Testing for ARCH--------------------------------------------------------------------------------The residuals from a preliminary OLS estimation can be tested for ARCH behaviour. Testing approaches are as follows. Tests for non-normality can be considered. If the normality assumption is used to describe the conditional error distribution then a property of ARCH is that the unconditional error distribution will be non-normal with high values for kurtosis. The autocorrelation structure of the residuals and the squared residuals can be inspected. An indication of ARCH is that the residuals will be uncorrelated but the squared residuals will show autocorrelation. Test statistics are given by Ljung-Box-Pierce portmanteau tests on the residuals and the squared residuals. A test based on the Lagrange multiplier (LM) principle can be applied. Consider the null hypothesis of no ARCH errors versus the alternative hypothesis that the conditional error variance is given by an ARCH(q) process. The test approach proposed in Engle [1982] is to regress the squared residuals on a constant and q lagged values of the squared residuals. From the results of this auxiliary regression, a test statistic is calculated as: (N-q)·R2 There is evidence to reject the null hypothesis if the test statistic exceeds the critical value from a chi-square distribution with q degrees of freedom. ExampleThe SHAZAM commands (filename: ARCH1.SHA) below generate some statistics for the exchange rate data set. SAMPLE 1 1974READ (DMBP.txt) Y DAYDUM* Estimation results: Table 2, column 1 (Bollerslev and Ghysels, 1996)* The GF option provides coefficients of skewness and kurtosis as well* as the Jarque-Bera test for non-normality.OLS Y / RESID=E GF* Inspect the autocorrelation structure of the residualsARIMA E * Inspect the autocorrelation structure of the squared residualsGENR E2=E*EARIMA E2* Calculate Lagrange multiplier test statistics for ARCH errorsSET NODOECHO NOOUTPUTGEN1 NLAG=10DIM LM NLAG PVALUE NLAGDO #=1,NLAG* Calculate the test statisticOLS E2 E2(1.#)GEN1 TESTVAL=$N*$R2* Calculate a p-valueDISTRIB TESTVAL / TYPE=CHI DF=# CDF=CDF GEN1 LM:#=TESTVALGEN1 PVALUE:#=1-CDFENDO* Print the resultsSAMPLE 1 NLAGGENR P=TIME(0)FORMAT(F8.0,F10.2,F10.4)PRINT P LM PVALUE / FORMAT STOP The above commands show the use of a DO-loop for calculating LM test statistics for ARCH(q), q=1,2,...,10. The SHAZAM output can be viewed. The results show the following test statistics based on the OLS residuals. Statistic p-value Skewness -0.25 Excess kurtosis 3.64 Jarque-Bera test 1102.9 < 0.0005 Q(20) 27.8 0.113 Q2(20) 507.6 < 0.0005 LM ARCH(1) 96.2 < 0.0005 LM ARCH(8) 185.3 < 0.0005 Note: Q(20) and Q2(20) are the Ljung-Box-Pierce portmanteau tests for up to twentieth order serial correlation in the residuals and the squared residuals respectively. The Jarque-Bera test statistic provides clear evidence to reject the null hypothesis of normality for the unconditional distribution of the daily percentage exchange rate changes. The high value for excess kurtosis indicates that the distribution is characterized by leptokurtosis. Note that the SHAZAM calculations for skewness and kurtosis incorporate small sample adjustments (see the SHAZAM User's Reference Manual). The sample autocorrelation function of the residuals shows no autocorrelation. The Q(20) test statistic does not reject the null hypothesis of uncorrelated price changes. However, the sample autocorrelation function of the squared residuals tells a different story. The high value for the Q2(20) test statistic suggests that conditional homoskedasticity can be rejected. The slow decline of the autocorrelation function of the squared residuals suggests that a GARCH(1,1) process may be suitable for describing the errors. That is, a low order ARCH process may not fully capture the time-varying volatility in the data. The LM tests for ARCH(1) and ARCH(8) errors confirm the presence of ARCH effects in the data. -------------------------------------------------------------------------------- [SHAZAM Guide home] --------------------------------------------------------------------------------SHAZAM output --------------------------------------------------------------------------------|_SAMPLE 1 1974|_READ (DMBP.txt) Y DAYDUMUNIT 88 IS NOW ASSIGNED TO: DMBP.txt 2 VARIABLES AND 1974 OBSERVATIONS STARTING AT OBS 1|_* Estimation results: Table 2, column 1 (Bollerslev and Ghysels, 1996)|_* The GF option provides coefficients of skewness and kurtosis as well|_* as the Jarque-Bera test for non-normality.|_OLS Y / RESID=E GF OLS ESTIMATION 1974 OBSERVATIONS DEPENDENT VARIABLE = Y...NOTE..SAMPLE RANGE SET TO: 1, 1974 R-SQUARE = 0.0000 R-SQUARE ADJUSTED = 0.0000VARIANCE OF THE ESTIMATE-SIGMA**2 = 0.22113STANDARD ERROR OF THE ESTIMATE-SIGMA = 0.47024SUM OF SQUARED ERRORS-SSE= 436.29MEAN OF DEPENDENT VARIABLE = -0.16427E-01LOG OF THE LIKELIHOOD FUNCTION = -1311.10VARIABLE ESTIMATED STANDARD T-RATIO PARTIAL STANDARDIZED ELASTICITY NAME COEFFICIENT ERROR 1973 DF P-VALUE CORR. COEFFICIENT AT MEANSCONSTANT -0.16427E-01 0.1058E-01 -1.552 0.121-0.035 0.0000 1.0000DURBIN-WATSON = 1.9805 VON NEUMANN RATIO = 1.9815 RHO = 0.00937RESIDUAL SUM = -0.14766E-13 RESIDUAL VARIANCE = 0.22113SUM OF ABSOLUTE ERRORS= 648.23R-SQUARE BETWEEN OBSERVED AND PREDICTED = 0.0000RUNS TEST: 963 RUNS, 1028 POS, 0 ZERO, 946 NEG NORMAL STATISTIC = -1.0508COEFFICIENT OF SKEWNESS = -0.2497 WITH STANDARD DEVIATION OF 0.0551COEFFICIENT OF EXCESS KURTOSIS = 3.6399 WITH STANDARD DEVIATION OF 0.1101JARQUE-BERA NORMALITY TEST- CHI-SQUARE(2 DF)= 1102.8823 P-VALUE= 0.000 GOODNESS OF FIT TEST FOR NORMALITY OF RESIDUALS - 60 GROUPSOBSERVED 23.0 7.0 8.0 5.0 2.0 1.0 3.0 5.0 3.0 13.0 9.0 10.0 13.0 8.0 14.0 16.0 27.0 19.0 17.0 30.0 26.0 34.0 48.0 59.0 52.0 72.0 74.0106.0114.0128.0125.0122.0 89.0104.0 78.0 80.0 66.0 62.0 41.0 31.0 37.0 32.0 25.0 26.0 19.0 14.0 13.0 6.0 4.0 11.0 6.0 3.0 5.0 4.0 2.0 1.0 3.0 1.0 4.0 14.0EXPECTED 3.8 1.4 1.8 2.4 3.0 3.9 4.9 6.3 7.9 9.7 11.6 14.2 17.2 20.1 23.7 27.6 31.6 36.1 40.7 45.4 50.1 54.9 59.4 63.8 67.5 71.3 74.0 76.2 78.0 78.6 78.6 78.0 76.2 74.0 71.3 67.5 63.8 59.4 54.9 50.1 45.4 40.7 36.1 31.6 27.6 23.7 20.1 17.2 14.2 11.6 9.7 7.9 6.3 4.9 3.9 3.0 2.4 1.8 1.4 3.8CHI-SQUARE = 438.8139 WITH 57 DEGREES OF FREEDOM, P-VALUE= 0.000|_* Inspect the autocorrelation structure of the residuals|_ARIMA E ARIMA MODELNUMBER OF OBSERVATIONS =1974...NOTE..SAMPLE RANGE SET TO: 1, 1974 IDENTIFICATION SECTION - VARIABLE=ENUMBER OF AUTOCORRELATIONS = 24NUMBER OF PARTIAL AUTOCORRELATIONS = 12 0 0 0SERIES (1-B) (1-B ) E NET NUMBER OF OBSERVATIONS = 1974MEAN= -0.56921E-17 VARIANCE= 0.22113 STANDARD DEV.= 0.47024 LAGS AUTOCORRELATIONS STD ERR 1 -12 0.01 -.03 0.03 0.02 0.02 0.00 -.02 0.02 0.02 0.01 -.04 0.00 0.02 13 -24 0.00 0.07 -.01 0.01 -.04 -.01 0.03 -.05 -.02 0.03 0.00 0.07 0.02 MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE 1 0.17 1 .677 13 9.75 13 .714 2 1.44 2 .486 14 18.78 14 .174 3 3.75 3 .289 15 19.06 15 .211 4 4.54 4 .338 16 19.26 16 .256 5 5.15 5 .398 17 21.94 17 .187 6 5.16 6 .524 18 22.10 18 .228 7 5.68 7 .577 19 23.44 19 .218 8 6.21 8 .624 20 27.84 20 .113 9 6.73 9 .665 21 28.31 21 .132 10 6.97 10 .728 22 30.02 22 .118 11 9.75 11 .553 23 30.05 23 .148 12 9.75 12 .638 24 39.56 24 .024 LAGS PARTIAL AUTOCORRELATIONS STD ERR 1 -12 0.01 -.03 0.03 0.02 0.02 0.00 -.02 0.01 0.01 0.01 -.04 0.00 0.02|_* Inspect the autocorrelation structure of the squared residuals|_GENR E2=E*E|_ARIMA E2 ARIMA MODELNUMBER OF OBSERVATIONS =1974...NOTE..SAMPLE RANGE SET TO: 1, 1974 IDENTIFICATION SECTION - VARIABLE=E2NUMBER OF AUTOCORRELATIONS = 24NUMBER OF PARTIAL AUTOCORRELATIONS = 12 0 0 0SERIES (1-B) (1-B ) E2 NET NUMBER OF OBSERVATIONS = 1974MEAN= 0.22102 VARIANCE= 0.27504 STANDARD DEV.= 0.52445 LAGS AUTOCORRELATIONS STD ERR 1 -12 0.22 0.18 0.14 0.12 0.19 0.09 0.09 0.10 0.09 0.12 0.05 0.06 0.02 13 -24 0.08 0.07 0.11 0.08 0.08 0.06 0.07 0.07 0.07 0.07 0.08 0.09 0.03 MODIFIED BOX-PIERCE (LJUNG-BOX-PIERCE) STATISTICS (CHI-SQUARE) LAG Q DF P-VALUE LAG Q DF P-VALUE 1 96.42 1 .000 13 416.61 13 .000 2 157.16 2 .000 14 426.78 14 .000 3 196.75 3 .000 15 452.89 15 .000 4 227.47 4 .000 16 466.87 16 .000 5 297.74 5 .000 17 479.10 17 .000 6 314.13 6 .000 18 486.82 18 .000 7 328.68 7 .000 19 497.13 19 .000 8 347.55 8 .000 20 507.59 20 .000 9 364.77 9 .000 21 518.58 21 .000 10 392.98 10 .000 22 529.09 22 .000 11 397.53 11 .000 23 540.34 23 .000 12 404.93 12 .000 24 554.81 24 .000 LAGS PARTIAL AUTOCORRELATIONS STD ERR 1 -12 0.22 0.13 0.08 0.06 0.13 0.00 0.02 0.04 0.03 0.05 -.02 0.01 0.02|_* Calculate Lagrange multiplier test statistics for ARCH errors|_SET NODOECHO NOOUTPUT|_GEN1 NLAG=10 |_DIM LM NLAG PVALUE NLAG|_DO #=1,NLAG|_* Calculate the test statistic|_OLS E2 E2(1.#)|_GEN1 TESTVAL=$N*$R2|_* Calculate a p-value|_DISTRIB TESTVAL / TYPE=CHI DF=# CDF=CDF|_GEN1 LM:#=TESTVAL|_GEN1 PVALUE:#=1-CDF|_ENDO|_* Print the results|_SAMPLE 1 NLAG|_GENR P=TIME(0)|_FORMAT(F8.0,F10.2,F10.4)|_PRINT P LM PVALUE / FORMAT P LM PVALUE 1. 96.24 0.0000 2. 129.30 0.0000 3. 142.18 0.0000 4. 149.70 0.0000 5. 182.43 0.0000 6. 182.27 0.0000 7. 182.58 0.0000 8. 185.28 0.0000 9. 187.18 0.0000 10. 192.38 0.0000|_STOP