Long Term Forecasting of El Niño Events via Dynamic Factor Simulations

Transcription

1 Long Term Forecasing of El Niño Evens via Dynamic Facor Simulaions Mengheng Li (a) Siem Jan Koopman (a,b) Ruger Li (a) Desislava Perova (c) (a) Vrije Universiei Amserdam and Tinbergen Insiue, The Neherlands (b) CREATES, Aarhus Universiy, Denmark (c) Caalan Insiue for Climae Science (IC3), Barcelona, Spain May 2, 2017 Corresponding auhor: S.J. Koopman, Deparmen of Economerics, Vrije Universiei Amserdam, De Boelelaan 1105, 1081 HV Amserdam, The Neherlands. 1

2 Long Term Forecasing of El Niño Evens via Dynamic Facor Simulaions by M. Li, S.J. Koopman, R. Li & D. Perova Absrac We propose a new forecasing procedure for he Niño3.4 ime series ha is linked wih he well-known El Niño phenomenon. This imporan climae ime series is subjec o an inricae serial correlaion srucure and is relaed o many oher relevan and relaed variables. Alhough he forecasing procedure is valid for all lead imes, i is paricularly developed for medium o long erm forecasing of El Niño. The procedure consiss of hree phases and relies on he subsequen use of an univariae ime series mehod for producing predicion errors, he formulaion of a dynamic facor model for hese predicion errors and he explanaory variables, he simulaion of many signal pahs from he dynamic facor model condiional on all explanaory variables, and he reconsrucion of arificial univariae ime series of he variable of ineres for which forecass can be generaed. The sample average of hese ensemble forecass is our final forecas. We jusify each sep of he procedure in general erms and relae he procedure o well-known conceps in economerics. Our proposed forecasing procedure is specially developed for he imporan and challenging problem of forecasing he warm El Niño evens. We provide evidence ha our procedure is superior in he forecasing of El Niño when compared o oher economeric forecasing mehods. Some key words: Climae research, Dynamic models, Kalman filer, Simulaion smoohing. JEL classificaion: C32, C42. 2

3 1 Inroducion El Niño is he imporan and well-known phenomenon of having higher han average sea surface emperaures in he cenral and easern equaorial Pacific. I has an enormous impac on he climae in many pars of he world. Therefore, i has been given much coverage in he popular media, and i is he subjec of exensive research in he scienific world. The occurrence of El Niño ypically causes changes in weaher paerns relaed o emperaure, pressure and rainfall. An El Niño even may no only have a negaive impac on local economies, bu can also have negaive consequences for public healh, as in some regions hese changes increase subsanially he risk of waer-borne and vecor-borne diseases of which dengue is an example. Given is huge impac especially on developing counries in he proximiy of he Pacific Ocean, i is self-eviden ha a imely forecas of he nex El Niño even is of crucial imporance. Much scienific research has been devoed o he developmen of forecasing mehods for El Niño. The oscillaion is characerized by an irregular period of beween 2 and 7 years. Currenly, forecass are issued on a regular basis for up o hree seasons in advance, bu he long erm of more han one year ahead forecass remain a real challenge. Only in a few heoreical sudies such long erm forecass are documened. A he same ime he physics underlying El Niño implies ha i is a self-susaining climaic flucuaion ha is quasi-periodic, wih several dominan peaks in is specrum, he main one being a abou every 4-5 years, a secondary a abou 2 years, and a hird one a abou 1.5 years. This suggess ha i may be predicable a lead imes on he order of several years. In his paper we propose a new forecasing procedure for El Niño (for he Niño3.4 ime series, in paricular) aimed a medium o long lead imes. The forecasing mehods are based on sae-of-he-ar developmens in ime series economerics including sae space models and simulaion mehods. In his paper we address he classical forecasing problem in ime series analysis where we need o forecas a single ime series of ineres ha is subjec o a possibly inricae serial correlaion srucure and for which a possibly large se of explanaory variables is available o us. The sandard approach is o consider a linear model ha simulaneously reas he dynamic srucure in he ime series and he explanaory variables which may be endogenous or exogenous. Boh in economerics and saisics, much aenion is given 3

4 o he selecion of he appropriae se of explanaory variables. The adoped mehodology depends on wheher he purpose is in-sample fi or ou-of-sample predicion. In he laer case, i is ypically argued ha more parsimonious models are more successful in forecasing. There is a vas lieraure on forecasing in his seing and in he main paper we provide an overall discussion wih appropriae references o he key forecasing mehodologies in his seing and we will indicae heir relevance o our novel approach. We denoe he variable of ineres by he scalar y and he explanaory variables by he vecor X for which we have observaions a ime poins = 1,..., T bu we do no have observaions available a fuure ime poins T + 1, T + 2,.... We assume ha he ime series y and hose in X are saionary bu persisen processes. We absrac ourselves from non-saionary and coinegraion issues in he curren sudy. We furher noice ha X may also conain lagged explanaory variables. The aim is o produce an esimae for y T +h where h is he forecas horizon, based on he available observaions for y and X up o ime T. For small values of h, we refer o he forecas made a ime T, denoed by ŷ T +h T, as a shor erm forecas while for larger values of h we refer o he forecas as a long erm forecas. I depends on he ime series and he purpose of he sudy of wha forecas horizon is associaed wih shor, medium or long erm forecass. Our focus is mosly direced owards long erm forecasing bu our proposed procedure is also valid for shor erm forecasing. To produce a forecas for y T +h, he informaion in X T +1,..., X T +h may be highly relevan for i bu we do no have heir observaions available a ime T. A possible soluion o his problem is o produce forecass for he explanaory vecor X T +1,..., X T +h in an ad-hoc way from which he acual forecas ŷ T +h T can be compued. Given ha hese forecass for he explanaory variables may be inaccurae, especially for larger values of h, he forecas ŷ T +h T is ofen less accurae, even when compared o univariae forecass for which no use is made of any explanaory variable; see, for example, he discussion in Ashley (1988). Anoher soluion for he handling of explanaory variables in he forecasing problem is o joinly analyze y and X wihin, for example, a vecor auoregressive (VAR) model. Alhough here is evidence of heir effeciveness in forecasing key variables, here is a surge in he number of parameers ha need o be esimaed when he number of variables in X increases. The esimaion errors can negaively affec he forecas accuracy for he variable of ineres, especially in he case of long erm forecasing; see, for example, he discussion 4

5 in Lierman (1986). In recen years much aenion is given o shrinkage mehods applied o large ses of explanaory variables including principal componens and empirical Bayes mehods. In many sudies where shrinkage procedures are adoped, convincing evidence of improved forecas accuracy is presened; see also he discussion in Sock and Wason (2012). 2 A new forecasing procedure In his secion we provide he deails of our forecasing procedure based on dynamic facor simulaions. This developmen is designed for he long-erm forecasing of a key variable by making use of many predicor variables ha have non-rivial dynamic srucures of heir own and have poenially a long-lasing, and ypically a cyclical, impac on he key variable. Our paricular case of forecasing El Niño evens is a clear illusraion bu such cases also occur in ime series relaed o climae change generally. Alhough he sea surface emperaure is direcly associaed wih he El Niño even, he emperaure ime series variable is coninuous while he El Niño even is a binary even ha is defined in erms of a consecuive sequence of emperaure exceedances. The La Niña even is defined in a similar way. A complexiy arises when we model he sea surface emperaure for he purpose of forecasing discree evens which in our case will be El Niño, La Niña and nohing of he wo. Apar from he probabiliy forecas, we also develop a simulaion-based procedure for compuing confidence inervals of he probabiliy forecass. A final challenge is ha he forecas horizon sreches ou o 30 monhs and where forecass for one year and beyond are of major ineres. Our forecasing procedure consiss of hree modelling sages: (i) a univariae ime series model for key variable of ineres, in our case Niño3.4, o produce one-sep ahead predicion errors; (ii) a join analysis of he predicion errors and all poenial predicor variables based on a dynamic facor model; (iii) simulaion and exrapolaion of dynamic facors for he consrucion of muliple forecass and confidence inervals for he probabiliy of he El Niño and La Niña evens. In his secion we provide he deails of hese hree seps and discuss he moivaion of he seps in words. More formal moivaions and explanaions are provided in he Appendix C in which we argue ha our simulaion-based procedure have beer properies for medium-erm forecasing when compared o he forecasing based on a dynamic facor model. Some simulaion evidence is also provided. 5

6 2.1 Sep (i): univariae ime series analysis The ime series properies of he key variable can be well esablished in a linear dynamic model. In our sudy we consider he class of unobserved componens (UC) ime series of Harvey (1989) bu oher linear dynamic models can be considered as well including sraighforward auoregressive moving average models (ARMA), see Tsay (2010) for an inroducion o ARMA models. In Appendix B.1 we provide he deails for a ime series analysis based on he UC model for a monhly ime series y ha is given by 3 y = µ + γ + ψ j, + ε, = 1,..., T, (1) j=1 where µ represens he rend componen, γ represens he monhly seasonal componen, ψ j, represens a sochasic cyclical process wih cerain ampliude and frequency, for j = 1, 2, 3, and ε represens he disurbance or noise componen. In a linear Gaussian model, we assume ha hese six unobserved componens are generaed by linear dynamic sochasic processes depending on Gaussian independen noise erms. In paricular, he rend µ is modelled as a random walk process, he seasonal γ consiues 11 dummy variables which are modelled as muual independen random walk processes, he cycles ψ j,, for j = 1, 2, 3, are modelled as ARMA processes wih low-order lag polynomials bu wih complex roos imposed on he auoregressive polynomial, and he disurbance componen ε is assumed o be whie noise. The variances of he noise erms and he parameers associaed wih he cycle componens are esimaed by he mehod of maximum likelihood. Once he parameer esimaes are given, he unobserved componens can be esimaed, which is referred o as signal exracion. As par of he esimaion procedure, muliple-seps ahead forecass for y can be made, ogeher wih sandard errors. This mehodology relies on he sae space approach o ime series analysis ha has been explored in Durbin and Koopman (2012) and has been considered for he modelling and forecasing of he Niño3.4 ime series in Perova, Koopman, Balleser, and Rodó (2016). In his recen sudy i is shown ha he UC model, exended by a selecion of predicor variables, can produce relaively highly accurae forecass, especially for he medium-erm, say from 1.5 o 2.5 years. The high forecas skill is o a large exen explained by he selecion of he predicor variables, which are dynamically 6

7 relevan o he El Niño evoluion specially in he ocean subsurface. The one-sep ahead predicion error of he UC model is defined as v uc = y E uc (y y 1,..., y 1 ), (2) where E uc refers o expecaion wih respec o he Gaussian densiy for y as implied by he univariae UC model (1) where he unknown parameers are replaced by heir maximum likelihood esimaes. The predicions errors for he UC model are compued by he Kalman filer which is a linear recursive esimaion process for he unobserved componens. I implies ha, for a given UC model, he predicion error v uc and pas observaions {y 1,..., y }, or in marix form is a linear combinaion of he concurren v uc = L y, where v uc = (v uc 1,..., v uc T ), y = (y 1,..., y T ) and L is a T T lower-riangular marix wih uniy values on he main diagonal, see Durbin and Koopman (2012, Secion 4.13) for a more deailed discussion. The lower riangular elemens of marix L = L(ψ) is a funcion implied by he UC model and wih he parameer vecor ψ as is only argumen. Since marix L is inverible, he reverse relaionship is also rue, ha is y = L 1 v uc. where L 1 is also a lower riangular marix. The laer relaionship is imporan for he developmen below since we require he expression ha y for a given UC model can be reconsruced as y = g (v uc 1,..., v uc ), = 1,..., T, (3) for a linear funcion g ( ) ha simply reflecs he h row of marix L Sep (ii): dynamic facor model analysis In he nex sep we joinly consider he predicion errors of he univariae UC model (for our key variable y ) and he predicor variables in X. The one-sep ahead predicion errors v uc 7

8 are compued by he Kalman filer applied o he UC model. We assume ha he number of variables in X equals N 1. We hen formulae he dynamic facor model (DFM) as vuc X = Λf + ξ, f = Φ 1 f Φ p f p + ν, (4) for = 1,..., T, where Λ is he N r facor loading marix, f is he r 1 vecor wih laen dynamic facors, ξ is he N 1 observaion disurbance vecor, which is assumed o be normally disribued, Φ i is an r r auoregressive coefficien marix, for i = 1,..., p, wih p being he order of he vecor auoregressive process for f, ha is VAR(p), and ν is he r 1 normally disribued facor disurbance vecor. The r dynamic facors in f represen he common dynamic variaions in all ime series variables in v uc and X. The dynamic process for f is specified as a sricly saionary vecor auoregressive process. Hence he underlying assumpions for (4) include he saionariy of v uc and all ime series variables in X. In case of he predicion errors, when he univariae UC model is an adequae model represenaion for y, i is implied ha all dynamic informaion in he ime series y 1,..., y T is accouned for. The predicion errors can herefore be regarded as whie noise and cerainly saionary. We furher assume ha all variables in X are saionary, possibly afer a ransformaion such as differencing or derending. The esimaion of he unknown parameers in he marices Λ, Σ ξ and Φ i, for i = 1,..., p, is carried by he wo-sep mehod of Doz, Giannone, and Reichlin (2011), see Appendix B.2 for he relevan deails. In he dynamic facor model as presened in equaion (4), he dynamic facors are ypically modelled parsimoniously wih, say p = 1 or p = 2, ha is a VAR(1) or VAR(2). However his may no be sufficien o capure he persisen and cyclical dynamics in X. We herefore inroduce an alernaive and parsimonious way o inroduce X and heir lags in he forecasing funcion of y. To fully capure he ineracions beween v uc and X, we can focus on he smooh densiy p dfm (v uc X 1,..., X T ), (5) for = 1,..., T. In case he univariae UC model is he rue daa generaion process for y, he predicion errors are whie noise and should no be affeced by he variables in X. We 8

9 hen simply have p dfm (v uc X 1,..., X T ) = p dfm (v uc ) wih Λ = 0, which implies a whie noise process for v uc. Any univariae model misspecificaion due o he omission of he informaion in X 1,..., X T will come o ligh when considering he smooh densiy (5). Alhough he DFM is mos likely o be misspecified wih respec o dynamic specificaion for X, i is of lesser imporance when our main focus is on he forecasing of he key variable y. We are only ineresed in a more appropriae model represenaion for X when i increases he forecas accuracy for y. We are no seeking he rue daa generaion process for X. Hence model (4) is a basic and sraighforward model represenaion for connecing he informaion in X ha is relevan for he par of y ha canno be explained by is own pas. The simulaion smoohing mehod of Durbin and Koopman (2002) allows he generaion of samples from he smooh densiy (5). Each realised sample for v uc is a linear funcion of X 1,..., X T while a series of samples from (5) can visualise he amoun of variaion ha is implied by (4) and explained by he collecion of predicors in X. We denoe a simulaed predicion residual series from (5) by v (i) series ha we ypically se o M = 200. for i = 1,..., M where M is he number of simulaed 2.3 Sep (iii): forecasing via simulaion and esimaion On he basis of he simulaed series v (i), for i = 1,..., M, from he DFM (4), we can generae he corresponding ime series y on he basis of he UC model (1) since from (3) we have ha y = g (v1 uc,..., v uc ). In his way we obain a sequence of M ime series for y which we refer o as he se of ensemble ime series and is denoed and generaed by y (i) = g (v (i) 1,..., v (i) ), v (i) p dfm (v uc X 1,..., X T ), = 1,..., T, (6) for i = 1,..., M. The ensemble ime series y (i) is he resul of an ineracion beween he UC and DFM models. This is a key noion of our forecasing mehod. The forecas for y T +h, for h = 1, 2,..., from he univariae UC model can be wrien as ŷ uc T +h = E uc (y T +h y 1,..., y T ) = E uc (y T +h v uc 1,..., v uc T ), (7) since v uc = g (y 1,..., y ) for = 1,..., T. For each ensemble ime series (6), we compue he 9

10 forecass as in (7) and we denoe hese by ŷ (i) T +h = E uc(y T +h y (i) 1,..., y (i) T ) = E uc i (y T +h v (i) 1,..., v (i) T ), h = 1, 2,..., (8) for i = 1,..., M, where E uci is expecaion wih respec o he UC model (1) bu wih he parameer vecor esimaed for he ensemble ime series y (i) 1,..., y (i). The final forecas from our hree-sep procedure involving dynamic facor simulaion (DFS) is obained by T ŷ dfs T +h = 1 M M i=1 ŷ (i) T +h. This forecas effecively evaluaes he forecas funcion ŷ T +h = y T +h (v uc )p dfm (v uc X)dv uc. This procedure requires he maximum likelihood esimaion of he univariae UC model for he original and M ensemble ime series. The DFM model is used o generae ensemble ime series o incorporae he predicive conribuion of X 1,..., X T for he key variable y. The acual forecasing of he X variables is no needed for he forecasing of y. 2.4 Discussion and review of he new forecasing procedure The incorporaion of predicor variables in he UC model (1) is sraighforward since we can simply add a regression erm such as X δ, where δ is he (N 1) 1 vecor of regression coefficiens, o he righ-handside of (1). The esimaion of δ, ogeher wih he parameer vecor ψ, can be carried ou wihin he sae space approach of ime series analysis. However, in cases where we have many variables in X and he dependence of y on X is srong and lass over many lags of X. The number of parameers o esimae increases rapidly and i is well known ha heavy parameerised models are ypically no superior in producing accurae forecass, and especially no for longer-erm forecass. Variable selecion and lagselecions procedures are hazardous wihou good knowledge of he srong dependencies amongs he variables and heir inricae and srong dynamic iner-linkages. Finally, in he curren conex in which he UC model wih X s will also be used for he forecasing of 10

11 y, i clearly also requires forecass for he variables in X. The effor o provide accurae forecass a each ime a forecas of y is required can be regarded as a severe ask and will no necessarily lead o a good forecas for y. To circumven a heavy parameerised UC model bu sill allowing predicor variables (including heir lags) o conribue in he forecasing of he key variable, various mehodologies have been developed. For example, he use of principal componens wihin a dynamic facor analysis have been heavily explored in he macroeconomic forecasing lieraure wih key conribuions by Sock and Wason (2002) and Doz e al. (2011). Alhough he use of he principal componens addresses he challenge of variable selecion, i does no address he lag-selecions for each variable. To capure he srong serial dependencies amongs he variables y and hose in X in a dynamic facor analysis, we sill require he empirical idenificaion of he dynamic iner-linkages beween he key variable y and he seleced dynamic facors as well as he dynamic srucures wihin he dynamic facors. Hence, we only have parially addressed he heavy parameerisaion issue. We herefore have developed he DFS soluion o address hese issues by relying boh on a univariae ime series analysis, for example based on he UC model, and on dynamic facor analysis. This simulaion mehod is simple and also provides a general soluion for he problem of forecasing non-linear signals. Our proposed procedure circumven he need o forecas X s compleely while he explanaory variables sill play an imporan role in he forecasing of y. I is a parsimonious procedure and i is based on he reconsrucion of a ime series y ha akes ino accoun he informaion of explanaory variables. The consrucion of hese ensemble ime series is compuaionally fas since we adop he simulaion smooher of Durbin and Koopman (2002). We emphasise ha he simulaed predicion errors ṽ (i) 1,..., ṽ (i) T from he DFM are by consrucion funcions of he explanaory variables X 1,..., X T, hese are heir lag, concurren and lead values for y wih 1 T. The univariae UC model of he iniial analysis is revisied for each ensemble ime series o re-esimae he parameers and o produce forecass for he ensemble y (i) T +h, for h = 1, 2,.... Our final forecas is simply he average of hese M forecass. The procedure is raher general and can be implemened using differen choices of models and esimaion mehods. For example, a simple auoregressive model can be considered in he firs sep while he second sep can be based on a few explanaory variables and one 11

12 principal componen. The dynamic facor model in he second sep is adoped as a daa reducion echnique o handle poenially many explanaory variables and as a framework o generae pahs of he predicion errors condiional on pas and concurren observaions of he explanaory variables. Oher shrinkage mehods can be considered bu we require a model represenaion o enable he simulaion from p(v X 1,..., X T ). The predicion errors are a key ingredien in he procedure as hey are consruced wih he aemp o remove he serial dependence in he ime series variable of ineres. Hence we separae he dependence of y from is own pas (serial correlaion) and he dependence of y on he informaion presen in he curren and pas observaions of he explanaory variables. We firs adop an appropriae univariae ime series and obain he predicion errors and, subsequenly, we joinly model he predicion errors and he explanaory variables wihin a dynamic facor model. The nex sep of reconsrucing predicion errors condiional on, or as a funcion of, X 1,..., X T, is an efficien way o incorporae his informaion ino he forecas for ŷ T +h. A more formal jusificaion of his mehod is presened in Appendix C. Our proposed forecasing mehod can also be moivaed from a model misspecificaion argumen since i provides he ingrediens o carry ou a Durbin-Wu-Hausman es as formulaed by Hausman (1978) wih respec o he forecas for ŷ T +h. I can be shown ha he forecass obained in he firs and hird seps are consisen while in he hird sep he forecass are more efficien. Since we adop he dynamic facor model in he second sep, we can also carry ou a es wheher he X s are endogenous. 3 Empirical in-sample resuls for he case of El Niño 3.1 The daa Our daa se includes he monhly ime series of emperaure values which is referred o as he Niño3.4 ime series and which is defined as he area-averaged sea surface emperaure in he region (5 N - 5 S, 170 W W). In his area he El Niño evens are idenified, see also he discussion in Barnson, Chelliah, and Goldenberg (1997). The Naional Ceners for Environmenal Informaion (NOAA) defines an El Niño or La Niña even as a phenomenon in he equaorial Pacific Ocean characerised by a five consecuive 3-monh running mean 12

13 of sea surface emperaure (SST) anomalies in he Niño 3.4 region ha is above (below) he hreshold of +0.5 C (-0.5 C). This sandard of measure is known as he Oceanic Niño Index (ONI). The ONI index is calculaed as he average sea surface emperaure in he Niño 3.4 region for each monh, and hen averaged wih values from he previous and following monhs. This running hree-monh average is compared o a 30-year average. The observed difference from he average emperaure in ha region, wheher warmer or cooler, is he ONI value for ha 3-monh season. This definiion is obained from he websie of NOAA, hps:// In our empirical sudy we consider he Niño3.4 ime series as he variable of key ineres and denoe i as y. This ime series is observed from January 1982 o he end of 2015, ha is 34 years of daa and in oal 408 monhly observaions. For his period, we also have all observaions for 24 explanaory variables which consis of physical measures of zonal wind sress and sea emperaures a differen dephs in he ocean and a differen locaions. These variables are seleced because hey are relevan o he dynamical processes for he generaion and evoluion of El Niño evens; see Perova e al. (2016) for furher deails. We do have observaions available for beyond 2015 bu only for a selecion of variables, no for all of hem. 3.2 Some compuaional deails In he implemenaion of our new forecasing procedure, all seps rely on he linear Gaussian sae space model as discussed in Durbin and Koopman (2012). In he firs sep, we consider he unobserved componens ime series model (1) wih sochasic componens consising of a level specified as a random walk, a ime-varying dummy seasonal, hree sochasically ime-varying cycles, and an irregular specified as a whie noise process. The parameers are esimaed by he mehod of maximum likelihood by means of he numerical maximisaion of he loglikelihood funcion ha is evaluaed by he Kalman filer. The same Kalman filer is also used for evaluaing he in-sample predicion errors v uc and for compuing he forecass for y T +h for h = 1, 2,.... In he second sep, he dynamic facor model is formulaed in sae space form wih he sysem marices obained from he Doz e al. (2011) wo-sep procedure based on principal componens. The number of principal componens (equals 13

14 he number of facors) is equal o wha number is necessary o capure 95% of he sample variaion in (v uc, X ). The dynamic facors are specified as he VAR(p) process wih p = 2. Once he dynamic facor model is presened in sae space form, he simulaed values ṽ (i) from p dfm (v uc X 1,..., X T ) are obained afer a sligh modificaion of he simulaion smooher of Durbin and Koopman (2002). In he hird sep, we repea he univariae forecas procedure from sep 1 for all ensemble ime series for which he parameers are re-esimaed by maximum likelihood for each series. Since he dynamic properies of he ensemble ime series are very similar, good saring values are available and he repeaed esimaions do no ake much compuer ime. The sample means of he forecass for each horizon h are recorded as our final forecass. We will provide empirical evidence ha hese forecass ouperform he forecass from a range of relevan and compeiive benchmark models. 3.3 Descripive daa analysis To gain some iniial insighs ino he dynamic properies of he Niño3.4 ime series, we presen in Figure 1 he ime series, is sample auocorrelaion funcion, he sample specrum (compued from he sample periodogram and subjec o some smoohing consan), and he yearly ime series plos for each monh and he corresponding averages over all 34 years. We may conclude ha he ime series has saionary feaures bu is highly persisen and cyclical srucures appear o be presen in he dynamics of he ime series. In paricular, we clearly observe a leas hree local peaks in he sample specrum. This may indicae ha hree cyclical processes wih differen periodiciies are imporan ingrediens in he daa generaion process of he Niño3.4 ime series. We may also conclude ha he ime series conains clear seasonal variaion in he mean. I is ineresing o noice ha he seasonal variaion in he lowes and highes perceniles are boh differen from he mean and from each oher. 3.4 Decomposiion of he Niño3.4 ime series As our univariae ime series model, we adop he unobserved componens ime series model as given by (1) and discussed in deail in Appendix B.1. The iniial findings above sugges ha we may decompose he Niño3.4 ime series variable ino sochasic processes for rend, 14

15 Figure 1 Niño3.4 ime series and is dynamic properies π 2π The figure shows he Niño3.4 ime series and some diagnosic graphs. In clock-wise order: monhly ime series plo y from 1982 unil 2015, he sample auocorrelaion funcion of y, he yearly plos of y for each monh and wih is periodic average; he sample esimae of he specrum of y (smoohed version of he sample periodogram). seasonal, hree cycles, and irregular componens. The parameers of he decomposiion model are esimaed by maximum likelihood as his is discussed in Appendix B.1. The differen ime series componens are exraced from he ime series using he Kalman filer and smoohing mehods, see Durbin and Koopman (2012). The maximum likelihood esimae of he variance ση 2 is numerically very close o zero so ha he sochasic rend effecively reduces o a fixed inercep. I indicaes ha he Niño3.4 ime series does no have nonsaionary properies (weakly saionary). Similarly, he seasonal componen γ also has reduced o a se of 11 fixed monhly seasonal dummy effecs during he esimaion process. Also he esimae of variance σε 2 for he disurbance erm is very close zero. The esimaed variances of he hree cyclical processes are all non-zero, alhough he variance for he second cycle is relaively close o zero. Hence, afer he esimaion procedure, he decomposiion model for he Niño3.4 ime series has 15

16 reduced o a linear regression model wih an inercep, 11 fixed monhly seasonal dummy effecs and auoregressive moving average (ARMA) disurbances for which he auoregressive polynomial has a muliple of complex roos. For each of he hree sochasic cycle componens in he decomposiion model we have esimaed parameers for persisence, frequency and variance. Their values for persisence are 0.96, 0.99 and 0.98 while he esimaes for frequency are 1.45, 2.46 and 4.44, respecively. These correspond approximaely o he hree local modes of he sample specrum for he Niño3.4 ime series. In Figure 2 we presen all esimaed componens, including he esimaed inercep and seasonal dummies, bu in a ime series plo. The sum of hese esimaed componens equals he original ime series y. The hird cycle componen wih he lowes frequency appears o accoun for mos of he variaion in he ime series since i has he highes ampliude. The firs cycle wih he highes frequency can be associaed wih he shor-erm cyclical dynamics which appears o accoun for more variaion han he seasonal componen. The second cycle wih he inermediae frequency is smooh and accouns for some bi-annual sysemaic variaion in he ime series bu is ampliude is relaively small compared o he seasonal and oher cyclical componens. The sandardised one-sep ahead predicion errors should be whie noise and normally disribued wih mean zero and uniy variance, when he model is well-specified. We can verify hese assumpions hrough a range of diagnosic ess. In Figure 3 we provide an insigh of he residual properies by presening a ime series plo of he sandardised residuals ogeher wih he correlogram, specrum and hisogram. These diagnosic plos confirm he whie noise properies of he errors o some exen alhough some weak cyclical behaviour appears o be unaccouned for. From he hisogram we learn ha he Gaussian disribuional assumpion does no appear o be seriously violaed despie he oulying value a he beginning of The ensemble residuals To provide some more insighs ino he role of he ensemble ime series in he hird sep of our forecasing procedure, we presen some feaures of he ensemble residuals. The simulaed one-sep ahead predicion errors from he smooh densiy p dfm (v uc X 1,..., X T ) are referred 16

17 Figure 2 Decomposiion of Niño3.4 ime series The figure presens he esimaed componens from he univariae decomposiion model applied o he Niño3.4 ime series. In he firs row, from lef o righ, we presen he esimaed rend componen µ, ogeher wih he Niño3.4 ime series, and he esimaed monhly seasonal componen γ, respecively. The variances for boh of hese componens are esimaed o be zero and hence he componens have reduced o fixed effecs. In he oher rows, he hree esimaed cycle componens are presened. The disurbance variance is esimaed o be zero and hence he disurbance componen has vanished from he model. o as ensemble residuals and hey are generaed by he simulaion smooher of Durbin and Koopman (2002) in he hird sep of he procedure. In Figure 4 we presen a sequence of 100 ensemble predicion error ime series, ogeher wih he original one-sep ahead predicion errors v uc. I is ineresing o see ha his se of ensemble residuals form a band ha goes hrough he one-sep ahead predicion errors, despie he fac ha uncondiionally he residuals can be regarded as whie noise. as poenial signals ha are relevan for v uc The ensemble residuals can be considered and are consruced from he informaion in X 1,..., X T. The main pah of he ensemble residuals reflecs he signal while he band-widh reflecs he amoun of variaion implied by he X s. To illusrae he significan conribuion of he X s on he predicion residuals v uc, we compue for each ensemble series he F-es for he fi in he regression model wih v uc as he dependen variable and he ih ensemble 17

18 Figure 3 Diagnosic graphics for decomposiion Niño3.4 ime series sandardised one-sep ahead predicion residuals 1.0 sample auocorrelaion funcion sample specrum hisogram The figure presens four diagnosic graphs for he univariae ime series analysis based on he UC model (1) applied o he Niño3.4 ime series. In he firs row, from lef o righ, we presen he sandardised one-sep ahead predicion residuals and is sample auocorrelaion funcion (upo 20 lags). In he second row, from lef o righ, we presen he corresponding sample specrum (smooh esimae of he sample periodogram) and he hisogram of he sandardised predicion errors. residual v (i) as he explanaory variable. We record he corresponding p-value of he F-es and presen he hisogram of he 100 p-values in Figure 4. I is clear ha each individual ensemble residual series conribue significanly in he fi of he predicion residuals v uc all p-values smaller han However, i also shows ha some ensembles produce a beer fi han ohers. wih To invesigae he individual role of a variable in X and o furher illusrae he rich informaion ha is conribued by X o he ensemble residuals, we presen in Figure 5 he average correlaion of ṽ (i) wih a selecion of variables in X and heir lagged values, up o lag 30. The correlaions are compued for each ensemble residual series ṽ (i) and hen he average is aken over i = 1,..., M. This procedure is similar in design and has he same aim as he R 2 plos of Sock and Wason (2002, see heir Figure 1) beween principal componens and individual explanaory variables. Some of he panels of Figure 5 clearly show a seasonal 18

19 Figure 4 Predicion and Ensemble residuals for Niño3.4 ime series 2 original residual Densiy F es p-value The figure presens he one-sep ahead predicion errors from he UC model and obained as par of he firs sep of our forecasing procedure. The predicion errors are compued by he Kalman filer applied o he unobserved componens ime series model wih rend, seasonal, and hree cycles for he Niño3.4 ime series. In addiion, M = 200 ensemble predicion errors sampled from he condiional densiy p dfm (v uc X 1,..., X T ) implied by he second sep dynamic facor model are ploed in he op panel. They appear as a band hrough he one-sep ahead predicion errors which is obained hrough X 1,..., X T. The boom panel shows he hisogram of p-values of he F -es for M regressions v uc = c+βṽ (i) +ɛ. The p-values are very low indicaing ha he esimaed parameers are highly significan which means ha a significan par of he variaion in is explained by he simulaed signal ṽ (i). v uc paern wih roughly one year period. Hence i is eviden ha he seasonaliy in he Niño3.4 ime series covariaes wih he seasonaliy of some of he explanaory variables as expeced, and hus X helps o capure his seasonal variabiliy beer during forecasing. On he oher hand, we are mainly ineresed in explaining hose pars of v uc ha are no seasonal, as El Niño is an oscillaion on iner-annual ime scales. Therefore, we are especially ineresed in he high correlaions beween X and v uc in Figure 5, hose ha do no correspond wih seasonal periodiciies. For example, in he case of he 100fin2 variable we observe ha i coninuously explains variabiliy for he lags 10-30, which is expeced as ypically before El Niño during his ime here is a subsurface anomalous warming of he ocean in he area where his variable is defined, which warming laer propagaes and plays an imporan role 19

20 in he generaion of El Niño in he easern Pacific. Similarly, in he case of he 500fin2 variable here is a high correlaion for he lags 16-30, again corresponding o his early subsurface warming of he ocean a greaer dephs. Finally, in he case of he wind sress variable we noice ha here is a coninuous high correlaion beween lags 0-16, which corresponds o a ime before El Niño when ypically weserly wind anomalies occur in he cenral Pacific and assis in he developmen of he phenomenon. All of hese variables were originally consruced wih he purpose o capure imporan dynamical informaion abou El Niño a is very early developmen sages and corresponding o very long lead imes. Wih our procedure we aim o exrac every bi of his informaion, and use i efficienly and parsimoniously in our forecasing of he El Niño evens. Figure 5 The correlaions beween he ensemble residual series and X fin fin fin2 WPAC fin windsress_ _ The figure presens he average correlaions beween ṽ (i) and a selecion of variables in X, conemporaneously and for is lags from 1 o 30. Some seasonal paerns can be deeced bu also i is shown ha some variables show a high correlaion a cerain lags bu no a ohers. The seleced explanaory variables are indicaed in he graphs. We have seleced 6 ou of 24 explanaory variables. 20

21 4 Forecasing Niño3.4 ime series and El Niño evens 4.1 Design of forecasing sudy In he firs par of our empirical forecasing sudy we consider he ime series Niño3.4 as he variable of ineres y ha we wan o forecas h-seps ahead for h = 1, 2,..., 30. Hence he maximum forecas window is 2.5 years. The ime series is observed from January 1982 o he end of 2015, ha is 34 years of daa and in oal 408 monhly observaions. For his period, we also have all observaions for he 24 explanaory variables; see Perova e al. (2016) for more deails and see also he discussion in he previous secion. We collec all 24 variables in he vecor of explanaory variables X. For producing he forecass of y we adop differen forecas modelling approaches and mehods, including our hree-sep forecasing procedure, indicaed as DFS. To evaluae he qualiy of he differen forecas sraegies we carry ou a rolling-window forecasing sudy. We consider an esimaion window of 275 monhly observaions. The firs window sars in May 1982 and we use he observaions in he esimaion window o carry ou he forecas procedure o obain ŷ T +h T for h = 1, 2,..., 30. The second window sars in June 1982 and we use again he nex 275 observaions o produce he nex 30 forecass by our procedure (all esimaion asks are repeaed). In his seing we can repea his 99 imes and hence obain 99 ou-of-sample forecas errors for each horizon h = 1, 2,..., 30. We hen compue measures of forecas precisions such as he roo mean squared error (RMSE), based on hese 99 forecas errors, for each h. The resuls for our DFS mehod are he focus of our presenaions. We compare differen measures of predicive accuracy of he DFS mehod wih hose obained from alernaive model-based forecasing mehods. We have considered he following models. (i) The seasonal auoregressive moving average (SARMA) model is specified wih seasonal auoregressive and moving average lag polynomials of lengh 12 and wih he corresponding regular polynomials of lengh 2 and 1, respecively. This specificaion is based on he Bayesian informaion crierion (BIC) for which hese polynomial lenghs have shown o have he smalles in-sample BIC values. Afer parameer esimaion by maximum likelihood, he model is represened in sae space form such ha he Kalman filer is used o produce all muli-sep ahead forecass. (ii) The unobserved componens (UC) ime series model or UCM as specified in he firs sep of our DFS forecasing procedure. (iii) The VAR forecass are based on an unresriced 21

22 VAR(2) esimaion for all 25 variables in he observaion vecor (y, X ) ). The lag lengh of 2 is also deermined by he smalles in-sample BIC value. Afer esimaion via regression, he VAR(2) model is formulaed in sae space form such ha he Kalman filer is used o produce all muli-sep ahead forecass. (iv) The forecass based direcly on he principal componens used in he second sep of DFS are compued as for he Sock and Wason (2002) approach. The number of principal componens ha are used is deermined by having 95% explained in he daa marix for X ; his is indicaed by he eigenvalues of he sample variance marix. We have ypically used 5 o 7 principal componens as a resul. The inclusion of lags for y and for he principal componens in he Sock and Wason procedure for forecasing is deermined by in-sample BIC. (v) The dynamic facor model (DFM) is he basic version wih he observaion vecor (y, X ) being linearly dependen on a se of five independen dynamic facors, each modelled as a sochasic cycle, see he discussion in Appendix B.2 and Durbin and Koopman (2012, Chaper 3). The model is cas direcly in sae space form and he loading, persisence and variance marices are esimaed by numerically maximizing he loglikelihood funcion ha is obained from Kalman recursions. (vi) The leas absolue shrinkage and selecion operaor (LASSO) is used o selec predicors from he collecion (X, X 1,..., X 36) which is 888-dimensional wih he lag of 36 suggesed by Perova e al. (2016). The LASSO hreshold or shrinkage parameer is chosen by BIC and he ypical number of nonzero coefficiens is 25. (vii) The collapsed dynamic facor model (CDFM) by Bräuning and Koopman (2014) summarizes informaion from he 889-dimensional vecor (y, X, X 1,..., X 36) using principal componens and form a DFM wih y and exraced principal componens. Based on he IC p 1 crierion of Bai and Ng (2002), he ypical number of principal componens is 30, close o he number of predicors seleced by LASSO. For our DFS forecasing procedure and he above-menioned model-based forecasing mehods, we repor he RMSE, MFLE and MRPS defined in he nex secion for each forecas horizon h = 1, 2,..., 30 wih he Diebold and Mariano (1995) es o verify he superior predicive accuracy of he forecass from our proposed mehod. Finally, we carry ou a forecas exercise for he probabiliy of El Niño evens using he DFS mehod. 22

23 4.2 Forecasing loss funcions and precision crieria We measure he precisions of ou-of-sample forecasing using differen loss funcions and from differen perspecive. The forecasing exercises are based on muliple seps ahead forecasing in a rolling window seing. In his way we obain a specific number of forecas errors for each mehod and for each forecas window lengh. We indicae a paricular rolling window by i and a forecasing lengh by h. To measure he predicive accuracy, we define he following loss differenial funcion for a rolling window i and a forecas lengh h as d i,h = L(e (j) i,h ) L(e(k) i,h ), for some loss funcion L( ) and h-sep ahead forecas errors e (j) i,h and e(k) i,h obained from models j and k, respecively. In our forecasing design, we have M = 99 rolling windows wih each conaining T = 275 daa poins, Harvey, Leybourne, and Newbold (1997) suggess he following Diebold-Mariano (DM) es saisic correced for small sample size M DMh h2 3h = DM h Suden s M 1 M where DM h is he Diebold and Mariano (1995) es saisic for he h-sep ahead forecas loss differenial as given by DM h = d 1 h / M ˆσ2 T,M, wih average loss differenial d h = 1 M M i=1 d i,h and where he heeroskedasic and auocorrelaion (HAC)-consisen variance esimaor of Var( M d i,h ) is given by P ˆσ T,M 2 = ˆγ(0) + 2 α pˆγ(p), see Giacomini and Whie (2006) for a discussion, wih he Barle weighs α p = 1 p/p wih P being he ineger par of M 1/4, for p = 1,..., P. When we adop he quadraic specral weighs, he values for our forecas precision measures have only changed marginally in our sudy. p=1 23

24 by Tables 1 and 2 presen he roo mean squared forecas error (RMSE) of model j as given RMSE (j) h = 1 M M i=1 L(e (j) i,h ), h = 1,..., 30, which summarizes he predicive accuracy based on he squared error loss funcion L(e (j) i,h ) = e (j) 2 i,h which we use in he HLN DM es. Table 3 and 4 shows he h-sep ahead mean linex forecas error MLF E (j) h of model j defined as MLF E (j) h = 1 M M i=1 L(e (j) i,h ), h = 1,..., 30, (9) which summarizes he predicive accuracy based on he linex loss funcion L(e (j) i,h ) = exp (βe(j) i,h ) βe (j) i,h 1; his loss funcion has been originally proposed by Varian (1975). The linex loss funcion is a combined measure of loss in poin forecas and loss in he forecasing direcion of change. Given he asymmery of he linex loss funcion, i is widely used in financial risk managemen where underesimaing value-a-risk (VaR) is more cosly han overesimaing i. The parameer β measures he aversion owards eiher negaive (β > 0) or posiive (β < 0) forecas errors. We choose β = 1 since underesimaing he Niño3.4 forecas increases he probabiliy of missing an El Niño even. Table 5 shows mean ranked probabiliy score defined in he same way as (9) o summarise he loss from some chosen models in forecasing he evens of El Niño and La Niña. The rank probabiliy score (RPS) saisic, Epsein (1969), is given by RP S = 1 2 ( 3 k 2 (ˆp j,i,h e j,i,h )), k=1 j=1 where j = 1 indicaes an El Niño even, j = 3 a La Niña even, and j = 2 corresponds o no even. The probabiliy ˆp j,i,h is he forecased probabiliy for each of he hree caegories j = 1, 2, 3 for window i and lag h. The binary variable e j,i,h is he acual oucome (0,1) for each of he hree caegories for window i and lag h. This scoring rule is sensiive o disance by aking ino accoun he probabiliy mass assigned o all caegories and no only he caegory of he observed oucome as for example in a log loss funcion. We average 24

25 he RP S over all forecass (for a specific forecas horizon h) from he rolling windows. This average RPS provides a good indicaor of he qualiy of our even forecass. The probabiliies ˆp j,i,h are obained by simulaion. For each of he simulaed series ν (i) obained in Sep (ii) of our proposed mehod, we draw N = 100 series wih he simulaion smooher. We exend he series ν (i) wih 30 missing values o obain a cloud of h = 1,..., 30 sep ahead forecass. Once we simulae a cloud of forecass we coun he number of imes an El Niño even, La Niña even, or no even occurs wihin a simulae 30-sep ahead forecas. The simulaed probabiliies ˆp j,i,h are he number of imes each even occurs divided by he oal number of draws. The confidence inerval for each probabiliy can be deermined by ˆp j,i,h Z ˆp j,i,h (1 ˆp j,i,h )/N, ˆp j,i,h + Z ˆp j,i,h (1 ˆp j,i,h )/N, wih Z he value corresponding o he desired inerval. We refer o Figures 9, 10, and 11 for he visualisaion of simulaed probabiliies for seleced El Niño evens. 4.3 Forecasing resuls for he 3.4Niño ime series Le us firs briefly discuss he srucure and conen of he Tables 1 o 5. A cell wih only a value and wihou any symbol (excep for he column wih header UC DFS) implies ha our DFS mehod ouperforms he oher model a a 5% level of significance. A shaded cell indicaes ha our DFS mehod underperforms anoher model a he 5% level. Single aserisk indicaes ha he DFS mehod ouperforms he oher model a a significance level of only 10% level. Double aserisks indicaes no significan difference in predicive accuracy beween US DFS and anoher model. Our DFS mehod is caegorised as a mulivariae forecasing mehod in Table 1 for he apparen reason ha i uilises relevan variables colleced in X. In erms of squared forecas errors, which is a measure for poin forecas accuracy, DFS significanly ouperforms all oher mulivariae models from 5-sep ahead on, excep for LASSO and VAR models which are occasionally inferior o DFS only a 10% level, such as for h = 7, 8, 9 and 29. The VAR model does well in he shor horizon, surpassing he DFS mehod a h = 1 and 3, and especially for h = 1, he RMSE of VAR is only , he lowes among all mulivariae models considered and 33% lower han ha of DFS which is A reason for his is ha 25

26 Table 1 RMSE of Mulivariae Models The able repors he roo mean squared forecas error RMSE h for mulivariae benchmark models as explained in Secion 4.1. A cell wihou a symbol (excep for he column wih header UC DFS) means he DFS mehod ouperforms anoher model a he 5% level. A shaded cell indicaes ha he UC DFS underperforms compared o anoher model a he 5% level. Single aserisk indicaes ha he UC DFS model ouperforms anoher model only a he 10% level. Double aserisks indicaes no significan difference in predicive accuracy beween US DFS and anoher model. Sep h DFM CDFM LASSO S&W VAR UC DFS