Benchmark comparisons of coefficients and standard errorsThe 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 An end product of the estimation routine is that different methods
are available for calculating estimates of standard errors for the
coefficient estimates.
The
The above calculations are implemented with the coefficient estimates that are available at the final iteration of the estimation algorithm. The SHAZAM commands (filename:
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 = 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 |_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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