ECT Instruction

ECT list of equations (describing the error correction terms)

ECT is a subcommand of SYSTEM used to add error correction term(s) to a vector autoregression, converting the system into a VECM. These are input using equations, one per needed term.

Wizard

The Time Series—VAR (Setup/Estimate) Wizard includes support for defining and estimating error-correction models.

Parameters

list of equations

lists equations describing the "stationary" relationships among variables in the VAR model. You can use a VECTOR of equations if you might need to vary the number. The equations can also include exogenous variables (such as CONSTANT or a trend) if desired.

Form of the Equations

The equations listed can either have one of the endogenous variables of the VAR as the dependent variable, or it can have a constructed variable (or no dependent variable at all), in which case the explanatory part of the equation shows the stationary relationship. For example,

equation ecteq y1

# y2 y3
...

system(model=cointmodel)

variables y1 y2 y3

lags 1 2 3 4

ect ecteq

end(system)

linreg(equation=ecteq)

uses the first form. The cointegrating relationship in this case will be \({\beta _2}{y_2} + {\beta _3}{y_3} - {y_1}\) where the \(\beta\) are coefficients from the preliminary LINREG. Note how the equation is renormalized to have a –1 coefficient on the dependent variable.

Suppose that, instead, a RECTANGULAR matrix B has been estimated which lists the cointegrating vectors. The example below creates a VECTOR of EQUATIONS to hold the equations. In this case, the dependent variables of the equations are ignored in forming the stationary conditions.

dec vect[equations] ecteqs(r)

do i=1,r

equation(coeffs=%xrow(b,i)) ecteqs(i)

# y1 y2 y3

end do i

system(model=cointmodel)

variables y1 y2 y3

lags 1 2 3 4

ect ecteqs

end(system)

Examples

See ECT.RPF for examples of the use of ECT for a single and for two cointegrating vectors.

The following excerpt is taken from the example program KPSW5.RPF, which reproduces Table 5 from King, Plosser, Stock and Watson (1991). You can find the program file in the "Paper Results/KPSW AER 1991" subdirectory. This is a six variable VAR with three cointegrating relations.

The coefficients (BETAY, etc.) in the error correction equations have already been estimated using a separate process.

equation(coeffs=||-betay,0.0,0.0,1.0,-betar,0.0||) mdemand

# y c in mp r dp

equation(coeffs=||-1.0,1.0,0.0,0.0,-phi1,phi1||) cratio

# y c in mp r dp

equation(coeffs=||-1.0,0.0,1.0,0.0,-phi2,phi2||) iratio

# y c in mp r dp

system(model=varmodel)

variables y c in mp r dp

lags 1 to 9

det constant

ect mdemand cratio iratio

end(system)

estimate(noprint) 1954:1 *

This excerpt estimates a first-stage cointegrating relation, then constructs and estimates a VECM.

linreg(define=cointeq) aus

# usa

system(model=vecm)

variables aus usa

lags 1 2

det constant

ect cointeq

end(system)

estimate

Sample Output

ECT itself produces no output. This shows how the output from the VAR is altered when you use an ECT term. (From the second example). Note that the explanatory variables shown are listed in differenced form with one fewer lag than listed on the LAGS. The loadings on the error correction term are shown in the coefficients for the EC1{1} term. (If you have more than one error correction term, they will be labeled as EC1, EC2, ...).

VAR/System - Estimation by Cointegrated Least Squares

Quarterly Data From 1970:03 To 2000:04

Usable Observations 122

Dependent Variable AUS

Mean of Dependent Variable 0.4993901639

Std Error of Dependent Variable 0.6522035928

Standard Error of Estimate 0.6243302485

Sum of Squared Residuals 45.995014591

Durbin-Watson Statistic 2.0773

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. D_AUS{1} 0.0403397064 0.1032347207 0.39076 0.69668133

2. D_USA{1} 0.3125731910 0.1262474970 2.47588 0.01470988

3. Constant 0.3127120679 0.0824498314 3.79276 0.00023645

4. EC1{1} 0.0886382527 0.0480838335 1.84341 0.06777834

Dependent Variable USA

Mean of Dependent Variable 0.5110459016

Std Error of Dependent Variable 0.5163721843

Standard Error of Estimate 0.4938444163

Sum of Squared Residuals 28.778112282

Durbin-Watson Statistic 2.1387

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. D_AUS{1} 0.041457402 0.081658530 0.50769 0.61261684

2. D_USA{1} 0.292991740 0.099861606 2.93398 0.00402214

3. Constant 0.344654008 0.065217710 5.28467 0.00000058

4. EC1{1} -0.041035755 0.038034250 -1.07892 0.28282573