Exercise: Building an Error Correction Model of Private Consumption. Part II Testing for Cointegration 1

Transcription

1 Bo Sjo Exercise: Building an Error Correcion Model of Privae Consumpion. Par II Tesing for Coinegraion Learning objecives: This lab inroduces esing for he order of inegraion and coinegraion. The lab requires ha you are somewha familiar wih he ADF es, he 2-sep procedure and Johansen s VAR approach. I is an exercise for sudens who have elemenary heoreical knowledge in he spurious regression problem, inegraed variables and coinegraion. These insrucions will be improved in he fuure I hope. A he momen hey are being ransformed from PcGive o Eviews, which migh cause some problems.

2 . Error Correcion Models and he Long-run Seady Sae In he firs par of his lab we esimaed a ypical Keynesian aggregae consumpion of he ype found in basic macroeconomic exbooks. We had a basic relaion ha we waned o esimae and we looked for a good way of consrucing an economeric model. The basic equaion said ha privae aggregaed consumpion in real erms where a funcion of real aggregaed income in economy and he ineres rae, c = β 0 + βy + β 2i where c is he aggregaed privae consumpion, y aggregaed income, and i is he ineres rae. All variables are in naural logarihms, which makes he parameers elasicises ha are easier o undersand. The ineres rae should be ransformed as ln (+ r) where r is he rae of ineres expressed a he same basis as he period of measuring income and consumpion. 2 The problems in esimaing his equaion where basically wo () he variables are non-saionary which will ruin sandard inference. There is no saisical foundaion for esing significance. The esimaed relaion migh be oally spurious. (2) The equaion, from he exbook, describes a long-run relaion no necessary a relaion ha will hold in each individual period. So, we have o expec some dynamic adjusmen in he equaion. In he firs par of he lab we solved his in raher ad hoc way. We esimaed he saic relaion, we argued ha he residual from his model was saionary and represened an error correcion erm. We could hen move on o esimae an error correcion model. In his model we had boh longrun and shor run ogeher. We argued ha we could use his model o represen a number of phenomenons in real life. In his exercise, we focus on modelling he dynamic srucure a bi more closely, and es if he underlying assumpions of regression model are fulfilled. In paricular if he variables are rending in he same way and share a long-run seady sae. If he relaion saed above is o make any sense, deviaion from he equaion should be emporary and no permanen. Or, in oher words, he variables should coinegrae, he residual should be a saionary process. The equaion above can be argued o represen a long run seady sae, or economic equilibrium. To prove his i will be necessary o es for coinegraion (common rends) among he variables, since i is no possible o use ordinary significance ess. 2. The Daa The daa series in his lab are he same as for he firs par of he lab Variable Cons GDP R60c CPI Explanaion Privae consumpion nominal value Real Gross Domesic Produc, volume index variable, base year = 995 The hree monh reasury bill rae, per cen p.a. Consumer price price index 2 You should recall he basic inerpreaion of he parameers in a mulivariae linear regression model. 2

3 All daa are from Sweden, yearly daa , in he file ecmdaa.xls. You need o calculae he real consumpion, ake logs and firs differences as before. 3. How o Tes? 3. Tes for he order of inegraion. If you apply he ADF es wih and wihou he rend, he conclusions are (IMO) CPI ~ I(2) Inegraed of order wo GDP ~I() Inegraed of order one r ~ I(), and ln(+r) Inegraed of order one. The laer ransformaion creaes he coninuously compounded rae of ineres, and is he beer rae use in an economeric equaion. Compare wih graphs of levels and firs log differences. 3. Tes for coinegraion using Johansen s VAR Approach. A beer way o es for coinegraion, and invesigae VAR approach. c β 0 + βy + β 2i = is o use Johansen s ) For VAR he principle is ha for each variable in he model, se up an OLS equaion where he variable is explained as a funcion of lagged variables of all variables in he sysem. The number of lags should be he same across all equaions. Eviews has a special menu (look under quick) for seing up and esimaing VAR models. You can choose 2 lags for each variables. 2) Afer esimaing model, look a he residual ess. You need o have no auocorrelaion in he residuals, and preferable a normal disribuion in he residuals. The laer can ypically be achieved wih a careful selecion of dummies (hi a he bigges ouliers) 3) If he Tes summary looks ok, proceed o es for coinegraion. In Eviews, under VAR, indicae ECM, and chose he suggesed alernaive under he coinegraion menu. If you sared wih 3 lags and removed insignifican lags, your Coinegraing es will sar a 967 and you will ge he following resuls (oupu from PcGive), 3

4 I() coinegraion analysis, 967 o 999 eigenvalue loglik for rank H0:rank<= Trace es [Prob] [0.57] [0.747] [0.702] This ells you ha here is no significan eigenvalues in he long-run marix, ad herefore no coinegraing vecor in he sysem. You rejec coinegraion and a long run seady sae relaion among your daa. The H 0 : r = 0, canno be rejeced. If you wan you can sop here. You rejec ha c = β 0 + βy + β 2i is a meaningful longrun relaionship. You can hen look for oher models or more variables o add o he equaion. Alernaively, you can analyse he residual erms for ouliers ha migh influence he esimaes in a bad way. If you look a he residual of he VAR, you see no dramaic ouliers according o he rule of humb ±3.5. There however, big ouliers in wo equaions a 977. Crae a dummy for 977 in he Calculaor. Add he dummy o he VAR model. Be careful. The program assumes by defaul ha you add anoher variable o be modelled in he sysem. You need o change he saus of he dummy variable from Y = Endogenous, o U = unresriced. Once ha is done, re-esimae he model as an unresriced sysem. Look a he es summary, check ha i is ok. Tes for coinegraion, and you ge somehing like I() coinegraion analysis, 967 o 999 eigenvalue loglik for rank H0:rank<= Trace es [Prob] [0.073] [0.355] [0.690] 4

5 Afer adding a dummy for an exreme oulier in he daa, he resul changes. Now we have one significan vecor a 0%. We migh accep ha as significan. 3 Nex, re-esimae he sysem and impose resricions or specific parameers on he coinegraing vecor. Under Var specificaion, afer esimaing he VAR, and chosing Vecor Error Correcion selec VEC resricions. Resricions can now be imposed on he alpha and/or he bea coefficiens in he sysem. To impose he value of.0 on he GDP you need o use B(i,j), where i is he row in he Bea marix corresponding o he coinegraing vecor. In his case i=. j is he row in column (i). In his case here wo explanaory variables, and 2. resrining hem o be and -0.5 (as a supid example) means B(,) =.0, B(,2) = When your resricion works you will se i in he oupu. You will also se a chi-square es wheher hese resricions are valid or no. Are he resricions in line wih he null of a saionary vecor? A rejecion means ha he resricions are no valid. The oupu gives you he esimaed vecor as Π x = αβ ' x in a VECM. The resuls should be esimaed alpha-vecor and he sandard errors of alpha, from PcGive bea Lrcons.0000 LGDPvol Lrb alpha Lrcons LGDPvol Lrb Sandard errors of alpha Lrcons LGDPvol Lrb The associaed -saisics for alpha is esimae/s.err. The esimaed long-run vecor reads [.0, -.009, ]. The program has auomaically by defaul normalized he esimaed vecor around he firs variable in he firs vecor. Thus, we see.0 as he firs parameer. If we wrie ou he equaion we have esimaed c =.0y 0. 38i. This equaion is our new error correcion erm. We can use his equaion in he Error correcion 3 Addiionally, you migh also add second impulse dummy for he second big error ha you idenify from he graphs of he esimaed residual from he VAR wih wo lags. The second bigges residual is for 97, confirm his in he graphs of he esimaed residual creae an impulse dummy for 97 and include i as an unresriced dummy in he model. Re-esimae and es for coinegraion. Wha is he oucome of he es? I is beer. 5

6 model in he firs par of his lab. Consruc he ECM variable as, ECM = c.0y i and he lag of his erm. I will represen a beer long-run seady sae. This esimaed long- run relaionship is no ha differen from he one we go esimaing he saic OLS. The difference is ha his mehod is a saisically beer mehod for esing for coinegraion. Noice he sign changes in he equaion. In marix form we have esimaed, ** 0.57 Lrcons αβ LGDPvol, ** 0.30 Lrb00 [ ' x ] = 0.0 [.0,.00, 0.38] where ** indicaes significance a he 5% level. Or in erms of he sysem, leaving he shor run dynamics ou, **' Lrcons 0.57 ε Lrcons LGDPvol = 2 ** Lrb ε Lrb00 [ αβ ' x ] ε = 0.0 [.0,.00, 0.38] LGDPvol... + We can see ha he coinegraing vecor predics changes in consumpion and changes in he real ineres rae. I does no predic changes in GDP, however. We know ha coinegraion will imply ha a leas one alpha coefficien mus be significan. The conclusion is ha we need o model a leas boh consumpion and he ineres rae o ge a correcly esimaed long-run vecor. The single equaion esimaion will be biased. The consumpion growh will adjus o follow he income in he long run. We can also see ha he ineres rae will change when we are ou of he seady sae. Of course, we expec he ineres rae o adjus during he business cycles. 3.3 How did I know he lag order of he VAR model? I didn know, I esimaed and esed for i. When esimaing VARs and esing for coinegraion he lag order will ofen be crucial for he resuls. Sar wih four lags on all variables. Tha gives you a huge number of parameers o esimae in he model and few degrees of freedom for he ess. Going beyond his migh be oo much given he lengh of he sample. If you sar wih 4 lags, and reduce lags sep by sep, and es for misspecificaion a he same ime you will be able o reduce he lag lengh in his model from 4 lags o 2 lags. In comparison wih he single model, he es resuls will now also conain so called vecor ess. The program pools he ess resuls of he individual equaion ogeher and ess if he vecor is significan or no. This make sense since we are esimaing a sysem, we wan he sysem o perform well. Hopefully, boh he individual equaion ess and he vecor ess will come ou insignifican. 6

7 Furhermore, in he Johansen s approach we will make use of he Full Informaion Maximum Likelihood procedure for geing he esimaes we need. This requires ha he residual process is normally disribued whie noise. We really need o es hese models more carefully han single equaion saionary ARMA models. The program will es for he significan of he parameers, and of he lags. You migh ask how his work? Is he spurious regression problem, presen? The answer is ha F-es of he variables, lags and heir parameers will work because he variables are inegraed of a mos order one I(), herefore when you condiion on he lag order in he model will he F-es be performed on saionary relaions. As a consequence of esing saionary relaions he F-es will work. The oucome of analysing he lag srucure in his model is a 2 lags will do. This is a rule of humb, 2 lags is ofen opimal. Bu someimes you need o impose a few impulse dummies o ge whie noise residuals. 4. The Dynamic Single Equaion All models are inerrelaed in various ways. The saic esimaion was no oo differen from he coinegraing VAR esimaion. Le us look a a dynamic OLS model. We know ha he parameers will be a bi biased from he VAR resuls. I migh be ineresing o compare an ADL model. Esimae a single dynamic equaion wih wo lags on each variable. Ask for Saic long run soluion and you will ge he following Solved saic long run equaion for Lrcons Coefficien Sd.Error -value -prob Consan LGDPvol Lrb Long-run sigma = ECM = Lrcons *LGDPvol *Lrb00; The saic long run is found be wriing ou he model using he lag operaor. Solve for y =... Nex se he lag operaor o uniy, L=, and calculae he resul. This is wha he program is doing. The resuls are similar o wha go before. There are some differences, especially for he ineres rae coefficien. Tha is an expeced bias. From he VAR we learned ha he ineres rae was imporan for he long-run. NEXT? Idenificaion and esing of he coinegraing vecor. The Johansen es only ells us ha here is saionary relaion among he variables, i does no ells us exacly wha i is, and wha he parameers are. We need o do impose some resricions and do some ess on he vecor o 7

8 idenify exacly wha we have found. For one significan vecor his is relaively easy. The normalizaion will jus idenify he vecor for us. We can hen es consrains on he oher parameers. If we have more han one vecor, hings ge a lile bi more complex. Finding r vecors imply r normalizaions and r(r-) resricions o achieve a leas exac idenificaion. Appendix Tesing for coinegraion using he Engle and Granger Two Sep Procedure. The Engle and Granger wo sep procedure was an early developed es for coinegraion. I is no longer used because i is no a good es, i is found no be consisen ec.. However, i has a nice inuiion o i, and we show i here as an addiional exercise. Sep one esimae he coinegraing regression y = β + β x β xn + ε 0 N Save he esimaed residual, ( εˆ ) and es if he esimaed residual is saionary by running he following regression, ˆ ε = α + π k ˆ ε + γ i ˆ ε i + i= v In his equaion you perform a one-sided es if πˆ is less han zero. The choice of augmenaion, lag lengh of he lagged dependen variable is imporan. The criical values will depend on ) number of variables in he coinegraing regression and 2) number of variables on he sample. To some exen he number of lags in he augmenaion will also be imporan. Here: ) Esimae he saic model from above (no lags), and save he residual (under Tes, Sore..) 2) Run he second sep o es if he residual from he firs sep equaion is saionary. Use an ADF-ype of es bu remember ha he criical values are hose of an ADF es. You will find he correc criical values in Sjöö (2008), for N=3 (=hree variables in he model), 5% risk level, 00 obs, lag in he 2-sep equaion, he criical value is = 4.0. If you end up wih one lag you migh argue a he 0% level ha you rejec he null of no coinegraion (inegraed residual erm). If you accep coinegraion, you are allowed o check he esimaed parameers of he model. Do hey make sense? Remember ha hey can be a bi biased. You canno and should no look a he -values. They will ypically sill be biased. If you find argumens for coinegraion, you have found a seady sae long run equilibrium and argumens for why your ECM variable will work in he Error Correcion model from he firs par of he lab. One message here, going back o he firs lab is he following. If you have a heoreical relaion ha you believe in, he ECM model is a nice way o approach modelling regardless of esing for inegraion and coinegraion. The model will make some sense, and be oally spurious. A lile knowledge in esing for he order of inegraion (uni ro ess) will help you along he way. 8