Unobserved Component Model with Observed Cycle Use of BTS Data for Short-Term Forecasting of Industrial Production

Transcription

1 Sławomir Dudek Dawid Pachucki Research Insiue for Economic Developmen (RIED) Warsaw School of Economics (WSE) Unobserved Componen Model wih Observed Cycle Use of BTS Daa for Shor-Term Forecasing of Indusrial Producion Absrac In he paper we are checking he explanaory power of business endency survey daa (BTS) in shor-erm forecass of indusrial producion wihin he framework of he unobserved componen model (UCM). I is assumed ha he "unobserved cyclical componen" is common for reference quaniaive variable and qualiaive variable. In ha sense he cyclical flucuaion of indusrial producion can be approximaed by he flucuaions of BTS indicaors. We call such a specificaion of srucural ime series model he Unobserved componen model wih observed cycle" (UCM-OC). To esimae he sysem we are using he Kalman filer echnique. Then we compare he model recursive one-period ahead forecass o he hisorical pah of he reference series o check is ou-of-sample daa fi. The forecasing properies are also evaluaed agains alernaive models, i.e. "pure" UCM and ARIMA model. The analysis was performed for Poland and seleced European Union counries. Key Words: indusrial producion, business endency survey, shor-erm forecasing, unobserved componen model

2 8 Sławomir Dudek, Dawid Pachucki. Inroducion Business endency survey daa (BTS) is ofen used as an indicaor of he cyclical flucuaions in he real economy. The oucome of many empirical sudies is ha he survey daa is usually leading or coinciden wih he quaniaive one. In our paper we are using his propery of he BTS o make shor-erm forecass of indusrial producion. For ha purpose, he unobserved componen model (UCM), also known as he srucural ime series model was used. Wihin his model he ime series of indusrial producion is decomposed ino unobserved componens: he rend and he cycle. I was assumed ha he rend is approximaed wih an univariae ime series model. As o he "unobserved cyclical componen" i was assumed ha i is common for reference quaniaive variable and qualiaive variable. Therefore, he cyclical flucuaion can be approximaed by he flucuaions of BTS indicaors. Such specificaion can be called Unobserved componen model wih observed cycle" (UCM-OC). Then he model was used for making recursive one-period ahead forecass o check is ou-of-sample daa fi. In addiion he forecasing properies were evaluaed agains alernaive models, i.e., "pure" UCM and ARIMA model. The analysis was performed for Poland and seleced European Union counries: Germany, France, Ialy and he Unied Kingdom. The reference variable is index of indusrial producion, seasonally adjused. As qualiaive variables here are used hree BTS indicaors: ICI indusrial confidence indicaor IPT balance on quesion regarding producion rend observed in recen monhs IPE balance on quesion regarding producion expecaions. 2. General mehodology The main purpose of our research is o assess wheher he informaion included in he qualiaive daa (BTS daa) allows for improving he forecas of he quaniaive variables. To carry ou he analysis firs a benchmark model was seleced, as a comparison for he forecas. I erms of forecass of an indusrial producion ime series an univariae model such as an ARIMA model seemed suiable. Afer shor research and ess of various alernaives we decided o use as a benchmark he ARIMA (,,) model which on average provided us wih he bes forecass for all he analysed counries. The nex sep was he selecion of a mulivariae model o include he informaion from qualiaive variables. In his case we decided o use he unobserved componen models (UCM), also known as he srucural ime series model, which seem o have

3 Unobserved Componen Model wih Observed Cycle Use of BTS Daa for 85 some advanages compared o oher possible ime series specificaions, like for example ARIMA, ARIMAX models. The mos imporan is he direc economic inerpreaion of componens in he model. Such models can also deal wih mulivariae series, some daa irregulariies like srucural brakes and missing observaions. Flexibiliy of he sae space models in erms of suiable formulaion of paricular componens, possibiliy of work wih non-saionary ime-series and he soluion algorihm offered by he recursive procedure which is Kalman filer, makes hem quie a powerful ool for economic analysis. The main disadvanage of he mehodology seems o be relaively high sensiiviy of he soluion o he iniial parameers used for compuing. We used UCM specificaion which allow o decompose he ime series of indusrial producion ino unobserved componens: he rend, he cycle. Taking ino accoun fac ha business endency survey daa (BTS) are ofen used as indicaors of he cyclical flucuaions in he real economy he specificaion is assuming ha "unobserved cyclical componen" can be exraced basing on he behaviour of qualiaive indicaor. In ha sense unobserved cyclical flucuaion are in fac observed in flucuaion of qualiaive indicaor. So we decided o call our model: "Unobserved componen model wih observed cycle" (UCM-OC). Nex o he work of Planas, Roeger and Rossi (29) we decided for following sae space represenaion of our model: y = + c c BTS = µ = µ c BTS = φ * c + β * c + φ c2 * c + a µ = ω *( ρ) + ρ * µ 2 BTS + a + a The firs wo equaions are so called signal or measuremen equaions which describe relaionship beween observed: counry X indusrial producion ( c µ () y ) and counry X seleced BTS indicaor ( BTS ), and unobserved rend ( ) and cycle ( c ). The nex ree equaions of he sysem () named in lieraure he sae or ransiion equaions, describes he behaviour of unobserved componens. In erms of he cycle which in he model is kind of common componen for he indusrial oupu and he BTS indicaor, an AR(2) process defined wih he (ϕ c ) and (ϕ c2 ) parameers is assumed. For he rend (hird and fourh equaions of he above sysem) he dumped rend process is being considered wih he slope defined wih (µ ) facor being

4 86 Sławomir Dudek, Dawid Pachucki depended on he slope form previous period and some consan (ω), boh conneced wih he damped parameer ( ρ ). We esed differen rend specificaion in he sysem () (Pedregal 22), however he damped rend proposed above seems o fi he bes all he analyzed ime series. The smoohing behaviour of he rend for ( ρ ) being consrained o ake values beween and (if here is no addiional shock o he sysem) is a quie good approximaion of he behaviour of economic ime series (Gardner, McKenzie 29). The a BTS, a µ, a c are whie noise processes. As a qualiaive variable we used separaely hree indicaors: he indusrial confidence indicaor (ICI) wih monh lead o he common cycle, he balance on quesion regarding producion expecaions (IPE) wih monh lead, and he balance on quesion regarding producion rend observed in recen monhs (IPT) as coinciden. Hence we esimaed hree models respecively: UCM-ICI, UCM-IPE and UCM-IPT. Bearing in mind he above menioned advanages of he unobserved componen models over ARIMA models, as an alernaive we also esed univariae version of he sysem (), where he only observed signal is for indusrial producion, i.e. specificaion wihou second equaion. This model will be indicaed as UCM. For ou-of-sample analysis purposes from he whole ime sample las P=39 observaions were excluded o compare forecasing properies. The exclude sample covers period 27:M-2:M3 o check o assess he models reacion o he las global financial crisis. Thus he saring esimaion sample include T=8 observaions, i covers period 992:M-26:M2 (for Poland sample sars from 992:M3). Using defined above models (ARIMA(,,), UCM, UCM-ICI, UCN-IPE, UCM-IPT), 39 poin (one monh ahead) forecass were calculaed recursively wih re-esimaion of ha models. A each recursion he esimaion sample was increased by one monh forward and forecased poin (monh) also. For all models here were calculaed forecas errors for ou-of sample and average measures like roo mean squared error (RMSE) and mean absolue error (MAE). RMSE = P MAE = P P = P = e 2 = e = P P P f ( y y ) = P = f y y In order o check wheher he forecass from UCM-OC models are superior o he forecass from reference benchmark, here were calculaed relaive RMSEs and MAEs, i.e. raios of he roo mean squared errors and mean absolue errors of he UCM-OC 2 (2) (3)

5 Unobserved Componen Model wih Observed Cycle Use of BTS Daa for 87 models o he reference ARIMA model. The relaive RMSE is called also as a Theil s raio (called in some papers as a U saisic). If Theil s U saisic or relaive MAE is smaller han one, hen he forecass based on he BTS indicaors are superior o he forecass of he benchmark model. To check wheher forecas superioriy is saisically significan we focus on he es of equal predicive accuracy of Diebold and Mariano (995), which is widely used for comparing forecass of wo compeing models. We use Diebold-Mariano es wih squared error loss funcion and wih absolue error loss funcion. The loss differenials for ou-of-sample are calculaed as: UCM OC where e, e ARIMA d sqr d = abs UCM OC ( ) 2 ARIMA e ( e ) 2 = e UCM OC e ARIMA are forecas errors from compeing models. Two forecass have equal accuracy if and only if he loss differenial ( or 5) has zero expecaion for all. Thus he null hypohesis of equal predicive accuracy is H : E( ) versus he alernaive hypohesis H : E( ) = µ d = d () (5) differen from zero. When module of Diebold-Mariano es saisics (used for or 5) is higher han criical value wih given significance level han null hypohesis of equal predicive accuracy have o be rejeced. When Diebold-Mariano es saisics is negaive and empirical p- value is less hen assumed significance level (e.g. 5% or %) han forecass received form UCM-OC models are significanly superior o he forecass from ARIMA model. I should be underlined ha all he forecas errors used o calculae above saisics for each period in ou-of-sample have he same weigh, henceforh we call hem unweighed. Bu in many pracical siuaions precise forecass for some periods are more imporan han for ohers. For example, accurae forecasing of he beginning of a recession is of special imporance. In case of indusrial producion, which is srongly affeced by cyclical flucuaions i is especially imporan. Very ofen he sar of a recession correspond o a large decrease in indusrial producion. Hence, when selecing among compeing forecasing models, i makes sense o focus on hese crucial observaions and o pu more weigh on he errors in his periods. For his purpose, we use approach proposed by van Dijk e al. (23). To compare forecas accuracy hey proposed modified Diebold-Mariano saisic by using a weighed average loss differenial. As an examples of sensible weighing funcion hey proposed o use empirical cumulaive densiy funcion of forecased variable. Basing on CDF we can consruc lef ail (LT) weighing funcion and righ ail (RT)

6 88 Sławomir Dudek, Dawid Pachucki weighing funcion. The former is puing more weigh on periods when high rae of growh of reference variable is observed, he laer opposie, when rae of change is largely negaive. Formally, he weigh funcions for he lef ail and righ ail are given by: where ( ) y LT : RT : w w LT RT = Φ = Φ ( y ) ( y ) Φ denoes he empirical cumulaive densiy funcion of forecased variable. (6) DLOG_DE_IP_SA DLOG_FR_IP_SA Probabiliy.6. Probabiliy DLOG_IT_IP_SA DLOG_PL_IP_SA Probabiliy.6. Probabiliy DLOG_UK_IP_SA..8 Probabiliy Figure. Empirical CDFs for dlog of reference variable. Source: Own calculaion; DLOG_ firs difference of logarihm of reference variable.

7 Unobserved Componen Model wih Observed Cycle Use of BTS Daa for 89 In our paper we use disribuion of log-change of reference variable because forecasing models are consruced on levels. Figure depics he empirical cumulaive densiy funcions of reference variable for analyzed counries which are used o consruc weighs. Using above defined weighs (6), weighed forecas errors are calculaed: e w = w e (7) This weighed errors are used o calculae relaive RMSEs, MAEs and loss differenials ( and 5) for Diebold-Mariano es. In all experimens, he compeing forecass are evaluaed using unweighed and weighed (lef ail and righ ail weighs) versions of he Diebold-Mariano es saisic and weighed and unweighed relaive RMSEs and MAEs. 3. Predicive power of UCM-OC models wih BTS indicaors (ou-ofsample analysis). The UCM models, boh univariae and mulivariae versions, where idenified for all he counries. The only excepion was Poland, where he univariae UCM idenified he cycle wih really srange behaviour. The model had some problems wih differeniaing he rend and he componen of business cycle frequencies. The only possibiliy o deal wih his issue was o pu some addiional consrains in he UCM, which made he model for Poland differen form he ohers. As we decided o no differeniae he sysems for paricular counries, in he furher analysis he univariae unobserved componen model for Poland is skipped. This is a good example ha his kind of models is quie sensiive o he assumed parameers and formulaion of paricular componens. On he oher hand, he mulivariae specificaion allowed for solving he problem. As i was menioned in he mehodological par, o assess wheher he UCM-OC model ouperforms he benchmark we are looking for lower han one values for relaives MAE and RMSE or negaive values for Diebold-Mariano saisics (DM--sqr and DM--abs). All he resuls are presened in Tables a-c. Comparison of unweighed forecas errors does no provide clear answer (see also Figure 2) o he key quesion posed in he paper. On average, he forecas errors seem o be lower in he UCM-OC ype of models, however he Diebold-Mariano ess do allow for he saemen ha he difference is saisically significan. On he oher hand, in case of he UK, he ARIMA(,,) model provides significanly beer forecass excep he UCM-OC model where he IPT index was used as indicaor for he cycle. In his case here was no significan difference in he qualiy of he forecas beween

8 9 Sławomir Dudek, Dawid Pachucki UCM-IPT and ARIMA model. Anoher finding is ha in general he mulivariae version of UCM provides lower forecass errors han he univariae one. However for paricular counries he confidence indicaor which allowed for reducion of he error was differen. In case of Germany i was IPT, for France - IPE, Ialy IPT, Poland ICI, and he UK IPT. Afer giving weigh for he periods of high growhs (Table b righ ail weighing funcion) or high drops (Table c lef ail weighing funcion) he conclusions changed. In case of high posiive growh rae (expansion phase) he simple ARIMA model seem o ouperform he UCM approach. I is possible o idenify a leas one UCM model for Germany, Poland and he UK where a % significance level he ARIMA forecass where beer hen he UCM. In case of France i is 2% significan level. In case of Ialy or oher no idenified above models, he differences ware saisically no significan, which proves ha forecass accuracy is he same for boh ypes of he models. From he Figure 2 i is visible ha on general UCM models wih qualiaive indicaors in erms of forecas accuracy performed beer during he period of inensificaion of global finance crisis (end of 28). Beer performance of he UCM models when lef ail weighing funcion is used is also proved by analysis of relaive RMSE sans MAEs (Table c). For all counries (excep for he UK) he relaive average error saisics are lower han one. In he case of Poland and Ialy UCM-OC models were able o provide significanly beer (according o DM saisics) forecas han he benchmark. In oher cases, lef ail weighing does no significanly improve he UCM performance compared o he conclusions for unweiged errors, however relaive RMSEs and MAEs are lower han one.

9 Unobserved Componen Model wih Observed Cycle Use of BTS Daa for 9 Table a Evaluaion of he ou-of-sample forecasing power of BTS indicaors (unweighed). Counry Model MAE Rank RMSE Rank DM- sqr DM- prob sqr DM- abs DM- prob abs DE UCM UCM-ICI UCM-IPE UCM-IPT FR UCM UCM-ICI UCM-IPE UCM-IPT IT UCM UCM-ICI UCM-IPE UCM-IPT PL UCM UCM-ICI UCM-IPE UCM-IPT UK UCM UCM-ICI UCM-IPE UCM-IPT a) RMSE, MAE relaive roo means square error and mean absolue error of UCM-OC models over benchmark ARIMA(,,) model, value less han indicae superioriy of UCM-OC model, Rank ranking of models based on relaive RMSE, MAE. b) DM- sqr, DM- prob sqr Diebold-Mariano -saisics and empirical p-value for es wih squared error loss funcion, ) DM- abs, DM- prob abs Diebold-Mariano -saisics and empirical p-value for es wih absolue error loss funcion, negaive DM- saisics means ha on average forecas errors from UCM-OC models are lees han from benchmark model, p-value empirical significance level, if less han 5 or % han accuracy of one model is significanly beer han rival one.

10 92 Sławomir Dudek, Dawid Pachucki DE FR I II III IV I II III IV I II III IV I I II III IV I II III IV I II III IV I ARIMA UCM UCM-ICI UCM-IPE UCM-IPT ARIMA UCM UCM-ICI UCM-IPE UCM-IPT IT PL I II III IV I II III IV I II III IV I I II III IV I II III IV I II III IV I ARIMA UCM UCM-ICI UCM-IPE UCM-IPT ARIMA UCM-IPE UCM-ICI UCM-IPT UK I II III IV I II III IV I II III IV I ARIMA UCM UCM-ICI UCM-IPE UCM-IPT Figure 2. Forecas errors for paricular models and counries.

11 Unobserved Componen Model wih Observed Cycle Use of BTS Daa for 93 Table b Evaluaion of he ou-of-sample forecasing power of BTS indicaors (weighed righ ail). Coun -ry Model MAE Rank RMSE Rank DM- sqr DM- prob sqr DM- abs DM- prob abs DE UCM UCM-ICI UCM-IPE UCM-IPT FR UCM UCM-ICI UCM-IPE UCM-IPT IT UCM UCM-ICI UCM-IPE UCM-IPT PL UCM UCM-ICI UCM-IPE UCM-IPT UK UCM UCM-ICI UCM-IPE UCM-IPT a) RMSE, MAE relaive roo means square error and mean absolue error of UCM-OC models over benchmark ARIMA(,,) model, value less han indicae superioriy of UCM-OC model, Rank ranking of models based on relaive RMSE, MAE. b) DM- sqr, DM- prob sqr Diebold-Mariano -saisics and empirical p-value for es wih squared error loss funcion, ) DM- abs, DM- prob abs Diebold-Mariano -saisics and empirical p-value for es wih absolue error loss funcion, negaive DM- saisics means ha on average forecas errors from UCM-OC models are lees han from benchmark model, p-value empirical significance level, if less han 5 or % han accuracy of one model is significanly beer han rival one.

12 9 Sławomir Dudek, Dawid Pachucki Table c Evaluaion of he ou-of-sample forecasing power of BTS indicaors (weighed lef ail). Counry Model MAE Rank RMSE Rank DM- sqr DM- prob sqr DM- abs DM- prob abs DE UCM UCM-ICI UCM-IPE UCM-IPT FR UCM UCM-ICI UCM-IPE UCM-IPT IT UCM UCM-ICI UCM-IPE UCM-IPT PL UCM UCM-ICI UCM-IPE UCM-IPT UK UCM UCM-ICI UCM-IPE UCM-IPT a) RMSE, MAE relaive roo means square error and mean absolue error of UCM-OC models over benchmark ARIMA(,,) model, value less han indicae superioriy of UCM-OC model, Rank ranking of models based on relaive RMSE, MAE. b) DM- sqr, DM- prob sqr Diebold-Mariano -saisics and empirical p-value for es wih squared error loss funcion, ) DM- abs, DM- prob abs Diebold-Mariano -saisics and empirical p-value for es wih absolue error loss funcion, negaive DM- saisics means ha on average forecas errors from UCM-OC models are lees han from benchmark model, p-value empirical significance level, if less han 5 or % han accuracy of one model is significanly beer han rival one.

13 Unobserved Componen Model wih Observed Cycle Use of BTS Daa for 95. Conclusions I is possible o find UCM-OC model which provides a leas as good forecass as ARIMA benchmark. In general he UCM-OC models had lower forecas errors, however he prioriy over simple ARIMA ones was no saisically significan. The op idenified UCM-OC models were as follows: for Germany: UCM-IPT, for France: UCM-IPE, for Ialy: UCM-IPT, for Poland: UCM-ICI, for he Unied Kingdom: UCM- IPT. The bes indenified models under UCM-OC mehodology significanly (%) over performed he ARIMA models for Ialy and Poland in he downward phase of he cycle. As was menioned in he paper, in case of univariae UCM for Poland i was no possible o idenify reasonable cycle and rend under common model. On he oher hand, some addiional consrains le o received saisfacory resuls. This allow us o expec ha he relaxaion of one ype of model for all couriers consrain can addiionally improve he qualiy of forecass from he UCM-OC represenaion. This hypohesis was no a subjec of presen analysis, however i appears o be a good exension o he paper in he near fuure.

14 96 Sławomir Dudek, Dawid Pachucki References Diebold, F.X. and R.S. Mariano (995), Comparing Predicive Accuracy, Journal of Business & Economic Saisics 3, Dijk, van D. and P. H. Franses (23), Selecing a Nonlinear Time Series Model using Weighed Tess of Equal Forecas Accuracy, Oxford Bullein of Economics and Saisics, 65, European Commission DG ECFIN (997), The Join Harmonised EU Programme of Business and Consumer Surveys, European Economy Repor and Sudies, No 6, Brussels. European Commission DG-ECFIN (27), The Join Harmonised EU Programme of Business and Consumer Surveys - User guide. Gardner, E.S. Jr. and McKenzie, E. (29), Why he damped rend works, Working Paper Harvey, A.C. (985), Trends and Cycles in Macroeconomic Time Series, Journal of Business and Economic Saisics, Vol. 3(3), Harvey, A.C. (989), Forecasing, Srucural Time Series Models and he Kalman Filer, Cambridge Universiy Press, Cambridge, New York and Melbourne. Kuner, K. (99), Esimaing poenial oupu as a laen variable, Journal of Business and Economic Saisics,2,3, Pedegral, D.J. (22), Trend models for predicion of economic cycles, Universiy of Casilla- La Mancha Working Paper. Planas, C. and Roeger, W. and Rossi, A. (29), Improving real-ime TFP cycle esimaes by using capaciy uilizaion European Commission, Join Research Cenre.

15 Unobserved Componen Model wih Observed Cycle Use of BTS Daa for 97 Appendix : Definiions and daa sources All variables used in he paper are encoded in a uniform manner. The synax of he variable code is as follows: [counry code]_[variable code]_sa where: _SA means seasonal adjusmen. Counry codes according o EUROSTAT: Germany DE France FR Ialy IT Poland PL Unied Kingdom - UK Index of indusrial producion (IP) Descripion Index of indusrial producion (NACE Rev.2), monhly frequency, , single-base index 25=, seasonally adjused. Source EUROSTAT: on-line daabase: hp://epp.eurosa.ec.europa.eu/poral/page/poral/saisics/search_daabase Indusry producion index - monhly daa - (25=) (NACE Rev.2) (ss_inpr_m) OECD: on-line daabase only for Poland for years hp://sas.oecd.org Daase: Producion and Sales (MEI)/Producion in oal indusrial sa, 25= Business endency survey indusry (ICI, IPE, IPT) Descripion Business endency survey indusry, monhly frequency, , seasonally adjused. ICI indusrial confidence indicaor is he arihmeic average of he balances (in percenage poins) of he answers o he quesions on producion expecaions, order books and socks of finished producs (he las wih invered sign). according o EU definiion, balances for quesions from harmonized quesionnaire (see EC DG-ECFIN 27). IPT Producion rend observed in recen monhs - balance IPE Producion expecaions for he monhs ahead balance. Source EUROSTAT: on-line daabase: hp://epp.eurosa.ec.europa.eu/poral/page/poral/saisics/search_daabase Business surveys (Source: DG ECFIN)/ Business surveys - NACE Rev../Indusry - monhly daa (bsin_m) The Research Insiue for Economic Developmen (RIED), The Warsaw School of Economics (WSE): Business Aciviy in Indusrial Indusry periodic survey.

16 98 Sławomir Dudek, Dawid Pachucki Appendix 2: Graphs DE DE DE_ICI_SA DE_IPICI_CYCLE DE_IPE_SA DE_IPIPE_CYCLE DE FR DE_IPT_SA DE_IPIPT_CYCLE FR_ICI_SA FR_IPICI_CYCLE FR FR FR_IPE_SA FR_IPIPE_CYCLE FR_IPT_SA FR_IPIPT_CYCLE Figure 3a. Unobserved common cycle componen and BTS indicaors. Source: Own calculaions.

17 Unobserved Componen Model wih Observed Cycle Use of BTS Daa for 99 IT IT IT_ICI_SA IT_IPICI_CYCLE IT_IPE_SA IT_IPIPE_CYCLE IT PL IT_IPT_SA IT_IPIPT_CYCLE PL_ICI_SA PL_IPICI_CYCLE PL PL PL_IPE_SA PL_IPIPE_CYCLE PL_IPT_SA PL_IPIPT_CYCLE Figure 3b. Unobserved common cycle componen and BTS indicaors. Source: Own calculaions.

18 Sławomir Dudek, Dawid Pachucki UK UK UK_ICI_SA UK_IPICI_CYCLE UK_IPE_SA UK_IPIPE_CYCLE UK UK_IPT_SA UK_IPIPT_CYCLE Figure 3c. Unobserved common cycle componen and BTS indicaors. Source: Own calculaions.