MONTEVECM.RPF is an example of Monte Carlo integration for the impulse responses for a VECM (Vector Error Correction Model). This requires RATS 9.2 or later. This is based upon the ECT.RPF example which does a more thorough analysis of the unit root and cointegration properties of the data. The analysis is of three series of US Treasury yields (FTBS3 is the 3-month T-bill rate, FTB12 is the 12-month T-bill rate and FCM7 is the 7-year bond rate).
If the cointegration vector is taken as fixed (it's quite difficult to allow it also to be unknown), the VECM is just another example of a multivariate linear system with the same right side variables in each equation. As such, the same methods for drawing coefficients can be employed as are used in a standard Vector Autoregression. The following estimates the (one) cointegrating vector and creates an error correction term based upon it. This uses the DET=RC option on @JOHMLE, which allows for a constant restricted to the cointegrating vector (so the series won't trend).
@johmle(lags=6,det=rc,cv=cvector)
# ftbs3 ftb12 fcm7
equation(coeffs=cvector) ecteq *
# ftbs3 ftb12 fcm7 constant
This sets up the VECM based upon that:
system(model=ectmodel)
variables ftbs3 ftb12 fcm7
lags 1 to 6
ect ecteq
end(system)
and this estimates it:
estimate
The number of estimated coefficients per equation is 16: 5 each on the lagged differences for each of the three variables, plus the one coefficient on the lagged error correction term. You don't have to create a separate series for the error correction, nor do you have to difference the data yourself⎯that's all done internally by the ESTIMATE instruction.
The draws for the impulse responses can be done using the stock @MCVARDODRAWS procedure:
@mcvardodraws(model=ectmodel,draws=mcdraws,steps=nsteps)
and graphs can be done with @MCGRAPHIRF (or you can use @MCPROCESSIRF to produce customized graphics or use @MCFEVDTABLE for error decompositions). Note that you do not use the ACCUM option on any of those procedures⎯the impulse responses for ECTMODEL already take into account the structure of the models, and produce responses of the levels, not the differences. This does the median responses with two layers of error bands, with 68% on the inner and 95% on the outer.
@mcgraphirf(model=ectmodel,varlabels=vlabels,shocklabels=vlabels,$
center=median,percent=||.025,.16,.84,.975||,$
footer="Pointwise 68 and 95% Posterior Bands")
Full Program
compute mcdraws=1000 compute nsteps=36 * cal(m) 1975:1 open data haverate.rat data(format=rats) 1975:1 2001:6 ftbs3 ftb12 fcm7 * @johmle(lags=6,det=rc,cv=cvector) # ftbs3 ftb12 fcm7 equation(coeffs=cvector) ecteq * # ftbs3 ftb12 fcm7 constant * * Set up the model with the error correction term * system(model=ectmodel) variables ftbs3 ftb12 fcm7 lags 1 to 6 ect ecteq end(system) * dec vect[strings] vlabels compute vlabels=||"3-Month T-Bills","1-Year T-Bills","7-year T-Bonds"|| estimate * @mcvardodraws(model=ectmodel,draws=mcdraws,steps=nsteps) @mcgraphirf(model=ectmodel,varlabels=vlabels,shocklabels=vlabels,$ center=median,percent=||.025,.16,.84,.975||,$ footer="Pointwise 68 and 95% Posterior Bands")
Output
Likelihood Based Analysis of Cointegration
Variables: FTBS3 FTB12 FCM7
Estimated from 1975:07 to 2001:06
Data Points 312 Lags 6 with Constant restricted to Cointegrating Vector
Unrestricted eigenvalues and -T log(1-lambda)
Rank EigVal Lambda-max Trace Trace-95% LogL
0 20.2392
1 0.0818 26.6264 41.2013 35.0700 33.5524
2 0.0333 10.5567 14.5750 20.1600 38.8307
3 0.0128 4.0183 4.0183 9.1400 40.8399
Cointegrating Vector for Largest Eigenvalue
FTBS3 FTB12 FCM7 Constant
-3.154123 3.132882 -0.321838 0.619010
VAR/System - Estimation by Cointegrated Least Squares
Monthly Data From 1975:07 To 2001:06
Usable Observations 312
Dependent Variable FTBS3
Mean of Dependent Variable -0.005929487
Std Error of Dependent Variable 0.572477918
Standard Error of Estimate 0.491510475
Sum of Squared Residuals 71.508433993
Durbin-Watson Statistic 1.9616
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. D_FTBS3{1} 0.143892238 0.183591916 0.78376 0.43380689
2. D_FTBS3{2} 0.192186442 0.183710031 1.04614 0.29634985
3. D_FTBS3{3} -0.092909875 0.191610137 -0.48489 0.62811289
4. D_FTBS3{4} 0.408247277 0.174242798 2.34298 0.01979251
5. D_FTBS3{5} -0.212660827 0.169932295 -1.25144 0.21176048
6. D_FTB12{1} 0.290088325 0.260050689 1.11551 0.26553809
7. D_FTB12{2} -0.362872849 0.253978857 -1.42875 0.15412986
8. D_FTB12{3} 0.063761999 0.259907946 0.24533 0.80637455
9. D_FTB12{4} -0.577081483 0.246168719 -2.34425 0.01972637
10. D_FTB12{5} 0.151509694 0.247373031 0.61247 0.54069404
11. D_FCM7{1} 0.218441482 0.191699666 1.13950 0.25541657
12. D_FCM7{2} -0.278771615 0.195277984 -1.42756 0.15447181
13. D_FCM7{3} 0.165902869 0.196382770 0.84479 0.39890849
14. D_FCM7{4} 0.162678086 0.199755989 0.81438 0.41607968
15. D_FCM7{5} 0.233716355 0.196062364 1.19205 0.23419602
16. EC1{1} 0.062282194 0.027826301 2.23825 0.02594825
Dependent Variable FTB12
Mean of Dependent Variable -0.008685897
Std Error of Dependent Variable 0.561943004
Standard Error of Estimate 0.478368229
Sum of Squared Residuals 67.735504190
Durbin-Watson Statistic 1.9627
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. D_FTBS3{1} -0.196215312 0.178682946 -1.09812 0.27304455
2. D_FTBS3{2} 0.609142341 0.178797903 3.40688 0.00074806
3. D_FTBS3{3} -0.220795394 0.186486772 -1.18397 0.23737345
4. D_FTBS3{4} 0.510834851 0.169583809 3.01229 0.00281676
5. D_FTBS3{5} -0.357619776 0.165388563 -2.16230 0.03139593
6. D_FTB12{1} 0.468313663 0.253097328 1.85033 0.06526232
7. D_FTB12{2} -0.744190992 0.247187847 -3.01063 0.00283170
8. D_FTB12{3} -0.015874824 0.252958401 -0.06276 0.95000264
9. D_FTB12{4} -0.618901400 0.239586540 -2.58321 0.01026839
10. D_FTB12{5} 0.359172305 0.240758650 1.49184 0.13680725
11. D_FCM7{1} 0.385581182 0.186573907 2.06664 0.03963750
12. D_FCM7{2} -0.250214867 0.190056546 -1.31653 0.18901517
13. D_FCM7{3} 0.438478573 0.191131793 2.29412 0.02248400
14. D_FCM7{4} 0.088147039 0.194414816 0.45340 0.65059530
15. D_FCM7{5} 0.114017121 0.190819954 0.59751 0.55062274
16. EC1{1} 0.018519738 0.027082268 0.68383 0.49461586
Dependent Variable FCM7
Mean of Dependent Variable -0.008173077
Std Error of Dependent Variable 0.382826930
Standard Error of Estimate 0.323888686
Sum of Squared Residuals 31.051548720
Durbin-Watson Statistic 1.9830
Variable Coeff Std Error T-Stat Signif
************************************************************************************
1. D_FTBS3{1} -0.200394764 0.120980828 -1.65642 0.09869691
2. D_FTBS3{2} 0.439301740 0.121058662 3.62883 0.00033528
3. D_FTBS3{3} -0.184196931 0.126264563 -1.45882 0.14567571
4. D_FTBS3{4} 0.279448031 0.114820077 2.43379 0.01553315
5. D_FTBS3{5} -0.361887479 0.111979603 -3.23173 0.00136925
6. D_FTB12{1} 0.196795311 0.171364559 1.14840 0.25173008
7. D_FTB12{2} -0.408312647 0.167363429 -2.43968 0.01528740
8. D_FTB12{3} -0.109147013 0.171270497 -0.63728 0.52443602
9. D_FTB12{4} -0.240149459 0.162216813 -1.48042 0.13982405
10. D_FTB12{5} 0.461874003 0.163010414 2.83340 0.00492198
11. D_FCM7{1} 0.461530253 0.126323560 3.65356 0.00030584
12. D_FCM7{2} -0.257426149 0.128681550 -2.00049 0.04636053
13. D_FCM7{3} 0.444830857 0.129409566 3.43739 0.00067154
14. D_FCM7{4} -0.022530435 0.131632402 -0.17116 0.86421351
15. D_FCM7{5} -0.116804509 0.129198430 -0.90407 0.36669313
16. EC1{1} -0.034984890 0.018336586 -1.90793 0.05736794
Graphs
