Estimation of a GARCH(1,1) modelThe
Pitfalls are:
The following warning messages give a signal that the estimation has failed.
The above message indicates that negative coefficients for the conditional variance equation were calculated as the default starting values.
The default iteration limit is 100 iterations. The above message indicates that the estimation algorithm failed to reach a solution within the iteration limit.
The above message indicates that the algorithm was unsuccessful in finding a solution.
The above message indicates that the variance-covariance matrix of the parameter estimates could not be computed.
The above message indicates that the coefficients of the conditional variance equation are not in the region required for stationarity. ExampleThe SHAZAM commands (filename:
The SHAZAM output can be viewed. The results show that the estimated GARCH(1,1) conditional variance function is:
where the êt are the residuals. The standardized residuals are defined as:
If the GARCH(1,1) model describes the data then the standardized
residuals should have zero mean and unit variance and be
independently and identically distributed.
In the SHAZAM program, the standardized residuals are saved with the
Note: Q(20) and Q2(20) are the Ljung-Box-Pierce portmanteau tests for up to twentieth order serial correlation in the standardized and the squared standardized residuals respectively. The Q(20) and Q2(20) statistics indicate no serial correlation in either the standardized residuals or the squared standardized residuals. This suggests that the GARCH(1,1) model has some success in explaining the time-varying volatility in the data. However, the sample excess kurtosis for the standardized residuals shows strong evidence of leptokurtosis. An alternative estimation strategy presented in Baillie and Bollerslev [1989] is to use conditional leptokurtic distributions (for example, the t-distribution) for modelling ARCH processes. A way of presenting the results is to plot the estimated conditional
standard deviations. Emphasis to the pattern of volatility can be
obtained by smoothing the data.
The SHAZAM program used the LOWESS locally weighted regression
method that is implemented with the
[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 |_* Preliminary OLS estimation |_OLS Y / DN GF OLS ESTIMATION 1974 OBSERVATIONS DEPENDENT VARIABLE = Y ...NOTE..SAMPLE RANGE SET TO: 1, 1974 R-SQUARE = 0.0000 R-SQUARE ADJUSTED = 0.0000 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.22102 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.47013 SUM OF SQUARED ERRORS-SSE= 436.29 MEAN OF DEPENDENT VARIABLE = -0.16427E-01 LOG OF THE LIKELIHOOD FUNCTION = -1311.10 ASYMPTOTIC VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR -------- P-VALUE CORR. COEFFICIENT AT MEANS CONSTANT -0.16427E-01 0.1058E-01 -1.552 0.121-0.035 0.0000 1.0000 DURBIN-WATSON = 1.9805 VON NEUMANN RATIO = 1.9815 RHO = 0.00937 RESIDUAL SUM = -0.14766E-13 RESIDUAL VARIANCE = 0.22102 SUM OF ABSOLUTE ERRORS= 648.23 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.0000 RUNS TEST: 963 RUNS, 1028 POS, 0 ZERO, 946 NEG NORMAL STATISTIC = -1.0508 COEFFICIENT OF SKEWNESS = -0.2497 WITH STANDARD DEVIATION OF 0.0551 COEFFICIENT OF EXCESS KURTOSIS = 3.6399 WITH STANDARD DEVIATION OF 0.1101 JARQUE-BERA NORMALITY TEST- CHI-SQUARE(2 DF)= 1102.8823 P-VALUE= 0.000 GOODNESS OF FIT TEST FOR NORMALITY OF RESIDUALS - 60 GROUPS OBSERVED [statistics not shown] EXPECTED [statistics not shown] CHI-SQUARE = 438.3392 WITH 57 DEGREES OF FREEDOM, P-VALUE= 0.000 |_* Estimation results: Table 2, column 4 (Bollerslev and Ghysels, 1996) |_HET Y / GARCH=1 PRESAMP STDRESID=E RESID=E1 ...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 = 0.142 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 |_* Diagnostic Testing |_STAT E NAME N MEAN ST. DEV VARIANCE MINIMUM MAXIMUM E 1974 -0.17749E-01 0.99902 0.99803 -6.7716 5.2625 |_OLS E / GF OLS ESTIMATION 1974 OBSERVATIONS DEPENDENT VARIABLE = E ...NOTE..SAMPLE RANGE SET TO: 1, 1974 R-SQUARE = 0.0000 R-SQUARE ADJUSTED = 0.0000 VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.99803 STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.99902 SUM OF SQUARED ERRORS-SSE= 1969.1 MEAN OF DEPENDENT VARIABLE = -0.17749E-01 LOG OF THE LIKELIHOOD FUNCTION = -2798.54 VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 1973 DF P-VALUE CORR. COEFFICIENT AT MEANS CONSTANT -0.17749E-01 0.2249E-01 -0.7894 0.430-0.018 0.0000 1.0000 DURBIN-WATSON = 1.8975 VON NEUMANN RATIO = 1.8985 RHO = 0.05065 RESIDUAL SUM = -0.44409E-13 RESIDUAL VARIANCE = 0.99803 SUM OF ABSOLUTE ERRORS= 1431.1 R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.0000 RUNS TEST: 963 RUNS, 1022 POS, 0 ZERO, 952 NEG NORMAL STATISTIC = -1.0711 COEFFICIENT OF SKEWNESS = -0.3474 WITH STANDARD DEVIATION OF 0.0551 COEFFICIENT OF EXCESS KURTOSIS = 3.5342 WITH STANDARD DEVIATION OF 0.1101 JARQUE-BERA NORMALITY TEST- CHI-SQUARE(2 DF)= 1060.0264 P-VALUE= 0.000 GOODNESS OF FIT TEST FOR NORMALITY OF RESIDUALS - 60 GROUPS OBSERVED [statistics not shown] EXPECTED [statistics not shown] CHI-SQUARE = 233.9221 WITH 57 DEGREES OF FREEDOM, P-VALUE= 0.000 |_ARIMA E ARIMA MODEL NUMBER OF OBSERVATIONS =1974 ...NOTE..SAMPLE RANGE SET TO: 1, 1974 IDENTIFICATION SECTION - VARIABLE=E NUMBER OF AUTOCORRELATIONS = 24 NUMBER OF PARTIAL AUTOCORRELATIONS = 12 0 0 0 SERIES (1-B) (1-B ) E NET NUMBER OF OBSERVATIONS = 1974 MEAN= -0.17749E-01 VARIANCE= 0.99803 STANDARD DEV.= 0.99902 LAGS AUTOCORRELATIONS STD ERR 1 -12 0.05 -.01 0.02 0.03 0.02 0.00 -.01 0.02 0.02 0.01 -.04 -.01 0.02 13 -24 0.00 0.04 0.00 -.01 -.01 0.00 0.01 -.03 0.00 0.01 0.01 0.06 0.02 MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE 1 5.06 1 .024 13 14.17 13 .362 2 5.27 2 .072 14 17.04 14 .254 3 6.14 3 .105 15 17.04 15 .316 4 7.68 4 .104 16 17.12 16 .378 5 8.19 5 .146 17 17.32 17 .433 6 8.19 6 .224 18 17.33 18 .500 7 8.39 7 .299 19 17.38 19 .564 8 9.28 8 .319 20 19.30 20 .503 9 9.94 9 .355 21 19.33 21 .564 10 10.12 10 .430 22 19.52 22 .613 11 13.79 11 .245 23 19.71 23 .659 12 14.16 12 .291 24 27.45 24 .284 LAGS PARTIAL AUTOCORRELATIONS STD ERR 1 -12 0.05 -.01 0.02 0.03 0.01 0.00 -.01 0.02 0.02 0.01 -.04 -.01 0.02 |_GENR E2=E*E |_ARIMA E2 ARIMA MODEL NUMBER OF OBSERVATIONS =1974 ...NOTE..SAMPLE RANGE SET TO: 1, 1974 IDENTIFICATION SECTION - VARIABLE=E2 NUMBER OF AUTOCORRELATIONS = 24 NUMBER OF PARTIAL AUTOCORRELATIONS = 12 0 0 0 SERIES (1-B) (1-B ) E2 NET NUMBER OF OBSERVATIONS = 1974 MEAN= 0.99784 VARIANCE= 5.5236 STANDARD DEV.= 2.3502 LAGS AUTOCORRELATIONS STD ERR 1 -12 0.04 -.01 -.03 0.00 0.00 -.04 -.03 -.01 0.00 0.02 -.01 -.02 0.02 13 -24 -.03 0.02 0.04 0.01 -.02 -.02 -.01 0.00 -.01 0.00 -.01 0.01 0.02 MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE 1 2.51 1 .113 13 11.44 13 .574 2 2.60 2 .273 14 12.54 14 .563 3 4.27 3 .234 15 16.08 15 .377 4 4.27 4 .370 16 16.15 16 .443 5 4.27 5 .511 17 16.79 17 .469 6 6.72 6 .347 18 17.44 18 .493 7 7.97 7 .335 19 17.49 19 .556 8 8.35 8 .400 20 17.51 20 .620 9 8.36 9 .498 21 17.87 21 .657 10 9.06 10 .526 22 17.87 22 .714 11 9.44 11 .581 23 18.26 23 .743 12 9.99 12 .617 24 18.33 24 .787 LAGS PARTIAL AUTOCORRELATIONS STD ERR 1 -12 0.04 -.01 -.03 0.00 0.00 -.04 -.02 -.01 0.00 0.02 -.02 -.02 0.02 |_* Get the conditional standard deviation |_GENR HSE=E1/E |_GENR DAY=TIME(0) |_* Smooth the results |_NONPAR HSE DAY / METHOD=LOWESS PREDICT=HSMTH SMOOTH=0.1 1974 OBSERVATIONS DEPENDENT VARIABLE = HSE ...NOTE..SAMPLE RANGE SET TO: 1, 1974 NONPARAMETRIC SMOOTHING USING LOWESS METHOD SMOOTH PARAMETER= 0.10000 ITERATIONS= 0 DELTA= 0.000 NUMBER OF NEAREST NEIGHBORS - R = 197 R-SQUARE = 0.4305 R-SQUARE ADJUSTED = 0.4246 ERROR VARIANCE SIGMA**2 = 0.16193E-01 STANDARD ERROR SIGMA = 0.12725 SUM OF SQUARED ERRORS SSE = 31.622 EQUIVALENT NUMBER OF PARAMETERS - K1 = 21.179 - K2 = 15.135 - K3 = 18.157 CROSS-VALIDATION MEAN SQUARE ERROR = 0.16310E-01 MODEL SELECTION TESTS - SEE JUDGE ET AL. (1985,P.242) AKAIKE (1969) FINAL PREDICTION ERROR - FPE = 0.16367E-01 (FPE IS ALSO KNOWN AS AMEMIYA PREDICTION CRITERION - PC) AKAIKE (1973) INFORMATION CRITERION - LOG AIC = -4.1125 SCHWARZ (1978) CRITERION - LOG SC = -4.0526 MODEL SELECTION TESTS - SEE RAMANATHAN (1992,P.167) CRAVEN-WAHBA (1979) GENERALIZED CROSS VALIDATION - GCV = 0.16368E-01 HANNAN AND QUINN (1979) CRITERION = 0.16731E-01 RICE (1984) CRITERION = 0.16370E-01 SHIBATA (1981) CRITERION = 0.16363E-01 SCHWARZ (1978) CRITERION - SC = 0.17378E-01 AKAIKE (1974) INFORMATION CRITERION - AIC = 0.16367E-01 |_* Plot the conditional standard deviation |_GRAPH HSE HSMTH / TIME LINEONLY 1974 OBSERVATIONS SHAZAM WILL NOW MAKE A PLOT FOR YOU NO SYMBOLS WILL BE PLOTTED, LINE ONLY |_STOP
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