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TomDoan wrote: ↑Tue Feb 27, 2024 12:14 pm A VECM will have a certain number forcibly on the unit circle. It generally only makes sense if you are unsure of the rank of cointegration, as you might see roots that are just barely inside the unit circle, which might indicate that, whether they are "statistically" significantly different from unit roots, they may not be practically different.
Code: Select all
compute companion=%modelcompanion(varmodel)
eigen(cvalues=cv) companionCode: Select all
*
* 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)
estimateCode: Select all
inverse AR roots:
Real Imag Modulus Period
1.00000 0.00000 1.00000
0.95722 -0.00000 0.95722
-0.47818 0.58345 0.75437 2.78342
-0.47818 -0.58345 0.75437
0.73863 0.05914 0.74099 78.64495
0.73863 -0.05914 0.74099
0.34590 -0.64913 0.73554
0.34590 0.64913 0.73554 5.81133
-0.21104 -0.68605 0.71778
-0.21104 0.68605 0.71778 3.36138
0.53261 0.41448 0.67489 9.50113
0.53261 -0.41448 0.67489
0.03510 -0.58437 0.58542
0.03510 0.58437 0.58542 4.15884
-0.50067 -0.00000 0.50067 2.00000
0.42779 0.00000 0.42779
0.18694 -0.00000 0.18694
-0.08272 0.00000 0.08272 2.00000Code: Select all
*===============================
* Equivalent Fit and OOS forecasts
* --------------------------------
comp nsteps = 10
* VAR
* ---
system(model=ratemodel)
variables ftbs3 ftb12 fcm7
lags 1 to 6
det constant
end(system)
estimate(print,noftests) 1975:07 2001:06-nsteps
*
comp N = %NREGSYSTEM
comp aic = %nobs*%logdet + 2*N
comp sbc = %nobs*%logdet + N*log(%nobs)
dis 'aic = ' aic 'sbc = ' sbc
*
* characteristic roots
compute companion=%modelcompanion(ratemodel)
eigen(cvalues=cv1) companion
disp cv1
*
forecast(model=ratemodel,results=f_ratemodel,from=2001:6-nsteps+1,steps=nsteps)
prin / f_ratemodel
@uforeErrors ftbs3 f_ratemodel(1)
@uforeErrors ftb12 f_ratemodel(2)
@uforeErrors fcm7 f_ratemodel(3)
*===============================
* VECM
* ----
@johmle(lags=6,det=rc,vectors=cvectors,print) 1975:07 2001:06-nsteps
# 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(print,noftests) 1975:07 2001:06-nsteps
*
comp N = %NREGSYSTEM
comp aic = %nobs*%logdet + 2*N
comp sbc = %nobs*%logdet + N*log(%nobs)
dis 'aic = ' aic 'sbc = ' sbc
*
* characteristic roots
compute companion=%modelcompanion(%modelsubstect(ect2model))
eigen(cvalues=cv2) companion
disp cv2
*
forecast(model=ect2model,results=f_ect2model,from=2001:6-nsteps+1,steps=nsteps)
prin / f_ect2model
@uforeErrors ftbs3 f_ect2model(1)
@uforeErrors ftb12 f_ect2model(2)
@uforeErrors fcm7 f_ect2model(3)TomDoan wrote: ↑Tue Mar 05, 2024 8:34 pm A full set of ECT's in a 3 variable model has 3 not 2. It's of no practical interest since it's identical to the simpler VAR in levels. If you add a 3rd component, you'll see that the log likelihoods match. You won't be able to match the AIC and SBC because the two step VECM's have extra parameters embedded in the first step that aren't included in %NREGSYSTEM.
TomDoan wrote: ↑Wed Mar 06, 2024 9:31 pm That's the correct value for N, but not for the correct reasons. (The PI matrix is decomposed as alpha beta' where there are redundant parameters in those). %NREGSYSTEM will includes the alpha's in the VECM, but not the beta's. With 3 ECT's (and with restricted constant), beta' is dimension 3 x 4. But there are 3 x 3 restrictions to identify the alpha beta' combinations, so it adds just an extra 3 parameters.
Note, however, that the logic behind AIC and SBC don't apply to choosing rank in a VECM's because of the asymptotic growth rate of unit root processes.
Yes, there are two tests to determine the rank of PI i.e. the number of cointegrating vectors, and the test statistics are based around characteristic roots i.e. eigenvalues (in order) as here https://estima.com/ratshelp/index.html? ... edure.html:TomDoan wrote: ↑Fri Mar 08, 2024 9:46 am The beta's are the coefficients that determine the cointegrating vectors. (PI = alpha beta'). And you have to allow for the fact that they need to be normalized, and so aren't all free. But it really doesn't matter because you can't use the AIC or SBC to choose the rank of the cointegrating space.
Yes, that's useful as it's straightforward to be in levels and not experience the problems here https://estima.com/forum/viewtopic.php?t=1724
That's exactly what it says in does in the description of the ECT instruction.ac_1 wrote: ↑Mon Mar 18, 2024 2:59 am
Questions
(i) I have noticed if I have, e.g.
equation(coeffs=%xcol(cvectors,1)) D_ect1 *
# ftbs3 ftb12 fcm7 constant
equation(coeffs=%xcol(cvectors,2)) D_ect2 *
# ftbs3 ftb12 fcm7 constant
they will appear as just EC1{1} EC2{1} in the VECM after ESTIMATE regardless of the D_ in front. Why?
Presumably, if after the dynamic (D_) forecasts, within the same RPF file, I am generating static 1-step ahead VECM static forecasts and defining the ECT's as S_ECT1, S_ECT2 using EQUATION, and ESTIMATE, there will be an overwrite, and the latest ECT's i.e. S_ECT1 S_ECT2 will be used?
DET=RC and DET=RTREND are the only ones that are more complicated, and that is only because they anticipate a restriction on the intercept (or trend). The adjusted "PI" matrix in rearranging the unrestricted VAR simply adds a column of the estimates on the constant (DET=RC) and trend term (DET=RTREND).ac_1 wrote: ↑Mon Mar 18, 2024 2:59 am (ii) To aid my understanding, given a VAR-in-levels with DET=NONE or DET=CONSTANT I can manually calculate the PI matrix
PI = - (I - GAMMA(1) - GAMMA(2) - ... - GAMMA(p))
where
I is the identity matrix
GAMMA's are the matrices of parameters up to the AR(pth) lag.
How to 'by-hand' calculate PI including the deterministic terms: DET=RC, DET=TREND, and SEASONAL?
For a VAR estimated in levels.
You are misunderstanding that. You have to normalize (somehow) in order to fully estimate the model. It's just that the normalization washes out of forecasts and anything related to that.ac_1 wrote: ↑Mon Mar 18, 2024 2:59 am (iv) I'll accept that I do not need to normalize the cointegrating vectors prior to forecasting, but I do not understand the normalization as in https://www.estima.com/ratshelp/index.h ... edure.html, enders4p389.rpf and Enders (1996) RATS Handbook for Econometric Time Series Chapter 6? Why normalize?
alpha depends upon beta. Different normalizations of beta leads to different alphas and different interpretations of alpha. (It's also possible to normalize alpha (the loadings) rather than beta, though that is less common).ac_1 wrote: ↑Mon Mar 18, 2024 2:59 am (vi) Should the values of PI (long-run matrix of coefficients) be interpreted or it's factor matrices:
- alpha (speed of adjustment to equilibrium coefficients) and
- beta (long-run matrix of coefficients).
What do the sign and magnitude of the coefficients mean?
Further, how to interpret the PI*Y(t-1) term in the VECM UG-247 eqn(31) ?