SINDH UNIVERSITY RESEARCH JOURNAL (SCIENCE SERIES) Diagnostics for GARCH-Type Models under Symmetric and Asymmetric Errors F. IQBAL ++ , Y. Z. JAFRI, G. H. TALPUR* Department of Statistics, University of Balochistan, Quetta – Pakistan Received 21 st December 2012 and Revised 28 th May 2013 1. INTRODUCTION The autoregressive conditional heteroscedastic (ARCH) model of Engle (1982) and the generalized ARCH (GARCH) model of Bollerslev (1986) have been found to be successful in capturing the volatility or the conditional variance structure of many financial time series. There is a huge literature on modelling these conditional heteroscedastic time series, but not much work has been done on model checking or model selection. Diagnostic is one of the important stages of model building. Residual autocorrelations are used to identify possible departure from the assumption that the white noise disturbances in the specified model are uncorrelated (see Box and Jenkins, 1970). To check the model adequacy, the asymptotic distribution of the squared and absolute residual autocorrelations derived from such models might be useful. One option is to build a test statistic to test the null hypothesis that the residuals are independent up to a lag M. The test statistic can be applied to check for non-linearity in mean and also for nonlinearity in variance. The test statistics usually used are called portmanteau statistics. One of the widely used portmanteau statistic is the one proposed by Box and Pierce (1970). This statistic is used to test the null hypothesis that the first M autocorrelations of a covariance stationary time series are zero. If significant autocorrelation is not found in the residuals from the model, then the model is declared to be adequate. Ljung and Box (1978) discussed the finite sample properties and conservative behaviour of the Box-Pierce statistic. In financial time series analysis, it is particularly important to check serial correlations of squared series. McLeod and Li (1983) derived a portmanteau test for model adequacy based on the squared residual autocorrelations in ARMA models. In practice, many researchers apply the Ljung- Box or McLeod-Li tests to the squares of the estimated standardised residuals when testing the adequacy of an ARCH/GARCH model. A  2 distribution with M degrees of freedom, as the large sample distribution for these statistics is found misleading and using the squared residual autocorrelations a correct portmanteau test is proposed by Li and Mak (1994). Tse and Zuo (1997) reported some Monte Carlo results for the finite sample performance of some commonly used diagnostics used in literature and found that the Li-Mak test based on the asymptotic variance under the Gaussian assumption performs favourably among other versions of statistics. Tsui (2004) through Monte Carlo simulation studied the empirical size and power of various tests and found that Li-Mak diagnostics is more powerful. Jianhong and Lixing (2009) proposed a new approach for checking the adequacy of GARCH-type models. Their tests involved weight functions, which provide them with the flexibility in choosing scores to enhance power performance. Carbon and Francq (2011) derived the asymptotic distribution of a vector of autocorrelation of squared residuals for asymmetric power GARCH models. Portmanteau tests are derived and results are obtained under weak moment assumptions. In this paper, we aim to study the empirical size and power of two important diagnostic tests; the Ljung-Box and Li-Mak tests used for univariate autoregressive conditional heteroscedastic models under Abstract: In this paper, the size and power of Ljung-Box and Li-Mak diagnostic tests for univariate autoregressive conditional heteroscedastic models were studied under both symmetric and asymmetric distributions for errors. Monte Carlo simulations with 1000 independent replications are conducted to generate conditional variances with standard normal, Students-t and Skewed-t distributions. It was found that though the Li-Mak test has higher empirical size than the nominal size of 5% but can be considered a better alternative to the Ljung-Box test in case of asymmetric errors. The empirical power of the Li-Mak test was also found slightly better for asymmetric heavy-tailed data. Keywords: ARCH-GARCH models, Portmanteau tests, Autocorrelation ++ Corresponding author email: F. IQBAL farhatiqb@gmail.com *Department of Statistics, University of Sindh, Jamshoro –Pakistan. Sindh Univ. Res. Jour. (Sci. Ser.) Vol. 45 (3) 529-533 (2013)