FORECASTING THE UK UNEMPLOYMENT RATE: MODEL COMPARISONS FLOROS, Christos *

Transcription

1 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.-(005) FORECASTING THE UK UNEMPLOYMENT RATE: MODEL COMPARISONS FLOROS, Chrisos * Absrac This paper compares he ou-of-sample forecasing accuracy of ime series models using he Roo Mean Square, Mean Absolue and Mean Absolue Percen Errors. We evaluae he performance of he compeing models covering he period January 97 o December 00. The forecasing sample (January 99 December 00) is divided ino four sub-periods. Firs, for oal forecasing sample, we find ha MA()-ARCH() provides superior forecass of unemploymen rae. On he oher hand, wo forecasing samples show ha he MA() model performs well, while boh MA() and AR() prove o be he bes forecasing models for he oher wo forecasing periods. The empirical evidence derived from our invesigaion suggess a close relaionship beween forecasing heory and labour marke condiions. Our findings bring forecasing mehods nearer o he realiies of UK labour marke. JEL classificaion: C53, E7 Keywords: UK, Unemploymen, Forecasing, AR, MA, GARCH. Inroducion A number of research papers have used ime series models for forecasing macroeconomic variables. Predicing he unemploymen rae is of grea imporance o many economic decisions. Various echniques, from he simple OLS mehod o he GARCH models, have been used o explain he forecasing performance of US and UK unemploymen raes. Recen invesigaions of forecasing unemploymen rae are Proiei (00), Gil-Alana (00), Peel and Speigh (000), Johnes (999), Peel and Speigh (995) and Rohman (99). * Chrisos Floros is Lecurer of Finance and Banking a he Universiy of Porsmouh (UK), Chrisos.Floros@por.ac.uk 57

2 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.-(005) Rohman (99) compares he ou-of-sample forecasing accuracy of six nonlinear models, while Parker and Rohman (99) model he quarerly adjused rae wih AR() model. Koop and Poer (999) use hreshold auoregressive (TAR) for modelling and forecasing he US monhly unemploymen rae. Recenly, Proiei (00) examines he ou-of-sample forecasing for he US monhly unemploymen rae by using seven forecasing models (linear and nonlinear). The sudy shows ha linear models are characerised by higher persisence perform significanly bes. As a general conclusion, Proiei (00) argues ha srucural ime series models are more parsimonious. Research papers for modelling and forecasing he UK unemploymen rae are Johnes (999), Peel and Speigh (000) and Gil-Alana (00). Johnes (999) repors he forecasing compeiion beween AR(), AR()-GARCH(,), SETAR(3;,,), Neural nework and Naïve forecas of UK monhly unemploymen rae. The sample covers he period January 90 o Augus 99. The resuls indicae ha SETAR model dominaes he ohers for shor period forecass, while non-lineariies are presen in he daa. Peel and Speigh (000) es wheher nonlinear ime series models of simple SETAR form are able o provide superior ou-of-sample forecass of UK unemploymen daa (February 97 Sepember 99). The resuls show evidence for superior ou-of-sample SETAR forecasing performance relaive o AR models (in erms of RMSE). Furhermore, Gil-Alana (00) uses a Bloomfield exponenial specral model for modelling UK unemploymen, as an alernaive o he ARMA models. The resuls indicae ha his model is a feasible way of modelling UK unemploymen rae. In his paper, we focus on modelling unemploymen using daa from he UK. In paricular, we re-examine he evidence for forecasing by using ime-series models. We compare hese forecass o several mehods based on he work of recen sudies. The main purpose of his paper is o es and repor he forecasing compeiion beween differen models and forecasing periods (horizons). Our approach is much in he same spiri of Proiei 5

3 Floros, Ch. Forecasing he UK unemploymen rae: model comparisons (00), Peel and Speigh (000) and Johnes (999) in ha i focuses on an in-deph comparison of forecasing models. We exend heir analysis and compare he performance of weny-hree models for UK unemploymen using recen daa. Johnes (999) compares five models (linear auoregressive, GARCH, hreshold auoregressive and neural nework), while Proiei (00) applies seven forecasing models for he levels of he unemploymen rae. In his paper, we use he RMSE, MAE and MAPE crieria, and compare various models. For comparison purposes, we esimae differen ARMA and (G)ARCH models, namely AR(p), MA(q), ARMA(p,q), GARCH(p,q), EGARCH(,) and TGARCH(,). Finally, we produce dynamic and saic forecass. The paper is organised as follows: Secion provides he daa and mehodology, while Secion 3 presens he main empirical resuls from various economeric models. Finally, Secion concludes he paper and summarises our findings.. Daa and Mehodology We employ monhly observaions of UK unemploymen rae covering he period January 97 o December 00. The daa are sourced from Daasream. The firs 300 observaions (January 97 December 995) are used for parameer esimaion, while he nex observaions (January 99 December 00) are used for forecas evaluaion. Figure presens he graph of he UK monhly unemploymen rae series for he period January 97 o December 00. In he Table, summary saisics for unemploymen rae are presened. Descripive Saisics show a mean of six and posiive values of skewness and kurosis. Also, he Jarque-Bera saisics sugges ha he normaliy is rejeced a 5% level. ADF es fails o rejec he null hypohesis of a uni roo in he unemploymen rae series. 59

4 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.-(005) Figure. The UK Unemploymen Rae (January 97 December 00) U K U n e m p l o y m e n R a e Table. Descripive Saisics UK Unemploymen Rae Mean.009 Median Maximum Minimum Sd. Dev Skewness Kurosis.590 Jarque-Bera Probabiliy Observaions 3 ADF- level Probabiliy ADF- s diff. Probabiliy (0.) (0.0053) The following ime-series models are employed: AR(p) model is one where he curren value of a variable depends on he values ha he variable ook in previous periods plus an error erm. AR(p) model can be expressed as:

5 Floros, Ch. Forecasing he UK unemploymen rae: model comparisons Y + = c + a Y + ay a py p u () where u is a whie noise disurbance erm. A moving average (MA) process is one in which he sysemaic componen is a funcion of pas innovaions. MA(q) model can be expressed as: Y = c + b ε + bε bpε p + ε () By combining he AR(p) and MA(q) models, an ARMA(p,q) model is a model ha he curren value of some series depends linearly on is own previous values plus a combinaion of curren and previous values of a whie noise error erm. The ARMA(p,q) specificaion is given by equaion (3): Y = c + a Y b ε + ay a py p + ε + bε + bε q q The GARCH model of Engle (9) and Bollerslev (9) requires join esimaion of mean and variance equaions. The curren condiional variance of a ime series depends on pas squared residuals of he process and on pas he condiional variances. The equaions of GARCH(p,q) are given by: Y ε σ = µ + ε ~ N(0,) = c + P Q aiε i + i= j= β j σ j (Mean equaion) (Variance Equaion) () The Variance equaion of he Exponenial- GARCH(,) model is ε ε given by: log ( σ ) = c + a + a + a3 log( σ ) (5) σ σ

6 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.-(005) The EGARCH model of Nelson (99) capures he volailiy clusering and measures he asymmeric effec. The main advanage over he GARCH model, proposed by Bollerslev (9), is ha now he leverage a is exponenial and variances are posiive. The Threshold-GARCH(,) model of Zakoian (990) and Glosen, Jagannahan and Runkle (993) is given by: Y ε σ = µ + ε ~ N(0,) = c + a ε + aε d + a σ 3 (Mean Equaion) (Variance Equaion) () Good news ( ε <0) and bad news ( ε >0) have an impac equal o a and a + a, respecively. In oher words, a negaive innovaion has a greaer impac han a posiive innovaion. The asymmery effec is capured by he use of he dummy variable d. Furhermore, we produce boh dynamic and saic forecass using he seleced models over he sample period. Dynamic mehod calculaes muli-sep forecass saring from he firs period in he forecas sample. Saic mehod calculaes a sequence of one-sep ahead forecass, using acual raher han forecased values for lagged dependen variables. If S is he firs observaion in he forecas sample, hen he dynamic forecas is given by: y ˆ s = cˆ() + cˆ() xs + cˆ(3) zs + cˆ() ys. On he oher hand, saic forecas is calculaed using he acual value of he lagged endogenous variable as: y ˆ S+ k = cˆ() + cˆ() xs+ k + cˆ(3) z S+ k + cˆ() ys + k. Following Brailsford and Faff (99) and Johnes (999), we compare he forecas performance of each ime-series model hrough he error saisics (crieria). Three error saisics are employed o measure he performance of he forecasing models. Namely, he

7 Floros, Ch. Forecasing he UK unemploymen rae: model comparisons Roo Mean Squared Error (RMSE), he Mean Absolue Error (MAE), and he Mean Absolue Percen Error (MAPE). Suppose ha he forecas sample is = S, S +,.., S + h and denoe he acual and forecased value in period as y and ŷ, respecively. The repored forecas error saisics are compued as follows: RMSE = h S+ h ( yˆ y ) + = S MAE = h + S+ h = S yˆ y (7) MAPE = h + S+ h = S yˆ y y The RMSE and MAE error saisics depend on he scale of he dependen variable. We use hem o compare forecass for he same series and sample across differen ime series models. The beer forecasing abiliy of he model is ha wih he smaller RMSE and MAE error saisics. 3. Empirical Resuls To ge a clear view and in-deph comparison of forecasing models, we divide he forecasing period (January 99 - December 00) ino four sub-periods. The forecasing sub-periods are as follows: (i) January 99 - Sepember 997, (ii) Ocober June 999, (iii) July March 00 and (iv) April 00 December 00. We apply four forecasing models for AR(p) and MA(q) models (wih p=q=,,3,) using he Leas Squares mehod, as well as four forecasing models for ARMA(p,q) (wih p=q=,). For 3

8 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.-(005) GARCH(p,q), we esimae four models (p=0, and q=,) using he Marquard algorihm. We also apply EGARCH(,) and TGARCH(,) models. Furhermore, we esimae a fourh order auoregressive/moving average model in which he residual variance is allowed o vary over ime following ARCH, EGARCH and TGARCH. Table shows he seleced models for each of he forecasing periods. We presen he bes forecasing models for UK unemploymen rae (i.e. he model wih he smaller forecas error saisics) using boh saic and dynamic mehods. In he case of he seleced RMSE, he error saisics vary from o.. Forecasing period provides he smalles RMSE for AR() model, while he larges RMSE is from oal forecasing period for MA()-ARCH(). Hence, in erms of RMSE, AR() model is he bes forecasing model. The forecasing resuls of he seleced MAE measures show a minimum value of for forecasing period, and a maximum value of for oal forecasing period. The smalles MAE value indicaes ha AR() model is superior han he oher ime series models. On he oher hand, MA()-ARCH() model proves o be he wors forecasing model wih he larges MAE value. In he case of MAPE, we find ha forecasing period provides he smalles value ( ), while oal forecasing period shows he larges value (9.73). The resuls show ha AR() provides superior forecass of unemploymen rae, while MA()-ARCH() model shows a poor forecasing performance. Appendix presens he parameers of he seleced forecasing models. For oal forecasing period, we selec MA()-ARCH() as he bes forecas model because i provides small RMSE, MAE and MAPE. For he same reason, we selec AR() for forecasing period, and MA() for forecasing period 3 as well as forecasing period. The resuls from oher models are available upon reques.

9 Floros, Ch. Forecasing he UK unemploymen rae: model comparisons For forecasing period, we selec he MA() model because i provides he smalles RMSE. Since mos of he parameers are significan a 5% level, we believe ha hese models can be used for poenial forecasing resuls. Table : Forecasing Performance for Seleced ARMA and GARCH models Model RMSE MAE MAPE Forecasing Period : January 99-Sepember 997 MA()- Dynamic Saic MA()- Dynamic Saic Forecasing Period : Ocober 997-June 999 AR()- Dynamic Saic Forecasing Period 3: July 999-March 00 MA()- Dynamic Saic Forecasing Period : April 00-December 00 MA()- Dynamic Saic Forecasing Period: January 99-December 00 MA()-ARCH()- Dynamic Saic Furhermore, we produce saic and dynamic forecass using he seleced models over he sample. Dynamic forecasing performs a muli-sep forecas of unemploymen rae Y, while saic forecasing performs a series of one-sep ahead forecass of he dependen variable. Appendix shows graphs of dynamic and saic forecass for UK unemploymen rae.. Conclusions Macroeconomic modelling, and forecasing, is a widely researched area in he applied economic lieraure. Predicing he unemploymen rae is one of he mos imporan applicaions for economiss and

10 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.-(005) policymakers. The accuracy of differen forecasing mehods is a opic of coninuing ineres and research. In his paper we repor he forecasing compeiion beween Auoregressive (AR), Moving Average (MA), GARCH, EGARCH and TGARCH models of he UK monhly unemploymen rae series. We es he ou-of-sample forecasing accuracy of weny-hree models. Specifically, we compare he forecasing echniques based on he following symmeric error saisics: Roo Mean Square Error (RMSE), Mean Absolue Error (MAE) and Mean Absolue Percen Error (MAPE). The resuls from he comparisons of saic and dynamic forecass by he ime series models show ha he simples models are he mos appropriae for forecasing. Our findings are basically as follows: (i) For oal forecasing period (January 99 December 00) he MA()-ARCH() model is a more appropriae approach han he oher models, (ii) For forecasing period January 99 - Sepember 997, he simple moving average MA() model produces he lowes RMSE. However, boh MAE and MAPE sugges ha MA() is he mos appropriae model, (iii) For forecasing period Ocober June 999, a fourh order linear auoregressive AR() is he seleced forecasing model, (iv) For forecasing periods July March 00 and April 00 - December 00, we find ha a fourh order moving average MA() model shows a good fi in our daa. The above resuls sugges ha AR and MA models perform well in erms of forecasing, in conras wih oher research papers, see Johnes (999). One possible explanaion for he forecasing superioriy of hese models is ha radiional, simple ime-series models capure he dynamical srucure generaing he unemploymen levels. However, a highly daa se is possible o affec he qualiy of he forecass. In addiion, forecasing resuls may change due o he forecasing periods (horizon) as well as he selecion of in-sample and forecas daa. Our findings bring economeric heory nearer o he realiies of UK labour marke. Addiional research is required o explain he forecasing superioriy of simple and highly approaches The selecion of he sar of he forecas sample is imporan for dynamic forecasing.

11 Floros, Ch. Forecasing he UK unemploymen rae: model comparisons using monhly and quarerly daa. Fuure research should seek o invesigae more complex forecasing mehods o predic European and Asian unemploymen series. References Bollerslev, T. (9): Generalized Auoregressive Condiional Heeroskedasiciy, Journal of Economerics, 3, Brailsford, T. J. and Faff, R.W. (99): An Evaluaion of Volailiy Forecasing Techniques, Journal of Banking and Finance, 0, 9-3. Engle, R. F. (9): Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50, Gil-Alana, L. A. (00): A fracionally Inegraed Exponenial Model for UK Unemploymen, Journal of Forecasing, 0, Glosen, L.R., Jagannahan, R., and Runkle, D.E. (993): On he Relaion Beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks, The Journal of Finance,, Johnes, G. (999): Forecasing Unemploymen, Applied Financial Economics,, Koop, G. and Poer, S. M. (999): Dynamic Asymmeries in U.S. Unemploymen, Journal of Business and Economic Saisics, 7, 9-3. Nelson, D. B. (99): Condiional Heeroskedasiciy in Asse Reurns: A New Approach, Economerica, 59, Parker, R. E. and Rohman, P. (99): The Curren Deph-of- Recession and Unemploymen Rae Forecass, Sudies in Nonlinear Dynamics and Economerics,,

12 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.-(005) Peel, D. A. and Speigh, A. E. H. (995): Nonlinear Dependence in Unemploymen, Oupu and Inflaion: Empirical Evidence for he UK, in The Naural Rae of Unemploymen: Reflecions on 5 Years of he Hypohesis, (Ed.) R. Cross, Cambridge Universiy Press, Cambridge. Peel, D. A. and Speigh, A. E. H. (000): Threshold Nonlineariies in Unemploymen Raes: Furher Evidence for he UK and G3 Economies, Applied Economics, 3, Proiei, T. (00): Forecasing he US Unemploymen Rae, Working Paper, Diparimeno di Scienze Saisiche, Universia di Udine, Udine, Ialy. Rohman, P. (99): Forecasing Asymmeric Unemploymen Raes, Review of Economics and Saisics, 0, -. Zakoian, J. M. (990): Threshold heeroskedasic models, Manuscrip, CREST, INSEE, Paris. Appendix. I.Toal Forecasing Period: January 99 December 00 MA()-ARCH() Mehod: ML - ARCH (Marquard) Mean Equaion Coefficien Sd. Error z-saisic Prob. c * MA() * MA() * MA(3) * MA() * Variance Equaion c * ARCH() * * Significan a he 5% level

13 Floros, Ch. Forecasing he UK unemploymen rae: model comparisons II. Forecasing Period : January 99-Sepember 997 MA() Mehod: Leas Squares Variable Coefficien Sd. Error -Saisic Prob. C * MA() * MA() Mehod: Leas Squares Variable Coefficien Sd. Error -Saisic Prob. C * MA() * MA() * * Significan a he 5% level III. Forecasing Period : Ocober 997-June 999 AR() Mehod: Leas Squares Variable Coefficien Sd. Error -Saisic Prob. c * AR() * AR() AR(3) AR() * * Significan a he 5% level IV: Forecasing Periods: July 999-March 00, April 00- December 00 MA() Mehod: Leas Squares Variable Coefficien Sd. Error -Saisic Prob. C * MA() * MA() * MA(3) * MA() * * Significan a he 5% level 9

14 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.-(005) Appendix. Graphs: Dynamic and Saic Forecass Forecasing Period M A( )- D Y N AM IC F O R EC A S T M A ( )- ST A T I C F O R EC A S T M A ( )- D Y N A M I C F O R E C A S T M A ( )- S T A T I C F O R E C A S T Forecasing Period A R ( )- D YN A MIC F O R E C AS T AR ( )- ST A T I C F O R EC A S T Forecasing Period M A ( )- D Y N A M I C F O R E C A S T M A ( )- S T A T I C F O R E C A S T 70

15 Floros, Ch. Forecasing he UK unemploymen rae: model comparisons Forecasing Period M A( )- D Y N AM IC F O R EC A S T M A ( )- S T A T I C F O R E C A S T Toal Forecasing Period M A ( )-A R C H ( )- D Y N A M I C F O R E C A S T.E+07.E+07.0E+07.E+07.E+07.0E+0.0E+0 0.0E F orec a s o f V a rian c e 7

16 Inernaional Journal of Applied Economerics and Quaniaive Sudies. Vol.-(005) M A ( )-A R C H ( ) - S T A T I C F O R E C A S T F o re c a s o f V a ria n c e Journal published by he Euro-American Associaion of Economic Developmen Sudies: hp:/ 7