# ARCH with Exogenous Variables

Shazam and SAS produce very different results when running an ARCH model with an exogenous variable. I am using Proc autoreg in SAS with a hetero statement. What can explain the difference? Thanks.

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Can you publish the sample data, commands and output for each please. Note that SHAZAM has numerous options on the HET command including 3 optimization methods. Suggest you also confirm the nature of the heteroskedasticity specified. e.g. multiplicative, dependent variable heteroskedascity etc

I am running a model where the index variable is in the ARCH equation for the variance. How would you call that type of het specification? Thank you for your help!

In SHAZAM I am running

het price index interaction lag1_pri (index) / arch=1


In SAS I am running

PROC AUTOREG DATA=test;
m1: model price=lag1_pri index interaction/ garch=(p=1) MAXITER = 1000 NOMISS;
hetero index ;

run;


OUTPUT SHAZAM:

FINAL STATISTICS :

TIME =     0.0290 SEC.   ITER. NO.    28 FUNCTION EVALUATIONS    48
LOG-LIKELIHOOD FUNCTION=   -161.1532
COEFFICIENTS
0.8023219     -0.4499052E-01  0.7363933       1.043197      0.4605022
1.850979     -0.3225361     -0.3241526
0.1081076E-01  0.1721445      0.1343623     -0.1907942E-02 -0.3892096E-01
-0.1332209E-02 -0.4898605E-01 -0.2773277E-02

...WARNING..STATIONARITY CONSTRAINTS NOT SATISFIED

SQUARED CORR. COEF. BETWEEN OBSERVED AND PREDICTED   0.77349

ASY. COVARIANCE MATRIX OF PARAMETER ESTIMATES IS ESTIMATED USING
THE INFORMATION MATRIX
LOG OF THE LIKELIHOOD ...
(more)

I should probably make it a very specific question. I want to run an ARCH(1) model where a have an independent indicator variable in the variance ARCH equation as a linea term i.e. h= const+ARCH(1)+b*indicator. So far I have been using het price index interaction lag1_pri (index) /arch=1 Is this the right syntax? I cannot assign model=varlin for a linear term with ARCH. What is the proper way to do it? Thanks.

Thank you for your reply.How about my second question regarding the functional form of the exogenous variable?

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For an ARCH(1) model, the SHAZAM statement simply needs the option ARCH=1 which has been set correctly. The specification of exogenous variable variance with index by listing (index) after the list of independent variables is also correct.

In the SAS model it appears from their online documentation that an ARCH(1) model would need the option GARCH=(P=0,Q=1) not GARCH=(P=1) which appears to fit a GARCH(1,1) model. The SAS documentation does not clearly indicate how to specify the same form of exogenous variable variance with the hetero statement.

Note that in SHAZAM you can check for the presence of heteroskedasticity using the DIAGNOS / HET statement following an OLS command to perform Lagrange Multipler Tests (including a test for ARCH=1 errors). Page 246 of the SHAZAM 11 manual has further details.

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Thank you so much for your reply. A few more follow-up questions. 1) Is there anything in SHAZAM that does not allow me to have the exogenous and the independent variable be the same? I am trying to replicate a study which uses the model that I was referring to. 2) In the variance equation, does the exogenous variable appear in a linear or exponential form. In other words, do we have sigma=ARCH(1) +bexog or sigma=ARCH(1) +expon(bexog). I know STATA has multiplicative heteroskedasticity in exponential form and SAS has it linear.

Thanks again!

Actually, it is possible to specify the same variable in the list of independent variables and in the list of exogenous variables. I have corrected the answer.

Multiplicative heteroskedasticity can be specified using the MODEL=MULT and the exact form of the error variance is specified on page 249 of the SHAZAM 11 manual while in a non-multiplicative form the exact form is specified on page 242.

In the multiplicative form the variance is specified as: h(t) = exp(Z'(t).alpha) as in Harvey [1976, 1990].

The reason I am asking is once you specify model=arch you can no longer use model=linear or model=multi. Thus, even though you can have an exogenous variable with arch it is not clear what the functional form is.

It is linear in that case (see pg 242). Yes, the MODEL= option switches between each form of heteroskedasticity.