Title: GARCH modelling applied to the Index of Industrial Production in Ireland

Transcription

1 Tile: GARCH modelling applied o he Index of Indusrial Producion in Ireland Candidae: Aoife Healy Masers projec Universiy Name: Naional Universiy of Ireland, Galway Supervisor: Dr. Damien Fay Monh and Year of submission: July 7 Number of volumes: 1 1

2 I hereby cerify ha his maerial, which I now submi for assessmen on he programme of sudy leading o he award of masers is enirely my own work and has no been aken from he work of ohers save and o he exen ha such work has been cied and acknowledged wihin he ex of my work. Signed: Candidae ID No: Dae:

3 TABLE OF CONTENTS PAGE 1. INTRODUCTION 3. LITERATURE REVIEW 5.1 PROPERTY PRICES 6. GARCH MODELLING 7 3. STATISTICAL ANALYSIS ASSUMPTIONS WHEN DEALING WITH DATA 9 3. MODELLING WITH THE RAW DATA SET Daa descripion and analysis Model of he raw daa Analysis of errors Analysis of variance Weighed Leas Squares (WLS Levenes Tes Saisics Resuls MODELLING TRANSFORMS Modelling wih he Log Transform of he Daa Modelling wih inverse ransform of he series Resuls from Box-Cox Transforms Models ERROR ANALYSIS OF CHOSEN MODEL UNIT ROOT PROCESS VARIANCE FORECASTING WITH GARCH MODELLING VARIANCE ARCH- Auoregressive condiional heeroskedasiciy Generalised ARCH GARCH GARCH(1, CRITERION OF FIT MODELLING THE VARIANCE OF IIP RESULTS CONCLUSION 51 APPENDICES 5 APPENDIX A 5 A.1 WEIGHTED LEAST SQUARES SOLUTIONS FIGURES 5 APPENDIX B 56 B.1 MODELLING TRANSFORMS 56 B.1.1 Raw daa 56 B.1. Squared Transform of he Daa 58 3

4 B.1.3 Square Roo Transform of he Daa 6 B.1.4 Inverse square roo ransform 61 APPENDIX C 63 C.1 STATISTICAL THEORY 63 C.1.1 Saisics 63 C.1. Correlaion 63 C.1.3 Auocorrelaion 64 C.1.4 Sample auocorrelaion funcion 64 C.1.5 Parial auocorrelaion funcion 64 C. HYPOTHESIS TESTING 64 C..1 Ljung-Box Tes 65 C.. Levene s Tes 65 C..3 ARCH Tes 65 C.3 STOCHASTIC PROCESS 66 C.3.1 Whie Noise 66 C.3. Saionariy 66 C.3.3 Moving Average Process 66 C.3.4 Auoregressive Process 67 C.3.5 Leas Square Esimaion 68 C.3.6 Maximum Likelihood Esimaion 68 C.4 UNIT ROOTS 68 C.5 FIGURES RE MODELLED VARIANCE 71 BIBLIOGRAPHY 75 4

5 1. INTRODUCTION The original idea for his hesis was o examine 3 imes series as follows: (i Average New House Price in Ireland (197 6 (ii Average House Price in he Unied Kingdom (197 6 (iii Average House Price in he Neherlands (197 6 A model for he condiional mean of Irish house prices was o be esablished and using General Auoregressive Heeroskedasisic models (herein afer referred o as GARCH models he volailiy in Irish house prices over he pas number years, ogeher wih forecased volailiy was o be examined. This was hen o be compared wih he volailiy in house prices in he UK & he Neherlands in he ime periods prior o propery price crashes. As a resul i was hoped ha a paern of volailiy would emerge which will allow one o predic a crash in propery prices. However, his sudy was unable o coninue due o he lack of available daa on he propery markes in quesion, especially he Irish Propery Marke. However, a large number of papers in he area of modelling propery prices had been examined in preparaion for his sudy and as such hese have been oulined in secion.1 of he lieraure review below. In he absence of available daa he echniques described above are applied o anoher ime series namely he Index of Indusrial Producion in Ireland, hereinafer referred o as IIP. This measure is similar o a Gross Domesic Produc (GDP index, however unlike GDP, IIP is available on a monhly basis and hus allows for beer modelling. A sudy of he IIP raw daa was carried ou in he form of some basic saisical analysis o ry and idenify any underlying paerns in he daa. I was found ha he series was nonsaionary afer his analysis a which sage a number of ransforms were aken o conver he series o a saionary one. The resuls of which are discussed in secion Following from hese resuls i was found ha he inverse ransform of he daa was mos successful, alhough did no iniially appear o conver he series o a saionary one. However on invesigaion i was believed ha he series was possibly uni roo. This was invesigaed and found o be rue. Finally GARCH modelling was underaken on he daa and a final GARCH model decided on o model he variance in IIP.. LITERATURE REVIEW Due o he naure of his hesis wo academic areas were researched; (i Propery Prices (ii GARCH modelling 5

6 .1 PROPERTY PRICES Upwards of 15 published papers in various Economic journals have been reviewed in order o undersand he main drivers of propery prices in a counry and wha ype of modelling echniques have been used o predic propery prices in he pas. From he review compleed he general consensus is ha he following facors are he main drivers of propery prices: (a Real Incomes: wha people earn aking inflaion ino accoun (b Demographics: he amoun of people in he -9 age bracke (c Availabiliy of Credi: how easy or difficul i is o ge a morgage (d Housing Sock: he basic economics of supply & demand (e Governmen Policy: ax & duy ec. (f User Cos: ineres raes & inflaion. The facors lised above are referred o in propery economics as fundamenals. Many of he papers reviewed addressed he problem in he Irish Propery Marke Does a speculaive bubble exis? ; i.e. are houses in Ireland overpriced? The general conclusion is ha alhough here may be a small elemen of inflaion due o Governmen Policy e.g. samp duy changes, a bubble does no exis and Irish propery prices can be modelled using fundamenals. An enormous amoun of differen models exis for predicing house prices. However here are a number of elemens in he curren siuaion which are unprecedened and make he modelling process difficul hese are: i. The size and duraion of he curren real house price increases. ii. The degree o which hey have ended o move ogeher across counries. iii. The exen o which hey have disconneced from he business cycle. Anoher facor o consider in modelling house prices is he Business Cycle. In he pas house prices and business cycle urning poins roughly coincided from The curren price boom worldwide is srikingly ou of sep wih he business cycle. 6

7 Fig.1: Real House Prices versus Business Cycle Noe: Real House Price Index: Calculaed using purchasing power pariy adjused GDP weighs. OECD (Organisaion for Economic Co-operaion & Developmen Oupu gap deermines he Business Cycle. As can be seen from figure 1 above propery prices in general around he world seem o be increasing (18 counries were sampled o produce figure. Of hese counries here appears o be a general consensus among he papers reviewed ha propery is overvalued in he following counries: (i Unied Kingdom (ii Ireland (iii Spain [1] Conrary o his opinion, many of he papers reviewed on he Irish marke sae ha a bubble does no exis due o he fac ha he models fail because he following canno be aken ino accoun: (i Changes in regulaory condiions (ii Tax changes (iii Demographics [5]. GARCH MODELLING There are wo major papers of ineres in his area. The firs is by Rober Engle who developed ARCH modelling in 198 [15]. The second by Tim Bollerslev who developed GARCH modelling and inroduced i in his paper eniled General Auoregressive Condiional Heeroskedasisic Modelling. In his paper Bollerslev inroduces he exension of he ARCH process inroduced by Engle (198 o he GARCH process, which bears 7

8 resemblance o he exension of he sandard ime series AR process o he general ARMA process. [8] Heeroskedasiciy refers o unequal variance in he regression errors, in oher words ime varying variance. Heeroskedasiciy can arise in a variey of ways and a number of ess have been proposed o es for is presence. Typically a es is designed o es he null hypohesis of homoskedasiciy (equal error variance agains some specific alernaive heeroskedasiciy specificaion. The paricular models of ineres here are GARCH (General Auoregressive Condiional Heeroskedasisic models. The bes way o describe how his modelling echnique will be applied is via an example. Suppose we look a he reurn on an asse or porfolio over ime and he variance of he reurn represens he risk level of hose reurns; his is a ime series applicaion where heeroskedasiciy is an issue. Looking a he reurns of an asse over ime will sugges ha some ime periods are riskier han ohers, ha is, he expeced value of he magniude of error erms a some imes is greaer han a ohers. Moreover hese risky imes are no scaered randomly across quarerly or annual daa, insead here is a degree of auocorrelaion in he 'riskiness' of reurns. The ampliude of he reurns varies over ime and his is described as volailiy clusering. The ARCH & GARCH models are designed o deal wih jus his se of issues. The goal of hese models is o provide a volailiy measure (like a sandard deviaion ha can be used o make decisions relaing o he daa se in quesion. The challenge is o specify how he informaion can be used o forecas he mean and variance of he reurn condiional on he pas informaion. Many mehods exis o use he mean o forecas fuure reurns, however unil he inroducion of ARCH/GARCH models virually no mehods were available involving variance. The primary descripive ool was he rolling sandard deviaion i.e. calculaed using a fixed number of he mos recen observaions e.g. he rolling sandard deviaion could be calculaed every day using he mos recen monh of observaions ( days his can be hough of as he firs ARCH model. This model assumes ha he variance of omorrows reurn is an equally weighed average of he squared residuals from he las days. The assumpion of equal weighs seems unaracive as one would hink ha more recen evens would be more relevan. Also he assumpion of zero weighs for observaions of more han a monh old is also unaracive. The ARCH model proposed by Engle (198 le hese weighs be parameers o be esimaed, hus he model allowed he daa o deermine he bes weighs o use in forecasing he variance. The main problem which arises wih he ARCH model is ha of he necessiy of high order ARCH models required o cach he dynamic of he condiional variance. This high order implies ha many parameers have o be esimaed and he calculaions of hese can become prohibiive. [9] 8

9 The GARCH model was inroduced by Bollerslev (1986. This model is also a weighed average of pas squared residuals bu i has declining weighs which never go compleely o zero. The mos widely used GARCH specificaion assers ha he bes predicor of he variance in he nex period is a weighed average of he long run average variance, he variance prediced for his period and he new informaion his period which is he mos recen squared residual The GARCH model is based on an infiniy ARCH specificaion and i allows reducion of he number of parameers o be esimaed from an infinie number o jus a few. In Bollerslev s GARCH model he condiional variance is a linear funcion of pas squared innovaions and earlier calculaed condiional variances. [8] 3. STATISTICAL ANALYSIS This secion deals wih background saisics in relaion he raw series of daa o ry and esablish any underlying paerns which will allow for beer modelling. 3.1 ASSUMPTIONS WHEN DEALING WITH DATA The following assumpions are used when modelling a ime series: 1. Lineariy in he parameers: E(Y is a linear funcion of he parameer β regardless of he relaionship beween Y and ime. (E(Y = α + β.. Homoskedasiciy of ε : his means ha he error erms have equal variance. I implies ha ou independen variable Y also possesses equal variance as i is a linear funcion of ε. 3. Normaliy of residuals: ε is an independen idenically disribued random variable ha is approximaely normal wih E(ε = and VAR(ε =1, alhough his assumpion is ofen violaed in ime series analysis we will show ha wih a large enough sample size we do no need normaliy of he error erms. 4. Independence: Any value of he dependen variable Y is saisically independen from any lagged value Y -k for k=1,,. 3. MODEL BUILDING WITH THE RAW DATA SET 9

10 3..1Daa descripion and analysis Daa No. of Sample Sample Minimum Maximum observaions (n Mean Variance Y Table 3.1 Where Sample Mean = Y n 1 Y n (3.1 n ( Y Y and Sample Variance = Var( Y (3. n 1 1 As can be seen from he above he IIP index has been highly volaile during he period observed Fig 3.1: Plo of IIP As can be seen from he above plo here is quie obviously a rend in he daa, i appears o be ime dependen i.e. i is increasing wih ime. From his we would deduce ha he Y observed are no independen and hus non-saionary. 1

11 Fig 3.: Hisogram of IIP We noe here he median is 5.65 which is o he lef of he mean. 3.. Model of he raw daa We will es for auocorrelaion by examining he sample auocorrelaion funcion (SACF which is given by SACF nk ( Y Y ( Y k Y k 1 (3.3 Var( Y Var( Y k We will also es for auocorrelaion by visual inspecion of he SACF graphed agains differen lags. 11

12 1 SACF for lags.95.9 SACF Lag Fig 3.3: SACF for lags of IIP The above graph would sugges ha he series is saionary and ha an Auoregressive model wih lag 1 (AR(1 is suiable for modelling he series. There does appear o be a kick a he lag 1 which needs furher examinaion. We now examine he Sample Parial ACF (SPACF a a lag of τ given by: rˆ x (1, 1 1 rˆ x ( ˆ ( 1, ˆ ( 1 ˆ (, rx j rx r x j1,,3,... ( rˆ ( 1, ˆ x j rx ( j j1 where r ˆ (, is he SPACF a a lag of τ, r ˆ ( is he SACF a a lag of τ, and rˆ x (, j, x x for j is defined (Bowerman & O Connell, 1987 as: rˆ (, j rˆ ( 1, j rˆ (, rˆ ( 1, j for j=1,,., τ-1 (3.5 x x x x 1

13 1 PACF, nx=16 alpha = Fig 3.4: PACF of IIP Our PACF figure also shows a significan peak a lag 1, i.e. a year lag. As such we now use leas squares o solve he following AR model for θ 1 & θ : x(k = θ 1 x(k-1 + θ x(k-1 + ε(k (3.6 Using our ime series for IIP we le his represen x(k and call i Y (a vecor. We hen obain x(k-1 by delaying Y by one and call his X1 (a vecor. We obain x(k-1 by delaying Y by welve and call his X. We hen se X = [X1, X] a marix. θ is obained as follows; θ = (X T X -1 X T Y, where θ = [θ 1, θ ] (3.7 For he above daa θ 1 =.479 and θ = Analysis of errors We now proceed o errors beween he acual IIP daa and he fied model. 13

14 Fig 3.5: Acual IIP (dashed line versus forecased IIP Fig 3.6: Plo of error beween acual IIP and forecased IIP 14

15 From he above plo we can see ha he error varies wih ime i.e. i exhibis heeroskedasiciy Fig 3.7: Plo of error squared

16 Fig 3.8: Plo of absolue value of error We now wan o model he error based on he acual value of he ime series and as such we now examine he ACF of he error squared; Fig 3.9: ACF of error squared From he above plo i appears ha he error looks random (which is good as i means our original forecas model of IIP performs well. However as no paricular lag is eviden we canno use his as a means of building a model for he error. The following accuracy measures are used; ( Y ˆ ˆ Y / Y (i Mean Absolue Percenage Error (MAPE: 1 n 1 n (3.8 (ii Mean Absolue Deviaion (MAD: n 1 Y Yˆ n (3.9 16

17 (iii Mean Squared Deviaion (MSD: n 1 Y Yˆ (3.1 n The MAPE measures he accuracy of he fied ime series values as a percenage value and here i equals The MAD expresses accuracy in he same unis as he daa, which helps concepualize he amoun of error, here i is Finally he MSD is always compued using he same denominaor, n, regardless of he model, so you can compare MSD values across models. MSD is a more sensiive measure of an unusually large forecas error han MAD. Here i equals From Fig 5 above we see he volailiy of he series ends o increase over ime, paricularly owards he laer sages, his paern suggess heerogeneiy of variance may be presen in our ime series, which violaes one of our main assumpions. The series does appear o be ime saionary hough as he expeced value doesn appear o vary wih ime in any significan way Analysis of variance We will now look a hree differen residual plos as hese will help us deermine wheher or no any more of our assumpions have been violaed. The hree graphs are as follows; (i Residuals versus he fied value (Fig 3.1 (ii Hisogram of he residuals (Fig

18 Fig 3.1: Residual versus he Fied Values From Fig 9 we can see a fanning ou effec among he residuals which as before would lead us o conclude ha our series possesses non consan variance. 18

19 Fig 3.11: Hisogram of Residuals From Fig 1 we may be emped o conclude ha he disribuion of e is approximaely normal, however since we have already seen ha our series possesses non-consan variance we can conclude ha he disribuion is no normal. We will now use he mehod of weighed leas squares o deal wih non consan variance. Due o he series e possessing non-consan variance some of he observaions in our sample provide more reliable informaion abou he regression funcion han ohers. Observaions wih small variance are more reliable han observaions wih large variance and should herefore be given more weigh. Denoe he variance of he error erm e by E[e ] = σ e. By assigning a weigh w = 1/ σ e o our model we have a new regression funcion expressed in marix noaion of he form: Y = B T TW + E & E(Y = B T TW since E(E =. (3.11 Where; 19

20 Y Y Y.. 1 Y n B T 1, 1 1, , n W w1,,,..., w,, ,,... w n 1 E. (3.1. n Such a model should recify our violaion of he assumpion of consan variance and yield a variance which is consan over ime. σ e is unknown and mus be esimaed from our daa. By assumpion σ e =E( s. e Unforunaely e s is unknown and mus be esimaed. We have observed ha e s ends o increase over ime (Fig 3.6; his means ha he fis obained from an ordinary leas squares regression of e using ime as predicor variable are unbiased esimaor of s e, provided ha he regression funcion is appropriae, meaning ha i accuraely describes he relaionship beween ε and ime. The validiy of his resul can be easily shown by observing ha se E( e ( E( e (3.13 e and since E(e = i follows ha s E( e. Finally since E( e E( ˆ i follows ha E eˆ can be used o esimae e ( e s. Similarly, one can esimae he sandard deviaion of he residual by regressing he absolue value of he error erm agains ime. We mus decide carefully wheher o esimae for variance or sandard deviaion in our weighs. The variance is less forgiving of observaions wih relaively large variances and is more appropriae when he discrepancies in he variance appear o be small. Since our residual plo agains ime (Fig 5 displayed a fanning ou effec where some residuals displayed a variance of considerable greaer magniude han ohers he esimae of he sandard deviaion of he residuals is a more appropriae choice of weigh in his case. Now we need o esimae he sandard deviaion of he residuals by choosing a funcional form which bes describes he relaionship beween he squared value of he residuals and ime.

21 Since i is hard o jusify any heoreical relaionship beween he populaion variance and ime i is bes ha we proceed by observing a plo of e agains ime and a fied regression line. As expeced he series does become more volaile over ime. I is obvious from he graph in Fig 3.8 ha he absolue value of he residuals is probably an exponenial or quadraic funcion of ime. However, we won compleely discoun he possibiliy of he relaionship being linear. Firs we fi a linear model o he error squared he resuling model is y = , where y is he error squared and is ime. 45 Plo of Error Squared versus fied line Fig 3.1: Plo of error squared versus fied line Nex we fi a quadraic model o he error squared and he resuling model is; y = , again y is he error squared and is ime. 1

22 45 Plo of Error Squared versus fied quadraic Fig 3.13: Plo of error squared versus fied quadraic We now fi an exponenial curve o he error squared of he form: y = ae b. The model hen becomes log(y = log(a + b. The fied model is: log(y = We hen ake he exponenial of he log(y o recover he fied curve.

23 45 Plo of Error Squared versus fied exponenial Fig 3.14: Plo of error squared versus fied exponenial Finally we aemp o model he error squared as a funcion of acual IIP i.e. x(k = a + by(k where x is he error squared and y is acual IIP. 3

24 45 Plo of Error Squared versus fied model based on IIP Plo 3.15: Plo of error squared versus fied model based on IIP 3..5 Weighed Leas Squares (WLS We now ry o esimae a new AR model on he basis of WLS. We will each of he fied models of he error above as our esimaes of and use his o calculae he W marix as oulined in equaion 3.1. We will hen es he resuling new error for homogeneiy of variance using Levenes es as oulined above. In he firs case we use he esimaes of new fied model afer carrying ou WLS is: as produced by he linear fied model above. The y(k =.388y(k y(k-1. (3.14 4

25 45 Plo of Error Squared verus esimaed sandard deviaion esimae Fig 3.16: Plo of error square versus esimae sandard deviaion esimae 14 Plo of acual IIP(dashed line versus forecased IIP using WLS Fig 3.17: Plo of acual IIP versus fied IIP using WLS 5

26 As can be seen from Fig 3.17 our model is much worse han he original model see plo of he new error squared below. 18 Plo of error squared beween IIP & forecased IIP using WLS Fig 3.18: Plo of error squared. The magniude of he error squared is much greaer han our original model. I is also noiceable from he plo ha using WLS has no solved he problem of heerogeneiy of variance. Please able of Levenes Saisics under resuls secion 3..6 below. In he second case we use he esimaes of as produced by he quadraic fied model above. The new fied model afer carrying ou WLS is y(k =.38798y(k y(k-1 (3.15 6

27 45 Plo of Error Squared verus esimaed sandard deviaion esimae Fig 3.19: Plo of error squared versus esimaed sandard deviaion The resuls are very similar o hose produced above wih he forecas model much worse han before, please see plo of he error squared below: 7

28 18 Plo of error squared beween IIP & forecased IIP using WLS Fig 3. Plo of error squared Again he magniude of he error appears much larger and he problem of heerogeneiy of variance remains. In he hird case we use he esimaes of as produced by he exponenial fied model above. The new fied model afer carrying ou WLS is y(k =.3845y(k y(k-1 (3.16 We again receive very similar resuls o hose achieved above, again wih he magniude of he error having increased and he problem of heerogeneiy remaining unsolved. In he final case we use he esimaes of as produced by he fied model based on IIP. The new fied model afer carrying ou WLS is: y(k =.3878y(k y(k-1 (3.17 Once again he same resuls emerge. Please see appendix 8

29 3..6 Levenes Tes Saisics Resuls There are four cases o be discussed as follows; (i Error squared modelled as line (ii Error squared modelled as quadraic (iii Error squared modelled as exponenial (iv Error squared modelled wih acual IIP Please noe ha H assumes ha here is equal variance i.e. he series is homoskedasisic Case Sample Sample Variance F Saisic Significance Size Level i Rejec ii Rejec iii Rejec iv Rejec Table 3. We conclude ha in all cases WLS has failed o solve he problem of heeroskedasiciy. This is due o he fac ha he original model was no a good enough fi in he firs place. We now aemp o ransform he series in an effor o achieve a series which can be modelled beer han he raw daa. H 3.3 MODELLING TRANSFORMS As saed a he ouse of secion 3. he raw daa series is non-saionary, as such we now underake some ransforms of he daa and examine he resuls. A well documened mehod of ransforming he daa is o use Box-Cox ransforms. The Box-Cox procedure auomaically idenifies a ransformaion from he family of power ransformaions on Y. The family of power ransformaions is of he form Y =Y λ, where λ is a parameer o be deermined from 9

30 he daa. Noe ha his family encompasses he following simple ransformaion which I have underaken in analysing he daa [13]: λ Y Y ½ Log e Y (by definiion - ½ -1 Table 3.3 The mehod I used o idenify he bes value for λ was as follows; 1. Conver he ime series according o he above-menioned ransforms Y 1 1 Y. Model he convered series using echniques oulined above i.e. using he ACF & PACF o esablish which lags o use for an AR or MA model. 3. Using he series X generaed by he relevan model, I ook he inverse of he ransform o ge he modelled series X. 4. Calculaed he residuals: r=y-x. 5. Calculaed he SSE and he MSE of he residuals in each case. Y A model was consruced for each of he above ransforms, in he secions below namely & 3.3. we deal wih he log ransform and inverse ransform respecively as hese had he mos promising resuls. The remainder of he models are discussed in Appendix A. The resuls and comparisons from all he models are deal wih in secion Log Transform of he Daa Given ha he underlying disribuion of he IIP is possibly exponenial we now ake he log values of he daa o cover he series o a linear series. A lile more ime was aken wih his ransform due o he belief of he possible underlying disribuion. 3

31 5 Log Value IIP versus Time 4.5 Log Value IIP Time Fig 3.1: Log Value of IIP versus Time We look a he ACF & PACF of his o ry and esablish a model for he series: 1. ACF plo for Log(IIP Fig 3.: ACF of Log(IIP 31

32 1. PACF, nx=16 alpha = Fig 3.3: PACF of Log(IIP From he above figures he process appears o be AR(1: 14 Acual IIP v's forecased IIP using AR(1 model on ransformed daa IIP Time 3

33 Fig 3.4: Plo of IIP versus model obained using log ransform 4 Plo of Errors 3 Magniude of Errors Time Fig 3.5: Plo of he error beween acual IIP & forecased IIP using AR(1 model on ransformed daa. As can be seen here are quie a large magniude of errors we inspec he ACF o esablish a beer model: 33

34 1 ACF Plo of Error Values Fig 3.6: ACF of errors Afer numerous aemps wih differen forms of models i was decided ha his series could no be modelled wih any degree of accuracy using hese mehods. Noe oher models aemped: AR model: lags 1,6,1 AR model: lags 1,,3,6,1,36 AR model: lags 1,1. Afer concluding he above series couldn be modelled adequaely a new series was looked a i which was he differences of he log(iip; 34

35 .3 Differences of Log(IIP..1 Differences Time Fig 3.7: Plo of difference of he Log(IIP The ACF & PACF of his series are hen examined o ry and esablish a model: 35

36 1 ACF of he Difference of Log(IIP Fig 3.8: ACF of differences of Log(IIP.5 PACF, nx=15 alpha = Fig 3.9: PACF of differences of Log(IIP 36

37 There appears o be reason o use an AR model wih lags a 1, 6 and 1. Firsly we will look a an AR(1 model: Acual IIP v's forecased IIP using AR(1 model wih log differences ransform 15 1 IIP Time Fig 3.3: Plo of acual IIP versus forecased IIP using an AR(1 model on ransformed daa using log differences. 37

38 3 Plo of Errors 1 Magniude of Errors Time Fig 3.31: Plo of he error beween acual IIP & forecased IIP using AR(1 model on ransformed daa. Now he AR model wih lags a 1, 6 & 1 is examined; 38

39 15 Acual IIP v's forecased IIP using AR(1,6,1 model on ransformed daa 1 IIP Time Fig 3.3: Plo of acual IIP versus forecased IIP based on AR(1,6,1 model on he ransformed daa 3 Plo of Errors 1 Magniude of Errors Time 39

40 Fig 3.33: Plo of he error beween acual IIP & forecased IIP using AR(1,6,1 model on ransformed daa. Again he resuls from each of hese echniques are discussed in secion Inverse ransform of he series 14 Acual IIP v's forecased IIP using AR(1 model using inverse ransform IIP Time Fig 3.34: Plo of IIP versus series modelled using inverse ransform 4

41 4 Plo of Error 3 Magniude of Error Time Fig 3.35: Plo of he error beween acual IIP & forecased IIP using AR(1 model on ransformed daa. Again he resuls will be discussed in secion below Box-Cox Transforms Resuls Model Transform Predicive SSE Predicive MSE Training Validaion Tes Training Validaion Tes AR(1 No applicable AR(1 Y AR(1 Y AR(1 Log e Y AR(1 1/Y AR(1 1/ Y AR(1 AR(1,6,1 Table 3.4 Log Differences Log Differences

42 From he above able of resuls, i appears ha he model which bes fis he daa is he one calculaed afer he inverse ransform is aken i.e. 1/Y and he AR(1 model. However wha is noeworhy abou he resuls is he sharp increase in he SSE & PMSE for he es and validaion ses as compared wih he raining ses. The reason for his we suspec is ha he es and validaion ses are aken from laer on in he daa. As discussed a he sar of his secion he daa was divided ino 3 ses, he raining se being he firs 13 poins, he es se he nex 44 poins and he validaion se he final 4 poins. As he series progresses he ime series is larger and as such he errors are larger. To invesigae his furher we calculae he predicion mean absolue percenage error (PMAPE which is defined as error( e( PMAPE x1 x1 (3.18 iip( Y ( The average of PMAPE for he raining, validaion and es ses are oulined in he able below: Model Transform PMAPE Training Validaion Tes AR(1 No applicable AR(1 Y AR(1 Y AR(1 Log e Y AR(1 1/Y AR(1 1/ Y AR(1 Log Differences AR(1,6,1 Log Differences Table 3.5 As can be seen from he above able he purpored cause of he large variaion in he PSSE & PMSE is rue and he PMAPE does no differ significanly over he hree daa ses. On he basis of he above resuls we proceed wih he model developed on he basis of he log ransform due he combinaion of he PSSE, PMSE and PMAPE resuls. 3.4 ERROR ANALYSIS OF CHOSEN MODEL 4

43 We now calculae he Ljung-Box Q saisic o es for model adequacy. The above AR(1 model based on he log ransform of he raw daa has been fi, we now wish o es he errors beween he fied model and he acual ransformed daa o see if here is a deparure from randomness of he errors. We are esing he null hypohesis ha he model fi is adequae, i.e. here is no serial correlaion a he corresponding elemen of lags. The resuls are as follows; Error Lags H P-Value Q-Saisic Criical Value Model Error Model Error Model Error Training Se Error Training Se Error Training Se Error Tes Se Error Tes Se Error Tes Se Error Validaion Se Error Validaion Se Error Validaion Se Error Table 3.6 Noes o Table 3.6 H = indicaes accepance of he null hypohesis, H=1 rejecs he null hypohesis. Significance level of all ess = 1% P-values (significance levels a which he null hypohesis of no serial correlaion a each lag in Lags is rejeced. Criical value is of he Chi-square disribuion for comparison wih he corresponding elemen of Qsa. As can be seen from he above able he null hypohesis has been rejeced a every level. So for he momen we will accep ha he errors from he model are serially correlaed. We now es he errors for ARCH effecs. This es will es for he null hypohesis ha he ime series of residuals (errors is i.i.d. Gaussian i.e. homoskedasisic. The resuls for he residuals from he fied model are as follows; 43

44 Lags Significance Level H P-Value ARCH-sa Criical Value 15 5% x Table 3.7 To summarise we are now in he following siuaion; We have ransformed he original series Y o X=ln(Y Modelled his using he following AR(1 model: X(k =.99685X(k-1 + e(k (3.19 We have esed he model errors for serial correlaion and i appears ha e(k is serially correlaed. Furhermore we have esed he model errors for ARCH effecs and we have found ha hey exhibi heeroskedasisciy. 3.5 UNIT ROOT PROCESS A his sage we suspec ha he ransformed series X(k is a uni roo process I(1, ha is he rue model of he daa is of he form: X(k = X(k-1+e(k. (3. Please see Appendix 7. for furher deails of his. We will use one of he Phillips-Perron ess for uni roos of which here are four cases. The case which applies here is he es done under he assumpion ha he rue process is of he form: X(k = α + X(k-1+e(k, α any value (3.1 and he modelled process is of he form: X ( k ˆ ˆ X ( k 1 ( k e( k (3. i.e. here is a drif in he modelled process. We have confirmed from he Ljung-Box es and he ARCH es ha he errors of he chosen model are serially correlaed and heeroskedaisic and he Phillips-Perron ess allow for hese siuaions and deal wih hem in esing for he uni roo. As such we calculae he Philips-Perron ρ es and he Phillips-Perron es saisics. 1 ( ˆ 1 (. ˆ / ( ˆ es T T ˆ ˆ s (3.3 ˆ.5 ˆ es ˆ / {.5( ˆ ( T. ˆ / s / ˆ} (3.4 ( ˆ where; T = sample size ˆ = auoregressive coefficien 44

45 ˆ ˆ =sandard error of he auoregressive coefficien s =sandard deviaion of he error (e(k ˆ ˆ {.8( ˆ.6( ˆ.4( ˆ.( ˆ } ˆ i= i h auocovariance = ( ˆ -1/ ˆ ˆ The following resuls were obained: 1 Phillips-Perron Value Significance Level Uni Roo ρ es >-1 Yes es >-3.44 Yes Table VARIANCE FORECASTING WITH GARCH 4.1 MODELLING VARIANCE When modelling using leas squares i is assumed ha he expeced value of all error erms when squared is he same a any given poin, his assumpion is known as homoskedasiciy. Daa in which he variances of he error erms is no equal, i.e. he variance varies over ime is said o be heeroskedasisic. In he presence of unequal variance and when using ordinary leas squares regression, he regression coefficiens remain unbiased, however he sandard errors and confidence inervals esimaed by convenional procedures will be oo narrow giving a false sense of precision ARCH- Auoregressive condiional heeroskedasiciy The ARCH-model was firs inroduced by Rober Engle in 198. Firs consider an ordinary AR(p model of he sochasic process y. y c y... y u 1 1 p p (4.1 Where u is whie noise. The basic AR(p-model is now exended so ha he condiional variance of u could change over ime. One exension is ha u iself follows an AR(m- process u u... u w ( where w is a new whie noise process and u is he error in forecasing y. This is he general ARCH(m-process. (Engle 198. For ease of calculaions and for esimaion, a sronger assumpion abou he process is added. m m u h v (4.3 45

46 where v is an i.i.d. Gaussian process wih zero mean and a variance equal o one, v ~N(,1 and he whole model variance is now where >, h 1 ~ (, N q i1 i h i >, i=1,..q and 1 i (4.4 is he informaion available a ime -1. Finally, when he process for modelling he variance is defined, we use an addiional equaion for modelling y. y c (4.5 his means ha is innovaions from a linear regression Generalised ARCH - GARCH This secion describes he generalisaion of he ordinary ARCH model. Bollerslev inroduced he GARCH(p,q model in Since hen a number of developmens on he basic modelled have been invened. The exising models can be divided ino wo caegories: symmeric and asymmeric models. However he basic GARCH model inroduced by Bollerslev is as follows and is comparable o he exension of an AR(p model o an ARMA(p,q model. Formally he GARCH process is wrien as h 1 ~ (, N q i1 i h i p i1 h i i (4.6 where p,q inegers, >, i, i=1,.q,, i=1, p. This addiional feaure is ha he process now also includes lagged h -I values. For p= he process is an ARCH(q process. For p=q= (an exension allowing q= if p=, is whie noise. (Bollerslev Theorem 4.1: The GARCH(p,q process as defined in equaion B.3 is wide sense saionary wih E( =, var( = (1-A(1-B(1-1 and cov( s A(1+B(1<. Proof: See Bollerslev (1986 page 33 q = for s if and only if In heorem 4.1 A(1= i and B(1= i. In mos cases he number of parameers i1 is raher small, e.g. GARCH(1,1 which is he focus of he nex secion. p i GARCH(1,1 46

47 In he case where p=q=1 he model becomes h 1 ~ N(, h 1 1 h 1 1 (4.7 where >,,, CRITERION OF FIT A good performance measure of he second momen can be hard o find as variance is no direcly observable. One way of dealing wih his is no o rely on one measure bu raher a number of measures. For he purpose of his hesis hree differen measures are used for evaluaing he performance of variance forecass from differen GARCH models: 1. Mean Square Error (MSE. Mean Absolue Error (MAE 3. Adjused Mean Absolue Percenage Error (AMAPE The MSE is calculaed as follows 1 MSE h 1 s h s ( ˆ (4.8 where h = number of seps ahead we wish o predic (he case here is normally 1, forecas volailiy, = rue volailiy ( and S = sample size. The MAE is: ˆ = The AMAPE is: 1 MAE h 1 s h s AMAPE 1 ˆ h 1 ˆ ˆ (4.9 s h s ( MODELLING THE VARIANCE OF IIP In secion 3 we modelled he underlying process of he IIP daa. We found ha by ransforming he original series by aking he log ransform we could model he series quie well on he basis of an AR(1 model wih he rue process shown o be uni roo. Now ha we have proved ha he process is uni roo we can ake he firs difference of he ransformed series and by definiion of a uni roo process his differenced series is saionary. In order o model he variance using he sofware package available (Malab, GARCH Toolbox we mus use a mean saionary ime series which we now have. 47

48 This sofware allows he series o be inpu ogeher wih a specificaion for he condiional mean model. The package will model he series for boh condiional mean and condiional variance, reurning suggesed coefficiens for each model. The log differenced series we now use was invesigaed in secion 3.3 and we discovered i can be modelled quie well using an AR(1 model. As such we specify an AR(1 model as he model for our condiional mean when using he GARCH oolbox. Please noe ha from here on in he ransformed differenced series is referred o as y(k. We now move on o discuss he ypes of GARCH modelling o be underaken: Case I: Condiional mean model specified as AR(1, allows he sofware package o produce esimaes of he condiional mean and condiional variance. Cases II involves an exernal variable no previously discussed. As discussed in he inroducion IIP is similar o a monhly measure of GDP and numerous economic variables are combined o produce he IIP measure. One of hese is elecriciy demand, we now use he ime series of elecriciy demand for he same period of he IIP ime series and build an ARMAX model o include his exogenous variable. Case II: Condiional mean model specified, we specify he underlying ARX model and allow he sofware o esimae he coefficiens of he condiional mean and condiional variance models. 4.4 RESULTS Case I: Condiional Mean Model Specified as AR(1 In his case we specified ha an AR(1 model be fied o he daa. As wih he mehod oulined in secion 3 he above is modelled on he raining se which is he firs 13 poins of he daa. Various GARCH specificaion models wih differen orders were run on he daa and i appears ha GARCH(1,1 model for he purpose of he sofware fis he daa bes and reurns an ARCH(1 model. The following model was reurned for he condiional mean of he inpu series: y(k= y(k-1 +ε(k (

49 Fied Condiional Sandard Deviaions, Specified AR(1 Condiional Mean Model, GARCH(1, Fig 4.1: Fied condiional variance versus uncondiional variance Case II: Condiional Mean Model Specified as ARX(1,1 We now aemp o model he IIP daa using an exogenous variable i.e. we wish o use an ARX(1,1 model o model he condiional mean. The variable used in elecriciy demand in he same ime period. Various GARCH specificaion models wih differen orders were run on he daa and i appears ha GARCH(?,? model for he purpose of he sofware fis he daa bes. The following model was reurned for he condiional mean of he inpu series: y(k= y(k x(k+ε(k (4.1 49

50 Fied Condiional Sandard Deviaions, Specified ARX(1,1 Condiional Mean Model, GARCH(1, Fig 4.: Fied condiional variance versus uncondiional variance GARCH(P,Q Case I; (1,1 Case II; (1,1 GARCH α α 1 β 1 α α 1 β 1 coefficiens Sandard Error T Saisic LLF MSE MAE AMAPE Condiional Mean Model Measures Daa Se Training Validaion Tes Training Validaion Tes MSE SSE PMAPE R c Table

51 Noes o able 4.1: 1. The T-Saisic a he 5% significance level for he sample size involved is relevan for T< Log Likelihood funcion (LLF: relaes o he use of Maximum Likelihood Esimaion (MLE in he calculaion of he parameers for he GARCH model, his is discussed in Appendix B The Mean Square Error (MSE, Mean Absolue Error (MAE & Adjused Mean Absolue Percenage Error (AMAPE are discussed above in secion 4.. These measures relae o he variance and are used in [14]. The use of hese measures is quesionable as hey are a measure of how well we are forecasing he variance which is aken o be he error, which is drawn from a cerain disribuion. We are rying o measure how well we can forecas his disribuion. 4. R c is he cenred sample muliple correlaion coefficien: The fi of an ordinary leas squares model is described by he sample muliple correlaion coefficien R. For R c, when he regression includes a consan erm as i does in all he above cases he value mus fall beween & 1. R c 1 y' X ( X ' X X ' y Ty y' y Ty (4.13 where y=he modelled ime series, X he marix of regression variable and T = size of he sample. 5. CONCLUSION The ime series of IIP was invesigaed on a number of levels, via Ordinary Leas Squares (OLS, Weighed Leas Squares (WLS and GARCH modelling. I can be concluded ha he mehod of GARCH modelling performed bes. The raw series was modelled firs via OLS, and he model did no perform well. The errors from he model were invesigaed and heeroskedasiciy was found o be an issue. As such we modelled he errors on a ime basis (fiing linear, quadraic & exponenial models, and on a level basis (Secion 3..4 o carry ou a WLS soluion. None of he WLS soluions appeared o deal wih he issue of heeroskedasiciy. As such a number of ransforms of he series were invesigaed and modelled via OLS (Secion 3.3. The log ransform of he IIP series was hypohesised o be a uni roo process and proved as such (Secion 3.5. Thus he firs difference of his series is a saionary ime 51

52 series. In oher words we have found a mehod of convering he IIP ime series o a saionary series which can be modelled. The series of he log differences was invesigaed and found o be an AR(1 process. The sofware used o model he variance and condiional mean (Malab, GARCH Toolbox required a saionary ime series o model and also allowed he specificaion of he condiional mean model. The able of resuls in secion 4.4 shows he wo cases as discussed. From he able we can see ha Case I, which specifies he condiional mean model as AR(1 performs bes wih he R c measure of.991 and accepable measures of MSE, SSE and PMAPE for he hree differen daa ses. I was hough ha by exending he AR(1 model o an ARX(1 model by including elecriciy demand as an exogenous variable may furher improve he fi i.e. reduce he error. As can be seen from he resuls in he able his was no he case, however i did no significanly reduce he fi. The R c in his case being and very similar resuls being reurned for MSE, SSE and PMAPE. In conclusion of he hree mehods used o model he ime series in quesion he GARCH modelling produced he bes model, he reason for his being he way in which heeroskedasiciy is deal wih. In he mehod of WLS he ime based and level based mehods were used o esimae he variance. As he series grows wih ime he WLS mehod using he ime based models assumes ha he variance grows wih ime, similarly he level based model does he same. GARCH modelling allows he variance o be modelled similar o an ARMA model and as such allows for heeroskedasiciy. Empirical evidence seems o show ha volailiy clusering which is he only effec ha GARCH akes ino accoun ha he oher mehods don allows he GARCH model o perform beer in forecasing our ime series. Furher research could be underaken by performing a WLS calculaion on he firs difference of he log ransformed series, which would allow for beer comparison beween he GARCH modelling echniques and he WLS mehods in forecasing his ime series. APPENDICES APPENDIX A A.1 WEIGHTED LEAST SQUARES SOLUTIONS FIGURES In case 3 in his secion we used he exponenial fied model of he error squared as our esimae of. The following are he figures produced from hose calculaions. 5

53 45 Plo of Error Squared verus esimaed sandard deviaion esimae Fig A.1: Plo of error squared versus esimaed sandard deviaion 14 Plo of acual IIP(dashed line versus forecased IIP using WLS Fig A.: Plo of acual IIP versus forecased IIP 53

54 18 Plo of error squared beween IIP & forecased IIP using WLS Fig A.3: Plo of error squared In he final case of modelling using WLS we used a model fied on acual IIP o fi he error squared, he following are he figures from he resuls: 54

55 45 Plo of Error Squared verus esimaed sandard deviaion esimae Fig A.4: Plo of error squared versus esimaed sandard deviaion 14 Plo of acual IIP(dashed line versus forecased IIP using WLS Fig A.5: Plo of acual IIP versus forecased IIP 55

56 18 Plo of error squared beween IIP & forecased IIP using WLS Fig A.6: Plo of error squared. APPENDIX B B.1 MODELLING TRANSFORMS B.1.1 Raw Daa For comparison purposes an AR(1 model is examined using he raw daa. I should be noed o consruc his model and all ohers he daa was divided ino hree sub-caegories per sandard modelling procedures: Daa Type Number of Daa Poins % of he daa Training Se 13 6 Tes Se 44 Validaion Se 4 The models were buil on he basis of he raining se, i.e. θ was calculaed using he leas squares mehod on his par of he daa only. The model buil on he raining se was hen used o forecas up o 16 seps ahead. These forecass were hen compared wih he es and validaion ses. 56

57 14 Acual IIP v's Forecased IIP using AR(1 model on raw daa IIP Time Fig B.1: Acual IIP versus forecased IIP using an AR(1 model wih he raw daa 4 Plo of Error 3 Magniude of Error Time 57

58 Fig B.: Plo of he error beween acual IIP & forecased IIP using AR(1 model on raw daa. As can be seen from he plo he errors grow larger he furher ahead we forecas which is o be expeced as he model is buil on he firs 13 daa poins. The resuls such as SSE and MSE are discussed in more deail in secion B.1. Modelling wih he Squared Transform of he Daa Using he mehod oulined above in seps 1-5 he following was observed: x 14 Squared Value of IIP verus Time Squared Value Time Fig B.3: Plo of Squared value of IIP ime series 58

59 14 Plo of acual IIP v's forecased IIP using AR(1 on ransformed daa IIP Time Fig B.4: Plo of IIP ime series versus modelled series using squared ransform 4 Plo of error 3 Magniude of Error Time 59

60 Fig B.5: Plo of he error beween acual IIP & forecased IIP using AR(1 model on ransformed daa. The resuls from each of hese echniques are discussed in secion B.1.3 Modelling wih he Square Roo Transform of he Daa Using he mehod oulined above in seps 1-5 he following was observed: 14 Acual IIP v's forecased IIP using AR(1 model on ransformed daa IIP Time Fig B.6: Plo of IIP versus model obained using square roo ransform 6

61 4 Plo of Error 3 Magniude of Error Time Fig B.7: Plo of he error beween acual IIP & forecased IIP using AR(1 model on ransformed daa. The resuls from each of hese echniques are discussed in secion B.1.4 Inverse square roo ransform 61

62 14 Acual IIP v's modelled IIP using AR(1 model on ransformed daa IIP Time Fig B.8: Plo of IIP versus modelled series using inverse square roo ransform 4 Plo of Error 3 Magniude of Error Time 6

63 Fig B.9: Plo of he error beween acual IIP & forecased IIP using AR(1 model on ransformed daa. Again, please see resuls secion above for discussion APPENDIX C C.1 STATISTICAL THEORY C.1.1 Saisics A brief descripion of he saisics used in he course of his hesis are presened below. In all cases below, X is a discree valued sochasic variable, k is he summaion index and p x (k is he probabiliy ha X akes he value k. [1] Ch 3. The firs momen is he populaion mean and is defined as E X kp x k The non-cenral second momen is hen defined as k X k p ( k Var( X E E ( X k Non-cenral momens are hen in he general case defined as r X r E k px k k x ( (C.1 (C. ( r = 1,,3. (C.3 Skewness is defined as 3 E(( X (C.4 3 / ( Var( X A variable wih posiive skewness is more likely o have values far above he mean value han far below. For a normal disribuion he skewness is zero. Kurosis is defined as 4 E(( X (C.5 ( Var( X For a normal disribuion he kurosis is 3. A disribuion wih a kurosis greaer han 3 has more probabiliy mass in he ails, so called fa ails or lepokuric. C.1. Correlaion The populaion correlaion beween wo differen random variables X and Y is defined by Cov( X, Y Corr( X, Y (C.6 Var( X Var( Y 63