Estimation of VAR(1)-GARCH(1,1) model

[TomDoan]

You can't print a FRML, but you can create a series with its values:

set r1 = e_r1
set r2 = e_r2
print / r1 r2

[irfanawan]

Dear Tom,
I wants to estimate the VAR(1)-GARCH(1,1) model with following command

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system(model=V1)
variables rkse  rwti
lags 1 
det constant
end(system)
*estiamte

garch(model=V1, p=1,q=1,mv=cc,variance=varma,mvhseries=hhkse_wti,stdresids=reskse_wti,pmethod=simplex,piteration=13, robust,method=bfgh)

the number of observations are 359. The model did not converge with different values of pre iterations and I try this with BHHH and BFGH methods.

How to resolve this problem of estimation? Please help me in resolving this issue of convergence.

Regards, Irfan

[TomDoan]

Have you looked at your data? The RKSE is basically dead flat for almost 20% of the data set (in the middle no less). It's not surprising that you have a hard time fitting a multivariate GARCH model to the pair.

[irfanawan]

Dear Tom Hello again.
I write the codes for EGARCH model of Nelson(1991) given below.

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nonlin c1 c2 c3 c4 c5
frml e = dollar(t)-c1
set h = .01
frml var = c2+c3*(abs(e(t-1)/h(t-1))-sqrt(2/%pi))+c4*log(h(t-1))+c5*(e(t-1)/h(t-1))

frml gln = h=exp(var(t)), %logdensity(h, e)
smpl 3 3388
compute c1= .17, c2 = 0.13, c3 = 0.8, c4 = .1, c5 = .0003
maximize(method=bfgh,iterations=500,rob) gln

Then I estimate the model and the results are

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MAXIMIZE - Estimation by BFGS
Convergence in    41 Iterations. Final criterion was  0.0000026 <=  0.0000100
With Heteroscedasticity/Misspecification Adjusted Standard Errors
Usable Observations                      3386
Function Value                     -2308.0730

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  C1                           -0.001845425  0.007264174     -0.25404  0.79946099
2.  C2                            0.002808800  0.002303969      1.21911  0.22280117
3.  C3                            0.035622732  0.003410383     10.44537  0.00000000
4.  C4                            1.019540332  0.002793323    364.99189  0.00000000
5.  C5                            0.010143690  0.002605036      3.89388  0.00009865

Again I estimate the same model by using built-in command of RATS

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GARCH(P=1,Q=1,EXP,ASYMMETRIC,rob) / DOLLAR

the results are as under

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GARCH Model - Estimation by BFGS
Convergence in    26 Iterations. Final criterion was  0.0000000 <=  0.0000100
With Heteroscedasticity/Misspecification Adjusted Standard Errors
Dependent Variable DOLLAR
Usable Observations                      3386
Log Likelihood                     -2297.7150

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  Mean(DOLLAR)        -0.003623162  0.007336290     -0.49387  0.62139904
2.  C                            -0.059493771  0.008272032     -7.19216  0.00000000
3.  A                             0.073091975  0.009842794      7.42594  0.00000000
4.  B                             0.995051320  0.001739255    572.11338  0.00000000
5.  D                             0.016531933  0.006827503      2.42137  0.01546200

I have estimated the same model using Eviews for same data the results are identical with the built-in command results of RATS. Why the results are so much different with each other? If you look at the likelihood value they are almost identical. Is there any mistake in my coding? What should I do to get identical results?
The data file is attached with this post.

[TomDoan]

You're dividing e by the variance, not the standard deviation. (You also are using an unrealistically small pre-sample value for h, though that will have only a minor effect on a data set this large).

[irfanawan]

Dear Tom,
Thanks for your reply and I got the point and restimate the model got almost identical results.
Now I have another problem; I write the codes of Bivariate EGARCH model (Koutmos). When I estimate the model with my codes and built-in codes the results are not identical. Please check my codes. I shall be very much pleased with your guidance. I have tried different per-iterations but still discrepancy in results...

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open data d-spcscointc.txt
data(format=free,org=columns,top=2) 1 2275 sp500 cisco intel
set r1 = sp500
set r2 = cisco
*

*
stats(noprint) r1
set h1 = %variance
stats(noprint) r2
set h2 = %variance
*
nonlin a10 a20 a11 a12 a21 a22 c10 c20 c11 c12 c21 c22  d1 d2 d11  d22 rho

frml e_r1 = r1(t)-a10-a11*r1(t-1)-a12*r2(t-1)      
frml e_r2 = r2(t)-a20-a21*r1(t-1)-a22*r2(t-1)

frml fr1 = (abs(e_r1/sqrt(h1))-sqrt(2/%pi))    
frml fr2 = (abs(e_r2/sqrt(h2))-sqrt(2/%pi))    

frml asy1 = e_r1/sqrt(h1)    * asymetric codes
frml asy2 = e_r2/sqrt(h2)    * asymetric codes


frml var_r1 = exp(c10+c11*(fr1(t-1)+d11*asy1(t-1))+c12*(fr2(t-1)+d22*asy2(t-1))+d1*log(h1(t-1))
frml var_r2 = exp(c20+c21*(fr1(t-1)+d11*asy1(t-1))+c22*(fr2(t-1)+d22*asy2(t-1))+d2*log(h2(t-1))
frml gln = h1 = var_r1(t), h2 = var_r2(t), %logdensity(||h1|rho*sqrt(h1*h2),h2||,||r1,r2||)
smpl 3 2275
compute a10 = 01, a20 = .2, a11 = .03, a12 = .04, a21 = .09, a22 = .08
compute c10 = .01, c20 = .04, c11= .04, c22 = .014, c12=.08, c21=.06, rho=.05
compute d1 = .01, d2 = .08, d11 = .04, d12 = .08, d21 = .088, d22 = .03
maximize(method=bfgh,ite=500) gln

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GARCH(P=1,Q=1,MV=CC,VARIANCES=KOUTMOS,REGRESSORS) / R1 R2
# Constant R1{1} R2{1}

My codes result

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MAXIMIZE - Estimation by BFGS
Convergence in   371 Iterations. Final criterion was  0.0000018 <=  0.0000100
Usable Observations                      2273
Function Value                     -7806.3133

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
1.  A10                            0.94654129   0.01186322     79.78789  0.00000000
2.  A20                           -0.96897829   0.01572007    -61.63958  0.00000000
3.  A11                            1.79161573   0.02272365     78.84367  0.00000000
4.  A12                           -1.70074997   0.01836631    -92.60160  0.00000000
5.  A21                          -14.10914474   0.06518259   -216.45571  0.00000000
6.  A22                            0.83065515   0.00371228    223.75862  0.00000000
7.  C10                           -0.14594409   0.01654922     -8.81879  0.00000000
8.  C20                            0.03537544   0.01832068      1.93090  0.05349513
9.  C11                            0.00717189   0.00179060      4.00529  0.00006194
10. C12                            0.04181039   0.00430738      9.70668  0.00000000
11. C21                            0.02385552   0.00251985      9.46704  0.00000000
12. C22                            0.02444275   0.00346553      7.05312  0.00000000
13. D1                             0.92168316   0.00871321    105.78001  0.00000000
14. D2                             0.91104776   0.01202741     75.74760  0.00000000
15. D11                            0.13514001   0.22268733      0.60686  0.54394389
16. D22                           -0.54256036   0.05862743     -9.25438  0.00000000
17. RHO                            0.52085735   0.01427604     36.48473  0.00000000

Built-in Code Result

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MV-CC GARCH  with Koutmos EGARCH Variances - Estimation by BFGS
Convergence in    71 Iterations. Final criterion was  0.0000087 <=  0.0000100
Usable Observations                      2273
Log Likelihood                     -7779.2762

    Variable                        Coeff      Std Error      T-Stat      Signif
************************************************************************************
Mean Model(R1)
1.  Constant                      0.046929299  0.014274026      3.28774  0.00100995
2.  R1{1}                         0.003722419  0.023733139      0.15684  0.87536719
3.  R2{1}                         0.015022205  0.005620426      2.67279  0.00752238
Mean Model(R2)
4.  Constant                      0.215860800  0.050275648      4.29355  0.00001758
5.  R1{1}                        -0.035825474  0.067177772     -0.53329  0.59383037
6.  R2{1}                         0.048751335  0.021288114      2.29007  0.02201708

7.  C(1)                         -0.094504902  0.013451492     -7.02561  0.00000000
8.  C(2)                         -0.006286595  0.016493744     -0.38115  0.70309173
9.  A(1,1)                        0.117605462  0.015763261      7.46073  0.00000000
10. A(1,2)                       -0.020509391  0.012259182     -1.67298  0.09433085
11. A(2,1)                       -0.005126020  0.007676322     -0.66777  0.50428020
12. A(2,2)                        0.105497797  0.017962805      5.87312  0.00000000
13. B(1)                          0.980616989  0.004045214    242.41409  0.00000000
14. B(2)                          0.969452251  0.006773487    143.12454  0.00000000
15. D(1)                         -0.537680096  0.097963254     -5.48859  0.00000004
16. D(2)                         -1.007991409  0.183890841     -5.48147  0.00000004
17. R(2,1)                        0.519248141  0.014852537     34.96023  0.00000000

Data file is attached...............

[TomDoan]

Your Dxx's aren't right in this---I think you mean D11 then D22 in the second line.

frml var_r1 = exp(c10+c11*(fr1(t-1)+d11*asy1(t-1))+c12*(fr2(t-1)+d22*asy2(t-1))+d1*log(h1(t-1))
frml var_r2 = exp(c20+c21*(fr1(t-1)+d22*asy1(t-1))+c22*(fr2(t-1)+d22*asy2(t-1))+d2*log(h2(t-1))

[irfanawan]

Dear Tom,
I edit my last post and correct the error in program that you have mentioned in your comment. Still the results are different from each other i.e. results of my codes and built-in codes.I have tried to run my codes with different pre-iterations but same problem.

What should I do to get almost identical results?

Regards, Irfan Malik

[TomDoan]

This is using the data rather than the residuals in evaluating the likelihood

frml gln = h1 = var_r1(t), h2 = var_r2(t), %logdensity(||h1|rho*sqrt(h1*h2),h2||,||r1,r2||)

[irfanawan]

dear Tom,
I wants to estimate the bi-variate EGARCH models which I have discussed with you in previous post but with dynamic correlation structure. Mathematics for dynamic correlation is given file attached with this post.
I write the codes for this mathematics and modified my likelihood function. The complete codes for bi-variate EGRCH model with dynamic correlation structure as

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open data d-spcscointc.txt
data(format=free,org=columns,top=2) 1 2275 sp500 cisco intel
set r1 = sp500
set r2 = cisco
*
* This replaces your initialization of variances to zero.
*
stats(noprint) r1
set h1 = %variance
stats(noprint) r2
set h2 = %variance
*
nonlin a10 a20 a11 a12 a21 a22 c10 c20 c11 c12 c21 c22  d1 d2 d11  d22 c30 c31 c32

frml e_r1 = r1(t)-a10-a11*r1(t-1)-a12*r2(t-1)
frml e_r2 = r2(t)-a20-a21*r1(t-1)-a22*r2(t-1)

frml fr1 = (abs(e_r1/sqrt(h1))-sqrt(2/%pi))
frml fr2 = (abs(e_r2/sqrt(h2))-sqrt(2/%pi))

frml asy1 = e_r1/sqrt(h1)
frml asy2 = e_r2/sqrt(h2)

frml var_r1 = exp(c10+c11*(fr1(t-1)+d11*asy1(t-1))+c12*(fr2(t-1)+d22*asy2(t-1))+d1*log(h1(t-1)))
frml var_r2 = exp(c20+c21*(fr1(t-1)+d11*asy1(t-1))+c22*(fr2(t-1)+d22*asy2(t-1))+d2*log(h2(t-1)))

frml u = c30+c31*fr1(t-1)*fr2(t-1)+c32*u(t-1)
frml rho = (1-exp(-u(t)))/(1+exp(-u(t)))

frml gln = h1 = var_r1(t), h2 = var_r2(t),correl=rho(t), %logdensity(||h1|correl*sqrt(h1*h2),h2||,||e_r1,e_r2||)
smpl 3 2275
compute a10 = 01, a20 = .2, a11 = .03, a12 = .04, a21 = .09, a22 = .08
compute c10 = .01, c20 = .04, c11= .04, c22 = .014, c12=.08, c21=.06
compute d1 = .01, d2 = .08, d11 = .04, d12 = .08, d21 = .088, d22 = .03
compute c30=.04, c31=.004,c32=0.08
maximize(method=bfgh,ite=500,pit=30,pmet=sim) gln

when I run my codes with same data which was uploaded in previous post, the RATS exe stop working. What I should do to overcome this problem?

Regards, Irfan

[TomDoan]

This is recursively defined, which doesn't work:

frml u = c30+c31*fr1(t-1)*fr2(t-1)+c32*u(t-1)

U(t) can't be computed without u(t-1) which can't be computed without u(t-2), ...

You need to create a series of the values for U and use that in the definition of U. You need to look carefully at our examples and see how different series are used. You are using the FRML references way too many times. Anything that's lagged should have been saved into a series at T-1 so you don't have to recompute it, even if there isn't an issue with a self-referencing formula, as it will be much more efficient.

[irfanawan]

Dear Tom,
Hello.
I am bit confused with the multivariate GARCH models outputs (I have also try different VARIANCE options). I estiamte the model using following comand with options

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GARCH(MODEL=V1, P=1,Q=1,MV=CC,VARIANCE=VARMA,MVHSERIES=HH_1_2,asy, $
STDRESIDS=RES_1_2,PMETHOD=SIMPLEX,PITERATION=02,METHOD=BFGh,hmatrices=hh,rvectors=rd)

I used the following comand to estimat the standarized residuals

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set z1 = rd(t)(1)/sqrt(hh(t)(1,1))
set z2 = rd(t)(2)/sqrt(hh(t)(2,2))

I have the standarized residuals series in (n*2) martix obtained by using builtin command

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STDRESIDS=RES_1_2

and

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SET RES_1 = RES_1_2(1)
SET RES_2 = RES_1_2(2)

When I Print the standardized residuals, the RES_1 and Z1 and the RES_2 and Z2 they are different from each other. The RES_1 and Z1 are slightly different from each other but RES_2 and Z2 are very much different from each other. This difference have an effect on the results of test of auto-correlation and ARCH test

By construction they should be identical with each other.

Please guide me in this regards

Irfan Malik
Pakistan

[TomDoan]

STDRESIDS produces jointly standardized residuals, which theoretically have covariance matrix equal to the identity (if the model is correct). Simply dividing through by the standard deviations gives you univariate standardization, where the variances are theoretically 1, but there is no prediction for the covariances.

[irfanawan]

Dear Tom,
Thanks for your comments and now my question is for diagnostics testing which standardized residuals I have to use? Because the choice may alter the results.


Regards, Irfan

[TomDoan]

From the help for GARCH:

STDRESIDS=(output) VECT[SERIES] of (jointly) standardized residuals
FACTORBY=[CHOLESKY]/EIGENDECOMP
If the model is correct, then the jointly standardized residuals should be serially uncorrelated with mean zero and covariance matrix equal to the identity matrix with no residual GARCH effects. Given the model residuals and the series of estimated covariances, there are many (that is, an infinite number) of ways to transform them into jointly standardized residuals. You have the option of a Cholesky factorization or an eigen-based one. Note that different ways of doing the standardization will result in different standardized residuals and thus different diagnostics.


There's nothing you can do to avoid that. You should decide which you plan to use in advance, and stick with it, rather than trying different standardizations to get the results you want.