Cointegration and Implications for Forecasting

Transcription

1 Coinegraion and Implicaions for Forecasing

2 Two examples (A) Y Y (B) Y Example B: Coinegraion Y and coinegraed wih coinegraing vecor [1, 0.9] because Y is a saionary process even hough Y and Example A: No such ha Y are I(1) is saionary

3 Behaviour of OLS behaves very differenly depending on siuaion A or B [A] Spurious Regression Y Y Simulaed -saisics in regression of Y on, 1000 replicaions

4 Comparison wih saionary case Disribuion on righ: simulaed -sas when and Y are saionary AR(1)

5 Siuaion is worse wih random walks wih drif Y Y Simulaed -saisics in regression of Y on, 1000 replicaions A c y n e u q re F

6 1 Y Y

7 [B] In Coinegraed Case: Superconsisency Y Y Simulaed -Saisics

8 Y Y Simulaed Sandard Errors

9 Summary: OLS behaves differenly when variables are saionary / no-saionary In non-saionary case: we saw Y Y 1 1 Y ~~ 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 is saionary even hough Y and are I(1) Firs example no such ha Y is saionary

10 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 (0.059)(0.006) Y The propensiy o consume is beween zero and one as expeced. The -saisic on 1 ˆ is , 2 R is 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)

11 L O G (C O N S ) L O G (G D P ) S N O C L LGDP

12 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 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

13 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

14 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 CLI R-sqr Mean dependen var Adj R-sqr S.D. dependen var S.E. of reg Akaike info crierion Sum sqrd resid Schwarz crierion Log likelihood Hannan-Quinn crier F-saisic Durbin-Wason sa Prob(F-sa) Residual Acual Fied

15 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 *MacKinnon (1996) p-values. Engle-Granger Tes Equaion: Variable Coefficien Sd. Error -Saisic Prob. RESID(-1) D(RESID(-1)) 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 Rejec null ha he wo series are no coinegraed

16 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 LOG(NOD) R-squared Mean dep. var Adj R-sqr S.D. dep var S.E. of reg Akaike info crierion Sum sqr resid Schwarz crierion Log likelihood H-Q crierion F-saisic Durbin-Wason sa Prob(F-saisic) Residual Acual Fied 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 *MacKinnon (1996) p-values. Engle-Granger Tes Equaion: Variable Coefficien Sd. Error -Sa Prob. RESID(-1) We do no rejec he null ha here is no coinegraing relaionship beween he wo variables.

17 8.3 Implicaion for Forecasing -- he Error Correcion Form If Y and are coinegraed, hen here exiss an error correcion form: Y Y () Y 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

18 Example Suppose Y u, u ~(0) I u, u ~(0) I Subracing boh sides by Y 1 and making a subsiuion for gives Rearranging: Y Y () u Y u y Y Y u u ()() This is a simple error-correcion form for no lags of 1 Y or Y

19 In his sysem When here is a posiive error ( Y ) hen Y () Y y ends o be negaive Y ends o fall owards he coinegraing relaionship When here is a negaive error ( Y ) hen Y () Y 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

20 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

21 Example The following is a Vecor Auoregression of order 1 Y 0.9Y y 0.3Y x i.e., Y Y 1 y, 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 y x () Y 0.6()() Y 1 1 y x

22 Noe ha Y Y 1 y, x, Y Y 1 Y 1 y, x, Y Y Y 1 y, x, ζ Y 1 y, x, α β 0.1 () y, Y x, y, x, The wo equaions are herefore Y 0.1() Y 1 1 y 0.3() Y 1 1 x

23 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

24 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 y perhaps wih more lags of he firs differences of Y and

25 Example We build an Error Correcion Model for log( RGDP _) SG using is coinegraing relaionship wih CLI esimaed earlier log( RGDP _) SG CLI The error correcion represenaion is hen of he form d(log( RGDP _)) SG d(log( RGDP _))() SG... d CLI (log( RGDP _) SG ) CLI wih possibly more lags of d(log( RGDP _)) SG and d() CLI 1 1

26 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)) *cli(-1) ) 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 @seas(4) d(cli(-1))

27 Dependen Variable: D(LOG(RGDP_SG)) Sample: 1980Q1 2004Q4 Included observaions: 100 Variable CoefficienSd. Error-Saisic Prob. C LOG(Y(-1)) *CLI(-1) D(CLI(-1)) Incidenally, he 2 R for his regression is

28 We can forecas from his regression in he usual way: smpl 2005q1 2011q4 eq3.fi(f=na) y_f y_se D(LOG(Y)) Y_UP Y_F Y_DOWN The forecas 2 R is jus over 0.15

29 We can incorporae lag BIZEP ino his model equaion eq3.ls d(log(rgdp_sg)) c @seas(4) d(cli(-1)) bizexp(-1) Y Y_F Y_UP Y_DOWN This model generaes a slighly higher forecas-r of