The Journal of Middle East and North Africa Sciences 2018; 4(01) http://www.jomenas.org 32 Bootstrapping the Li-Mak and McLeod-Li Portmanteau Tests for GARCH Models Gul Nisa 1* • Farhat Iqbal 2 1 Balochistan University of Information Technology, Engineering and Management Sciences, Quetta, Pakistan 2 Department of Statistics, University of Balochistan, Quetta, Pakistan farhatiqb@gmail.com Abstract: In this paper, blocks-of-blocks (BOB) bootstrap method is employed for the commonly used diagnostic tests for generalized autoregressive conditional heteroscedastic (GARCH) models. More specifically, the single block-of-blocks and double blocks-of-blocks bootstrap techniques, using three different block lengths of size 4, 10, and 20, are implemented for bootstrapping the Li-Mak and Mcleod-Li portmanteau tests. Using Monte Carlo simulations, the size and power of both tests under the standard normal and Student-t errors are investigated. It was found that the discrepancy between the true and nominal probability of rejection was reduced for both the tests using single block-of-blocks and double blocks-of-blocks bootstrap methods. The power of the Li-Mak test for the GARCH (1, 1) model was found slightly better than the Mcleod-Li test. An empirical example using the monthly data of currency exchange rate (US $ per Pak Rupees) is also reported. To cite this article [Nisa, G., & Iqbal, F. (2018). Bootstrapping the Li-Mak and McLeod-Li Portmanteau Tests for GARCH Models. The Journal of Middle East and North Africa Sciences, 4(01), 32-38]. (P-ISSN 2412- 9763) - (e-ISSN 2412-8937). www.jomenas.org. 4 Keywords: Blocks-of-blocks bootstrap; GARCH; Portmanteau tests; bootstrapped p-values 1. Introduction: In the classical linear regression model, one of the common assumptions is that the residuals of the estimated regression line are stochastically independent from each other. To determine the adequacy of the fitted model, various diagnostic tests are used based on the autocorrelation function (ACF) of the residuals. Box and Pierce (1970) developed one of the most commonly used portmanteau tests. Ljung and Box (1978) proposed a modification of the Box-Pierce test. The Ljung-Box test gives good approximations and is adequate for many practical purposes. Granger and Andersen (1978) suggested that for autoregressive moving average (ARMA) models if the statistical dependence in the residuals was found to be non- linear than squared residual autocorrelations may be useful. It was observed from some of the time series models that squared values of residuals are highly correlated as compared to the residuals itself. McLeod and Li (1983) analyzed autocorrelation of squared residuals of ARMA models. They developed a test based on the square residual autocorrelations. For large sample size, the autocorrelation of squared residuals is asymptotically normally distributed with a mean of zero and unit covariance matrix. One of the assumptions of an econometric model is that the model has constant forecast variance. Engle (1982) developed autoregressive conditional heteroscedastic (ARCH) a process which allows the conditional variance varying with time. The conditional variance, also known as volatility, in ARCH models is a function of the past squared errors. Bollerslev (1986) introduced the generalized ARCH (GARCH) model in which the current conditional variance equation also includes the past conditional variance as explanatory variables. Li and Mak (1994) developed the portmanteau statistic which depends on the squared standardized residuals autocorrelations. This test is considered useful for the diagnostic testing of nonlinear time series with conditional heteroscedasticity. Bootstrap is a technique that can be used for determining the distribution of test statistic or an estimator by resampling the available data. The bootstrap work as an alternative method in that situation in which the asymptotic distributions are difficult to obtain. For finite samples, the first-order asymptotic theory often does not gives an accurate approximation to the distributions of test statistic as the bootstrap method. As a consequence, the test depends on the asymptotic critical values can lead to the nominal levels very different from the true levels (Horowitz; 1997). A simulation study conducted by Chen (2002) indicated that for heavily tailed data the size and power performance of the Ljung-Box and Mcleod-Li tests are not