Multivariate EGARCH-X with spillovers

[bekar]

I am a pretty new RATS user. So, please forgive me in advance if my questions are so trivial.

I am trying to analyze spillover effects between the volatility of three markets. The mean equation for each market includes a constant and 5 lags of the return. The variance equations have arch, garch, and egarch parameters along with exogenous variables. I tried to estimate the model with the following code but I am not able to obtain convergence. Any suggestions would be appreciated. Thanks in advance!

EQUATION CL R_CL
# CONSTANT RLAG1_CL RLAG2_CL RLAG3_CL RLAG4_CL RLAG5_CL
EQUATION HO R_HO
# CONSTANT RLAG1_HO RLAG2_HO RLAG3_HO RLAG4_HO RLAG5_HO
EQUATION NG R_NG
# CONSTANT RLAG1_NG RLAG2_NG RLAG3_NG RLAG4_NG RLAG5_NG
GROUP MEANM CL HO NG

GARCH(P=1,Q=1,MODEL=MEANM,MV=bekk,asymmetric, XREGRESSORS,METHOD=BHHH,PMETHOD=SIMPLEX,PITERS=10, HMATRICES=H, MVHSERIES=MVH, RVECTORS=U,ITERATIONS=500)
# DCPI CPINEG DIP IPNEG DTCMSPRD DPCINVCL INVCLNEG DPCINVHO INVHONEG DPCINVNG INVNGNEG ASNFC OPEC SEP9 IRQINV KTRN LEHMN $
MON TUE WED THU JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV

[moderator]

That seems like an awfully large number of "xregressors", and a fairly large number of parameters over all. How many observations do you have?

In any case, if you are having convergence problems, I'd recommend that you start with a simpler form of the model to see if you can get convergence. For example, you might try estimating without the XREGRESSORS option (and list) first, just to see how that works.

Regards,
Tom Maycock

[TomDoan]

In addition to starting with a simpler model, I wouldn't recommend using BHHH. BFGS is a better choice unless you're fairly sure that you're starting close to the optimum, and it sounds as if you aren't.

[bekar]

Thanks Tom. I have 4276 observations. I tried the following code (changed to BFGS and increased convergence criteria) and got convergence in 472 iterations. I saw several other posts about using "maximize" rather than "garch." Do you think what I get from this "garch" instruction should be ok or should I try "maximize" somehow? Also, I have another question about the output. I have 3x3 C, A, B, and D matrices. I assume, for instance A(1,1) is the ARCH effect on the 1st variance, D(1,1) is the asymmetric effect on the 1st variance. If this is true then is the spillover effect from market 1 to market 2 is A(1,2) or B(1,2)? I would think if I am looking at the spillover in the variances then B(1,2). Am I right? Also, for the xregressors I have 6 coefficient estimates. What is the order on those? Is it upper triangular with the elements ordered in the output as X(1,1), X(1,2), X(1,3), X(2,2), X(2,3), X(3,3), where X(i,i) is the effect of the exogenous variable on the variance of i, and X(i,j) is the effeect on the covariance between i and j?

GARCH(P=1,Q=1,MODEL=MEANM,MV=bekk,asymmetric, XREGRESSORS,METHOD=BFGS,PMETHOD=SIMPLEX,PITERS=10, HMATRICES=H, MVHSERIES=MVH, RVECTORS=U,ITERATIONS=500, cvcrit=0.00005)
# DCPI CPINEG DIP IPNEG DTCMSPRD DPCINVCL INVCLNEG DPCINVHO INVHONEG DPCINVNG INVNGNEG ASNFC OPEC SEP11 IRQINV KTRN LEHMN $
MON TUE WED THU JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV

Thanks a lot.
Berna

[bekar]

Tom, another question is (if you don't mind):
how the exogenous variables enter into the BEKK model with asymmetry if we have trivariate model? Is it like the following:

H(t)=C'C + A'u(t-1)u(t-1)'A + B'H(t-1)B + D'v(t-1)v(t-1)'D + G1' X1(t) X1(t)' G1 + G2' X2(t) X2(t)' G2

where C, A, B, D, G1, and G2 are 3x3 matrices, u(t-1), v(t-1)=u(t-1).*I(u(t-1)), X1(t), and X2(t) are 3x1 vectors

Another issue is how to interpret the results. Especially if the xregressors enter the variance equations squared then how do we interpret the effect of variable X1 and X2 on H? And if the above specification is correct then the t-stat would be for G1'G1 I suppose.

Sorry to bother with so many questions but I couldn't find any of these in the manuals.

Thanks!

[TomDoan]

Thanks Tom. I have 4276 observations. I tried the following code (changed to BFGS and increased convergence criteria) and got convergence in 472 iterations. I saw several other posts about using "maximize" rather than "garch." Do you think what I get from this "garch" instruction should be ok or should I try "maximize" somehow?

If you have a model (such as this) that can be estimated with GARCH, use GARCH. The only advantage to MAXIMIZE is that it can handle models which aren't in a form that GARCH includes.

Also, I have another question about the output. I have 3x3 C, A, B, and D matrices. I assume, for instance A(1,1) is the ARCH effect on the 1st variance, D(1,1) is the asymmetric effect on the 1st variance. If this is true then is the spillover effect from market 1 to market 2 is A(1,2) or B(1,2)? I would think if I am looking at the spillover in the variances then B(1,2). Am I right?

The terms are in the form A'u(t-1)u(t-1)'A and B'H(t-1)B so in the A, B and D matrices, an (i,j) subscript has j as the target of the effect and i as the source so from 1 to 2 is at (1,2) in any of the three.

Also, for the xregressors I have 6 coefficient estimates. What is the order on those? Is it upper triangular with the elements ordered in the output as X(1,1), X(1,2), X(1,3), X(2,2), X(2,3), X(3,3), where X(i,i) is the effect of the exogenous variable on the variance of i, and X(i,j) is the effeect on the covariance between i and j?

Similar to the constant, those are in Cholesky factor form so X'X times the dummy is the effect. The only one of those that has an immediate interpretation is X(3,3), which is the square root of the variance effect for variable 3. All the others interact through the matrix multiplication to give the variances and covariances of the effect.

[bekar]

Thanks Tom. One more thing. When you say Cholesky factor form is it uppe or lower triangular in RATS? That is:

is the constant matrix C (and any coefficient matrix of an exogenous variable in the variance equation) as follows (with Matlab's matrix notation): [c11 0 0 ; c21 c22 c23; c31 c32 c33]

And the BEKK representation is

H(t)= C' C+ A' u(t-1) u(t-1)' A + B' H(t-1) B + D' v(t-1) v(t-1)' D + G' G X(t)

Thanks!

[bekar]

And in this case I don't have only dummy variables in the variance equation. Some of the X(t) are actually economic variables. So, if G is upper or lower triangular then G' G will be 3x3. Then X(t) should be also 3x3. But I don't think that's the case. The variable X has the same value for each market at time t. So, like residual vector u(t-1), X(t) will be 3x1 vector. That's why I'm confused how these exogenous variables are introduced into the BEKK model.

[bekar]

Tom, please ignore my question about the dimension of X(t). If it is the same value for all variables then it is like a dummy variable and I can multiply G'G with X(t) through elementwise multiplication. Right?

But still, can you please clarify the Cholesky factor form? Thanks.

[TomDoan]

Thanks Tom. One more thing. When you say Cholesky factor form is it uppe or lower triangular in RATS? That is:

is the constant matrix C (and any coefficient matrix of an exogenous variable in the variance equation) as follows (with Matlab's matrix notation): [c11 0 0 ; c21 c22 c23; c31 c32 c33]

And the BEKK representation is

H(t)= C' C+ A' u(t-1) u(t-1)' A + B' H(t-1) B + D' v(t-1) v(t-1)' D + G' G X(t)

Thanks!

The Cholesky form is lower triangular (your c23 should be zero), but is multiplied as C'C rather than CC'.

[TomDoan]

Tom, please ignore my question about the dimension of X(t). If it is the same value for all variables then it is like a dummy variable and I can multiply G'G with X(t) through elementwise multiplication. Right?

That's correct. The BEKK form is really not designed for exogenous variables in the variance equation because the parameters come in as matrix "squares". A way to handle these that would be more in keeping with the idea behind BEKK would be to use (C+GX(t))'(C+GX(t)) where C and G are triangular. That would be forcibly p.s.d. while making the sign of X(t) irrelevant. However, it would break the relationship between BEKK and VECH representations.

[bekar]

Thanks Tom. I guess I'll just figure out the coefficients for the variables from BEKK and use SUMMARIZE to test the combination of those.