The Analysis of Czech Macroeconomic Time Series (L'analyse des séries temporelles macroéconomiques tchèque)

Transcription

1 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p.6270 The Analysis of Czech Macroeconomic Time Series (L'analyse des séries emporelles macroéconomiques chèque) Marek, Lubos Deparmen of Saisics and Probabiliy, Universiy of Economics, Prague W. Churchill sq. 4 Prague, , Czech Republic marek@vse.cz Inroducion In his paper, we will analyze he ime series of Czech Republic expors and impors. The paper is focused on applicaions; is goal is o idenify models suiable for he ime series of Czech Republic expors (impor) and uilizaion of such models for predicions. Daa of he Czech Republic foreign rade ime series are published monhly by he Czech Saisical Office. However, he published daa are no acual a he firs sage, only esimaes are published, which are subsequenly made more accurae (in several seps). Moreover, even hese esimaes are published wih a ime delay of wo o hree monhs. For example, a he ime of wriing his paper (May 2011), he values of he above-menioned ime series are only known unil February On he oher hand, values of cerain series (such as he exchange raes of CZK wih respec o oher currencies) are published more or less immediaely and are exac. Hence, if we are able o idenify a model describing he dependence beween he above-menioned ime series, such a model can, a leas, considerably speed up consrucion of preliminary esimaes, hus significanly reducing he waiing ime for one of he mos imporan and mos closely wached macroeconomics ime series in he Czech Republic. In his paper, we will analyze he ime series of expors (impors) from he Czech Republic as a whole. Then we will ry o prove is muual dependence wih he CZK/USD (or CZK/EUR) exchange rae ime series. A suiable model will be sough by uilizing he heory of sochasic models for ime series [1], [2], [3], [4], [7], as well as he heory of ransfer funcion models [5], [6]. Our effor will be focused on consruing predicions of he CR expor ime series for several fuure periods. The source of our daa is he Czech Saisical Office (ime series of Czech Republic expors, expressed in million CZK in fixed prices) and he Czech Naional Bank (ime series of he CZK/EUR and CZK/USD exchange raes expressed as monhly averages). All of hese series have monhly values and were racked in he following periods of ime: CR expors, January 1996 February 2011 (182 observaions); CZK/USD exchange rae, January 1996 April 2011 (184 observaions); and CZK/EUR exchange rae, January 1999 April 2011 (148 observaions). The analysis and all calculaions were carried ou using he SCA sofware. Mehodology We applied he SARIMA models in he general form o he ime series analysis. L d D L (B) (B )(1- B ) (1- B ) Y = (B) (B ) p P q Q In addiion o he usual means (ACF, PACF), oher funcions (EACF, IACF) and mehods (SCAN, corner able) were used for model idenificaion. All analyzed series were nonsaionary and had o be ransformed (by sandard and seasonal differences) in order o

2 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p.6271 achieve saionariy. The saionariy was esed wih he aid of several differen crieria uni roo ess, homoscedasiciy ess, ec. We calculaed he cross-correlaion funcion (CCF) in order o esablish linear dependence beween he ransformed (i.e., already saionary) series. This value confirmed he linear dependence beween he Expors and CZK exchange rae series. Afer ha, a model wih ransfer funcion in he general form was se up, 1 Y c X X X... X K K L (1 1(B) )(1 1( B )) in which he oupu series Y sands for he expors (afer he respecive ransformaions) and he inpu series X sands for he CZK exchange rae (also afer he respecive ransformaions), and he noise series is he las componen of he model N. We used he LTF mehod for esimaing he parameers cf. e.g. [5] and [6]. The resuling model was used for calculaing predicions. Analysis of he CR expors ime series Model 1 Firs, we will find a suiable SARIMA model for expor series. Afer a horough analysis and sudying ACF, PACF, EACF, IACF, uni roo ess and homoscedasiciy ess, we derived a raher complicaed model (cf. he compuer oupu): ( B 0.249B B ) Y ( B)( B ) where Y (1 B)(1 B ) Expor, is he whie noise. This model has been successfully verified and proven as fully adequae; his fac is also indicaed by he index of deerminaion, which is equal o EXPORTN RANDOM ORIGINAL (1-B ) (1-B ) 1 TH1 EXPORTN MA 1 1 NONE TH EXPORTN MA 2 NONE PHI3 EXPORTN AR 1 3 NONE PHI5 EXPORTN AR 1 5 NONE PHI10 EXPORTN AR 1 10 NONE EFFECTIVE NUMBER OF OBSERVATIONS R-SQUARE RESIDUAL STANDARD ERROR E+04 Le us now se up a model for he ime series of he CZK/USD exchange rae (hereinafer he USD series). The corresponding SARIMA for his series is: (1 B) X ( B ) where X = USD and he index of deerminaion is equal o Boh he model and he char show ha he USD series had o be differeniaed boh sandard and seasonal differeniaion ook place.

3 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p USDN RANDOM ORIGINAL (1-B ) 1 TH1 USDN MA 1 1 NONE EFFECTIVE NUMBER OF OBSERVATIONS R-SQUARE RESIDUAL STANDARD ERROR E+00 On he SCA sofware oupu we will have a look a he cross-correlaion funcion, now beween saionary series (1 B)(1 B ) Y and (1 BX ) ; he funcion deermines boh he inensiy and he direcion of linear dependence. Le us calculae he value of he crosscorrelaion funcion (hereinafer CCF) of boh differeniaed series. NAMES OF THE SERIES USDN EXPORTN EFFECTIVE NUMBER OF OBSERVATIONS STANDARD DEVIATION OF THE SERIES MEAN OF THE (DIFFERENCED) SERIES STANDARD DEVIATION OF THE MEAN T-VALUE OF MEAN (AGAINST ZERO) CORRELATION BETWEEN EXPORTN AND USDN IS -.09 CROSS CORRELATION BETWEEN USDN(T) AND EXPORTN(T-L) ST.E CROSS CORRELATION BETWEEN EXPORTN(T) AND USDN(T-L) CCF char: ST.E I IX XI IXX I IX IX I IXX XI I XXI IXXX+XX XXI IX XI XI IX XI I I XI XI IXX XXXI +. + IXXX+ Boh he calculaed values and he char of CCF (a he 95% confidence inerval) imply ha here is only one significan value of CCF, namely, a ime -1. This means here is a significan linear dependence beween he (differeniaed) Expors ime series a ime and (again differeniaed) USD imes series a ime -1. Hence we can ry o idenify a ransfer funcion model. We uilize he LTF mehod [5] for such idenificaion. The only significan weigh which occurs is ν 1 ; ha is, he oupu (Expors) series depends on he inpu (USD) series' value wih a ime delay equal o 1. This was already indicaed by he CCF char. All oher weighs are insignifican. Afer laborious idenificaion we ge he following model for he error series:

4 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p EXPORT RANDOM ORIGINAL (1-B ) (1-B ) 1 USDN RANDOM ORIGINAL (1-B ) (1-B ) 1 V1 USDN NUM. 1 1 NONE TH1 EXPORT MA 1 1 NONE TH EXPORT MA 2 NONE PHI3 EXPORT AR 1 3 NONE PHI5 EXPORT AR 1 5 NONE PHI10 EXPORT AR 1 10 NONE EFFECTIVE NUMBER OF OBSERVATIONS R-SQUARE RESIDUAL STANDARD ERROR E+04 wih he resuling ransfer funcion model wrien as ( B 0.25B B ) Y * X-1 ( B)( B ) where Y (1 B)(1 B )* Vyvoz and X (1 B)* USD This model has been successfully verified and proven as fully adequae. Qualiy of he model is also indicaed by he index of deerminaion, which is equal o The values of cross correlaion funcion beween residuals of SARIMA inpu series and residuals of ransfer funcion model are no signican as we can see from he folloving oupus: CCF chars: I -. + IXXX I IX XI XXI XI IX XI IX I XXI I XXI IXXXX XXXI XI XXXXI IX XI IXX IX IX XI IXX IX + CORRELATION BETWEEN RESY3 AND RESY2 IS -.08 CROSS CORRELATION BETWEEN RESY2(T) AND RESY3(T-L) ST.E CROSS CORRELATION BETWEEN RESY3(T) AND RESY2(T-L) ST.E We can herefore observe ha he Expors ime series (afer sandard and seasonal differeniaion) a ime depends on is pas values (wih he ime shif equal o 3, 5 and 10), he values of he CZK/USD exchange rae ime series (afer sandard and seasonal

5 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p.6274 differeniaion) a ime -1, and he pas values of he error series (wih he seasonal parameer). Le us have a look a he predicions for which he model was sough in he firs place. A he ime of wriing his paper (beginning of May 2011), he Expors ime series values were only known unil February 2011, bu he CZK/USD exchange rae ime series values were also known ill April 2011 (CNB publishes he exchange rae values on is websie a 16:00 hours every day). We are hus able o pu he currem exchange rae values ino he model and esimae he CR expors for he nex hree period much more accurae way han on he basis of he inpu series values only. The reason is ha we do no uilize esimaes of fuure values of he inpu series. Tab. 1: Predicions of he CR expors ime series (Model 1) Predicions Monh SARIMA model TFM March 246,953 (8318) 245,283 (8171) April 235,986 (9200) 234,640 (9162) The abled values show predicions for he original seasonal ARIMA model and for he ransfer funcion model; he sandard deviaion values of he predicions are given in parenheses afer he esimaed value. We can see ha he TFM predicions and SARIMA predicions are very similar. We can furher observe ha he SARIMA predicions for boh monhs are higher han he TFM ones. Analysis of he CR expor ime series Model 2 Le us now idenify a similar ransfer funcion model, bu he CZK/EUR exchange rae (EUR series) will be aken for he inpu series. The mos suiable model is idenified as (1 B) X (1 214 B ) wih he index of deerminaion equal o 0.987, where X = Euro. We can clearly see from he model ha sandard differeniaion of he EUR series was necessary o achieve saionariy. EURN RANDOM ORIGINAL (1-B ) 1 THI1 EURN MA 1 1 NONE EFFECTIVE NUMBER OF OBSERVATIONS R-SQUARE RESIDUAL STANDARD ERROR E+00 We have o realize ha we have daa from a differen period of ime for he Expors ime series, namely, years We apply he previous model bu wih differen esimaes of is parameers: ( ) ( )( ) B B B B Y B B and he index of deerminaion is equal o

6 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p EXPORTN RANDOM ORIGINAL (1-B ) (1-B ) 1 TH1 EXPORTN MA 1 1 NONE TH EXPORTN MA 2 NONE PHI3 EXPORTN AR 1 3 NONE PHI5 EXPORTN AR 1 5 NONE PHI10 EXPORTN AR 1 10 NONE PHI EXPORTN AR 1 NONE EFFECTIVE NUMBER OF OBSERVATIONS.. 1 R-SQUARE RESIDUAL STANDARD ERROR E+04 The cross-correlaion funcion values are now calculaed beween saionary series (1 B)(1 B ) Y (denoed by EXPND1 in he compuer oupu) and (1 BX ) (denoed by EURND1). Le us idenify he inensiy and direcion of he linear dependence. TIME PERIOD ANALYZED TO 146 NAMES OF THE SERIES EURND1 EXPND1 EFFECTIVE NUMBER OF OBSERVATIONS STANDARD DEVIATION OF THE SERIES MEAN OF THE (DIFFERENCED) SERIES STANDARD DEVIATION OF THE MEAN T-VALUE OF MEAN (AGAINST ZERO) CORRELATION BETWEEN EXPND1 AND EURND1 IS -.13 CROSS CORRELATION BETWEEN EURND1(T) AND EXPND1(T-L) ST.E CROSS CORRELATION BETWEEN EXPND1(T) AND EURND1(T-L) ST.E I IXX IX IXXX IXX XXI IXXX IXX XI IXX I XXXI IXXX+XXX XXXI XXI I XXXI IXX I XI IXX I XXXI IX I XXI + Boh he calculaed values and he char of CCF (a he 95% confidence inerval) imply ha here is only one significan value of CCF, namely, a ime -1. This means here is a significan linear dependence beween he (differeniaed) Expors ime series a ime and (again, differeniaed) EUR ime series a ime -1. Le us idenify a ransfer funcion model. We again uilize he LTF mehod [5] for such idenificaion. The only significan weighs which occur are ν 0 and ν 1, ha is, he oupu (Expors) series depends on he inpu (EUR) series value wih a ime delay equal o 1 (where boh series have been differeniaed as

7 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p.6276 specified above). This was already indicaed by he CCF char. All oher weighs are insignifican. Afer laborious idenificaion we ge he following model for he error series: 1 EXPORTN RANDOM ORIGINAL (1-B ) (1-B ) 1 EURN RANDOM ORIGINAL (1-B ) 1 V0 EURN NUM. 1 0 NONE V1 EURN NUM. 1 1 NONE TH1 EXPORTN MA 1 1 NONE TH EXPORTN MA 2 NONE PHI3 EXPORTN AR 1 3 NONE PHI5 EXPORTN AR 1 5 NONE PHI10 EXPORTN AR 1 10 NONE PHI EXPORTN AR 1 NONE EFFECTIVE NUMBER OF OBSERVATIONS.. 1 R-SQUARE RESIDUAL STANDARD ERROR E+04 The resuling ransfer funcion model is ( B 0.252B 0.228B B ) Y -5328* X +6421* X-1 ( B)( B ) where Y (1 B)(1 B )* Expor and X (1 B)* Euro This model has been successfully verified and proven as fully adequae. Qualiy of he model is also indicaed by he index of deerminaion, which is equal o The values of cross correlaion funcion beween residuals of SARIMA inpu series and residuals of ransfer funcion model are no signican as we can see from he folloving oupus: CORRELATION BETWEEN RESY3 AND RESY1 IS -.05 CROSS CORRELATION BETWEEN RESY1(T) AND RESY3(T-L) ST.E CROSS CORRELATION BETWEEN RESY3(T) AND RESY1(T-L) ST.E I I IXXXXX XI I IXX IX IX IXX I IX XXXI IXX XI XXXXI XXXI I IXXX XI IXXX I XXI XI XI I XXI +

8 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p.6277 We can herefore observe ha he Expors ime series (afer sandard and seasonal differeniaion) a ime depends on is pas values (wih he ime shif equal o 3, 5,10 and ), he values of he CZK/EUR exchange rae ime series (afer sandard differeniaion) a ime poin -1, and he value of he error series. Le us compare he predicions obained by he SARIMA and TFM models: Tab. 2: Predicions of he CR expors ime series (Model 2) Predicions Monh SARIMA model TFM March 246,713 (8749) 242,861 (8519) April 235,422 (9638) 233,944 (9335) The abled values show predicions for he original seasonal ARIMA model and for he ransfer funcion model; he sandard deviaion values of he predicions are given in parenheses afer he esimaed value. The SARIMA predicions for all hree monhs are higher han he TFM predicions. Now we can compare predicions wihin he SARIMA and TFM models for Model 1 and Model 2. Tab. 3: Predicions of he CR expors Model 1 vs. Model 2 Model 1 (USD) Model 2 (EUR) SARIMA TFM SARIMA TFM March 246, , , ,861 April 235, , , ,944 Analysis of he CR impor ime series Model 3 We repea whole analysis for impor series. We build TFM model wih wih oupu series Impor and inpu series USD (EUR). The will work by he same way and we do no wrie he deails of our analysis. Only final resuls are published wihou compuer oupus. The mos suiable SARIMA model for Impor series is ( B 0.273B B B ) Y ( B)( B ) where Y (1 B)(1 B ) Impor, is he whie noise. This model has been successfully verified and proven as fully adequae; his fac is also indicaed by he index of deerminaion, which is equal o The corresponding SARIMA for his USD series we know from previous analysis: (1 B) X ( B ) where X = USD and he index of deerminaion is equal o The final TFM model is where ( ) B B B B Y = 2102* X * X-2 ( B)( B ) Y (1 B)(1 B )* Impor and (1 )* X B USD

9 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p.6278 This model has been successfully verified and proven as fully adequae. Qualiy of he model is also indicaed by he index of deerminaion, which is equal o The abled values show predicions for he original seasonal ARIMA model and for he ransfer funcion model Tab. 4: Predicions of he CR impors ime series (Model 3) Predicions Monh SARIMA model TFM March 227,050 (8421) 228,402 (8109) April 216,590 (9197) 213,865 (9089) Analysis of he CR impor ime series Model 4 The mos suiable model is known from previous analysis (1 B) X (1 214 B ) wih he index of deerminaion equal o 0.987, where X = Euro. The mos suiable SARIMA model for Impor series is ( ) ( )( ) B B B B Y B B wih index of deerminaion The resuling ransfer funcion model is where ( ) B B B B Y = -4216* X +7146* X * X-2 ( B)( B ) Y (1 B)(1 B )* Impor and (1 )* X B Euro Tab. 5: Predicions of he CR impors ime series (Model 4) Predicions Monh SARIMA model TFM March 227,601 (9093) 231,341 (8846) April 216,687 (9987) 217,492 (9959) Now we can compare predicions wihin he SARIMA and TFM models for Model 3 and Model 4. Tab. 6: Predicions of he CR expors Model 3 vs. Model 4 Model 3 (USD) Model 4 (EUR) Monh SARIMA TFM SARIMA TFM March 227, , , ,341 April 216, , , ,492

10 In. Saisical Ins.: Proc. 58h World Saisical Congress, 2011, Dublin (Session CPS020) p.6279 Conclusions We have successfully proven a muual relaionship beween he ime series of Czech Republic expors (impors), expressed in million CZK in fixed prices, and he ime series of he CZK/USD (CZK/EUR) exchange rae. The analysis led us o creaing a model based on a ransfer funcion. I urned ou ha wih he growing values of he CZK exchange rae (weakening Czech Crown), he expors are also growing. Since he values of he CZK exchange rae are known wo o hree monhs before he expor (impor) values, such models enable us o predic he expor (impor) ime series immediaely afer he end of he respecive monh. In comparison, he SARIMA predicions are higher han he predicions wihin he TFM models. Sandard deviaions for TFM forecass are mosly smaller han sandard deviaions for SARIMA forecass. An advanage of he TFM models is he fac ha acual, observed values are used in he inpu series raher han he usual esimaes. REFERENCES [1] Abraham, B. Ledoler, J.: Saisical Mehods for Forecasing, John Wiley & Sons, New York, [2] Anderson, O. D.: Time series analysis and forecasing Box-Jenkins approach. London, Buerworh [3] Box, G. E. P. Jenkins, G. M. Reinsel, G.C.: Time series analysis, forecasing and conrol. Third ediion. Prenice-Hall, Inc. Englewood Cliffs, New Jersey, Forecasing and Time Series Analysis using he SCA Saisical Sysem. Volume 1. Scienific Associaes Corp. Oak Brook, Illinois, USA, [4] Granger, C. W. J., Newbold, P.: Forecasing Economic Time Series. Academic Press. New York, [5] Pankraz, Alan: Forecasing wih dynamic regression models. John Wiley & Sons inc., New York, [6] The SCA Saisical Sysem. Reference manual for general saisical analysis. Scienific Associaes Corp. Oak Brook, Illinois, USA, [7] Wei, W. W. S.: Time series analysis Univariae and mulivariae mehods. Redwood Ciy, California, Addison-Wesley Publishing Company Absrac The goal of his paper is o prove a muual relaionship beween he ime series of he Czech Republic expors (impors), expressed in million CZK in fixed prices, and he ime series of he CZK/USD (or CZK/EUR) exchange rae. This relaionship can be described using a ransfer funcion model. Such models can be uilized in predicing he CR expor (impor) ime series. There is a ime-delayed muual dependence beween he expor (impor) ime series and he CZK/USD (or CZK/EUR) exchange rae ime series. Since he values of he CZK exchange rae are known wo o hree monhs before he expor values, such models enable us o predic he expor ime series immediaely afer he end of he respecive monh. Moreover, he model has an addiional advanage he predicion is based on acual values of he inpu series (and no prediced values, as is he usual case).