cointegration and error correction model

Error Correction Model And Its Application To Agricultural

Economics Research.PresenterAditya K.S., PALB (1094)Sr. M.Sc. (Agricultural Economics)

Major Adviser: Dr. T.N. Prakash Kammardi

Flow of presentation

Department Of Agricultural Economics, Bangalore

2

Concepts and definitions.Cointegration.Residual based test for cointegration.Johansen’s cointegration test.Introduction to ECM.Engle – Granger two step ECM.Market integration of Arecanut in Karnataka state: An ECM approach.Final outcome.Concluding remarks.References.

3Department Of Agricultural Economics, Bangalore

Concept and definitions

Stationary v/s non stationary

• If a time series is stationary, its mean and variance remain the same no matter at what point we measure them;

That is, they are time invariant.



Figure 1: Monthly prices of Arecanut in Mangalore from 2005 to 2011

Pure Random Walk


Random Walk with Drift

Order of Integration Differencing is a way to convert non stationary data into stationary. If the data has to be differenced d times to make it stationary then series

said to be integrated of order (d) and represented as I(d) I(1) processes are fairly common in economic time series data



Figure 2: 1st difference of monthly prices of Arecanut in Mangalore from 2005 to 2011

Price series is I(1)

UNIT ROOT• If ρ = 1 it becomes a pure random walk. • If ρ is in fact 1, we face what is known as the unit root

problem, that is, a situation of nonstationary; • The name unit root is due to the fact that ρ = 1.

Yt = ρYt −1 + ut


Synonymous

Testing for unit rootsAugmented dickey fuller test(ADF) – Include the lagged terms.

Phillip Perron tests (PP) – Non parametric method.


NH: Series contains unit root

AH : Series does not contain unit roots

Decision rule: Reject NH if P<0.05

Spurious Regression


Suppose that Yt and Xt are two non stationary time series variables

Yt = βXt + error:

β significant β not significant

Yt and Xt are independentDue to trend(non stationarity)

Due to actual relationship

Spurious regressionCointegration R2 >D.W stat

Cointegration• Economic theory often suggests that certain

subset of variables should be linked by a long-run equilibrium relationship.

• Although the variables under consideration may drift away from equilibrium for a while, economic forces or government actions may be expected to restore equilibrium.



Observe that two series follow each

other closely

Figure 3: Monthly prices of Arecanut in Mangalore and Kundapura from 2005 to 2011

Short run disequilibrium

Residual-based Test for Cointegration

• One of most popular tests for (a single) co integration has been suggested by Engle and Granger (1987, Econometrica).


Consider the multiple regression: Yt = βXt + ut;

• for yt and xt to be cointegrated, ut must be I(0).• Otherwise it is spurious. Thus, a basic idea

behind is to test whether ut is I(0) or I(1).

Ut is stationary

Ut is not stationary

Cointegration

Spurious regression


Residual plot of regression Bantwala V/S kundapura


Johansen's procedure

• Johansen's procedure builds cointegrated variables directly on maximum likelihood estimation

• Tests for determining the number of cointegrating vectors.

• Multivariate generalization of the Dickey-Fuller test. • Two different likelihood ratio tests namely the Trace

test and the Maximum Eigen value test.


Two time series are cointegrated if

Cointegrated data are never drift too far away from each other

Both are integrated of the same order.

There is a linear combination of the two time series that is I(0) - i.e. - stationary.


An Introduction to ECMs• Error Correction Models (ECMs) multiple time

series models that estimate the speed at which a dependent variable - Y - returns to equilibrium after a change in an independent variable - X. i.e SPEED OF ADJUSTMENT


ECMs can be appropriate whenever

time series data

Interested in both short and

long term relationships

Non stationary

Integrated of same order Cointegrated


• Yt = βXt + Ut

• Here, Ut represents the portion of Y (in levels) that is not attributable to X.

• In short, Ut will capture the error correction relationship by capturing the degree to which Y and X are out of equilibrium.


• Ut-1 = Yt-1 - Xt-1

• When Ut-1 = 0 the system is in its equilibrium state.

• So ECM can be built as ∆Yt = C + Φ ∆Xt + αUt-1


Engle and Granger Two-Step ECM


• Engle and Granger (1987) suggested an appropriate model for Y, based two or more time series that are cointegrated.

• First, we can obtain an estimate of Ut by regressing Y on X.

• Second, we can regress ∆ Yt on Ut-1 plus any relevant short term effects as ∆ Xt.


• Market integration of Arecanut in Karnataka state: An ECM approach.

(Source: Author)


Market Integration

• Spatial market integration refers to co-movements or a long run relationship of prices.

• It is defined as the smooth transmission of price signals and information across spatially separated markets



28

Integrated markets:EfficiencyEqualityStabilityMaximize social welfare

• Study of market integration is very important though neglected.

• Knowledge of market integration would be vital to know the market efficiency, and to device measures to overcome imperfections.


• Traditional method of study employs correlation matrix to study the market integrations.

• Since the data are non stationary results may not be accurate and hence criticized.


30

Data and Methodology• For the purpose of analyzing the integration of

arecanut markets, monthly prices of arecanut from 2005 to 2011 in 7 major arecanut markets in Karnataka was used.

• Data was collected from Agmarknet.


Table 1:Markets selected for study

Sl no WCT RBT

1 Mangalore Shimoga

2 Bantwala Sagara

3 Kundapura Davangeree

4 Sirsi


Methodology


Unit root testingNH: Series is non stationary


Table 2: Results of Unit root test for arecanut price in major RBT markets from 2005 to 2011

At level

PP P value ADF P value

Sagara -1.90949 0.3259 -1.53207 0.5105

Shimoga -2.59777 0.0991 -2.69163 0.0815

Davangeree -2.39903 0.1464 -1.59475 0.4787

Sirsi -2.14473 0.2285 -1.13264 0.6969

After first difference

PP P value ADF P value

Sagara -10.8727 0 -10.1247 0

Shimoga -14.8105 0 -6.57014 0

Davangeree -11.1522 0 -8.09634 0

Sirsi -11.37 0 -8.307 0


Table 3: Results of Unit root test for arecanut price in major WCT markets from 2005 to 2011

Both tests indicate that prices are integrated of order (1)


At levelADF P PP p

Mangalore -1.75041 0.4024 -1.75041 0.4024

Kundapura -2.09198 0.2484 -2.13241 0.2328

Bantwala -0.56366 0.8719 -0.64773 0.8531

At 1st differenceADF P PP p

Mangalore -7.89198 0 -7.94013 0

Kundapura -7.02788 0 -8.49619 0

Bantwala -12.1208 0.0001 -12.3691 0.0001

Testing for cointegration


Engle Granger test -Decision rule

• Engle Granger critical value at 1% LOS is -3.96


Ut= ΏUt-1 + e

Table 4. Engle Granger cointegration testfor major arecanut markets in Karnataka

Kundapura MangaloreBantwala -6.2580 -2.57891Kundapura -6.47711

Sagara Shimoga SirsiDavangeree -5.264 -6.16165 -5.4227

Sagara -5.7895 -5.5994

Shimoga -5.4529

There is cointegration among all markets under consideration except Bantwala and Mangalore


Johansen cointegration test


Table 5. Johansen’s cointegration test for RBT arecanut markets

Shimoga Davangeree Sirsi

No of coint

equations trace stat p trace stat p trace stat p

Sagara R=0 20.68967 0.0075 26.24133 0.0008 22.90293 0.0032

R ≤ 1 2.148919 0.1427 2.197354 0.1382 2.391261 0.122

Shimoga R=0 29.09037 0.0003 18.48941 0.0171

R≤ 1 4.906882 0.0267 2.71361 0.0995

Davangeree R=0 29.16382 0.0003

R≤ 1 2.9382 0.0865


Contd…………….

Shimoga Davangere SirsiNumber of

coint

equations

Max eigen

valuep

Max

eigen

value

p

Max

eigen

value

p

SagaraR=0 18.54075 0.0099 24.04398 0.0011 20.51167 0.0045

R≤ 1 2.148919 0.1427 2.197354 0.1382 2.391261 0.122

ShimogaR=0 24.18349 0.001 15.7758 0.0286

R≤ 1 4.906882 0.0267 2.71361 0.0995

DavangereeR=0 26.22562 0.0004

R≤ 1 2.9382 0.0865


Table :6 Johansen’s cointegration test for WCT arecanut markets

trace stat Max eigen value

Dependent IndependentNo. of coint

equationvalue P value P

Bantwala Mangalore R=0 10.29579 0.3888 7.901239 0.255

R ≤ 1 2.394551 0.1218 2.394551 0.1218

Kundapura Bantwala R=0 23.32457 0.0027 23.26433 0.0015

R ≤ 1 0.060234 0.8061 0.060234 0.8061

Kundapura Mangalore R=0 16.93599 0.0301 13.84253 0.05

R ≤ 1 3.093461 0.0786 3.093461 0.0786


Table 7 : Error correction models for RBT arecanut markets

Error Correction model results for RBT.

∆ Dav = -9.73171+0.8484∆ sag – 0.64371 et-1

(0.90) (0) (0)

∆ Dav = -12.3961 + 0.8104 ∆ sir – 0.64273 et-1

(0.8967) (0) (0)

∆ Dav = -6.92457 + 0.7822 ∆ shiv – 0.73867 et-1

(0.9249) (0) (0)

∆ Sag = - 2.2253 + 0.65523 ∆ shiv – 0.6073 et-1

(0.98) (0) (0)

∆ Sag = - 3.9302 + 0.8762 ∆ shir– 0.6453 et-1

(0.95) (0) (0)

∆ shiv = - 7.6146 + 1.007 ∆ shir– 0.6719 et-1

(0.93) (0) (0)44Department Of Agricultural Economics,

Bangalore

Figures in parenthesis indicate the probability values.

Model estimated: ∆ Yt= C + Φ ∆Xt+ α Ut-1

Error Correction model results for WCT.

∆ kund = 3.79 + 0.83 ∆mang -0.66 et-1

( 0.98) ( 0.001) (0)

∆ bant = 22.75 +0.75 ∆kund -0.72 et-1

(0.97) (0.002) (0)

Figures in parenthesis indicate the probability values.


Table 8 : Error correction models for RBT arecanut markets

Table 9: Speed of error correction

Sagara Shimoga Sirsi

Davang

eree

64 73 64

Sagara 60 64

Shimog

a

67

Mangalore Kundapura

Bantwala 66 72


Final outcome• Arecanut markets are highly cointegrated may be

because of better marketing infrastructure, existence of cooperatives, easy flow of market information and non perishability.

• Price volatility observed during last few years has nothing to do with the inefficiency of domestic markets.

• If the government wants to stabilize the prices of arecanut, then it can be done by stabilizing the prices in one important market.


Concluding remarks• Most valuable contribution of concept of cointegration is to

force us to test for Stationarity of the residuals.• Cointegration can be thought as pre test to avoid spurious

regression situation. • Cointegrated variables will always have a built in error

correction mechanism, estimation of which will be helpful to know short run dynamics of the system.


48

• Though theoretically appealing, practically simple, ECM cannot be used in complex situations involving more number of non stationary variables.

• In such situations one can go for vector error correction models (VECM) which are nothing but multivariate specification of ECM.


49


50

cointegration and error correction model

Documents

stationary ut

stationary data

stationary vs

unit root yt

johansens cointegration

price series

moretime series

estimate of ut byregressing