Error Correction Model

Example: Error Correction Model in TSMT

The difference equation for a stationary VAR(P) model can be written as (Hamilton, 1994):

Δyt = ζ1 Δyt-1 + ζ2 Δyt-2 + ... + ζp-1 Δyt-p+1 + ζ0 yt-1 + εt

where:

ζ0 ≡ -(Ιn - Φ1 - Φ2 - ... Φp)

The error-correction form of a cointegrated VAR(P) system can be represented by:

Δyt = ζ1 Δyt-1 + ζ2 Δyt-2 + ... + ζp-1 Δyt-p+1 + α - BA'yt-1 + εt

For a model with h cointegrating relationships, we can define:

zt ≡ A'yt-1 (a stationary hx1 vector).

we can rewrite the equation as:

Δyt = ζ1 Δyt-1 + ζ2 Δyt-2 + ... + ζp-1 Δyt-p+1 + α - Bzt-1 + εt

The time series module provides a number of routines for performing pre-estimation data analysis, model parameter estimation, and post-estimation diagnosis of cointegrated time series.

Load and plot time series

For this example, we will use a data set that comes with TSMT, named rpie.xlsx. This file contains data for seasonally adjusted real personal income and expenditures, quarterly. The first column contains the dates and the second and third columns contain the data.

//Get path to GAUSSHOME directory
path = getGAUSSHome();

//Add file name to path
fname = path $+ "examples/rpie.xlsx";

//Load data from the second and third columns
//of a space separated text file
y = xlsReadM(fname, "B2:C277");

//Load first date
first_date = xlsReadM(fname, "A2:A2");

//Declare 'myPlot' to be a plotControl structure
//and fill with default settings for XY plots
struct plotControl myPlot;
myPlot = plotGetDefaults("xy");

//Set text and location for legend
plotSetLegend(&myPlot, "Consumption" $| "Income", "top left inside");

//Place x-tics every 10 years (4 quarters/year * 10 years = 40)
//and place the first tic at 1950
plotSetXTicInterval(&myPlot, 40, 1950);

//Only print 4 digit year at x-tic labels
plotSetXTicLabel(&myPlot, "YYYY");

//Plot data using settings applied to 'myPlot'
plotTS(myPlot, first_date, 1, y);

The above code should produce a plot that looks similar to this:

Plot of quarterly personal consumption and income.

The two series definitely seem to have a relationship. Their change over time seems to be superlinear. In this tutorial, we will model the logged percent change of these variables. We can make this data transformation with the following code:

//Make TSMT library functions available
library tsmt;

//Transform data by taking first difference of each
//of our variables after taking their natural log
y_chg = vmdiffmt(ln(y), 1);

Let's now plot our transformed data.

//Continuing with 'myPlot' structure made above
//Add a plot title and set its: text, font and font size
plotSetTitle(&myPlot, "Log percent change", "Helvetica", 18);

//Plot transformed data
plotTS(myPlot, first_date, 4, y_chg);

Which should produce a graph similar to this:

Plot of log change of personal income and expenditures.

This graph looks like what we would expect. The variance is not constant and any wild swings are corrected back towards normal quickly. The only problem here is the scale of the data. It appears that we have only one or, maybe two, data points with an absolute value greater than 0.05. The function we will use to estimate the parameters of this error-correction model is a Full Information Maximum Likelihood (FIML) method. The numerical optimization procedures can be sensitive to scaling problems with the data. In order to improve the scale of our data for the estimation, we will multiply it by 100 so that the range of our data will be from about -5 to 5, rather than -0.05 to 0.05.

Estimating the Error Correction Model

The ecmmt function is a convenient tool for estimating the parameters of ECM models with or without exogenous variables.  The ecmmt function takes two required arguments:
  1. A varmamtControl structure.
  2. An N x K data vector, y, where each column of y is a different time series.
Note that ecmmt will produce an error if the data is not stationary and invertible.

The varmamtControl Structure

This structure handles the matrices and strings that control the estimation process such as specifying whether to include a constant, output appearance, and estimation parameters.  Continuing with the example above,
//Declare 'ctl' to be a varmamtControl structure
//and fill with default values
struct varmamtControl ctl;
ctl = varmamtControlCreate();

//Declare 'out' to be an varmamtOut structure,
//to hold the results from 'ecmmt'
struct varmamtOut out;

//No constraints on the optimization problem--for simplicity
ctl.setConstraints = 0;

//Specify the AR order to 2
ctl.ar = 2;

//No constant in model
ctl.const = 0;
This tells ecmmt that the model is an VAR(2) model and does not include a constant.  The remaining control elements (the ones that we did not set above), left at the default values, specify preferences for output display and estimation. With the control structure options set, we can now perform the estimation:
//Scale data
y_chg = 100 * y_chg;

//Perform estimation with 'ecmmt'
out = ecmmt(ctl, y_chg);
which will return output similar to what is shown below. Note that this output uses the parameter Pi, Π, to represent BA' from above and that the data is internally demeaned so that the model has an intercept, α equal to 0. This creates the equation:

Δyt = ζ1 Δyt-1 + ζ2 Δyt-2 + ... + ζp-1 Δyt-p+1 - Πyt-1 + εt

=====================================================
 ECM Version 2.1.5                                   
=====================================================


Residual Covariance Matrix

 150.73574   73.83026 
  73.83026  249.96929 


Zeta


Plane [1,.,.] 

  -0.29990    0.10971 
  -0.24282    0.12436 

Plane [2,.,.] 

  -0.14146    0.08794 
  -0.15634    0.18373 


Pi

   1.33370   -0.15140 
   0.00141    0.90544 

Augmented Dickey-Fuller UNIT ROOT Test for    Y1
                                                          Critical Values
                               ADF Stat        1%       5%       10%      90%      95%     99%
   No Intercept                -3.6411     -2.5897  -1.9567  -1.6179   0.9025   1.3221   2.0538
   Intercept                   -7.0128     -3.4404  -2.8697  -2.5829  -0.4516  -0.0789   0.6319
   Intercept and Time Trend    -7.2323     -3.9542  -3.4305  -3.1392  -1.2333  -0.9390  -0.3755

Augmented Dickey-Fuller UNIT ROOT Test for    Y2
                                                          Critical Values
                               ADF Stat        1%       5%       10%      90%      95%     99%
   No Intercept                -4.6936     -2.5897  -1.9567  -1.6179   0.9025   1.3221   2.0538
   Intercept                   -8.9575     -3.4404  -2.8697  -2.5829  -0.4516  -0.0789   0.6319
   Intercept and Time Trend    -9.5108     -3.9542  -3.4305  -3.1392  -1.2333  -0.9390  -0.3755


Phillips-Perron UNIT ROOT Test for    Y1
                                  PPt      1%      5%
   No Intercept                -9.7015  -2.6349  -1.9678
   Intercept                  -15.5683  -3.4456  -2.8418
   Intercept and Time Trend   -15.7258  -3.9914  -3.4154

Phillips-Perron UNIT ROOT Test for    Y2
                                  PPt      1%      5%
   No Intercept               -12.4127  -2.6349  -1.9678
   Intercept                  -17.2373  -3.4456  -2.8418
   Intercept and Time Trend   -17.7911  -3.9914  -3.4154


Augmented Dickey-Fuller COINTEGRATION Test for    Y1    Y2

                                                          Critical Values
                               ADF Stat        1%       5%       10%      90%      95%     99%
   No Intercept               -10.4949     -3.3567  -2.7752  -2.4659  -0.2541   0.1961   1.0722
   Intercept                   -7.3543     -3.9243  -3.3804  -3.0821  -1.0076  -0.6342   0.0946
   Intercept and Time Trend    -7.4509     -4.4817  -3.8340  -3.5454  -1.6020  -1.3184  -0.7343



Johansen's  Trace  and  Maximum  Eigenvalue  Statistics. r = # of CI Equations
                                                                      Critical Values
                               r    Trace    Max. Eig          1%       5%       10%      90%
   No Intercept                0  170.8718  153.6202
                               1   17.2516   17.2516         1.0524   1.7046   2.1927   9.3918

   Intercept                   0  213.8515  156.7800
                               1   57.0716   57.0716         2.2515   3.3599   4.0975  12.8635

   Intercept and Time Trend    0  221.3177  160.5904
                               1   60.7273   60.7273         4.0389   5.3796   6.1879  16.1762

Dep. Variable(s)    :       D(Y1      )       D(Y2      )
No. of Observations :            275               275
Degrees of Freedom  :            260               260
Mean of Y           :         0.8163            0.8157
Std. Dev. of Y      :         0.8202            1.0107
Y Sum of Squares    :       184.3448          279.8773
SSE                 :       150.7957          250.0293
MSE                 :         0.5637            0.9347
sqrt(MSE)           :         0.7508            0.9668
R-Squared           :         0.1820            0.1066
Adjusted R-Squared  :         0.1379            0.0585


Model Selection (Information) Criteria
......................................
Likelihood Function :     -1933.9647
Akaike AIC          :      3837.9293
Schwarz BIC         :      3952.1809
Likelihood Ratio    :      3867.9293




Characteristic Equation(s) for Stationarity and Invertibility
   AR Roots and Moduli:
   Real :  1.95 1.23 1.23 
   Imag.:  0.00 1.63 -1.63 
   Mod. :  1.95 2.04 2.04 



                                 MULTIVARIATE ACF

               LAG01                    LAG02                    LAG03
      0.008238   -0.04828       0.02109   0.009994       0.05089    0.05083
       0.01015    -0.0364      0.002671  -0.009055       0.04582   -0.03054


               LAG04                    LAG05                    LAG06
      -0.06684    0.09705       -0.1203   -0.09942      -0.07708    0.01643
      -0.06681    -0.1699      -0.09808    -0.0361       0.01039    0.06907


               LAG07                    LAG08                    LAG09
       0.05776    0.04065       -0.1715   -0.07736        0.1374    0.08797
        0.1593    0.05391      -0.03936    -0.0111       0.08368    0.01126


               LAG10                    LAG11                    LAG12
       0.08925    0.06825       0.03497    0.08071      -0.02322  -0.006081
     -0.006279     0.1147       0.03267    0.05381        0.0337   0.005717


    ACF INDICATORS: SIGNIFICANCE = 0.95 
  (using Bartlett's large sample standard errors)

LAG01     LAG02     LAG03     LAG04     LAG05     LAG06     
-+-+      -+-+      -+-+      -+-+      -+-+      -+-+      
-+-+      -+-+      -+-+      -+-+      -+-+      -+-+      


LAG07     LAG08     LAG09     LAG10     LAG11     LAG12     
-+-+      -+-+      -+-+      -+-+      -+-+      -+-+      
-+-+      -+-+      -+-+      -+-+      -+-+      -+-+      


Multivariate Goodness of Fit Test
Lag       Qs     P-Value
3      3.8602    0.4253
4     24.4700    0.0019
5     30.5424    0.0023
6     34.2922    0.0050
7     41.6454    0.0031
8     50.2697    0.0013
9     56.8432    0.0010
10    64.2697    0.0006
11    66.3653    0.0015
12    67.1565    0.0046

The Command File

Finally we put it all together in the command file below, which will reproduce all output shown above.
//Load TSMT library functions
library tsmt;

/*******************************************
**                                        **
**     Load and view original data        **
**                                        **
*******************************************/

//Get path to GAUSSHOME directory
path = getGAUSSHome();

//Add file name to path
fname = path $+ "examples/rpie.xlsx";

//Load data from the second and third columns
//of a space separated text file
y = xlsReadM(fname, "B2:C277");

//Load first date
first_date = xlsReadM(fname, "A2:A2");

//Declare 'myPlot' to be a plotControl structure
//and fill with default settings for XY plots
struct plotControl myPlot;
myPlot = plotGetDefaults("xy");

//Set text and location for legend
plotSetLegend(&myPlot, "Consumption" $| "Income", "top left inside");

//Place x-tics every 10 years (4 quarters/year * 10 years = 40)
//and place the first tic at 1950
plotSetXTicInterval(&myPlot, 40, 1950);

//Only print 4 digit year at x-tic labels
plotSetXTicLabel(&myPlot, "YYYY");

//Plot data using settings applied to 'myPlot'
plotTS(myPlot, first_date, 4, y);

/*******************************************
**                                        **
**            Transform data              **
**                                        **
*******************************************/
//Transform both columns (variables) in one step
y = 100 * vmdiffmt(ln(y), 1);

/*******************************************
**                                        **
**        Plot transformed data           **
**                                        **
*******************************************/
//Create new graph tab so we do not
//overwrite our last plot
plotOpenWindow();

//Continuing with 'myPlot' structure made above
//Add a plot title and set its: text, font and font size
plotSetTitle(&myPlot, "Log percent change", "Helvetica", 18);

//Plot transformed data
plotTS(myPlot, first_date, 4, y);

/*******************************************
**                                        **
**             Estimate                   **
**                                        **
*******************************************/

//Declare 'ctl' to be a varmamtControl structure
struct varmamtControl ctl;

//Fill 'ctl' with default values
ctl = varmamtControlCreate();

//Specify a VAR(2) model
ctl.ar = 2;

//Do not set constraints for internal optimization model
ctl.setConstraints = 0;

//Declare 'out' to be a varmamtOut structure
//to hold the return values from 'out'
struct varmamtOut out;

//Estimate the parameters of the model
//and print diagnostic information
out = ecmmt(ctl, y); 

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