OUT-OF-SAMPLE FORECASTING PERFORMANCE OF A ROBUST NEURAL EXCHANGE RATE MODEL OF RON/USD

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1 6. OUT-OF-SAMPLE FORECASTING PERFORMANCE OF A ROBUST NEURAL EXCHANGE RATE MODEL OF RON/USD Corina SAMAN 1 Absrac This paper aims o explore he forecasing accuracy of RON/USD exchange rae srucural models wih moneary fundamenals. I used robus regression approach for consrucing robus neural models less sensiive o conaminaion wih ouliers and I sudied is predicabiliy on 1 o 6-monh horizon agains nonrobus linear and nonlinear regressions and, especially, random walk. The resuls show ha robus model wih low breakdown poin improve he forecas accuracy of RW and AR models on 1- and 4-monh horizon and performs beer han RW a all ime horizons. Keywords: exchange rae, forecasing, neural neworks, ouliers JEL Classificaion: E22, C22, C45, C53 I. Inroducion Esimaion of exchange rae models is a very imporan ask for any open small economy, especially because he developmen is based on compeiiveness and inernaional rade, and hus exposed he economy o exchange rae risk. Exchange raes shows large changes over ime as a resul of crises in he marke, changing in economic policies or due o economic cycles. The presence of ouliers in he daa series can disrup he predicion, bu heir neglec can lead erroneously inadequae or poorly specified models (van Dijk, Franses & Lucas, 1999; Preminger & Frank, 2007). Many sudies show ha exchange raes are unpredicable because he srucural models do no perform beer han random-walk (RW) models in predicing exchange rae. 1 Insiue for Economic Forecasing, Romanian Academy, csaman@ipe.ro Romanian Journal of Economic Forecasing XVIII (1)

2 Insiue for Economic Forecasing In a seminal paper, Meese and Rogoff (1983) compared srucural linear models for differen exchange raes wih RW for one o welve monh horizons and reached he same conclusion. Linear moneary and porfolio models are sudied and he resul on heir explanaory power are mixed (MacDonald & Taylor, 1991, 1994; Groen, 2000; Chinn, 2000; Alquis & Chinn, 2006). The lieraure suggess a leas wo imporan heoreical reasons for he poor resuls of srucural linear models o esimae he exchange rae: he need for a more dynamic han ha resuling from saic money demand equaion and a more complex mechanism for adjusing relaive prices. Thus o improve forecasing accuracy Somanh (1986) proposed dynamic srucural models wih variable delay and obained beer resuls denying he conjecure of inabiliy o bea he random walk model se by Meese & Rogof. Laer, more sudies dedicaed o his problem explained his negaive resul by he nonlineariy in he exchange raes and ried o capure his phenomenon using nonparameric models, such as kernel regression (Meese and Rose, 1991; Diebold and Nason, 1990), mulivariae GARCH model using M-esimaion (Boud and Croux, 2007) or Markov-swiching models (Engle, 1994; Yuan, 2011), bu wihou obaining more accurae models in erms of forecasing accuracy. Also, he soluion of neural models is applied o his problem wih higher ou-ofsamples forecas accuracy. Kuan and Liu (1995) use a model based on a neural nework for forecasing he daily exchange raes, which generaes smaller MSE forecasing errors han RW models. Anoher recen aricle - ha of Preminger and Frank (2007) - analyzes he forecasing performance of neural robus auoregressive models based on S-esimaors. The resuls are encouraging indicae ha robus models are beer han classical ones. The conclusions regarding he comparison of he neural models relaive o random walk are saisically significan in several cases wih he amendmen ha on sandard measures RMSE and MAE hey show beer resuls. Some nonlinear models wih moneary fundamenals are also employed. Qi and Wu (2003) sudied shor and medium erm predicabiliy of exchange raes using neural models wih macroeconomic fundamenals and found ou ha canno bea randomwalk bu occasionally exhibis limied marke-iming abiliy. In general hey concluded ha neural models presens poor predicabiliy when are more complex and when he horizon lenghens. Yuan (2011) sudies he dynamics of he exchange rae for four USD exchange rae using nonlinear models of Markov ype processes wih regime change in volailiy where he ransiion probabiliies from one sae o anoher are deermined by he fundamenal values resuling from srucural models based on macroeconomic variables (broad money, income, differences in ineres raes and foreign rade balance of he wo counries ha corresponded o he exchange rae) and finds ha he marke responds o macroeconomic changes forcing regime change in he ARCH process. In his paper, he idea of robus regression is also used in order o reduce he impac of ouliers on he esimaion. Due o major changes over ime caused by he economic crises, marke crashes or simply business cycles, he ouliers may have a major effec on he exchange raes. 94 Romanian Journal of Economic Forecasing XVIII (1) 2015

3 Ou-of-sample Forecasing Performance The research quesion of his paper is o invesigae he predicabiliy of he exchange rae movemens a differen forecas horizons (1-6 monhs) using linear and nonlinear specificaions, sandard and robus esimaion mehods. The ou-of-sample predicive abiliy of hese models is compared and i is found ou wheher he robus regressions may provide beer performance and wheher models wih moneary fundamenals work beer han he auoregressive models. The srucure of he paper is he following. Secion 2 presens a brief descripion of he specificaion of he models ha are compared; Secion 3 describes he neural models and robus esimaion procedure; Secion 4 inroduces he daa, he forecasing evaluaion mehods and he empirical resuls; and Secion 5 concludes. II. Exchange Rae Models II.1. The Forecasing Problem In his paper, an approach based on neural neworks robus o ouliers is conduced for predicing he monhly changes in he RON/USD exchange rae. I is based on srucural models ha use as deerminans variables ofen used in he empirical models. The moneary model wih flexible prices of Frenkel-Bilson, he model wih sicky prices of Dornbusch-Frankel or he Hooper-Moron model can be described by a reduced form specificaion (Meese and Rogoff, 1983): s a a ( m m) a ( y y) a ( r r) a ( ) a TB a TB u, (1) e e 0 1 f 2 f 3 f 4 f 5 f 6 where he variables are he logarihm of he exchange rae (s), he logarihm of he raio of he money supply o he foreign money supply ( m f m ), he logarihm of he raio of he wo counries real income ( ( r f r ), he expeced long-ime inflaion differenial ( y f y ), he shor-erm ineres rae differenial e f ) and f TB ; TB represen he foreign rade balances. The error erm u may have serial correlaion. The Frenkel-Bilson model assumes he purchasing power pariy, consraining o a 4=a 5=a 6=0. The Dornbusch-Frankel model allows for deviaion from he purchasing pariy assuming a 5=a 6=0, and he Hooper-Moron model does no impose any resricion on he coefficiens. Meese and Rogoff (1983) demonsraed ha here is no gain in he ou-of-sample forecas accuracy if he model considers separae coefficiens for he variables corresponding o he wo counries. II.2. The Hooper-Moron Model The Hooper-Moron model (as presened in Hooper and Moron, 1982) is based on boh moneary and porfolio balance heories. Romanian Journal of Economic Forecasing XVIII (1)

4 I is summarized by equaions (2a) - (2f) 2. Insiue for Economic Forecasing ( log e) e rf r (2a) e e e ( log e) (log( e) log e) ( log e) (2b) e e e f ( log e) (2c) e e e e e ( P f / P )* q (2d) M / P y exp r (2e) r f f f f M / P y exp (2f) The relaion (2a) represens he open ineres pariy in an open economy adding he risk rae. The relaion (2b) expresses he adjusmen afer a moneary shock (because prices are sicky) of he exchange rae. The difference beween he expeced values of he monhly changes in he exchange rae and he changes in he equilibrium value is proporional o he differenial beween he spo value and he marke expecaion value of he curren equilibrium exchange rae, being he velociy of adjusmen afer he shock The equaion (2d) expresses he equilibrium nominal rae as a produc of relaive price and real componen. Wih no changes in he equilibrium real exchange rae, relaion (2d) becomes he long-run purchasing pariy condiion. (2e) and (2f) represen he equilibrium relaions (consisen wih he moneary model) from he money marke in he home and foreign counry, assuming ha parameers alfa and bea in he demand equaion are idenical for boh counries. The Hooper-Moron model is consisen wih he long-run porfolio balance considering ha he equilibrium real exchange rae is defined as he value expeced o balance he curren accoun in he long-run. The model (1) for s=log(e) resuls from relaions (2a-2f), by considering he dependence of exchange rae on he curren accoun balance as well. III. Neural Models and Robus Esimaion Procedure The ype of models considered in his paper is nonlinear, based on feed-forward neural nework wih one hidden layer. ^ y y( x ; w) (3) 2 e denoes he expeced value and - denoes he equilibrium value. 96 Romanian Journal of Economic Forecasing XVIII (1) 2015

5 Ou-of-sample Forecasing Performance Due o he propery of universal approximaion (Cybenko, and Hornik, 1989) his ype of neworks can be used o solve nonlinear regressions of he form: ^ nh y f ( x) l( w h( w w x ) w ) o[ j,1] i[ n1, j ] i[ i, j] i o[ nh 1,1] j1 i1 n (4) where: h (.) is he sigmoid funcion (a smooh, monoonically increasing funcion x h( x) 1/(1 e ) ), w is he weighs associaed o each linked nodes, nh is he number of hidden nodes and l( x) x is he acivaion funcion associaed wih oupu node. The nonlinear sigmoidal funcion h (.) provides he compuaional flexibiliy allowing for he adjusmen of he model o minimum errors on he daa se. Usually, he esimaion of he neural model is based on minimizaion of he mean squared error (MSE): T 1 2 min ( y f ( x, )) (5) T 1 MSE, as a measure of dispersion of errors, is very sensiive o ouliers, because even a single one can cause non-informaive esimaions, which is he definiion of he breakdown poin. Neural models are especially suiable o encompass changes in he exchange rae ha are raher frequen and someimes dramaic due o major evens in he marke, business cycles or changes in he economic policy. These evens can be viewed as causing he presence of ouliers ha may lead o misspecificaion, bias parameer esimaes, deerioraion of forecasing qualiy of he models (Chen and Liu, 1993; van Dijk, Franses and Lucas, 1999; Preminger & Frank, 2007). The robus neural models ha I use in his paper are based on he S-esimaion mehod which minimizes a scale measure of he model errors, which is in he class of regular scales abou he origin (Sakaa and Whie, 2001): T 1 min S( ) sups R / (( y f ( x, )) / s) K (6) T 1 for a funcion ha is even, bounded, coninuously differeniable, has he propery ( 0 ) 0 and sricly increases for every posiive value before reaching supremum, 1 K (0, ). 2 In he ime series conex he lower bound for breakdown poin is deermined by he raio K / /( nlag 1), where nlag is he number of lags. In his paper, he funcion is he Tukey funcion: Romanian Journal of Economic Forecasing XVIII (1)

6 Insiue for Economic Forecasing 2 6 x x4 x if x c 2 4 ( x) 2 2c 6c (7) 2 c if x c 6 The S-esimaors have a high breakdown poin and he propery of asympoic normaliy (Rousseeuw and Zohai 1984) bu he algorihms are inensive. Anoher complicaion is caused by he fac ha he minimizaion funcion is no convex and has more han one local minimum, so global opimizaion algorihms mus be used (in his paper he adaped simulaed annealing algorihms of Ingher ASA as in Sakaa and Whie, 2001 and Preminger & Frank, 2007). This algorihm - and, implicily, he robus regression approach - is more compuaionally demanding han oher esimaion procedures, bu his is compensaed by he gain in forecas abiliy. IV. Empirical Neural Models for he RON/USD Exchange Rae IV.1. The Daa The sample sars in January 2000 and ends in Augus For Romania, he daa was colleced from he Romanian Naional Insiue of Saisics and for he USA he daa were colleced from he Federal Reserve Board daabase. All he variables are logarihms of he indices. Thus, for he exchange rae he significance of he variable is ha of coninuously compounded reurn or log reurn, so we can use measures of he sign predicabiliy ha is imporan o invesors who seek profi maximizaion and are no ineresed only in minimizing forecas errors. The esimaed relaion is: s f ( m m, y y, r r,, TB) u (8) f f f I used he indusrial producion index as proxy for oupu and he variable represening he curren accoun defici of he Hooper-Moron model was replaced by he ne curren expor of Romania and was no considered for he USA. These are he resricions imposed on he Hooper-Moron model due o he peculiariy of he model for Romania and o he availabiliy of daa. IV.2. The Models In his paper a robus neural models wih macrodeerminans is consruced based on (8) wih 3 lags. I employ wo values 0.05 and 0.10 for K o produce models resisan o ouliers wih a low and a high breakdown poin denoed RNNMD5% and RNNMD10%. For comparison reasons I also esimaed he naive random walk (RW) model, an auoregressive model (AR) and a robus neural auoregressive model (RNNARR5%) for K f 98 Romanian Journal of Economic Forecasing XVIII (1) 2015

7 Ou-of-sample Forecasing Performance The consruced neural neworks have one hidden layer wih 6 nodes (n h=6). A reasonable number (n=100) for he iniial random values was used in he process of selecion of he bes model. The forecas horizon h is chosen o be from one monh o 6 monhs. 37 experimens are conduced o esimae he models using a moving regression in a window wih a fixed size of 105 observaions. Based on each esimaed model, he h- sep forecas was generaed, so he ou-of-sample forecas consis of 37 observaions. For each window, he models used 3 lags of he curren reurn besides he chosen deerminans of he exchange rae based on he Schwarz informaional crierion, which was used o choose he lag order. This approach is used by Granger, King and Whie (1995), Swanson and Whie (1995, 1997). In he ime series conex he lower bound for he breakdown poin is deermined by he raio K / /( nlag 1), where nlag is he number of lags so we obain a model (RNNMD5%) resisan o 1.25% ouliers and respecively o 2.25% ouliers for RNNMD10%. The selecion of he hidden nodes was based on heurisic mehods of deerminaion, basically consising in he search conduced for he enire daase from he simple nework (wih one uni in he hidden layer) o a more complex one adding a uni unil he nework fails o improve he error funcion. This approach is moivaed by Occam s razor principle ha implies preference for simpler models from he se of beer ones. In our experimens I se he number of hidden nodes o 6 as resuled from he selecion procedure. The same choice of fixing he number of hidden nodes was made by Whie (1988) and Preminger and Franck (2007). IV.3. Forecas Evaluaion The ou-of-sample performances of he models are examined using measures based on RMSE (roo mean square error), MAE (mean absolue error), and MAD (median of absolue deviaion). RMSE 1 n n i1 ^ ( r i r i ) n 1 ^ MAE r i r i n i1 Also, he SR (success raio) was used o assess he predicion accuracy of he models 1 ^ regarding he sign of he reurn. n SR I ( r r i i ), where I(.) is he indicaor funcion n i 1 (I(a>0)=1; I(a<=0)=0). Diebold-Mariano es (1995) (DM) is used o es wheher he forecass of he wo models are equally accurae. This saisic is calculaed as follows. Suppose wo models produce forecas errors: 1 1 h/ y h y h/ ; and 2 2 h/ y h y h/ Romanian Journal of Economic Forecasing XVIII (1)

8 Insiue for Economic Forecasing and since he accuracy of he forecas is measured by a funcion L( null hypohesis H : E( L( )) E( L( )) h/ h/ h/ h/ h / ) we es he H : E( L( )) E( L( )) o he alernaive The Diebold-Mariano es is based on he difference of he cos funcion d L( ) L( ) and he saisics 1 2 h/ h/ mean of he values d and DM ^ d LVRd / T, where: d is he ^ LVR d is a consisen esimaor of he asympoic variance of T d (i is used because he series {d } is serially correlaed for h>1). LVR d 0 2, cov( d, d ) j j j. j1 Under he null hypohesis of equal predicive accuracy, DM is sandard normally disribued, N(0,1). IV.4. Empirical Resuls Table 1 repors he resuls of he ou-of-sample forecasing performance of he models for he movemens in he RON/USD exchange rae, one panel for each forecas horizon (1-6-monh horizon). Table 2 displays he DM saisics for he null of equal predicive repored o he RW. The forecas accuracy measures indicae ha he ou-of-sample performance of he models varies wih he forecas horizon. A he 1-monh horizon all models eiher liner or nonlinear, robus or non-robus ouperforms RW in erms of RMSE and MAE. The robus neural model wih low breakdown poin (RNNMD5%) shows he bes ou-of-sample accuracy in erms of boh sandard measures (RMSE and MAE) and we can rejec he hypohesis of equal accuracy (benchmark model is RW) a 10% significance level. Also i predics beer he direcion of change for he exchange rae. As he forecas horizon lenghens he linear models accuracy improved and performs beer han he neural models on 3-, 5- and 6-monh horizon. The resuls of he DM es (able 2) indicae ha all he models performs beer han RW, bu we canno rejec he hypohesis of equal forecas accuracy beween he RW and he oher models for all forecas horizons. Table 2 shows ha on he 2-, 3-, 4- and 5-monh horizon he AR model performs beer han random-walk a 5% significance level. A 4-monh forecas horizon he robus neural model wih low breakdown poin (RNNMD5%) shows beer accuracy han RW a 10% significance in erms of RMSE respecively a 5% level in erms of MAE and a 5-monh forecas horizon boh robus models wih macrodeerminans performs beer han RW a 10% level of significance in erms of RMSE. 100 Romanian Journal of Economic Forecasing XVIII (1) 2015

9 Ou-of-sample Forecasing Performance The resuls of he DM es show ha for one-monh horizon we canno rejec he null of equal predicive accuracy beween all models and RW models a 10%, bu he robus nonlinear models wih low breakdown poin is significanly beer han he random-walk model. For he 4- and 5-monh horizon he robus models wih macrodeerminans bea RW a 10% significance, bu so does he auoregressive model robus or nonrobus. The null of equal predicive abiliy canno be rejeced and for he 6-monh horizon for neural models and for he AR, which is o be expeced because he neural models mus be rerained o mainain heir performance. Table 1 Ou-of-sample Forecasing Performance Based on Forecas Accuracy Measures Model RMSE MAE SR 1-monh horizon RW ARR * RNNAR5% RNNMD5% * * RNNMD10% monh horizon RW ARR RNNAR5% RNNMD5% * * 0.541* RNNMD10% monh horizon RW AR * * RNNAR5% * RNNMD5% RNNMD10% monh horizon RW AR * 0.568* RNNAR5% RNNMD5% * RNNMD10% monh horizon RW AR * * RNNAR5% RNNMD5% RNNMD10% * Romanian Journal of Economic Forecasing XVIII (1)

10 Insiue for Economic Forecasing Model RMSE MAE SR 6-monh horizon RW AR * * RNNAR5% RNNMD5% RNNMD10% * Noe: RMSE, MAE are forecas accuracy measures, SR is he saisics regarding he sign of he acual reurn. NN5% and NN10% refer o robus neural models wih 5% and 10% ouliers, respecively. Aserisk (*) represens he lower value on he column for forecas accuracy measures and he highes value for SR. Table2 Ou-of-sample Diebold-Mariano Tes for Equal Accuracy wih Random Walk Model DM RMSE MAE 1-monh horizon ARR RNNAR5% RNNMD5% 0.934* RNNMD10% monh horizon ARR 0.985** 0.984** RNNAR5% 0.946* RNNMD5% RNNMD10% monh horizon ARR 0.984** 0.997* RNNAR5% RNNMD5% RNNMD10% monh horizon RW ARR 0.977** 0.903* RNNAR5% RNNMD5% 0.967** 0.991** RNNMD10% monh horizon ARR 0.994** RNNAR5% RNNMD5% 0.954** RNNMD10% 0.914* monh horizon ARR 0.928* RNNAR5% Romanian Journal of Economic Forecasing XVIII (1) 2015

11 Ou-of-sample Forecasing Performance Model DM RMSE MAE RNNMD5% RNNMD10% Noe: DM is p-values for Diebold-Mariano es (1995) for differen coss. P-values lower or equal o signifies ha he reference model (RW) yields a lower forecas error wih 5% significance. Values greaer or equal o 0.95 indicae ha RW yields higher errors wih 5% significance level..(*) and (**) represens beer performance han RW a 10% respecively 5% level. V. Conclusions The predicion of exchange rae movemens is a major subjec for economic policy applicaion. A number of researchers ried o capure he srucural breaks in he exchange rae due o marke evens, financial crises or changes in business cycles by incorporaing nonlineariies in heir models. They used non-parameric kernel regression, Markov-swiching models, bu found ha he models do no perform beer han he random-walk model. There are some oher researchers ha used neural neworks and obained some evidence of beer performance in ou-of-sample forecass. They applied he mehodology o he daily exchange raes (Kuan and Liu, 1995, Gencay, 1999) and found some improvemens in he mean squared forecas errors. Preminger and Franck (2007) used neural auoregressive models robus o ouliers and found ou ha hese models can provide beer ou-of-sample forecas in comparison wih non-robus models and beer marke iming abiliy a all ime horizons. However he hypohesis of equal accuracy in comparison o RW canno be rejeced in mos cases excep for JPY/USD a one-monh horizon, 15% level of significance. In his paper, we follow he same robus regression approach based on S-esimaors as in Preminger and Franck (2007) o forecasing exchange raes wih macro deerminans of Hooper-Moron's srucural model. We found ou ha in erms of forecas accuracy measures he robus neural models wih oher deerminans han he hisory of he exchange rae show beer performance han he random-walk model and he robus purely auoregressive neural model for 1-, 2-, 3- and 4-monh horizon. The predicive accuracy measures presened in Table 1 show ha he ou-of-sample measures of accuracy of each model vary wih he forecas horizon, bu he general conclusion is ha all models eiher linear or nonlinear are beer han RW for all 1-6 monh horizon. A one-monh horizon he bes forecass are hose provided by he robus nonlinear models wih moneary fundamenals (RNNMD5% is beer according o accuracy measures RMSE and MAE and also in erms of he forecas success rae, SR). We can conclude ha an imporan cause of lower performance of he models ha predic he exchange rae, besides nonlineariies, is he presence of ouliers in he daa. The soluion could be he robus neural models ha are rained frequenly o incorporae new informaion. Romanian Journal of Economic Forecasing XVIII (1)

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14 Insiue for Economic Forecasing Whie, H., Economic predicions using neural neworks: The case of IBM daily sock reurns. Proceedings of he second annual IEEE Conference on Neural Neworks. Yuan, C., The Exchange Rae and Macroeconomic Deerminans: Time-Varying Transiional Dynamics. UMBC Economics Deparmen Working Papers Zang, G. Pauwo, B.E. and Hu, M.Y., Forecasing wih arificial neural neworks: The sae of ar. Inernaional Journal of Forecasing, 14, pp Romanian Journal of Economic Forecasing XVIII (1) 2015