SHAZAM GARCH benchmark

Benchmark comparisons of coefficients and standard errors

The Bollerslev and Ghysels [1996] data set of daily exchange rate changes for the Deutschemark/British pound has been adopted as a benchmark data set by McCullough and Renfro [1999] (the "MR benchmark"). In this section, the SHAZAM results for GARCH(1,1) model estimation are compared with the MR benchmark results.

An end product of the estimation routine is that different methods are available for calculating estimates of standard errors for the coefficient estimates. The HET command in SHAZAM provides the following options.

  • The default method obtains a covariance matrix estimate based on the inverse of the information matrix.

  • The OPGCOV option generates a covariance matrix estimate based on the inverse of the sum of the outer product of the gradients evaluated at each observation.

  • The NUMCOV option calculates a covariance matrix estimate based on the negative of the inverse of the Hessian matrix evaluated by numerical derivatives.

  • Some applied workers report standard errors that are robust to the assumption of conditional normality of the errors (see Weiss [1986] and Bollerslev-Wooldridge [1992]). These are based on the matrix calculation:
          A-1BA-1
    where A is the information matrix and B is the outer product of the gradient. The COV= and GMATRIX= options can be used to save results for use in the calculations required for obtaining robust standard errors.

The above calculations are implemented with the coefficient estimates that are available at the final iteration of the estimation algorithm.

The SHAZAM commands (filename: ARCH3.SHA) below estimate the GARCH(1,1) model and calculate alternative estimates of the standard errors for the coefficient estimates. The results are then compared with the MR benchmark results. 

SAMPLE 1 1974
READ (DMBP.txt) Y DAYDUM

* Covariance matrix - Information matrix  
HET Y / GARCH=1 PRESAMP COEF=BETA STDERR=IM COV=AINV GMATRIX=G 

* Compute robust standard errors (see Weiss and Bollerslev-Wooldridge)
MATRIX V=AINV*(G'G)*AINV
MATRIX BW=SQRT(DIAG(V))

* Covariance matrix - outer product of gradient (OP)
?HET Y / GARCH=1 PRESAMP OPGCOV STDERR=OP

* Covariance matrix - Hessian estimation by numerical derivatives
?HET Y / GARCH=1 PRESAMP NUMCOV STDERR=HNUM

* List the coefficients and standard errors
SAMPLE 1 4
PRINT BETA HNUM OP IM BW 

* Compare with the benchmark results in McCullough and Renfro 
* Benchmark coefficients 
READ BMARK / BYVAR
 -0.00619041  0.0107613  0.153134  0.805974
* Calculate the log relative error -- the number of digits of accuracy
GENR LRE=-LOG(ABS(BETA-BMARK)/ABS(BMARK))/2.3026
PRINT BMARK BETA LRE
* Benchmark standard error estimates
READ BH BOP BIM BBW 
 0.00846212  0.00843359  0.00837628  0.00873092
 0.00285271  0.00132298  0.00192881  0.00312364
 0.0265228   0.0139737   0.0194012   0.0273219
 0.0335527   0.0165604   0.0218399   0.0301509
GENR LRE1=-LOG(ABS(HNUM-BH)/ABS(BH))/2.3026  
GENR LRE2=-LOG(ABS(OP-BOP)/ABS(BOP))/2.3026  
GENR LRE3=-LOG(ABS(IM-BIM)/ABS(BIM))/2.3026  
GENR LRE4=-LOG(ABS(BW-BBW)/ABS(BBW))/2.3026  
FORMAT(4F12.2)
PRINT LRE1 LRE2 LRE3 LRE4 / FORMAT
STOP

Note that the ? prefix instructs SHAZAM to suppress output generated by the command. This reduces the size of the output file.

The SHAZAM output can be viewed.

The SHAZAM results for the coefficients and standard errors can be compared with the MR benchmark results. The number of digits of accuracy can be measured by the log relative error calculated as:

      LRE = - log10(|x - c| / |c|)

where x is the SHAZAM estimate and c is the MR benchmark.

The results on the SHAZAM output show that all estimates for the coefficients and standard errors have 3 or more digits of accuracy.

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SHAZAM output 

|_SAMPLE 1 1974
|_READ (DMBP.txt) Y DAYDUM
UNIT 88 IS NOW ASSIGNED TO: DMBP.txt
   2 VARIABLES AND     1974 OBSERVATIONS STARTING AT OBS       1

|_* Covariance matrix - Information matrix

|_HET Y / GARCH=1 PRESAMP COEF=BETA STDERR=IM COV=AINV GMATRIX=G
...NOTE..SAMPLE RANGE SET TO:     1,  1974
      1974 OBSERVATIONS

ARCH     HETEROSKEDASTICITY MODEL    1974 OBSERVATIONS
         ANALYTIC DERIVATIVES
PRE-SAMPLE VARIANCE ESTIMATE =  0.22102

  QUASI-NEWTON METHOD USING BFGS U DATE FORMULA

INITIAL STATISTICS :

TIME =    0.021 SEC.   ITER. NO.     1 FUNCTION EVALUATIONS     1
LOG-LIKELIHOOD FUNCTION=   -1217.268
COEFFICIENTS
 -0.1642679E-01  0.1723165      0.2208491       0.000000
GRADIENT
   122.2147      -299.7114       84.31881       110.2291

FINAL STATISTICS :

TIME =    0.151 SEC.   ITER. NO.    16 FUNCTION EVALUATIONS    25
LOG-LIKELIHOOD FUNCTION=   -1106.607
COEFFICIENTS
 -0.6194411E-02  0.1075673E-01  0.1531225      0.8060014
GRADIENT
  0.2928278       2.689984      0.1493853      0.2967491

SQUARED CORR. COEF. BETWEEN OBSERVED AND PREDICTED   0.00000

ASY. COVARIANCE MATRIX OF PARAMETER ESTIMATES IS ESTIMATED USING
THE INFORMATION MATRIX

LOG OF THE LIKELIHOOD FUNCTION = -1106.61

                             ASYMPTOTIC
VARIABLE   ESTIMATED  STANDARD   T-RATIO        PARTIAL STANDARDIZED ELASTICITY
  NAME    COEFFICIENT   ERROR   --------   P-VALUE CORR. COEFFICIENT  AT MEANS
         MEAN EQUATION:
CONSTANT -0.61944E-02 0.8376E-02 -0.7396     0.460-0.017     0.0000     0.3771
         VARIANCE EQUATION:
ALPHA_    0.10757E-01 0.1928E-02   5.579     0.000 0.12
ALPHA_    0.15312     0.1940E-01   7.893     0.000 0.17
PHI_      0.80600     0.2184E-01   36.91     0.000 0.63

|_* Compute robust standard errors (see Weiss and Bollerslev-Wooldridge)
|_MATRIX V=AINV*(G'G)*AINV
|_MATRIX BW=SQRT(DIAG(V))

|_* Covariance matrix - outer product of gradient (OP)
|_?HET Y / GARCH=1 PRESAMP OPGCOV STDERR=OP

|_* Covariance matrix - Hessian estimation by numerical derivatives
|_?HET Y / GARCH=1 PRESAMP NUMCOV STDERR=HNUM

|_* List the coefficients and standard errors
|_SAMPLE 1 4
|_PRINT BETA HNUM OP IM BW
      BETA           HNUM           OP             IM             BW
 -0.6194411E-02  0.8469006E-02  0.8432776E-02  0.8375834E-02  0.8731003E-02
  0.1075673E-01  0.2850064E-02  0.1322336E-02  0.1928089E-02  0.3122941E-02
  0.1531225      0.2651373E-01  0.1397134E-01  0.1939880E-01  0.2732012E-01
  0.8060014      0.3353406E-01  0.1655609E-01  0.2183573E-01  0.3014772E-01

|_* Compare with the benchmark results in McCullough and Renfro
|_* Benchmark coefficients
|_READ BMARK / BYVAR
   1 VARIABLES AND        4 OBSERVATIONS STARTING AT OBS       1

|_* Calculate the log relative error -- the number of digits of accuracy
|_GENR LRE=-LOG(ABS(BETA-BMARK)/ABS(BMARK))/2.3026
|_PRINT BMARK BETA LRE
      BMARK          BETA           LRE
 -0.6190410E-02 -0.6194411E-02   3.189496
  0.1076130E-01  0.1075673E-01   3.371813
  0.1531340      0.1531225       4.122623
  0.8059740      0.8060014       4.467921

|_* Benchmark standard error estimates
|_READ BH BOP BIM BBW
   4 VARIABLES AND        4 OBSERVATIONS STARTING AT OBS       1

|_GENR LRE1=-LOG(ABS(HNUM-BH)/ABS(BH))/2.3026
|_GENR LRE2=-LOG(ABS(OP-BOP)/ABS(BOP))/2.3026
|_GENR LRE3=-LOG(ABS(IM-BIM)/ABS(BIM))/2.3026
|_GENR LRE4=-LOG(ABS(BW-BBW)/ABS(BBW))/2.3026
|_FORMAT(4F12.2)
|_PRINT LRE1 LRE2 LRE3 LRE4 / FORMAT
      LRE1           LRE2           LRE3           LRE4
       3.09        4.02        4.27        5.02
       3.03        3.31        3.43        3.65
       3.47        3.77        3.91        4.18
       3.26        3.58        3.72        3.98
|_STOP
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