ECT.RPF

ECT.RPF demonstrates the analysis of a Vector Error Correction model. It analyzes a set of three interest rate yields, first testing for cointegration, then imposing it.

RATS does only a limited form of cointegration analysis. For a more complete treatment, the program CATS is available from Estima as an add-on. CATS provides more sophisticated tests for cointegrating rank, model diagnostics and tests on the structures of both the cointegrating vectors and their loadings (and much more).

cal(m) 1975:1

open data haverate.rat

data(format=rats) 1975:1 2001:6 ftbs3 ftb12 fcm7

FTBS3 is the 3 month US Treasury bill yield, FTB12 is the 12 month T-bill yield and FCM7 is the 7 year T-Bond yield.

The first step is to test that the three series have unit roots. (It’s strongly accepted for all).

@dfunit(lags=6) ftbs3

@dfunit(lags=6) ftb12

@dfunit(lags=6) fcm7

There are several ways to estimate the cointegrating vector. The simplest is to run an “Engle-Granger” regression (one of the yields on the other two). While the estimates from that are “super-consistent” if the series are cointegrated, there are other ways to get better smaller sample performance. Here we’ll use the @JOHMLE procedure, which does maximum likelihood estimation. Other procedures for estimating a cointegrating vector or vectors are @FM (does fully-modfied least squares) and @SWDOLS (does dynamic OLS). These are all included in the Time Series—Cointegration Estimation wizard.

Note that it is far from obvious (a priori) that the cointegrating rank will be one. It might very well be zero, or, if there were a single stochastic trend linking the yields, then the cointegrating rank would be two.

Because the series don’t have a trend, the appropriate choice for the deterministic component is DET=RC, which doesn’t include a constant in the individual equations (where it would cause the series to drift because of the unit roots), but restricts it to the cointegrating vector.

The components of the cointegrating vector produced by @JOHMLE with this choice will have four components: the three variables + the constant. If we had used the default DET=CONSTANT, it would only be a 3-vector, but we would also include DET CONSTANT in the SYSTEM definition. Be careful with your choice for the DET option—it’s a common error to select the wrong one. Note that there is no reason that the error correction term(s) will have mean zero—the requirement is that they be stationary, which includes processes with non-zero means.

The cointegrating rank is analyzed with

@johmle(lags=6,det=rc,cv=cvector)

# ftbs3 ftb12 fcm7

The output from @JOHMLE would have us reject rank=0 (41.2013 is bigger than the critical value 34.8) but accept rank=1. (Read the tests from top down). The CV option on the procedure returns (a) cointegrating vector corresponding to the largest eigenvalue. (This is subject to a scale factor—note that the "normalization" in this is inherited from the embedded eigenvalue problem and has no economic meaning).

This makes the cointegrating vector and creates an EQUATION for it for use in the Vector Error Correction Model (VECM). Note that this is defined without a dependent variable (the * parameter) since it's a weighted sum of all four components. The values (where the coefficients on FTBS3 and FTB12 are nearly equal and opposite sign and much larger than the coefficient on FCM7) would suggest that the stationary component might be the yield spread between the two shorter term rates.

equation(coeffs=cvector) ecteq *

# ftbs3 ftb12 fcm7 constant

This sets up the model with the error correction term. Note that, as described above, there is no CONSTANT as the constant is included in the cointegrating vector.

system(model=ectmodel)

variables ftbs3 ftb12 fcm7

lags 1 to 6

ect ecteq

end(system)

This estimates the model and computes the decomposition of variance:

estimate

compute sigma=%sigma

errors(model=ectmodel,steps=36)

ESTIMATE with a model with ECT elements defines the matrices %VECMPI and %VECMALPHA. %VECMPI is the estimated value of \(\Pi\) and %VECMALPHA are the loadings on the cointegrating vectors. Note well that the lag sums (used in the Blanchard-Quah and the short-and-long run restrictions) are much more complicated for a VECM model than a regular VAR as the long-run response matrix isn’t full rank. (If x and y are cointegrated, then if x has a zero long-run response, y must as well). See the SHORTANDLONGVECM.RPF example for the proper way to handle this.

If we needed two cointegrating vectors rather than one (in this case, we really don’t), we would need to use the VECTORS option on @JOHMLE to get the full set of eigenvectors, then %XCOL to pull out the coefficients for each of the two equations that we would need to define. For instance, here

@johmle(lags=6,det=rc,vectors=cvectors)

# ftbs3 ftb12 fcm7

equation(coeffs=%xcol(cvectors,1)) ect1 *

# ftbs3 ftb12 fcm7 constant

equation(coeffs=%xcol(cvectors,2)) ect2 *

# ftbs3 ftb12 fcm7 constant

system(model=ect2model)

variables ftbs3 ftb12 fcm7

lags 1 to 6

ect ect1 ect2

end(system)

estimate

errors(model=ect2model,steps=36)

Full Program

cal(m) 1975:1
open data haverate.rat
data(format=rats) 1975:1 2001:6 ftbs3 ftb12 fcm7
*
* Test for unit roots in the original variables
*
@dfunit(lags=6) ftbs3
@dfunit(lags=6) ftb12
@dfunit(lags=6) fcm7
*
* Use @JohMLE to test the cointegrating rank and estimate the
* cointegrating vector corresponding to the largest eigenvalue. Note
* that it is far from obvious (a priori) that the cointegrating rank
* will be one. It might very well be zero, or, if there were a single
* stochastic trend linking the yields, then the cointegrating rank would
* be two.
*
* Because the series don't have a trend, the appropriate choice for the
* deterministic component is det=rc, which doesn't include a constant in
* the individual equations (where it would cause the series to drift
* because of the unit roots), but restricts it to the cointegrating
* vector. The components of the cointegrating vector produced by JohMLE
* with this choice will have four components: the three variables + the
* constant. If we had done the default DET=CONSTANT, it would only be a
* 3-vector. We would also include DET CONSTANT in the SYSTEM definition.
*
@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)
*
* Estimate it and compute the decomposition of variance
*
estimate
compute sigma=%sigma
errors(model=ectmodel,steps=36)
*
* Set up the model without the error correction term
*
system(model=ratemodel)
variables ftbs3 ftb12 fcm7
lags 1 to 6
det constant
end(system)
*
* Estimate and compute the decomposition of variance. This uses the same
* covariance matrix of residuals as above.
*
estimate
errors(model=ratemodel,steps=36,cv=sigma)
*
* Allowing for two cointegrating vectors. (The results of the
* cointegration test suggest that one is adequate).
*
@johmle(lags=6,det=rc,vectors=cvectors)
# ftbs3 ftb12 fcm7
equation(coeffs=%xcol(cvectors,1)) ect1 *
# ftbs3 ftb12 fcm7 constant
equation(coeffs=%xcol(cvectors,2)) ect2 *
# ftbs3 ftb12 fcm7 constant
*
system(model=ect2model)
variables ftbs3 ftb12 fcm7
lags 1 to 6
ect ect1 ect2
end(system)
estimate
errors(model=ect2model,steps=36)

Output

Dickey-Fuller Unit Root Test, Series FTBS3

Regression Run From 1975:08 to 2001:06

Observations 312

With intercept

Using fixed lags 6

Null is unit root. Reject in left tail.

Sig Level Crit Value

1%(**) -3.45303

5%(*) -2.87096

10% -2.57175

T-Statistic -1.35838

Dickey-Fuller Unit Root Test, Series FTB12

Regression Run From 1975:08 to 2001:06

Observations 312

With intercept

Using fixed lags 6

Null is unit root. Reject in left tail.

Sig Level Crit Value

1%(**) -3.45303

5%(*) -2.87096

10% -2.57175

T-Statistic -1.28267

Dickey-Fuller Unit Root Test, Series FCM7

Regression Run From 1975:08 to 2001:06

Observations 312

With intercept

Using fixed lags 6

Null is unit root. Reject in left tail.

Sig Level Crit Value

1%(**) -3.45303

5%(*) -2.87096

10% -2.57175

T-Statistic -1.29649

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, -T log(1-lambda) and Trace Test

Roots Rank EigVal Lambda-max Trace Trace-95%

3 0 0.0818 26.6264 41.2013 34.8000

2 1 0.0333 10.5567 14.5750 19.9900

1 2 0.0128 4.0183 4.0183 9.1300

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

Decomposition of Variance for Series FTBS3

Step Std Error FTBS3 FTB12 FCM7

1 0.47874179 100.000 0.000 0.000

2 0.86063935 97.772 2.116 0.112

3 1.09462944 97.074 2.808 0.118

4 1.25645195 97.202 2.682 0.117

5 1.37615925 97.065 2.518 0.417

6 1.48812665 96.120 2.855 1.025

7 1.60547581 94.152 4.401 1.447

8 1.71750515 92.209 6.000 1.791

9 1.82196997 90.851 6.974 2.175

10 1.92093202 89.760 7.692 2.548

11 2.01392368 88.902 8.306 2.792

12 2.10309100 88.180 8.909 2.911

13 2.18921891 87.498 9.505 2.997

14 2.27217662 86.921 10.000 3.079

15 2.35211841 86.451 10.394 3.155

16 2.42881809 86.046 10.722 3.232

17 2.50284549 85.696 11.002 3.302

18 2.57492782 85.384 11.258 3.358

19 2.64517901 85.100 11.494 3.406

20 2.71370940 84.846 11.704 3.450

21 2.78057990 84.620 11.891 3.489

22 2.84578756 84.416 12.057 3.526

23 2.90949666 84.232 12.208 3.560

24 2.97185890 84.063 12.347 3.591

25 3.03295481 83.906 12.475 3.618

26 3.09286395 83.762 12.594 3.644

27 3.15163700 83.628 12.703 3.668

28 3.20932402 83.505 12.805 3.691

29 3.26599021 83.390 12.899 3.712

30 3.32169102 83.282 12.986 3.731

31 3.37647710 83.182 13.069 3.749

32 3.43039249 83.088 13.146 3.766

33 3.48347293 82.999 13.218 3.782

34 3.53575532 82.916 13.287 3.798

35 3.58727546 82.838 13.351 3.812

36 3.63806599 82.763 13.411 3.825

Decomposition of Variance for Series FTB12

Step Std Error FTBS3 FTB12 FCM7

1 0.46594096 84.525 15.475 0.000

2 0.84691391 77.897 21.680 0.422

3 1.07662900 79.623 19.744 0.632

4 1.23966782 81.185 17.750 1.065

5 1.36371021 81.484 16.391 2.124

6 1.48693943 81.284 15.895 2.822

7 1.62088228 79.825 17.081 3.094

8 1.74816531 78.508 18.102 3.390

9 1.86704620 78.026 18.279 3.695

10 1.97802859 77.780 18.210 4.010

11 2.08089794 77.696 18.061 4.243

12 2.18015027 77.695 17.971 4.334

13 2.27630977 77.662 17.952 4.386

14 2.36897668 77.682 17.876 4.441

15 2.45854705 77.755 17.751 4.494

16 2.54443962 77.843 17.604 4.553

17 2.62725405 77.946 17.447 4.607

18 2.70787407 78.046 17.307 4.646

19 2.78639010 78.138 17.182 4.680

20 2.86296187 78.231 17.061 4.709

21 2.93764307 78.321 16.943 4.735

22 3.01041212 78.409 16.829 4.761

23 3.08146879 78.494 16.721 4.785

24 3.15097104 78.573 16.621 4.805

25 3.21901494 78.647 16.528 4.824

26 3.28570185 78.717 16.442 4.841

27 3.35108044 78.783 16.360 4.857

28 3.41520830 78.845 16.283 4.872

29 3.47816324 78.903 16.211 4.886

30 3.54000688 78.958 16.143 4.899

31 3.60079986 79.010 16.079 4.911

32 3.66059465 79.058 16.019 4.922

33 3.71943210 79.104 15.963 4.933

34 3.77735554 79.148 15.910 4.943

35 3.83440655 79.189 15.859 4.952

36 3.89062308 79.228 15.812 4.961

Decomposition of Variance for Series FCM7

Step Std Error FTBS3 FTB12 FCM7

1 0.31547455 45.312 33.565 21.122

2 0.56629789 43.452 35.774 20.774

3 0.71962294 48.782 29.636 21.582

4 0.84182334 50.656 24.765 24.579

5 0.95048963 50.969 21.185 27.846

6 1.05262204 52.943 19.222 27.835

7 1.15401281 54.424 18.703 26.874

8 1.24999908 55.585 17.924 26.491

9 1.34153123 57.063 16.716 26.221

10 1.42725499 58.339 15.532 26.129

11 1.50760607 59.450 14.462 26.087

12 1.58496383 60.557 13.575 25.868

13 1.65974848 61.561 12.826 25.614

14 1.73239941 62.474 12.129 25.397

15 1.80306830 63.306 11.487 25.207

16 1.87128456 64.046 10.901 25.052

17 1.93730046 64.719 10.371 24.910

18 2.00148069 65.329 9.900 24.770

19 2.06396059 65.882 9.479 24.639

20 2.12490396 66.389 9.096 24.515

21 2.18435578 66.848 8.747 24.405

22 2.24234375 67.266 8.428 24.307

23 2.29896495 67.649 8.136 24.215

24 2.35430147 68.001 7.869 24.130

25 2.40844320 68.325 7.625 24.051

26 2.46146832 68.624 7.399 23.977

27 2.51342315 68.900 7.191 23.909

28 2.56435989 69.156 6.998 23.846

29 2.61433176 69.394 6.819 23.787

30 2.66338664 69.615 6.652 23.733

31 2.71157457 69.821 6.497 23.681

32 2.75893876 70.014 6.352 23.633

33 2.80551602 70.195 6.217 23.589

34 2.85134234 70.364 6.089 23.546

35 2.89645111 70.524 5.970 23.507

36 2.94087400 70.674 5.857 23.469

VAR/System - Estimation by Least Squares

Monthly Data From 1975:07 To 2001:06

Usable Observations 312

Dependent Variable FTBS3

Mean of Dependent Variable 6.7367948718

Std Error of Dependent Variable 2.7537935220

Standard Error of Estimate 0.4913307785

Sum of Squared Residuals 70.731938624

Durbin-Watson Statistic 1.9641

Variable Coeff Std Error T-Stat Signif

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

1. FTBS3{1} 0.949849690 0.173113672 5.48686 0.00000009

2. FTBS3{2} 0.051481560 0.268026287 0.19208 0.84781531

3. FTBS3{3} -0.282233594 0.281940441 -1.00104 0.31763357

4. FTBS3{4} 0.501041691 0.275806862 1.81664 0.07029398

5. FTBS3{5} -0.623065201 0.253321054 -2.45959 0.01448731

6. FTBS3{6} 0.208500797 0.169911615 1.22711 0.22076509

7. FTB12{1} 0.476067063 0.247464217 1.92378 0.05535061

8. FTB12{2} -0.657054273 0.371246505 -1.76986 0.07779049

9. FTB12{3} 0.420523195 0.373870710 1.12478 0.26160183

10. FTB12{4} -0.644083928 0.363017481 -1.77425 0.07706014

11. FTB12{5} 0.730043756 0.351141259 2.07906 0.03848154

12. FTB12{6} -0.127395657 0.247929464 -0.51384 0.60775233

13. FCM7{1} 0.192032517 0.191984750 1.00025 0.31801545

14. FCM7{2} -0.488028875 0.296027721 -1.64859 0.10030307

15. FCM7{3} 0.448184371 0.300335289 1.49228 0.13670166

16. FCM7{4} -0.003945419 0.302960913 -0.01302 0.98961841

17. FCM7{5} 0.079191893 0.301378882 0.26277 0.79291620

18. FCM7{6} -0.273733240 0.197287068 -1.38749 0.16634767

19. Constant 0.216859820 0.120690747 1.79682 0.07339366

F-Tests, Dependent Variable FTBS3

Variable F-Statistic Signif

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

FTBS3 21.5872 0.0000000

FTB12 1.8715 0.0855076

FCM7 1.2319 0.2897412

Dependent Variable FTB12

Mean of Dependent Variable 7.4486538462

Std Error of Dependent Variable 2.9034485730

Standard Error of Estimate 0.4769099191

Sum of Squared Residuals 66.640819779

Durbin-Watson Statistic 1.9625

Variable Coeff Std Error T-Stat Signif

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

1. FTBS3{1} -0.230517263 0.168032680 -1.37186 0.17115660

2. FTBS3{2} 0.805261296 0.260159551 3.09526 0.00215653

3. FTBS3{3} -0.818231176 0.273665316 -2.98990 0.00302745

4. FTBS3{4} 0.727931362 0.267711762 2.71909 0.00693678

5. FTBS3{5} -0.863706088 0.245885926 -3.51263 0.00051369

6. FTBS3{6} 0.353386040 0.164924605 2.14271 0.03295991

7. FTB12{1} 1.482998727 0.240200991 6.17399 0.00000000

8. FTB12{2} -1.216676944 0.360350193 -3.37637 0.00083368

9. FTB12{3} 0.711994659 0.362897375 1.96197 0.05071211

10. FTB12{4} -0.607768593 0.352362696 -1.72484 0.08561120

11. FTB12{5} 0.975179282 0.340835047 2.86115 0.00452492

12. FTB12{6} -0.341483748 0.240652582 -1.41899 0.15696465

13. FCM7{1} 0.392957590 0.186349880 2.10871 0.03581834

14. FCM7{2} -0.622034655 0.287339126 -2.16481 0.03120948

15. FCM7{3} 0.694582808 0.291520264 2.38262 0.01782746

16. FCM7{4} -0.347590427 0.294068824 -1.18200 0.23816264

17. FCM7{5} 0.035785764 0.292533227 0.12233 0.90272106

18. FCM7{6} -0.157107875 0.191496572 -0.82042 0.41264290

19. Constant 0.167572496 0.117148399 1.43043 0.15365941

F-Tests, Dependent Variable FTB12

Variable F-Statistic Signif

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

FTBS3 3.3921 0.0029936

FTB12 19.6640 0.0000000

FCM7 1.8813 0.0838181

Dependent Variable FCM7

Mean of Dependent Variable 8.3178205128

Std Error of Dependent Variable 2.4713398464

Standard Error of Estimate 0.3236191070

Sum of Squared Residuals 30.685692641

Durbin-Watson Statistic 1.9852

Variable Coeff Std Error T-Stat Signif

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

1. FTBS3{1} -0.086590214 0.114022761 -0.75941 0.44821688

2. FTBS3{2} 0.641575287 0.176537745 3.63421 0.00032919

3. FTBS3{3} -0.620815798 0.185702418 -3.34307 0.00093625

4. FTBS3{4} 0.463264389 0.181662485 2.55014 0.01127699

5. FTBS3{5} -0.642211692 0.166852021 -3.84899 0.00014564

6. FTBS3{6} 0.359067373 0.111913699 3.20843 0.00148259

7. FTB12{1} 0.078120663 0.162994367 0.47928 0.63209381

8. FTB12{2} -0.607883255 0.244524601 -2.48598 0.01347685

9. FTB12{3} 0.294169849 0.246253055 1.19458 0.23321590

10. FTB12{4} -0.133321061 0.239104486 -0.55758 0.57755358

11. FTB12{5} 0.702639074 0.231282113 3.03802 0.00259594

12. FTB12{6} -0.446105406 0.163300805 -2.73180 0.00668116

13. FCM7{1} 1.470147847 0.126452354 11.62610 0.00000000

14. FCM7{2} -0.712357751 0.194981122 -3.65347 0.00030645

15. FCM7{3} 0.704831977 0.197818338 3.56303 0.00042779

16. FCM7{4} -0.467556535 0.199547727 -2.34308 0.01979403

17. FCM7{5} -0.088610150 0.198505709 -0.44639 0.65564824

18. FCM7{6} 0.089477242 0.129944769 0.68858 0.49163307

19. Constant 0.097078520 0.079493965 1.22121 0.22298971

F-Tests, Dependent Variable FCM7

Variable F-Statistic Signif

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

FTBS3 5.7206 0.0000121

FTB12 4.0263 0.0006814

FCM7 212.3250 0.0000000

Decomposition of Variance for Series FTBS3

Step Std Error FTBS3 FTB12 FCM7

1 0.47874179 100.000 0.000 0.000

2 0.85683563 97.855 2.039 0.106

3 1.08410680 97.293 2.594 0.114

4 1.23825037 97.494 2.386 0.120

5 1.34957665 97.373 2.167 0.460

6 1.45145169 96.486 2.379 1.135

7 1.55616864 94.719 3.711 1.570

8 1.65315259 93.093 5.036 1.871

9 1.74054038 92.083 5.747 2.170

10 1.82041509 91.338 6.233 2.430

11 1.89289892 90.805 6.644 2.551

12 1.96051244 90.377 7.062 2.561

13 2.02356589 89.976 7.478 2.546

14 2.08173752 89.671 7.800 2.530

15 2.13542251 89.451 8.041 2.509

16 2.18471515 89.275 8.238 2.488

17 2.23038050 89.131 8.408 2.461

18 2.27306570 89.003 8.572 2.425

19 2.31280410 88.888 8.725 2.386

20 2.34978204 88.790 8.864 2.347

21 2.38419364 88.703 8.990 2.307

22 2.41617403 88.625 9.108 2.267

23 2.44596834 88.552 9.220 2.228

24 2.47375431 88.482 9.329 2.190

25 2.49964794 88.414 9.433 2.153

26 2.52377954 88.348 9.534 2.118

27 2.54625894 88.285 9.631 2.084

28 2.56719750 88.223 9.725 2.053

29 2.58670654 88.161 9.816 2.023

30 2.60487734 88.100 9.905 1.995

31 2.62179606 88.039 9.991 1.970

32 2.63754302 87.979 10.074 1.946

33 2.65219134 87.920 10.155 1.925

34 2.66581321 87.861 10.234 1.906

35 2.67847541 87.802 10.310 1.888

36 2.69023897 87.744 10.383 1.873

Decomposition of Variance for Series FTB12

Step Std Error FTBS3 FTB12 FCM7

1 0.46594096 84.525 15.475 0.000

2 0.84175537 78.133 21.408 0.458

3 1.06326430 80.047 19.212 0.741

4 1.21758832 81.661 17.025 1.314

5 1.33230611 81.863 15.518 2.619

6 1.44419802 81.617 14.869 3.514

7 1.56289301 80.303 15.833 3.864

8 1.67100255 79.253 16.573 4.174

9 1.76814249 79.035 16.520 4.445

10 1.85555904 79.031 16.281 4.687

11 1.93387648 79.172 16.011 4.817

12 2.00749440 79.383 15.818 4.798

13 2.07619072 79.559 15.695 4.746

14 2.13956348 79.777 15.526 4.697

15 2.19836693 80.025 15.330 4.644

16 2.25245256 80.266 15.140 4.595

17 2.30267452 80.499 14.963 4.538

18 2.34976263 80.712 14.817 4.470

19 2.39370587 80.905 14.694 4.401

20 2.43474798 81.086 14.583 4.331

21 2.47306806 81.252 14.487 4.261

22 2.50879634 81.402 14.404 4.193

23 2.54221242 81.538 14.336 4.126

24 2.57348316 81.658 14.281 4.060

25 2.60273024 81.765 14.239 3.997

26 2.63009619 81.859 14.206 3.936

27 2.65568354 81.941 14.182 3.877

28 2.67960906 82.014 14.165 3.821

29 2.70199036 82.076 14.156 3.768

30 2.72291833 82.129 14.154 3.717

31 2.74248345 82.174 14.157 3.669

32 2.76076877 82.212 14.164 3.624

33 2.77784886 82.242 14.176 3.582

34 2.79379929 82.267 14.191 3.542

35 2.80868927 82.285 14.209 3.506

36 2.82258223 82.299 14.230 3.471

Decomposition of Variance for Series FCM7

Step Std Error FTBS3 FTB12 FCM7

1 0.31547455 45.312 33.565 21.122

2 0.56356422 43.337 35.738 20.924

3 0.71249839 48.522 29.524 21.955

4 0.82987740 50.194 24.622 25.184

5 0.93261483 50.296 21.080 28.625

6 1.02718578 52.108 19.219 28.673

7 1.11941584 53.541 18.811 27.648

8 1.20418019 54.780 18.098 27.123

9 1.28290896 56.404 16.952 26.644

10 1.35481434 57.868 15.850 26.282

11 1.42083928 59.207 14.874 25.919

12 1.48339537 60.558 14.078 25.364

13 1.54233880 61.797 13.414 24.788

14 1.59797434 62.941 12.801 24.258

15 1.65068194 63.999 12.245 23.756

16 1.70022575 64.962 11.752 23.286

17 1.74704653 65.852 11.320 22.828

18 1.79149072 66.671 10.950 22.379

19 1.83361667 67.425 10.628 21.947

20 1.87360247 68.122 10.345 21.533

21 1.91153033 68.763 10.098 21.139

22 1.94749279 69.354 9.882 20.764

23 1.98164640 69.901 9.695 20.404

24 2.01408187 70.406 9.535 20.059

25 2.04489113 70.874 9.397 19.729

26 2.07416379 71.307 9.279 19.414

27 2.10196185 71.708 9.178 19.113

28 2.12836293 72.081 9.093 18.826

29 2.15344077 72.426 9.022 18.552

30 2.17725501 72.746 8.963 18.291

31 2.19986729 73.044 8.916 18.041

32 2.22133291 73.320 8.877 17.803

33 2.24170279 73.576 8.848 17.576

34 2.26102871 73.814 8.826 17.360

35 2.27935833 74.035 8.811 17.154

36 2.29673665 74.240 8.802 16.958

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 0.492145429

Sum of Squared Residuals 71.451101355

Durbin-Watson Statistic 1.9599

Variable Coeff Std Error T-Stat Signif

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

1. D_FTBS3{1} 0.138523739 0.184159956 0.75219 0.45253504

2. D_FTBS3{2} 0.184842417 0.184565657 1.00150 0.31740614

3. D_FTBS3{3} -0.094836473 0.191898528 -0.49420 0.62153192

4. D_FTBS3{4} 0.404134436 0.174672569 2.31367 0.02137319

5. D_FTBS3{5} -0.212580681 0.170151901 -1.24936 0.21252444

6. D_FTB12{1} 0.303745337 0.261895294 1.15980 0.24706935

7. D_FTB12{2} -0.349298761 0.255832826 -1.36534 0.17318648

8. D_FTB12{3} 0.071058136 0.260675425 0.27259 0.78535727

9. D_FTB12{4} -0.570728462 0.246832365 -2.31221 0.02145449

10. D_FTB12{5} 0.155224089 0.247810227 0.62638 0.53154821

11. D_FCM7{1} 0.213056337 0.192266177 1.10813 0.26870739

12. D_FCM7{2} -0.281287770 0.195598634 -1.43809 0.15146946

13. D_FCM7{3} 0.164867481 0.196647981 0.83839 0.40249113

14. D_FCM7{4} 0.163747663 0.200026123 0.81863 0.41365754

15. D_FCM7{5} 0.235929471 0.196368339 1.20146 0.23053498

16. EC1{1} 0.062282194 0.027862248 2.23536 0.02614182

17. EC2{1} -0.013555754 0.027862248 -0.48653 0.62695437

Dependent Variable FTB12

Mean of Dependent Variable -0.008685897

Std Error of Dependent Variable 0.561943004

Standard Error of Estimate 0.477777924

Sum of Squared Residuals 67.340164614

Durbin-Watson Statistic 1.9585

Variable Coeff Std Error T-Stat Signif

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

1. D_FTBS3{1} -0.210312648 0.178783661 -1.17635 0.24040233

2. D_FTBS3{2} 0.589857400 0.179177518 3.29203 0.00111560

3. D_FTBS3{3} -0.225854518 0.186296316 -1.21234 0.22635220

4. D_FTBS3{4} 0.500034792 0.169573244 2.94878 0.00344544

5. D_FTBS3{5} -0.357409317 0.165184551 -2.16370 0.03129021

6. D_FTB12{1} 0.504176107 0.254249623 1.98300 0.04829503

7. D_FTB12{2} -0.708546302 0.248364141 -2.85285 0.00463926

8. D_FTB12{3} 0.003284366 0.253065366 0.01298 0.98965385

9. D_FTB12{4} -0.602218772 0.239626435 -2.51316 0.01249861

10. D_FTB12{5} 0.368926069 0.240575750 1.53351 0.12622120

11. D_FCM7{1} 0.371440134 0.186653232 1.99000 0.04751367

12. D_FCM7{2} -0.256822132 0.189888402 -1.35249 0.17725447

13. D_FCM7{3} 0.435759710 0.190907116 2.28257 0.02316702

14. D_FCM7{4} 0.090955680 0.194186637 0.46839 0.63984940

15. D_FCM7{5} 0.119828622 0.190635637 0.62857 0.53011430

16. EC1{1} 0.018519738 0.027048849 0.68468 0.49408502

17. EC2{1} -0.035596545 0.027048849 -1.31601 0.18919251

Dependent Variable FCM7

Mean of Dependent Variable -0.008173077

Std Error of Dependent Variable 0.382826930

Standard Error of Estimate 0.324238836

Sum of Squared Residuals 31.013592693

Durbin-Watson Statistic 1.9815

Variable Coeff Std Error T-Stat Signif

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

1. D_FTBS3{1} -0.204762861 0.121329604 -1.68766 0.09253385

2. D_FTBS3{2} 0.433326250 0.121596891 3.56363 0.00042642

3. D_FTBS3{3} -0.185764514 0.126427986 -1.46933 0.14280873

4. D_FTBS3{4} 0.276101604 0.115079054 2.39923 0.01704976

5. D_FTBS3{5} -0.361822268 0.112100714 -3.22765 0.00138871

6. D_FTB12{1} 0.207907385 0.172543765 1.20495 0.22918661

7. D_FTB12{2} -0.397268045 0.168549646 -2.35698 0.01907763

8. D_FTB12{3} -0.103210487 0.171740082 -0.60097 0.54832202

9. D_FTB12{4} -0.234980302 0.162619896 -1.44497 0.14952866

10. D_FTB12{5} 0.464896233 0.163264138 2.84751 0.00471570

11. D_FCM7{1} 0.457148612 0.126670203 3.60897 0.00036103

12. D_FCM7{2} -0.259473428 0.128865717 -2.01352 0.04496807

13. D_FCM7{3} 0.443988410 0.129557055 3.42697 0.00069710

14. D_FCM7{4} -0.021660170 0.131782667 -0.16436 0.86955809

15. D_FCM7{5} -0.115003800 0.129372819 -0.88893 0.37476323

16. EC1{1} -0.034984890 0.018356409 -1.90587 0.05763919

17. EC2{1} -0.011029684 0.018356409 -0.60086 0.54839272

Decomposition of Variance for Series FTBS3

Step Std Error FTBS3 FTB12 FCM7

1 0.47854983 100.000 0.000 0.000

2 0.85969982 97.828 2.048 0.124

3 1.09269978 97.209 2.637 0.153

4 1.25371447 97.374 2.445 0.181

5 1.37319529 97.158 2.235 0.607

6 1.48464269 96.140 2.467 1.393

7 1.59981922 94.246 3.791 1.963

8 1.70793799 92.432 5.115 2.454

9 1.80745104 91.163 5.834 3.003

10 1.90046141 90.138 6.316 3.545

11 1.98675427 89.351 6.713 3.936

12 2.06877811 88.718 7.105 4.177

13 2.14710794 88.134 7.484 4.383

14 2.22144487 87.651 7.759 4.590

15 2.29210471 87.264 7.942 4.794

16 2.35915914 86.927 8.071 5.002

17 2.42327465 86.634 8.166 5.200

18 2.48502500 86.376 8.245 5.379

19 2.54445596 86.144 8.307 5.549

20 2.60174852 85.938 8.349 5.713

21 2.65705527 85.754 8.374 5.872

22 2.71047057 85.586 8.387 6.027

23 2.76220435 85.433 8.391 6.175

24 2.81240454 85.291 8.390 6.319

25 2.86116314 85.158 8.383 6.459

26 2.90858619 85.034 8.371 6.595

27 2.95475441 84.917 8.355 6.728

28 2.99974817 84.807 8.335 6.858

29 3.04365089 84.702 8.313 6.985

30 3.08652917 84.602 8.289 7.109

31 3.12844519 84.507 8.264 7.230

32 3.16945529 84.415 8.237 7.348

33 3.20960899 84.328 8.208 7.464

34 3.24895543 84.243 8.179 7.578

35 3.28753927 84.162 8.149 7.689

36 3.32540071 84.083 8.119 7.798

Decomposition of Variance for Series FTB12

Step Std Error FTBS3 FTB12 FCM7

1 0.46457923 84.689 15.311 0.000

2 0.84282455 78.298 21.210 0.492

3 1.07030014 80.166 19.003 0.831

4 1.23220604 81.724 16.794 1.482

5 1.35583936 81.843 15.257 2.899

6 1.47776908 81.554 14.561 3.885

7 1.60726262 80.216 15.431 4.353

8 1.72724915 79.041 16.100 4.859

9 1.83743794 78.623 15.997 5.380

10 1.93902084 78.387 15.701 5.912

11 2.03240381 78.298 15.366 6.336

12 2.12180161 78.321 15.103 6.576

13 2.20720247 78.330 14.901 6.769

14 2.28827713 78.385 14.645 6.970

15 2.36570552 78.479 14.354 7.167

16 2.43928924 78.571 14.060 7.369

17 2.50977952 78.670 13.772 7.558

18 2.57784776 78.765 13.510 7.725

19 2.64349154 78.852 13.264 7.884

20 2.70694342 78.938 13.027 8.035

21 2.76834682 79.018 12.800 8.182

22 2.82778650 79.091 12.584 8.324

23 2.88549785 79.160 12.380 8.460

24 2.94161785 79.221 12.189 8.591

25 2.99624356 79.276 12.007 8.717

26 3.04948945 79.325 11.835 8.840

27 3.10142711 79.370 11.671 8.959

28 3.15214054 79.409 11.515 9.075

29 3.20171651 79.445 11.367 9.188

30 3.25021898 79.476 11.226 9.297

31 3.29771274 79.504 11.092 9.404

32 3.34425472 79.529 10.963 9.508

33 3.38989365 79.551 10.840 9.610

34 3.43467991 79.570 10.722 9.709

35 3.47865846 79.586 10.608 9.805

36 3.52186979 79.600 10.500 9.900

Decomposition of Variance for Series FCM7

Step Std Error FTBS3 FTB12 FCM7

1 0.31528168 45.271 33.927 20.802

2 0.56573636 43.423 35.917 20.660

3 0.71927237 48.718 29.587 21.696

4 0.84222325 50.511 24.617 24.871

5 0.95142750 50.752 21.026 28.222

6 1.05348608 52.696 19.084 28.219

7 1.15430822 54.178 18.566 27.256

8 1.24900555 55.330 17.772 26.898

9 1.33884264 56.786 16.560 26.655

10 1.42294394 58.039 15.389 26.572

11 1.50223099 59.147 14.342 26.510

12 1.57897959 60.272 13.480 26.248

13 1.65303215 61.296 12.753 25.952

14 1.72471285 62.230 12.074 25.696

15 1.79438043 63.086 11.451 25.463

16 1.86173495 63.856 10.889 25.255

17 1.92716045 64.566 10.387 25.047

18 1.99099297 65.218 9.945 24.837

19 2.05327641 65.814 9.552 24.634

20 2.11416422 66.365 9.196 24.439

21 2.17371587 66.871 8.874 24.255

22 2.23198683 67.337 8.583 24.079

23 2.28909681 67.771 8.321 23.908

24 2.34511349 68.172 8.084 23.743

25 2.40010622 68.546 7.870 23.584

26 2.45413872 68.895 7.674 23.431

27 2.50724727 69.221 7.495 23.284

28 2.55948316 69.526 7.332 23.142

29 2.61089448 69.813 7.183 23.004

30 2.66151753 70.083 7.046 22.871

31 2.71139152 70.337 6.920 22.743

32 2.76054905 70.577 6.804 22.619

33 2.80901839 70.803 6.697 22.499

34 2.85682920 71.018 6.599 22.383

35 2.90400765 71.222 6.507 22.271

36 2.95057795 71.416 6.422 22.162