Methodology. -ratios are biased and that the appropriate critical values have to be increased by an amount. that depends on the sample size.

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1 Mehodology. Uni Roo Tess A ime series is inegraed when i has a mean revering propery and a finie variance. I is only emporarily ou of equilibrium and is called saionary in I(0). However a ime series ha has o be differenced before i is saionary has an infinie variance. The series is saionary a a higher order I(d) e.g. an I() series is saionary a firs difference.. Dickey-Fuller (DF) Tes As a es for saionariy Dickey and Fuller (979) referred a uni roo es o us which is well esablished. To show he mean conens Alexander (200) uses an AR() model y c + ˆ α y + uˆ ˆ = where û i.i.d.(0 σ 2 ). If αˆ < he model is assumed o be saionary and he characerisic polynomial of he AR() process lies inside he uni circle oherwise i is nonsaionary and he variance increases wih ime. Unforunaely i is no useful o es wheher αˆ = and hen use a simple -es because hey are biased. I is more efficien o ake he firs difference of an AR() process y ˆ = ϕ y + û where ϕˆ = (αˆ - ). Now one can use a one sided -es wih H 0 : ϕˆ = 0 and H a : ϕˆ < 0 and compare i wih he criical values from Dickey and Fuller because hey showed ha sandard ˆ ϕ -raios are biased and ha he appropriae criical values have o be increased by an amoun ha depends on he sample size.

2 More recenly MacKinnon (99) implemens a much larger se of simulaions han hose abulaed by Dickey and Fuller. In addiion MacKinnon esimaes response surfaces for he simulaion resuls permiing he calculaion of Dickey-Fuller criical values and criical values for arbirary sample sizes. His resuls are summarised in he MacKinnon able. Sequenial es procedure:. Saring wih a relaively high number of 0 lags 2. Subsequenly reduce he number of lags unil he las coefficien is significan differen from zero on he 0 % level. 3. Compare he hree differen models (wihou drif and rend wih drif and wih drif and rend) by looking a he Akaike crierion. Choose he model wih he lowes Akaike crierion. 4. If he value of he es saisic is greaer han (or in absolue values lesser han) he criical value you canno rejec he I() null hypohesis on convenional significan levels. 2. Box-Jenkins Approach a) Idenifying Graphical display (srucural breaks oulier ec.) Deerminaion of he inegraion level (ADF- or PP-es) Comparison beween he heoreical and empirical ACF and PACF o deermine model ype and he order of he process: for AR models PACF in order deerminaion because PACF cus off a lag p for an AR(p) process; for MA models ACF is useful specifying he order because ACF cus off a lag q for an MA(q) series. For a MA() process he significan coefficien mus be lesser han 0.5 if he process should be inverible whereas he PACF decays. ARMA processes are difficul o idenify by looking a he ACF. Hence we have o esimae he differen model ypes and have o choose he lag lengh by looking a he

3 Akaike or Schwarz informaion crieria. Boh he coefficiens have o be significan and we mus selec he lowes crieria. b) Esimaion The comparison of he esimaed ACF and PACF wih he heoreical ARMA processes could be argue for several models hence selec he model by aking ino accoun: Parsimony Fulfilling he saionariy and inveribiliy condiion Evaluaion of he goodness of fi (see AIC and SIC). c) Diagnosic (Inspecion) Inspecion of he residual plo: Are here ouliers in periods when he model do no fi well? The residuals have o be whie noise by checking he ACF and PACF of he residuals 95 % confidence inerval (here are no sysemaic effecs or any srucures in he error erms (Chow break-poin es)) d) Forecas In-Sample forecas o conrol for he accuracy of he model: o Deecion of urning poins (MA-erm) o Raios: Roo Mean squared Error (RMSE) Mean Absolue Error (MAE) Inequaliy Coefficiens of Theil Theil s U (comparison wih Random Walk) Ou-of-Sample Forecas o Make only shor-erm forecass o One has o recalculae he acual values afer aking he logs or afer differeniaion o EViews can no do an appropriae Ou-of-Sample forecas hus one have o calculae he model in an Excel file 3. Coinegraion Consider we have wo I() variables y as dependen and x as explanaory variable for simpliciy wihou a consan. Generally if we make a linear combinaion ou of hem or y = ˆ α x + uˆ uˆ = y αˆ x he combinaion û will normally sill be I() since hey boh have infinie variance. However if he consan αˆ is herefore such ha he bulk of he long run componens of y and x cancel ou he combinaion could be I(0) more precisely he difference û would be I(0). If a linear combinaion of I() variables is saionary hen he variables are said o be coinegraed..

4 2. The Engle-Granger wo Sep Approach Engle and Granger (987) sugges a coinegraion es which consiss of esimaing he coinegraion regression by OLS obaining he residual û and applying uni roo es for û. To es an equilibrium asserion hey propose esing he null ha û has a uni roo agains he alernaive ha i has a roo less han uniy. Since û are hemselve esimaes new criical values need o be abulaed. Thus one has o use he correced MacKinnon criical values. We have he equaion uˆ = y αˆ x where û follows an auoregressive progress u ˆ ρ ˆ + ε = u wih εˆ i.i.d.(0 σ 2 ). One could assume hree possibiliies ha ρˆ is smaller equal or higher han one: If ρˆ > : y ~ I() and x ~ I() hen u ~ I(2) If ρˆ = : y ~ I() and x ~ I() hen u ~ I() If ρˆ < : y ~ I() and x ~ I() hen u ~ I(0). Only if ρˆ < a coinegraion relaionship exiss. If one wans o derive more informaion abou he dynamic behaviour of he variables he will have o apply an Error-Correcion model. Engle and Weiß (983) demonsraed ha if a se of coinegraed variables exis hey can be regarded as being generaed by an Error-Correcion model which is called he Granger Represenaion Theorem. 2.2 Error-Correcion model (ECM) Coinegraion is concerned wih long run equilibrium. On he oher hand Granger causaliy (see below) is concerned wih shor run forecasabiliy. These wo differen models can be considered in an error correcion model. The name error-correcion model is derived from he fac ha i has a self regulaing mechanism. Tha means i reurns afer deviaions auomaically o is long run equilibrium. The ECM has a long run equilibrium and uses pas disequilibrium as explanaory variables in he dynamic behaviour of curren variables. One can esimae a Vecor Auoregressive (VAR) process Tsay (2002)

5 y x = ˆ α y = ˆ α y 2 + ˆ α x 2 + ˆ α x 22 + ˆ ε + ˆ ε 2 where y no only depends on is own pas lags bu also on he pas lags of x and where x no only depends on is own pas lags bu also on he pas lags of y. If one akes he differences of a VAR model he will receive he Error-Correcion model: y x = λ zˆ = λ zˆ i= i= ( cˆ ( cˆ i 2 i y y i i + cˆ + cˆ 2 i 22 i x i x i ) + ˆ ε ) + ˆ ε 2 wih zˆ = y αˆ x which is exacly our residual series and wih λ 0 and λ 0. As one 2 can easily see if y - is oo high y will be reduced again over z and λ. The same holds for ˆ x over z and λ 2. However ẑ regulaes only he long run equilibrium wih λ and λ 2 as adjusmen speed bu if one wans o derive informaion abou he shor run adjusmen you will have o pay aenion o he second par of he equaion. The ECM shows how significan he lagged variables are by using simple -ess. If one wans o know how srong he influences of all lagged values ogeher are you will have o apply a es for Granger Causaliy. 2.3 Granger Causaliy Granger (988) showed ha in he case of a bivariae sysem wih he ime series x and y which are inegraed a he same order when he pas and presen value of y provides some useful informaion o forecas x + a ime i is said ha y Granger causes x. The normal esing procedure for Granger causaliy is esing for he significance of he coefficiens of lagged y which are used as he explanaory variables for x in he regression conex. If one looks a he second par of he Error-Correcion Model he es for Granger causaliy from y o x is an F-es for he join significance of ˆ ˆ (i = -). Similarly c 2 i he es for Granger causaliy from y o x is an F-es for he join significance of ˆ c 2 i. The srengh of Granger causaliy can change over ime he direcion of causaliy can change depending on ime ha is measured or here can be bidirecional causaliy. Granger causaliy

6 means ha a lead-lag relaionship beween variables in a mulivariae ime series is eviden. However his does no mean ha if we make a srucural change in one series he oher will change as well bu he urning poin in one series precede he urning poins of he oher 2. 2 See Maddala (998) p.88

7 Number of variables M ype of es Source: Response Surface esimaions o calculae he criical values of Dickey-Fuller and Engle-Granger ess wih a consan. Propabiliy value β β β 2 wihou consan wihou rend % % % wihou rend % % % wih rend % % % wihou rend % % % wih rend % % % wihou rend % % % wih rend % % % wihou rend % % % wih rend % % % wihou rend % % % wih rend % % % wihou rend % % % wih rend % % % MacKinnon (99 Table ). Excep for he value in he firs row all values refer o equaions wih a drif (consan (inercep)). K = β + β T - + β 2 T -2