Co-integration

Contents :

Definition of Co-integration . Different Approaches of Co-integration. Johansen and Juselius (J.J) Co-integration. Error Correction Model (ECM). Interpretation of ECM term. Long – Run Co-integration Equation.

Definition of Co-integration

The concept of cointegration was first introduced by Granger (1981) and elaborated further by Engle and Granger (1987), Engle and Yoo (1987), Phillips and Ouliaris (1990), Stock and Watson (1988), Phillips (1986 and 1987) and johansen (1988, 1991, 1995a).Time series Yt and Xt are said to be cointegrated of order d, where d > 0, written as Yt, Xt ~ CI (d). If (a) Both series are integrated of order d, (b) There exists a linear combination of these variables.

Examples :

The old woman and the boy are

unrelated to one another, except

that they are both on a random

walk in the park.  Information

about the boy's location tells us

nothing about the old woman's

location.

The old man and the dog are joined by one of

those leashes that has the cord rolled up inside

the handle on a spring. Individually, the dog and

the man are each on a random walk. They cannot

wander too far from one another because of the

leash.  We say that the random processes

describing their paths are cointegrated.

Approaches of Co-integration :

Engle-Granger (1987)

Used when only one co integrating vector is under consideration

Johansen and Juselius (1990)

Used when more than one co integrating vector are under

consideration

Conditions Of Co-integration :

If all variables are stationary on level , we use OLS method of estimation. If all variables or single variable are stationary on first difference , we use Co-integration Method. If all the variables are stationary on first difference , we use Johnson Co-integration and ARDL also. If some variables are stationary on level and some are stationary on first difference , we only use ARDL model.

Johansen and Juselius (1990) J.J Co-integration : If all the variables are stationary on first difference , we use

Johnson Co-integration. Although Johansen’s methodology is typically used in a

setting where all variables in the system are I(1), having stationary variables in the system is theoretically not an issue and Johansen (1995) states that there is little need to pre-test the variables in the system to establish their order of integration.

Johansen Co-integration :

Johansen, Is a procedure for testing cointegration of several I(1) time series. This test permits more than one cointegrating relationship so is more generally applicable than the engle–granger test .

Yt = α0 + α1x1t + α2x2t + et 

Yt = α0 + α1x1t + α2x1t-1 + α3x2t + α4x2t-1 + et

Steps For Johnson Co-integration : STEP 1:-

Check stationarity take only those variables which are stationary at 1st difference.

STEP 2:- File/new workfile/structured and dated/start date & end

date CLICK OK. Paste the data. STEP 3:- Quick/Group statistic/Co-integration test Write variables name CLICK OK

Steps of j-j cointegrationDate: 05/06/14 Time: 07:04    Sample (adjusted): 1981 2010    Included observations: 30 after adjustments  

Trend assumption: Linear deterministic trend  Series: LPGDP LINV LATAX LPS     Lags interval (in first differences): 1 to 1  

         Unrestricted Cointegration Rank Test (Trace)  

Hypothesized   Trace 0.05  No. of CE(s) Eigenvalue Statistic Critical Value Prob.**

None *  0.620080  53.12601  47.85613  0.0147At most 1  0.376331  24.09216  29.79707  0.1966

At most 2  0.265635  9.928096  15.49471  0.2863At most 3  0.021943  0.665631  3.841466  0.4146

 Trace test indicates 1 cointegrating eqn(s) at the 0.05 level

Definition of Error Correction Model If, then, Yr and Xt are cointegrated, by definition ftr ~ /(0).

Thus, we can express the relationship between Yt and Xr with an ECM specification as:

∆Yt= a0 + b1∆Xt-µ^t-1 + Yt

In this model, b1 is the impact multiplier (the short-run effect) that measures the immediate impact that a change in Xt will have on a change in Yt . On the other hand πt is the feedback effect, or the adjustment effect, and shows how much of this disequilibrium is being corrected.

Steps For VAR Estimate :

STEPS :-Quick /Estimate VARVAR type: Vector Error Correction.Endogenous variables:- All variables nameLag intervals:-1 ,1 CLICK OK

 Vector Error Correction Estimates     Date: 05/26/14 Time: 22:36     Sample (adjusted): 1981 2010     Included observations: 30 after adjustments   Standard errors in ( ) & t-statistics in [ ]  

Cointegrating Eq:  CointEq1                        

LPGDP(-1)  1.000000               

LINV(-1) -4.620559         (0.47459)        [-9.73587]               

LATAX(-1) -3.350165         (1.20384)        [-2.78289]               

LPS(-1)  1.274220         (0.49822)        [ 2.55755]               C -3.861790                        

Error Correction: D(LPGDP) D(LINV) D(LATAX) D(LPS)                  

CointEq1 -0.011599  0.167417 -0.004750 -0.060848   (0.02160)  (0.04974)  (0.02840)  (0.03017)  [-0.53695] [ 3.36570] [-0.16725] [-2.01682]

Estimation of ECM value :

If T value is 1.67 or more than 1.70 then we conclude that variable is significant…. OR when Tcal is > 1.70 or when Tcal = 1.67 We conclude variable is significant… Where there’s –ve sign we consider it +ve as the value of Linv is -4.62 we consider it +ve and conclude that the there is +ve relationship between lpgdp and linv…… In Coint Equ 1 the value of Lpgdp is -0.01 which shows Convergence to equilibrium and 1 % convergance in one year

Lag Length Criteria : STEPS :-

Go to The view of result window of VAR Estimate. Go to Lag Length Structure and select Lag Length

Criteria. In Lag specification Select the lags to include as 3. Click OK

VAR Lag Order Selection Criteria        

Endogenous variables: LPGDP LINV LATAX LPS       

Exogenous variables: C         

Date: 05/06/14 Time: 08:50        

Sample: 1979 2010          

Included observations: 29        

 Lag LogL LR FPE AIC SC HQ

0  145.5371 NA   6.78e-10 -9.761179 -9.572586 -9.702114

1  273.4307   211.6859*   3.06e-13*  -17.47798*  -16.53501*  -17.18265*

2  288.0984  20.23135  3.60e-13 -17.38610 -15.68876 -16.85451

3  295.8181  8.518318  7.70e-13 -16.81504 -14.36334 -16.04720

 * indicates lag order selected by the criterion      

 LR: sequential modified LR test statistic (each test at 5% level)    

 FPE: Final prediction error        

 AIC: Akaike information criterion        

 SC: Schwarz information criterion shows the lag length 1 .        

 HQ: Hannan-Quinn information criterion      

Long Run Equation For Results :

LPGDP = α + β1 LINV + β2 LATAX + β3 LPS

LPGDP = 3.86 +4.62 LINV + 3.35 LATAX – 1.27 LPS.

REFRENCES : http://www.google.com.pk/url?

sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCkQFjAA&url=http%3A%2F%2Fwww.eco.uc3m.es%2Fjgonzalo%2Fteaching%2FtimeseriesMA%2Fspuriousregandcointegration.ppt&ei=3IWDU8KIFKXm7Aa1q4GgAQ&usg=AFQjCNEMeDrxYrVbmOezMTPaUHqr7Fmsbw&sig2=oDcwQxDEQ64CZxFKeobvkg&bvm=bv.67720277,d.ZGU

powershow.com/view/9d319-MzYzM/TIME_SERIES_REGRESSION_COINTEGRATION_powerpoint_ppt_presentation

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https://www.imf.org/external/pubs/ft/wp/2007/wp07141.pdf Applied Economics By Esterio. Basic Econometrics By Damodar Gujrati.