dollar invesmens in Russian asses afer he spike in Ocober-November 6 fell a large MICEX indices. Esimaes ˆβ, ˆβ and ˆ γ vary lile. The applicaion of his adapive KF o oher economic problems can be found in []. 5. Conclusion A mehod is proposed for forecasing he exchange rae hrough he modeling of he moneary masses behavior of wo respecive currencies paricipaing in he exchange. Along wih he use of convenional KF, an adapive exension of KF has been developed and applied, in which forecass are made wih model parameers opimized for he immediaely preceding window. I gives more adequae (ouof-sample) forecass and reveals he dynamics of economic sysems inner srucure. References [] Michele Ca Zorzi, Marcin Kolasa and Michał Rubaszek, Exchange rae forecasing wih DSGE models, Journal of Inernaional Economics, vol. 7, pp. 7-46, 7. [] Marin S. Eichenbaum, Benjamin K. Johannsen and Sergio Rebelo, Moneary Policy and he Predicabiliy of Nominal Exchange Raes, Naional Bureau of Economic Research Working Paper Series, No. 58, 7. [] S. M. Borodachev, Predicion of he dollar o he ruble rae. A sysem-heoreic approach, AIP Conference Proceedings, vol. 86, p. 565, 7. [4] M. Aguiar, S. Chaerjee, H. Cole and Z. Sangebye, Quaniaive Models of Sovereign Deb Crises, in Handbook of Macroeconomics, vol., Amserdam: Norh-Holland, 6, pp [5] Marín Uribe and Sephanie Schmi-Grohé, Open Economy Macroeconomics, Princeon Universiy Press, 7. [6] S. M. Borodachev, GDP and efficiency of Russian economy, AIP Conference Proceedings, vol. 96, p., 8. [7] Yemei Qin, Hui Peng, Yanhui Xi, Wenbiao Xie, Yapeng Sun and Xiaohong Chen. An adapive modeling and asse allocaion approach o financial markes based on discree microsrucure mode, Applied Sof Compuing, vol. 4, pp. 9-45, 6. [8] S. M. Borodachev, Unobservable characerisics of currency and sock markes sysem in Russia, paper presened in 4h Inernaional Mulidisciplinary Scienific Conference on Social Sciences and Ars (SGEM 7), Conference Proceedings Book, vol., pp [9] A. P. Dempser, N. M Laird, and D. B. Rubin, "Maximum Likelihood from Incomplee Daa via he EM Algorihm," Journal of he Royal Saisical Sociey, Series B, vol. 9 (), pp. -8, 977. [] S. M. Borodachev, Economic applicaions of Adapive Kalman filer, paper presened in 7h Inernaional Scienific Conference on Economic and Social Developmen (Rome, - March 8), Conference Proceedings Book, pp