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
