Forecasting the Polish zloty with non-linear models
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1 Forecasing he Polish zloy wih non-linear models Michał Rubaszek Paweł Skrzypczyński Grzegorz Koloch WNE UW Research Seminar Oc. 14, 2010
2 Ouline 1. Moivaion 2. Relevan lieraure 3. Compeing models 4. Resuls 5. Conclusions 2
3 1. Moivaion Moivaion Shorage of empirical sudies for he Polish zloy in he lieraure on exchange rae forecasing. Our aricle aims o fill his gap. Wha we do We compare he accuracy of forecass for PLN agains EUR, USD, GBP, CHF and CZK from a random walk, Markov Swiching models and Arificial Neural Neworks. The main quesion Can he non-linear models ouperform a random walk in forecasing he zloy? 3
4 2. Relevan lieraure Saring from Meese and Rogoff (1983), who show ha moneary models canno ouperform a naive random walk in ou-of-sample ER forecasing, ER forecasing is an exensively discussed opic in he lieraure. The empirical work evolved in many direcions: oher economic models were esed, differen economeric echniques were used and analyses were conduced for various currencies, ime samples or daa frequencies. The relevan lieraure can be divided arbirarily ino wo lines of research: emphasis on he underlying macroeconomic heory: fundamenals help predic fuure ER movemens, emphasis on economeric echniques applied in he analysis: exchange raes are generaed by a sophisicaed, non-linear process. 4
5 2. Relevan lieraure Macroeconomic fundamenals help predic fuure ER movemens Wolff (1987); Canova (1993): Applicaion of moneary models wih ime-varying parameers. No able o ouperform RW in ER forecasing. Mark (1995); Chinn and Meese (1995): Long-erm ER forecass from moneary models are superior o RW forecass. Criics of hese resuls: assumpion abou coinegraion of he ER wih fundamenals no horoughly esed (Berkowiz and Giorgianni, 2001), robusness of he resuls wih respec o a change in a ime or counry sample (Kilian, 1999), revisions of macroeconomic daa no aken ino accoun (Faus e al. 2001). Cheung e al. (2002): Analysis of he mos popular ER models (moneary, UIP, HBS, BEER) for many ER and ime periods. The main finding is ha no model is able o consisenly ouperform a random walk in ER forecasing. 5
6 2. Relevan lieraure Applicaion of non-linear models Engel and Hamilon (1990): ER forecass generaed by a univariae wo-sae regimes MS process end o be more accurae han hose from a random walk model. Furher research: Engel (1994) no confirmaion for a larger group of currencies, Kirikos (2000) no confirmaion for longer ime sample. Dacco and Sachell (1999) MS models fi ER daa relaively well, bu hey do no produce superior forecass o a random walk. 6 Yu e al. (2007) The survey of 45 journal aricles using ANNs for ER forecasing shows ha he relaive success of ANNs depends on he ime sample, he frequency of daa and he group of currencies under consideraion.
7 2. Relevan lieraure Cenral & Easern European counries Crespo-Cuaresma and Hlouskova (2005); Ardic e al. (2008) The only wo aricles (according o our bes knowledge) ha invesigae he accuracy of model-based forecass for he currencies of CEE counries. The resuls of hese aricles, which are using linear models, show ha a random walk model ends o be a very difficul benchmark o bea in he case of he CEE currencies, including he Polish zloy. Our conribuion o he lieraure The firs empirical work ha ess he usefulness of MS models and ANNs in forecasing he Polish zloy agains EUR, USD, CHF, GBP and CZK. We address he relaive shorage of he empirical work for he Polish zloy in he lieraure on exchange rae forecasing. 7
8 3. Compeing models We compare forecass of he PLN rae agains EUR, USD, GBP, CHF and CZK from he following models: i. linear models: random walk (RW) fracional random walk (FRW) ii. non-linear models: Markov swiching models (MS) 3 specificaions: MS(2), MS(3) i MS(TS) arificial neural neworks (ANN) 2 specificaions: ANN-L(arge) and ANN-S(mall) A random walk is used as a benchmark model in forecass evaluaion. 8
9 3. Compeing models Random Walk & Fracional Random Walk Random walk model (1 L) y ~ 2 NID(0, ) Fracional random walk (1 L) d y 2 ~ NID(0, ) The fracional inegraion parameer d is esimaed wih he Geweke and Porer-Hudak (1983) algorihm of he log periodogram regression. 9
10 3. Compeing models Markov Swiching models: MS(2), MS(3) & MS(TS) MS(K) model for K=2,3: S K-regime Markov chain wih he ransiion probabiliies p ij = P(S =j S -1 =i). For S = s he log growh rae of he nominal exchange rae Δy equals o: y s, where 2 ~ NID(0, s ) MS(TS) model (TS denoes rading sraegy) Regime 1 describes he marke dominaed by fundamenaliss : y 2 ( y, where 1 y 1) ~ NID(0, 1 ) Regime 2 describes he marke dominaed by chariss : y y 1, where ~ NID(0, 2 2 ) 10 Parameers esimaion: Expecaion-maximizaion algorihm
11 3. Compeing models Arificial neural neworks: ANN-L & ANN-S Arificial neural nework: p hidden layers, q neurons in each hidden layer, m lags of y in he inpu layer. For ANN-L: p=3, q=3, m=10 For ANN-S: p=1, q=2, m=3 The parameers of ANNs are compued by minimizing he in-sample sum of squared errors. We use he backpropagaion echnique wih he Levenberg-Marquard algorihm. 11
12 4. The resuls We use weekly, end-of-period daa for he raes EUR/PLN, USD/PLN, GBP/PLN, CHF/PLN and CZK/PLN from he period Each model is esimaed on he se of he recursive samples saring in 1999:1 and ending in one of he weeks from he period (312 weeks), denoed by T. Then ou-of-sample forecass are generaed for periods T+1,T+2,,T+52. Due o few badly-behaved forecass, we conrol for he ouliers in he forecass by allowing he forecased pahs of he exchange raes o vary wihin he ±5% band in reference o he las observaion T. 12
13 4. The resuls Forecass for EUR/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S 13
14 4. The resuls Forecas evaluaion mehods Evaluaed forecass horizons, h: 1, 4, 8, 12, 26 and 52 weeks. Forecas evaluaion sample: 2004:2-2009:52 (312 weeks). Number of observaions for evaluaing h-sep ahead forecass: 313-h. Forecass are evaluaed on he basis of hree ess: i. forecas unbiasedness es ii. equal forecas accuracy es iii. forecas encompassing es 14
15 4. The resuls Forecas unbiasedness es For a given ER, model and forecas horizon h we es: H0: Mean Forecas Error = 0 To es he null we use he p-value of he coefficien of he forecas errors regression on a consan. To correc for heeroskedasiciy and auocorrelaion we use he HAC covariance marix esimaes obained via he modified Barle kernel in line wih Newey and Wes (1987), where he runcaion lag is se auomaically as proposed by Newey and Wes (1994). 15
16 4. The resuls Forecas unbiasedness es (EUR/PLN) EUR/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S Noes: 1/ Shaded figures indicae minimal absolue value of he MFE for a given forecas horizon. 2/ A posiive MFE indicaes ha on average forecass are below he acual values. 3/ Symbols ***, ** and * indicae he rejecion of he null a 1%, 5% and 10% significance levels, respecively. 16
17 4. The resuls Forecas unbiasedness es (USD, GBP, CHF, CZK) 17 USD/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S * * *** *** * * ** * * * ** * * * * * GBP/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S * * * * * ** ** * ** ** ** *** *** ** *** *** *** *** *** *** *** ** *** *** *** *** *** CHF/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S CZK/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S * *** *** ** **
18 4. The resuls Equal forecas accuracy es For a given ER, model and forecas horizon h we es: 2 2 H0 : E(e M, erw, ) 0 which means ha he Roo Mean Squared Forecas Error from model M is no significanly differen from he RMSFE from a random walk The null is verified wih he Harvey-Leybourne-Newbold (1997) modificaion of he Diebold-Mariano (1995) es 2 2 The long-run variance of he series em, erw, is esimaed via he modified Barle kernel, where he runcaion lag is se o h-1. 18
19 4. The resuls Equal forecas accuracy es (EUR/PLN) EUR/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S ** *** 1.05* * * 1.15*** 1.07* *** * Noes: 1/ For a RW model RMSFEs are repored in levels, for he oher models as a raio o a RW RMSFE. 2/ A raio below uniy indicaes ha he a given model ouperforms a RW (shaded figures). 3/ Symbols ***, ** and * indicae he rejecion of he null of he HLN-DM es a 1%, 5% and 10% significance levels. 19
20 4. The resuls Equal forecas accuracy es (USD, GBP, CHF, CZK) 20 USD/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S * 1.26*** 1.08*** ** 1.21*** 1.14*** * 1.13*** 1.17*** *** 1.13*** ** 1.10*** * GBP/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S *** *** 1.12*** *** *** 1.14*** ** *** 1.14** ** 1.13** * CHF/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S *** ** 1.14*** * 1.11*** *** *** CZK/PLN RW FRW MS(2) MS(3) MS(TS) ANN-L ANN-S *** 1.10*** * *** 1.14** ** ** ** ** ** * **
21 4. The resuls Forecas encompassing es For a given ER, model and forecas horizon h we es wheher forecass from model M encompass forecass from a RW The hypohesis is verified by running he regression: y c β M y F M, β RW y F RW, ε and esing wheher β M 0 and β RW = 0. To correc for heeroskedasiciy and auocorrelaion we use he HAC covariance marix esimaes obained via he modified Barle kernel in line wih Newey and Wes (1987), where he runcaion lag is se auomaically as proposed by Newey and Wes (1994). 21
22 4. The resuls Forecas encompassing es (EUR/PLN) EUR/PLN FRW MS(2) MS(3) MS(TS) ANN-L ANN-S *** 1.02** ** * *** *** *** 1.14** *** * 2.26*** *** *** ** 1.25** ** ** 2.89*** *** -0.32** 1.13*** ** ** ** ** 2.91*** *** -0.35** 1.05*** *** -1.12** 2.04*** -1.94*** 1.97*** -1.84*** -1.45* 1.68*** *** -0.35** 0.64*** *** -1.92*** 1.53*** -1.83*** 2.15*** -2.50*** Noes: 1/ Shaded figures indicae cases where he coefficien represening a given model forecas is significanly differen from zero and he coefficien represening a random walk forecas is insignifican. These cases indicae he evidence of forecas encompassing by a given model in reference o a random walk. 2/ Symbols ***, ** and * indicae he rejecion of he null ha he given coefficien is equal o zero a 1%, 5% and 10% significance levels, respecively. 22
23 4. The resuls Forecas encompassing es (USD, GBP, CHF, CZK) USD/PLN FRW MS(2) MS(3) MS(TS) ANN-L ANN-S *** *** *** *** *** *** * * ** *** *** ** ** *** -0.40* 1.27*** * ** *** *** * ** -0.97*** 1.39*** *** -2.10*** * -1.10*** 1.24*** GBP/PLN FRW MS(2) MS(3) MS(TS) ANN-L ANN-S * 1.27*** * *** *** ** 1.43*** *** *** *** *** 1.63*** ** *** *** *** *** 1.65*** *** *** *** ** 1.35*** * -0.79* 1.49*** 0.96* *** *** *** -1.17** *** -1.95*** -0.58* 1.03*** -0.50* 0.96*** CHF/PLN FRW MS(2) MS(3) MS(TS) ANN-L ANN-S * ** ** 0.24*** 0.75*** *** *** *** *** *** ** *** * 2.25*** *** *** ** ** ** 2.32*** -0.29* 1.04*** *** ** *** -2.52** 2.66*** -2.50*** -1.26* 1.63*** -0.40** 0.78*** -0.53* 0.88*** *** -2.30*** 4.25*** -4.83*** 3.35*** -3.82*** -1.29*** 1.30*** -0.50** 0.44** -0.48** 0.40* CZK/PLN FRW MS(2) MS(3) MS(TS) ANN-L ANN-S ** *** *** *** 1.57** ** *** *** *** 2.47*** -1.70* 2.62*** -1.80* * *** *** *** 2.78*** -2.10** 1.45* ** *** *** * * 3.92** -2.81*** 2.80*** *** *** *** 7.96*** -8.61*** 10.79*** -4.54*** 2.91*** -0.79*** 0.84*** -0.57** 0.57*
24 5. Conclusions 1. MS models performed somewha beer han ANNs in forecasing he zloy (his needs furher invesigaion). 2. MS models are well suied o describe in-sample dynamics of he PLN raes, bu hey were unable o predic he ou-of-sample urning poins. 3. We found a general endency ha small ANNs ended o produce smaller ou-of-sample errors han large ANNs. 4. If exchange rae series are characerised by highly non-linear, selfrepeaing paerns, hen, since larger neworks are capable of capuring hese dynamics, one would expec hem o produce more accurae forecass. Our experimen suggess however, ha his is no he case. 5. The main conclusion of our sudy is ha he analysed models were no able o consisenly ouperform he random walk in forecasing he Polish zloy. 24
Forecasting the Polish Zloty with Non-Linear Models
Central European Journal of Economic Modelling and Econometrics Forecasting the Polish Zloty Non-Linear Models Michał Rubaszek, Paweł Skrzypczyński, Grzegorz Koloch Submitted: 17.08.2010, Accepted: 3.11.2010
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