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.

[SHAZAM Guide home]

#### SHAZAM output

```|_SAMPLE 1 1974
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
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
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
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
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
```