Analysis And Evaluation of Econometric Time Series Models: Dynamic Transfer Function Approach

Transcription

1 1 Appeared n Proceedng of he 62 h Annual Sesson of he SLAAS (2006) pp 96. Analyss And Evaluaon of Economerc Tme Seres Models: Dynamc Transfer Funcon Approach T.M.J.A.COORAY Deparmen of Mahemacs Unversy of Morauwa Morauwa e mal cooray@mah.mr.ac.lk

2 2 Analyss And Evaluaon of Economerc Tme Seres Models: Dynamc Transfer Funcon Approach T.M.J.A.COORAY Deparmen of Mahemacs Unversy of Morauwa Morauwa Ths paper presened how o apply se of response varables and se of explanaory varables by usng sysems of equaons of regresson models as well as same se of varables model led by Dynamc Transfer Funcon (DTF) model, n order o compare degrees of accuracy and ner relaon beween such varables. we sudy a macro economerc forecasng sysem employed by he Accounng and Sascs of Annul Bullen publshed by Cenral Bank of Sr Lanka. We shall use hs sudy o llusrae he problems wh he ncluson of conemporaneous npu varables n he denfcaon of a ransfer funcon model and her effecs on he forecasng performance of he model. Followng he mehods are developed by he auhor. The daa used n hs sudy began n he frs quarer of 1991

3 3 Analyss And Evaluaon of Economerc Tme Seres Models: Dynamc Transfer Funcon Approach Absrac: In he applcaon of large scale economerc models for forecasng, he sngle-equaon ordnary leas squares (OLS) mehod s ofen used o esmae parameers n each model equaon. Ths paper nvesgaes he properes of he parameer esmaes under sngle-equaon esmaon mehods. Snce he resduals of a me seres regresson model s seldom a whe nose process, s found ha bas s almos nevable as long as endogenous varables are presen n a model equaon. Ths paper proposes a model denfcaon mehod based on reduced form Dynamc ransfer funcon (DTF) models ha can avod he bas of ransfer funcon wegh esmaes under raher praccal assumpons. I s found ha forecass can be grealy mproved f approprae models are denfed and employed. Inroducon The developmen and applcaon of me seres analyss n economerc forecasng has occurred rapdly durng he pas wo decades. In recen years, he focus n hs area has shfed from unvarae or sngle equaon o mulvarae and smulaneous equaon models. Despe vas advancemens n he developmen of economerc me seres modellng, "classcal" economerc models are sll one of he major ools used by many commercal economc forecasng frms o provde naonal economc forecass. In hs secon "classcal" economerc models are consdered as he smulaneous equaon sysems and Klen (1950), and suded exensvely by a number of economercans. In ypcal applcaons of he classcal economerc models, a smulaneous equaon sysem ofen consss of a se of lnear lag regresson equaons wh whe nose. For ypcal naonal economc forecasng sysems, he number of varables and equaons ncluded n he sysems s ofen large. Therefore s mpossble o perform a jon parameer esmaon of he full sysem as recommended n modern me seres economerc models. Even hough he classcal economerc model s ofen referred o as a sysem of equaons, s mporan o noe ha n ypcal applcaons of large scale economerc models, he use of "sysem" or "jon model" comes n a he forecasng sage, raher han a he model esmaon sage. In erms of model esmaon, he ordnary leas squares (OLS) mehod s usually appled o each equaon n he sysem ndvdually. In such a case, a large sysem of equaons s merely a collecon of sngle-equaon lnear regresson or lag regresson models as far as model esmaon s concerned. Snce he sngle equaon OLS esmaes may have serous bas, he accuracy of forecass based on such based esmaes s dubous. A number of alernave mehods, such as ARIMA and DTF have been suded exensvely. However, such mehods n pracce can only be appled o a small sysem of equaons wh whe nose.

4 4 The usefulness of large scale economerc models has been subjec o many crcsms, parcularly, he valdy of he models and her forecasng performance. In hs paper we consder an exenson of classcal economerc models ha may avod some drwabacks n modellng and mprove he forecasng performance of he models. In addon, we nvesgae he ssue of enave model denfcaon n economerc me seres modellng. Snce he properes of parameer esmaes and he accuracy of forecass based on he assumpon ha he form of he model s correcly specfed, he mporance of model denfcaon s apparen. In hs paper, The use of Dynamc ransfer funcon (DTF) analyss n he denfcaon of a reduced form economerc model s proposed. Classcal Economerc Models We brefly oulne he framework of classcal economerc models and her exensons n lgh of me seres models. We wll consder a sysem of equaons ha consss of wo varables, y and x where y and x may be ner-relaed and boh can be endogenous varables n he sysem. A general form of a classcal economerc model for such a sysem can be expressed as ω θ = + x + z (1) 1 1 y c1 1 δ 1( φ1( ω θ = + y + z (2) 2 2 x c2 2 δ 2 φ2 In classcal economerc models, s ofen assumed ha z 1 and z 2 are ndependenly and θ dencally dsrbued as mulvarae normal N ( 0, ), are he auoregressve-movng φ ω average (ARMA) operaors of he dsurbance erm, and 's are he ransfer funcons δ (Box and Jenkns (1976)), where φ( = 1 φ B φ B 2... φ B p 1 2 p (3) θ = 1 θ B θ B 2... θ B q 1 2 q (4) ω( = ω ω B ω B (5) ω B g g B q q δ = 1 δ B δ B 2... δ 1 2 (6) The model n (1) and (2) s also referred o as a raonal srucural form (RSF) model (Wall (1976)), or a smulaneous ransfer funcon (STF) model. In general, a k-equaon STF model can be wren n a compac marx form. The Proposed DTF Mehod Based on he resuls n sepwse auoregresson, we may employ DTF analyss. We shall presen he resuls based on he maxmum lag order of p. In he frs sep of he DTF esmaon, an AR(1) dsurbance model s used for all hree equaons. In examnng he

5 5 sample auocorrelaon funcon (ACF) and paral auocorrelaon funcon (PACF) of he resdual of he above wo equaons, The proposed DTF models s appled o he Naonal economc me seres of Sr Lanka. We sudy a macro economerc forecasng sysem employed by he Accounng and Sascs of Annul Bullen publshed by Cenral Bank of Sr Lanka. We shall use hs sudy o llusrae he problems wh he ncluson of npu varables n he denfcaon of a ransfer funcon model and her effecs on he forecasng performance of he model. Followng he mehods are developed by he auhor. The daa used n hs sudy began n he frs quarer of 1991 and ended n he fourh quarer of 2004, a oal of 56 observaons for each varable. Among he 56 observaons n each seres, he frs 52 observaons wll be used for model denfcaon and parameer esmaon. The las 4 observaons wll be used solely for he comparson of he forecasng performance of he models. One of he mos mporan applcaons of hs economerc model s o forecas he quarerly economc me seres of Sr Lanka. Among he dependen varables n he behavoral equaons of he economerc model, we shall only nclude 15 of hem n hs sudy. For he varables excluded n he sudy, four of hem are prce deflaon varables, wo of hem are ax varables, and hree oher varables are no easly nerpreable. Snce he wholesale prce ndex (WPI) and consumer prce ndex (CPI) are ncluded n he sudy, he behavor of hese wo varables s represenave for oher prce ndex varables excluded from hs sudy. The ax varables are no ncluded n he sudy due o he rregular varably of her seasonal paerns, whch s manly caused by changes of ax regulaons. For convenence of reference, he abbrevaons and defnons of he 6 dependen varables and her relevan explanaory varables are lsed below. The me seres plos for he 6 dependen varables (orgnal seres). All he seres o be suded are nonsaonary and possess srong seasonaly (excep a few). Noe ha resuls and me seres plos are delberaely omed because of no enough space. Those wll be ncluded n he full paper CONCLUSIONS The analyses shown above reveal several neresng pons ha are worh furher dscusson. When he DTF analyss s employed, a number of ransfer funcon models degenerae o ARIMA models. Ths resul ndcaes ha he assocaon beween he explanaory varables and he dependen varable s no as srong as he orgnal hypoheses of he models suggesed (or wha he classcal regresson models ndcaed). Ths s no oo surprsng f we ake he economc envronmen of Sr Lanka no consderaon. Sr Lankan has a hghly regulaed economy. A number of foregn and domesc evens also have had mporan mpacs on Sr Lankan s economy. All hese facors conrbue o major dsurbances whch mgh weaken he poenal relaonshps beween he dependen varables and her explanaory

6 6 varables. As he free economc envronmen becomes more maure and he polcal suaon becomes more sable, we may fnd ransfer funcon models more useful n modellng Sr Lankan economc me seres. References Chen, T. (1987). A quck glance a four economerc models. An Execuve's Gude o Economerc Forecasng (Eded by A. Mglaro and C. L. Jan). Graceway Publshng Company, Flushng, New York. Jenkns, G. M. and Alav, A. S. (1981). Some aspecs of modellng and forecasng mulvarae me seres. J. Tme Ser. Anal. 2, Johnson, J. (1984). Economerc Mehods. McGraw-Hll, New York. Klen, L. R. (1950). Economc flucuaons n he Uned Saes, Cowless Commsson Monograph 11. John Wley, New York. Tao, G. C. and Box, G. E. P. (1981). Modelng mulple me seres wh applcaons. J. Amer. Sas. Assoc. 76,