Coinegraion and Implicaions for Forecasing
Two examples (A) Y Y 1 1 1 2 (B) Y 0.3 0.9 1 1 2 Example B: Coinegraion Y and coinegraed wih coinegraing vecor [1, 0.9] because Y 0.9 0.3 is a saionary process even hough Y and Example A: No such ha Y are I(1) is saionary
Behaviour of OLS behaves very differenly depending on siuaion A or B [A] Spurious Regression Y Y 1 1 1 2 Simulaed -saisics in regression of Y on, 1000 replicaions
Comparison wih saionary case Disribuion on righ: simulaed -sas when and Y are saionary AR(1)
Siuaion is worse wih random walks wih drif Y Y 10 1 1 20 1 2 Simulaed -saisics in regression of Y on, 1000 replicaions A 90 80 70 60 c y n e u q re F 50 40 30 20 10 0-10 0 10 20 30 40 50 60 70 80 90
1 Y 1 100 60 80 50 40 60 30 40 20 20 75 100 125 150 175 200 10 75 100 125 150 175 200 6 0 5 0 4 0 Y1 3 0 2 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1
[B] In Coinegraed Case: Superconsisency Y 0.3 0.9 1 Y 0.3 0.9 1 2 1 2 Simulaed -Saisics
Y 0.3 0.9 1 Y 0.3 0.9 1 2 1 2 Simulaed Sandard Errors
Summary: OLS behaves differenly when variables are saionary / no-saionary In non-saionary case: we saw Y Y 1 1 Y 1 2 0.3 0.9 1 1 2 ~~ Spurious Regressions: huge -sas! ~~ Superconsisency: huge -sas! Phenomenon is second example is called Coinegraion Y and coinegraed wih coinegraing vecor [1, 0.9] because Y 0.9 0.3 is saionary even hough Y and are I(1) Firs example no such ha Y is saionary
Example funcions of he form Early macro-economeric work, common o esimae consumpion C Y 0 1 Using Singapore daa from 1975Q1 o 2011Q2 (we use logs of he variables hroughou), we ge he regression All seems good Cˆ 0.753 0.843 (0.059)(0.006) Y The propensiy o consume is beween zero and one as expeced. The -saisic on 1 ˆ is 143.623, 2 R is 0.993 bu is his regression spurious? if no, how o use coinegraing relaionship for forecasing? if yes, does i mean a log(cons) canno be used o forecas log(gdp)
L O G (C O N S ) L O G (G D P ) 10.5 10.0 9.5 9.0 8.5 8.0 1975 1980 1985 1990 1995 2000 2005 2010 11.5 11.0 10.5 10.0 9.5 9.0 8.5 1975 1980 1985 1990 1995 2000 2005 2010 1 0.4 1 0.0 9.6 S N O C L 9.2 8.8 8.4 8.0 8.5 9.0 9.5 1 0.0 1 0.5 1 1.0 1 1.5 LGDP
Tesing for Coinegraion [Engle-Granger] Check if he residual series from regression is I (1) a. Regress Y 0 1 by OLS b.tes ˆ for uni roo using ADF: collec residuals ˆ Y ()ˆ ˆ 0 1 ˆ ˆ ˆ... ˆ u 1 1 1 p1 p1 If 0 - uni roo in - and Y no coinegraed If 0 - hen no uni roo in - and Y coinegraed Imporan: usual Dickey-Fuller criical values do no apply, because we are esing on he residuals, no he acual noise erms
Mos economeric sofware packages include he appropriae criical values, which have been obained using simulaion mehods. Example Tes for coinegraing relaionship beween log( RGDP _) SG and Composie Leading Index CLI for Singapore. L O G (R G D P _ S G ) C L I 11.5 11.0 10.5 10.0 9.5 9.0 8.5 1975 1980 1985 1990 1995 2000 2005 2010 140 120 100 80 60 40 20 1975 1980 1985 1990 1995 2000 2005 2010
We begin by regressing log(rgdp_sg) on CLI: Dependen Variable: LOG(RGDP_SG) Mehod: Leas Squares Sample: 1981Q1 2011Q4 Included observaions: 124 Variable Coefficien Sd. Error -Saisic Prob. C 8.291 0.026 313.541 0.000 CLI 0.025 0.000 79.197 0.000 R-sqr 0.981 Mean dependen var 10.304 Adj R-sqr 0.981 S.D. dependen var 0.586 S.E. of reg. 0.081 Akaike info crierion -2.167 Sum sqrd resid 0.805 Schwarz crierion -2.122 Log likelihood 136.377 Hannan-Quinn crier. -2.149 F-saisic 6272.215 Durbin-Wason sa 0.316 Prob(F-sa) 0.000.3.2.1.0 -.1 -.2 82 84 86 88 90 92 94 96 98 00 02 04 06 08 10 Residual Acual Fied 11.5 11.0 10.5 10.0 9.5 9.0
The Engle-Granger es resuls are: Coinegraion Tes - Engle-Granger Specificaion: LOG(RGDP_SG) CLI C Auomaic lag specificaion (lag=1 based on Schwarz Info Crierion, maxlag=12) Value Prob.* Engle-Granger au-sa -4.234 0.005 *MacKinnon (1996) p-values. Engle-Granger Tes Equaion: Variable Coefficien Sd. Error -Saisic Prob. RESID(-1) -0.210 0.05-4.234 0.00 D(RESID(-1)) 0.257 0.09 2.93 0.00 The eviews commands are: smpl 1981q1 2004q4 equaion eq1.coinreg log(rgdp_sg) c cli eq1.coin(mehod=eg) The -saisic on he esimaed ˆ has p-value 0.005 Rejec null ha he wo series are no coinegraed
Example: log( RGDP _) SG and log()nod Dependen Variable: LOG(RGDP_SG) Mehod: Leas Squares Sample: 1981Q1 2011Q4 Included obss: 124 Variable Coef. Sd. Error -Sa. Prob. C 3.874 0.128 30.189 0.000 LOG(NOD) 0.664 0.013 50.292 0.000 R-squared 0.954 Mean dep. var 10.304 Adj R-sqr 0.954 S.D. dep var 0.586 S.E. of reg 0.126 Akaike info crierion -1.287 Sum sqr resid 1.941 Schwarz crierion -1.242 Log likelihood 81.796 H-Q crierion -1.269 F-saisic 2529.303 Durbin-Wason sa 0.105 Prob(F-saisic) 0.000.3.2.1.0 -.1 -.2 -.3 82 84 86 88 90 92 94 96 98 00 02 04 06 08 10 Residual Acual Fied 11.5 11.0 10.5 10.0 9.5 9.0 Coinegraion Tes - Engle-Granger Specificaion: LOG(RGDP_SG) LOG(NOD) C Auo lag specificaion (lag=0 based on SIC) Value Prob.* Engle-Granger au-saisic -1.345 0.818 *MacKinnon (1996) p-values. Engle-Granger Tes Equaion: Variable Coefficien Sd. Error -Sa Prob. RESID(-1) -0.040 0.030-1.345 0.181 We do no rejec he null ha here is no coinegraing relaionship beween he wo variables.
8.3 Implicaion for Forecasing -- he Error Correcion Form If Y and are coinegraed, hen here exiss an error correcion form: Y Y () Y 0 1 1 2 1 1 0 1 1 Inuiion: if Y 0 1 holds in he long run, hen fuure changes in Y mus end oward Y 0 1 Change in Y should reac o pas deviaions
Example Suppose Y u, u ~(0) I 0 1 1 1 u, u ~(0) I 1 2 2 Subracing boh sides by Y 1 and making a subsiuion for gives Rearranging: Y Y () u Y u 1 0 1 1 1 1 y Y Y u u ()() 1 0 1 1 1 1 2 This is a simple error-correcion form for no lags of 1 Y or Y
In his sysem When here is a posiive error ( Y 1 0 1 1) hen Y () Y 1 0 1 1 y ends o be negaive Y ends o fall owards he coinegraing relaionship When here is a negaive error ( Y 1 0 1 1) hen Y () Y 1 0 1 1 y ends o be posiive Y rises owards he coinegraing relaionship Noe he implicaion of he error-correcion form for forecasing: - pas errors in he coinegraing relaionship helps o predic fuure changes in Y because of a endency o rever o he coinegraing relaionship
Remarks Because 1 here, he correcion is on average of he full amoun In general, he adjusmen facor need no be 1, i.e., he immediae response migh no be a correcion of he full amoun. Correcion migh ake place slowly. In general, here will be an error-correcion form for boh possible, however, ha only Y responds, or only Y and responds., hough i is
Example The following is a Vecor Auoregression of order 1 Y 0.9Y 0.1 1 1 y 0.3Y 0.7 1 1 x i.e., Y 0.9 0.1 Y 1 y, 0.3 0.7 1 x, In his example, boh Y and are I(1) (proof omied) They are also coinegraed wih coinegraing vecor [1, 1], i.e., Y is I(0) To see his, subracing he second equaion from he firs: Y 0.6Y 0.6 1 1 y x () Y 0.6()() Y 1 1 y x
Noe ha Y 0.9 0.1 Y 1 y, 0.3 0.7 1 x, Y 0.9 0.1 Y 1 Y 1 y, 0.3 0.7 1 1 x, 0.9 0.1 Y 1 1 0 Y 1 0.3 0.7 0 1 0 1 1 0.1 0.1 Y 1 y, 0.3 0.3 1 x, ζ 0.1 1 1 Y 1 y, 0.3 1 x, α β 0.1 () y, 1 1 0.3 Y x, y, x, The wo equaions are herefore Y 0.1() Y 1 1 y 0.3() Y 1 1 x
These equaions show ha - boh Y and have error-correcion forms, and boh adjus - If a ime 1 here is posiive error, i.e., Y 1 1, hen Y decreases ( Y 0) and increases ( 0) o move he variables back owards he relaionship Y - The adjusmen facors are 0.1 in α - The vecor β gives he coinegraing vecor and 0.3 are called he adjusmen facors, given - Noice ha he marix ζ 0 was wrien as he ouer produc of he vecor of adjusmen facors and he coinegraing vecor - The VAR ransformed ino equaions involving Y (and is lags) and Y 1 he vecor error-correcion form is called
Warning: i is emping o hink ha if wo variables are no coinegraed, hen here is no connecion beween he wo. However, here may sill be a relaionship beween and, even if Y and are I (1), bu no coinegraed. Y In paricular, here may sill be a relaionship of he form Y Y 0 1 1 2 1 y perhaps wih more lags of he firs differences of Y and
Example We build an Error Correcion Model for log( RGDP _) SG using is coinegraing relaionship wih CLI esimaed earlier log( RGDP _) SG 8.178 0.027 CLI The error correcion represenaion is hen of he form d(log( RGDP _)) SG d(log( RGDP _))() SG... d CLI 0 1 1 2 1 (log( RGDP _) SG8.178 0.027) CLI wih possibly more lags of d(log( RGDP _)) SG and d() CLI 1 1
We begin by regressing d(log( RGDP _)) SG on he error correcion erm only, using he command ls d(log(rgdp_sg)) c (log(rgdp_sg(-1))-0.0272470594171*cli(-1)-8.17790480233) and add lags of firs differences, seasonals, ec. We sele on he error-correcion form wih one lag of d() CLI and seasonals equaion eq3.ls d(log(rgdp_sg)) c (log(rgdp_sg(-1))-0.0272470594171*cli(-1)- 8.17790480233) @seas(2) @seas(3) @seas(4) d(cli(-1))
Dependen Variable: D(LOG(RGDP_SG)) Sample: 1980Q1 2004Q4 Included observaions: 100 Variable CoefficienSd. Error-Saisic Prob. C -0.017 0.003-5.753 0.000 LOG(Y(-1))-0.0272470594171*CLI(-1)-8.17790480233-0.073 0.022-3.283 0.001 @SEAS(2) 0.043 0.004 10.281 0.000 @SEAS(3) 0.044 0.004 10.682 0.000 @SEAS(4) 0.034 0.004 8.266 0.000 D(CLI(-1)) 0.004 0.001 3.858 0.000 Incidenally, he 2 R for his regression is 0.685.
We can forecas from his regression in he usual way:.100.075.050.025 smpl 2005q1 2011q4 eq3.fi(f=na) y_f y_se.000 -.025 -.050 -.075 -.100 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 D(LOG(Y)) Y_UP Y_F Y_DOWN The forecas 2 R is jus over 0.15
We can incorporae lag BIZEP ino his model equaion eq3.ls d(log(rgdp_sg)) c (log(rgdp_sg(-1))-0.0272470594171*cli(-1)- 8.17790480233) @seas(2) @seas(3) @seas(4) d(cli(-1)) bizexp(-1).12.08.04.00 -.04 -.08 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 Y Y_F Y_UP Y_DOWN This model generaes a slighly higher forecas-r of 0.26 2