VECMCAUSE.RPF

VECMCAUSE.RPF is an example of testing for causality in a VECM. This has separate tests for causality and long-run causality.

The data are 3 and 6 month US TBill yields. Because these don't trend, the appropriate choice for the Johansen test (done with @JOHMLE) is DET=RC (constant restricted to the cointegrating vector):

@johmle(lags=nlags,det=rc,cv=cv)

# tb3mo tb6mo

Based upon the output, we conclude that the series are cointegrated. The further analysis is done with the estimated cointegrating vector:

set z = cv(1)*tb3mo+cv(2)*tb6mo+cv(3)

The CV(3) is there because of the use of DET=RC. The constant is included there and not in the error correction representation. Error correction regressions are estimated for each of the series, regressing the difference on lagged "Z" and the lags of both differenced series. This does the regression for the 3 month rate:

set d3mo = tb3mo-tb3mo{1}

set d6mo = tb6mo-tb6mo{1}

linreg d3mo

# z{1} d3mo{1 to nlags-1} d6mo{1 to nlags-1}

(Note: there is no CONSTANT because the cointegrating model is DET=RC). The overall causality test for the 6 month to 3 month is testing jointly for zeros on Z{1} and the lags of D6MO:

exclude(title="Test for 6MO causing 3MO")

# z{1} d6mo{1 to nlags-1}

and the test for long-run causality looks at Z{1} only:

exclude(title="Test for 6MO long-run causing 3MO")

# z{1}

The results show long-run causality from 6MO to 3MO

The reverse direction tests are done with the analogous:

linreg d6mo

# z{1} d3mo{1 to nlags-1} d6mo{1 to nlags-1}

exclude(title="Test for 3MO causing 6MO")

# z{1} d3mo{1 to nlags-1}

exclude(title="Test for 3MO long-run causing 6MO")

# z{1}

The results here show causality but not long-run causality from 3MO to 6MO.

Full Program

open data w-tb3n6ms.txt
calendar(w) 1958:12:12
data(format=free,top=2,org=columns) 1958:12:12 2004:8:6 tb3mo tb6mo
*
* Test for cointegration. (We should test for unit roots first, but it's
* fairly well-known that these are non-stationary).
*
compute nlags=10
*
@johmle(lags=nlags,det=rc,cv=cv)
# tb3mo tb6mo
*
* We conclude that the two series are cointegrated. The rest of this
* uses the estimated cointegrating vector, though a theoretical value
* (in this case (1,-1)) could also be used.
*
set z = cv(1)*tb3mo+cv(2)*tb6mo+cv(3)
*
set d3mo = tb3mo-tb3mo{1}
set d6mo = tb6mo-tb6mo{1}
*
* 6 causing 3
*
linreg d3mo
# z{1} d3mo{1 to nlags-1} d6mo{1 to nlags-1}
exclude(title="Test for 6MO causing 3MO")
# z{1} d6mo{1 to nlags-1}
exclude(title="Test for 6MO long-run causing 3MO")
# z{1}
*
* 3 causing 6
*
linreg d6mo
# z{1} d3mo{1 to nlags-1} d6mo{1 to nlags-1}
exclude(title="Test for 3MO causing 6MO")
# z{1} d3mo{1 to nlags-1}
exclude(title="Test for 3MO long-run causing 6MO")
# z{1}

Output

Likelihood Based Analysis of Cointegration

Variables: TB3MO TB6MO

Estimated from 1959:02:20 to 2004:08:06

Data Points 2373 Lags 10 with Constant restricted to Cointegrating Vector

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

Roots Rank EigVal Lambda-max Trace Trace-95%

2 0 0.0259 62.3562 67.0737 19.9900

1 1 0.0020 4.7175 4.7175 9.1300

Cointegrating Vector for Largest Eigenvalue

TB3MO TB6MO Constant

5.319718 -5.375225 1.128813

Linear Regression - Estimation by Least Squares

Dependent Variable D3MO

Weekly Data From 1959:02:20 To 2004:08:06

Usable Observations 2373

Degrees of Freedom 2354

Centered R^2 0.1334503

R-Bar^2 0.1268242

Uncentered R^2 0.1334558

Mean of Dependent Variable -0.000535188

Std Error of Dependent Variable 0.212736745

Standard Error of Estimate 0.198789464

Sum of Squared Residuals 93.023608385

Log Likelihood 475.9999

Durbin-Watson Statistic 1.9979

Variable Coeff Std Error T-Stat Signif

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

1. Z{1} -0.017101378 0.004080792 -4.19070 0.00002884

2. D3MO{1} 0.052434528 0.050425930 1.03983 0.29852445

3. D3MO{2} -0.181149552 0.050481690 -3.58842 0.00033948

4. D3MO{3} -0.068750983 0.051100938 -1.34540 0.17862706

5. D3MO{4} 0.097109180 0.050711964 1.91492 0.05562422

6. D3MO{5} -0.067968381 0.050557673 -1.34437 0.17895727

7. D3MO{6} -0.112841890 0.050401548 -2.23886 0.02525828

8. D3MO{7} 0.098750478 0.050231057 1.96592 0.04942467

9. D3MO{8} -0.067073655 0.048796087 -1.37457 0.16939552

10. D3MO{9} 0.106837695 0.048705265 2.19356 0.02836486

11. D6MO{1} 0.261721396 0.055864536 4.68493 0.00000296

12. D6MO{2} 0.209471878 0.056306025 3.72024 0.00020369

13. D6MO{3} 0.096165935 0.056789132 1.69339 0.09051428

14. D6MO{4} -0.060031557 0.056530885 -1.06192 0.28837869

15. D6MO{5} 0.097089164 0.056315322 1.72403 0.08483417

16. D6MO{6} 0.165910068 0.055984745 2.96349 0.00307226

17. D6MO{7} -0.225042143 0.055952339 -4.02203 0.00005952

18. D6MO{8} 0.104141828 0.054811972 1.89998 0.05755741

19. D6MO{9} -0.157050531 0.054790232 -2.86640 0.00418862

Test for 6MO causing 3MO

Null Hypothesis : The Following Coefficients Are Zero

Z Lag(s) 1

D6MO Lag(s) 1 to 9

F(10,2354)= 12.75506 with Significance Level 0.00000000

Test for 6MO long-run causing 3MO

Null Hypothesis : The Following Coefficients Are Zero

Z Lag(s) 1

t(2354)= -4.190700 or F(1,2354)= 17.561969 with Significance Level 0.00002884

Linear Regression - Estimation by Least Squares

Dependent Variable D6MO

Weekly Data From 1959:02:20 To 2004:08:06

Usable Observations 2373

Degrees of Freedom 2354

Centered R^2 0.1141866

R-Bar^2 0.1074132

Uncentered R^2 0.1141963

Mean of Dependent Variable -0.000627897

Std Error of Dependent Variable 0.189616152

Standard Error of Estimate 0.179143300

Sum of Squared Residuals 75.545325958

Log Likelihood 722.9349

Durbin-Watson Statistic 1.9983

Variable Coeff Std Error T-Stat Signif

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

1. Z{1} -0.003605806 0.003677492 -0.98051 0.32693671

2. D3MO{1} -0.028232969 0.045442386 -0.62129 0.53446798

3. D3MO{2} -0.009897222 0.045492635 -0.21756 0.82779347

4. D3MO{3} -0.043994419 0.046050684 -0.95535 0.33949994

5. D3MO{4} 0.127715368 0.045700151 2.79464 0.00523789

6. D3MO{5} -0.095529392 0.045561109 -2.09673 0.03612386

7. D3MO{6} -0.058179627 0.045420414 -1.28091 0.20035019

8. D3MO{7} 0.052914689 0.045266772 1.16895 0.24254128

9. D3MO{8} -0.072366890 0.043973619 -1.64569 0.09996135

10. D3MO{9} 0.064662009 0.043891773 1.47321 0.14082682

11. D6MO{1} 0.307452051 0.050343500 6.10709 0.00000000

12. D6MO{2} 0.054514688 0.050741357 1.07436 0.28276968

13. D6MO{3} 0.076150520 0.051176719 1.48799 0.13688709

14. D6MO{4} -0.079513381 0.050943994 -1.56080 0.11870539

15. D6MO{5} 0.079280714 0.050749736 1.56219 0.11837773

16. D6MO{6} 0.092612714 0.050451829 1.83567 0.06653312

17. D6MO{7} -0.157359766 0.050422625 -3.12082 0.00182538

18. D6MO{8} 0.108200135 0.049394960 2.19051 0.02858506

19. D6MO{9} -0.110082952 0.049375368 -2.22951 0.02587389

Test for 3MO causing 6MO

Null Hypothesis : The Following Coefficients Are Zero

Z Lag(s) 1

D3MO Lag(s) 1 to 9

F(10,2354)= 2.54456 with Significance Level 0.00471672

Test for 3MO long-run causing 6MO

Null Hypothesis : The Following Coefficients Are Zero

Z Lag(s) 1

t(2354)= -0.980507 or F(1,2354)= 0.961394 with Significance Level 0.32693671